Existence and uniqueness of solutions for coupled system of fractional differential equations involving proportional delay by means of topological degree theory

Advances in Difference Equations

article

Ali, Anwar¹  Sarwar, Muhammad¹  Zada, Mian Bahadur¹  Abdeljawad, Thabet² 

[1] Department of Mathematics, University of Malakand;Department of Mathematics and General Sciences, Prince Sultan University;Department of Medical Research, China Medical University;Department of Computer Science and Information Engineering, Asia University

关键词: Fractional differential equations;    Boundary value problems;    Existence results;    Topological degree theory;   

DOI  :  10.1186/s13662-020-02918-0

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

In this manuscript, we obtain sufficient conditions required for the existence of solution to the following coupled system of nonlinear fractional order differential equations:$$ \begin{gathered} D^{\gamma}\omega(\ell)= \mathcal{F} \bigl( \ell,\omega(\lambda\ell), \upsilon(\lambda\ell) \bigr), \\ D^{\delta}\upsilon(\ell)=\mathcal{\overline{F}} \bigl(\ell,\omega ( \lambda\ell), \upsilon(\lambda\ell) \bigr), \end{gathered} $$ with fractional integral boundary conditions$$ \begin{gathered} \mathfrak{a}_{1}\omega(0)- \mathfrak{b}_{1}\omega(\eta)-\mathfrak {c}_{1}\omega(1)= \frac{1}{\varGamma(\gamma)} \int_{0}^{1}(1-\rho )^{\gamma-1} \phi \bigl( \rho, \omega(\rho) \bigr)\, d\rho\quad\text{and} \\ \mathfrak{a}_{2}\upsilon(0)-\mathfrak{b}_{2} \upsilon (\xi)-\mathfrak{c}_{2}\upsilon(1)=\frac{1}{\varGamma(\delta)} \int _{0}^{1}(1-\rho)^{\delta-1} \psi \bigl( \rho, \upsilon(\rho) \bigr) \,d\rho, \end{gathered} $$ where$\ell\in\mathfrak{Z}=[0,1]$ ,$\gamma, \delta\in(0,1]$ ,$0<\lambda<1$ , D denotes the Caputo fractional derivative (in short CFD),$\mathcal{F}, \mathcal{\overline{F}}: \mathfrak{Z}\times \mathfrak{R}\times\mathfrak{R} \rightarrow\mathfrak{R}$ and$\phi , \psi:\mathfrak{Z}\times\mathfrak{R}\rightarrow\mathfrak{R}$ are continuous functions. The parameters η, ξ are such that$0<\eta, \xi<1$ , and$\mathfrak{a}_{i}, \mathfrak{b}_{i}, \mathfrak {c}_{i}$ ( $i=1, 2$ ) are real numbers with$\mathfrak{a}_{i}\neq\mathfrak {b}_{i}+\mathfrak{c}_{i}$ ( $i=1, 2$ ). Using topological degree theory, sufficient results are constructed for the existence of at least one and unique solution to the concerned problem. For the validity of our result, an appropriate example is presented in the end.

【 授权许可】

CC BY