The Belle II Collaboration

# Measurement of the branching fraction of the decay $B^{-}\rightarrow
D^{0}\rho(770)^{-}$ at Belle II

I. Adachi  0000-0003-2287-0173 L. Aggarwal  0000-0002-0909-7537 H. Aihara
0000-0002-1907-5964 N. Akopov  0000-0002-4425-2096 A. Aloisio
0000-0002-3883-6693 N. Anh Ky  0000-0003-0471-197X D. M. Asner
0000-0002-1586-5790 H. Atmacan  0000-0003-2435-501X V. Aushev
0000-0002-8588-5308 M. Aversano  0000-0001-9980-0953 R. Ayad
0000-0003-3466-9290 V. Babu  0000-0003-0419-6912 H. Bae  0000-0003-1393-8631
S. Bahinipati  0000-0002-3744-5332 P. Bambade  0000-0001-7378-4852 Sw.
Banerjee  0000-0001-8852-2409 S. Bansal  0000-0003-1992-0336 M. Barrett
0000-0002-2095-603X J. Baudot  0000-0001-5585-0991 A. Baur
0000-0003-1360-3292 A. Beaubien  0000-0001-9438-089X F. Becherer
0000-0003-0562-4616 J. Becker  0000-0002-5082-5487 J. V. Bennett
0000-0002-5440-2668 V. Bertacchi  0000-0001-9971-1176 M. Bertemes
0000-0001-5038-360X E. Bertholet  0000-0002-3792-2450 M. Bessner
0000-0003-1776-0439 S. Bettarini  0000-0001-7742-2998 F. Bianchi
0000-0002-1524-6236 T. Bilka  0000-0003-1449-6986 D. Biswas
0000-0002-7543-3471 A. Bobrov  0000-0001-5735-8386 D. Bodrov
0000-0001-5279-4787 A. Bolz  0000-0002-4033-9223 A. Bondar
0000-0002-5089-5338 A. Boschetti  0000-0001-6030-3087 A. Bozek
0000-0002-5915-1319 M. Bračko  0000-0002-2495-0524 P. Branchini
0000-0002-2270-9673 R. A. Briere  0000-0001-5229-1039 T. E. Browder
0000-0001-7357-9007 A. Budano  0000-0002-0856-1131 S. Bussino
0000-0002-3829-9592 M. Campajola  0000-0003-2518-7134 L. Cao
0000-0001-8332-5668 G. Casarosa  0000-0003-4137-938X C. Cecchi
0000-0002-2192-8233 J. Cerasoli  0000-0001-9777-881X M.-C. Chang
0000-0002-8650-6058 P. Cheema  0000-0001-8472-5727 B. G. Cheon
0000-0002-8803-4429 K. Chilikin  0000-0001-7620-2053 K. Chirapatpimol
0000-0003-2099-7760 H.-E. Cho  0000-0002-7008-3759 K. Cho  0000-0003-1705-7399
S.-J. Cho  0000-0002-1673-5664 S.-K. Choi  0000-0003-2747-8277 S. Choudhury
0000-0001-9841-0216 L. Corona  0000-0002-2577-9909 J. X. Cui
0000-0002-2398-3754 F. Dattola  0000-0003-3316-8574 E. De La Cruz-Burelo
0000-0002-7469-6974 S. A. De La Motte  0000-0003-3905-6805 G. de Marino
0000-0002-6509-7793 G. De Nardo  0000-0002-2047-9675 M. De Nuccio
0000-0002-0972-9047 G. De Pietro  0000-0001-8442-107X R. de Sangro
0000-0002-3808-5455 M. Destefanis  0000-0003-1997-6751 S. Dey
0000-0003-2997-3829 R. Dhamija  0000-0001-7052-3163 A. Di Canto
0000-0003-1233-3876 F. Di Capua  0000-0001-9076-5936 J. Dingfelder
0000-0001-5767-2121 Z. Doležal  0000-0002-5662-3675 I. Domínguez Jiménez
0000-0001-6831-3159 T. V. Dong  0000-0003-3043-1939 M. Dorigo
0000-0002-0681-6946 D. Dorner  0000-0003-3628-9267 K. Dort
0000-0003-0849-8774 D. Dossett  0000-0002-5670-5582 S. Dreyer
0000-0002-6295-100X S. Dubey  0000-0002-1345-0970 K. Dugic
0009-0006-6056-546X G. Dujany  0000-0002-1345-8163 P. Ecker
0000-0002-6817-6868 M. Eliachevitch  0000-0003-2033-537X P. Feichtinger
0000-0003-3966-7497 T. Ferber  0000-0002-6849-0427 T. Fillinger
0000-0001-9795-7412 C. Finck  0000-0002-5068-5453 G. Finocchiaro
0000-0002-3936-2151 A. Fodor  0000-0002-2821-759X F. Forti
0000-0001-6535-7965 A. Frey  0000-0001-7470-3874 B. G. Fulsom
0000-0002-5862-9739 A. Gabrielli  0000-0001-7695-0537 E. Ganiev
0000-0001-8346-8597 M. Garcia-Hernandez  0000-0003-2393-3367 R. Garg
0000-0002-7406-4707 G. Gaudino  0000-0001-5983-1552 V. Gaur
0000-0002-8880-6134 A. Gaz  0000-0001-6754-3315 A. Gellrich
0000-0003-0974-6231 G. Ghevondyan  0000-0003-0096-3555 D. Ghosh
0000-0002-3458-9824 H. Ghumaryan  0000-0001-6775-8893 G. Giakoustidis
0000-0001-5982-1784 R. Giordano  0000-0002-5496-7247 A. Giri
0000-0002-8895-0128 A. Glazov  0000-0002-8553-7338 B. Gobbo
0000-0002-3147-4562 R. Godang  0000-0002-8317-0579 O. Gogota
0000-0003-4108-7256 P. Goldenzweig  0000-0001-8785-847X W. Gradl
0000-0002-9974-8320 S. Granderath  0000-0002-9945-463X E. Graziani
0000-0001-8602-5652 D. Greenwald  0000-0001-6964-8399 Z. Gruberová
0000-0002-5691-1044 T. Gu  0000-0002-1470-6536 K. Gudkova  0000-0002-5858-3187
S. Halder  0000-0002-6280-494X Y. Han  0000-0001-6775-5932 T. Hara
0000-0002-4321-0417 K. Hayasaka  0000-0002-6347-433X H. Hayashii
0000-0002-5138-5903 S. Hazra  0000-0001-6954-9593 C. Hearty
0000-0001-6568-0252 M. T. Hedges  0000-0001-6504-1872 A. Heidelbach
0000-0002-6663-5469 I. Heredia de la Cruz  0000-0002-8133-6467 M. Hernández
Villanueva  0000-0002-6322-5587 T. Higuchi  0000-0002-7761-3505 M. Hoek
0000-0002-1893-8764 M. Hohmann  0000-0001-5147-4781 P. Horak
0000-0001-9979-6501 C.-L. Hsu  0000-0002-1641-430X T. Humair
0000-0002-2922-9779 T. Iijima  0000-0002-4271-711X K. Inami
0000-0003-2765-7072 N. Ipsita  0000-0002-2927-3366 A. Ishikawa
0000-0002-3561-5633 R. Itoh  0000-0003-1590-0266 M. Iwasaki
0000-0002-9402-7559 W. W. Jacobs  0000-0002-9996-6336 D. E. Jaffe
0000-0003-3122-4384 E.-J. Jang  0000-0002-1935-9887 S. Jia
0000-0001-8176-8545 Y. Jin  0000-0002-7323-0830 H. Junkerkalefeld
0000-0003-3987-9895 D. Kalita  0000-0003-3054-1222 A. B. Kaliyar
0000-0002-2211-619X J. Kandra  0000-0001-5635-1000 S. Kang
0000-0002-5320-7043 G. Karyan  0000-0001-5365-3716 T. Kawasaki
0000-0002-4089-5238 F. Keil  0000-0002-7278-2860 C. Kiesling
0000-0002-2209-535X C.-H. Kim  0000-0002-5743-7698 D. Y. Kim
0000-0001-8125-9070 K.-H. Kim  0000-0002-4659-1112 Y.-K. Kim
0000-0002-9695-8103 H. Kindo  0000-0002-6756-3591 K. Kinoshita
0000-0001-7175-4182 P. Kodyš  0000-0002-8644-2349 T. Koga  0000-0002-1644-2001
S. Kohani  0000-0003-3869-6552 K. Kojima  0000-0002-3638-0266 T. Konno
0000-0003-2487-8080 A. Korobov  0000-0001-5959-8172 S. Korpar
0000-0003-0971-0968 E. Kovalenko  0000-0001-8084-1931 R. Kowalewski
0000-0002-7314-0990 T. M. G. Kraetzschmar  0000-0001-8395-2928 P. Križan
0000-0002-4967-7675 P. Krokovny  0000-0002-1236-4667 T. Kuhr
0000-0001-6251-8049 Y. Kulii  0000-0001-6217-5162 J. Kumar
0000-0002-8465-433X M. Kumar  0000-0002-6627-9708 R. Kumar
0000-0002-6277-2626 K. Kumara  0000-0003-1572-5365 T. Kunigo
0000-0001-9613-2849 A. Kuzmin  0000-0002-7011-5044 Y.-J. Kwon
0000-0001-9448-5691 S. Lacaprara  0000-0002-0551-7696 K. Lalwani
0000-0002-7294-396X T. Lam  0000-0001-9128-6806 L. Lanceri
0000-0001-8220-3095 J. S. Lange  0000-0003-0234-0474 M. Laurenza
0000-0002-7400-6013 R. Leboucher  0000-0003-3097-6613 F. R. Le Diberder
0000-0002-9073-5689 M. J. Lee  0000-0003-4528-4601 P. Leo  0000-0003-3833-2900
C. Lemettais  0009-0008-5394-5100 D. Levit  0000-0001-5789-6205 P. M. Lewis
0000-0002-5991-622X L. K. Li  0000-0002-7366-1307 S. X. Li
0000-0003-4669-1495 Y. Li  0000-0002-4413-6247 Y. B. Li  0000-0002-9909-2851
J. Libby  0000-0002-1219-3247 Z. Liptak  0000-0002-6491-8131 M. H. Liu
0000-0002-9376-1487 Q. Y. Liu  0000-0002-7684-0415 Z. Q. Liu
0000-0002-0290-3022 D. Liventsev  0000-0003-3416-0056 S. Longo
0000-0002-8124-8969 T. Lueck  0000-0003-3915-2506 C. Lyu  0000-0002-2275-0473
Y. Ma  0000-0001-8412-8308 M. Maggiora  0000-0003-4143-9127 R. Maiti
0000-0001-5534-7149 S. Maity  0000-0003-3076-9243 G. Mancinelli
0000-0003-1144-3678 R. Manfredi  0000-0002-8552-6276 E. Manoni
0000-0002-9826-7947 M. Mantovano  0000-0002-5979-5050 D. Marcantonio
0000-0002-1315-8646 S. Marcello  0000-0003-4144-863X C. Marinas
0000-0003-1903-3251 C. Martellini  0000-0002-7189-8343 A. Martens
0000-0003-1544-4053 T. Martinov  0000-0001-7846-1913 L. Massaccesi
0000-0003-1762-4699 M. Masuda  0000-0002-7109-5583 K. Matsuoka
0000-0003-1706-9365 D. Matvienko  0000-0002-2698-5448 S. K. Maurya
0000-0002-7764-5777 F. Mawas  0000-0002-7176-4732 J. A. McKenna
0000-0001-9871-9002 R. Mehta  0000-0001-8670-3409 F. Meier
0000-0002-6088-0412 M. Merola  0000-0002-7082-8108 C. Miller
0000-0003-2631-1790 M. Mirra  0000-0002-1190-2961 S. Mitra
0000-0002-1118-6344 G. B. Mohanty  0000-0001-6850-7666 S. Mondal
0000-0002-3054-8400 S. Moneta  0000-0003-2184-7510 H.-G. Moser
0000-0003-3579-9951 M. Mrvar  0000-0001-6388-3005 R. Mussa
0000-0002-0294-9071 I. Nakamura  0000-0002-7640-5456 M. Nakao
0000-0001-8424-7075 Y. Nakazawa  0000-0002-6271-5808 A. Narimani Charan
0000-0002-5975-550X M. Naruki  0000-0003-1773-2999 D. Narwal
0000-0001-6585-7767 Z. Natkaniec  0000-0003-0486-9291 A. Natochii
0000-0002-1076-814X L. Nayak  0000-0002-7739-914X M. Nayak
0000-0002-2572-4692 G. Nazaryan  0000-0002-9434-6197 M. Neu
0000-0002-4564-8009 M. Niiyama  0000-0003-1746-586X S. Nishida
0000-0001-6373-2346 A. Novosel  0000-0002-7308-8950 S. Ogawa
0000-0002-7310-5079 Y. Onishchuk  0000-0002-8261-7543 H. Ono
0000-0003-4486-0064 G. Pakhlova  0000-0001-7518-3022 S. Pardi
0000-0001-7994-0537 K. Parham  0000-0001-9556-2433 H. Park
0000-0001-6087-2052 S.-H. Park  0000-0001-6019-6218 B. Paschen
0000-0003-1546-4548 A. Passeri  0000-0003-4864-3411 S. Patra
0000-0002-4114-1091 T. K. Pedlar  0000-0001-9839-7373 R. Peschke
0000-0002-2529-8515 R. Pestotnik  0000-0003-1804-9470 M. Piccolo
0000-0001-9750-0551 L. E. Piilonen  0000-0001-6836-0748 P. L. M. Podesta-Lerma
0000-0002-8152-9605 T. Podobnik  0000-0002-6131-819X S. Pokharel
0000-0002-3367-738X C. Praz  0000-0002-6154-885X S. Prell  0000-0002-0195-8005
E. Prencipe  0000-0002-9465-2493 M. T. Prim  0000-0002-1407-7450 H. Purwar
0000-0002-3876-7069 G. Raeuber  0000-0003-2948-5155 S. Raiz
0000-0001-7010-8066 N. Rauls  0000-0002-6583-4888 M. Reif  0000-0002-0706-0247
S. Reiter  0000-0002-6542-9954 I. Ripp-Baudot  0000-0002-1897-8272 G. Rizzo
0000-0003-1788-2866 M. Roehrken  0000-0003-0654-2866 J. M. Roney
0000-0001-7802-4617 A. Rostomyan  0000-0003-1839-8152 N. Rout
0000-0002-4310-3638 D. A. Sanders  0000-0002-4902-966X S. Sandilya
0000-0002-4199-4369 L. Santelj  0000-0003-3904-2956 Y. Sato
0000-0003-3751-2803 V. Savinov  0000-0002-9184-2830 B. Scavino
0000-0003-1771-9161 M. Schnepf  0000-0003-0623-0184 C. Schwanda
0000-0003-4844-5028 Y. Seino  0000-0002-8378-4255 A. Selce
0000-0001-8228-9781 K. Senyo  0000-0002-1615-9118 J. Serrano
0000-0003-2489-7812 M. E. Sevior  0000-0002-4824-101X C. Sfienti
0000-0002-5921-8819 W. Shan  0000-0003-2811-2218 C. P. Shen
0000-0002-9012-4618 X. D. Shi  0000-0002-7006-6107 T. Shillington
0000-0003-3862-4380 J.-G. Shiu  0000-0002-8478-5639 D. Shtol
0000-0002-0622-6065 B. Shwartz  0000-0002-1456-1496 A. Sibidanov
0000-0001-8805-4895 F. Simon  0000-0002-5978-0289 J. B. Singh
0000-0001-9029-2462 J. Skorupa  0000-0002-8566-621X R. J. Sobie
0000-0001-7430-7599 M. Sobotzik  0000-0002-1773-5455 A. Soffer
0000-0002-0749-2146 A. Sokolov  0000-0002-9420-0091 E. Solovieva
0000-0002-5735-4059 S. Spataro  0000-0001-9601-405X B. Spruck
0000-0002-3060-2729 M. Starič  0000-0001-8751-5944 P. Stavroulakis
0000-0001-9914-7261 S. Stefkova  0000-0003-2628-530X R. Stroili
0000-0002-3453-142X M. Sumihama  0000-0002-8954-0585 K. Sumisawa
0000-0001-7003-7210 W. Sutcliffe  0000-0002-9795-3582 N. Suwonjandee
0009-0000-2819-5020 H. Svidras  0000-0003-4198-2517 M. Takahashi
0000-0003-1171-5960 M. Takizawa  0000-0001-8225-3973 U. Tamponi
0000-0001-6651-0706 K. Tanida  0000-0002-8255-3746 F. Tenchini
0000-0003-3469-9377 A. Thaller  0000-0003-4171-6219 O. Tittel
0000-0001-9128-6240 R. Tiwary  0000-0002-5887-1883 D. Tonelli
0000-0002-1494-7882 E. Torassa  0000-0003-2321-0599 K. Trabelsi
0000-0001-6567-3036 I. Tsaklidis  0000-0003-3584-4484 I. Ueda
0000-0002-6833-4344 K. Unger  0000-0001-7378-6671 Y. Unno  0000-0003-3355-765X
K. Uno  0000-0002-2209-8198 S. Uno  0000-0002-3401-0480 S. E. Vahsen
0000-0003-1685-9824 R. van Tonder  0000-0002-7448-4816 K. E. Varvell
0000-0003-1017-1295 M. Veronesi  0000-0002-1916-3884 A. Vinokurova
0000-0003-4220-8056 V. S. Vismaya  0000-0002-1606-5349 L. Vitale
0000-0003-3354-2300 V. Vobbilisetti  0000-0002-4399-5082 R. Volpe
0000-0003-1782-2978 M. Wakai  0000-0003-2818-3155 S. Wallner
0000-0002-9105-1625 E. Wang  0000-0001-6391-5118 M.-Z. Wang
0000-0002-0979-8341 Z. Wang  0000-0002-3536-4950 A. Warburton
0000-0002-2298-7315 M. Watanabe  0000-0001-6917-6694 S. Watanuki
0000-0002-5241-6628 O. Werbycka  0000-0002-0614-8773 C. Wessel
0000-0003-0959-4784 X. P. Xu  0000-0001-5096-1182 B. D. Yabsley
0000-0002-2680-0474 S. Yamada  0000-0002-8858-9336 W. Yan  0000-0003-0713-0871
S. B. Yang  0000-0002-9543-7971 J. H. Yin  0000-0002-1479-9349 K. Yoshihara
0000-0002-3656-2326 C. Z. Yuan  0000-0002-1652-6686 Y. Yusa
0000-0002-4001-9748 L. Zani  0000-0003-4957-805X F. Zeng  0009-0003-6474-3508
B. Zhang  0000-0002-5065-8762 Y. Zhang  0000-0003-2961-2820 V. Zhilich
0000-0002-0907-5565 Q. D. Zhou  0000-0001-5968-6359 X. Y. Zhou
0000-0002-0299-4657 V. I. Zhukova  0000-0002-8253-641X R. Žlebčík
0000-0003-1644-8523

