# SOLUTION PROPERTIES FOR PERTURBED LINEAR AND NONLINEAR INTEGRAL EQUATIONS

M.Z. TSOUKALAS1 Computational Mechanics Laboratory, School of Pedagogical and
Technological Education, Athens, Greece
∗ Corresponding author  P.G. ASTERIS1,∗
<EMAIL_ADDRESS>Computational Mechanics Laboratory, School of
Pedagogical and Technological Education, Athens, Greece
∗ Corresponding author<EMAIL_ADDRESS> Computational Mechanics
Laboratory, School of Pedagogical and Technological Education, Athens, Greece
∗ Corresponding author

###### Abstract

In this study we consider perturbative series solution with respect to a
parameter $\epsilon>0$. In this methodology the solution is considered as an
infinite sum of a series of functional terms which usually converges fast to
the exact desired solution. Then we investigate perturbative solutions for
kernel perturbed integral equations and prove the convergence in an
appropriate ranges of the perturbation series. Next we investigate
perturbation series solutions for nonlinear perturbations of integral
equations of Hammerstein type and formulate conditions for their convergence.
Finally we prove the existence of a maximal perturbation range for non linear
integral equations.

Keywords:  perturbation, Fredholm, Hammerstein integral equation

##  1 Introduction

Integral equations arise in many fields in mathematical physics, biology,
chemical kinetics, mechanics, etc.. In recent years there is a literature
dealing with homotopy perturbation methods in integral equations ([1, 3, 2, 4,
5, 6, 9, 8, 10, 13, 14, 18]). However despite its fast numerical convergence,
the convergence of the OHAM (optimal homotopy asymptotic method) is not proved
formally in ([1, 2, 4, 5, 3, 8, 9, 14, 18]). In this work we deal mainly with
existence theorems for nonlinear integral equations for solutions in $C[0,1]$
or $L^{2}[0,1]$ spaces. Existence theorems are given in ([12, 16, 17]) but
here we give versions of these theorems for perturbed integral equations by
formulating and proving basic existence theorems for non linear integral
equations considering perturbative series solution with respect to a parameter
$\epsilon>0$. In this methodology the solution is considered as an infinite
sum of a series of functional terms which usually converges fast to the exact
desired solution. Particularly we investigate perturbative solutions for
kernel perturbed integral equations and prove the convergence in appropriate
ranges of the perturbation series. In the next section 3 we investigate
perturbation series solutions for nonlinear perturbations of non linear
integral equations of Hammerstein type and formulate conditions for their
convergence. Finally we prove the existence of a maximal perturbation range
for non linear integral equations. The methods used here can be exploited to
implement numerical procedures for perturbation series solutions.

##  2 Series Solutions for perturbed kernels

Here we consider ’small’ in a sense as we shall see perturbations in a non
linear sense of the kernel $K$ of the integral equation and we prove that the
perturbative series solution converges absolutely in an appropriate range for
$\epsilon$, the perturbation parameter.
In theorem (2.1) we deal with perturbations of the kernel in the linear
Fredholm equation of the second kind and prove that if the unperturbed
original equation has a $C[0,1]$ or $L^{2}[0,1]$ solution then the perturbed
equation has a corresponding $C[0,1]$ or $L^{2}[0,1]$ solution. Next in
theorem (2.2) we prove the corresponding result dropping the boundedness
assumption for the perturbed kernel. Continuing we prove in theorem (2.3) that
the corresponding result holds under suitable assumptions for a non linear
Hammerstein integral equation. Following we prove the corresponding result in
theorem (2.4) for a linear Fredholm equation of the second kind. with
boundedly differentiable kernel. Next we prove the theorem (2.5) stating the
result for $L^{2}$ integrable perturbed kernel in a Hammerstein equation. We
prove the same kind of result for equations with boundedly differentiable
kernel in theorem (2.6). We proceed with theorem (2.7) stating the
corresponding result for $L^{1}$ integrable kernel. We denote by
$\phi(\epsilon,x)=\phi_{(0,0)}(x)$ the unknown sought function and by
$\phi_{(0,\nu)}(x)=\frac{\partial^{\nu}\phi(\epsilon,x)}{\partial\epsilon^{\nu}}|_{\epsilon=0},\nu=0,1,2,...$
its various order derivatives at $\epsilon=0$.

###### Theorem 2.1

Let’s consider the perturbed integral equation

$\displaystyle\phi(\epsilon,x)-\omega\int_{0}^{1}\Gamma(\epsilon,x,y)\phi(\epsilon,y)dy=f(x),\;\;\;\;x\in[0,1],\epsilon\geq
0$ (2.1)

where $\Gamma(\epsilon,x,y)=\Gamma_{0}(x,y)+\epsilon\Gamma_{1}(x,y)$ is
continuous and $L^{2}$ integrable respectively functions of their variables
and $\int_{0}^{1}\int_{0}^{1}|\Gamma(\epsilon,x,y|dxdy<\infty$. Suppose that

$\displaystyle|\Gamma_{0}(\epsilon,x,y)|\leq C,\;\;\;\;\;\;0\leq x,y\leq
1,\epsilon\geq 0.$ (2.2)

Assuming that the unperturbed equation for $\epsilon=0$ has a solution in
$C([0,1])$ or $L^{2}([0,1])$ respectively then the perturbation series

$\displaystyle\phi_{(0,0)}(x)=\omega\int_{0}^{1}\Gamma_{0},x,y)\phi(0,y)dy+f(x)$
(2.3)
$\displaystyle\phi_{(0,1)}(x)=\omega\int_{0}^{1}\Gamma_{1}(x,y)\phi(0,y)dy+\omega\int_{0}^{1}\Gamma_{0}(x,y)\phi_{(0,1)}(y)dy$
(2.4) $\displaystyle\cdot\cdot\cdot$
$\displaystyle\phi_{(0,\nu)}(x)=\frac{\partial^{\nu}\phi(\epsilon,x)}{\partial\epsilon^{\nu}}|_{\epsilon=0}=\omega\int_{0}^{1}\Gamma_{1}((x,y)\phi_{(0,\nu-1)}(y)dy$
$\displaystyle+\omega\int_{0}^{1}\Gamma(0,x,y)\phi_{(0,\nu)}(y)dy,\nu=1,2,....$
(2.5)

converges absolutely in $||\cdot(x)||_{0}=\sup_{x\in[0,1]}|\cdot(x)|$ norm and
$||\cdot||_{2}$ norm to a solution

$\displaystyle\phi(\epsilon,x)=\sum_{j=0}^{\infty}\phi_{(0,j)}(x)\frac{\epsilon^{j}}{j!}$
(2.6)

which is continuous with respect to $(\omega,\epsilon,x)$.

Proof: As $\Gamma_{0}(x,y)\leq C$ and $\Gamma_{1}(x,y)\leq C$ we have that
$||\Gamma_{0}||_{1}\leq C,||\Gamma_{1}||_{1}\leq C$, which implies

$\displaystyle||\phi_{0}^{(\nu)}||_{0}=||\phi(0,x)||_{0}\leq|\omega|||\Gamma_{1}||_{1}\cdot||\phi_{0}^{(\nu-1)}||_{0}+|\omega|||\Gamma_{0}||_{1}\cdot||\phi_{0}^{(\nu)}||_{0}$
(2.7)

leading to

$\displaystyle||\phi_{0}^{(\nu)}||_{0}\leq\frac{|\omega|||\Gamma_{1}||_{1}}{1-|\omega|||\Gamma_{0}||_{1}}||\phi^{(\nu-1)}||_{2}$
(2.8)

which implies that for

$\frac{|\omega|||\Gamma_{1}||_{1}}{1-|\omega|||\Gamma_{0}||_{1}}=\rho_{0},$

we have that

$\displaystyle||\phi(\epsilon,x)||_{0}=\sum_{j=0}^{\infty}||\phi_{(0,j)}(x)||_{0}\frac{\epsilon^{j}}{j!}$
(2.9)
$\displaystyle\leq\sum_{j=0}^{\infty}\rho_{0}^{j}\frac{\epsilon^{j}}{j!}||\phi_{0}||_{0}$
(2.10)

the solution converges in $||\cdot||_{0}$ for any value of
$\omega,\rho_{0},\epsilon\geq 0$.

