On solvability of fractional coupled hybrid systems
From the introduction and summary: \[ \begin{aligned} \lambda_1 D^\alpha_{0^+}(u(t)-f(t,u(t)))-\lambda_2 u(t)= g_1(t,u(t), v(t)),\quad & 1<\alpha\leq 2,\\ \mu_1 D^\beta_{0^+}(v(t)-f(t,v(t)))-\mu_2 v(t)= g_2(t, u(t),v(t)),\quad & 1<\beta\leq 2,\end{aligned} \] \[ \begin{aligned} u(0)-f*0,u(0))= \gamma_1(u(1)-\lambda_1 f(1,u(1))),\quad & u'(0)= f'(0, u(0))),\\ v(0)-f(0, v(0))= \gamma_1(v(1)-\mu_1 f(1,v(1))),\quad & v'(0)= f'(0,v(0)),\end{aligned} \] where \(t\in{\mathcal I}= (0,1)\) and \(D^\alpha_{0^+}\) denotes the fractional Caputo derivative of order \(\alpha>0\). Additionally assume that \(\lambda_i,\mu_i\in \mathbb{R}^+\), \(\gamma_i\in (0,1/2\delta)\) for \(i= 1,2\), where \(\delta\) is an upper bound for the generalized Mittag-Leffler function \(E_{\alpha,1}(t)\). We study the existence of multiple positive solutions. To this aim we apply fixed point theorems.
scientific article; zbMATH DE number 6637973