\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 94, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/94\hfil Existence of solutions] {Existence of solutions for second-order differential equations and systems on infinite intervals} \author[T. Moussaoui, R. Precup\hfil EJDE-2009/94\hfilneg] {Toufik Moussaoui, Radu Precup} % in alphabetical order \address{Toufik Moussaoui \newline Department of Mathematics, E.N.S.\\ P.O. Box 92, 16050 Kouba, Algiers, Algeria} \email{moussaoui@ens-kouba.dz} \address{Radu Precup \newline Department of Applied Mathematics \\ Babe\c{s}-Bolyai University\\ 400084 Cluj, Romania} \email{r.precup@math.ubbcluj.ro} \thanks{Submitted October 7, 2008. Published August 6, 2009.} \subjclass[2000]{34B40} \keywords{Boundary value problem; fixed point theorem} \begin{abstract} We study the existence of nontrivial solutions to the boundary-value problem \begin{gather*} -u''+cu'+\lambda u = f(x,u),\quad -\infty 0$, we will prove the uniform convergence of$(Tu_{n})_{n}$to$Tu_{0}$on the interval$[-a,a]$. Let$\varepsilon >0$and choose some$b>a$large enough. By the uniform convergence of the sequence$(u_{n})_{n}$on$[-b,b]$, there exists an integer$N=N(\varepsilon ,b)$satisfying $I_1: =\sup_{x\in {\mathbb{R}} }\int_{-b}^{+b}G(x,s)|p(s)||f(u_{n}(s)) -f(u_{0}(s))|\,ds<\frac{\varepsilon }{2}\quad \text{for all }n\geq N.$ For$x\in [-a,a]$, we have that$|(Tu_{n})(x)-(Tu_{0})(x)|\leq I_1+I_2+I_{3}$, where $I_2: =\sup_{x\in {\mathbb{R}}}\int_{{\mathbb{R}} -[-b,+b]}G(x,s)|p(s)f(u_{0}(s))|\,ds\leq \frac{\varepsilon }{4}$ (by the Cauchy Convergence Criterion and$\lim_{|s|\to +\infty }p(s)f(u_{0}(s))=0$), $I_{3}: =\sup_{x\in {\mathbb{R}}}\,\int_{{\mathbb{R}} -[-b,+b]}G(x,s)|p(s)f(u_{n}(s))|\,ds\leq \frac{\varepsilon }{4}$ (by the Lebesgue Dominated Convergence Theorem). This proves the uniform convergence of the sequence$(Tu_{n})_{n}$to$Tu_{0}$on$[-a,a]$. \textbf{Claim 3:} For any$M>0$, the set$\{Tu,\,\| u\| _{0}\leq M\}$is relatively compact in$C_{0}({\mathbb{R}},{\mathbb{R}})$. By the Ascoli-Arzela Theorem, it is sufficient to prove that all the functions of this set are equicontinuous on every subinterval$[-a,a]$and that there exists a function$\gamma \in C_{0}({\mathbb{R}},{ \mathbb{R}})$such that for any$x\in {{\mathbb{R}}}$,$|Tu(x)|\leq \gamma (x)$. Let$x_1,x_2\in [-a,a]. We have successively the estimates \begin{align*} |Tu(x_2)-Tu(x_1)| &\leq \int_{-\infty }^{+\infty }|G_1(x_2,s)-G_1(x_1,s)||p(s)f(u(s))|\,ds \\ &\leq \max_{y\in \lbrack -M,M]}|f(y)|\int_{-\infty }^{+\infty }|G_1(x_2,s)-G_1(x_1,s)||p(s)|\,ds. \end{align*} By