Solvability of a State–Dependence Functional Integro-Differential Inclusion with Delay Nonlocal Condition

: There is great focus on phenomena that depend on their past history or their past state. The mathematical models of these phenomena can be described by differential equations of a self-referred type. This paper is devoted to studying the solvability of a state-dependent integro-differential inclusion. The existence and uniqueness of solutions to a state-dependent functional integro-differential inclusion with delay nonlocal condition is studied. We, moreover, conclude the existence of solutions to the problem with the integral condition and the inﬁnite-point boundary one. Some properties of the solutions are given. Finally, two examples illustrating the main result are presented.


Introduction
A functional equation is an equation involving an unknown function at more than an argument value. In functional equations, argument deviations are the variations between the argument values of an unknown function and an independent variable t. The functional differential equation, or differential equations with diverging arguments, is created by combining the concepts of differential and functional equations. Functional differential equations are used to describe many phenomena in different sciences, see [1]. Nonlocal problems in mathematical physics are problems in which, unlike traditional boundary value conditions, the desired function's values at distinct places of the boundary (and/or its values at the boundary and outside it) are related. In fact, it is reasonable to treat the theory of functional differential equations and the theory of nonlocal problems as one indivisible theory.
For various reasons, many researchers have been interested in researching the nonlocal problem of functional differential equations with infinite point conditions; see for z (t) = z(z(t)) + z(t).
In this study, the initial value problem of the functional differential inclusion with self-dependence on a nonlinear delay integral operator dy dt ∈ Ψ t, y µ(t) 0 ψ(u, y(u))du a.e. t ∈ (0, B], with the nonlocal condition q y(µ(τ )) = y 0 , q > 0, τ ∈ (0, B), was investigated. We study the existence of the absolutely continuous solution y ∈ AC[0, B] and demonstrate the continuous dependence on y 0 and ψ. Moreover, as applications, we study the nonlocal problem of Equation ( and with the infinite-point boundary condition q y(µ(τ )) = y 0 , if ∑ ∞ =1 q is convergent.
The paper is organized as follows: In Section 2, the equivalence of the functional differential inclusion with state-dependence on the nonlinear delay integral operator (1) with the nonlocal condition (2) is given. In Section 3, we study the existence of absolutely continuous solutions to problem (1) and (2), and conclude the existence of solutions to problem (1) with the integral condition (3) and the infinite-point boundary condition (4). In Section 4, we establish the existence of exactly one solution for (1) and (2). In Section 5, the continuous dependence of the solution is studied. Finally, in Section 6, two examples are given to corroborate the main existence result and a numerical example is given to demonstrate the difference between the exact solution and numerical solution.

Auxiliary Results
Consider the following assumptions: From the assumptions (a)-(d), we can deduce that there exists a r ∈ Ψ(t, y), such that the following is satisfied: There exist a bounded measurable function c(t) and a positive constant b > 0, such that |r(t, )| ≤ |c(t)| + b| |, |c(t)| ≤ M. and the functional r satisfies the integro-differential equation (2) ψ : [0, B] × R → R + satisfies Carathéodory condition: - (1), we can deduce that every solution of (1) is also a solution of (5).

Remark 1. From (a) and
The equivalence of (5)-(2) and the integral equation is given in the following lemma.
are equivalent.

Existence of Solution
In the following theorem, using Schauder's fixed point theorem, we establish the existence of at least one solution of (1) and (2). Proof. First, we define the operator A associated with Equation (6) and set P L by Then we have for y ∈ P, and so Thus, But Hence From (11) and (12), we obtain From (12) and (13), we obtain This proves that A : P L → P L ; the class of functions {Ay} is uniformly bounded and equi-continuous in P L .

•
For the nonlocal integral condition, we present the following theorem.
then, as κ → ∞ the nonlocal condition (2) will be As κ → ∞, the solution of the nonlocal problem (1), (3) will be This completes the proof.
• For the infinite-point boundary condition, we present the following theorem.

Uniqueness of the Solution
Consider the following assumptions: ( a) The set Ψ(t, y) is nonempty, convex and closed ∀(t, y) ∈ [0, B] × R.
-Ψ satisfies the Lipschitz condition with a positive constant b such that where H(t, y) is the Hausdorff metric between the two subsets A, B ∈ [0, B] × E.

Remark 2.
From this assumptions we can deduce that there exists a function r ∈ Ψ(t, y), such that (1 * ) r : [0, B] × R → R is measurable in t for any y ∈ R and satisfies the Lipschitz condition |r(t, y) − r(t, q)| ≤ b|y − q|.
In the following theorem, we establish existence of exactly one solution of (1) and (2).  (1) and (2) is unique.
Proof. Let x, y be two the solutions of (1) and (2). Then Therefore, we obtain Thus, we arrive at Hence, which implies x(t) = y(t) and the solution of the nonlocal problem (1) and (2) is unique.
Proof. Let y * be a solution of the integral equation such that then |y 0 − y * 0 | < δ. Then Thus, we arrive at Therefore, we obtain Hence Then the solution of the nonlocal problem (1) and (2) depends continuously on y 0 . Theorem 6. If (4)-(6) hold, then the solution of the nonlocal problem (1) and (2) continuously depends on the function ψ.
Thus, we find Therefore, we have Then the solution of the nonlocal problem (1) and (2) continuously depends on the function ψ.

Examples
Example 1. Consider the differential equation with condition Set r t, y Then r t, y and also |ψ(u, y(u))| ≤ 1.

Example 2. Consider the differential equation
with condition The integral equation and also |ψ(u, y(u))| ≤ 1.

Conclusions
In this work, the existence of an absolutely continuous solution using Schauder's fixed point theorem, the uniqueness solution and the continuous dependence of the functional differential inclusion with self-dependence on a nonlinear delay integral operator were studied. Some examples were introduced to illustrate the benefits of our results. Lastly, the Picard method was used to estimate the solution of a given example and plot the solution.