Abstract
By establishing a comparison theorem and applying the monotone iterative technique combined with the method of lower and upper solutions, we investigate the existence of extremal solutions of the initial value problem for fractional q-difference equation involving Caputo derivative. An example is presented to illustrate the main result.
Highlights
The quantum calculus is not of recent appearance.It appeared as a connection between mathematics and physics
The quantum difference operator has a lot of applications in different mathematical areas, such as number theory, combinatorics, special functions, basic hyper-geometric functions, the calculus of variations, control theory, mechanics, and the theory of relativity
The topic of quantum calculus has attracted the attention of several researchers and a variety of new results can be found in [ – ] and references cited therein
Summary
The quantum calculus (calculus without limits or q-calculus) is not of recent appearance.It appeared as a connection between mathematics and physics. Many authors developed the upper and lower solutions methods to solve fractional differential equations; for examples, see [ – ]. Motivated by the above-mentioned work, we investigate the existence of extremal solutions for the following initial value problem of a nonlinear fractional quantum difference equation: The aim of this paper is to extend the method of upper and lower solutions coupled with the monotone iterative technique to fractional q-difference equations.
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