Existence of solutions for some two-point fractional boundary value problems under barrier strip conditions

Zhiyu Li,Zhanbing Bai

doi:10.1186/s13661-019-01307-1

Zhiyu Li, Zhanbing Bai

Open Access

https://doi.org/10.1186/s13661-019-01307-1

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Abstract

In this paper, we are dedicated to researching the boundary value problems (BVPs) for equation D^{alpha }x(t)=f(t,x(t),D^{alpha -1}x(t)), with the boundary value conditions to be either: x(0)=A, D^{alpha -1}x(1)=B or D^{alpha -1}x(0)=A, x(1)=B. Let the nonlinear term f satisfy some sign conditions, then by making use of the Leray–Schauder nonlinear alternative, some existence results are obtained. In the end, an example is given to verify the main results.

Highlights

1 Introduction In the last several years, because the fractional calculus theory has been extensively used in non-Newtonian fluid mechanics, diffusion and transportation theory, engineering, biology, image processing, and other fields [9, 12, 15,16,17,18,19, 21, 24, 25, 35, 36, 38, 41,42,43,44,45,46,47], the fractional differential equations (FDEs) have been researched with different methods by many scholars
The barrier strips technique was used by many researchers to study integerorder boundary value problems (BVPs) and IVPs
By making use of the barrier strips technique, the existence results for integer p-Laplacian BVP and first order IVP have been obtained by Kelevedjiev and Tersian, see [29] and [28]

Summary

Introduction

In the last several years, because the fractional calculus theory has been extensively used in non-Newtonian fluid mechanics, diffusion and transportation theory, engineering, biology, image processing, and other fields [9, 12, 15,16,17,18,19, 21, 24, 25, 35, 36, 38, 41,42,43,44,45,46,47], the fractional differential equations (FDEs) have been researched with different methods by many scholars. We are dedicated to researching the boundary value problems (BVPs) for equation Dαx(t) = f (t, x(t), Dα–1x(t)), with the boundary value conditions to be either: x(0) = A, Dα–1x(1) = B or Dα–1x(0) = A, x(1) = B. Let the nonlinear term f satisfy some sign conditions, by making use of the Leray–Schauder nonlinear alternative, some existence results are obtained.

Results

Conclusion