###### Abstract

We measure the branching fraction of the decay $B^{-}\rightarrow
D^{0}\rho(770)^{-}$ using data collected with the Belle II detector. The data
contain 387 million $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ pairs
produced in $e^{+}e^{-}$ collisions at the $\mathchar 28935\relax{(4S)}$
resonance. We reconstruct $8360\pm 180$ decays from an analysis of the
distributions of the $B^{-}$ energy and the $\rho(770)^{-}$ helicity angle. We
determine the branching fraction to be
$({0.939}\pm{0.021}\mathrm{(stat)}\pm{0.050}\mathrm{(syst)})\%$, in agreement
with previous results. Our measurement improves the relative precision of the
world average by more than a factor of two.

The Belle II experiment uses $e^{+}e^{-}$ collisions at center-of-mass
energies of about 10.6$\mathrm{\,Ge\kern-1.00006ptV}$ to access an abundant
sample of $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ pairs produced
through $\mathchar 28935\relax{(4S)}$ decays and search for non-standard-model
physics in weak decays of $B$ mesons [1, 2]. A large part of this physics
program depends on reconstructing the decays of both $B$ mesons: one is the
signal, the other (tag decay) is used to infer signal properties or to
suppress background from continuum production of light quarks. This analysis
technique is known as tagging. Belle II tagging algorithms use multivariate
classifiers trained with simulated events [3]. The algorithms are calibrated
on control data to correct for simulation mismodeling, which originates mainly
from inaccurate or missing information on branching fractions and decay models
of some tag decays [4]. Improved knowledge of these decays would yield better
training, resulting in enhanced and more reliable tagging performance. This
will improve Belle II physics reach.

Decays into fully hadronic final states offer the best signal-to-background
ratio for tagging. The dominant hadronic tag channels are Cabibbo-favored
decays into a charm meson and several pions. Among these, $B^{-}\rightarrow
D^{0}\rho(770)^{-}$ is one of the most effective. (Charge conjugation is
implied throughout the letter unless otherwise stated.) However, the tagging
efficiency differs significantly between data and simulation for this tag
decay, strongly suggesting the need to revisit the measurement of its
branching fraction. The current world average, $(1.34\pm 0.18)\%$ [5], is
dominated by a single measurement from 1994 by the CLEO collaboration, which
employed a sample of $e^{+}e^{-}$ collisions at the $\mathchar
28935\relax{(4S)}$ resonance with an integrated luminosity of
$0.9\mbox{\,fb}^{-1}$ [6]. We note that the world average from Ref. [5] is not
updated to the latest branching fractions of $D^{0}$ and $\mathchar
28935\relax{(4S)}$ decays; scaling them to the current values, the average
would be about 10% smaller.

Branching-fraction measurements of the decays $B^{-}\rightarrow
D^{0}\rho^{-}$, $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\rho^{-}$, and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{0}\rho^{0}$, also
provide tests of calculations of hadronic decay rates based on the heavy-quark
limit and factorization models [7, 8]. Hereafter, we refer to $\rho(770)$
mesons as $\rho$. Using isospin symmetry, their decay amplitudes can be
expressed in terms of amplitudes for two isospin eigenstates, $A_{1/2}$ and
$A_{3/2}$ [9, 10, 11, 12, 13, 14]. The ratio $R$ and strong-phase difference
$\delta$ between these amplitudes are related to the branching fractions
$\mathcal{B}(D\rho)$ of the three decays and the ratio $\tau_{+}/\tau_{0}$ of
the $B^{+}$ and $B^{0}$ lifetimes, as

$\displaystyle R$
$\displaystyle=\left(\frac{3}{2}\frac{\tau_{+}}{\tau_{0}}\frac{\mathcal{B}(D^{0}\rho^{0})+\mathcal{B}(D^{+}\rho^{-})}{\mathcal{B}(D^{0}\rho^{-})}-\frac{1}{2}\right)^{\frac{1}{2}}\,,$
(1) $\displaystyle\cos\delta$
$\displaystyle=\frac{1}{2R}\left(\frac{3}{2}\frac{\tau_{+}}{\tau_{0}}\frac{\mathcal{B}(D^{0}\rho^{0})-2\mathcal{B}(D^{+}\rho^{-})}{\mathcal{B}(D^{0}\rho^{-})}+\frac{1}{2}\right)\,.$
(2)

In the heavy-quark limit, factorization models predict
$R=1+\mathcal{O}(\mathchar 28931\relax_{\rm QCD}/m_{b})$ and
$\delta=\mathcal{O}(\mathchar 28931\relax_{\rm QCD}/m_{b})$, where $m_{b}$ is
the $b$ quark mass and $\mathchar 28931\relax_{\rm QCD}$ the QCD scale. The
LHCb collaboration reported $R=0.69\pm 0.15$ and
$\cos\delta=0.984^{+0.113}_{-0.048}$ [15]. The accuracy of the test is limited
by the 13% and 16% fractional uncertainties on the branching fractions of the
$B^{-}\rightarrow D^{0}\rho^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\rho^{-}$ decays,
respectively [5].

In this letter, we report an improved measurement of the $B^{-}\rightarrow
D^{0}\rho^{-}$ branching fraction using data collected at Belle II. We
reconstruct the $D^{0}$ meson in its two-body favored decay $D^{0}\rightarrow
K^{-}\pi^{+}$ and restrict the $\pi^{-}\pi^{0}$ invariant mass to a
300${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ range centered at the $\rho^{-}$
mass pole. In this range, approximately twice the $\rho^{-}$ natural width, we
expect the $B^{-}\rightarrow D^{0}\rho^{-}$ decay to nearly saturate the
$D^{0}\pi^{-}\pi^{0}$ final state, with only a small contribution from the
three-body $B^{-}\rightarrow D^{0}\pi^{-}\pi^{0}$ decay [6]. To determine the
$B^{-}\rightarrow D^{0}\rho^{-}$ yield, we analyze the background-subtracted
distribution of the helicity angle of the $\rho^{-}\rightarrow\pi^{-}\pi^{0}$
decay. The latter is obtained by fitting the $B^{-}$ energy distribution in
nine independent intervals of the helicity angle. The $B^{-}\rightarrow
D^{0}\rho^{-}$ yield is divided by the reconstruction efficiency and the
number of $B^{-}$ mesons produced to determine the branching fraction. The
data sample has an integrated luminosity of 362$\mbox{\,fb}^{-1}$, which
enables a more precise measurement than the current world average. The main
uncertainty is systematic and related to the calculation of the signal
efficiency, and to the fit model. To reduce the systematic uncertainty, the
selection is chosen to minimize background contamination while retaining a
few-percent fractional statistical precision on the branching-fraction
measurement. We designed the analysis using simulated and experimental control
data before examining the signal region.

The Belle II experiment [16] is located at SuperKEKB, which collides electrons
and positrons at and near the $\mathchar 28935\relax(4S)$ resonance [17]. The
Belle II detector [16] has a cylindrical geometry and includes a two-layer
silicon-pixel detector surrounded by a four-layer double-sided silicon-strip
detector and a 56-layer central drift chamber. These detectors reconstruct
tracks of charged particles. Only one sixth of the second layer of the
silicon-pixel detector was installed for the data analyzed here. The symmetry
axis of these detectors, defined as the $z$ axis, is almost coincident with
the direction of the electron beam. Surrounding the drift chamber, which also
provides $\mathrm{d}E/\mathrm{d}x$ energy-loss measurements, is a time-of-
propagation counter in the central region and an aerogel-based ring-imaging
Cherenkov counter in the forward region. These detectors provide charged-
particle identification. Surrounding them is an electromagnetic calorimeter
(ECL) based on CsI(Tl) crystals that primarily provides energy and timing
measurements for photons and electrons. Outside of the ECL is a
superconducting solenoid magnet. Its flux return is instrumented with
resistive-plate chambers and plastic scintillator modules to detect muons,
$K^{0}_{L}$ mesons, and neutrons. The solenoid magnet provides a 1.5 T
magnetic field parallel to the $z$ axis.