Also we observe that as $||\Gamma_{j}||_{2}<\infty$ for $j=0,1$,

$\displaystyle||\phi_{0}^{(\nu)}||_{2}$ $\displaystyle=$
$\displaystyle||\phi_{(0,0)}(x)||_{2}\leq|\omega|||\Gamma_{1}||_{2}\cdot||\phi_{(0,\nu-1)}||_{2}$
(2.11) $\displaystyle+$
$\displaystyle|\omega|||\Gamma_{0}||_{2}\cdot||\phi_{(0,\nu)}||_{2}$

leading to

$\displaystyle||\phi_{(0,\nu)}||_{2}\leq\frac{|\omega|||\Gamma_{1}||_{2}}{1-|\omega|||\Gamma_{0}||_{2}}||\phi_{(0,\nu-1)}||_{2}$
(2.12)

which implies that for

$\frac{|\omega|||\Gamma_{1}||_{2}}{1-|\omega|||\Gamma_{0}||_{2}}=\rho,$

we have that

$\displaystyle||\phi(\epsilon,x)||_{2}\leq\sum_{j=0}^{\infty}||\phi_{(0^{,}j)}(x)||_{2}\left[\frac{\epsilon^{j}}{j!}\right]$
(2.13)
$\displaystyle\leq\sum_{j=0}^{\infty}|\rho^{j}\left[\frac{\epsilon^{j}}{j!}\right]||\phi_{0}||_{2}\leq||\phi_{0}||_{2}e^{|\rho\epsilon|}$
(2.14)

the solution converges for any value of $\omega,\rho,\epsilon\geq 0$.

As a convergent power series everywhere it is continuous with respect to
$(x,\epsilon)$ as one can see immediately from uniform convergence. Also we
have that is continuous with respect to $\omega$.

###### Theorem 2.2

Let’s consider the perturbed integral equation

$\displaystyle\phi_{(\epsilon)}(x)-\omega\int_{0}^{1}\Gamma(\epsilon,x,y)\phi_{(\epsilon)}(y)dy=f(x),\;\;\;\;x\in[0,1],\epsilon\geq
0$ (2.15)

where $\Gamma(\epsilon,x,y)=\Gamma_{0}(x,y)+\epsilon\Gamma_{1}(x,y)$ is
$L^{2}$ integrable respectively functions of their variables and
$\int_{0}^{1}\int_{0}^{1}|\Gamma(\epsilon,x,y)|dxdy<\infty$. Assuming that the
unperturbed equation for $\epsilon=0$ has a solution in $C([0,1])$ then the
perturbation series

$\displaystyle\phi_{(0,0)}(x)=\omega\int_{0}^{1}\Gamma_{0},x,y)\phi_{(0,0)}(y)dy+f(x)$
(2.16)
$\displaystyle\phi_{(0,1)}(x)=\omega\int_{0}^{1}\Gamma_{1}(x,y)\phi_{(0,0)}(y)dy+\omega\int_{0}^{1}\Gamma_{0}(x,y)\phi_{(0,1)}(y)dy$
(2.17) $\displaystyle\cdot\cdot\cdot$
$\displaystyle\phi_{(0,\nu)}(x)=\frac{\partial^{\nu}\phi(x)}{\partial
x^{\nu}}|_{x=0}=\omega\int_{0}^{1}\Gamma_{1}((x,y)\phi_{(0,\nu-1)}(y)dy$
$\displaystyle+\omega\int_{0}^{1}\Gamma(0,x,y)\phi_{(0,\nu)}(y)dy,\nu=1,2,....$
(2.18)

converges absolutely in $||\cdot||_{0}=\sup_{x\in[0,1]}|\cdot(x)|$ norm and
$||\cdot||_{2}$ norm to a solution

$\displaystyle\phi(\epsilon,x)=\sum_{j=0}^{\infty}\phi_{(0,j)}(x)\frac{\epsilon^{j}}{j!}$
(2.19)

which is continuous with respect to $(\omega,\epsilon,x)$.

Proof: As a result of the $L^{2}$ integrability we have that we have
converging in $L^{2}$ sequences of kernels to $\Gamma_{0},\Gamma_{1}$
respectively such that $\Gamma_{0,n}(x,y)\leq C_{n}\rightarrow
C_{|infty}\leq\infty$ and $\Gamma_{1,n}(x,y)\leq C_{n}\rightarrow
C_{\infty,1}\leq\infty$ and we have that $||\Gamma_{0,n}||_{1}\leq
C_{n},||\Gamma_{1,n}||_{1}\leq C_{n}$, which implies corresponding
inequalities to (2.7,2.8), and therefore there exists a solution sequence

$\displaystyle||\phi_{(n,\epsilon)}(x)||_{0}=\sum_{j=0}^{\infty}||\phi_{(n,0,j)}(x)||_{0}\frac{\epsilon^{j}}{j!}$
(2.20)
$\displaystyle\leq\sum_{j=0}^{\infty}\rho_{0,n}^{j}\frac{\epsilon^{j}}{j!}||\phi_{0,n}||_{0},n=0,1,2,...$
(2.21)

the solution converges in $||\cdot||_{0}$ for any value of
$\omega,\rho_{0},\epsilon\geq 0$.

As $\rho_{0,n}\rightarrow\rho_{0}<\infty$ we have that the limit solution
$||\phi(\epsilon,x)||_{0}<\infty$ is bounded everywhere for
$(\omega,x,\epsilon)$. As the convergence is uniform we have that
$\phi(\epsilon,x)$ is continuous.

In the same manner we have as a result that $\phi(\epsilon,x)$ belongs to
$L_{2}[0,1]$.

###### Theorem 2.3

Let’s consider the perturbed integral equation

$\displaystyle\phi(\epsilon,x)-\omega\int_{0}^{1}\Gamma(\epsilon,x,y)\psi(y,\phi(\epsilon,y))dy=f(x),\;\;\;\;x\in[0,1],\epsilon\geq
0$ (2.22)

where $\Gamma(\epsilon,x,y)=\Gamma_{0}(x,y)+\epsilon\Gamma_{1}(x,y)$ is
$L^{2}$ integrable respectively function of its variables and
$\int_{0}^{1}\int_{0}^{1}|\Gamma(\epsilon,x,y)|^{2}dxdy<\infty$. Suppose that

$\displaystyle|\Gamma_{j}(\epsilon,x,y)|\leq C_{j},\;\;\;\;\;\;0\leq x,y\leq
1,\epsilon\geq 0,j=0,1.$ (2.23)

and $\psi^{(0,\nu)}(y,s)=\frac{\partial^{\nu}\psi(y,s)}{\partial
s^{\nu}}\leq\frac{b^{\nu}}{E(\nu,\nu)},0\leq b\leq 1,\;\;\;\;\;\nu=0,1,2,...$
where $E(n,k)\leq{{2n-1}\choose{n-1}}$ is the number of integer solutions of
the equation

$\displaystyle\sum_{j=1}^{k}r_{j}\cdot s_{j}=n,0\leq r_{j},s_{j}\leq
n,j=1,2,..,k.$ (2.24)

Assuming that the unperturbed equation for $\epsilon=0$ has a solution in
$C([0,1])$ then the perturbation series