the continuity of the Green functionG$, the last term tends to$0$, as$ x_2$tends to$x_1$, whence comes the equicontinuity of the functions from$\{Tu;\| u\| _{0}\leq M\}. Analogously we have \begin{align*} |Tu(x)| &\leq \int_{-\infty }^{+\infty }G(x,s)|p(s)f(u(s))|\,ds \\ &\leq \max_{y\in \lbrack -M,M]}|f(y)|\int_{-\infty }^{+\infty }G(x,s)|p(s)|\,ds: =\gamma (x). \end{align*} Clearly,\gamma \in C_{0}({\mathbb{R}},{\mathbb{R}})$. Now, we consider the boundary-value problem \begin{gathered} -u''+cu'+\lambda u = f_{\infty }p(x)u(s),\quad -\infty 0$, there exists $R>0$ such that $|f(u)-f_{\infty }u|<\varepsilon |u|,\quad |u|>R.$ Set $R^{\ast }=\max_{|u|\leq R}|f(u)|$ and select $M>0$ such that $R^{\ast }+|f_{\infty }|R<\varepsilon M$. Denote $I_1=\{x\in {\mathbb{R}}:|u(x)|\leq R\},\quad I_2=\{x\in {\mathbb{R}}:|u(x)|>R\}.$ Thus for any $u\in C_{0}({\mathbb{R}},{\mathbb{R}})$ with $\| u\|_{0}>M$, when $x\in I_1$, we have $|f(u(x))-f_{\infty }u(x)|\leq |f(u(x))|+|f_{\infty }||u(x)| \leq R^{\ast}+|f_{\infty }|R<\varepsilon M<\varepsilon \| u\| _{0}.$ Similarly, we conclude that for any $u\in C_{0}({\mathbb{R}},{\mathbb{R}})$ with $\| u\| _{0}>M$, when $x\in I_2$, we also have that $|f(u(x))-f_{\infty }u(x)|<\varepsilon \| u\| _{0}.$ We conclude that for any $u\in C_{0}({\mathbb{R}},{\mathbb{R}})$ with $\|u\| _{0}>M$, we have $|f(u(x))-f_{\infty }u(x)|<\varepsilon \| u\| _{0}.$ Then for any $u\in C_{0}({\mathbb{R}},{\mathbb{R}})$ with $\| u\|_{0}>M$, one has \begin{align*} |f(u(x))| &\leq |f(u(x))-f_{\infty }u(x)|+|f_{\infty }u(x)| \\ &\leq \varepsilon \| u\| _{0}+|f_{\infty }|\| u\| _{0} \\ &\leq (f_{\infty }|+\varepsilon )\| u\| _{0}. \end{align*} Hence we obtain \begin{align*} |Tu(x)| &= \Big|\int_{-\infty }^{+\infty }G(x,s)p(s)f(u(s))\,ds-f_{\infty }\int_{-\infty }^{+\infty }G(x,s)p(s)u(s)\,ds\Big| \\ &\leq \frac{1}{k}\int_{-\infty }^{+\infty }|p(s)||f(u(s))-f_{\infty }u(s)|\,ds \\ &< \frac{1}{k}|p|_1\varepsilon \| u\| _{0}. \end{align*} Then we have $\lim_{\| u\| _{0}\to \infty }\frac{\| Tu-Au\| _{0}}{ \| u\| _{0}}=0.$ Theorem \ref{KZ} now guarantees that Problem \eqref{nonl1} has a nontrivial solution. This completes the proof. \end{proof} \section{Existence result for a generalized Fisher-like system} In this section, we study the system \begin{gathered} -u''+c_1u'+\lambda _1u = p(x)f(u,v),\quad -\infty 0$, we will prove the uniform convergence of$(T(u_{n},v_{n}))_{n}$to$T(u_{0},v_{0})$on$[-a,a]$. Let$\varepsilon >0$and choose some$b>a$large enough. By the uniform convergence of the sequence$\big((u_{n},v_{n})\big)_{n}$on$[-b,b]$, there exists