Abstract
In this paper, we study a sequential Caputo fractional q-integrodifference equation with fractional q-integral and Riemann–Liouville fractional q-derivative boundary value conditions. Our problem contains 2(M+N+1) different orders and six different numbers of q in derivatives and integrals. The problem contains separate nonlinear functions. To examine existence and uniqueness results of the problem, Banach’s contraction principle and the Leray–Schauder nonlinear alternative are employed. An illustrative example is also provided.
Highlights
In the 20th century, q-difference calculus and fractional q-difference calculus play an important role in the areas of mathematics and applications [1,2,3] such as the applications to orthogonal polynomials and mathematical control theories
There is a lack of research in boundary value problem of nonlinear q-difference equations
In 2014, Ahmad et al [15] studied the existence of solutions for the Caputo fractional q-difference integral equation with nonlocal boundary conditions
Summary
In the 20th century, q-difference calculus and fractional q-difference calculus play an important role in the areas of mathematics and applications [1,2,3] such as the applications to orthogonal polynomials and mathematical control theories. In 2014, Ahmad et al [15] studied the existence of solutions for the Caputo fractional q-difference integral equation with nonlocal boundary conditions,. Patanarapeelert et al [33] considered a sequential q-integrodifference boundary value problem involving two different orders and six different numbers of q in derivatives and integrals of the form. Definition 2.1 ([6]) For q ∈ (0, 1), the q-derivative of a real function f is defined by f (t) – f (qt). D0qf (x) = f (x), the fractional q-derivative of the Caputo type of order α is defined by CDαq f (x) := Iqm–αDmq f (x) =. Lemma 2.1 ([6]) Let α, β ≥ 0 and f be a function defined on [0, T].
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