On Existence and Uniqueness of Solutions of a Nonlinear Integral Equation

The purpose of this paper is to study the existence of fixed point for a nonlinear integral operator in the framework of Banach space 𝑋∶=𝐶([𝑎,𝑏],ℝ𝑛). Later on, we give some examples of applications of this type of results.


Introduction
In this paper, we intend to prove the existence and uniqueness of the solutions of the following nonhomogeneous nonlinear Volterra integral equation: where x, t ∈ a, b , −∞ < a < b < ∞, f : a, b → R n is a mapping, and F is a continuous function on the domain D : { x, t, u : x ∈ a, b , t ∈ a, x , u ∈ X}.The solutions of integral equations have a major role in the fields of science and engineering 1, 2 .A physical event can be modeled by the differential equation, an integral equation, an integrodifferential equation, or a system of these 3, 4 .Investigation on existence theorems for diverse nonlinear functional-integral equations has been presented in other references such as 5-10 .

Note 1.
As ϕ is a bounded linear mapping on X, then ϕ x λx, where λ does not depend on x ∈ X. Definition 2.1.Let S denote the class of those functions α : 0, ∞ → 0, 1 satisfying the condition lim sup

Existence and Uniqueness of the Solution of Nonlinear Integral Equations
In this section, we will study the existence and uniqueness of the nonlinear functional-integral equation 1.1 on X. iii there exists a integrable function p Proof.Consider the iterative scheme

3.3
As the function φ is increasing then so, we obtain

3.6
and so the sequence {d u n 1 , u n } is nonincreasing and bounded below.Thus, there exists τ ≥ 0 such that lim n → ∞ d u n 1 , u n τ.Since lim sup s → τ α s < 1 and α τ < 1, then there exist r ∈ 0, 1 and > 0 such that α s < r for all s ∈ τ, τ . We can take ν ∈ N such that τ ≤ d u n 1 , u n ≤ τ for all n ∈ N with n ≥ ν.On the other hand, we have and hence, {u n } is a Cauchy sequence.Since X, d is a complete metric space, then there exists a u ∈ X such that lim n → ∞ u n u.Now, by taking the limit of both sides of 3.2 , we have F x, t, u t dt .

3.9
So, there exists a solution u ∈ X such that Tu u.It is clear that the fixed point of T is unique.
Note 2. Theorem 3.1 was proved with the condition i , but there exist some nonlinear examples ϕ, such that by the analogue method mentioned in this theorem, the existence, and uniqueness can be proved for those.For example ϕ x sin x .

Theorem 3 . 1 .
Consider the integral equation 1.1 such that i ϕ : X → X is a bounded linear transformation, ii F : D → R n and f : a, b → R n are continuous,

Remark 4 . 4 .
The unique solution u ∈ C 0, 1 , R of the Volterra integral 4.5 is given byu x E 1−α λΓ 1 − α x 1−α u 0 , x ∈ 0, 1 , -Leffler function.The Mittag-Leffler function was introduced early in the 20th century by the Swedish mathematician whose name it bears.Additional properties and applications can be found, for example, in Erdélyi 13 and, especially, in the survey paper by Mainardi and Gorenflo 14 .
this section, for efficiency of our theorem, some examples are introduced.For Examples 4.1 and 4.2, 5 is used.Maleknejad et al. presented some examples that the existence of their solutions can be established using their theorem.Generally, Examples 4.1 and 4.2 are introduced for the first time in this work.On the other hand, for Example 4.3, 12 is applied.In Chapter 6 of this reference, the existence theorems for Volterra integral equations with weakly singular kernels is discussed.Example 4.1 is extracted from this chapter.≤ 1, then by applying the result obtained in Theorem 3.1, we deduce that 4.1 has a unique solution in Banach space C 0, 1 , R .