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

TL;DR

This paper investigates the existence of a unique positive solution for a class of nonlinear Caputo-type fractional q-difference equations with nonlocal boundary conditions using fixed point theorems and properties of Green's functions.

Contribution

It introduces a new analysis for generalized nonlinear Caputo fractional q-difference equations with nonlocal boundary conditions, establishing conditions for unique positive solutions.

Findings

Existence of a unique positive solution under certain conditions.

Development of properties for the Green's function in this context.

Application of fixed point theorem in cones to fractional q-difference equations.

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 integral $\tilde{\alpha}[x]:=\int_{0}^{1}~x(t)d\Lambda(t).$ By applying the fixed point theorem in cones, we investigate an existence of a unique positive solution depends on $\lambda>0.$ We present some useful properties related to the Green's function for $m-$point boundary value problem.