Existence and uniqueness of positive solutions for a new class of coupled system via fractional derivatives

Abstract

AbstractIn this paper we study the existence of unique positive solutions for the following coupled system: $$\begin{aligned} \textstyle\begin{cases} D_{0^{+}}^{\alpha }x(\tau )+f_{1}(\tau ,x(\tau ),D_{0^{+}}^{\eta }x( \tau ))+g_{1}(\tau ,y(\tau ))=0, \\ D_{0^{+}}^{\beta }y(\tau )+f_{2}(\tau ,y(\tau ),D_{0^{+}}^{\gamma }y( \tau ))+g_{2}(\tau ,x(\tau ))=0, \\ \tau \in (0,1),\qquad n-1< \alpha ,\beta < n; \\ x^{(i)}(0)=y^{(i)}(0)=0,\quad i=0,1,2,\ldots ,n-2; \\ [D_{0^{+}}^{\xi }y(\tau ) ]_{\tau =1}=k_{1}(y(1)),\qquad [D_{0^{+}}^{\zeta }x(\tau ) ]_{\tau =1}=k_{2}(x(1)), \end{cases}\displaystyle \end{aligned}$$ {D0+αx(τ)+f1(τ,x(τ),D0+ηx(τ))+g1(τ,y(τ))=0,D0+βy(τ)+f2(τ,y(τ),D0+γy(τ))+g2(τ,x(τ))=0,τ∈(0,1),n−1<α,β<n;x(i)(0)=y(i)(0)=0,i=0,1,2,…,n−2;[D0+ξy(τ)]τ=1=k1(y(1)),[D0+ζx(τ)]τ=1=k2(x(1)), where the integer number $n>3$n>3 and $1\leq \gamma \leq \xi \leq n-2$1≤γ≤ξ≤n−2, $1\leq \eta \leq \zeta \leq n-2$1≤η≤ζ≤n−2, $f_{1},f_{2}:[0,1]\times \mathbb{R^{+}}\times \mathbb{R^{+}} \rightarrow \mathbb{R^{+}}$f1,f2:[0,1]×R+×R+→R+, $g_{1},g_{2}:[0,1]\times \mathbb{R^{+}}\rightarrow \mathbb{R^{+}}$g1,g2:[0,1]×R+→R+ and $k_{1},k_{2}:\mathbb{R^{+}}\rightarrow \mathbb{R^{+}}$k1,k2:R+→R+ are continuous functions, $D_{0^{+}}^{\alpha }$D0+α and $D_{0^{+}}^{\beta }$D0+β stand for the Riemann–Liouville derivatives. An illustrative example is given to show the effectiveness of theoretical results.