Abstract
The aim of this paper is to investigate the boundary value problem of a fractional q-difference equation with ϕ-Laplacian, where ϕ-Laplacian is a generalized p-Laplacian operator. We obtain the existence and nonexistence of positive solutions in terms of different eigenvalue intervals for this problem by means of the Green function and Guo–Krasnoselskii fixed point theorem on cones. Finally, we give some examples to illustrate the use of our results.
Highlights
The theory of q-calculus has been developed for more than 100 years; see [1]
We aim to obtain the existence of at least one or two positive solutions in terms of different eigenvalue intervals using the Green function and Guo–Krasnoselskii fixed point theorem on cones
5 Conclusions This research establishes the existence of at least one or two positive solutions in terms of different eigenvalue intervals for the boundary value problems (BVPs) of φ-Laplacian fractional q-difference equation, by applying the Green function and Guo–Krasnoselskii fixed point theorem on cones
Summary
The theory of q-calculus has been developed for more than 100 years; see [1]. Fractional q-calculus has a wide range of applications in many fields, such as cosmic strings and black holes, quantum theory, aerospace dynamics, biology, economics, control theory, medicine, electricity, signal processing, image processing, biophysics, blood flow phenomenon, and so on; see [4,5,6,7,8,9,10] and the references therein. The fractional q-difference equations are very important, and their basic theory has been continuously developed. As a new research direction, the solvability of boundary value problems (BVPs) of fractional q-difference equations have been widely concerned by scholars at home and abroad, and some conclusions have been obtained; see [11,12,13,14]. There are a few studies of eigenvalue problems for fractional q-difference equations with φ-Laplacian operator, and lots of work should be done
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