an integer$N=N(\varepsilon ,b)satisfying \begin{align*} I_1: &= \sup_{x\in {\mathbb{R}} }\int_{-b}^{+b}G_1(x,s)|p(s)f(u_{n}(s),v_{n}(s))-p(s)f(u_{0}(s),v_{0}(s))| \,ds \\ &< \frac{\varepsilon }{2}\quad \text{for }n\geq N. \end{align*} Forx\in [-a,a]$, we have that$|T_1(u_{n},v_{n})(x)-T_1(u_{0},v_{0})(x)|\leq I_1+I_2+I_{3}$with $I_2: =\sup_{x\in {\mathbb{R}}}\,\int_{{\mathbb{R}} -[-b,+b]}G_1(x,s)|p(s)f(u_{0}(s),v_{0}(s))|\,ds \leq \frac{\varepsilon }{4}$ (by the Cauchy Convergence Criterion and$\lim_{|s|\to +\infty }p(s)f(u_{0}(s),v_{0}(s))=0$) and $I_{3}: =\sup_{x\in {\mathbb{R}}}\,\int_{{\mathbb{R}} -[-b,+b]}G_1(x,s)|p(s)f(u_{n}(s),v_{n}(s))|\,ds \leq \frac{\varepsilon }{4}$ (by the Lebesgue Dominated Convergence Theorem). This proves the uniform convergence of$(T_1(u_{n},v_{n}))_{n}$to$T_1(u_{0},v_{0})$on$[-a,a]$. Similarly, one can prove the uniform convergence of$(T_2(u_{n},v_{n}))_{n} $to$T_2(u_{0},v_{0})$on$[-a,a]$. \textbf{Claim 3:} For any$M>0$, the set$\{T(u,v);\,\| (u,v)\| \leq M\}$is relatively compact in$C_{0}({\mathbb{R}},{\mathbb{R}}^2)$. By the Ascoli-Arzela Theorem, it is sufficient to prove that the functions of this set are equicontinuous on every subinterval$[-a,a]$and that there exist functions$\gamma _1,\gamma _2\in C_{0}({\mathbb{R}},{\mathbb{R}})$such that for any$x\in {{\mathbb{R}}}$,$|T_1(u,v)(x)|\leq \gamma _1(x)$and$|T_2(u,v)(x)|\leq \gamma _2(x)$. Let$x_1,x_2\in [-a,a]. Then \begin{align*} &|T_1(u,v)(x_2)-T_1(u,v)(x_1)| \\ &\leq \int_{-\infty }^{+\infty }|G_1(x_2,s)-G_1(x_1,s)||p(s)f(u(s),v(s))|\,ds \\ &\leq \max_{y+z\in \lbrack -M,M]}|f(y,z)|\int_{-\infty }^{+\infty }|G_1(x_2,s)-G_1(x_1,s)||p(s)|\,ds. \end{align*} By the continuity of the Green functionG_1$, the last term tends to$0$, as$x_2$tends to$x_1. Similarly, \begin{align*} &|T_2(u,v)(x_2)-T_2(u,v)(x_1)| \\ &\leq \max_{y+z\in \lbrack -M,M]}|g(y,z)|\int_{-\infty }^{+\infty }|G_2(x_2,s)-G_2(x_1,s)||q(s)|\,ds. \end{align*} By the continuity of the Green functionG_2$, the last term tends to$0$, as$x_2$tends to$x_1$, whence comes the equicontinuity of$\{T(u,v);\| (u,v)\| \leq M\}. Now, we check analogously the second statement: \begin{align*} |T_1(u,v)(x)| &\leq \int_{-\infty }^{+\infty }G_1(x,s)|p(s)f(u(s),v(s))|\,ds \\ &\leq \max_{y+z\in \lbrack -M,M]}|f(y,z)|\int_{-\infty }^{+\infty }G_1(x,s)|p(s)|\,ds: =\gamma _1(x) \end{align*} and\gamma _1\in C_{0}({\mathbb{R}},{\mathbb{R}})$. Also $|T_2(u,v)(x)|\leq \max_{y+z\in \lbrack -M,M]}|g(y,z)|\int_{-\infty }^{+\infty }G_2(x,s)|q(s)|\,ds: =\gamma _2(x)$ and$\gamma _2\in C_{0}({\mathbb{R}},{\mathbb{R}})$. Now