Existence of solutions for equations and inclusions of multiterm fractional q-integro-differential with nonseparated and initial boundary conditions

Mohammad Esmael Samei,Vahid Hedayati,Ghorban Khalilzadeh Ranjbar

doi:10.1186/s13660-019-2224-2

Abstract

The goal of this paper is to investigate existence of solutions for the multiterm nonlinear fractional q-integro-differential {}^{c}D_{q}^{alpha } u(t) in two modes equations and inclusions of order alphain(n -1, n], with non-separated boundary and initial boundary conditions where the natural number n is more than or equal to five. We consider a Carathéodory multivalued map and use Leray–Schauder and Covitz–Nadler famous fixed point theorems for finding solutions of the inclusion problems. Besides, we present results whenever the multifunctions are convex and nonconvex. Lastly, we give some examples illustrating the primary effects.

Highlights

1 Introduction Fractional calculus and q-calculus are the significant branches in mathematical analysis
The field of fractional calculus has countless applications, and the subject of fractional differential equations ranges from the theoretical views of existence and uniqueness of solutions to the analytical and mathematical methods for finding solutions
There has been an intensive development in fractional differential equations and inclusion

Summary

Introduction

Fractional calculus and q-calculus are the significant branches in mathematical analysis. In 2013, Baleanu, Rezapour and Mohammadi et al, by using fixed-point methods, studied the existence and uniqueness of a solution for the nonlinear fractional differential equation boundary-value problem Dαu(t) = f (t, u(t)) with a Riemann–Liouville fractional derivative via the different boundary-value conditions: u(0) = u(δ), as well as the threepoint boundary condition u(0) = β1u(η) and u(δ) = β2u(η), where δ > 0, t ∈ I = [0, δ], α ∈ (0, 1) η ∈ (0, δ) and 0 < β1 < β2 < 1 [12]. In 2017, Baleanu, Mousalou and Rezapour presented a new method to investigate some fractional integro-differential equations involving the Caputo–Fabrizio derivative.

Iqα f

Define the multifunction