Existence of Solutions to a Class of Nonlinear Arbitrary Order Differential Equations Subject to Integral Boundary Conditions

Ananta Thakur,Javid Ali,Rosana Rodríguez-López

doi:10.3390/fractalfract5040220

Abstract

We investigate the existence of positive solutions for a class of fractional differential equations of arbitrary order δ>2, subject to boundary conditions that include an integral operator of the fractional type. The consideration of this type of boundary conditions allows us to consider heterogeneity on the dependence specified by the restriction added to the equation as a relevant issue for applications. An existence result is obtained for the sublinear and superlinear case by using the Guo–Krasnosel’skii fixed point theorem through the definition of adequate conical shells that allow us to localize the solution. As additional tools in our procedure, we obtain the explicit expression of Green’s function associated to an auxiliary linear fractional boundary value problem, and we study some of its properties, such as the sign and some useful upper and lower estimates. Finally, an example is given to illustrate the results.

Highlights

The need for fractional order differential equations stems in part from the fact that many phenomena cannot be modeled by differential equations with integer derivatives
The work [4] gives some fundamental ideas on initial value problems for fractional differential equations from the point of view of Riemann–Liouville operators, discussing local and global existence, or extremal solutions, and the monograph [5] includes different theoretical results as well as developments related to applications in the field of fractional calculus
This section of the paper is focused on the study of the existence of at least one positive solution to the nonlinear boundary value problem specified in expression (1)

Summary

Introduction

The need for fractional order differential equations stems in part from the fact that many phenomena cannot be modeled by differential equations with integer derivatives. The existence results for solutions to fractional differential equations have received considerable attention in recent years. Some relevant monographs on fractional calculus and fractional differential equations are, for instance [1,2,3]. The work [4] gives some fundamental ideas on initial value problems for fractional differential equations from the point of view of Riemann–Liouville operators, discussing local and global existence, or extremal solutions, and the monograph [5] includes different theoretical results as well as developments related to applications in the field of fractional calculus

Methods

Results

Conclusion