we consider the boundary-value problem \begin{gathered} -u''+c_1u'+\lambda _1u = p(x)(u(x)+v(x)), \quad -\infty 0$, there exists $R>0$ such that \begin{gather*} |f(u,v)-f_{\infty }(u+v)|<\varepsilon |u+v|,\quad \text{for}\;|u+v|>R, \\ |g(u,v)-g_{\infty }(u+v)|<\varepsilon |u+v|,\quad \text{for}\;|u+v|>R. \end{gather*} Set $R^{\ast }=\max \Big\{\max_{|u+v|\leq R}|f(u,v)|,\max_{|u+v|\leq R}|g(u,v)|\Big\}$ and select $M>0$ such that $R^{\ast }+\max \{|f_{\infty }|,|g_{\infty }|\}R<\varepsilon M$. Denote $I_1=\{x\in {\mathbb{R}}:|u(x)+v(x)|\leq R\},\quad I_2=\{x\in {\mathbb{R}}:|u(x)+v(x)|>R\}.$ Thus for any $(u,v)\in C_{0}({\mathbb{R}},{\mathbb{R}}^2)$ with $\|(u,v)\| >M$, when $x\in I_1$, we have \begin{align*} |f(u(x),v(x))-f_{\infty }(u(x)+v(x))| &\leq |f((u(x),v(x))|+|f_{\infty }||u(x)+v(x)| \\ &\leq R^{\ast }+|f_{\infty }|R\\ &<\varepsilon M<\varepsilon \| (u,v)\| . \end{align*} Similarly, we conclude that for any $(u,v)\in C_{0}({\mathbb{R}},{\mathbb{R}}^2)$ with $\| (u,v)\| >M$, when $x\in I_2$, we also have that $|f(u(x),v(x))-f_{\infty }(u(x)+v(x)|<\varepsilon \| (u,v)\| .$ We conclude that for any $(u,v)\in C_{0}({\mathbb{R}},{\mathbb{R}}^2)$ with $\| (u,v)\| >M$, one has $|f(u(x),v(x))-f_{\infty }(u(x)+v(x))|<\varepsilon \| (u,v)\| .$ Then for any $(u,v)\in C_{0}({\mathbb{R}},{\mathbb{R}}^2)$ with $\|(u,v)\| >M$, we have \begin{align*} |f(u(x),v(x))| &\leq |f(u(x),v(x))-f_{\infty }(u(x)+v(x))|+|f_{\infty }(u(x)+v(x))| \\ &\leq \varepsilon \| (u,v)\| +|f_{\infty }|\| (u,v)\| \\ &\leq (|f_{\infty }|+\varepsilon )\| (u,v)\| . \end{align*} In the same way, we find that for any $(u,v)\in C_{0}({\mathbb{R}},{\mathbb{R }}^2)$ with $\| (u,v)\| >M$, we have $|g(u(x),v(x))|\leq (|g_{\infty }|+\varepsilon )\| (u,v)\| .$ Hence we obtain \begin{align*} &|T_1(u,v)(x)-A_1(u,v)(x)|\\ &= |\int_{-\infty }^{+\infty }G_1(x,s)p(s)f(u(s),v(s))\,ds-f_{\infty }\int_{-\infty }^{+\infty }G_1(x,s)p(s)(u(s)+v(s))\,ds| \\ &\leq \frac{1}{k_1}\int_{-\infty }^{+\infty }|p(s)||f(u(s),v(s)))-f_{\infty }(u(s)+v(s))|\,ds \\ &< \frac{1}{k_1}|p|_1\varepsilon \| (u,v)\| \\ &= \frac{1}{k_1}|p|_1\varepsilon (\| u\| _{0}+\| v\| _{0}). \end{align*} Similarly, $|T_2(u,v)(x)-A_2(u,v)(x)|<\frac{1}{k_2}|q|_1\varepsilon (\| u\| _{0}+\| v\| _{0}).$ Then we have \begin{gather*} \lim_{\| u\| _{0}+\| v\| _{0}\to \infty }\frac{\| T_1(u,v)-A_1(u,v)\| _{0}}{\| u\| _{0}+\| v\| _{0}} = 0, \\ \lim_{\| u\| _{0}+\| v\| _{0}\to \infty }\frac{\| T_2(u,v)-A_2(u,v)\| _{0}}{\| u\| _{0}+\| v\| _{0}} = 0 \end{gather*} and hence $\lim_{\| u+v\| \to \infty }\frac{\| T(u,v)-A(u,v)\| }{\| u+v\| }=0.