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Solvability of \(p, q\)-Laplacian systems with potential boundary conditions. (English) Zbl 1189.34040
Summary: The aim of this article is to establish a general existence result for the \(p,q\)-Laplacian system
\[ \begin{cases} -[h_p(u')]'=f(t,u,v)+\alpha(t)\\ -[h_qv')]'=g(t,u,v)+\beta(t)\quad & (t\in (0,T)),\end{cases}\tag{1} \]
subjected to the boundary conditions
\[ \begin{cases} (h_p(u')(0),\;-h_p(u')(T))\in \partial j(u(0),u(T)),\\ (h_q(v')(0),\;-h_q(v')(T))\in \partial k(v(0),v(T)).\end{cases}\tag{2} \]
Here \(h_p\), \(h_q\) are homeomorphisms of \(\mathbb R^n\) and \(\mathbb R^m\) defined, respectively, by \(h_p(x)=|x|^{p-2}x\) \((x\in\mathbb R^n)\), \(h_q(y)=|y|^{q-2}y\) \((y\in\mathbb R^m)\), for some fixed \(p,q\in(1,\infty)\); \(\alpha\in L^1 (0,T;\mathbb R^n)\), \(\beta\in L^1(0,T;\mathbb R^m)\) and \(f:(0,T)\times \mathbb R^n\times \mathbb R^m\to\mathbb R^n\), \(g:(0,T)\times \mathbb R^n\times \mathbb R^m\to\mathbb R^m\) are Carathéorody mappings satisfying some hypotheses.
The approach relies on Schaefer fixed point theorem, combined with a technique involving matrices convergent to zero.

34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
Full Text: DOI
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