Existence and uniqueness criteria for the higher-order Hilfer fractional boundary value problems at resonance

Yousef Gholami

doi:10.1186/s13662-020-02941-1

Abstract

This investigation is devoted to the study of a certain class of coupled systems of higher-order Hilfer fractional boundary value problems at resonance. Combining the coincidence degree theory with the Lipschitz-type continuity conditions on nonlinearities, we present some existence and uniqueness criteria. Finally, to practically implement the obtained theoretical criteria, we give an illustrative application.

Highlights

The general assumptions of the coupled system (1.1)–(1.2) are as follows: (A1) Daα+,β stands for the Hilfer fractional derivative of order n – 1 < α ≤ n, n ∈ N2, and type 0 ≤ β ≤ 1
We study the coupled systems of the higher-order Hilfer fractional differential equations
The general assumptions of the coupled system (1.1)–(1.2) are as follows: (A1) Daα+,β stands for the Hilfer fractional derivative of order n – 1 < α ≤ n, n ∈ N2, and type 0 ≤ β ≤ 1. (A2), : [a, b] × Rn → R are continuous functions governing the nonlinearities. (A3) Iaj + g(t) = 0, j = 1, 2, . . . , n, and Ia1+–β(n–α)g(t) = 0, g = u, v, where Iaμ+ stands for the Riemann–Liouville fractional integral of order μ

Summary

Introduction

The general assumptions of the coupled system (1.1)–(1.2) are as follows: (A1) Daα+,β stands for the Hilfer fractional derivative of order n – 1 < α ≤ n, n ∈ N2, and type 0 ≤ β ≤ 1. Combining the coincidence degree theory with some controls on the nonlinearities and , we prove the existence of at least one solution of the fractional coupled system (1.1)–(1.2), and imposing another conditions on these nonlinearities, we present a uniqueness criterion. We can conclude that the homogeneous Hilfer fractional system (2.15)– (2.16) has a nontrivial solution of the form u(t), v(t) = cn–1(t – a)n–(n–α)(1–β)–1, cn–1(t – a)n–(n–α)(1–β)–1 .

Results

Conclusion