$\displaystyle\phi_{(0,0)}(x)=\phi(0,x)=\omega\int_{0}^{1}\Gamma_{0}(x,y)\psi(y,\phi_{(0,0)}(y))dy$
$\displaystyle\phi_{(0,1)}(x)=\omega\int_{0}^{1}\Gamma_{1}(x,y)\psi(y,\phi_{(0,0)}(y))dy$
$\displaystyle+$
$\displaystyle\omega\int_{0}^{1}\Gamma_{0}(x,y)\psi^{(0,1)}(y,\phi_{(0,0)}(y))\phi_{(0,1)}(y)dy\;\;\;\;\;\;\;\;$
$\displaystyle\cdot\cdot\cdot\cdot$
$\displaystyle\phi_{(0,\nu)}(x)=\omega\int_{0}^{1}\Gamma_{0}(x,y)\psi^{(0,\nu)}(y,\phi_{(0,0)}(y))P_{\nu+1}(y)dy$
$\displaystyle+$
$\displaystyle\omega\int_{0}^{1}\Gamma_{1}(x,y)\psi^{(0,\nu-1)}(y,\phi_{(0,0)}(y))P_{\nu}(y)dy\;\;\;\;\;\;\;\;\;\;\;$
$\displaystyle P_{\nu}(y)=$ $\displaystyle\sum_{{\sum_{j=1}^{\nu-1}r_{j}\cdot
s_{j}=\nu-1}\atop{r_{j},s_{j}=0,j=1,...,\nu-1}}^{\nu-1}$
$\displaystyle{{\nu-1}\choose{r_{1},s_{1};\cdot\cdot\cdot;r_{\nu-1},s_{\nu-1}}}\prod_{j=1}^{\nu-1}\phi_{(0,s_{j})}^{r_{j}}(y),\;\;\;\nu=2,3,...\;\;\;\;\;$
(2.31)
$\displaystyle{m\choose{r_{1},s_{1};\cdot\cdot;r_{m},s_{m}}}=\frac{m!}{\prod_{j=1}^{m}(s_{j}!)^{r_{j}}}$
$\displaystyle\psi^{(k,\nu)}(s,\phi(\epsilon,y))=\frac{\partial^{k}\partial^{\nu}\psi(s,\phi(\epsilon,y))}{\partial
s^{k}\partial\epsilon^{\nu}},\;\;\;\;\;k,\nu=0,1,2,...\;\;\;\;\;\;\;\;$

converges absolutely in $||\cdot||_{0}=\sup_{x\in[0,1]}|\cdot(x)|$ norm and
$||\cdot||_{2}$ norm to a solution

$\displaystyle\phi(\epsilon,x)=\sum_{j=0}^{\infty}\phi_{(0,j)}(x)\frac{\epsilon^{j}}{j!}$
(2.32)

which is continuous with respect to $(\omega,\epsilon,x)$ for suitable ranges.

Proof: As a result of the $L^{2}$ integrability we have that we have
converging in $L^{2}$ sequences of kernels to $\Gamma_{0},\Gamma_{1}$
respectively such that $\Gamma_{0,n}(x,y)\leq C_{n}^{\prime}\rightarrow
C_{0,}\leq\infty$ and $\Gamma_{1,n}(x,y)\leq C_{n}^{\prime}\rightarrow
C_{1}\leq\infty$ and we have that $||\Gamma_{0,n}||_{1}\leq
C_{n},||\Gamma_{1,n}||_{1}\leq C_{n}^{\prime}$, which implies that inductively
we can show that
$||\phi_{n}(0,\cdot)||_{0}\leq\left[\frac{\omega(C_{0}}{1-\omega
C_{0}}+\frac{\omega C_{1}}{1-\omega C_{0}}\right](n)!D^{n+1},n=1,2,...$, so
for $\omega\leq g_{\pm}(C_{0},C_{1},D)$ we have the bound
$||\phi_{n}(0,\cdot)||_{0}\leq n!D^{n},n=1,2,...$, where if
$\Delta(C_{0},C_{1},D)=[2DC_{0}+C_{0}+C_{1}]^{2}-4(DC_{1}+2C_{0}^{2})<0$ we
have for any $\omega$ the bound and otherwise for $\Delta(C_{0},C_{1},D)>0$ we
have for

$\displaystyle\omega<g_{\pm}(C_{0},C_{1},D)=\frac{2DC_{0}+C_{0}+C_{1}\pm\sqrt{\Delta(C_{0},C_{1},D)}}{2(DC_{1}+2C_{0}^{2})}>0$
(2.33)

The induction is based on the inequality

$\displaystyle|P_{\nu}(y)|\leq E(\nu-1,\nu-1)(\nu-1)!$ (2.34)

and

$\displaystyle||\phi_{0,\nu}(\cdot)||_{0}\leq|\omega|\int_{0}^{1}|\Gamma_{0}(x,y)||\psi^{(0,\nu)}(y,\phi_{0,0}(y))||P_{\nu+1}(y)|dy$
$\displaystyle+$
$\displaystyle|\omega|\int_{0}^{1}|\Gamma_{1}(x,y)||\psi^{(0,\nu-1)}(y,\phi_{0,0)}(y))||P_{\nu}(y)|dy$
$\displaystyle\leq$
$\displaystyle|\omega|C_{0}||\phi_{\nu}(0,\cdot)||_{0}\frac{b^{\nu}}{E(\nu,\nu)}+|\omega|C_{0}|\frac{b^{\nu}}{E(\nu,\nu)}E(\nu-1,\nu-1)\nu!D^{\nu}$
$\displaystyle+$
$\displaystyle|\omega|C_{1}||_{0}\frac{b^{\nu}}{E(\nu,\nu)}(\nu-1)!D^{\nu-1}.$
(2.36)

and which leads to the convergent series

$\displaystyle||\phi_{(n,\epsilon)}(x)||_{0}=\sum_{j=0}^{\infty}||\phi_{n,0,j)}(x)||_{0}\frac{\epsilon^{j}}{j!}$
(2.37)
$\displaystyle\leq\sum_{j=0}^{\infty}kj!D^{j}\frac{\epsilon^{j}}{j!}||\phi_{0,n}||_{0},n=0,1,2,...$
(2.38)

converging in $||\cdot||_{0}$ for any value of $\omega\leq
g_{\pm}(C_{0},C_{1},D),1/D>\epsilon\geq 0$.

So we have that the limit solution $||\phi(\epsilon,x)||_{0}<\infty$ is
bounded everywhere for $(\omega,x,\epsilon)$ in the allowed ranges.
As the convergence is uniform we have that $\phi(\epsilon,x)$ is continuous on
the specified ranges.

In the same manner we have as a result that $\phi(\epsilon,x)$ belongs to
$L_{2}[0,1]$.

###### Corrolary 2.1

Let $f$, $\Gamma_{0}$ be continuous and $\Gamma_{1}$ to be $L^{2}$ integrable.
Then $\Gamma(\epsilon,x,y)$ defines a continuous , bounded solution for
$\epsilon<\epsilon_{0}>0$.

Proof: If $\Gamma_{0}$ is continuous the solution $\phi_{(0)}(x)$ as in
$L^{2}[0,1]$ it implies that
$\phi_{(0)}(x)=f(x)+\omega\int_{0}^{1}\Gamma_{0}(x,y)\phi_{(0)}(y)dy$ as a sum
of a continuous function and an integral with a continuous kernel is
continuous. Applying theorem (2.3) we have the sought implication.

###### Corrolary 2.2

Let $f$ continuous and $\Gamma_{1}$ to be $L^{2}$ integrable. Then
$\Gamma(\epsilon,x,y)=\epsilon I+\epsilon(\Gamma_{1}-I)$ defines a continuous
, bounded solution for $\epsilon<\epsilon_{0}>0$.

Proof: Immediate from the above corollary (2.1). Next we consider the behavior
of the solution of the perturbed integral equation when the kernel has bounded
uniformly derivative with respect to $x$. Then the perturbation series has a
bounded derivative and uniformly convergent solution.

