Nonlocal q-fractional boundary value problem with Stieltjes integral conditions

Chengbo Zhai,Jing Ren

doi:10.15388/na.2019.4.6

Abstract

In this paper, we are dedicated to investigating a new class of one-dimensional lower-order fractional q-differential equations involving integral boundary conditions supplemented with Stieltjes integral. This condition is more general as it contains an arbitrary order derivative. It should be pointed out that the problem discussed in the current setting provides further insight into the research on nonlocal and integral boundary value problems. We first give the Green's functions of the boundary value problem and then develop some properties of the Green's functions that are conductive to our main results. Our main aim is to present two results: one considering the uniqueness of nontrivial solutions is given by virtue of contraction mapping principle associated with properties of u0-positive linear operator in which Lipschitz constant is associated with the first eigenvalue corresponding to related linear operator, while the other one aims to obtain the existence of multiple positive solutions under some appropriate conditions via standard fixed point theorems due to Krasnoselskii and Leggett–Williams. Finally, we give an example to illustrate the main results.

Highlights

We discuss the existence of unique solution and multiple positive solutions for the following fractional q-differential equation: Dqαu(t) + f t, u(t) = 0, t ∈ [0, 1], (1)
Where Dqα is the standard Riemann–Liouville fractional q-derivative of order α, 2 < α 3, α − 1 − β > 0, 0 < q < 1, φ ∈ L1[0, 1] is nonnegative, α[u] is a linear functional given by α[u] = u(t) dA(t) involving the Stieltjes integral with respect to the function A : [0, 1] → R, A(t) is rightcontinuous on [0, 1), left-continuous at t = 1
We first give two lemmas that will be used to show the existence of multiple positive solutions

Summary

Introduction

In connection with broad research on the mathematical modeling of systems, the description of hereditary properties of various materials and the optimal control theory, it has become necessary to investigate boundary value problems of fractional differential equations as the nonlocal characteristics of the corresponding fractional-order operators [5,6,8,9,10,18,20,24,25,26,27,29,31,32]. Inspired by the previous works, different from [19, 20], we consider problem (1)–(2) in which the boundary conditions involve the Stieltjes integral, β is an arbitrary order derivative, (2) is more general boundary condition Up to now, this case of nonlocal boundary conditions for fractional q-differential equations is relatively rare to be done. 0, where α[tα−1] > 0, and α[1] > 0

Auxiliary results

Unique solution

Multiple positive solutions

An example