Boundary Value Problem of Nonlinear Hybrid Differential Equations with Linear and Nonlinear Perturbations

Saïd Melliani ,Abdelati El Allaoui,Lalla Saâdia Chadli

doi:10.1155/2020/9850924

Abstract

The aim of this paper is to study a boundary value problem of the hybrid differential equation with linear and nonlinear perturbations. It generalizes the existing problem of second type. The existence result is constructed using the Leray–Schauder alternative, and the uniqueness is guaranteed by Banach’s fixed-point theorem. Towards the end of this paper, an example is provided to illustrate the obtained results.

Highlights

Sultan Moulay Slimane University, Department of Mathematics, Laboratory of Applied Mathematics & Scientific Calculus, P.O
Introduction e application of differential equations in different real domains has increased the importance of this theory which is still under development
Hybrid differential equations are a subfield of differential equations which has enough importance

Summary

Preliminaries

We recall some basic results used in this paper. Let Π: Y ⟶ Y be a completely continuous operator and PΠ 􏼈y ∈ Y: y δΠy for some 0 < δ < 1􏼉. We recall the following lemmas on which we will base ourselves to build the solution of our problem. Lemma 2 (see [6]). Suppose that x↦x − g(t, x) is increasing in R for each t ∈ I. en, for any h: I ⟶ R+, the function x ∈ C(J, R+) is a solution of the hybrid differential equation g(t, x(t))] h(t),. X(0) x0 ∈ R, if and only if x satisfies the following hybrid integral equation:

Existence Result

Conclusion