Existence of solutions for integral boundary value problems of singular Hadamard-type fractional differential equations on infinite interval

Advances in Difference Equations

article

Liu, Weiwei¹  Liu, Lishan¹  Wu, Yonghong² 

[1] School of Mathematics Sciences, Qufu Normal University;Department of Mathematics and Statistics, Curtin University

关键词: Hadamard-type fractional differential equation;    Carathéodory condition;    Infinite interval;    Fixed point theory;   

DOI  :  10.1186/s13662-020-02726-6

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

We consider the existence of solutions for the following Hadamard-type fractional differential equations:$$ \textstyle\begin{cases} {}^{H}D^{\alpha }u(t)+q(t)f(t,u(t), {}^{H}D^{\beta _{1}}u(t),{}^{H}D^{ \beta _{2}}u(t))=0,\quad 1< t< +\infty , \\ u(1)=0, \\ {}^{H}D^{\alpha -2}u(1)=\int ^{+\infty }_{1}g_{1}(s)u(s)\frac{ds}{s}, \\ {}^{H}D^{\alpha -1}u(+\infty )=\int ^{+\infty }_{1}g_{2}(s)u(s) \frac{ds}{s}, \end{cases} $$ where$2<\alpha \leq 3$ ,$0<\beta _{1}\leq \alpha -2<\beta _{2}\leq \alpha -1$ ,$f:J \times \mathbb{R}^{3}\rightarrow \mathbb{R}$ satisfies the q-Carathéodory condition,$q,g_{1},g_{2}:J\rightarrow \mathbb{R}^{+}$ are nonnegative, where$J=[1,+\infty )$ . Nonlinear term f is dependent on the fractional derivative of lower order$\beta _{1}$ ,$\beta _{2}$ , which creates additional complexity to verify the existence of solutions. The singularity occurring in our problem is associated with${}^{H}D^{\beta _{2}}u\in C(1,+\infty )$ at the left endpoint$t=1$ (if$\beta _{2}<\alpha -1$ ).

【 授权许可】

CC BY