Solutions of a Class of Deviated-Advanced Nonlocal Problems for the Differential Inclusion

We study the existence of solutions for deviated-advanced nonlocal and integral condition problems for the di ﬀ erential inclusion x 1 (cid:2) t (cid:3) ∈ F (cid:2) t,x (cid:2) t (cid:3)(cid:3) As an application, we deduce the existence of a solution for the nonlocal problem of the di ﬀ erential inclusion with the deviated-advanced integral condition It must be noticed that the following nonlocal and integral conditions are special cases of our nonlocal and integral conditions


Introduction
Problems with nonlocal conditions have been extensively studied by several authors in the last two decades. The reader is referred to 1-12 and references therein. Consider the deviated-advanced nonlocal problem b j x ψ η j , a k , b j > 0, 1.2 where τ k , η j ∈ 0, 1 , α > 0 is a parameter, and ψ and φ are, respectively, deviated and advanced given functions.
Our aim here is to study the existence of at least one absolutely continuous solution x ∈ AC 0, 1 for the problem 1.1 -1.2 when the set-valued function F : R → P R is L 1 -Carathéodory.

Abstract and Applied Analysis
As an application, we deduce the existence of a solution for the nonlocal problem of the differential inclusion 1.1 with the deviated-advanced integral condition 1 0 x φ s ds α 1 0 x ψ s ds. 1.3 It must be noticed that the following nonlocal and integral conditions are special cases of our nonlocal and integral conditions x ψ s ds 0.

Preliminaries
The following preliminaries are needed.
x is upper semicontinuous for almost all t ∈ 0, 1 , for almost all t ∈ 0, 1 .

Abstract and Applied Analysis
3 ii x → f t, x is continuous for almost all t ∈ 0, 1 , iii there exists m ∈ L 1 0, 1 , D , D ⊂ R such that |f| ≤ m.
is called the set of selections of the set-valued function F.
has at least one solution x ∈ AC 0, T .

Existence of Solution
Consider the following assumptions.
Now we have the following lemma.

Abstract and Applied Analysis
Proof. From the assumption that the set-valued function F : 0, 1 × R → P R is L 1 -Carathéodory, then Theorem 2.4 there exists a single-valued selection f : Let t φ τ k . Then For the existence of the solution, we have the following theorem. Proof. Define a subset Q r ⊂ C 0, 1 by Clearly, the set Q r is nonempty, closed, and convex. Let H be an operator defined by

3.11
By assumptions i -ii and the Lebesgue dominated convergence theorem, we deduce that lim n → ∞ Hx n t Hx t .

3.12
Then H is continuous. Now, letting x ∈ Q r , then φ t ≤ t and ψ t ≥ t , we obtain

3.16
Then the integral equation 3.2 has at least one continuous solution x ∈ C 0, 1 .
Abstract and Applied Analysis 7 The following theorem proves the existence of at least one solution for the nonlocal problem 1.1 -1.2 . To complete the proof, we prove that the integral equation 3.2 satisfies nonlocal problem 1.1 -1.2 .

3.17
Letting t φ τ k in 3.2 , we obtain

3.18
Also, letting t ψ η j in 3.2 , we obtain

3.19
And from 3.19 from 3.18 , we obtain