Degree theory and existence of positive solutions to coupled systems of multi-point boundary value problems

By Kamal Shah, Amjad Ali, Rahmat Ali Khan. Boundary Value Problems, 2016

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In this article, we investigate existence and uniqueness of positive solutions to coupled systems of multi-point boundary value problems for fractional order differential equations of the form {Dαx(t)=ϕ(t,x(t),y(t)),t∈I=[0,1],Dβy(t)=ψ(t,x(t),y(t)),t∈I=[0,1],x(0)=g(x),x(1)=δx(η),0<η<1,y(0)=h(y),y(1)=γy(ξ),0<ξ<1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left \{ \textstyle\begin{array}{l} D^{\alpha} x(t)=\phi(t,x(t),y(t)), \quad t\in I=[0,1], \\ D^{\beta} y(t)=\psi(t,x(t),y(t)),\quad t\in I=[0,1], \\ x(0)=g(x),\qquad x(1)=\delta x(\eta),\quad 0< \eta< 1, \\ y(0)=h(y),\qquad y(1)=\gamma y(\xi),\quad 0< \xi< 1, \end{array}\displaystyle \right . $$\end{document} where α,β∈(1,2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha, \beta\in(1,2]$\end{document}, D denotes the Caputo fractional derivative, 0<δ,γ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<\delta, \gamma<1$\end{document} are parameters such that δηα<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\delta\eta^{\alpha}<1$\end{document}, γξβ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma\xi^{\beta}<1$\end{document}, h,g∈C(I,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$h, g\in C(I,\mathbb{R})$\end{document} are boundary functions and ϕ,ψ:I×R×R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\phi,\psi:I\times\mathbb{R} \times \mathbb{R} \rightarrow\mathbb{R}$\end{document} are continuous. We use the technique of topological degree theory to obtain sufficient conditions for existence and uniqueness of positive solutions to the system. Finally, we provide an example to illustrate our main results.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/5092390d9f56c851d8dca09643eb616c7f175606