$ Theorem \ref{KZ} now guarantees that Problem \eqref{nonl2} has at least one nontrivial solution. This completes the proof. \end{proof} \section{Further results} In this section, we study the system \begin{gathered} -u''+c_1u'+\lambda _1u = f(x,u,v),\quad -\infty 0$such that $$\begin{gathered} |f(x,u,v)|\leq p(x)\varphi (|u|,|v|),\quad\text{for } (x,u,v)\in {{\mathbb{R}}}^{3}, \\ |g(x,u,v)|\leq q(x)\psi (|u|,|v|),\quad \text{for } (x,u,v)\in {{\mathbb{R}}}^{3}, \\ \max \Big\{\frac{1}{k_1}|p|_1\varphi (M_{0},M_{0}),\frac{1}{k_2} |q|_1\psi (M_{0},M_{0})\Big\}\leq M_{0}. \end{gathered} \label{S}$$ Then Problem \eqref{nonl3} admits at least one nontrivial solution$(u,v)\in C_{0}({\mathbb{R}},{\mathbb{R}}^2)$. \end{theorem} \begin{proof} Define the mapping$T: \,C_{0}({\mathbb{R}},{\mathbb{R}} ^2)\to \,C_{0}({\mathbb{R}},{\mathbb{R}}^2)$by$T=(T_1,T_1)$where \begin{gather*} T_1(u,v)(x)=\int_{-\infty }^{+\infty }G_1(x,s)f(s,u(s),v(s))\,ds,\\ T_2(u,v)(x)=\int_{-\infty }^{+\infty }G_2(x,s)g(s,u(s),v(s))\,ds. \end{gather*} In view of Schauder's fixed point theorem, we look for fixed points of$T$in the Banach space$C_{0}({\mathbb{R}},{\mathbb{R}}^2)$. The proof is split into four steps. \textbf{Claim 1:} The mapping$T$is well defined. Indeed, for any$(u,v)\in C_{0}({\mathbb{R}},{\mathbb{R}}^2), we get, by Assumptions \eqref{S}, the estimate \begin{align*} |T_1(u,v)(x)| &\leq \int_{-\infty }^{+\infty }G_1(x,s)|f(s,u(s),v(s))|\,ds \\ &\leq \int_{-\infty }^{+\infty }G_1(x,s)p(s)\varphi (|u(s)|,|v(s)|)\,ds \\ &\leq \varphi (\| u\| ,\| v\| )\int_{-\infty }^{+\infty }G_1(x,s)p(s)\,ds,\;\forall \,x\in {{\mathbb{R}}} \\ &\leq \frac{1}{k_1}|p|_1\varphi (\| u\| ,\| v\| ). \end{align*} In the same way, one can prove that $|T_2(u,v)(x)\leq \frac{1}{k_2}|q|_1\psi (\| u\| ,\| v\| ).$ The convergence of the integrals definingT(u,v)(x)$is then established. In addition for any$s\in {\mathbb{R}}$,$G_1(\pm \infty ,s)=0,\;G_1(\pm \infty ,s)=0$, and then, taking the limit in$ T(u,v)(x)=(T_1(u,v)(x),T_2(u,v)(x))$, we obtain$T(u,v)(\pm \infty )=0$. Therefore, the mapping$T:C_{0}({\mathbb{R}},{\mathbb{R}}^2) \to C_{0}({\mathbb{R}},{\mathbb{R}}^2)$is well defined. \textbf{Claim 2:} As in Section 3, one can prove easily that the operator$T=(T_1,T_2)$is continuous. \textbf{Claim 3:} As in Section 3, one can prove easily that for any$M>0$, the set$\{T(u,v)=(T_1(u,v),T_2(u,v));\,\| (u,v)\| \leq M\}$is relatively compact in$C_{0}({\mathbb{R}},{\mathbb{R}}^2)$. \textbf{Claim 4:} There exists a nonempty closed bounded convex$K$such that$T$maps$K$into itself. Let $K=\left\{ (u,v)\in C_{0}({\mathbb{R}},{\mathbb{R}}^2):\| u\| _{0}\leq M_{0},\| v\| _{0}\leq M_{0}\right\} .