Large samples of simulated data are generated by modeling the physics
processes resulting from $e^{+}e^{-}$ collisions and propagating the final-
state particles through a detailed simulation of the detector. We use the
EVTGEN [18], PYTHIA8 [19], and KKMC [20] software libraries to model particle
production and decay, PHOTOS [21] for photon radiation, and GEANT4 [22] for
material interaction and detector response. The simulation includes beam-
induced backgrounds [23]. Collision and simulation data are processed using
the BASF2 [24, 25] software.

The event selection starts online with criteria based on the total energy and
charged-particle multiplicity, which are fully efficient for signal and
strongly suppress low-multiplicity events. Offline, we select tracks with
loose requirements on their radial ($\delta
r<$0.5\text{\,}\mathrm{c}\mathrm{m}$$) and longitudinal ($|\delta
z|<$3.0\text{\,}\mathrm{c}\mathrm{m}$$) displacements from the average
$e^{+}e^{-}$ interaction point. We require tracks to be in the polar-angle
acceptance of the drift chamber
($$17\text{\,}\mathrm{\SIUnitSymbolDegree}$<\theta<$150\text{\,}\mathrm{\SIUnitSymbolDegree}$$).

Photons are reconstructed from ECL energy clusters that are not matched to
tracks. Photons reconstructed in the central region of the ECL
($$32.2\text{\,}\mathrm{\SIUnitSymbolDegree}$<\theta<$128.7\text{\,}\mathrm{\SIUnitSymbolDegree}$$)
are required to have energies greater than 50 MeV, and those in the forward
($$12.4\text{\,}\mathrm{\SIUnitSymbolDegree}$<\theta<$31.4\text{\,}\mathrm{\SIUnitSymbolDegree}$$)
and backward
($$130.7\text{\,}\mathrm{\SIUnitSymbolDegree}$<\theta<$155.1\text{\,}\mathrm{\SIUnitSymbolDegree}$$)
end-caps greater than 60 and 100 MeV, respectively. To suppress beam
background, photon clusters must include more than one ECL crystal and have a
signal time within 200 ns of the collision time. We reconstruct $\pi^{0}$
candidates by combining pairs of photons. We require the angle between the two
photons to be smaller than 1 rad, the difference in their azimuthal angles to
be smaller than 2.2 rad, and the diphoton mass to be consistent with the known
$\pi^{0}$ mass within approximately two times the resolution. We also exclude
extreme values of the cosine of the $\pi^{0}$ helicity angle. In the $\pi^{0}$
rest frame, this is the angle between the photon direction and the $\pi^{0}$
boost direction from the lab frame. We remove candidates that have cosine
values larger than 0.98 to suppress combinatorial background from collinear
low-momentum photons. We train a boosted-decision-tree (BDT) classifier with
14 variables associated with cluster shapes to suppress photons
misreconstructed from hadronic clusters and hadronic-shower splitting [26]. We
choose a threshold on the BDT output optimized to select $\pi^{0}$ candidates
from $\rho^{-}\rightarrow\pi^{0}\pi^{-}$ decays. A requirement that the
$\pi^{0}$ momentum be larger than 240${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$
suppresses low-energy $\pi^{0}$ candidates from decays of excited charm
states.

Tracks are assumed to be charged pions. We combine a neutral and a charged
pion to form a $\rho^{-}$ candidate. We reconstruct $D^{0}$ candidates by
combining oppositely charged kaon and pion candidates with invariant masses
between 1.85 and 1.88${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, a range about
six units of mass resolution wide. The kaon candidate is a track that
satisfies a threshold on the ratio
$\mathcal{L}_{K}/(\mathcal{L}_{\pi}+\mathcal{L}_{K})$, where the likelihood
$\mathcal{L}_{\pi,K}$ for a pion or kaon hypothesis combines particle-
identification information from all subdetectors except the pixel detector.
The requirement retains 95% of kaons and rejects about 90% of pions
misidentified as kaons. We combine $D^{0}$ and $\rho^{-}$ candidates through a
kinematic fit of the entire decay chain to form $B^{-}\rightarrow
D^{0}\rho^{-}$ decays, constraining the $B^{-}$ to originate from the
$e^{+}e^{-}$ interaction region and the $\pi^{0}$ mass to its known value to
improve $B$ energy and momentum resolutions. For each $B$ candidate, we
calculate the beam-constrained mass, $M_{\rm bc}=\sqrt{E_{\rm
beam}^{*2}/c^{4}-|\vec{p}_{B}^{\ *}|^{2}/c^{2}}$, and the energy difference,
$\Delta E=E_{B}^{*}-E_{\rm beam}^{*}$, where $E_{\rm beam}^{*}$ is the beam
energy, $E_{B}^{*}$ is the $B$ energy, and $\vec{p}_{B}^{\ *}$ its momentum
vector, all computed in the center-of-mass frame. We require $M_{\rm
bc}>5.27{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and $-0.18<\Delta E<0.2$
GeV.

We define the $\rho$ helicity angle $\theta_{\rho}$ as the angle between the
$\pi^{-}$ momentum and the direction opposite to the $B^{-}$ momentum in the
$\rho^{-}$ rest frame. We select the region $\cos\theta_{\rho}<0.7$, which
excludes events with invariant masses $m(D^{0}\pi^{0})$ smaller than about
$2.6$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. This removes background from
$B$ decays into $D^{0}\pi^{-}\pi^{0}$ through $D_{0}^{*0}\rightarrow
D^{0}\pi^{0}$, $D_{2}^{*0}\rightarrow D^{0}\pi^{0}$, which could interfere
with the signal. Rates for these processes are not known, but simulation
studies show that our selection suppresses to negligible levels any excited
$D^{*}$ contribution regardless of the unknown decay amplitudes.

To further suppress background, we use all tracks not associated with the
signal to reconstruct the decay vertex of the other $B$ meson and to identify
its flavor with a dedicated algorithm [27]. This information, along with
event-shape variables (modified Fox-Wolfram moments [28, 29], energy flows
[30], sphericity-related quantities [31], and thrust-related quantities [32]),
is used as input to a BDT classifier. The BDT is trained with simulation to
distinguish signal from continuum background: $e^{+}e^{-}\rightarrow q\bar{q}$
processes, where $q=u,d,s,c$. The most discriminating inputs are the ratio
between the second and zeroth Fox-Wolfram moments, the magnitude of the signal
$B$ thrust, and the cosine of the angle between the thrust axis of the signal
$B$ and that of all the remaining charged and neutral particles in the event.
We cut on the BDT output to reduce the continuum background to a negligible
level. There are multiple candidates in 4% of events (2% in the signal region
$|\Delta E|<50$$\mathrm{\,Me\kern-1.00006ptV}$) and are all retained.

After the selection, simulation shows that $64\%$ of the total background is
due to $B$ decays other than signal, the majority being Cabibbo-favored
semileptonic decays. We refer to these decays as $B\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ background. The remaining $36\%$
originates from misreconstructed signal decays (self-cross-feed), where one of
the final-state particles is mistakenly taken from the decay of the
accompanying $B$ meson. Such misreconstruction only occurs for the
$\rho^{-}\rightarrow\pi^{-}\pi^{0}$ decay. The neutral pion is
misreconstructed using a photon not belonging to the signal in half of the
cases of self-cross-feed, and a charged or neutral pion from the other
$B$-meson decay is used in the other half.

We determine the sample composition with a maximum likelihood fit to the
unbinned distribution of $\Delta E$, including three components: signal,
$B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ background, and self-cross-
feed. At this stage, we treat possible contamination of the three-body decay
$B^{-}\rightarrow D^{0}\pi^{-}\pi^{0}$, where the $\pi^{-}\pi^{0}$ system is
nonresonant, as part of the signal, as its $\Delta E$ distribution is
indistinguishable from that of $B^{-}\rightarrow D^{0}\rho^{-}$ decay. The
$\Delta E$ fit is performed in nine independent intervals of
$\cos\theta_{\rho}$ to reconstruct the background-subtracted distribution of
this variable and separate $B^{-}\rightarrow D^{0}\rho^{-}$ from
$B^{-}\rightarrow D^{0}\pi^{-}\pi^{0}$ decays. As the $\rho^{-}$ is fully
longitudinally polarized in $B^{-}\rightarrow D^{0}\rho^{-}$ decays, the
helicity angle distribution is proportional to $\cos^{2}\theta_{\rho}$; by
contrast, in nonresonant $B^{-}\rightarrow D^{0}\pi^{-}\pi^{0}$ decays the
$\pi^{-}\pi^{0}$ system is in $S$-wave with a uniform $\cos\theta_{\rho}$
distribution. Interference between the amplitudes of the two decays modifies
these distributions. To maximize the sensitivity at $\cos\theta_{\rho}\simeq
0$, where $B^{-}\rightarrow D^{0}\rho^{-}$ decays are suppressed, we use a
variable interval width so that the expected $B^{-}\rightarrow D^{0}\rho^{-}$
distribution is remapped into a uniform distribution. With this choice, the
$B^{-}\rightarrow D^{0}\pi^{-}\pi^{0}$ shape peaks in the central region. The
interval scheme is chosen by using simulated data and accounts for the
efficiency variation as a function of $\cos\theta_{\rho}$. We require the same
number of $B^{-}\rightarrow D^{0}\rho^{-}$ candidates in each of the nine
intervals and we check that the $\cos\theta_{\rho}$ resolution is negligible
compared to the interval width.

Figure 1: Distribution of $\Delta E$ with fit projection overlaid for (top)
the lowest and (bottom) the highest $\cos\theta_{\rho}$ intervals.

The $\Delta E$ distribution changes significantly across the
$\cos\theta_{\rho}$ intervals for each fit component. Examples in the two
extremal intervals are shown in Fig. 1. The signal is modeled with a Johnson
$S_{U}$ function [33], which peaks close to zero in each interval and has a
tail at negative $\Delta E$. However, variations of the peak position, width,
and skewness are observed because the energy resolution depends on the momenta
of the pions in the $\rho^{-}$ decay. When $\cos\theta_{\rho}$ approaches
$-1$, the energy resolution is dominated by the ECL energy resolution on the
fast $\pi^{0}$; for $\cos\theta_{\rho}$ values closer to $1$, the largest
contribution to the energy resolution comes from the measurement of the
momentum of the fast $\pi^{-}$. Self-cross-feed also peaks around zero, but
with a broad structure that is modeled by a Gaussian function; in the extremal
$\cos\theta_{\rho}$ intervals, an exponential function is also added. The
shape of the $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ background is
exponential in each $\cos\theta_{\rho}$ interval and peaks at the lower edge
of the $\Delta E$ range with different slopes; intervals closer to
$\cos\theta_{\rho}=-1$ have a smaller slope. All shapes are determined by
fitting simulated data and allow for adjustments across the
$\cos\theta_{\rho}$ intervals. For each interval, the free parameters of the
fits to the experimental data are the yields of the signal and $B\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ background, the exponential slope of
the $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ background, and shift- and
width-correction parameters common to the signal and self-cross-feed peaks.
The ratio of self-cross-feed to signal is also a fit parameter that is
Gaussian constrained to the value found in simulation; the uncertainty on this
constraint is discussed below. The $\Delta E$ distribution is modeled well in
each $\cos\theta_{\rho}$ interval, with $p$ values from 0.07 to 0.96. The $p$
value of a fit is calculated as the $\chi^{2}$ probability from the pull
distribution of the fit projection with 26 bins in $-0.18<\Delta E<0.08$ GeV,
with degrees of freedom equaling the number of bins minus the free parameters
of the fit. Two example fits are shown in Fig. 1.

Figure 2: (Top) Background-subtracted $\cos\theta_{\rho}$ distribution with
fit projection overlaid. The horizontal bar on the markers represents the
$\cos\theta_{\rho}$ interval. The $y$-axis scale is zero-suppressed to show
the small $B^{-}\rightarrow D^{0}\pi^{-}\pi^{0}$ contribution. (Bottom)
Background-subtracted distribution of the $\pi^{-}\pi^{0}$ invariant mass,
with overlaid simulated data reflecting the $B^{-}\rightarrow D^{0}\rho^{-}$
and $B^{-}\rightarrow D^{0}\pi^{-}\pi^{0}$ proportions from the
$\cos\theta_{\rho}$ fit.

The background-subtracted $\cos\theta_{\rho}$ distribution is shown in Fig. 2
(top). We carry out a $\chi^{2}$ fit to this distribution to extract the
$B^{-}\rightarrow D^{0}\rho^{-}$ signal yield. Since we adopt an interval
scheme with equiprobable $B^{-}\rightarrow D^{0}\rho^{-}$ yields, we use a
uniform probability density function to model $B^{-}\rightarrow D^{0}\rho^{-}$
candidates and a peaking template obtained from simulation to model
$B^{-}\rightarrow D^{0}\pi^{-}\pi^{0}$ candidates. We find $8360\pm 180$
$B^{-}\rightarrow D^{0}\rho^{-}$ decays and a $(1.9\pm 1.8)\%$ fraction of
$B^{-}\rightarrow D^{0}\pi^{-}\pi^{0}$ decays. The $p$ value of the fit is
0.074. Figure 2 (bottom) shows the background-subtracted distribution of the
$\pi^{-}\pi^{0}$ invariant mass, overlaid with the simulated distribution
using the fractions of $B^{-}\rightarrow D^{0}\rho^{-}$ and $B^{-}\rightarrow
D^{0}\pi^{-}\pi^{0}$ decays resulting from the fit. Background is subtracted
using the sPlot method [34] with per-candidate weights calculated from the
$\Delta E$ fits in the nine $\cos\theta_{\rho}$ intervals. Simulation
reproduces the data distribution well. We check the stability of the obtained
results by separately fitting the candidates with $\pi^{-}\pi^{0}$ invariant
masses above and below 770${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. We
obtain compatible $\cos\theta_{\rho}$ distributions, with fractions of
nonresonant $B^{-}\rightarrow D^{0}\pi^{-}\pi^{0}$ decays again consistent
with zero and branching fractions of $B^{-}\rightarrow D^{0}\rho^{-}$ decays
that agree within half a standard deviation considering only the statistical
uncertainties. These results indicate that interference effects between the
two amplitudes are negligible.

