Abstract
In this paper, we consider a nonlinear sequential q-difference equation based on the Caputo fractional quantum derivatives with nonlocal boundary value conditions containing Riemann–Liouville fractional quantum integrals in four points. In this direction, we derive some criteria and conditions of the existence and uniqueness of solutions to a given Caputo fractional q-difference boundary value problem. Some pure techniques based on condensing operators and Sadovskii’s measure and the eigenvalue of an operator are employed to prove the main results. Also, the Ulam–Hyers stability and generalized Ulam–Hyers stability are investigated. We examine our results by providing two illustrative examples.
Highlights
In several areas of sciences, such as biology, chemistry, economics, physics, and engineering, fractional calculus and its relevant differential equations and BVPs have been used extensively [1,2,3]
References [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] are available for some improvements on the fractional differential equations theory
By virtue of developments in fractional quantum calculus (q-FC), a number of scientists and researchers [19, 20] were attracted to a study of fractional q-difference equations, beginning in the nineteenth century, and wide interest lately [21,22,23]
Summary
In several areas of sciences, such as biology, chemistry, economics, physics, and engineering, fractional calculus and its relevant differential equations and BVPs have been used extensively [1,2,3]. Fractional derivatives are a generalization of ordinary derivatives, and they explain dynamical behavior of various physical processes and effectively (real life phenomena) in contrast to integer order derivatives. References [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] are available for some improvements on the fractional differential equations theory. In 2007, Atici et al [24] studied some notions in relation to fractional q-calculus on time scales. In 2012, Annaby and Mansour presented their investigations by pub-
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