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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:
[1] DOI: 10.1007/BF02829608 · Zbl 1052.34022
[2] Jebelean P, Adv. Diff. Eqns 13 pp 273– (2008)
[3] DOI: 10.3934/cpaa.2008.7.267 · Zbl 1147.34010
[4] DOI: 10.1016/j.jmaa.2005.04.022 · Zbl 1105.34008
[5] Granas A, Fixed Point Theory (2003)
[6] Couchouron J-F, Topol. Methods Nonlin. Anal. 30 pp 157– (2007)
[7] DOI: 10.1016/j.mcm.2008.04.006 · Zbl 1165.65336
[8] DOI: 10.1080/00036810802307553 · Zbl 1160.47049
[9] Precup R, Methods in Nonlinear Integral Equations (2002)
[10] DOI: 10.1007/s00013-007-2330-0 · Zbl 1146.34014
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