Table 1: Summary of the fractional systematic uncertainties. The total is the sum in quadrature of the individual contributions. Source | Fractional uncertainty (%)
---|---
$N_{B\kern 1.25995pt\overline{\kern-1.25995ptB}{}}$ | 1.5
$f^{+-}$ | 2.4
$\mathcal{B}_{\rm sub}$ | 0.8
Fit modeling | 1.7
$\pi^{0}$ efficiency | 3.7
Particle-identification efficiency | 0.6
Continuum-suppression efficiency | 1.5
Tracking efficiency | 0.7
Total | 5.3

The branching fraction of the $B^{-}\rightarrow D^{0}\rho^{-}$ decay is given
by

$\mathcal{B}(B^{-}\rightarrow D^{0}\rho^{-})=\frac{N}{2\,N_{B\kern
1.25995pt\overline{\kern-1.25995ptB}{}}\,f^{+-}\,\varepsilon\,\mathcal{B}_{\rm
sub}}\,,$ (3)

where $N$ is the $B^{-}\rightarrow D^{0}\rho^{-}$ yield, $N_{B\kern
1.25995pt\overline{\kern-1.25995ptB}{}}=(387\pm 6)\times 10^{6}$ the number of
$B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ pairs in the sample,
$f^{+-}=0.516\pm 0.012$ the branching fraction of the decay $\mathchar
28935\relax{(4S)}\rightarrow B^{+}B^{-}$ [35], $\varepsilon=(5.71\pm 0.03)\%$
the signal efficiency, and $\mathcal{B}_{\rm sub}=(3.90\pm 0.03)\%$ the
product of the branching fractions of the sub-decays $D^{0}\rightarrow
K^{-}\pi^{+}$ and $\pi^{0}\rightarrow\gamma\gamma$ [5]. The result is

$\mathcal{B}(B^{-}\rightarrow
D^{0}\rho^{-})=({0.939}\pm{0.021}\pm{0.050})\%\,,$ (4)

where the first uncertainty is statistical and the second systematic. This
value already includes the simulation corrections for the efficiency described
below.

Contributions to the systematic uncertainty are reported in Table 1. The
uncertainties on the number $N_{B\kern
1.25995pt\overline{\kern-1.25995ptB}{}}$ of $B\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ pairs, on the branching fraction
$f^{+-}$, and on the sub-decay branching fraction $\mathcal{B}_{\rm sub}$
contribute systematic uncertainties of 1.5%, 2.4%, and 0.8%, respectively.

We calculate systematic uncertainties associated with the choice of the
$\Delta E$ fit model, for each component, by drawing one thousand samples of
pseudodata from an alternative model and fitting them with the model used in
the analysis (nominal model). The alternative models consist of different
probability density functions that fit simulated data with the same quality as
the nominal models. In addition, we generate samples with variations of
parameters that are fixed to the values found from the fit to simulated data
and fit them with the default model, to mimic potential data-simulation
discrepancies. We assign as a systematic uncertainty the observed average
residual, i.e., the difference between the average fit value and the true
value, of the signal $B^{-}\rightarrow D^{0}\rho^{-}$ yield. We sum in
quadrature the systematic uncertainties related to the model choices and to
the fixed parameters to obtain the systematic uncertainty associated with the
fit modeling reported in Table 1.

We validate the background composition by inspecting several distributions in
two sidebands of the diphoton invariant mass (110 –120 and 145
–160${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$), which are enriched in self-
cross-feed and $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ background. A
10% relative uncertainty is assigned to the fraction of self-cross-feed
determined from simulated data in each $\cos\theta_{\rho}$ interval. This
uncertainty is calculated in sideband data by comparing the simulated and
experimental distributions of the $\pi^{0}$ momentum, which is the kinematic
variable most sensitive to variation of the self-cross-feed fraction. Since
this fraction is Gaussian constrained in the $\Delta E$ fit, the corresponding
systematic uncertainty is already included in the statistical uncertainty on
the signal yields.

The signal efficiency is calculated from simulated data; the quoted $0.03\%$
uncertainty is statistical. Control data are used to validate the efficiency
and to correct for data-simulation differences. We propagate uncertainties of
all efficiency corrections as systematic uncertainties on the branching
fraction.

The correction for the $\pi^{0}$-reconstruction efficiency is obtained by
measuring the ratio of the yields of the decays $D^{*+}\rightarrow
D^{0}(\rightarrow K^{-}\pi^{+}\pi^{0})\pi^{+}$ and $D^{*+}\rightarrow
D^{0}(\rightarrow K^{-}\pi^{+})\pi^{+}$, scaled by the inverse of the
branching-fraction ratio. We measure the yield ratio in experimental and
simulated data to determine a per-candidate correction as a function of the
momentum and polar angle of the $\pi^{0}$ candidate. The corrections span a
range between $0.7$ and $1.1$ with an average of $1.011\pm 0.037$. The
uncertainty is dominated by that on the ratio of $D^{0}$-decay branching
fractions [5].

The correction for the efficiency of the kaon selection is obtained using an
abundant control sample of $D^{*+}\rightarrow D^{0}(\rightarrow
K^{-}\pi^{+})\pi^{+}$ decays. We measure the data-to-simulation efficiency
ratio from the control sample and scale the signal efficiency using per-
candidate corrections as a function of the momentum and polar angle of the
kaon candidate. The corrections span a range between $0.7$ and $1.2$ with an
average of $1.009\pm 0.005$. The uncertainty is dominated by that on the
background subtraction of the control channels.

The efficiency of the continuum-suppression requirement is validated using
$B^{-}\rightarrow D^{0}(\rightarrow K^{-}\pi^{+})\pi^{-}$ decays. We use a
signal-enriched region (signal purity of 95%) and a continuum-background-
dominated region (continuum-background fraction of 83%) to compare
distributions of the BDT input variables and BDT output between experimental
and simulated data. No significant discrepancy is found. The ratio of
efficiency of the continuum-suppression requirement in experimental and
simulated control data is $0.987\pm 0.015$, where the uncertainty is
statistical. We use this ratio to correct the signal efficiency and propagate
the corresponding uncertainty to the signal branching fraction.

The tracking efficiency is validated using
$e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events, where one $\tau$ decays
leptonically, $\tau^{+}\rightarrow\ell^{+}\nu_{\ell}\overline{\nu}_{\tau}$
with $\ell=e,\mu$, and the other hadronically,
$\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$. The efficiency to
reconstruct a track is found to be the same for experimental and simulated
data with an uncertainty of $0.24\%$. No correction is applied to the signal
efficiency.

Finally, we validate the overall efficiency and its corrections by measuring
the branching fraction of the $B^{-}\rightarrow D^{0}(\rightarrow
K^{-}\pi^{+}\pi^{0})\pi^{-}$ decay, selected as the signal (except for
requirements on $\pi^{-}\pi^{0}$ invariant-mass and $\cos\theta_{\rho}$).
Although the final-state particles are the same, the kinematics of the control
decay differs from that of the signal decay. This results in a selection
efficiency for the control channel of $(2.98\pm 0.03)$%, where the difference
with that of the signal is mainly due to the charged pion accompanying the
$D^{0}$ candidate in the $B^{-}\rightarrow D^{0}\pi^{-}$ decay. However, for
the other particles, the $B^{-}\rightarrow D^{0}(\rightarrow
K^{-}\pi^{+}\pi^{0})\pi^{-}$ decay covers most of the relevant
$B^{-}\rightarrow D^{0}\rho^{-}$ phase-space. In particular, the neutral pion
from the $B^{-}\rightarrow D^{0}(\rightarrow K^{-}\pi^{+}\pi^{0})\pi^{-}$
decay has a softer momentum distribution that peaks in the region where the
efficiency corrections are larger. As in the signal analysis, the control
sample is categorized by three components: signal $B^{-}\rightarrow
D^{0}(\rightarrow K^{-}\pi^{+}\pi^{0})\pi^{-}$ (75% fraction), $B\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ background (21%), and self-cross-feed
(3%). Continuum background is negligible. To determine the signal yield, we
use a maximum likelihood fit to the unbinned $\Delta E$ distribution using
probability density functions similar to those described for the signal
$\Delta E$ fit. We obtain $7700\pm 90$ $B^{-}\rightarrow D^{0}(\rightarrow
K^{-}\pi^{+}\pi^{0})\pi^{-}$ decays, which yields a branching fraction of
$(0.454\pm 0.022)$% in agreement with the world average value, $(0.461\pm
0.010)$% [5]. The result includes the efficiency corrections and their
uncertainties, as well as those due to the number of $B\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ pairs, in $f^{+-}$, and in the sub-
decay branching fractions.

Using our result, we update the value of $R$ and $\cos\delta$ in Eqs (1) and
(2). Taking the branching fractions $\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\rho^{-})=(0.76\pm
0.12)\%$ and $\mathcal{B}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{0}\rho^{0})=(0.0321\pm 0.0021)\%$, and the ratio of lifetimes
$\tau_{+}/\tau_{0}=1.076\pm 0.004$ [5], we obtain $R=0.93^{+0.11}_{-0.12}$ and
$\cos\delta=0.919^{+0.012}_{-0.009}$, which agree with and are significantly
more precise than previous determinations [15]. These results confirm
expectations from factorization in the heavy-quark limit.

In conclusion, we measure the branching fraction of the $B^{-}\rightarrow
D^{0}\rho^{-}$ decay using $e^{+}e^{-}$-collision data collected by the Belle
II detector at the $\mathchar 28935\relax{(4S)}$ resonance and containing 387
million $B\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ meson pairs. The
result is $({0.939}\pm{0.021}\mathrm{(stat)}\pm{0.050}\mathrm{(syst)})\%$, in
agreement with previous determinations. Our measurement improves the
fractional precision of the world average [5] by more than a factor of two and
will also significantly ameliorate the calibration factor of the Belle II
hadronic-tagging algorithm.

This work, based on data collected using the Belle II detector, which was
built and commissioned prior to March 2019, was supported by Higher Education
and Science Committee of the Republic of Armenia Grant No. 23LCG-1C011;
Australian Research Council and Research Grants No. DP200101792, No.
DP210101900, No. DP210102831, No. DE220100462, No. LE210100098, and No.
LE230100085; Austrian Federal Ministry of Education, Science and Research,
Austrian Science Fund No. P 31361-N36 and No. J4625-N, and Horizon 2020 ERC
Starting Grant No. 947006 “InterLeptons”; Natural Sciences and Engineering
Research Council of Canada, Compute Canada and CANARIE; National Key R&D
Program of China under Contract No. 2022YFA1601903, National Natural Science
Foundation of China and Research Grants No. 11575017, No. 11761141009, No.
11705209, No. 11975076, No. 12135005, No. 12150004, No. 12161141008, and No.
12175041, and Shandong Provincial Natural Science Foundation Project
ZR2022JQ02; the Czech Science Foundation Grant No. 22-18469S; European
Research Council, Seventh Framework PIEF-GA-2013-622527, Horizon 2020 ERC-
Advanced Grants No. 267104 and No. 884719, Horizon 2020 ERC-Consolidator Grant
No. 819127, Horizon 2020 Marie Sklodowska-Curie Grant Agreement No. 700525
“NIOBE” and No. 101026516, and Horizon 2020 Marie Sklodowska-Curie RISE
project JENNIFER2 Grant Agreement No. 822070 (European grants); L’Institut
National de Physique Nucléaire et de Physique des Particules (IN2P3) du CNRS
and L’Agence Nationale de la Recherche (ANR) under grant ANR-21-CE31-0009
(France); BMBF, DFG, HGF, MPG, and AvH Foundation (Germany); Department of
Atomic Energy under Project Identification No. RTI 4002, Department of Science
and Technology, and UPES SEED funding programs No. UPES/R&D-SEED-
INFRA/17052023/01 and No. UPES/R&D-SOE/20062022/06 (India); Israel Science
Foundation Grant No. 2476/17, U.S.-Israel Binational Science Foundation Grant
No. 2016113, and Israel Ministry of Science Grant No. 3-16543; Istituto
Nazionale di Fisica Nucleare and the Research Grants BELLE2; Japan Society for
the Promotion of Science, Grant-in-Aid for Scientific Research Grants No.
16H03968, No. 16H03993, No. 16H06492, No. 16K05323, No. 17H01133, No.
17H05405, No. 18K03621, No. 18H03710, No. 18H05226, No. 19H00682, No.
20H05850, No. 20H05858, No. 22H00144, No. 22K14056, No. 22K21347, No.
23H05433, No. 26220706, and No. 26400255, and the Ministry of Education,
Culture, Sports, Science, and Technology (MEXT) of Japan; National Research
Foundation (NRF) of Korea Grants No. 2016R1D1A1B02012900, No.
2018R1A2B3003643, No. 2018R1A6A1A06024970, No. 2019R1I1A3A01058933, No.
2021R1A6A1A03043957, No. 2021R1F1A1060423, No. 2021R1F1A1064008, No.
2022R1A2C1003993, and No. RS-2022-00197659, Radiation Science Research
Institute, Foreign Large-Size Research Facility Application Supporting
project, the Global Science Experimental Data Hub Center of the Korea
Institute of Science and Technology Information and KREONET/GLORIAD;
Universiti Malaya RU grant, Akademi Sains Malaysia, and Ministry of Education
Malaysia; Frontiers of Science Program Contracts No. FOINS-296, No. CB-221329,
No. CB-236394, No. CB-254409, and No. CB-180023, and SEP-CINVESTAV Research
Grant No. 237 (Mexico); the Polish Ministry of Science and Higher Education
and the National Science Center; the Ministry of Science and Higher Education
of the Russian Federation and the HSE University Basic Research Program,
Moscow; University of Tabuk Research Grants No. S-0256-1438 and No.
S-0280-1439 (Saudi Arabia); Slovenian Research Agency and Research Grants No.
J1-9124 and No. P1-0135; Agencia Estatal de Investigacion, Spain Grant No.
RYC2020-029875-I and Generalitat Valenciana, Spain Grant No.
CIDEGENT/2018/020; National Science and Technology Council, and Ministry of
Education (Taiwan); Thailand Center of Excellence in Physics; TUBITAK ULAKBIM
(Turkey); National Research Foundation of Ukraine, Project No. 2020.02/0257,
and Ministry of Education and Science of Ukraine; the U.S. National Science
Foundation and Research Grants No. PHY-1913789 and No. PHY-2111604, and the
U.S. Department of Energy and Research Awards No. DE-AC06-76RLO1830, No. DE-
SC0007983, No. DE-SC0009824, No. DE-SC0009973, No. DE-SC0010007, No. DE-
SC0010073, No. DE-SC0010118, No. DE-SC0010504, No. DE-SC0011784, No. DE-
SC0012704, No. DE-SC0019230, No. DE-SC0021274, No. DE-SC0021616, No. DE-
SC0022350, No. DE-SC0023470; and the Vietnam Academy of Science and Technology
(VAST) under Grants No. NVCC.05.12/22-23 and No. DL0000.02/24-25.

