Positive solutions of infinite coupled system of fractional differential equations in the sequence space of weighted means

Abstract

We first discuss the existence of solutions of the infinite system of $ (n-1, n) $-type semipositone boundary value problems (BVPs) of nonlinear fractional differential equations $ \begin{equation*} \begin{cases} D^{\alpha}_{0_+}u_i(\rho)+\eta f_i(\rho,v(\rho)) = 0,& \rho\in(0,1), \\ D^{\alpha}_{0_+}v_i(\rho)+\eta g_i(\rho,u(\rho)) = 0,& \rho\in(0,1), \\u_i^{(j)}(0) = v_{i}^{(j)}(0) = 0,& 0\leq j\leq n-2, \\ u_{i}(1) = \zeta\int_0^1 u_i(\vartheta)d\vartheta, \ v_{i}(1) = \zeta\int_0^1 v_i(\vartheta)d\vartheta,& i\in\mathbb{N},\\ \end{cases} \end{equation*} $ in the sequence space of weighted means $ c_0(W_1, W_2, \Delta) $, where $ n\geq3 $, $ \alpha\in(n-1, n] $, $ \eta, \zeta $ are real numbers, $ 0 < \eta < \alpha, $ $ D^{\alpha}_{0_+} $ is the Riemann-Liouville's fractional derivative, and $ f_i, g_i, $ $ i = 1, 2, \ldots $, are semipositone and continuous. Our approach to the study of solvability is to use the technique of measure of noncompactness. Then, we find an interval of $ \eta $ such that for each $ \eta $ lying in this interval, the system of $ (n-1, n) $-type semipositone BVPs has a positive solution. Eventually, we demonstrate an example to show the effectiveness and usefulness of the obtained result.

Keywords

• darbo's theorem
• difference sequence space of weighted means
• infinite system of boundary value problems
• measure of noncompactness