$ From assumption \eqref{S}, we know that $\frac{1}{k_1}|p|_1\varphi (M_{0},M_{0})\leq 1,\quad \frac{1}{k_2}|q|_1\psi (M_{0},M_{0})\leq 1.$ If$\| u\| \leq M_{0}$and$\| v\| \leq M_{0}, then \begin{align*} \| T_1(u,v)\| _{0} &\leq \sup_{x\in {{\mathbb{R}}}}\int_{-\infty }^{+\infty }G_1(x,s)p(s)\varphi (|u(s)|,|v(s)|)\,ds \\ &\leq \frac{1}{k_1}|p|_1\varphi (M_{0},M_{0})\\ &\leq M_{0}. \end{align*} Similarly,\| T_2(u,v)\| _{0}\leq M_{0}$. Therefore, the operator$T$maps$K$into itself. The proof of Theorem \ref{thm2} then follows from Schauder's fixed point theorem. \end{proof} Using of Schauder's theorem, one can also prove the existence of a positive solution under some integral conditions on the nonlinear terms: \begin{theorem}\label{thm4} Suppose that the functions$f$and$g$are positive with$f(x,0,0)\not\equiv 0$or$g(x,0,0)\not\equiv 0$, and satisfy the following two mean growth assumptions: \begin{gather*} |f(x,u,v)|\leq \varphi (x,|u|,|v|),\\ |g(x,u,v)|\leq \psi (x,|u|,|v|) \end{gather*} where$\varphi ,\psi :{\mathbb{R}}\times {\mathbb{R}}_{+}\times {\mathbb{R} }_{+}\to {\mathbb{R}}_{+}$are continuous, nondecreasing with respect to their two last arguments and verify $$\int_{-\infty }^{+\infty }\varphi (x,M_{0},M_{0})\,dx\leq k_1M_{0},\quad \int_{-\infty }^{+\infty }\psi (x,M_{0},M_{0})\,dx\leq k_2M_{0} \label{ST}$$ for some constant$M_{0}>0$. Then Problem \eqref{nonl3} has a positive solution$(u,v)\in C_{0}({\mathbb{R}},{\mathbb{R}}_{+}\times { \mathbb{R}}_{+})$. \end{theorem} \begin{proof} To prove Theorem \ref{thm4}, we proceed as in Theorem \ref{thm3} by taking the closed convex subset$K$of$C_{0}({\mathbb{R}},{\mathbb{R}}^2)$defined by: $K=\{(u,v)\in C_{0}({\mathbb{R}},{\mathbb{R}}^2):0\leq u(x)\leq M_{0},\;0\leq v(x)\leq M_{0}\text{ on }{\mathbb{R}}\}.$ Using Assumption (\ref{ST}) and the fact that the mapping$\varphi $and$\psi $are nondecreasing with respect to their two last arguments, we find that$T$maps$Kinto itself. Indeed, we derive the estimates: \begin{align*} 0&\leq T_1(u,v)(x) \leq \int_{-\infty }^{+\infty }G_1(x,s)\varphi (s,|u(y)|,|v(y)|)\,ds \\ &\leq \frac{1}{k_1}\int_{-\infty }^{+\infty }\varphi (s,M_{0},M_{0})\,ds \leq M_{0} \end{align*} and $0\leq T_2(u,v)(x)\leq M_{0}.