\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 26, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/26\hfil A fixed point method] {A fixed point method for nonlinear equations involving a duality mapping defined on \\ product spaces} \author[J. Cr\^ inganu, D. Pa\c sca \hfil EJDE-2013/26\hfilneg] {Jenic\u a Cr\^ inganu, Daniel Pa\c sca} % in alphabetical order \address{Jenic\u a Cr\^ inganu \newline Department of Mathematics, University of Gala\c ti, Str. Domneasc\u a 47, Gala\c ti, Romania} \email{jcringanu@ugal.ro} \address{Daniel Pa\c sca \newline Department of Mathematics and Informatics, University of Oradea, University Street 1, 410087 Oradea, Romania} \email{dpasca@uoradea.ro} \thanks{Submitted July 31, 2012. Published January 27, 2013.} \subjclass[2000]{58C15, 35J20, 35J60, 35J65} \keywords{Duality mapping; Leray-Schauder degree; $(q,p)$-Laplacian} \begin{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 $\varphi_1(t)=t^{q-1}$, $\varphi_2(t)=t^{p-1}$, 1q \\ +\infty & \text{if } N \leq q \end{cases} $$and$$ 1 < p_1 < p^* = \begin{cases} \frac{Np}{N-p} & \text{if } N>p \\ +\infty & \text{if } N \leq p \end{cases} $$and q^*, p^* are the critical Sobolev exponents of q,p respectively; \item [(H3)] Let i=1,2. For any gauge functions \varphi_i :\mathbb{R}_+ \to \mathbb{R}_+, the corresponding duality mapping J_{\varphi_i}:X_i \to X_i^* (see the precise definition in Section 2.1 below) is continuous and satisfies the (S_+) condition: if x_{in} \rightharpoonup x_i (weakly) in X_i and \limsup_{n \to \infty} \langle J_{\varphi_i} x_{in}, x_{in} - x_i \rangle \leq 0 then x_{in} \to x_i (strongly) in X_i; \item [(H4)] J_{q,p} : X_1 \times X_2 \to X_1^* \times X_2^*, J_{q,p} =(J_q, J_p), where J_q, J_p are the duality mappings corresponding to the gauge functions \varphi_1(t) = t^{q-1}, t \geq 0, \varphi_2(t) = t^{p-1}, t \geq 0 respectively; \item [(H5)] N_{f,g} :L^{q_1}(\Omega) \times L^{p_1}(\Omega) \to L^{q_1'}(\Omega) \times L^{p_1'}(\Omega), where \frac{1}{q_1} + \frac{1}{q_1'} = 1, \frac{1}{p_1} + \frac{1}{p_1'} = 1 defined by N_{f,g}(u_1,u_2)(x) = \big( f(x,u_1(x),u_2(x)), g(x,u_1(x),u_2(x)) \big), is the Nemytskii operator generated by the Carath\'{e}odory functions f,g: \Omega \times \mathbb{R} \times \mathbb{R} \to \mathbb{R}, which satisfies the growth conditions \begin{gather}\label{i2} | f(x,s,t)| \leq c_1|s|^{q_1-1}+ c_2|t|^{(q_1-1)\frac{p_1}{q_1}} + b_1(x),\quad \text{for } x\in \Omega, (s,t) \in \mathbb{R}\times \mathbb{R},\\ \label{i3} | g(x,s,t)| \leq c_3|s|^{(p_1-1)\frac{q_1}{p_1}} + c_4 |t|^{p_1-1}+ b_2(x),\quad \text{for } x\in \Omega, (s,t) \in \mathbb{R} \times \mathbb{R}, \end{gather} where c_1, c_2,c_3, c_4 > 0 are constants, b_1\in L^{q_1'}(\Omega),b_2\in L^{p_1'}(\Omega), \frac{1}{q_1}+\frac{1}{q_1'} =1, \frac{1}{p_1}+\frac{1}{p_1'} =1. \end{itemize} We make the convention that in the case of a Carath\'{e}odory function, the assertion x\in\Omega'' is understood in the sense a.e. x\in\Omega''. To prove the existence of the solutions of the problem \eqref{i1} we use topological methods via Leray-Schauder degree. We note that equality \eqref{i1} is