###### Theorem 2.4

Let’s consider the perturbed integral equation

$\displaystyle\phi(\epsilon,x)-\omega\int_{0}^{1}\Gamma(\epsilon,x,y)\phi(\epsilon,y)dy=f(x),\;\;\;\;x\in[0,1],\epsilon\geq
0$ (2.39)

where $\Gamma(\epsilon,x,y)=\Gamma_{0}(x,y)+\epsilon\Gamma_{1}(x,y)$ is
continuous and $L^{2}$ integrable respectively functions of their variables
and $\int_{0}^{1}\int_{0}^{1}\Gamma(\epsilon,x,y)dxdy<\infty$. Let

$\sup_{0\leq x\leq 1}\frac{\partial\Gamma_{j}(x,y)}{\partial x}\leq
C_{1}>0,\;\;j=0,1.$

Suppose that

$\displaystyle|\Gamma_{0}(\epsilon,x,y)|\leq C,\;\;\;\;\;\;0\leq x,y\leq
1,\epsilon\geq 0.$ (2.40)

Assuming that the unperturbed equation for $\epsilon=0$ has a solution in
$C^{1}([0,1])$ or $L^{2}([0,1])$ respectively then the perturbation series

$\displaystyle\phi_{(0,0)}(x)=\omega\int_{0}^{1}\Gamma_{0},x,y)\phi_{(0,0)}(y)dy+f(x)$
(2.41)
$\displaystyle\phi_{(0,1)}(x)=\omega\int_{0}^{1}\Gamma_{1}(x,y)\phi_{(0,0)}(y)dy+\omega\int_{0}^{1}\Gamma_{0}(x,y)\phi_{(0,1)}(y)dy$
(2.42) $\displaystyle\cdot\cdot\cdot$
$\displaystyle\phi_{(0,\nu)}(x)=\frac{\partial^{\nu}\phi(x)}{\partial
x^{\nu}}|_{x=0}=\omega\int_{0}^{1}\Gamma_{1}((x,y)\phi_{(0,\nu-1)}(y)dy$
$\displaystyle+\omega\int_{0}^{1}\Gamma(0,x,y)\phi_{(0,\nu)}(y)dy,\nu=1,2,....$
(2.43)

converges absolutely in $||\cdot||_{0}=\sup_{x\in[0,1]}|\cdot(x)|$ norm and
$||\cdot||_{2}$ norm to a solution

$\displaystyle\phi(\epsilon,x)=\sum_{j=0}^{\infty}\phi_{(0,j)}(x)\frac{\epsilon^{j}}{j!}$
(2.44)

which is continuous with respect to $(\omega,\epsilon,x)$ and has a convergent
series of derivatives.

Proof: As $\Gamma_{0}(x,y)\leq C$ and $\Gamma_{1}(x,y)\leq C$ we have that
$||\Gamma_{0}||_{1}\leq C,||\Gamma_{1}||_{1}\leq C$, which implies

$\displaystyle||\phi_{(0,\nu)}||_{0}$ $\displaystyle=$
$\displaystyle||\phi_{(0,0)}^{\prime}(x)||_{0}\leq|\omega|||\Gamma_{1}^{\prime}||_{1}\cdot||\phi_{(0,\nu-1)}||_{0}$
(2.45) $\displaystyle+$
$\displaystyle|\omega|||\Gamma_{0}^{\prime}||_{1}\cdot||\phi_{(0,\nu)}||_{0}$

leading to

$\displaystyle||\phi_{(0,\nu)}^{\prime}||_{0}\leq\frac{|\omega|||\Gamma_{1}^{\prime}||_{1}}{1-|\omega|||\Gamma_{0}^{\prime}||_{1}}||\phi_{(0,\nu-1)}^{\prime}||_{2}$
(2.46)

which implies that for

$\frac{|\omega|||\Gamma_{1}^{\prime}||_{1}}{1-|\omega|||\Gamma_{0}^{\prime}||_{1}}=\rho_{0},$

we have that

$\displaystyle||\phi^{\prime}_{(\epsilon)}(x)||_{0}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}||\phi_{(0,j)}^{\prime}(x)||_{0}\frac{\epsilon^{j}}{j!}$
(2.48)
$\displaystyle\leq\sum_{j=0}^{\infty}\rho_{0}^{j}\frac{\epsilon^{j}}{j!}||\phi_{0}^{\prime}||_{0}$

the solution converges in $||\cdot||_{0}$ for any value of
$\omega,\rho_{0},\epsilon\geq 0$.

Also we observe that as $||\Gamma_{j}^{\prime}||_{2}<\infty$ for $j=0,1$,

$\displaystyle||\phi_{(0,\nu)}^{\prime}||_{2}$ $\displaystyle=$
$\displaystyle||\phi^{\prime}_{(0,0)}(x)||_{2}\leq|\omega|||\Gamma_{1}^{\prime}||_{2}\cdot||\phi_{(0,\nu-1)}^{\prime}||_{2}$
(2.49) $\displaystyle+$
$\displaystyle|\omega|||\Gamma_{0}^{\prime}||_{2}\cdot||\phi_{(0,\nu)}^{\prime}||_{2}$

leading to

$\displaystyle||\phi_{(0,\nu)}^{\prime}||_{2}\leq\frac{|\omega|||\Gamma_{1}^{\prime}||_{2}}{1-|\omega|||\Gamma_{0}^{\prime}||_{2}}||\phi_{(0,\nu-1)}^{\prime}||_{2}$
(2.50)

which implies that for

$\frac{|\omega|||\Gamma_{1}^{\prime}||_{2}}{1-|\omega|||\Gamma_{0}^{\prime}||_{2}}=\rho,$

we have that

$\displaystyle||\phi^{\prime}_{(\epsilon,0)}(x)||_{2}\leq\sum_{j=0}^{\infty}||\phi_{(0,\nu)}^{\prime}(x)||_{2}\left[\frac{\epsilon^{j}}{j!}\right]$
$\displaystyle\leq\sum_{j=0}^{\infty}|\rho^{j}\left[\frac{\epsilon^{j}}{j!}\right]||\phi_{0}^{\prime}||_{2}\leq||\phi_{0}^{\prime}||_{2}e^{|\rho\epsilon|}$
(2.51)

the derivative of the solution converges for any value of
$\omega,\rho,\epsilon\geq 0$.

As a convergent power series everywhere it is continuous with respect to
$(x,\epsilon)$ as one can see immediately from uniform convergence. Also we
have that is continuous with respect to $\omega$.

Next we continue with the following theorem.

###### Theorem 2.5

Let’s consider the perturbed integral equation

$\displaystyle\phi(\epsilon,x)-\omega\int_{0}^{1}\Gamma(\epsilon,x,y)\phi(\epsilon,y)dy=f(x),\;\;\;\;x\in[0,1],\epsilon\geq
0$ (2.52)

where $\Gamma(\epsilon,x,y)=\Gamma_{0}(x,y)+\epsilon\Gamma_{1}(x,y)$ is
$L^{2}$ integrable respectively functions of their variables and
$\int_{0}^{1}\int_{0}^{1}\Gamma(\epsilon,x,y)dxdy<\infty$. Suppose that

$\displaystyle|\Gamma_{j}(x,y)|\leq C,\;\;\;\;\;\;0\leq x,y\leq 1,\epsilon\geq
0,j=0,1.$ (2.53)

Assuming that the unperturbed equation for $\epsilon=0$ has a solution in
$C^{1}([0,1])$ then the perturbation series

$\displaystyle\phi_{(0,0)}(x)=\omega\int_{0}^{1}\Gamma_{0},x,y)\phi_{(0,0)}(y)dy+f(x)$
(2.54)
$\displaystyle\phi_{(0,1)}(x)=\omega\int_{0}^{1}\Gamma_{1}(x,y)\phi_{(0,0)}(y)dy+\omega\int_{0}^{1}\Gamma_{0}(x,y)\phi_{(0,1)}(y)dy$
(2.55) $\displaystyle\cdot\cdot\cdot$
$\displaystyle\phi_{(0,\nu)}(x)=\frac{\partial^{\nu}\phi(x)}{\partial
x^{\nu}}|_{x=0}=\omega\int_{0}^{1}\Gamma_{1}((x,y)\phi_{(0,\nu-1)}(y)dy$
$\displaystyle+\omega\int_{0}^{1}\Gamma(0,x,y)\phi_{(0,\nu)}(y)dy,\nu=1,2,....$
(2.56)

converges absolutely in $||\cdot(x)||_{0}=\sup_{x\in[0,1]}|\cdot(x)|$ norm and
$||\cdot||_{2}$ norm to a solution

$\displaystyle\phi(\epsilon,x)=\sum_{j=0}^{\infty}\phi_{(0,j)}(x)\frac{\epsilon^{j}}{j!}$
(2.57)

which is continuous with respect to $(\omega,\epsilon,x)$.

