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

Advances in Difference Equations

Dumitru Baleanu¹  Hojjat Afshari²  Mojtaba Sajjadmanesh² 

[1] Department of Mathematics, Cankaya University;Department of Mathematics, Faculty of Basic Science, University of Bonab;

关键词: Fractional differential equation;    Mixed monotone operator;    Normal cone;    Coupled system;   

DOI  :  10.1186/s13662-020-02568-2

来源: DOAJ

【 摘 要 】

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

【 授权许可】

Unknown