understood in the sense of X_1^* \times X_2^*, where the norm on this product space is \|(x_1^*,x_2^*)\|_{X_1^*\times X_2^*} = \|x_1^*\|_{X_1^*} + \|x_2^*\|_{X_2^*}. More precisely, let i_1 : X_1 \to L^{q_1}(\Omega) and i_2 : X_2 \to L^{p_1}(\Omega) be the identity mappings on X_1, X_2 respectively and i_1^* : L^{q_1'}(\Omega) \to X_1^* and i_2^* : L^{p_1'}(\Omega) \to X_2^* be the corresponding dual:$$ i_1^* u_1^* = u_1^* \circ i_1 \text{ for } u_1^* \in L^{q_1'}(\Omega) \quad i_2^* u_2^* = u_2^* \circ i_2 \text{ for } u_2^* \in L^{p_1'}(\Omega). $$We define i : X_1 \times X_2 \to L^{q_1}(\Omega) \times L^{p_1}(\Omega) given by i(u_1,u_2)=(i_1(u_1), i_2(u_2)) and its dual i^* : L^{q_1'}(\Omega) \times L^{p_1'}(\Omega) \to X_1^* \times X_2^* given by$$ i^*(u_1^*, u_2^*) = (i_1^* u_1^*, i_2^* u_2^*) = (u_1^* \circ i_1, u_2^* \circ i_2). We say that (u_1,u_2)\in X_1\times X_2 is a {\it solution} of \eqref{i1} if and only if $$\label{i4} J_{q,p}(u_1,u_2) = i^*N_{f,g}(i(u_1,u_2))$$ or equivalently \label{i5} \begin{aligned} &\langle J_{q,p}(u_1,u_2), (v_1,v_2) \rangle_{X_1^*\times X_2^*, X_1 \times X_2}\\ & =\langle i^*N_{f,g}(i(u_1,u_2)), i(v_1,v_2) \rangle_{L^{q_1'}(\Omega) \times L^{p_1'}(\Omega), L^{q_1}(\Omega) \times L^{p_1}(\Omega)} \\ &= \int_{\Omega} \big[ f(x,u_1(x),u_2(x)) v_1(x) + g(x,u_1(x),u_2(x)) v_2(x) \big]dx \end{aligned} for all (v_1,v_2) \in X_1 \times X_2. The rest of this article is organized as follows. The preliminary and abstract results are presented in Section 2. In Section 3 we prove the existence results for problem \eqref{i1} using the method mentioned above. Section 4 provides some examples. \section{Preliminary results} \subsection{Duality mappings} Let i=1,2, (X_i, \|\cdot \|_{X_i}) be real Banach spaces, X^*_i the corresponding dual spaces and \langle \cdot , \cdot \rangle the duality between X^*_i and X_i. Let \varphi_i : \mathbb{R}_+ \to \mathbb{R}_+ be gauge functions, such that \varphi_i are continuous, strictly increasing, \varphi_i (0)=0 and \varphi_i (t) \to \infty as t \to \infty. The duality mapping corresponding to the gauge function \varphi_i is the set valued mapping J_{\varphi_i}: X_i \to 2^{X_i^*}, defined by J_{\varphi_i} x = \big\{ x_i^* \in X_i^* : \langle x_i^*, x_i \rangle = \varphi_i (\|x_i\|_{X_i})\|x_i\|_{X_i},\, \|x_i^*\|_{X_i^*} = \varphi_i (\|x_i\|_{X_i}) \big\}. $$If X_i are smooth, then J_{\varphi_i} : X_i \to X_i^* is defined by$$ J_{\varphi_i}0 = 0, \quad J_{\varphi_i}x_i = \varphi_i (\|x_i\|_{X_i}) \|~ \|'_{X_i}(x_i), \quad x_i \neq 0, and the following metric properties being consequent: $$\label{d1} \|J_{\varphi_i} x_i\|_{X_i^*} = \varphi_i (\|x_i\|_{X_i}), \quad \langle J_{\varphi_i} x_i, x_i \rangle = \varphi_i(\|x_i\|_{X_i})\|x_i\|_{X_i}.