Mawhin’s Continuation Technique for a Nonlinear BVP of Variable Order at Resonance via Piecewise Constant Functions

Shahram Rezapour,Sina Etemad,Zoubida Bouazza,Mohammed Said Souid,Sotiris K Ntouyas,Suphawat Asawasamrit,Jessada Tariboon

doi:10.3390/fractalfract5040216

Abstract

In this paper, we establish the existence of solutions to a nonlinear boundary value problem (BVP) of variable order at resonance. The main theorem in this study is proved with the help of generalized intervals and piecewise constant functions, in which we convert the mentioned Caputo BVP of fractional variable order to an equivalent standard Caputo BVP at resonance of constant order. In fact, to use the Mawhin’s continuation technique, we have to transform the variable order BVP into a constant order BVP. We prove the existence of solutions based on the existing notions in the coincidence degree theory and Mawhin’s continuation theorem (MCTH). Finally, an example is provided according to the given variable order BVP to show the correctness of results.

Highlights

The initial idea of fractional calculus is taken from the powers of real or complex numbers in the order of differentiation and integration operators
Before the variable order systems, discussion of boundary value problems with fractional constant orders has attracted the attention of most researchers, and valuable findings have been established
The main novelty of this paper is that we use the Mawhin’s continuation technique for the first time for proving the existence of solutions of a Caputo boundary value problem at resonance equipped with variable order

Summary

Introduction

The initial idea of fractional calculus is taken from the powers of real or complex numbers in the order of differentiation and integration operators. Based on the aforementioned technique in relation to Mawhin’s method, in this paper, we shall investigate a nonlinear boundary value problem of variable order at resonance which takes the form as follows c D u+(t) φ(t) = g(t, φ(t)), t ∈ A,. The main novelty of this paper is that we use the Mawhin’s continuation technique for the first time for proving the existence of solutions of a Caputo boundary value problem at resonance equipped with variable order. Most papers apply this technique on the constant order systems, while we here try to derive the necessary conditions on a variable order system.

Auxiliary Concepts

Existence of Solutions

Example

Conclusions