These acknowledgements are not to be interpreted as an endorsement of any
statement made by any of our institutes, funding agencies, governments, or
their representatives.

We thank the SuperKEKB team for delivering high-luminosity collisions; the KEK
cryogenics group for the efficient operation of the detector solenoid magnet;
the KEK Computer Research Center for on-site computing support; the NII for
SINET6 network support; and the raw-data centers hosted by BNL, DESY, GridKa,
IN2P3, INFN, and the University of Victoria.

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# Gerghaty type results via simulation and $\mathcal{C}$-class functions with
application

Azhar Hussain, Muhammad Ishfaq Tanzeela Kanwal Stojan Radenović  Department of
Mathematics, University of Sargodha, Sargodha-40100, Pakistan. Email:
<EMAIL_ADDRESS>Department of Mathematics, University of Sargodha,
Sargodha-40100, Pakistan. Email: ishfaqahmad5632@gmail.com,Department of
Mathematics, University of Sargodha, Sargodha-40100, Pakistan. Email:
tanzeelakanwal16@gmail.com,University of Belgrade, Faculty of Mechanical
Engineering, Kraljice Marije 16, 11120 Beograd 35, Serbia, Department of
Mathematics, College of Science, King Saud University, Saudi Arabia, Email:
<EMAIL_ADDRESS>
Abstract: In this paper we study the notion of Gerghaty type contractive
mapping via simulation function along with $\mathcal{C}$-class functions and
prove the existence of several fixed point results in ordinary and partially
ordered metric spaces. An example is given to show the validity of our results
given herein. Moreover, existence of solution of two-point boundary value
second order nonlinear differential equation is obtain.

Mathematics Subject Classification: 54H25, 47H10

Keywords: Simulation functions, $\mathcal{C}$-class function, partially
ordered metric space.

##  1 Introduction

The Banach contraction principle [4] is one of the fundamental result in
metric fixed point theory. Because of its importance in nonlinear analysis,
number of authors have improved, generalized and extended this basic result
either by defining a new contractive mapping in the context of a complete
metric space or by investigating the existing contractive mappings in various
abstract spaces (see, e.g., [1, 5, 6, 7, 18, 26] and references therein).

In particularly, Geraghty [8] consider an auxiliary function and generalized
the Banach contraction in the frame work of complete metric space. Later on,
Amini-Harandi and Emami [26] obtained similar results in the setting of
partially ordered metric spaces. Using the concept of Samet [9], Cho et al.
[6] generalized Geraghty contraction to $\alpha$-Geraghty contraction and
prove a fixed point theorem for such contraction. On the other hand, Khojasteh
et al. [13] introduced the notion of $\mathcal{Z}$-contraction by using a
function called simulation function and proved a version of Banach contraction
principle.

##  2 Preliminaries

In this section we present some basic notions and results from the literature:
We denote by $\mathcal{F}$ the class of all functions
$\beta:[0,\infty)\rightarrow[0,1)$ satisfying $\beta(t_{n})\to 1$, implies
$t_{n}\rightarrow 0$ as $n\to\infty$.

######  Definition 2.1.

[8] Let $(X,d)$ be a metric space. A map $T:X\rightarrow X$ is called Geraghty
contraction if there exists $\beta\in\mathcal{F}$ such that for all $x,y\in
X$,

$d(Tx,Ty)\leq\beta(d(x,y))d(x,y).$

######  Theorem 2.1.

[8] Let $(X,d)$ be a complete metric space. Mapping $T:X\rightarrow X$ is
Geraghty contraction. Then $T$ has a fixed point $x\in X$, and
$\\{T^{n}x_{1}\\}$ converges to $x$.

In 2015, Khojasteh et al. [13] introduced simulation function
$\zeta:[0,\infty)\times[0,\infty)\rightarrow\mathbb{R}$, satisfying the
following assertions:

$(\zeta_{1})$

$\zeta(0,0)=0$;

$(\zeta_{2})$

$\zeta(t,s)<s-t$ for all $t,s>0$;

$(\zeta_{3})$

If $\\{t_{n}\\},\\{s_{n}\\}$ are sequences in $(0,\infty)$ such that
$\lim\limits_{n\rightarrow\infty}t_{n}=\lim\limits_{n\to\infty}s_{n}>0$ then

$\lim\limits_{n\rightarrow\infty}sup~{}\zeta(t_{n},s_{n})<0$

######  Definition 2.2.

[13] Let $(X,d)$ be a metric space, $T:X\rightarrow X$ a mapping and $\zeta$ a
simulation function. Then $T$ is called a $\mathcal{Z}$-contraction with
respect to $\zeta$ if it satisfies

$\zeta(d(Tu,Tv),d(u,v))\geq 0~{}~{}\text{for all}~{}u,v\in X.$ (1)

######  Theorem 2.2.

[13] Let $(X,d)$ be a complete metric space and $T:X\rightarrow X$ be a
$\mathcal{Z}$-contraction with respect to $\zeta$. Then $T$ has a unique fixed
point $u\in X$ and for every $x_{0}\in X$, the Picard sequence $\\{x_{n}\\}$
where $x_{n}=Tx_{n-1}$ for all $n\in\mathbb{N}$ converges to this fixed point
of $T$.

######  Example 2.1.

[13] Let
$\zeta_{i}:[0,\infty)\times[0,\infty)\rightarrow\mathbb{R},~{}~{}i=1,2,3$ be
defined by

$(i)$

$\zeta_{1}(t,s)=\lambda s-t$, where $\lambda\in(0,1)$;

$(ii)$

$\zeta_{2}(t,s)=s\varphi(s)-t$, where $\varphi:[0,\infty)\rightarrow[0,1)$ is
a mapping such that $\lim\limits_{t\rightarrow r^{+}}sup~{}\psi(t)<1$ for all
$r>0$;

$(iii)$

$\zeta_{3}=s-\psi(s)-t$, where $\psi:[0,\infty)\rightarrow[0,\infty)$ is a
continuous function such that $\psi(t)=0$ if and only if $t=0.$

Then $\zeta_{i}$ for $i=1,2,3$ are simulation functions.

Rold$\acute{a}$n-L$\acute{o}$pez-de-Hierro et al. [22] modified the notion of
a simulation function by replacing $(\zeta_{3})$ by $(\zeta^{\prime}_{3})$,

($\zeta^{\prime}_{3}$)

: if $\\{t_{n}\\},\\{s_{n}\\}$ are sequences in $(0,\infty)$ such that
$\lim_{n\rightarrow\infty}t_{n}=\lim\limits_{n\rightarrow\infty}s_{n}>0$ and
$t_{n}<s_{n}$, then

$\lim\limits_{n\rightarrow\infty}sup\zeta(t_{n},s_{n})<0.$

The function $\zeta:[0,\infty)\times[0,\infty)\rightarrow\mathbb{R}$
satisfying $(\zeta_{1}-\zeta{2})$ and $(\zeta^{\prime}_{3})$ is called
simulation function in the sense of Rold$\acute{a}$n-L$\acute{o}$pez-de-
Hierro.

######  Definition 2.3.

[2] A mapping $\mathcal{G}:[0,+\infty)^{2}\rightarrow\mathbb{R}$ is called a
$\mathcal{C}$-class function if it is continuous and satisfies the following
conditions:

(1)

$\mathcal{G}(s,t)\leq s$;

(2)

$\mathcal{G}(s,t)=s$ implies that either $s=0$ or $t=0$, for all
$s,t\in[0,+\infty)$.

######  Definition 2.4.

[16] A mapping $\mathcal{G}:[0,+\infty)^{2}\rightarrow\mathbb{R}$ has the
property $\mathcal{C}_{\mathcal{G}}$, if there exists and
$\mathcal{C}_{\mathcal{G}}\geq 0$ such that

(1)

$\mathcal{G}(s,t)>\mathcal{C}_{\mathcal{G}}$ implies $s>t$;

(2)

$\mathcal{G}(s,t)\leq\mathcal{C}_{\mathcal{G}}$, for all $t\in[0,+\infty)$.

Some examples of $\mathcal{C}$-class functions that have property
$\mathcal{C}_{\mathcal{G}}$ are as follows:

(a)

$\mathcal{G}(s,t)=s-t,~{}\mathcal{C}_{\mathcal{G}}=r,r\in[0,+\infty)$;

(b)

$\mathcal{G}(s,t)=s-\frac{(2+t)t}{(1+t)},\mathcal{C}_{\mathcal{G}}=0$;

(c)

$\mathcal{G}(s,t)=\frac{s}{1+kt},k\geq
1,\mathcal{C}_{\mathcal{G}}=\frac{r}{1+k},r\geq 2$.

For more examples of $\mathcal{C}$-class functions that have property
$\mathcal{C}_{\mathcal{G}}$ see [3, 7, 16].

######  Definition 2.5.

[16] A $\mathcal{C}_{\mathcal{G}}$ simulation function is a mapping
$\mathcal{G}:[0,+\infty)^{2}\rightarrow\mathcal{R}$ satisfying the following
conditions:

(1)

$\zeta(t,s)<\mathcal{G}(s,t)$ for all $t,s>0$, where
$\mathcal{G}:[0,+\infty)^{2}\rightarrow\mathbb{R}$ is a $\mathcal{C}$-class
function;

(2)

if $\\{t_{n}\\},\\{s_{n}\\}$ are sequences in $(0,+\infty)$ such that
$\lim\limits_{n\rightarrow\infty}t_{n}=\lim\limits_{n\rightarrow\infty}s_{n}>0$,
and $t_{n}<s_{n}$, then
$\lim\limits_{n\rightarrow\infty}\sup\zeta(t_{n},s_{n})<\mathcal{C}_{\mathcal{G}}$.

Some examples of simulation functions and
$\mathcal{C}_{\mathcal{G}}$-simulation functions are:

(1)

$\zeta(t,s)=\frac{s}{s+1}-t$ for all $t,s>0$.

(2)

$\zeta(t,s)=s-\phi(s)-t$ for all $t,s>0$, where
$\phi:[0,+\infty)\rightarrow[0,+\infty)$ is a lower semi continuous function
and $\phi(t)=0$ if and only if $t=0$.

For more examples of simulation functions and
$\mathcal{C}_{\mathcal{G}}$-simulation functions see [3, 22, 13, 16, 17, 25].

######  Definition 2.6.

[9] Let $T:A\rightarrow B$ be a map and $\alpha:X\times
X\rightarrow\mathbb{R}$ be a function. Then $T$ is said to be
$\alpha$-admissible if $\alpha(x,y)\geq 1$ implies $\alpha(Tx,Ty)\geq 1.$

######  Definition 2.7.

[12] An $\alpha$-admissible map $T$ is said to be triangular
$\alpha$-admissible if $\alpha(x,z)\geq 1$ and $\alpha(z,y)\geq 1$ implies
$\alpha(x,y)\geq 1$

Cho et al. [6] generalized the concept of Geraghty contraction to
$\alpha$-Geraghty contraction and prove the fixed point theorem for such
contraction.

######  Definition 2.8.

[6] Let $(X,d)$ be a metric space, and let $\alpha:X\times
X\rightarrow\mathbb{R}$ be a function. A map $T:X\rightarrow X$ is called
$\alpha$-Geraghty contraction if there exists $\beta\in\mathcal{F}$ such that
for all $x,y\in X$,

$\alpha(x,y)d(Tx,Ty)\leq\beta(d(x,y))d(x,y).$

######  Theorem 2.3.

[6] Let $(X,d)$ be a complete metric space, $\alpha:X\times
X\rightarrow\mathbb{R}$ be a function. Define a map $T:X\rightarrow X$
satisfying the following conditions:

  1. 1.

