New Sequential Fractional Differential Equations with Mixed-Type Boundary Conditions

Haiyan Zhang,Yaohong Li,Jingbao Yang

doi:10.1155/2020/6821637

Abstract

In this paper, we introduce new sequential fractional differential equations with mixed-type boundary conditions CDq+kCDq−1ut=ft,ut,CDq−1ut,t∈0,1,α1u0+β1u1+γ1Iruη=ε1,η∈0,1,α2u′0+β2u′1+γ2Iru′η=ε2, where q∈1,2 is a real number, k,r>0,αi,βi,γi,εi∈ℝ,i=1,2,CDq is the Caputo fractional derivative, and the boundary conditions include antiperiodic and Riemann-Liouville fractional integral boundary value cases. Our approach to treat the above problem is based upon standard tools of fixed point theory and some new inequalities of norm form. Some existence results are obtained and well illustrated through the aid of examples.

Highlights

In this paper, we focus on sequential fractional differential equations with mixed-type boundary conditions. 8 >>>< C Dq + kCDq−1 uðtÞ = f À t, uðtÞ, C Dq−1uðt Á Þ,>>>: α1uð0Þ + β1uð1Þ + α2u′ð0Þ + β2u′ð1Þ η ε2
We provide some necessary definitions and lemmas of the Caputo fractional calculus; for more information, see the books [1–3]
Let T : E → E be completely continuous

Summary

Introduction

We focus on sequential fractional differential equations with mixed-type boundary conditions. Motivated by the HIV infection model and its application background in [12], the existence and uniqueness of solutions for the following sequential fractional differential system are obtained by means of Leray-Schauder’s alternative and Banach’s contraction principle where q ∈ ð1, 2 is a real number and k, r > 0, αi, βi, γi, εi ∈ R, i = 1, 2, CDq is the Caputo fractional derivative of order q. In [20], the authors discussed the following fractional differential equation with integral boundary conditions given by Definition 1. 0, xð1Þ ð μ xðsÞds, Dα xð0Þ+C ð4Þ where CDα and CDβ are the Caputo fractional derivatives; 0 < α < 1, 1 < β ≤ 2, k > 0, and μ > 0 are real numbers; and f is a given continuous function. We shall discuss the problem [1] by using the standard tools of fixed point theory and some new inequalities of norm from

Preliminary and Lemmas

Main Results

Conclusions