\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). $$ We say that $(u_1,u_2)\in X_1\times X_2$ is a {\it solution} of \eqref{i1} if and only if \begin{equation}\label{i4} J_{q,p}(u_1,u_2) = i^*N_{f,g}(i(u_1,u_2)) \end{equation} or equivalently \begin{equation}\label{i5} \begin{aligned} &\langle J_{q,p}(u_1,u_2), (v_1,v_2) \rangle_{X_1^*\times X_2^*, X_1 \times X_2}\\ & =\langle i^*N_{f,g}(i(u_1,u_2)), i(v_1,v_2) \rangle_{L^{q_1'}(\Omega) \times L^{p_1'}(\Omega), L^{q_1}(\Omega) \times L^{p_1}(\Omega)} \\ &= \int_{\Omega} \big[ f(x,u_1(x),u_2(x)) v_1(x) + g(x,u_1(x),u_2(x)) v_2(x) \big]dx \end{aligned} \end{equation} for all $(v_1,v_2) \in X_1 \times X_2$. The rest of this article is organized as follows. The preliminary and abstract results are presented in Section 2. In Section 3 we prove the existence results for problem \eqref{i1} using the method mentioned above. Section 4 provides some examples. \section{Preliminary results} \subsection{Duality mappings} Let $i=1,2$, $(X_i, \|\cdot \|_{X_i})$ be real Banach spaces, $X^*_i$ the corresponding dual spaces and $\langle \cdot , \cdot \rangle $ the duality between $X^*_i$ and $X_i$. Let $\varphi_i : \mathbb{R}_+ \to \mathbb{R}_+$ be gauge functions, such that $\varphi_i$ are continuous, strictly increasing, $\varphi_i (0)=0$ and $\varphi_i (t) \to \infty$ as $t \to \infty$. The duality mapping corresponding to the gauge function $\varphi_i$ is the set valued mapping $J_{\varphi_i}: X_i \to 2^{X_i^*}$, defined by $$ 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: \begin{equation}\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}. \end{equation} 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 \begin{equation}\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, \end{equation} 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), $$ with $J_{q,p}^{-1} N_{f,g} : X_1 \times X_2 \to X_1 \times X_2 $ a compact operator. We define the operator $T = J_{q,p}^{-1} N_{f,g} = (T_1, T_2)$, where \begin{equation}\label{ls1} T_1(u_1,u_2) = J_q^{-1} N_f (u_1,u_2), \quad T_2(u_1,u_2) = J_p^{-1} N_g (u_1,u_2) \end{equation} and we shall prove that the compact operator $T$ has at least one fixed point using the Leray-Schauder fixed point theorem. For this it is sufficient to prove that the set $$ 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, $$ that is $S$ is bounded. We have obtained the following result. \begin{theorem}\label{tels1} Assume that $X_1$, $X_2$ are locally uniformly convex, $J_q : X_1 \to X_1^*$, $J_p : X_2 \to X_2^*$ and the Carath\`{e}odory functions $f$ and $g$ satisfy \eqref{i2} and \eqref{i3}, respectively with $q_1 \in (1,q)$ and $p_1 \in (1,p)$. Then the operator $T = J_{q,p}^{-1} N_{f,g}$ has one fixed point in $X_1 \times X_2$ or equivalently problem \eqref{i1} has a solution. Moreover, the set of solutions of problem \eqref{i1} is bounded in $X_1\times X_2$. \end{theorem} \section{Examples} \subsection{Dirichlet problem for systems with $(q,p)$-Laplacian} If $X_1 \times X_2 = W_0^{1,q}(\Omega) \times W_0^{1,p}(\Omega)$, then $J_{q,p} = ( -\Delta_q, -\Delta_p)$ and the solutions set of equation $J_{q,p}(u_1,u_2) = N_{f,g}(u_1, u_2)$ coincides with the solutions set of the Dirichlet problem \begin{equation}\label{e1} \begin{gathered} - \Delta_{q} u_1 = f(x,u_1,u_2) \quad\text{in } \Omega, \\ - \Delta_{p} u_2 = g(x,u_1,u_2) \quad \text{in } \Omega, \\ u_1 = u_2 = 0 \quad \text{on } \partial \Omega. \end{gathered} \end{equation} \subsection{Neumann problem for systems with $(q,p)$-Laplacian} We consider $X_1 \times X_2 = W^{1,q}(\Omega) \times W^{1,p}$, endowed with the norm $$ \|(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 \begin{equation} \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} \end{equation} we mean an element $(u_1,u_2) \in W^{1,q}(\Omega) \times W^{1,p}(\Omega)$ which satisfies \begin{equation}\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} \end{equation} 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}