Existence and Uniqueness of Positive Solutions for the Fractional Differential Equation Involving the <i>ρ</i>(<i>τ</i>)-Laplacian Operator and Nonlocal Integral Condition

Abstract

This paper aims to investigate the Caputo fractional differential equation involving the ρ(τ) Laplacian operator and nonlocal multi-point of Riemann–Liouville’s fractional integral. We also prove the uniqueness of the positive solutions for Boyd and Wong’s nonlinear contraction via the Guo–Krasnoselskii fixed-point theorem in Banach spaces. Finally, we illustrate the theoretical results and show that by solving the nonlocal problems, it is possible to obtain accurate approximations of the solutions. An example is also provided to illustrate the applications of our theorem.

Keywords

• Caputo fractional derivative
• Laplacian operator
• fixed point
• Guo–Krasnoselskii fixed-point theorem
• integral boundary value problems