\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 192, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/192\hfil Infinite elastic beam equations] {Existence of solutions to infinite elastic beam equations with unbounded nonlinearities} \author[H. Carrasco, F. Minh\'os \hfil EJDE-2017/192\hfilneg] {Hugo Carrasco, Feliz Minh\'os} \address{Hugo Carrasco \newline Departamento de Matem\'atica, Escola de Ci\^encias e Tecnologia, Universidade de \'Evora, Rua Rom\~{a}o Ramalho, 59, 7000-671 \'Evora, Portugal} \email{hugcarrasco@gmail.com} \address{Feliz Minh\'os \newline Departamento de Matem\'atica, Escola de Ci\^encias e Tecnologia, Centro de Investiga\c{c}\~{a}o em Matem\'atica e Aplica\c{c}\~{o}es (CIMA-UE), Instituto de Investiga\c{c}\~{a}o e Forma\c{c}\~{a}o Avan\c{c}ada, Universidade de \'Evora, Rua Rom\~{a}o Ramalho, 59, 7000-671 \'Evora, Portugal} \email{fminhos@uevora.pt} \thanks{Submitted January 19, 2017. Published August 1, 2017.} \subjclass[2010]{34B10, 34B15, 34B40} \keywords{Half line problem; Schauder fixed point theorem; \hfill\break\indent unbounded and nonordered upper and lower solutions; one-sided Nagumo condition} \begin{abstract} This article concerns the existence of unbounded solutions to fourth-order boundary-value problem on the half-line with two-point boundary conditions. One-sided Nagumo condition plays a special role as it allows an asymmetric unbounded behavior on the nonlinearity. The arguments are based on the Schauder fixed point theorem and lower and upper solutions method. As an application, an example is given with non-ordered lower and upper solutions, to prove our results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Fourth-order differential equations can model the bending of an elastic beam and, in this sense, we refer them as beam equations. They have received increased interest from several fields of science and engineering, either on bounded domains \cite{DA+FM, AC+GF, JF+FM, FM+TG+AIS, Song} either on the real line \cite{agarwal1, Choi, Jang1, Jang2, Ma, Mah}. We study the fully nonlinear beam equation on the half line, \begin{equation} u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)) ,\quad t\in [ 0,+\infty ), \label{edo} \end{equation} where $f:[0,+\infty )\times \mathbb{R}^{4}\to \mathbb{R}$ is an $L^{1}$-Carath\'eodory function, and the boundary conditions are of Sturm-Liouville type, \begin{equation} u(0)=A,u'(0)=B,\quad u''(0)+au'''(0)=C,\quad u'''(+\infty )=D, \label{cf} \end{equation} $A,B,C,D\in \mathbb{R}$ , $a<0$ and $u'''(+\infty ):= \lim_{t\to +\infty }u'''(t)$. The non-compactness of the interval requires delicate techniques to obtain sufficient conditions for the solvability of boundary value problems on the half-line. As examples, we refer the extension of continuous solutions on the corresponding finite intervals under a diagonalization process, fixed point theory in some Banach spaces and lower and upper solutions method (see \cite{agarwal2, Bai, Lian, Yan} and the references therein). We present here an approach based on an adequate space of weighted functions. Lower and upper solutions method is a very adequate technique to deal with boundary value problems as it provides not only the existence of bounded or unbounded solutions but also their localization and, from that, some qualitative data about solutions, their variation and behavior (see \cite {AC+FM, JG+LK+FM, Lian2, Lian3, FM}). In this paper we use not necessarily ordered lower an upper solutions, generalizing, in this way, the set of admissible lower and upper functions. As far as we know, it is the first time where such functions are applied to boundary value problems defined on the half line, to obtain unbounded solutions. An important tool is the Nagumo condition, useful to get \textit{a priori} estimates on some derivatives of the solution. The usual growth condition of the Nagumo type used in the literature is\ a bilateral one. However the same estimations hold with a similar one-sided assumption, which allows