Electron. J. Diff. Equ., Vol. 2013 (2013), No. 26, pp. 1-10.

A fixed point method for nonlinear equations involving a duality mapping defined on product spaces

Jenica Cringanu, Daniel Pasca

Abstract:
The aim of this paper is to obtain solutions for the equation
$$ J_{q,p} (u_1,u_2) =N_{f,g}(u_1,u_2), $$
where $J_{q,p}$ is the duality mapping on a product of two real, reflexive and smooth Banach spaces $X_1, X_2$, corresponding to the gauge functions $\varphi_1(t)=t^{q-1}$, $\varphi_2(t)=t^{p-1}$, $1<q, p<\infty$, $N_{f,g}$ being the Nemytskii operator generated by the Caratheodory functions f,g which satisfies some appropriate conditions. To prove the existence solutions we use a topological method via Leray-Schauder degree. As applications, we obtained in a unitary manner some existence results for Dirichlet and Neumann problems for systems with (q,p)-Laplacian, with (q,p)-pseudo-Laplacian or with $(A_q, A_p)$-Laplacian.

Submitted July 31, 2012. Published January 27, 2013.
Math Subject Classifications: 58C15, 35J20, 35J60, 35J65.
Key Words: Duality mapping; Leray-Schauder degree; (q,p)-Laplacian.

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Jenica Cringanu
Department of Mathematics, University of Galati
Str. Domneasca 47, Galati, Romania
email: jcringanu@ugal.ro
Daniel Pasca
Department of Mathematics and Informatics, University of Oradea
University Street 1, 410087 Oradea, Romania
email: dpasca@uoradea.ro

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