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Existence of solutions for a coupled system with \(\phi\)-Laplacian operators and nonlinear coupled boundary conditions. (English) Zbl 1391.34052
Summary: We study the existence of solutions of the system \[ \begin{cases}(\phi_1(u_1^\prime(t)))^\prime=f_1(t,u_1(t),u_2(t),u_1^\prime(t),u_2^\prime(t)),\quad\mathrm{a.e.}t\in[0,T],\\(\phi_2(u_2^\prime(t)))^\prime=f_2(t,u_1(t),u_2(t),u_1^\prime(t),u_2^\prime(t)),\quad\mathrm{a.e.}t\in[0,T],\end{cases} \] submitted to nonlinear coupled boundary conditions on \([0,T]\) where \(\phi_1,\phi_2\colon(-a,a)\rightarrow\mathbb{R}\), with \(0<a<+\infty\), are two increasing homeomorphisms such that \(\phi_1(0)=\phi_2(0)=0\), and \(f_i:[0,T]\times\mathbb{R}^{4}\rightarrow\mathbb{R},i\in\{1,2\}\) are two \(L^1\)-Carathéodory functions. Using some new conditions and Schauder fixed point Theorem, we obtain solvability result.
34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI
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