Abstract
In this work, we present a boundary value problem of hybrid functional differential inclusion with nonlocal condition. The boundary conditions of integral and infinite points will be deduced. The existence of solutions and its maximal and minimal will be proved. A sufficient condition for uniqueness of the solution is given. The continuous dependence of the unique solution will be studied.
Highlights
IntroductionModels of hybrid functional differential and integral equations have many applications (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14])
Models of hybrid functional differential and integral equations have many applications.Boundary value problems with nonlocal boundary conditions have been studied by some authors.Here, we assess the boundary value problem of hybrid nonlinear functional differential inclusion with nonlocal condition.d x ( t ) − x (0)∈ F (t, x (φ2 (t))), t ∈ (0, 1) (1)dt g(t, x (φ1 (t)))with the nonlocal boundary condition m∑ ak x(τk ) = x0, Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.ak > 0 τk ∈ [0, 1]. (2)
There exists at least one solution x ∈ C [0, 1] of the hybrid nonlinear functional differential Equation (5) with the nonlocal condition (2)
Summary
Models of hybrid functional differential and integral equations have many applications (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14]). We assess the boundary value problem of hybrid nonlinear functional differential inclusion with nonlocal condition. We deduce the same results for the boundary value problem of hybrid nonlinear functional differential inclusion (1) with a nonlocal integral condition. From assumptions (ii)–(iv), we can deduce that the set of selection S F of F is nonempty (see [1,2,5]), that there exists f ∈ F (t, x ) such that (vi) f : I × R −→ R is measurable in t for every x ∈ R and continuous in x for t ∈ [0, 1], there exists a bounded measurable function a2 : [0, 1] → R and a positive constant. Equation (5) with any of the nonlocal boundary conditions (2)–(4) is a solution of the nonlocal problem of the hybrid nonlinear functional differential inclusion with any one of the nonlocal conditions (1)–(4).
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