Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 26, pp. 1-10.

Title: A fixed point method for nonlinear equations involving a duality
       mapping defined on product spaces
        
Authors: Jenica Cringanu  (Univ.of Galati, Galati, Romania)
         Daniel Pasca  (Univ. of Oradea, Romania)
        
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.