Proof: As we have that $||\Gamma_{0}||_{1}\leq C,||\Gamma_{1}||_{1}\leq C$,
which implies corresponding inequalities to (2.45,2.46), and therefore there
exists a solution sequence

$\displaystyle||\phi_{(n,\epsilon)}(x)||_{0}=\sum_{j=0}^{\infty}||\phi_{(n,0,j)}(x)||_{0}\frac{\epsilon^{j}}{j!}$
(2.58)
$\displaystyle\leq\sum_{j=0}^{\infty}\rho_{0,n}^{j}\frac{\epsilon^{j}}{j!}||\phi_{0,n}||_{0}\leq||\phi_{0,n}||_{0}e^{|\rho_{0,n}\epsilon},n=0,1,2,...$
(2.59)

the solution converges in $||\cdot||_{0}$ for any value of
$\omega,\rho_{0},\epsilon\geq 0$.

As $\rho_{0,n}\rightarrow\rho_{0}<\infty$ we have that the limit solution
$||\phi(\epsilon,x)||_{0}<\infty$ is bounded everywhere for
$(\omega,x,\epsilon)$. As the convergence is uniform we have that
$\phi(\epsilon,x)$ is $C^{1}[0,1]$ continuous.

In the same manner we have as a result that $\phi^{\prime}_{(\epsilon)}(x)$
belongs to $L_{2}[0,1]$.

We give the respective theorem to (2.3) for kernels with uniformly bounded
derivative with respect to $x$.

###### Theorem 2.6

Let’s consider the perturbed integral equation

$\displaystyle\phi(\epsilon,x)-\omega\int_{0}^{1}\Gamma(\epsilon,x,y)\psi(y,\phi(\epsilon,y))dy=f(x),\;\;\;\;x\in[0,1],\epsilon\geq
0$ (2.60)

where $\Gamma(\epsilon,x,y)=\Gamma_{0}(x,y)+\epsilon\Gamma_{1}(x,y)$ is
$L^{2}$ integrable respectively function of its variables and
$\int_{0}^{1}\int_{0}^{1}|\Gamma(\epsilon,x,y)|^{2}dxdy<\infty$. Let

$\sup_{0\leq x\leq 1}\frac{\partial\Gamma_{j}(x,y)}{\partial x}\leq
C_{j}^{\prime}>0,\;\;j=0,1.$

Suppose that

$\displaystyle|\Gamma_{j}(x,y)|\leq C_{j},\;\;\;\;\;\;0\leq x,y\leq
1,\epsilon\geq 0,j=0,1.$ (2.61)

and and $\psi^{(0,\nu)}(y,s)=\frac{\partial^{\nu}\psi(y,s)}{\partial
s^{\nu}}\leq\frac{b^{\nu}}{E(\nu,\nu)},0\leq b\leq 1,\;\;\;\;\;\nu=0,1,2,...$
where $E(n,k)\leq{{2n-1}\choose{n-1}}$ is the number of integer solutions of
the equation

$\displaystyle\sum_{j=1}^{k}r_{j}\cdot s_{j}=n,0\leq r_{j},s_{j}\leq
n,j=1,2,..,k.$ (2.63)

Assuming that the unperturbed equation for $\epsilon=0$ has a solution in
$C([0,1])$ then the perturbation series

$\displaystyle\phi_{(0,0)}(x)=\omega\int_{0}^{1}\Gamma_{0}(x,y)\psi(y,\phi_{(0,0)}(y))dy$
$\displaystyle\phi_{(0,1)}(x)=\omega\int_{0}^{1}\Gamma_{1}(x,y)\psi(y,\phi_{(0,0)}(y))dy$
$\displaystyle+$
$\displaystyle\omega\int_{0}^{1}\Gamma_{0}(x,y)\psi^{(0,1)}(y,\phi_{(0,0)}(y))\phi_{(0,1)}(y)dy\;\;\;\;\;\;$
$\displaystyle.....$ $\displaystyle\phi_{(0,\nu)}(x)$
$\displaystyle=\omega\int_{0}^{1}\Gamma_{0}(x,y)\psi^{(0,\nu)}(y,\phi_{(0,0)}(y))P_{\nu+1}(y)dy$
$\displaystyle+$
$\displaystyle\omega\int_{0}^{1}\Gamma_{1}(x,y)\psi^{(0,\nu-1)}(y,\phi_{(0,0)}(y))P_{\nu}(y)dy\;\;\;\;\;\;$
(2.66) $\displaystyle P_{\nu}(y)=$
$\displaystyle\sum_{{\sum_{j=1}^{\nu-1}r_{j}\cdot
s_{j}=\nu-1}\atop{r_{j},s_{j}=0,j=1,...,\nu-1}}^{\nu-1}$
$\displaystyle{{\nu-1}\choose{r_{1},s_{1};\cdot\cdot\cdot;r_{\nu-1},s_{\nu-1}}}\prod_{j=1}^{\nu-1}\phi_{(0,s_{j})}(x)^{r_{j}}(y),\nu=2,3,...\;\;$
(2.69)
$\displaystyle{m\choose{r_{1},s_{1};\cdot\cdot;r_{m},s_{m}}}=\frac{m!}{\prod_{j=1}^{m}(s_{j}!)^{r_{j}}}$
$\displaystyle\psi^{(k,\nu)}(s,\phi(\epsilon,y))=\frac{\partial^{k}\partial^{\nu}\psi(s,\phi(\epsilon,y))}{\partial
s^{k}\partial\epsilon^{\nu}}k,\nu=0,1,2,...\;\;\;\;\;\;\;\;$

converges absolutely in $||\cdot||_{0}=\sup_{x\in[0,1]}|\cdot(x)|$ norm and
$||\cdot||_{2}$ norm to a solution

$\displaystyle\phi(\epsilon,x)=\sum_{j=0}^{\infty}\phi_{(0,j)}(x)\frac{\epsilon^{j}}{j!}$
(2.70)

which is continuous with respect to $(\omega,\epsilon,x)$ for suitable ranges.

Proof: As $\Gamma_{0}(x,y)\leq C$ and $\Gamma_{1}(x,y)\leq C$ we have that
$||\Gamma_{0}||_{1}\leq C,||\Gamma_{1}||_{1}\leq C_{1}$, which implies that
inductively we can show as in theorem (2.3) that
$||\phi_{(0,n)}(\cdot)||_{0}\leq\left[\frac{\omega C_{0}}{1-\omega
C_{0}}+\frac{\omega C_{1}}{1-\omega C_{0}}\right](n)!D^{n},n=1,2,...$, so for
$\omega\leq g_{\pm}(C_{0},C_{1},D)$ we have the bound
$||\phi_{(0,n)}(\cdot)||_{0}\leq n!D^{n+1},n=1,2,...$, where if
$\Delta(C_{0},C_{1},D)=[2DC_{0}+C_{0}+C_{1}]^{2}-4(DC_{1}+2C_{0}^{2})<0$ we
have for any $\omega$ the bound and otherwise for $\Delta(C_{0},C_{1},D)>0$ we
have for

$\displaystyle\omega<g_{\pm}(C_{0},C_{1},D)=\frac{2DC_{0}+C_{0}+C_{1}\pm\sqrt{\Delta(C_{0},C_{1},D)}}{2(DC_{1}+2C_{0}^{2})}>0$
(2.71)

This leads to

$\displaystyle||\phi_{(\epsilon,n)}(x)||_{0}\leq\sum_{j=0}^{\infty}||\phi_{(0,j,n)}(x)||_{0}\frac{\epsilon^{j}}{j!}$
(2.72)
$\displaystyle\leq\sum_{j=0}^{\infty}kj!D^{j}\frac{\epsilon^{j}}{j!}||\phi_{0,n}||_{0},n=0,1,2,...$
(2.73)

converging in $||\cdot||_{0}$ for any value of $\omega\leq
g_{\pm}(C_{0},C_{1},D),1/D>\epsilon\geq 0$.

So we have that the limit solution $||\phi(\epsilon,x)||_{0}<\infty$ is
bounded everywhere for $(\omega,x,\epsilon)$ in the allowed ranges.
As the convergence is uniform we have that $\phi(\epsilon,x)$ is continuous on
the specified ranges.