$T$ is continuous $\alpha$-Geraghty contraction;

  2. 2.

$T$ be a triangular $\alpha$-admissible;

  3. 3.

there exists $x_{1}\in X$ such that $\alpha(x_{1},Tx_{1})\geq 1$;

Then $T$ has a fixed point $x\in X$, and $\\{T^{n}x_{1}\\}$ converges to $x$.

######  Lemma 2.1.

[12] Let $T:X\rightarrow X$ be a triangular $\alpha$-admissible map. Assume
that there exists $x_{1}\subset X$ such that $\alpha(x_{1},Tx_{1})\geq
1$.Define a sequence $\\{x_{n}\\}$ by $x_{n+1}=Tx_{n}$. Then we have
$\alpha(x_{n},x_{m})\geq 1$ for all $m,n\in\mathbb{N}$ with $n<m$.

######  Lemma 2.2.

[20] Let $(X,d)$ be a metric space and let $\\{x_{n}\\}$ be a sequence in $X$
such that

$\lim\limits_{n\rightarrow\infty}d(x_{n},x_{n+1})=0.$ (2)

If $\\{x_{n}\\}$ is not a Cauchy sequence in $X$, then there exists
$\varepsilon>0$ and two sequences $x_{m(k)}$ and $x_{n(k)}$ of positive
integers such that $x_{n(k)}>x_{m(k)}>k$ and the following sequences tend to
$\varepsilon^{+}$ when $k\rightarrow\infty$:

$d(x_{m(k)},x_{n(k)}),d(x_{m(k)},x_{n(k)+1}),d(x_{m(k)-1},x_{n(k)}),$
$d(x_{m(k)-1},x_{n(k)+1}),d(x_{m(k)+1},x_{n(k)+1}).$

Motivated by the above results, we introduce the notion of Gerghaty type
$\mathcal{Z}_{(\alpha,\mathcal{G})}$-contraction and prove some fixed point
results in metric and partially ordered metric spaces. An example to prove the
validity and application to nonlinear differential equation for the usability
of our results is presented.

##  3 Fixed point results in usual metric space

We begin with the following notion:

######  Definition 3.1.

Let $(X,d)$ be a metric space and $\alpha:X\times X\rightarrow[0,\infty)$ be a
function. A mapping $T:X\rightarrow X$ is called a
$\mathcal{Z}_{(\alpha,\mathcal{G})}$-Geraghty contraction if there exists
$\beta\in\mathcal{F}$ such that for all $x,y\in X$

$\zeta(\alpha(x,y)d(Tx,Ty),\beta(M(x,y))M(x,y))\geq\mathcal{C}_{\mathcal{G}}$
(3)

where

$M(x,y)=\max\\{d(x,y),d(x,Tx),d(y,Ty)\\}.$

######  Remark 3.1.

Since the functions belonging to $\mathcal{F}$ are strictly smaller than 1,
$(\ref{E3.1})$ implies that

$d(Tx,Ty)<M(x,y)$

for any $x,y\in X$ with $x\neq y$ and for $\alpha(x,y)\geq 1$,
$\zeta(t,s)<\mathcal{G}(s,t)=s-t$.

######  Theorem 3.1.

Let $(X,d)$ be a complete metric space, $\alpha:X\times
X\rightarrow[0,\infty)$ and $T:X\rightarrow X$ be two functions. Suppose that
the following conditions are satisfied:

(1)

$T$ is continuous $Z_{(\alpha,\mathcal{G})}$-Geraghty contraction;

(2)

$T$ is triangular $\alpha$-admissible;

(3)

there exists $x_{1}\in X$ such that $\alpha(x_{1},Tx_{1})\geq 1$;

(4)

$T$ is continuous.

Then $T$ has a fixed point $x^{*}\in X$ and $T$ is a Picard operator that is,
${T^{n}x_{1}}$ converges to $x^{*}$.

###### Proof.

Let $x_{1}\in X$ be such that $\alpha(x_{1},Tx_{1})\geq 1$. Define a sequence
$\\{x_{n}\\}\subset X$ by $x_{n+1}=Tx_{n}$ for $n\in\mathbb{N}$. If
$x_{n_{0}}=x_{n_{0}+1}$ for some $n_{0}\in\mathbb{N}$, then $x_{n_{0}}$ is a
fixed point of $T$ and hence the proof is completed. Thus, we assume that
$x_{n}\neq x_{n+1}$ for all $n\in\mathbb{N}$. By Lemma 2.1, we have

$\alpha(x_{n},x_{n+1})\geq 1$ (4)

for all $n\in\mathbb{N}$. Then

$\displaystyle d(x_{n+1},x_{n+2})$ $\displaystyle=$ $\displaystyle
d(Tx_{n},Tx_{n+1})$ (5) $\displaystyle\leq$
$\displaystyle\alpha(x_{n},x_{n+1})d(Tx_{n},Tx_{n+1}).$

Since $T$ is a $\mathcal{Z}_{(\alpha,\mathcal{G})}$-Geraghty contraction, we
have

$\displaystyle\mathcal{C}_{\mathcal{G}}$ $\displaystyle\leq$
$\displaystyle\zeta(\alpha(x_{n},x_{n+1})d(Tx_{n},Tx_{n+1}),\beta(M(x_{n},x_{n+1}))M(x_{n},x_{n+1}))$
$\displaystyle<$
$\displaystyle\mathcal{G}(\beta(M(x_{n},x_{n+1}))M(x_{n},x_{n+1}),\alpha(x_{n},x_{n+1})d(Tx_{n},Tx_{n+1})).$

Using $(\mathcal{G}_{1})$, we obtain

$\alpha(x_{n},x_{n+1})d(Tx_{n},Tx_{n+1})<\beta(M(x_{n},x_{n+1}))M(x_{n},x_{n+1}).$
(6)

From (5) and (6), we have

$d(x_{n+1},x_{n+2})<\beta(M(x_{n},x_{n+1}))M(x_{n},x_{n+1})$ (7)

for all $n\in\mathbb{N}$, where

$\displaystyle M(x_{n},x_{n+1})$ $\displaystyle=$
$\displaystyle\max\\{d(x_{n},x_{n+1}),d(x_{n},Tx_{n}),d(x_{n+1},Tx_{n+1})\\}$
$\displaystyle=$
$\displaystyle\max\\{d(x_{n},x_{n+1}),d(x_{n},x_{n+1}),d(x_{n+1},x_{n+2})\\}$
$\displaystyle=$ $\displaystyle\max\\{d(x_{n},x_{n+1}),d(x_{n+1},x_{n+2})\\}.$

If $M(x_{n},x_{n+1})=d(x_{n+1},x_{n+2})$, then by definition of $\beta$, we
have

$\displaystyle d(x_{n+1},x_{n+2})$ $\displaystyle<$
$\displaystyle\beta(d(x_{n},x_{n+1})d(x_{n},x_{n+1}))$ $\displaystyle<$
$\displaystyle d(x_{n+1},x_{n+2})$

a contradiction. Thus we conclude that $M(x_{n},x_{n+1})=d(x_{n},x_{n+1})$ for
all $n\in\mathbb{N}$ and so the sequence $\\{d(x_{n},x_{n+1})\\}$ of real
numbers is decreasing and bounded below by zero. Hence there exists $r\geq 0$
such that

$\lim\limits_{n\rightarrow\infty}d(x_{n},x_{n+1})=r.$

We claim that $r=0$. Suppose on contrary that $r>0$, then by (7) we have

$\displaystyle\frac{d(x_{n+1},x_{n+2})}{d(x_{n},x_{n+1})}\leq\beta(d(x_{n},x_{n+1})<1$

which yields that $\lim\limits_{n\rightarrow\infty}\beta(d(x_{n},x_{n+1})=1$.
Since $\beta\in\mathcal{F}$, we get that

$\lim\limits_{n\rightarrow\infty}d(x_{n},x_{n+1})=0.$ (8)

We now show that $\\{x_{n}\\}$ is a Cauchy sequence. Suppose on contrary that
it is not. Thus there exists $\epsilon>0$ such that for all $k>0$,
$m(k)>n(k)>k$ with the (smallest number satisfying the condition below)
$d(x_{m(k)},x_{n(k)})\geq\epsilon$ and $d(x_{m(k)-1},x_{n(k)})<\epsilon$. Then
we have

$\displaystyle\epsilon$ $\displaystyle\leq$ $\displaystyle
d(x_{m(k)},x_{n(k)})$ $\displaystyle\leq$ $\displaystyle
d(x_{m(k)},x_{m(k)-1})+d(x_{m(k)-1},x_{n(k)})$ $\displaystyle\leq$
$\displaystyle d(x_{m(k)},x_{m(k)-1})+\epsilon.$

Letting $k\rightarrow\infty$ in the above inequality, we have

$\lim\limits_{k\to\infty}d(x_{m(k)},x_{n(k)})=\epsilon.$ (9)

By using (8) and (9), we obtain

$\lim\limits_{k\to\infty}d(x_{m(k)-1},x_{n(k)-1})=\epsilon.$

By Lemma 2.1, $\alpha(x_{m(k)-1},x_{n(k)-1})\geq 1$, thus

$\displaystyle d(x_{m(k)},x_{n(k)})$ $\displaystyle=$ $\displaystyle
d(Tx_{m(k)-1},Tx_{n(k)-1})$ (10) $\displaystyle\leq$
$\displaystyle\alpha(x_{m(k)-1},x_{n(k)-1})d(Tx_{m(k)-1},Tx_{n(k)-1}).$

Since $T$ is a $\mathcal{Z}_{(\alpha,\mathcal{G})}$-Geraghty contraction, we
have

$\displaystyle\zeta(\alpha(x_{m(k)-1},x_{n(k)-1})d(Tx_{m(k)-1},Tx_{n(k)-1}),\beta(M(x_{m(k)-1},x_{n(k)-1}))M(x_{m(k)-1},x_{n(k)-1}))\geq\mathcal{C}_{\mathcal{G}}.$

This implies

$\displaystyle\mathcal{C}_{\mathcal{G}}$ $\displaystyle\leq$
$\displaystyle\zeta(\alpha(x_{m(k)-1},x_{n(k)-1})d(Tx_{m(k)-1},Tx_{n(k)-1}),\beta(M(x_{m(k)-1},x_{n(k)-1}))M(x_{m(k)-1},x_{n(k)-1}))$
$\displaystyle<$
$\displaystyle\mathcal{G}(\beta(M(x_{m(k)-1},x_{n(k)-1}))M(x_{m(k)-1},x_{n(k)-1}),\alpha(x_{m(k)-1},x_{n(k)-1})d(Tx_{m(k)-1},Tx_{n(k)-1})).$

Using $(\mathcal{G}_{1})$, we obtain

$\alpha(x_{m(k)-1},x_{n(k)-1})d(Tx_{m(k)-1},Tx_{n(k)-1}))<\beta(M(x_{m(k)-1},x_{n(k)-1}))M(x_{m(k)-1},x_{n(k)-1}),$
(11)

where

$\displaystyle M((x_{m(k)-1},x_{n(k)-1}))$ $\displaystyle=$
$\displaystyle\max\\{d(x_{m(k)-1},x_{n(k)-1}),d((x_{m(k)-1},Tx_{m(k)-1}),d(x_{n(k)-1},Tx_{n(k)-1})\\}$
$\displaystyle=$
$\displaystyle\max\\{d(x_{m(k)-1},x_{n(k)-1}),d(x_{m(k)-1},x_{m(k)}),d(x_{n(k)-1},x_{n(k)})\\}.$

If $M(x_{m(k)-1},x_{n(k)-1})=d(x_{m(k)-1},x_{m(k)})$, we have

$\displaystyle\alpha(x_{m(k)-1},x_{n(k)-1})d(x_{m(k)},x_{n(k)})$
$\displaystyle<$
$\displaystyle\beta(d(x_{m(k)-1},x_{m(k)}))d(x_{m(k)-1},x_{m(k)})$
$\displaystyle<$ $\displaystyle d(x_{m(k)-1},x_{m(k)})$

a contradiction. Similarly, we have contradiction when
$M(x_{m(k)-1},x_{n(k)-1})=d(x_{n(k)-1},x_{n(k)})$. Thus we conclude that
$M(x_{m(k)-1},x_{n(k)-1})=d(x_{m(k)-1},x_{n(k)-1})$. So

$\displaystyle\frac{d(x_{m(k)},x_{n(k)})}{d(x_{m(k)-1},x_{n(k)-1})}\leq\beta(d(x_{m(k)-1},x_{n(k)-1})<1.$

Letting $k\rightarrow\infty$ in above inequality, we derive that

$\lim\limits_{k\rightarrow\infty}\beta(d(x_{m(k)-1},x_{n(k)-1}))=1.$

This implies

$\lim\limits_{k\rightarrow\infty}d(x_{m(k)-1},x_{n(k)-1})=0.$

Hence $\epsilon=o$, which is a contradiction. Thus we conclude that
$\\{x_{n}\\}$ is a Cauchy sequence. It follows from completeness of $X$ that
there exists $x^{*}\in X$ such that

$\lim\limits_{n\rightarrow\infty}x_{n}=x^{*}.$

Since $T$ is continuous, we get
$\lim\limits_{n\rightarrow\infty}x_{n+1}=Tx_{n}$ and so $x^{*}=Tx^{*}$. This
completes the proof. ∎

######  Theorem 3.2.

Let $(X,d)$ be a complete metric space, $\alpha:X\times
X\rightarrow[0,\infty)$ and $T:X\rightarrow X$ be two mappings. Suppose that
the following conditions are satisfied:

(1)

$T$ is $\mathcal{Z}_{(\alpha,\mathcal{G})}$-Geraghty contraction;

(2)

$T$ is triangular $\alpha$-admissible;

(3)

there exists $x_{1}\in X$ such that $\alpha(x_{1},Tx_{1})\geq 1$;

(4)

if $x_{n}$ is a sequence in $X$ such that $\alpha(x_{n},x_{n+1})\geq 1$ for
all $n$ and $x_{n}\rightarrow x^{\ast}\in X$ as $n\rightarrow\infty$, then
there exist a subsequence $\\{x_{n(k)}\\}$ of $\\{x_{n}\\}$ such that
$\alpha(x_{n(k)},x^{\ast})\geq 1$ for all $k$.

