الفهرس 
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Abstract
ABSTRACT
Differential and Integral equations is one of the most important fields in pure and applied mathematics . The integral equations appear in a variety of applications are often obtained from a differential equations , and the purpose of the application is to be the problem-solving easier and sometimes to be able to prove the fundamental results on the existence and uniqueness of the solutions.
Integral equations are classified into linear and nonlinear integral equations.
On the other hand we have the fractional calculus , which is an important role in the field of integral equations as ther e are many physical problems that are applicable to an integral equations with fractional orders , which is applied in various areas of engineering,
Science , applied mathematics , bioengineering and others.
Clarify that we take an example in physics illustrates the application of integral equations in physics .
The thesis consists of three chapters .
Chapter 1 {Preliminaries}
The first chapter begins with an introduction of the integral equations and including an applied example in physics .and it is then clarify all the definitions and theories that apply to finding solutions to the problems described in the following chapters.
Capter 2 {Existence and Uniqueness Solutions}
This chapter deals with the existence of a unique integrable solution and a unique continuous solutions by using Banach fixed point theorem in the L1[0,1] space and C [0,1] for the following equations:
1-Fredholm equation:
x(t)=g(t)+ ∫_0^1▒〖k(t,s ) f(s,x(s) ) ds〗 …………t∈L^1 [0,1]….(1)
We will study the solvability of the above equation in the space L1 on the interval [0,1] by Banach fixed point theorem ,under certain assumptions to prove that the equation (1) has a unique fixed point in the space L1[0,1].
As well as we will study the solvability of the same equation , but , in the space C on the interval [0,1] :
x(t)=g(t)+ ∫_0^1▒〖k(t,s ) f(s,x(s) ) ds〗 …………t∈C[0,1]
Which is has a unique fixed point.
2- Volterra equation:
x(t)=g(t)+∫_0^t▒〖k(t,s) f(s,x(s)) ds〗 ……t∈L^1 [0,T](2)
Now for the solvability of the equation (2), we will prove that the equation (2)has a unique fixed point in the space L1 on the interval [0,T] by Banach fixed point theorem .
As well as we will study the solvability of the same equation , but , in the space C on the interval [0,T] :
x(t)=g(t)+∫_0^t▒〖k(t,s) f(s,x(s)) ds〗 ……t∈C[0,T](2)
Which is has a unique fixed point.
Chapter 3 { Existence of Integrable Solutions }
In this chapter we will use Darbo fixed point theorem to study the existence of at least one solution for the Fredholm integral equation:
x(t)=g(t)+f(t,∫_0^1▒〖k(t,s) x(s) ds〗………………………t∈L^1 [0,1]..(1)
x(t)=g(t)+∫_0^1▒〖k( t,s) f(s,x(s) )ds…………t∈L^1 [0,1]…(2)〗
x(t)=g(t)+∫_0^1▒〖k_1 (t,s) f_1 (s〗,∫_0^1▒〖k_2 (s,τ)f_2 (τ,x(τ) dτ)ds〗 …..…….(3)
t∈L^1 [0,1]
Similarly as the last equation we can prove the existence theorem for the Volterra integral equation
x(t)=g(t)+∫_0^t▒〖k_1 (t,s) f_1 (s〗,∫_0^t▒〖k_2 (s,τ)f_2 (τ,x(τ) dτ)ds〗 …..…….(4)
t∈L^1 [0,T]
In the sequel, we can treat the solvability of the system
x^’=f_1 (t,y(t) )…….t∈[0,T]
〖y^’=f〗_2 (t,x(t))……..t∈[0,T]
Then we have:
x(t)=g_1 (t)+∫_0^t▒〖f_1 (s,y(s) )ds……t∈[0,T]..〗
y(t)=g_2 (t)+∫_0^t▒〖f_2 (τ,x(τ) )dτ………..〗 t∈[0,T]
So combine these two equations:
〖x(t)=g_1 (t)+∫_0^t▒〖f_1 (s,∫_0^t▒〖f_2 (τ,x(τ) )dτ)ds……………t∈[0,T].(5)〗〗〗_
Finally in this chapter we studied the Existence of at least a solution for Fredholem nonlinear integral equation with fractional order.
x(t)=g(t)+∫_0^1▒〖(t-s)^(α-1)/Γ(α) f_1 (s,∫_0^1▒(s-τ)^(γ-1)/Γ(γ) f_2 (τ,x(τ) )dτ)ds….(6)〗
t∈L^1 [0,1]
Such that 0<α≤1 and 0<γ≤1
We proved that there is at least one solution in the space L1[0, 1] by using Darbo fixed point theorem.