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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.

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