On nonlocal fractional q-integral boundary value problems of fractional q-difference and fractional q-integrodifference equations involving different numbers of order and q

Abstract

In this paper, we study some new class of nonlocal three-point fractional q-integral boundary value problems of a nonlinear fractional q-difference equation and a nonlinear fractional q-integrodifference equation. Our problems contain different numbers of order and q in derivatives and integrals. The existence and uniqueness results are based on Banach's contraction mapping principle and Krasnoselskii's fixed point theorem. In addition, some examples are presented to illustrate the importance of these results.

Existence Theory for a Fractional q-Integro-Difference Equation with q-Integral Boundary Conditions of Different Orders

“…represents a Riemann-Liouville fractional integral of order ξ ∈ (0, 1), f , g : [0, 1] × R → R are continuous functions, λ = 0 and p, k, α i , β i , σ i ∈ R, η i ∈ (0, 1), i = 1, 2. For some recent works on boundary value problems of fractional q-integro-difference equations, for instance, see [27][28][29][30][31].…”

Existence of positive solutions for two-point boundary value problems of nonlinear fractional q-difference equation

“…At the same time, the topic of the fractional quantum difference equation has also attracted the attention of many researchers in recent years (see [4][5][6] and the references therein). In recent years, some boundary value problems with fractional q-differences have aroused heated discussion among many authors [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. They obtained many results as regards the existence and multiplicity of nontrivial solutions, positive solutions, negative solutions and extremal solutions by applying some well-known tools of fixed point theory such as the Banach contraction principle, the Guo-Krasnosel'skii fixed point theorem on cones, monotone iterative methods and Leray-Schauder degree theory.…”

On four-point fractional q-integrodifference boundary value problems involving separate nonlinearity and arbitrary fractional order

“…In 2016, Sitthiwirattham [31] examined the existence results of solutions to a fractional q-difference equation and a fractional q-integrodifference equation …”