Multi-point boundary value problems for a coupled system of nonlinear fractional differential equations

Advances in Difference Equations

Mengru Hao¹  Chengbo Zhai¹ 

[1] School of Mathematical Sciences, Shanxi University, Taiyuan, P.R. China

关键词: existence and uniqueness;    solutions;    Riemann-Liouville fractional derivative;    multi-point boundary value problems;    Banach contraction principle;    Krasnoselskii’s fixed point theorem;    26A33;    34B18;    34B27;   

DOI  :  10.1186/s13662-015-0487-6

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

In this paper, we investigate the existence and uniqueness of solutions to the coupled system of nonlinear fractional differential equations{−D0+ν1y1(t)=λ1a1(t)f(y1(t),y2(t)),−D0+ν2y2(t)=λ2a2(t)g(y1(t),y2(t)),$$\left \{ \begin{array}{@{}l} -D^{\nu_{1}}_{0^{+}}y_{1}(t)=\lambda_{1}a_{1}(t)f(y_{1}(t),y_{2}(t)), \\ -D^{\nu_{2}}_{0^{+}}y_{2}(t)=\lambda_{2}a_{2}(t)g(y_{1}(t),y_{2}(t)), \end{array} \right . $$whereD0+ν$D^{\nu}_{0^{+}}$is the standard Riemann-Liouville fractional derivative of order ν,t∈(0,1)$t\in(0,1)$,ν1,ν2∈(n−1,n]$\nu_{1}, \nu_{2} \in(n-1,n]$forn>3$n>3$andn∈N$n \in\mathbf{N} $, andλ1,λ2>0$\lambda_{1}, \lambda_{2} > 0$, with the multi-point boundary value conditions:y1(i)(0)=0=y2(i)(0)$y^{(i)}_{1}(0)=0=y^{(i)}_{2}(0)$,0≤i≤n−2$0 \leq i \leq n-2$;D0+βy1(1)=∑i=1m−2biD0+βy1(ξi)$D^{\beta}_{0^{+}}y_{1}(1)=\sum^{m-2}_{i=1}b_{i}D^{\beta }_{0^{+}}y_{1}(\xi_{i})$;D0+βy2(1)=∑i=1m−2biD0+βy2(ξi)$D^{\beta}_{0^{+}}y_{2}(1)=\sum^{m-2}_{i=1}b_{i}D^{\beta }_{0^{+}}y_{2}(\xi_{i})$, wheren−2<β

【 授权许可】

CC BY