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

[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.

[学科分类] 航空航天科学

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