Perturbed integral equations in modular function

We focus our attention on a class of perturbed integral equations in modular spaces, by using xed point Theorem I.1 (see (1)).


Introduction
In the present work, we focus our attention on a class of perturbed integral equation which can be written as u(t) = exp(−tA)f 0 + t 0 exp((s − t)A)T u(s)ds (I) in the modular space C ϕ = C([0, b], L ϕ ) (see [1]), where L ϕ is the Musielak-Orlicz space, f 0 is a fixed element in L ϕ , A : L ϕ → L ϕ is a linear operator and T : L ϕ → L ϕ is ρ − c-Lipschitz, i.e. there exists k > 0 such that ρ(c(T x − T y)) ≤ kρ(x − y) for any x, y in L ϕ ( ρ being a modular ).Since ρ is not subadditive, then the sum of these operators is not necessarily ρ-Lipschitz and the convexity of the integral presents a more delicate problem.Therefore, it is natural in our study to introduce c 0 constant c 0 and assume some hypotheses on A, T , and b.
Recall that ρ has the Fatou property if: ρ(x−y) ≤ lim inf ρ(x n −y n ), whenever x n ρ → x and y n ρ → y.And we say that ρ satisfies the ∆ 2 -condition if: ρ(2x n )→0 as n→ + ∞ whenever ρ(x n )→0 as n→ + ∞, for any sequence (x n ) n∈IN in X ρ .

Perturbed integral equation class
In this section, we will study the existence of solution of the following perturbed integral equation: We present the general hypotheses of the equation (I).exp (−at)ρ(u(t)) is a modular of C([0, b], L ϕ ) with a > 0 ( see [1]).H 2 ) Let A : L ϕ → L ϕ be a linear application, assume that there exist α 0 > max(e −1 , eb 2 ) and M > 0 such that ρ(α 0 Ax) ≤ M ρ(x) for any x ∈ L ϕ .H 3 ) Let T : L ϕ → L ϕ be ρ − c-Lipschitz with c > 0, i.e there exists k > 0 such that ρ(c(T x − T y)) ≤ kρ(x − y) for any x, y ∈ L ϕ .H 4 ) Let f 0 be fixed element in L ϕ .

Remark.
If we restrict our attention to the Banach space (L ϕ , .ρ ).Then the equation (I) can be written as follows: Thus, if A ≡ I then ( * ) becomes But the latter equation has been treated before in [1] and [4].This let us to reduce the study to the case A ≡ I when ( * ) can be written in the form below: It follows from the fact that ρ is not subadditive that T + B is not necessarily ρ-Lipschitz contrary to the situation in [1] and [2].We cite first the theorem below which we shall use in the proof of Theorem 2.1.
Theorem 2.2 .(See [1]) Let X ρ be a ρ-complete modular space.Assume that ρ is an s-convex, satisfying the ∆ 2condition and having the Fatou property.Let B be a ρ-closed subset of X ρ and T : B → B a mapping such that Then T has a fixed point.
Proof of Theorem 2.1.

+∞ m=0
A m m! (x) make a sense. And 3 rd step.In this part, we show that We have And since We have α 0 ≥ eb c 0 > 0, and since α 0 > max(e −1 , eb 2 ), then α 0 > b.

th
Step.We have It suffices to take a > ke M −1 c 0 , then we have λk exp (M − 1) c 0 a (1 − e −ab ) < λ .By Theorem 2.2, S has a fixed point which is a solution of the equation (I).

Remark
In third step, instead of the combination convex , which gives the conclusion of theorem under the following hypotheses: is a linear application , and there exists M > 0 such that : Consider now the following perturbed integral equation.
The same techniques than in the proof of Theorem 2.1 are used to establish Theorem 2.3 below by taking care of the choose of λ in (1, 1  1−e −b ] , which gives ρ( t 0 λe s−t e (s−t)A x(s)ds) ≤ lim inf( n−1 i=0 λ(t i+1 − t i )e t i −t ρ(e (t i −t)A x(t i )) and Theorem 2.3 Assume that for α 1 ≥ eb, there exists M > 0 such that ρ(α 1 Ax) ≤ M ρ(x) for any x ∈ L ϕ and there exists k > 0 such that ρ(e(T x − T y)) ≤ kρ(x − y) for any x, y in L ϕ .Then, the perturbed integral equation (II) has a solution u ∈ C([0, b], L ϕ ).
Remark.By using the same technics as in the proof of Theorem 2.3, we can prove the existence of a solution of the equation below: Example of the equation (I).Let ϕ be a Musielak-Orlicz function on a measurable space ([0, 1], A, µ), ρ ϕ be a modular defined by In this example, we study the existence of a solution of the following integral equation a fixed element in L ϕ and the operator A is equal to k 0 I, where I is the identity function of L ϕ with k 0 ≤ 1 α 0 .Let T be a mapping from L ϕ into L ϕ defined by T u = 1 0 c 0 e K 1 (s, u)ds.