$ In addition, the mappingT$is continuous as can easily be seen and one can check that$T(K)$is relatively compact. Then the claim of Theorem \ref{thm4} follows. \end{proof} \section{Examples} In this section, we give some examples to illustrate our results. (1) Consider the boundary-value problem $$\label{e5.1} \begin{gathered} -u''+u'+2u = \frac{1}{\pi (x^2+1)}[2u+1+\lg (1+|u|)], \\ u(-\infty )=u(+\infty ) = 0. \end{gathered}$$ Here$p(x)=\frac{1}{\pi (x^2+1)}$and$f(u)=2u+1+\lg (1+|u|)$. Notice that$k=3$,$|p|_1=1$and$f_{\infty }=2$. Thus, by Theorem \ref{thm1}, Problem \eqref{e5.1} has at least one nontrivial solution$u\in C_{0}(R,R)$. (2) Consider the boundary-value system $$\label{e5.2} \begin{gathered} -u''+u'+2u = \frac{1}{2\pi (x^2+1)}\big[2(u+v)+1+\lg (1+|u+v|)\big], \\ -v''+\sqrt{5}v'+v = \frac{1}{2\sqrt{\pi }} e^{-x^2}\big[2(u+v)+1+\sqrt{1+|u+v|}\big], \\ u(-\infty )=u(+\infty ) = 0, \quad v(-\infty )=v(+\infty ) = 0. \end{gathered}$$ Set$p(x)=\frac{1}{2\pi (x^2+1)}$,$q(x)=\frac{1}{2\sqrt{\pi }}e^{-x^2}$,$f(u,v)=2(u+v)+1+\lg (1+|u+v|)$, and$g(u,v)=2(u+v)+1+ \sqrt{1+|u+v|}$. Notice that$k_1=k_2=3$,$\alpha =\beta=2$,$|p|_1=|q|_1=\frac{1}{2}$, and$f_{\infty }=g_{\infty }=2$. Thus, by Theorem \ref{thm2}, Problem \eqref{e5.3} has at least one nontrivial solution$(u,v)\in C_{0}(R,R^2)$. (3) Consider the boundary-value system $$\label{e5.3} \begin{gathered} -u''+u'+2u = \frac{1}{\pi (x^2+1)}\big( \left\vert v\right\vert ^{\mu }+1\big), \\ -v''+\sqrt{5}v'+v = \frac{1}{\sqrt{\pi }}\exp ^{-x^2}\big(\left\vert u\right\vert ^{\nu }+1\big), \\ u(-\infty )=u(+\infty ) = 0, \quad v(-\infty )=v(+\infty ) = 0. \end{gathered}$$ where$\mu $and$\nu $are real numbers such that$0<\mu <1$and$0<\nu <1$. Set$p(x)=\frac{1}{\pi (x^2+1)}$,$q(x)=\frac{1}{\sqrt{\pi }}e^{-x^2}$,$\varphi (y,z)=z^{\mu }+1$, and$\psi (y,z)=y^{\nu }+1$. Notice that$k_1=k_2=3$,$|p|_1=|q|_1=1$and if we choose any$M_{0}$large enough, then condition \eqref{S} is satisfied. Thus, by Theorem \ref{thm3}, Problem \eqref{e5.3} has at least one nontrivial solution$(u,v)\in C_{0}(R,R^2)\$. \begin{thebibliography}{0} \bibitem{DjeMou1} S. Djebali and T. Moussaoui; \emph{A class of second order BVPs on infinite intervals,} Electron. Journal of Qualitative Theory of Diff. Eqns. 4 (2006), 1-19. \bibitem{DjeMou2} S. Djebali and T. Moussaoui; \emph{Qualitative properties and existence of solutions for a generalized Fisher-like equation,} to appear. \bibitem{DjeMeb1} S. Djebali and K. Mebarki; \emph{Existence results for a class of BVPs on the positive half-line,} Commun. Appl. Nonl. Anal., (2007). \bibitem{KZ1} M. A. Krasnoselskii, P. P. Zabreiko; \emph{Geometrical methods of nonlinear analysis}, Springer-Verlag, New York, 1984. \bibitem{Ze} E. Zeidler; \emph{Nonlinear Functional Analysis,} T1, Fixed Point Theory, Springer, 1985. \end{thebibliography} \end{document}