Abstract
In this paper, we study the existence of solutions for a new class of fractional q-integro-difference equations involving Riemann-Liouville q-derivatives and a q-integral of different orders, supplemented with boundary conditions containing q-integrals of different orders. The first existence result is obtained by means of Krasnoselskii’s fixed point theorem, while the second one relies on a Leray-Schauder nonlinear alternative. The uniqueness result is derived via the Banach contraction mapping principle. Finally, illustrative examples are presented to show the validity of the obtained results. The paper concludes with some interesting observations.
Highlights
Introduction and PreliminariesFractional calculus, dealing with differential and integral operators of arbitrary order, serves as a powerful modelling tool for many real-world phenomena
We prove the following lemma which characterizes the structure of solutions for boundary value problems (1) and (2)
We have derived some new existence and uniqueness results for a nonlinear fractional q-integro-difference equation equipped with q-integral boundary conditions
Summary
Fractional calculus, dealing with differential and integral operators of arbitrary order, serves as a powerful modelling tool for many real-world phenomena. In 2014, Ahmad et al [13] derived some existence results for a nonlinear fractional q-difference equation with four-point nonlocal integral boundary conditions given by c D β Niyom et al [14] studied the following boundary value problem containing Riemann-Liouville fractional derivatives of different orders:. In [26], the authors studied a nonlocal four-point boundary value problem of nonlinear fractional q-integro-difference equations given by c. Where 0 < q < 1, 1 < α, β < 2, 0 < δ < 1, 0 < λ ≤ 1, 0 ≤ μ ≤ 1, α − β > 1 and Dqα denotes the Riemann-Liouville fractional q-derivative of order α and f , g : [0, 1] × R → R are continuous functions.
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