Boundary value problem for nonlinear fractional differential equations of variable order via Kuratowski MNC technique

Amar Benkerrouche,Dumitru Baleanu,Ali Hakem,Mustafa Inc,Mohammed Said Souid

doi:10.1186/s13662-021-03520-8

Abstract

In the present research study, for a given multiterm boundary value problem (BVP) involving the Riemann-Liouville fractional differential equation of variable order, the existence properties are analyzed. To achieve this aim, we firstly investigate some specifications of this kind of variable-order operators, and then we derive the required criteria to confirm the existence of solution and study the stability of the obtained solution in the sense of Ulam-Hyers-Rassias (UHR). All results in this study are established with the help of the Darbo’s fixed point theorem (DFPT) combined with Kuratowski measure of noncompactness (KMNC). We construct an example to illustrate the validity of our observed results.

Highlights

The idea of fractional calculus is replacing the natural numbers in the derivative order with rational ones
We introduce the solution to boundary value problem (BVP) (1)
6 Conclusion Our proposed multiterm BVP has been successfully investigated in this work via three theorems: The Darbo’s fixed point theorem (DFPT), the Kuratowski measure of noncompactness (KMNC), and the Ulam-Hyers-Rassias stability (UHR) to prove the existence and stability of solutions for our proposed BVP

Summary

Introduction

The idea of fractional calculus is replacing the natural numbers in the derivative order with rational ones. Where 1 < u(t) ≤ 2, f1 : J × X × X → X is a continuous function, and Du0+(t) and I0u+(t) are the Riemann–Liouville fractional derivative and integral of variable order u(t). The left Riemann–Liouville fractional integral (RLFI) of variable order u(t) for function h1(t) is [15,16,17]. The left Riemann–Liouville fractional derivative (RLFD) of variable-order v(t) for function h1(t) is [15,16,17]. 0 ≤ δ ≤ min u(t) , t∈J the variable-order fractional integral I0u+(t)h1(t) exists for any points on J. Lemma 2.4 ([28]) If U ⊂ C(J, X) is an equicontinuous and bounded set, : (i) the function ζ (U(t)) is continuous for t ∈ J, and ζ (U) = sup ζ U(t) ; t∈J (ii) ζ(. Du0+(t)z(t) – f t, z(t), I0u+(t)z(t) ≤ θ(t), t ∈ J, there exists a solution x ∈ C(J, X) of equation (1) with z(t) – x(t) ≤ cf θ(t), t ∈ J

Existence of solutions Let us introduce the following assumptions:

Conclusion