Unique solution for a new system of fractional differential equations

Abstract

Abstract In this article, we discuss a new system of fractional differential equations $$ \textstyle\begin{cases} D_{0^{+}}^{s_{1}} u(t) + f(t,u(t),v(t))=z_{1}(t), \quad 0< t< 1, \\ D_{0^{+}}^{s_{2}} v(t) + g(t,u(t),v(t))=z_{2}(t), \quad 0< t< 1, \\ u(0)=u(1)=u^{\prime }(0)=u^{\prime }(1)=0, \qquad D_{0^{+}}^{\beta _{1}} u(0)=0, \qquad D_{0^{+}}^{\beta _{1}} u(1)= b_{1} D_{0^{+}}^{\beta _{1}} u(\eta _{1}), \\ v(0)=v(1)=v^{\prime }(0)=v^{\prime }(1)=0, \qquad D_{0^{+}}^{\beta _{2}} v(0)=0, \qquad D_{0^{+}}^{\beta _{2}} v(1)= b_{2} D_{0^{+}}^{\beta _{2}} v(\eta _{2}), \end{cases} $$ { D 0 + s 1 u ( t ) + f ( t , u ( t ) , v ( t ) ) = z 1 ( t ) , 0 < t < 1 , D 0 + s 2 v ( t ) + g ( t , u ( t ) , v ( t ) ) = z 2 ( t ) , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ′ ( 0 ) = u ′ ( 1 ) = 0 , D 0 + β 1 u ( 0 ) = 0 , D 0 + β 1 u ( 1 ) = b 1 D 0 + β 1 u ( η 1 ) , v ( 0 ) = v ( 1 ) = v ′ ( 0 ) = v ′ ( 1 ) = 0 , D 0 + β 2 v ( 0 ) = 0 , D 0 + β 2 v ( 1 ) = b 2 D 0 + β 2 v ( η 2 ) , where $s_{i}=\alpha _{i}+\beta _{i}$ s i = α i + β i , $\alpha _{i} \in (1,2]$ α i ∈ ( 1 , 2 ] , $\beta _{i} \in (3,4]$ β i ∈ ( 3 , 4 ] , $z_{i} :[0,1]\rightarrow [0,+\infty )$ z i : [ 0 , 1 ] → [ 0 , + ∞ ) is continuous, $D_{0^{+}}^{\alpha _{i}}$ D 0 + α i and $D_{0^{+}}^{\beta _{i}}$ D 0 + β i are the standard Riemann–Liouville derivatives, $\eta _{i} \in (0,1)$ η i ∈ ( 0 , 1 ) , $b_{i} \in (0, {\eta _{i}}^{1-\alpha _{i}})$ b i ∈ ( 0 , η i 1 − α i ) , $i=1,2$ i = 1 , 2 , and $f,g\in C([0,1]\times \mathbf{R}^{2} , \mathbf{R})$ f , g ∈ C ( [ 0 , 1 ] × R 2 , R ) . We establish the existence and uniqueness of solutions for the problem by a recent fixed point theorem of increasing Ψ-$(h,e)$ ( h , e ) -concave operators defined on ordered sets. Furthermore, the results obtained are well proven by means of a specific example.