The Extremal Solution To Conformable Fractional Differential Equations Involving Integral Boundary Condition

Shuman Meng,Yujun Cui

doi:10.3390/math7020186

Abstract

In this article, by using the monotone iterative technique coupled with the method of upper and lower solution, we obtain the existence of extremal iteration solutions to conformable fractional differential equations involving Riemann-Stieltjes integral boundary conditions. At the same time, the comparison principle of solving such problems is investigated. Finally, an example is given to illustrate our main results. It should be noted that the conformal fractional derivative is essentially a modified version of the first-order derivative. Our results show that such known results can be translated and stated in the setting of the so-called conformal fractional derivative.

Highlights

Fractional calculus is an excellent tool for the description of the process of mathematical analysis in various areas of finance, physical systems, control systems and mechanics, and so forth [1,2,3,4,5]
Motivated by the above works, we consider the existence of solutions for the following nonlinear conformable fractional differential equation involving integral boundary condition, using the method of upper and lower solutions and its associated monotone iterative technique
We prove the existence of extremal solutions for conformable fractional differential equations involving integral boundary condition

Summary

Introduction

Fractional calculus is an excellent tool for the description of the process of mathematical analysis in various areas of finance, physical systems, control systems and mechanics, and so forth [1,2,3,4,5]. Khalil et al [24] gave a new simple fractional derivative called “the conformable fractional derivative” depending on the familiar limit definition of the derivative of a function and that break with other definitions. It is suitable for many extensions to the classical theorem of calculus, such as the derivative of the product and compound of two functions, the Rolle’s and the mean value theorem, conformable integration by parts, fractional power series expansion and many more. Mathematics 2019, 7, 186 infinitely differentiable at some points; where there is no Taylor power series expansion, in conformal calculus theory, they do exist. Motivated by the above works, we consider the existence of solutions for the following nonlinear conformable fractional differential equation involving integral boundary condition, using the method of upper and lower solutions and its associated monotone iterative technique. For applications of the method of upper and lower solutions and monotone iterative technique to differential equations and differential systems such as ordinary differential equations [35,36,37], ordinary differential systems [38], fractional differential equations [39,40,41,42], fractional differential systems [43]

Preliminaries

Main Results

Conclusions