On the solvability of a nonlinear functional integral equations via measure of noncompactness in L p ( R N )

Using the technique of a suitable measure of non-compactness and the Darbo ﬁxed point theorem, we investigate the existence of a nonlinear functional integral equation of Urysohn type in the space of Lebesgue integrable functions L p ( R N ). In this space, we show that our functional-integral equation has at least one solution. Finally an example is also discussed to indicate the natural realizations of our abstract result.


Preliminaries
We will collect in this section some definitions and basic results which will be used further on throughout the paper.
First, we denote L p (U ) (U ∈ R N ) as the space of Lebesgue integrable functions on U with the standard norm Theorem 2.1 [1,8,9] Let F be a bounded set in where τ h f (x) = f (x + h) for all x, h ∈ R N . Also for > 0 there is a bounded and measurable subset Next, we recall the concept of measure of noncompactness, let E be an infinite dimensional Banach space with norm . and zero element θ. Denote byM E the family of all nonempty and bounded subsets of E , N E and N W E the family of all nonempty relatively compact and weakly relatively compact sets, respectively. The symbolsX and ConvX stand for the closure and closed convex hull of a subset X of E, respectively. We use the standard notation X + Y and λX for algebraic operations on sets, while, we denote B r = B(θ, r) the closed ball centered at θ and with radius r.
(5) If X n ∈ M E , X n =X n and X n+1 ⊂ X n for n = 1, 2, . . . and if

Theorem 2.2 [1]
Suppose 1 ≤ p < ∞ and X is a bounded subset of (R N ). For x ∈ X and > 0 is a measure of non compactness on L p (R N ).
At the end of this section, we recall the fixed point theorem due to Darbo which enables us to prove the existence theorem for solutions of a several integral equations considered in nonlinear analysis. To quote this theorem we need the following definitions.

Definition 2.2 [12]
The function f : I × R → R satisfies Carathéodory condition if it satisfies the following two conditions: (1) f is measurable in t ∈ I for any x ∈ R.
(2) f is continuous in x ∈ R for almost all t ∈ I.

Definition 2.3 (Darbo condition)[11]
Let Ω be a nonempty subset of a Banach space E and let A : Ω → E be a continuous operator which transforms bounded sets onto bounded ones. We say that A satisfies the Darbo condition (with a constant k ≥ 0) with respect to a measure of noncompactness µ if for any bounded subset X of Note that, if A satisfies the Darbo condition with k < 1, then it is called a contraction operator with respect to µ.

Theorem 2.3 (Darbo fixed point theorem)[7]
Let Ω be a nonempty, bounded, closed and convex subset of E and let f : Ω → Ω be a continuous transformation which is a contraction with respect to the measure of noncompactness µ, i.e. there exists a constant k ∈ [0, 1) such that for any nonempty subset X of Ω. Then f has at least one fixed point in the set Ω.

Main result
This section is devoted to discuss the solvability of the following nonlinear functional integral equation Now, we will try to assume some assumptions under which we can prove our existence theorem.
Assume the following conditions are satisfied:

and continuous in
x for all t ∈ R ) and there exists a constant l ∈ [0, 1) and a i ∈ L p (R N ) such that for any u, v ∈ R and almost all x, y ∈ R N where i = 1, 2.
(v) The operator Q is bounded linear operator and continuously maps the space L p (R N ) into itself. Moreover, there exists a nondecreasing function ψ : for any u ∈ L p (R N ).
(vi) there exists a positive constant r 0 such that

y)u(y)dy
and Proof: First of all, we define the operator F : Next, we show that Now, we show that w 0 (F X) ≤ l(b 2 + 1)w 0 (X) for any nonempty set X ⊂ B r0 . To do this, we fix arbitrary T > 0 and > 0, let us choose u ∈ X and for x, h ∈ B T with h R N ≤ , we have Thus, we obtain Also, we have w T (f 2 , ) ,w T (f, ), and w T (a i , ) → 0 as → ∞ where i = 1, 2, 3 then, we obtain where l(b 2 + 1) ≤ 1.

(-13)
Next, let us fix an arbitrary number T > 0, then taking into account our assumptions, for an arbitrary function u ∈ X. We have .
and hence we obtain that

Consequentially we infer from equation -13, -16
w 0 (F X) ≤ l(b 2 + 1)w 0 (X), so, the operator F satisfies all conditions of Darbo fixed point theorem, which enables us to deduce that F has at least one solution inL p (R N ) . Thus the proof is finished.
Next, we will need the following theorem that help us in a coming example.