$$ Now we define J_{\varphi_1, \varphi_2} : X_1 \times X_2 \to 2^{X_1^*} \times 2^{X_2^*} by J_{\varphi_1, \varphi_2} (x_1,x_2) = (J_{\varphi_1} x_1, J_{\varphi_2} x_2). From \eqref{d1} we obtain \begin{gather}\label{d2} \begin{aligned} \|J_{\varphi_1, \varphi_2} (x_1,x_2)\|_{X_1^* \times X_2^*} &= \|J_{\varphi_1} x_1\|_{X_1^*} + \|J_{\varphi_2} x_2\|_{X_2^*} \\ &= \varphi_1 (\|x_1\|_{X_1}) + \varphi_2 (\|x_2\|_{X_2}), \end{aligned}\\ \label{d3} \begin{aligned} \langle J_{\varphi_1, \varphi_2} (x_1,x_2), (x_1,x_2) \rangle & =\langle J_{\varphi_1} x_1, x_1 \rangle + \langle J_{\varphi_2} x_2, x_2 \rangle\\ &= \varphi_1(\|x_1\|_{X_1})\|x_1\|_{X_1}+ \varphi_2(\|x_2\|_{X_2})\|x_2\|_{X_2}. \end{aligned} \end{gather} In what follows we consider the particular case when J_{\varphi_i}: X_i \to X_i^* are the duality mappings, assumed to be single-valued, corresponding to the gauge functions \varphi_1(t) = t^{q-1}, \varphi_2(t) = t^{p-1}, 10, \\ \lambda_2 = \inf \big\{ \frac{\|u_2\|_{X_2}^{p_1}}{\|i_2 (u_2)\|_{L^{p_1} (\Omega)}^{p_1}}: u_2\in X_2 \setminus \{0\} \big\}>0. \end{gather*} \begin{proposition}\label{dp6} \lambda_1, \lambda_2 are attained and \lambda_1^{-1/q_1} and \lambda_2^{-1/p_1} are the best constants C_1 and C_2, respectively in the writing of the embeddings of X_1 into L^{q_1}(\Omega) and X_2 into L^{p_1}(\Omega), respectively. \end{proposition} For a proof of the above proposition, see \cite[Proposition 4]{DJ01}. \subsection{Nemytskii operators} Let \Omega be an open subset in \mathbb{R}^N, N\geq 1 and f,g : \Omega \times \mathbb{R} \times \mathbb{R} \to \mathbb{R} be Carath\'{e}odory functions, i.e.: \begin{itemize} \item[(i)] for each (s,t)\in \mathbb{R} \times \mathbb{R}, the functions x\mapsto f(x,s,t), x\mapsto g(x,s,t) are Lebesgue measurable in \Omega; \item[(ii)] for a.e. x\in \Omega, the functions (s,t) \mapsto f(x,s,t), (s,t) \mapsto g(x,s,t) are continuous in \mathbb{R} \times \mathbb{R}. \end{itemize} Let \mathcal{M} be the set of all measurable functions u: \Omega \to \mathbb{R}. If f,g : \Omega \times \mathbb{R} \times \mathbb{R} \to \mathbb{R} are Carath\'{e}odory functions and (v_1,v_2) \in \mathcal{M} \times \mathcal{M} then the function x \mapsto (f(x,v_1(x),v_2(x)), g(x,v_1(x),v_2(x))) is measurable in \Omega. So, we can define the operator N_{f,g} : \mathcal{M} \times \mathcal{M} \to \mathcal{M} \times \mathcal{M} by N_{f,g}(v_1,v_2)(x)=(f(x,v_1(x),v_2(x)), g(x,v_1(x),v_2(x))) $$which we will be the Nemytskii operator. We need the following result: \begin{lemma}\label{nl1} Let r_1,r_2, k_1,k_2>0. Then there are the constants k_3,k_4>0 such that$$ k_1a^{r_1}+k_2b^{r_2}\leq k_3 (a+b)^{\max(r_1,r_2)}+k_4,\quad \text{for all } a,b>0. \end{lemma} \begin{proof} If a,b\geq 1 we have \begin{align*} k_1a^{r_1}+k_2b^{r_2} &\leq k_1a^{\max(r_1,r_2)}+k_2b^{\max(r_1,r_2)}\\ &\leq \max(k_1,k_2)(a^{\max(r_1,r_2)}+b^{\max(r_1,r_2)})\\ &\leq \max(k_1,k_2)(a+b)^{\max(r_1,r_2)}, \end{align*} and the proof is ready with k_3=\max(k_1,k_2) and k_4>0 arbitrary. If a,b < 1 then k_1a^{r_1}+k_2b^{r_2} \leq k_1+k_2 $$and we may take k_4=k_1+k_2, k_3>0, arbitrary. If a\geq 1, b<1,$$ k_1a^{r_1}+k_2b^{r_2}\leq k_1a^{r_1}+k_2 \leq k_1(a+b)^{r_1}+k_2 \leq k_1(a+b)^{\max(r_1,r_2)}+k_2, and similarly if a<1,b\geq1. \end{proof} Some properties of the Nemytskii operator that will be used in the sequel are contained in the following proposition. \begin{proposition}\label{np1} Let p_1,q_1>1, f,g : \Omega \times \mathbb{R} \times\mathbb{R} \to \mathbb{R} be a