Then $T$ has a fixed point $x^{*}\in X$ and $T$ is a Picard operator, that is,
${T^{n}x_{1}}$ converges to $x^{*}$ .

###### Proof.

Following the arguments those given in Theorem 3.1, we conclude that the
sequence ${x_{n}}$ defined by $x_{n+1}=Tx_{n}$ for all $n\geq 0$, converges to
$x^{*}\in X$. By condition (4) we deduce that there exists a subsequence
$\\{x_{n(k)}\\}$ of $\\{x_{n}\\}$ such that $\alpha(x_{n(k)},x^{*})\geq 1$ for
all $k$. Also

$\displaystyle d(x_{n(k)+1},Tx^{*})$ $\displaystyle=$ $\displaystyle
d(Tx_{n(k)},Tx^{*})$ (12) $\displaystyle\leq$
$\displaystyle\alpha(x_{n(k)},x^{*})d(Tx_{n(k)},Tx^{*}).$

Since $T$ is $\mathcal{Z}_{(\alpha,\mathcal{G})}$-Geraghty contraction, we
have

$\displaystyle\mathcal{C}_{\mathcal{G}}\leq\zeta(\alpha(x_{n(k)},x^{*})d(Tx_{n(k)},Tx_{*}),\beta(M(x_{n(k)},x^{*}))M(x_{n(k)},x^{*}))$
$\displaystyle<\mathcal{G}(\beta(M(x_{n(k)},x^{*}))M(x_{n(k)},x^{*})),\alpha(x_{n(k)},x^{*})d(Tx_{n(k)},Tx^{*})).$

By definition of $\mathcal{G}$, we get that

$\alpha(x_{n(k)},x^{*})d(Tx_{n(k)},Tx^{*})<\beta(M(x_{n(k)},x^{*}))M(x_{n(k)},x^{*})).$
(13)

From (12) and (13), we have

$d(x_{n(k)+1},Tx^{*})<\beta(M(x_{n(k)},x^{*}))M(x_{n(k)},x^{*})),$ (14)

where

$\displaystyle M(x_{n(k)},x^{*})$ $\displaystyle=$
$\displaystyle\max\\{d(x_{n(k)},x^{*}),d(x_{n(k)},Tx_{n(k)}),d(x^{*},Tx^{*})\\}$
$\displaystyle=$
$\displaystyle\max\\{d(x_{n(k)},x^{*}),d(x_{n(k)},x_{n(k)+1}),d(x^{*},Tx^{*})\\}.$

Letting $k\rightarrow\infty$ in the above inequality, we get that

$\lim_{k\rightarrow\infty}M(x_{n(k)},x^{*})=d(x^{*},Tx^{*}).$ (15)

Suppose $d(x^{*},Tx^{*})>0$. By definition of $\beta$ and (14), we have

$d(x_{n(k)+1},Tx^{*})<M(x_{n(k)},x^{*}).$

Letting $k\rightarrow\infty$ in the above inequality and using (15), we obtain
that

$d(x^{*},Tx^{*})<d(x^{*},Tx^{*}),$

a contradiction. Thus $d(x^{*},Tx^{*})=0$, that is, $x^{*}=Tx^{*}.$ ∎

For the uniqueness of fixed point, we consider the following hypothesis:

$(\mathcal{U})$ For all $x,y\in Fix(T)$, there exists $z\in X$ such that
$\alpha(x,z)\geq 1$ and $\alpha(y,z)\geq 1$. Here, $Fix(T)$ denotes the set of
fixed points of $T$.

######  Theorem 3.3.

Adding condition $(\mathcal{\mathcal{U}})$ to the hypothesis of Theorem 3.1
(resp. Theorem 3.2), we obtain that $x^{*}$ is the unique fixed point of $T$.

###### Proof.

From Theorem 3.1 (resp. Theorem 3.2), we have a fixed point, namely $x^{*}\in
X$ of $T$. For uniqueness, suppose there is another fixed point of $T$, say,
$y^{\ast}\in X$. Then, by assumption $(\mathcal{U})$, there exists $z\in X$
such that

$\alpha(x^{*},z)\geq 1~{}~{}and~{}~{}\alpha(y^{*},z)\geq 1.$

Since $T$ is a $\alpha$-admissible, we have

$\alpha(x^{*},T^{n}z)\geq 1~{}~{}and~{}~{}\alpha(y^{*},T^{n}z)\geq 1$

for all $n$. Hence we have

$\displaystyle d(x^{*},T^{n}z)$ $\displaystyle=$ $\displaystyle
d(Tx^{*},TT^{n-1}z)$ (16) $\displaystyle\leq$
$\displaystyle\alpha(x^{*},T^{n-1}z)d(Tx^{*},TT^{n-1}z).$ (17)

Since $T$ is a $\mathcal{Z}_{(\alpha,\mathcal{G})}$-Geraghty contraction, we
have

$\displaystyle\mathcal{C}_{\mathcal{G}}\leq\zeta(\alpha(x^{*},T^{n-1}z)d(Tx^{*},TT^{n-1}z),\beta(M(x^{*},T^{n-1}z))M(x^{*},T^{n-1}z)$
$\displaystyle<\mathcal{G}(\beta(M(x^{*},T^{n-1}z))M(x^{*},T^{n-1}z),\alpha(x^{*},T^{n-1}z)d(Tx^{*)},TT^{n-1}z)).$

By definition of $\mathcal{G}$, we have

$\displaystyle\alpha(x^{*},T^{n-1}z)d(Tx^{*},TT^{n-1}z)<\beta(M(x^{*},T^{n-1}z))M(x^{*},T^{n-1}z),$

where

$\displaystyle M(x^{*},T^{n-1}z)$ $\displaystyle=$
$\displaystyle\max\\{d(x^{*},T^{n-1}z),d(x^{*},Tx^{*}),d(T^{n-1}z,TT^{n-1}z)\\}$
$\displaystyle=$ $\displaystyle d(x^{*},T^{n-1}z).$

Hence we have

$\displaystyle\alpha(x^{*},T^{n-1}z)d(Tx^{*},TT^{n-1}z)$ $\displaystyle<$
$\displaystyle\beta(d(x^{*},T^{n-1}z))d(x^{*},T^{n-1}z).$ (18)

Inequality (16) together with (18) gives

$\displaystyle d(x^{*},T^{n}z)$ $\displaystyle<$
$\displaystyle\beta(d(x^{*},T^{n-1}z))d(x^{*},T^{n-1}z).$ (19)

By definition of $\beta$, (19) gives

$d(x^{*},T^{n}z)<d(x^{*},T^{n-1}z)$

for all $n\in\mathbb{N}$. Thus the sequence ${d(x^{*},T^{n}z)}$ is non
increasing, and so there exists $u\geq 0$ such that
$\lim\limits_{n\rightarrow\infty}d(x^{*},T^{n}z)=u$. From (19), we have

$\frac{d(x^{*},T^{n}z)}{d(x^{*},T^{n-1}z)}\leq\beta(d(x^{*},T^{n-1}z))$

and so $\lim\limits_{n\rightarrow\infty}\beta(d(x^{*},T^{n-1}z))=1$.
Consequently, we have $\lim\limits_{n\rightarrow\infty}(d(x^{*},T^{n-1}z))=0$,
and hence $\lim\limits_{n\rightarrow\infty}T^{n}z=x^{*}$. Similarly, we can
find that $\lim\limits_{n\rightarrow\infty}T^{n}z=y^{*}$. By uniqueness of
limit, we obtain $x^{*}=y^{*}.$ ∎

######  Example 3.1.

Let $X=[0,\infty)$ and $d:X\times X\rightarrow\mathbb{R}$ be defined by
$d(x,y)=|x-y|$ for all $x,y\in X$. Let
$\zeta(t,s)=\frac{8}{9}s-t,\mathcal{G}(s,t)=s-t$ for all $s,t\in[0,\infty)$,
$\mathcal{C}(\mathcal{G})=0$ and $\beta(t)=\frac{1}{1+t}$ for all $t\geq 0$.
Then it is clear that $\beta\in\mathcal{F}$. We define $T:X\rightarrow X$ by

$Tx=\begin{cases}\frac{1}{3}x&~{}if~{}0\leq x\leq 1,\\\
3x&otherwise.\end{cases}$

and $\alpha:X\times X\rightarrow[0,\infty)$ by

$\alpha(x,y)=\begin{cases}1&~{}if~{}0\leq x,y\leq 1,\\\
0&otherwise.\end{cases}$

Clearly, $T$ is continuous and condition $(3)$ of Theorem 3.1 is satisfied
with $x_{1}=1$. Let $x,y\in X$ such that $\alpha(x,y)\geq 1$. Then
$x,y\in[0,1]$, so $Tx,Ty\in[0,1]$ and thus $\alpha(Tx,Ty)=1$. Hence $T$ is
$\alpha$-admissible. Further, if $z=Ty$, then $\alpha(y,z)\geq 1$, this
implies $\alpha(x,z)\geq 1$. So $T$ is triangular $\alpha$\- admissible, hence
condition $(2)$ of Theorem 3.1 is satisfied. Finally, if $0\leq x,y\leq 1$,
then $\alpha(x,y)=1$, and we have

$\displaystyle\zeta(\alpha(x,y)d(Tx,Ty),\beta(M(x,y))M(x,y))$ $\displaystyle=$
$\displaystyle\frac{8}{9}\beta(M(x,y))M(x,y)-\alpha(x,y)d(Tx,Ty)$
$\displaystyle=$ $\displaystyle\frac{8(M(x,y))}{9(1+M(x,y))}-d(Tx,Ty),$

where

$\displaystyle M(x,y)$ $\displaystyle=$
$\displaystyle\max\\{d(x,y),d(x,Tx),d(y,Ty)\\},$

for all $x,y\in[0,1]$.
Case-I: If $M(x,y)=d(x,y)$, then

$\displaystyle\zeta(\alpha(x,y)d(Tx,Ty),\beta(M(x,y))M(x,y))$ $\displaystyle=$
$\displaystyle\frac{8d(x,y)}{9(1+d(x,y))}-d(\frac{x}{3},\frac{y}{3})$
$\displaystyle=$
$\displaystyle\frac{8|x-y|}{9(1+|x-y|)}-|\frac{x}{3}-\frac{y}{3}|$
$\displaystyle=$ $\displaystyle\frac{8|x-y|}{9(1+|x-y|)}-\frac{|x-y|}{3}$
$\displaystyle=$ $\displaystyle\frac{8|x-y|-3(1+|x-y|)|x-y|}{9(1+|x-y|)}$
$\displaystyle=$ $\displaystyle\frac{|x-y|(5-3|x-y|)}{9(1+|x-y|)}$
$\displaystyle\geq$ $\displaystyle 0.$

Case-II: If $M(x,y)=d(x,Tx)$, then

$\displaystyle\zeta(\alpha(x,y)d(Tx,Ty),\beta(M(x,y))M(x,y))$ $\displaystyle=$
$\displaystyle\frac{8d(x,Tx)}{9(1+d(x,Tx))}-d(\frac{x}{3},\frac{y}{3})$
$\displaystyle=$
$\displaystyle\frac{8|x-\frac{x}{3}|}{9(1+|x-\frac{x}{3}|)}-|\frac{x}{3}-\frac{y}{3}|$
$\displaystyle=$ $\displaystyle\frac{16|x|}{9(3+2|x|)}-\frac{|x-y|}{3}$
$\displaystyle\geq$ $\displaystyle\frac{16|x|}{9(3+2|x|)}-\frac{2|x|}{9}$
$\displaystyle=$ $\displaystyle\frac{|x|(10-4|x|)}{9(3+2|x|)}$
$\displaystyle\geq$ $\displaystyle 0.$

Similarly, if $M(x,y)=d(y,Ty)$, we have

$\zeta(\alpha(x,y)d(Tx,Ty),\beta(d(y,Ty))d(y,Ty))\geq 0.$

Hence for $0\leq x,y\leq 1$, $T$ is a generalized
$\mathcal{Z}_{(\alpha,\mathcal{G})}$-Geraghty contraction. In either case
$\alpha(x,y)=0$ and $T$ is a $\mathcal{Z}_{(\alpha,\mathcal{G})}$-Geraghty
contraction. Thus all the hypothesis of Theorem 3.1 are satisfied and $T$ has
a fixed point $x^{*}=0$.

##  4 Fixed point results in partially ordered metric space

Let $(X,d,\preceq)$ be a partially ordered metric space. Many authors has
proved the existence of fixed point results in the frame work of partially
order metric spaces (see for example [21, 1, 18]). In this section, we obtain
some new fixed point results in partially order metric spaces, as an
application of our results given in above section.