In the same manner we have as a result that $\phi(\epsilon,x)$ belongs to
$L_{2}[0,1]$.
We also have in the same manner that $\sup_{0\leq x\leq
1}|\frac{\partial\phi_{0,n}}{\partial x}|\leq n!D_{1}^{n},n=2,3,...$ for

$\omega\leq g_{\pm}(C_{0}^{\prime},C_{1}^{\prime},D_{1})$

holds and which leads to the uniformly convergent series

$\displaystyle||\phi_{(0,n)}^{\prime}(x)||_{0}\leq\sum_{j=0}^{\infty}||\phi_{(0,j,n)}(x)||_{0}\frac{\epsilon^{j}}{j!}$
$\displaystyle\leq\sum_{j=0}^{\infty}k_{1}j!D_{1}^{j}\frac{\epsilon^{j}}{j!}||\phi^{\prime}_{(0,n)}||_{0},n=0,1,2,...$
(2.74)

converging in $||\cdot||_{0}$ for any value of $\omega\leq
g_{\pm}(C_{0}^{\prime},C_{1}^{\prime},D_{1}),1/D_{1}>\epsilon\geq 0$.

So we have that the limit solution $||\phi^{\prime}(\epsilon,x)||_{0}<\infty$
is bounded everywhere for $(\omega,x,\epsilon)$ in the allowed ranges.
As the convergence is uniform we have that $\phi^{\prime}(\epsilon,x)$ is
continuous on the specified ranges.

In the same manner we have as a result that $\phi^{\prime}_{(\epsilon,0)}(x)$
belongs to $L_{2}[0,1]$.

###### Corrolary 2.3

Let $f$, $\Gamma_{0}$ be continuous and $\Gamma_{1}$ to be $L^{2}$ integrable.
Let

$\sup_{0\leq x\leq 1}\frac{\partial\Gamma_{j}(x,y)}{\partial x}\leq
C_{1}>0,\;\;j=0,1.$

Then $\Gamma(\epsilon,x,y)$ defines a continuous , bounded $C^{1}[0,1]$
solution for $\epsilon<\epsilon_{0}>0$.

Proof: If $\Gamma_{0}$ is continuous the solution $\phi(0,x)$ as in
$L^{2}[0,1]$ it implies that
$\phi(0,x)=f(x)+\omega\int_{0}^{1}\Gamma_{0}(x,y)\phi(0,y)dy$ as a sum of a
continuous function and an integral with a continuous kernel is continuous.
Applying theorem (2.6) we have the sought implication.

###### Corrolary 2.4

Let $f$ continuous and $\Gamma_{1}$ to be $L^{2}$ integrable. Let

$\sup_{0\leq x\leq 1}\frac{\partial\Gamma_{j}(x,y)}{\partial x}\leq
C_{1}>0,\;\;j=0,1.$

Then $\Gamma(\epsilon,x,y)=\epsilon I+\epsilon(\Gamma_{1}-I)$ defines a
continuous , $C^{1}[0,1]$ bounded solution for $\epsilon<\epsilon_{0}>0$.

Proof: Immediate from the above corollary (2.3). We have the respective
theorem as well.

###### Theorem 2.7

Let’s consider the perturbed integral equation

$\displaystyle\phi(\epsilon,x)-\omega\int_{0}^{1}\Gamma(\epsilon,x,y)\psi(y,\phi(\epsilon,y))dy=f(x),\;\;\;\;x\in[0,1],\epsilon\geq
0$ (2.75)

where $\Gamma(\epsilon,x,y)=\Gamma_{0}(x,y)+\epsilon\Gamma_{1}(x,y)$ is
$L^{1}$ integrable respectively functions of their variables and
$\int_{0}^{1}\int_{0}^{1}\Gamma(\epsilon,x,y)dxdy<\infty$. Let

$\sup_{0\leq x\leq 1}\frac{\partial\Gamma_{j}(x,y)}{\partial x}\leq
C_{2}>0,\;\;j=0,1.$

Suppose that

$\displaystyle|\Gamma_{j}(\epsilon,x,y)|\leq C_{j},\;\;\;\;\;\;0\leq x,y\leq
1,\epsilon\geq 0,j=0,1.$ (2.76)

Assuming that the unperturbed equation for $\epsilon=0$ has a solution in
$C([0,1])$ then the perturbation series

$\displaystyle\phi_{(0,0)}(x)=\phi(0,x)=\omega\int_{0}^{1}\Gamma_{0}(x,y)\psi(y,\phi_{(0,0)}(y))dy$
$\displaystyle\phi_{(0,1)}(x)=\omega\int_{0}^{1}\Gamma_{1}(x,y)\psi(y,\phi_{(0,0)}(y))dy$
$\displaystyle+$
$\displaystyle\omega\int_{0}^{1}\Gamma_{0}(x,y)\psi^{(0,1)}(y,\phi_{(0,0)}(y))\phi_{(0,1)}(y)dy\;\;\;\;\;\;\;\;$
$\displaystyle.....$ $\displaystyle\phi_{(0,\nu)}(x)$
$\displaystyle=\omega\int_{0}^{1}\Gamma_{0}(x,y)\psi^{(0,\nu)}(y,\phi_{(0,0)}(y))P_{\nu+1}(y)dy$
$\displaystyle+$
$\displaystyle\omega\int_{0}^{1}\Gamma_{1}(x,y)\psi^{(0,\nu-1)}(y,\phi_{(0,0)}(y))P_{\nu}(y)dy\;\;\;\;\;\;\;\;\;$
(2.79) $\displaystyle P_{\nu}(y)=$
$\displaystyle\sum_{{\sum_{j=1}^{\nu-1}r_{j}\cdot
s_{j}=\nu-1}\atop{r_{j},s_{j}=0,j=1,...,\nu-1}}^{\nu-1}$
$\displaystyle{{\nu-1}\choose{r_{1},s_{1};\cdot\cdot\cdot;r_{\nu-1},s_{\nu-1}}}\prod_{j=1}^{\nu-1}\phi_{(0,s_{j})}(x)^{r_{j}}(y),\nu=2,3,...\;\;\;$
(2.82)
$\displaystyle{m\choose{r_{1},s_{1};\cdot\cdot;r_{m},s_{m}}}=\frac{m!}{\prod_{j=1}^{m}(s_{j}!)^{r_{j}}}$
$\displaystyle\psi^{(k,\nu)}(s,\phi(\epsilon,y))=\frac{\partial^{k}\partial^{\nu}\psi(s,\phi(\epsilon,y))}{\partial
s^{k}\partial\epsilon^{\nu}},\;\;k,\nu=0,1,2,...\;\;\;\;\;\;\;\;$

converges absolutely and uniformly in
$||\cdot||_{0}=\sup_{x\in[0,1]}|\cdot(x)|$ norm and $||\cdot||_{1}$ norm to a
$C^{1}[0,1]$ solution

$\displaystyle\phi(\epsilon,x)=\sum_{j=0}^{\infty}\phi_{(0,j)}(x)\frac{\epsilon^{j}}{j!}$
(2.83)

which is continuous with respect to $(\omega,\epsilon,x)$ for suitable ranges.

Proof: Along the same lines as in theorem (2.6).

Note: We have corresponding results if

$\sup_{0\leq x\leq 1}\frac{\partial^{m}\Gamma_{j}(x,y)}{\partial x^{m}}\leq
C_{m}>0,\;\;j=0,1,m=1,2,...,p.$

Then the respective nonlinear Fredholm equations have a $C^{m}[0,1]$ solution
for $m=1,2,...,p.$

##  3 Solution procedure for nonlinear integral equations

Here we investigate ’small’ perturbations employed on the non linear term
where the unknown function is involved rather than the kernel of the integral
equation. Again we show in theorem (3.1) that the perturbative series
converges to a continuous solution. Then we show in therem (3.2) below next,
the existence of a maximal perurbation range for the perurbation parameter
using Zorn’s lemma on a appropriate partially ordered set of partitions.