Carath\'{e}odory functions which satisfy the growth conditions: \begin{gather}\label{n1} | f(x,s,t)| \leq c_1|s|^{q_1-1}+ c_2|t|^{(q_1-1)\frac{p_1}{q_1}} + b_1(x),\quad \text{for } x\in \Omega, (s,t) \in \mathbb{R}\times \mathbb{R}, \\ \label{n2} | g(x,s,t)| \leq c_3|s|^{(p_1-1)\frac{q_1}{p_1}} + c_4 |t|^{p_1-1}+ b_2(x),\quad \text{for } x\in \Omega, (s,t) \in \mathbb{R}\times \mathbb{R}, \end{gather} where c_1, c_2,c_3, c_4 > 0 are constants, b_1\in L^{q_1'}(\Omega), b_2\in L^{p_1'}(\Omega), \frac{1}{q_1}+\frac{1}{q_1'} =1, \frac{1}{p_1}+\frac{1}{p_1'} =1. Then N_{f,g} is continuous from L^{q_1}(\Omega)\times L^{p_1}(\Omega) into L^{q_1'}(\Omega)\times L^{p_1'}(\Omega) and maps bounded sets into bounded sets. Moreover, it holds $$\label{n3} \| {N_{f,g}(v_1,v_2)}\|_{L^{q_1'}(\Omega)\times L^{p_1'}(\Omega)} \leq c_8\|(v_1,v_2)\|_{L^{q_1}(\Omega) \times L^{p_1}(\Omega)}^{R_1-1}+c_9,$$ for all (v_1,v_2)\in L^{q_1}(\Omega) \times L^{p_1}(\Omega), where c_8,c_9>0 are constants and R_1=\max(q_1,p_1). \end{proposition} \begin{proof} From \eqref{n1} and \eqref{n2}, for (v_1,v_2)\in L^{q_1}(\Omega)\times L^{p_1}(\Omega) we have \begin{align*} &\| {N_{f,g}(v_1,v_2)}\|_{L^{q_1'}(\Omega)\times L^{p_1'}(\Omega)}\\ &= \| {N_f(v_1,v_2)}\|_{L^{q_1'}(\Omega)} + \| {N_g(v_1,v_2)}\|_{L^{p_1'}(\Omega)} \\ &\leq c_1\| |v_1|^{q_1-1}\|_{L^{q_1'}(\Omega)} +c_2 \Big\| |v_2|^{(q_1-1)\frac{p_1}{q_1}}\Big\|_{L^{q_1'}(\Omega)} + \|b_1\|_{L^{q_1'}(\Omega)}\\ &\quad + c_3\Big\| |v_1|^{(p_1-1)\frac{q_1}{p_1}}\Big\|_{L^{p_1'}(\Omega)} +c_4 \| |v_2|^{p_1-1}\|_{L^{p_1'}(\Omega)}+ \|b_2\|_{L^{p_1'}(\Omega)}\\ &= c_1 \|{v_1}\|_{L^{q_1}(\Omega)}^{q_1-1} +c_2\|{v_2}\|_{L^{p_1}(\Omega)}^{(q_1-1)\frac{p_1}{q_1}}+K_1 +c_3\|{v_1}\|_{L^{q_1}(\Omega)}^{(p_1-1)\frac{q_1}{p_1}} +c_4 \|{v_2}\|_{L^{p_1}(\Omega)}^{p_1-1} +K_2 . \end{align*} By Lemma \ref{nl1} there are the constants c_5,c_6,c_7>0, such that \begin{align*} &\| {N_{f,g}(v_1,v_2)}\|_{L^{q_1'}(\Omega)\times L^{p_1'}(\Omega)}\\ &\leq c_5(\|{v_1}\|_{L^{q_1}(\Omega)}+\|{v_2}\|_{L^{p_1}(\Omega)}) ^{\max(p_1-1,q_1-1)}\\ &\quad +c_6(\|{v_1}\|_{L^{q_1}(\Omega)}+ \|{v_2}\|_{L^{p_1}(\Omega)})^{\max\big((q_1-1)\frac{p_1}{q_1},(p_1-1) \frac{q_1}{p_1}\big)}+ c_7\\ &= c_5\|(v_1,v_2)\|_{L^{q_1}(\Omega) \times L^{p_1}(\Omega)}^{\max(p_1-1,q_1-1)} +c_6\|(v_1,v_2)\|_{L^{q_1}(\Omega) \times L^{p_1}(\Omega)}^{\max\big((q_1-1) \frac{p_1}{q_1},(p_1-1)\frac{q_1}{p_1}\big)} +c_7. \end{align*} Since \max\Big((q_1-1)\frac{p_1}{q_1},(p_1-1)\frac{q_1}{p_1}\Big) \leq \max(p_1-1,q_1-1) $$we obtain$$ \| {N_{f,g}(v_1,v_2)}\|_{L^{q_1'}(\Omega)\times L^{p_1'}(\Omega)} \leq c_8\|(v_1,v_2)\|_{L^{q_1}(\Omega) \times L^{p_1}(\Omega)}^{R_1-1}+c_9, $$for all (v_1,v_2)\in L^{q_1}(\Omega) \times L^{p_1}(\Omega), where c_8,c_9>0 are constants and R_1=\max(q_1,p_1). Now assume that (v_{1n},v_{2n})\to(v_1,v_2) in L^{q_1}(\Omega)\times L^{p_1}(\Omega) and claim that N_{f,g}(v_{1n},v_{2n})\to N_{f,g}(v_1,v_2) in L^{q_1'}(\Omega)\times L^{p_1'}(\Omega). Given any sequence of (v_{1n},v_{2n}) there is a further subsequence (call it again (v_{1n},v_{2n})) such that$$ |v_{1n}(x)|\leqslant h_1(x), |v_{2n}(x)|\leqslant h_2(x) $$for