######  Definition 4.1.

Let $(X,d,\preceq)$ be a complete partially ordered metric space and let
$x\preceq y$ for all $x,y\in X$. A map $T:X\rightarrow X$ is called
$\mathcal{Z}_{\mathcal{G}}$-Geraghty contraction if there exists
$\beta\in\mathcal{F}$ such that for all $x,y\in X$

$\zeta(d(Tx,Ty),\beta(M(x,y))M(x,y))\geq\mathcal{C}_{\mathcal{G}},$

where

$M(x,y)=\max\\{d(x,y),d(x,Tx),d(y,Ty)\\}.$

######  Theorem 4.1.

Let $(X,d,\preceq)$ be a complete partially ordered metric space with
$x\preceq y$ for all $x,y\in X$. Let $T:X\rightarrow X$ be a continuous
mapping satisfying

(1)

$T$ is $\mathcal{Z}_{\mathcal{G}}$-Geraghty contraction;

(2)

$T$ is increasing;

(3)

there exists $x_{1}\in X$ such that $x_{1}\preceq Tx_{1}$.

Then $T$ has a fixed point $x^{*}\in X$ and $T$ is a Picard operator that is,
${T^{n}x_{1}}$ converges to $x^{*}$.

###### Proof.

Define $\alpha:X\times X\rightarrow[0,+\infty)$ by

$\alpha(x,y)=\begin{cases}1,&~{}if~{}x\preceq y,\\\ 0,&otherwise.\end{cases}$

Since $T$ is a $Z_{\mathcal{G}}$-Geraghty contraction, we have

$\displaystyle\mathcal{C}_{\mathcal{G}}$ $\displaystyle\leq$
$\displaystyle\zeta(d(Tx,Ty),\beta(M(x,y))M(x,y))$ $\displaystyle<$
$\displaystyle\mathcal{G}(\beta(M(x,y))M(x,y),d(Tx,Ty)).$

By definition of $\mathcal{G}$, we get

$\displaystyle d(Tx,Ty)<\beta(M(x,y))M(x,y),$

so,

$\displaystyle\alpha(x,y)d(Tx,Ty)\leq d(Tx,Ty)<\beta(M(x,y))M(x,y).$

Hence $T$ is $\mathcal{Z}_{(\alpha,\mathcal{G})}$-Geraghty contraction. Since
$T$ is increasing, $\alpha(x,y)=1$ implies $\alpha(Tx,Ty)=1$ for all $x,y\in
X$. Further if $z=Ty$, then $\alpha(y,z)=1$, this implies $\alpha(x,z)=1$.
Thus, $T$ is triangular $\alpha$-admissible. Condition $(2)$ implies that
there exists $x_{1}\in X$ such that $\alpha(x_{1},Tx_{1})=1$, and so condition
$(3)$ of Theorem 3.1 is satisfied. Thus by Theorem 3.1, $T$ has a fixed point
in $X$. ∎

Continuity of the mapping can be omitted in Theorem 4.1 and fixed point result
can be obtain with an extra condition given in the following theorem:

######  Theorem 4.2.

Let $(X,d,\preceq)$ be a complete partially ordered metric space with
$x\preceq y$ for all $x,y\in X$. Let $T:X\rightarrow X$ be a mapping
satisfying

(1)

$T$ is $\mathcal{Z}_{\mathcal{G}}$-Geraghty contraction type mapping;

(2)

$T$ is increasing;

(3)

there exists $x_{1}\in X$ such that $x_{1}\preceq Tx_{1}$;

(4)

if $x_{n}$ is a sequence in $X$ such that $x_{n}\preceq Tx_{n+1}$ for all $n$
and $x_{n}\rightarrow x\in X$ as $n\rightarrow\infty$, then there exist a
subsequence $\\{x_{n(k)}\\}$ of $\\{x_{n}\\}$ such that $x_{n(k)}\preceq x$
for all $k$.

Then $T$ has a fixed point $x^{*}\in X$ and $T$ is a Picard operator that is,
${T^{n}x_{1}}$ converges to $x^{*}$.

###### Proof.

Define $\alpha:X\times X\rightarrow[0,+\infty)$ by

$\alpha(x,y)=\begin{cases}1&~{}if~{}x\preceq y,\\\ 0&otherwise.\end{cases}$

Since $T$ is a $\mathcal{Z}_{\mathcal{G}}$-Geraghty contraction, we have

$\displaystyle\mathcal{C}_{\mathcal{G}}$ $\displaystyle\leq$
$\displaystyle\zeta(d(Tx,Ty),\beta(M(x,y))M(x,y))$ $\displaystyle<$
$\displaystyle\mathcal{G}(\beta(M(x,y))M(x,y),d(Tx,Ty)).$

By the definition of $\mathcal{G}$, we obtain

$\displaystyle d(Tx,Ty)<\beta(M(x,y))M(x,y).$

This implies

$\displaystyle\alpha(x,y)d(Tx,Ty)\leq d(Tx,Ty)<\beta(M(x,y))M(x,y).$

Hence $T$ is generalized $\mathcal{Z}_{(\alpha,\mathcal{G})}$-Geraghty
contraction. Since $T$ is increasing, $\alpha(x,y)=1$ implies
$\alpha(Tx,Ty)=1$ for all $x,y\in X$. Further if $z=Ty$, then $\alpha(y,z)=1$,
this implies $\alpha(x,z)=1$. Thus, $T$ is triangular $\alpha$-admissible.
Condition $(2)$ implies that there exists $x_{1}\in X$ such that
$\alpha(x,Tx)=1$ and so condition $(3)$ of Theorem 3.2 is satisfied. Condition
$(4)$ implies that the condition $(4)$ of Theorem 3.2 is satisfied. Thus, all
the conditions of Theorem 3.2 are satisfied. Hence $T$ has a fixed point in
$X$. ∎

######  Remark 4.1.

Uniqueness of fixed point follows from Theorem 4.1 (respectively Theorem 4.2)
with the condition
$\mathcal{U}$: For all $x,y\in Fix(T)$ with $x\preceq y$, there exists $z\in
X$ such that $\alpha(x,z)\geq 1$ and $\alpha(y,z)\geq 1$.

##  5 Application to Differential Equations

Denote by $C([0,1])$ the set of all continuous functions defined on $[0,1]$
and let $d:C([0,1])\times C([0,1])\to\mathbb{R}$ be defined by

$d(x,y)=||x-y||_{\infty}=\max_{t\in[0,1]}|x(t)-y(t)|.$ (20)

It is well known that $(C([0,1]),d)$ is a complete metric space. Let us
consider the two-point boundary value problem of the second-order differential
equation:

$\displaystyle-\frac{d^{2}x}{dt^{2}}=f(t,x(t)),~{}~{}t\in[0,1];$
$\displaystyle x(0)=x(1)=1,$ (21)

where $f:[0,1]\times\mathbb{R}\to\mathbb{R}$ is continuous.The Green function
associated to (5) is defined by

$G(t,s)=\left\\{\begin{array}[]{cc}t(1-s)&\text{if}~{}0\leq t\leq s\leq 1,\\\
s(1-t)&\text{if}~{}0\leq s\leq t\leq 1.\end{array}\right.$

Assume that the following conditions hold:

  1. (i)

there exist a function $\xi:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ such that

$|f(t,a)-f(t,b)|\leq\max{\left\\{|a-b|,|a-Ta|,|b-Tb|\right\\}}$

for all $t\in[0,1]$ and $a,b\in\mathbb{R}$ with $\xi(a,b)>0$, where
$T:C[0,1]\to C[0,1]$ is defined by

$Tx(t)=\int_{0}^{1}G(t,s)f(s,x(s))ds;$

  2. (ii)

there exists $x_{0}\in C[0,1]$ such that $\xi(x_{0}(t),Tx_{0}(t))\geq 0$ for
all $t\in[0,1]$;

  3. (iii)

for each $t\in[0,1]$ and $x,y\in C[0,1]$, $\xi(x(t),y(t))>0$ implies
$\xi(Tx(t),Ty(t))>0$;

  4. (iv)

for each $t\in[0,1]$, if $\\{x_{n}\\}$ is e sequence in $C[0,1]$ such that
$x_{n}\to x$ in $C[0,1]$ and $\xi(x_{n}(t),x_{n+1}(t))>0$ for all
$n\in\mathbb{N}$, then $\xi(x_{n}(t),x(t))>0$ for all $n\in\mathbb{N}$.

We now prove that existence of a solution of the mentioned second-order
differential equation.

######  Theorem 5.1.

Under assumptions $(i)-(iv)$, (5) has a solution in $C^{2}([0,1])$.

###### Proof.

It is well known that $x\in C^{2}([0,1])$ is a solution of (5) is equivalent
to $x\in C([0,1])$ is a solution of the integral equation (see [9])

$x(t)=\int_{0}^{1}G(t,s)f(s,x(s))ds,~{}t\in[0,1].$ (22)

Let $T:C[0,1]\to C[0,1]$ be a mapping defined by

$Tx(t)=\int_{0}^{1}G(t,s)f(s,x(s))ds.$ (23)

Suppose that $x,y\in C([0,1])$ such that $\xi(x(t),y(t))\geq 0$ for all
$t\in[0,1]$. By applying (i), we obtain that

$\displaystyle|Tu(x)-Tv(x)|$ $\displaystyle=$
$\displaystyle\int_{0}^{1}G(t,s)f(s,x(s))ds-\int_{0}^{1}G(t,s)f(s,y(s))ds$
$\displaystyle=$ $\displaystyle\int_{0}^{1}G(t,s)[f(s,x(s))-f(s,y(s))]ds$
$\displaystyle\leq$
$\displaystyle\Bigg{(}\int_{0}^{1}G(t,s)ds\Bigg{)}\Bigg{(}\int_{0}^{1}|f(s,x(s))-f(s,y(s))|ds\Bigg{)}$
$\displaystyle\leq$
$\displaystyle\Bigg{(}\int_{0}^{1}G(t,s)ds\Bigg{)}\Bigg{(}\int_{0}^{1}(\max\\{|x(s)-y(s)|,|x(s)-Tx(s)|,|y(s)-Ty(s)|\\}ds\Bigg{)}$
$\displaystyle\leq$
$\displaystyle\sup_{t\in[0,1]}\Bigg{(}\int_{0}^{1}G(t,s)ds\Bigg{)}\Bigg{(}\int_{0}^{1}(\max\left\\{\sup_{s\in[0,1]}|x(s)-y(s)|,\right.$
$\displaystyle\left.\sup_{s\in[0,1]}|x(s)-Tx(s)|,\sup_{s\in[0,1]}|y(s)-Ty(s)|\right\\}ds\Bigg{)}$
$\displaystyle\leq$
$\displaystyle\sup_{t\in[0,1]}\Bigg{(}\int_{0}^{1}G(t,s)ds\Bigg{)}(\max\left\\{||x-y||_{\infty},||x-Tx||_{\infty},||y-Ty||_{\infty}\right\\})\int_{0}^{1}ds$
$\displaystyle\leq$
$\displaystyle\sup_{t\in[0,1]}\Bigg{(}\int_{0}^{1}G(t,s)ds\Bigg{)}(M(x,y)).$

Since $\int_{0}^{1}G(t,s)ds=-(t^{2}/2)+(t/2)$, for all $t\in[0,1]$, we have
$\sup_{t\in[0,1]}(\int_{0}^{1}G(t,s)ds)=1/8$. It follows that

$||Tx-Ty||_{\infty}\leq\frac{1}{8}M(x,y).$ (24)

Let $\zeta(t,s)=\frac{1}{4}s-t,\mathcal{G}(s,t)=s-t$ for all
$s,t\in[0,\infty)$, $\mathcal{C}(\mathcal{G})=0$ and $\beta(t)=\frac{1}{2}$
for all $t\geq 0$. Then it is clear that $\beta\in\mathcal{F}$. Also define

$\alpha(x,y)=\left\\{\begin{array}[]{cc}1&\text{if}~{}\xi(x(t),y(t))>0,t\in[0,1],\\\
0&\text{otherwise}.\end{array}\right.$

Now

$\displaystyle\zeta(\alpha(x,y)d(Tx,Ty),\beta(M(x,y))M(x,y))$ $\displaystyle=$
$\displaystyle\frac{1}{4}\beta(M(x,y))M(x,y)-\alpha(x,y)d(Tx,Ty)$ (25)
$\displaystyle=$ $\displaystyle\frac{1}{8}M(x,y)-d(Tx,Ty),$

Then from (24)

$\zeta(\alpha(x,y)d(Tx,Ty),\beta(M(x,y))M(x,y))\geq 0.$

Therefore the mapping $T$ is a $\mathcal{Z}_{(\alpha,\mathcal{G})}$-Geraghty
contraction.

From (ii) there exists $x_{0}\in C[0,1]$ such that $\alpha(x_{0},Tx_{0})\geq
1$. Next by using (iii), we get the following assertions holding for all
$x,y\in C[0,1]$

$\displaystyle\alpha(x,y)\geq 1$ $\displaystyle\Rightarrow$
$\displaystyle\xi(x(t),y(t))>0~{}~{}\text{for all}~{}t\in[0,1]$
$\displaystyle\Rightarrow$ $\displaystyle\xi(Tx(t),Ty(t))>0~{}~{}\text{for
all}~{}t\in[0,1]$ $\displaystyle\Rightarrow$ $\displaystyle\alpha(Tx,Ty)\geq
1,$

hence $T$ is $\alpha$-admissible.

Applying Theorem (3.1), we obtain that $T$ has a fixed point in $C([0,1])$;
say $x$. Hence, $x$ is a solution of (5). ∎

### Acknowledgments

This paper was funded by the University of Sargodha, Sargodha funded research
project No. UOS/ORIC/2016/54. The first author, therefore, acknowledges with
thanks UOS for financial support.

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