###### Theorem 3.1

Let’s consider the nonlinear integral equation

$\displaystyle\phi(\epsilon,x)-\omega\int_{0}^{1}\Gamma(x,y)\psi(\epsilon,y,\phi(\epsilon,y))dy=0,\;\;\;\;x\in[0,1],\epsilon\geq
0$ (3.1)

where $\Gamma(x,y)$ is $L^{2}$ integrable respectively function of its
variables and
$\int_{0}^{1}\int_{0}^{1}\Gamma(\epsilon,x,y)dxdy<\infty$. We have
$\psi(\epsilon,y,z)=z+\epsilon\Psi(y,z)$. Suppose that

$\displaystyle|\Gamma(x,y)|\leq C,\;\;\;\;\;\;0\leq x,y\leq 1,\epsilon\geq 0.$
(3.2)

and $\Psi^{(0,\nu)}(y,s)=\frac{\partial^{\nu}\Psi(y,s)}{\partial
s^{\nu}}\leq\frac{b^{\nu}}{E(\nu,\nu)},0\leq b\leq 1,\;\;\;\;\;\nu=0,1,2,...$
where $E(n,k)\leq{{2n-1}\choose{n-1}}$ is the number of integer solutions of
the equation

$\displaystyle\sum_{j=1}^{k}r_{j}\cdot s_{j}=n,0\leq r_{j},s_{j}\leq
n,j=1,2,..,k.$ (3.3)

If the integral equation

$\displaystyle\phi_{0}(0,x)=\phi(0,x)=\omega\int_{0}^{1}\Gamma(x,y)\phi_{0}(0,y)dy$
(3.4)

has continuous solutions in $C[0,1]$, then the integral equation (3.1) has a
solution in $C[0,1]$.

Proof: Differentiating eq.(3.1) with respect to $\epsilon$ at $0$, we get the
series of integral equations

$\displaystyle\phi_{(0,0)}(x)=\phi(0,x)=\omega\int_{0}^{1}\Gamma(x,y)\phi_{(0,0)}(y)dy$
$\displaystyle\phi_{(0,1)}(x)=\omega\int_{0}^{1}\Gamma(x,y)\phi_{(0,1)}(y)dy$
$\displaystyle+$
$\displaystyle\omega\int_{0}^{1}\Gamma(x,y)\Psi(y,\phi_{(0,0)}(y))dy\;\;\;\;\;\;$
$\displaystyle.....$
$\displaystyle\phi_{(0,\nu)}(x)=\omega\int_{0}^{1}\Gamma(x,y)\phi_{(0,\nu)}(y)dy$
$\displaystyle+$
$\displaystyle\omega\int_{0}^{1}\Gamma(x,y)\Psi^{(0,\nu-1)}(y,\phi_{(0,0)}(y))P_{\nu}(y)dy\;\;\;\;\;\;$
(3.8) $\displaystyle P_{\nu}(y)$ $\displaystyle=$
$\displaystyle\sum_{{\sum_{j=1}^{\nu-1}r_{j}\cdot
s_{j}=\nu-1}\atop{r_{j},s_{j}=0,j=1,...,\nu-1}}^{\nu-1}{{\nu-1}\choose{r_{1},s_{1};\cdot\cdot\cdot;r_{\nu-1},s_{\nu-1}}}\prod_{j=1}^{\nu-1}\phi_{(0,s_{j})}^{r_{j}}(y),\nu=2,3,...\;\;\;\;\;\;\;$
(3.10)
$\displaystyle{m\choose{r_{1},s_{1};\cdot\cdot;r_{m},s_{m}}}=\frac{m!}{\prod_{j=1}^{m}(s_{j}!)^{r_{j}}}$

which are linear Fredholm integral equations of the second kind after the
first. The first has solutions the eigenfunctions $\phi_{n},n=1,2,...$ of
$\Gamma(x,y)$ with associated eigenvalues $\mu_{n},n=1,2,...$ under suitable
assumptions on $K$. Equation (3.8) has solution as a second kind Fredholm
integral equation of the form

$\displaystyle\phi_{n}(0,x)=f_{n}(x)+\omega\sum_{j=1,\omega\mu_{j}\neq
1}\frac{\mu_{j}}{1-\omega\mu_{j}}(f_{n},\phi_{j})\phi_{j}(x)$ (3.11)
$\displaystyle
f_{n}(x)=\omega\int_{0}^{1}\Gamma(x,y)\Psi^{(0,n-1)}(y,\phi_{0}(0,y))P_{n}(y)dy,n=1,2,...$
(3.12)

Inductively we can show as in theorem (2.3) that
$||\phi_{(0,n)}(\cdot)||_{0}\leq\frac{\omega C}{1-\omega
C}(n-1)!D^{n-1},n=1,2,...$, so for $\omega\leq\frac{D}{C+CD}$ we have the
bound $||\phi_{(0,n)}(\cdot)||_{0}\leq n!D^{n},n=1,2,...$, This implies that
under the assumption for continuity of solution in eq.(3.4) then the
perturbation series solution converges absolutely and uniformly in
$||\cdot||_{0}=\sup_{x\in[0,1]}|\cdot(x)|$ norm and $||\cdot||_{1}$ norm to a
$C^{1}[0,1]$ solution

$\displaystyle\phi(\epsilon,x)=\sum_{j=0}^{\infty}\phi_{(0,j)}(x)\frac{\epsilon^{j}}{j!}$
(3.13)

which is continuous with respect to $(\omega,\epsilon,x)$ for suitable ranges.
More precisely it holds for $\epsilon<\frac{1}{D}$ and then the solution is
absolutely bounded by $\frac{1}{1-D\epsilon}$ for all $x\in[0,1]$.

The question arises that as given a perturbative series solution to a non
linear integral equation if the region of perturbation can be expanded further
so as to extend the region of convergence of the series.
The following theorem establishes the existence of a maximal perturbation
along a univariate family of perturbation functions.

###### Theorem 3.2

Let’s consider the nonlinear integral equation

$\displaystyle\phi(\epsilon,x)-\omega\int_{0}^{1}\Gamma(x,y)\psi(\epsilon,y,\phi(\epsilon,y))dy=0,\;\;\;\;x\in[0,1],\epsilon\geq
0$ (3.14)

where $\Gamma(x,y)$ is $L^{2}$ integrable respectively functions of their
variables and
$\int_{0}^{1}\int_{0}^{1}\Gamma(\epsilon,x,y)dxdy<\infty$. We have
$\psi(\epsilon,y,z)=z+\epsilon\Psi(y,z)$. Suppose that

$\displaystyle|\Gamma_{0}(\epsilon,x,y)|\leq C,\;\;\;\;\;\;0\leq x,y\leq
1,\epsilon\geq 0.$ (3.15)

and $\Psi^{(0,\nu)}(y,s)=\frac{\partial^{\nu}\Psi(y,s)}{\partial
s^{\nu}}\leq\frac{b^{\nu}}{E(\nu,\nu)},0\leq b\leq 1,\;\;\;\;\;\nu=0,1,2,...$
where $E(n,k)\leq{{2n-1}\choose{n-1}}$ is the number of integer solutions of
the equation

$\displaystyle\sum_{j=1}^{k}r_{j}\cdot s_{j}=n,0\leq r_{j},s_{j}\leq
n,j=1,2,..,k.$ (3.16)

If the integral equation

$\displaystyle\phi_{0}(0,x)=\phi(0,x)=\omega\int_{0}^{1}\Gamma(x,y)\phi_{0}(0,y)dy$
(3.17)

has continuous solutions in $C[0,1]$, then the integral equation (3.14) has a
maximal perturbation solution in $C[0,1]$.

