Positive solutions for generalized two-term fractional differential equations with integral boundary conditions

Hanan A Wahash,Satish K Panchal

doi:10.48185/jmam.v1i1.35

Abstract

In this paper, we consider a class of boundary value problems for nonlinear two-term fractional differential equations with integral boundary conditions involving two $\psi $-Caputo fractional derivative. With the help of the properties Green function, the fixed point theorems of Schauder and Banach, and the method of upper and lower solutions, we derive the existence and uniqueness of positive solution of a proposed problem. Finally, an example is provided to illustrate the acquired results.

Highlights

Fractional calculus can be thought of as a generalization of calculus with integer order
Despite the fact that inside the start, fractional calculus had an advancement as a purely mathematical idea, in current quite a while its utilization had unfurl into numerous fields such as physics, mechanics, chemistry, biology, engineering, bioengineering and electrochemistry, e.g., [1, 2, 3, 4, 5, 6, 7, 8, 9] and the references therein
We concentrate on the positivity of the solutions for the following nonlinear fractional differential equations (FDEs) with integral boundary conditions

Summary

Introduction

Fractional calculus can be thought of as a generalization of calculus with integer order. We concentrate on the positivity of the solutions for the following nonlinear fractional differential equations (FDEs) with integral boundary conditions. Xu and Han in [39] studied the positivity results of the following nonlinear two-term FDEs. ψ(s))α−β−1g(s, u(s))ds, in the Riemann–Liouville derivatives sense. To the best of our knowledge, no article has studied the existence of positive solutions for nonlinear FDEs with integral boundary conditions (1.1) This problem has two nonlinear terms and includes two generalized fractional derivatives. To show the existence and uniqueness of the positive solution, we transform (1.1) into a fractional integral equation with the aid of the Green function, and by the method of upper and lower solutions and use Schauder and Banach fixed point theorems we obtain our results.

Preliminaries

Main results

An example

Conclusions