some h_1 \in L^{q_1'}(\Omega),h_2\in L^{p_1'}(\Omega). It follows from \eqref{n1} and \eqref{n2} that \begin{gather*} |f(x,v_{1n}(x),v_{2n}(x))|\leqslant c_1|h_1(x)|^{q_1-1} +c_2|h_2(x)|^{(q_1-1)\frac{p_1}{q_1}}+b_1(x), \\ |g(x,v_{1n}(x),v_{2n}(x))|\leqslant c_3|h_1(x)|^{(p_1-1)\frac{q_1}{p_1}} +c_4|h_2(x)|^{p_1-1}+b_2(x). \end{gather*} Since f(x,v_{1n}(x),v_{2n}(x)) converges a.e. to f(x,v_1(x),v_2(x)), g(x,v_{1n}(x),v_{2n}(x)) converges a.e. to g(x,v_1(x),v_2(x)), the result follows from the Lebesgue Dominated Convergence Theorem and a standard result on metric spaces. \end{proof} \section{Existence of solutions for \eqref{i1} using a Leray-Schauder technique} We start we the statement of the Leray-Schauder fixed point theorem. \begin{theorem} Let T be a continuous and compact mapping of a Banach space X into itself, such that the set$$ \{ x\in X : x=\lambda Tx \text{ for some } 0\leq \lambda \leq 1 \} $$is bounded. Then T has a fixed point. \end{theorem} Since X_1 \to L^{q_1}(\Omega) and X_2 \to L^{p_1}(\Omega) are compact, the diagram$$ X_1 \times X_2 \stackrel{i}{\longrightarrow} L^{q_1}(\Omega) \times L^{p_1}(\Omega) \stackrel{N_{f,g}} \longrightarrow L^{q_1'}(\Omega) \times L^{p_1'}(\Omega) \stackrel{i^*} \longrightarrow X_1^* \times X_2^* $$show that N_{f,g} (by which we mean i^* N_{f,g} i) is compact. By Proposition \ref{dp5}, the operator J_{q,p} : X_1 \times X_2 \to X_1^* \times X_2^* is bijective with its inverse J_{q,p}^{-1}(u_1^*, u_2^*) = (J_q^{-1}u_1^*, J_p^{-1}u_2^*) bounded, continuous and monotone. Consequently \eqref{i1} can be equivalently written$$ (u_1,u_2) = J_{q,p}^{-1} N_{f,g}(u_1,u_2), $$with J_{q,p}^{-1} N_{f,g} : X_1 \times X_2 \to X_1 \times X_2 a compact operator. We define the operator T = J_{q,p}^{-1} N_{f,g} = (T_1, T_2), where $$\label{ls1} T_1(u_1,u_2) = J_q^{-1} N_f (u_1,u_2), \quad T_2(u_1,u_2) = J_p^{-1} N_g (u_1,u_2)$$ and we shall prove that the compact operator T has at least one fixed point using the Leray-Schauder fixed point theorem. For this it is sufficient to prove that the set$$ S = \big \{(u_1,u_2) \in X_1 \times X_2 : (u_1,u_2) = \alpha T (u_1,u_2) \text{ for some } \alpha \in [0,1] \big \} is bounded in X_1 \times X_2. By \eqref{ls1}, \eqref{i2} and \eqref{i3} for (u_1,u_2) \in X_1 \times X_2 we have \begin{align*} &\| T_1(u_1,u_2) \|_{X_1}^q \\ &= \langle J_q (T_1(u_1,u_2)), T_1(u_1,u_2) \rangle\\ &= \langle N_f (u_1, u_2), T_1(u_1,u_2) \rangle \\ &= \int_\Omega f(x,u_1(x),u_2(x)) T_1(u_1(x),u_2(x)) dx \\ &\leq \int_\Omega \Big( c_1 |u_1(x)|^{q_1-1} + c_2 |u_2(x)|^{(q_1-1)\frac{p_1}{q_1}} + |b_1(x)| \Big) |T_1(u_1(x), u_2(x))|dx, \end{align*} and similarly \begin{align*} &\| T_2(u_1,u_2) \|_{X_2}^p \\ &= \langle J_p (T_2(u_1,u_2)), T_2(u_1,u_2) \rangle \\ &= \langle N_g (u_1, u_2), T_2(u_1,u_2) \rangle \\ &= \int_\Omega g(x,u_1(x),u_2(x)) T_2(u_1(x),u_2(x)) dx \\ &\leq \int_\Omega \Big( c_3 |u_1(x)|^{(p_1-1)\frac{q_1}{p_1}} + c_4 |u_2(x)|^{p_1-1} + |b_2(x)| \Big) |T_2(u_1(x), u_2(x))|dx. \end{align*} If (u_1,u_2)\in S, that is (u_1,u_2)=\alpha T(u_1,u_2) =(T_1(u_1,u_2), T_2(u_1,u_2)) with \alpha \in [0,1], we have \begin{align*} &\| T_1(u_1,u_2) \|_{X_1}^q \\ &\leq \int_\Omega \Big( c_1 \alpha^{q_1-1} |T_1(u_1(x), u_2(x))|^{q_1-1} \\ &\quad +c_2 \alpha^{(q_1-1)\frac{p_1}{q_1}} |T_2(u_1(x),u_2(x))|^{(q_1-1) \frac{p_1}{q_1}} + |b_1(x)| \Big) |T_1(u_1(x), u_2(x))|dx \\ &\leq c_1 \alpha^{q_1-1} \| T_1(u_1,u_2) \|_{L^{q_1}(\Omega)}^{q_1} + c_2 \alpha^{(q_1-1)\frac{p_1}{q_1}} \| T_2(u_1,u_2) \|_{L^{p_1}(\Omega)}^{(q_1-1)\frac{p_1}{q_1}} \| T_1(u_1,u_2) \|_{L^{q_1}(\Omega)} \\ &\quad + \|b_1\|_{L^{q_1'}(\Omega)} \| T_1(u_1,u_2) \|_{L^{q_1}(\Omega)} \\ &\leq c_1 k_1^{q_1} \| T_1(u_1,u_2) \|_{X_1}^{q_1} + c_2 k_1 k_2^{(q_1-1)\frac{p_1}{q_1}} \| T_2(u_1,u_2) \|_{X_2}^{(q_1-1)\frac{p_1}{q_1}} \| T_1(u_1,u_2) \|_{X_1} \\ &\quad + k_1 \|b_1\|_{L^{q_1'}(\Omega)} \| T_1(u_1,u_2) \|_{X_1}, \end{align*} where k_1,k_2>0 are coming from the compact embeddings X_1 \to L^{q_1}(\Omega) and X_2 \to L^{p_1}(\Omega), respectively. In the same way we obtain \begin{align*} \| T_2(u_1,u_2) \|_{X_2}^p & \leq c_3 k_1^{(p_1-1)\frac{q_1}{p_1}} k_2 \| T_1(u_1,u_2) \|_{X_1}^{(p_1-1) \frac{q_1}{p_1}} \| T_2(u_1,u_2) \|_{X_2} \\ &\quad + c_4 k_2^{p_1} \| T_2(u_1,u_2) \|_{X_2}^{p_1} + k_2 \|b_2\|_{L^{p_1'}(\Omega)} \| T_2(u_1,u_2) \|_{X_2}. \end{align*} Consequently, for each (u_1,u_2)\in S it hold \begin{align*} &\| T_1(u_1,u_2) \|_{X_1}^q - c_5 \| T_1(u_1,u_2) \|_{X_1}^{q_1} \\ &- c_6 \| T_2(u_1,u_2) \|_{X_2}^{(q_1-1)\frac{p_1}{q_1}} \| T_1(u_1,u_2) \|_{X_1} -c_7 \| T_1(u_1,u_2) \|_{X_1} \leq 0 \end{align*} and \begin{align*} &\| T_2(u_1,u_2) \|_{X_2}^p - c_8 \| T_1(u_1,u_2) \|_{X_1}^{(p_1-1) \frac{q_1}{p_1}}\| T_2(u_1,u_2) \|_{X_2} \\ &- c_9 \| T_2(u_1,u_2) \|_{X_2}^{p_1} - c_{10} \| T_2(u_1,u_2) \|_{X_2} \leq 0, \end{align*} with c_5,\ldots, c_{10} positive constants. \begin{lemma}\label{lels1} Let q> p> 1, 10 such that \begin{gather*} a^q\leq c_5 a^{q_1} + c_6 ab^{(q_1-1)\frac{p_1}{q_1}} + c_7 a,\\ b^p\leq c_8 a^{(p_1-1)\frac{q_1}{p_1}}b + c_9 b^{p_1} + c_{10} b, \end{gather*} where c_5,\ldots, c_{10}>0 positve constants. Then there is the constant K>0 be such that a+b\leq K. \end{lemma} \begin{proof} We consider the following cases: \begin{enumerate} \item If a\leq 1, b\leq 1 then a+b\leq2. \item If a\leq 1, b > 1 we have b^p \leq c_8 b + c_9 b^{p_1} + c_{10} b and since p>p_1>1, there is a constant K_1>0 such that b\leq K_1. Consequently a+b\leq 1+K_1. \item If a>1, b\leq 1 we have a^q \leq c_5 a^{q_1} + c_6 a+c_7 a and since q>q_1>1, there is a constant K_2>0 such that a\leq K_2. Consequently a+b\leq 1+K_2. \item We consider a>1,b>1. Let us remark that \max\Big( (q_1-1)\frac{p_1}{q_1},(p_1-1)\frac{q_1}{p_1}\Big) \leq \max(p_1-1,q_1-1). $$\end{enumerate} If a\geq b we have a^q \leq c_5 a^{q_1} + c_6 ab^{\max(p_1-1,q_1-1)} + c_7 a \leq c_5 a^{q_1} + c_6 a^{\max(p_1,q_1)} + c_7 a, and since q>q_1, q>\max(p_1,q_1)>1, there is a constant K_3>0 such that a\leq K_3 and so a+b\leq 2K_3. If a\leq b we reasoning similarly. \end{proof} Now, by Lemma \ref{lels1}, there exists a constant K>0 such that \|T(u_1,u_2)\|_{X_1\times X_2} = \|T_1(u_1,u_2)\|_{X_1} + \|T_2(u_1,u_2)\|_{X_2} \leq K for (u_1,u_2)\in S and then$$ \|(u_1, u_2)\|_{X_1\times