Proof: Let
$\frac{\partial^{j}\psi(\epsilon,y,\phi(\epsilon,y))}{\partial\epsilon^{j}}\leq
B(j,\epsilon))$ where $B(j,\epsilon)\leq
Qb(\epsilon)^{j})/E(j,j),Q>0,j=1,2,...$. Let a convex, respectively concave
function $f_{1},f_{2}:[a,b]\rightarrow\mathbf{R}$ on its domain of definition.
Then we define the functional on its domain

$\displaystyle V_{1}(f_{1},P,a,b)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{n-1}\left|\frac{f_{1}(x_{j+1})-f_{1}(x_{j})}{x_{j+1}-x_{j}}\right|$
(3.19) $\displaystyle V_{2}(f_{2},P,a,b)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\left|\frac{f_{2}(x_{j+1})-f_{2}(x_{j})}{x_{j+1}-x_{j}}\right|$
(3.20) $\displaystyle P$ $\displaystyle=$
$\displaystyle\\{a=x_{0},x_{1},...,x_{k},..x_{n}=b\\}$ (3.21)

We observe that as $||P||=\max_{0\leq j\leq n-1}|x_{j+1}-x_{j}|\rightarrow 0$
as $n\rightarrow\infty$ we have that
$\lim_{n\rightarrow\infty}V_{j}(f_{j},P,a,b)=\int_{a}^{b}|f_{j}^{\prime}(x)|dx,\;\;j=1,2$.
So constructing the functional $V(f,a,b)$ for a function $f$ with convex or
concave on successive subintervals on its domain of definition $[a,b]$ as

$\displaystyle
V(f,P,a,b)=\sum_{j=0}^{k}V_{i_{j}}(f_{i_{j}},P_{j},a_{j},b_{j})$ (3.22)
$\displaystyle\bigcup_{j=0}^{k}[a_{j},b_{j})=[a,b)$ (3.23) $\displaystyle
f_{i_{j}}\;\;\;\mbox{convex on}[a_{j},b_{j}),\mod(j,2)=0$ (3.24)
$\displaystyle f_{i_{j}}\;\;\;\mbox{concave on}[a_{j},b_{j}),\mod(j,2)=1$
(3.25) $\displaystyle\mbox{ or reversely},\;\;\;j=0,1,...,k.$ (3.26)
$\displaystyle
P=\bigcup_{j=0}^{k}P_{j},\mbox{span}(P_{j})=b_{j}-a_{j},j=0,...,k.$ (3.27)

We have that
$\lim_{n\rightarrow\infty}V(f,P,a,b)=\int_{a}^{b}|f^{\prime}(x)|dx,\;\;$.
Moreover from the construction of the discrete functionals the convergence is
monotone decreasing. Additionally this allows monotonicity and convergence
with respect to refinement in partitions of the interval $[a,b)$ and shall be
used in the following construction.

Given a function $f:[0,\infty)\rightarrow\mathbf{R}$ we impose an order on the
partitions of $\mathbf{R}^{+}$ considering $P\preceq Q$ if
$\mbox{span}(P)<\mbox{span}(Q)$ or if $\mbox{span}(P)=\mbox{span}(Q)$ and
$V(f,P)>V(f,Q)$.

Thus as each chain of ascending partitions $P_{0}\preceq
P_{1}\preceq\cdot\cdot\cdot\preceq P_{m}$ has an upper bound with span
$\sup_{j=0}^{\infty}\mbox{span}(P_{j})$ or if spans are finally equal, with
value of the functional $\liminf_{j\rightarrow\infty}V(f,P_{j})$, by Zorn’s
lemma the partial ordered set of chains has a maximal element $P^{*}$. So we
have a maximal perturbation range $\mbox{span}(P^{*})$. Suppose we are
achieving a perturbation of magnitude $\epsilon_{0}>0$. And from this value we
achieve an additional perturbation of magnitude $\epsilon_{1}-\epsilon_{0}$
with nonlinear term $\psi(\epsilon_{1},y,\phi(\epsilon_{1},y))$. Continuing in
this manner we achieve perturbations with magnitudes
$\epsilon_{0}<\epsilon_{1}<\cdot\cdot\cdot<\epsilon_{m},m\in\mathbf{N}$. This
defines a partition
$P=\\{\epsilon_{0},\epsilon_{1},\cdot\cdot\cdot,\epsilon_{m}\\},m\in\mathbf{N}$of
$[0,\infty)$. We order in the same way as above with $\mbox{span}(P_{m})$ and
$V(\psi(\epsilon,\cot,\cdot),P)$ these partitions defined by successive
perturbations of the integral equation (3.14) and we have as a result that
there is a maximal element by Zorn’s lemma on this set of partitions
$P_{\psi}$ with maximal span and therefore range.

##  4 Conclusions

We investigate perturbative solutions for kernel perturbed integral equations
and prove the convergence in an appropriate ranges of the perturbation series.
Next we investigate perturbation series solutions for nonlinear perturbations
of integral equations of Hammerstein type and formulate conditions for their
convergence. Then applying Zorn’s lemma to a construction for the succesive
’small’ perturbations to the perturbing function $\psi(\epsilon,y,z)$ with
respect to the variable $\epsilon$ we get that there is a maximal span where
$\epsilon$ can vary and give finite solutions to the integral equation (3.14)
which are useful for the practitioner. Furthermore we note that these methods
can be extended to more general equations arising in mathematical physics and
mechanics.

## References

  * [1] Agarwal P., Akbar M., Rashid Nawaz R., Jleli M.,”Solutions of system of Volterra integro-differential equations using optimal homotopy asymptotic method”,Math Meth Appl Sci. 2021;44:2671–2681.
  * [2] Ahsan S., Nawaz R., Akbar M., Nisar K.S., Abualnaja K.M., Mahmoud E.E., Abdel-Aty A., ”Numerical solution of two-dimensional fractional order Volterra integro-differential equations”,AIP Advances 11, 035232 (2021).
  * [3] Hashmi M.S.,N.Khan N.,and Iqbal S.,“Optimal homotopy asymptotic method for solving nonlinear Fredholm integral equa-tions of second kind,”Applied Mathematics and Computation,vol. 218, no. 22, pp. 10982–10989, 2012.
  * [4] Almousa M., Ahmad I. ,”Optimal Homotopy Asymptotic Method for Solving the Linear Fredholm Integral Equations of the First Kind”,Abstract and Applied Analysis, Special Issue,Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models, vol. 2013.
  * [5] Almousa M., Ahmad I.,”Numerical solution of Fredholm-Hammerstein integral equations by using optimal homotopy asymptotic method andhomotopy perturbation method”,AIP Conference Proceedings 1605, 90 (2014); https://doi.org/10.1063/1.4887570.
  * [6] Cabada A., Infante G.,Tojo F.A., ”Nonzero solutions of perturbed hammerstein integral equations with deviated arguments and applications”, Topological Methods in Nonlinear Analysis, Volume 47, No. 1, 2016, 265–287.
  * [7] Delves L.M., Mohamed J.L., ”Computational Methods for Integral Equations”, Cambridge University Press, 1985.
  * [8] Yusufoglu (Agadjanov) E., ”A homotopy perturbation algorithm to solve a system of Fredholm–Volterra type integral equations”, Mathematical and Computer Modelling 47 (2008) 1099–1107.
  * [9] Eshkuvatov, Z. K. , Ismail, Sh. , Mamatova, H. X., Viscarra, D. S., and Aloev, R. D., ”Modified HAM for solving linear system of Fredholm-Volterra Integral Equations”, Malaysian Journal of Mathematical Sciences 16(1): 87–103 (2022).
  * [10] Asghar Ghorbani A., , Jafar Saberi-Nadjafi J., ”Exact solutions for nonlinear integral equations by a modified homotopy perturbation method”, Computers $\&$ Mathematics with Applications, Volume 56, Issue 4, August 2008, Pages 1032-1039.
  * [11] He J.H., ”The homotopy perturbation method for nonlinear oscillators with discontinuities”, Appl. Math. Comput. 151 (2004) 287–292.
  * [12] Hochstadt H., ”Integral Equations”, Wiley, New York, London , 1989.
  * [13] Infante G., Pietramala P.,”Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations”, Nonlinear Analysis 71 (2009) 1301–1310.
  * [14] Nawaz Y., Ahsan S., Akbar M., Farooq M., Sulaiman M.,Ullah H., Islam S., ”Semi Analytical Solutions of Second Type of Three-Dimensional Volterra Integral Equations”,Int. J. Appl. Comput. Math (2020) 6:109.
  * [15] Nayfeh A.H., ”Introduction to Perturbation Technique”, John Wiley and Sons, New York, 1981.
  * [16] Precup R., ”Methods in Nonlinear Integral Equations”,2002, Springer Science+Business Media Dordrecht.
  * [17] O’Regan D., Meehan M., ”Existence Theory for Nonlinear Integral and Integrodifferential Equations”, 1998, Springer Science+Business Media Dordrecht.
  * [18] Yildirim A., ”Homotopy perturbation method for the mixed Volterra–Fredholm integral equations”, Chaos, Solitons and Fractals, 42 (2009), 2760–2764.