Existence and Stability for a Nonlinear Coupled p -Laplacian System of Fractional Differential Equations

Abstract

In this paper, we study the nonlinear coupled system of equations with fractional integral boundary conditions involving the Caputo fractional derivative of orders ${\theta }_1\text{\hspace{0.17em}and\hspace{0.17em}}{\theta }_2$ and Riemann–Liouville derivative of orders ${\varrho }_1\text{\hspace{0.17em}and\hspace{0.17em}}{\varrho }_2$ with the $p$ -Laplacian operator, where $n-1<{\theta }_1,{\theta }_2,{\varrho }_1,{\varrho }_2\le n$ , and $n\ge 3$ . With the help of two Green’s functions $\left(G^{{\mathrm{\varrho }}_1}\left(w,\Im \right),G^{{\mathrm{\varrho }}_2}\left(w,\Im \right)\right)$ , the considered coupled system is changed to an integral system. Since topological degree theory is more applicable in nonlinear dynamical problems, the existence and uniqueness of the suggested coupled system are treated using this technique, and we find appropriate conditions for positive solutions to the proposed problem. Moreover, necessary conditions are highlighted for the Hyer–Ulam stability of the solution for the specified fractional differential problems. To confirm the theoretical analysis, we provide an example at the end.