Unique positive solution for nonlinear Caputo-type fractional q-difference equations with nonlocal and Stieltjes integral boundary conditions

Abstract

This paper contain a new discussion for the type of generalized nonlinear Caputo fractional q-difference equations with m-point boundary value problem and Riemann-Stieltjes integralα[x] := 1 0 x(t)dΛ(t). By applying the fixed point theorem in cones, we investigate an existence of a unique positive solution depends on λ > 0. We present some useful properties related to the Green's function for m−point boundary value problem.AMS classifications: 26A33; 34B15; 39A13; 33D05; 34B27.1 0 x(t)dΛ(t) is the Riemann-Stieltjes integral with respect to the bounded variation function Λ : [0, 1] → R.The fractional q-difference equations introduced and studied by Jackson [1], Adams [2], AlSalam [3] and Agarwal [4]. Due to its applicability in mathematical modeling in different branches like technical sciences, engineering, physics and biomathematics, it has drawn wide attention to many researchers.Recent studies on fractional calculus and boundary value problems of fractional q-difference equations indicate that it is an important topic of current research. There have appeared various articles covering these problems related to finding the existence positive or nontrivial solutions for different kinds of boundary value

Katugampola Fractional Calculus With Generalized k−Wright Function

“…They have made a qualitative contribution to fractional differential equations. For more details, see [1,[5][6][7][9][10][11][12][13][14] and references therein. Definition 1.1.…”

Existence of solutions to boundary value problems for a class of infinite point fractional q-difference equations

“…With the continuous research on fractional differential equations by researchers, fractional differential equations have been developed rapidly in recent decades, and many scientific research achievements have also been made [1,3,8]. In recent years, some scholars have introduced the ideas and methods of fractional differential equations into the study of fractional q-difference equations, and also made some research achievements [2,4,5,6,9,10]. However, the definition of fractional order q-difference equation is more complex, so it is difficult to judge the solution of boundary value problems, especially in the study of the properties of Green function.…”

Generalizations of fractional q-integrals involving Cauchy polynomials and some applications

“…Their treatment from the viewpoint of the fractional q-calculus can additionally open up new perspectives as it did, for example, in optimal control problems [7]. For further information about fractional q-integrals, we refer to [1,2,4,6,7,9,10,15,18,19,20,21,22,23] and the references therein.…”