X_2} = \alpha \|T(u_1,u_2)\| \leq \alpha K \leq K, \quad \text{for } (u_1,u_2)\in S, $$that is S is bounded. We have obtained the following result. \begin{theorem}\label{tels1} Assume that X_1, X_2 are locally uniformly convex, J_q : X_1 \to X_1^*, J_p : X_2 \to X_2^* and the Carath\{e}odory functions f and g satisfy \eqref{i2} and \eqref{i3}, respectively with q_1 \in (1,q) and p_1 \in (1,p). Then the operator T = J_{q,p}^{-1} N_{f,g} has one fixed point in X_1 \times X_2 or equivalently problem \eqref{i1} has a solution. Moreover, the set of solutions of problem \eqref{i1} is bounded in X_1\times X_2. \end{theorem} \section{Examples} \subsection{Dirichlet problem for systems with (q,p)-Laplacian} If X_1 \times X_2 = W_0^{1,q}(\Omega) \times W_0^{1,p}(\Omega), then J_{q,p} = ( -\Delta_q, -\Delta_p) and the solutions set of equation J_{q,p}(u_1,u_2) = N_{f,g}(u_1, u_2) coincides with the solutions set of the Dirichlet problem $$\label{e1} \begin{gathered} - \Delta_{q} u_1 = f(x,u_1,u_2) \quad\text{in } \Omega, \\ - \Delta_{p} u_2 = g(x,u_1,u_2) \quad \text{in } \Omega, \\ u_1 = u_2 = 0 \quad \text{on } \partial \Omega. \end{gathered}$$ \subsection{Neumann problem for systems with (q,p)-Laplacian} We consider X_1 \times X_2 = W^{1,q}(\Omega) \times W^{1,p}, endowed with the norm$$ \|(u_1,u_2)\| = \|u_1\|_{1,q} + \|u_2\|_{1,p} where \begin{gather*} \| u_1 \|_{1,q}^q = \| u_1 \|_{0,q}^q + \| |\nabla u_1| \|_{0,q}^q \quad \text{for all } u_1 \in W^{1,q}(\Omega), \\ \| u_2 \|_{1,p}^p = \| u_2 \|_{0,p}^p + \| |\nabla u_2| \|_{0,p}^p \quad \text{for all } u_2 \in W^{1,p}(\Omega), \end{gather*} which are equivalent with the standard norms on the spaces W^{1,q}(\Omega), W^{1,p}(\Omega) respectively (see \cite{C04}). In this case, the duality mappings J_q, J_p on \big(W^{1,q}(\Omega),\| \cdot \|_{1,q}\big), \big(W^{1,p}(\Omega),\| \cdot \|_{1,p}\big), respectively, corresponding to the gauge functions \varphi_1(t) = t^{q-1} and \varphi_2(t) = t^{p-1} are defined by \begin{gather}\label{e2} \begin{aligned} J_q : \big( W^{1,q}(\Omega), \| \cdot \|_{1,q} \big) \to \big( W^{1,q}(\Omega), \| \cdot \|_{1,q}\big)^* \\ J_q u_1 = - \Delta_q u_1 + |u_1|^{q-2} u_1 \text{ for all } u_1 \in W^{1,q} (\Omega) \end{aligned}\\ \label{e3} \begin{aligned} J_p : \big( W^{1,p}(\Omega), \| \cdot \|_{1,p} \big) \to \big( W^{1,p}(\Omega), \| \cdot \|_{1,p}\big)^* \\ J_p u_2 = - \Delta_p u_2 + |u_2|^{p-2} u_2 \quad \text{ for all } u_2 \in W^{1,p}(\Omega) \end{aligned} \end{gather} (see [5]). By a weak solution of the Neumann problem $$\label{e4} \begin{gathered} -\Delta_q u_1 + |u_1|^{q-2} u_1 = f(x,u_1,u_2) \quad\text{in } \Omega, \\ -\Delta_p u_2 + |u_2|^{p-2} u_2 = g(x,u_1,u_2) \quad\text{in } \Omega, \\ | \nabla u_1|^{q-2} \frac{\partial u_1}{\partial n} = 0 \quad \text{on } \partial \Omega, \\ | \nabla u_2|^{p-2} \frac{\partial u_2}{\partial n} = 0 \quad \text{on } \partial \Omega, \end{gathered}$$ we mean an element (u_1,u_2) \in W^{1,q}(\Omega) \times W^{1,p}(\Omega) which satisfies \label{e5} \begin{aligned} &\int_\Omega |\nabla u_1(x)|^{q-2} \nabla u_1(x) \nabla v_1(x) dx + \int_\Omega |u_1(x)|^{q-2} u_1(x) v_1(x) dx \\ &+ \int_\Omega |\nabla u_2(x)|^{p-2} \nabla u_2(x) \nabla v_2(x) dx + \int_\Omega |u_2(x)|^{p-2} u_2(x) v_2(x) dx\\ &=\int_\Omega f(x,u_1(x),u_2(x)) v_1(x) + g(x,u_1(x),u_2(x)) v_2(x) dx, \end{aligned} for all (v_1, v_2) \in W^{1,q}(\Omega) \times W^{1,p}(\Omega). It is easy to see that (u_1,u_2) \in W^{1,q}(\Omega) \times W^{1,p}(\Omega) is a solution of the problem \eqref{e4}, in the sense of \eqref{e5} if and only if J_{q,p} (u_1,u_2) = (i^* N_{f,g} i ) (u_1,u_2),$where$J_{q,p}(u_1,u_2)=(J_q u_1, J_p u_2)$and$J_q$,$J_p$are given by \eqref{e2} and \eqref{e3},$i(u_1,u_2)=(i_1u_1, i_2u_2)$, and$i_1 : W^{1,q}(\Omega) \to L^{q_1}(\Omega)$,$i_2 : W^{1,p}(\Omega) \to L^{p_1}(\Omega)$are the compact embeddings of$W^{1,q}(\Omega)$into$L^{q_1}(\Omega)$and of$W^{1,p}(\Omega)$into$L^{p_1}(\Omega)$, respectively. By$i^* : L^{q_1'}(\Omega) \times L^{p_1'}(\Omega) \to (W^{1,q}(\Omega), \|\cdot\|_{1,q})^* \times (W^{1,p}(\Omega), \|\cdot\|_{1,p})^*$we denoted the dual of$i$. So, we are in the functional framework described in introduction. Indeed, the spaces$\big( W^{1,q}(\Omega), \|\cdot\|_{1,q}\big)$and$\big( W^{1,p}(\Omega), \|\cdot\|_{1,p}\big)$are smooth reflexive Banach spaces, compactly embedded in$L^{q_1}(\Omega)$and$L^{p_1}(\Omega)$, respectively.$J_q : \big( W^{1,q}(\Omega), \| \cdot\|_{1,q} \big) \to \big( W^{1,q}(\Omega),\| \cdot\|_{1,q}\big)^*$and$J_p : \big( W^{1,p}(\Omega), \| \cdot\|_{1,p} \big) \to \big( W^{1,p}(\Omega),\| \cdot\|_{1,p}\big)^*$are single valued, continuous and satisfies the ($S_+$) condition (see \cite{C04}). Consequently, the existence result given in section 3 becomes the existence result for the Neumann problem \eqref{e4}. \begin{remark}\label{er1} \rm We note that using the same method it is possible to proved the existence of a solution for the Dirichlet and Neumann problems with$(q,p)$--pseudo-Laplacian or with$(A_q, A_p)$--Laplacian (see \cite{DJ01}). \end{remark} \begin{remark}\label{er2}\rm In \cite{DJ97} the authors used the same method to proved the existence of a solution for the Dirichlet problem with$p$-Laplacian. \end{remark} \begin{thebibliography}{10} \bibitem{B64} F. E. Browder; \emph{Probl\{e}mes Non-Lineaires}, Les Presses de l'Universit\'{e} de Montreal, 1964. \bibitem{C74} I. Cior\u anescu; \emph{Duality mapping in nonlinear functional analysis}, Publishing House of the Romanian Academy, Bucharest, 1974. \bibitem{C04} J. Cr\^inganu; \emph{Variational and topological methods for Neumann problems with$p$-Laplacian}, Communications on Applied Nonlinear Analysis 11 (2004), 1-38. \bibitem{DJM01} G. Dinc\u a, P. Jebelean, J. Mawhin; \emph{Variational and Topological methods for Dirichlet problems with$p$-Laplacian}, Portugaliae Mathematica vol. 58 no. 3 (2001), 339-378. \bibitem{DJ97} G. Dinc\u a, P. Jebelean; \emph{Une m\'{e}thode de point fixe pour le$p\$-Laplacien}, C.R. Acad. Sci. Paris t. 324, Serie I (1997), 165-168. \bibitem{DJ01} G. Dinc\u a, P. Jebelean; \emph{Some existence results for a class of nonlinear equations involving a duality mapping}, Nonlinear Analysis 46 (2001), 347-363. \end{thebibliography} \end{document}