unbounded nonlinearities in boundary value problems. Therefore, it generalizes the two-sided condition, as it is proved in \cite{JF+FM,MRG+FM+AS}. In short, this work has the following novelties related to the existent literature in this field: \begin{itemize} \item The nonlinearity $f$ is an $L^{1}$-Carath\'eodory function, allowing discontinuities in time; \item From the unilateral Nagumo growth condition assumed on $f$, equation \eqref{edo} can deal with unbounded nonlinearities; \item The lower and upper solutions do not need to be well ordered, or even ordered, and, moreover, their boundary conditions are more general, making easier to obtain lower and upper solutions to the problem. \item The non-compactness of the associated operator is overcome by considering an adequate space of weighted functions. \end{itemize} The paper is organized as it follows: In Section 2 some auxiliary result are defined such as the space, the weighted norms, the unilateral Nagumo condition and lower and upper solutions to be used. Section 3 contains the main result: an existence and localization theorem, where it is proved the existence of a solution, and some bounds on the first and second derivatives as well. Finally, an example, which is not covered by the existent results, shows the applicability of the main theorem. \section{Definitions and preliminary results} In this work we consider the space \[ X=\big\{ x\in C^3[0,+\infty ):\lim_{t\to +\infty }\frac{x^{(i)}(t)}{1+t^{3-i}} \text{ exists in }\mathbb{R},\; i=0,1,2,3\big\} \] with the norm $\| x\| _{X}:=\max \{\| x\| _0,\| x'\| _0,\| x''\| _0,\| x'''\| _0\} $, where \[ \| \omega ^{(i)}\| _0=\sup_{0\leq t<+\infty }| \frac{\omega ^{(i)}(t)}{1+t^{3-i}}| ,\; i=0,1,2,3. \] It can be proved that $(X,\| \cdot\| _{X})$ is a Banach space (see \cite{Lian3}). The following definition establishes the assumptions assumed on the nonlinearity. \begin{definition} \rm A function $f:[0,+\infty )\times \mathbb{R}^{4}\to \mathbb{R}$ is called an $L^{1}$-Carath\'eodory function if it satisfies: \begin{itemize} \item[(i)] for each $(x,y,z,w) \in \mathbb{R}^{4}$ , $t\mapsto f(t,x,y,z,w)$ is measurable on $[0,+\infty )$; \item[(ii)] for almost every $t\in [ 0,+\infty ),(x,y,z,w)\mapsto f(t,x,y,z,w)$ is continuous in $\mathbb{R}^{4}$; \item[(iii)] for each $\rho >0$, there exists a positive function $\varphi _{\rho }\in L^{1}[0,+\infty )$ such that for all $ (x,y,z,w)\in $ $\mathbb{R}^{4}$ with $\| (x,y,z,w)\| _{X}<\rho $, then \[ | f(t,x,y,z,w)| \leq \varphi _{\rho }(t),\quad\text{a.e. }t\in [0,+\infty ). \] \end{itemize} \end{definition} Solutions of the linear problem associated with \eqref{edo}-\eqref{cf} are defined with Green's function, which can be obtained by standard calculus. \begin{lemma} \label{lemma-equi} Let $t^3\eta \in L^{1}[0,+\infty )$. Then the linear boundary value problem \begin{equation} u^{(4)}(t)+\eta (t)=0,\quad t\in [ 0,+\infty ), \label{prob1} \end{equation} with boundary conditions \eqref{cf}, has a unique solution in $X$. Moreover, this solution can be expressed as \begin{equation} u(t)=A+Bt+\frac{C-aD}{2}t^{2}+\frac{D}{6}t^3+\int_0^{+\infty }G(t,s)\eta (s)ds \label{sol} \end{equation} where \[ G(t,s)= \begin{cases} \frac{s^3}{6}-\frac{s^{2}t}{2}+\frac{st^{2}}{2}-\frac{at^{2}}{2}, & 0\leq s\leq t \\ -\frac{at^{2}}{2}+\frac{t^3}{6}, & t\leq s<+\infty . \end{cases} \] \end{lemma} To apply a fixed point theorem it is important to have an \emph{a priori} estimation for $u'''$. In the literature this bound is obtained from a bilateral Nagumo-type growth. In this paper it is used a more general one-sided Nagumo condition, which allows unbounded nonlinearities on \eqref{edo} (for more details see \cite{MRG+FM+AS2, Song, Zhang}. Remark that, as it is proved in \cite{MRG+FM+AS} a function can verify an unilateral Nagumo condition but not a two-sided one. Let $\gamma _{i},\Gamma _{i}\in C[0,+\infty ),\gamma _{i}(t)\leq \Gamma _{i}(t),i=0,1,2$ and define the set \[ E=\big\{ (t,x_0,x_{1},x_{2},x_{3})\in [ 0,+\infty )\times \mathbb{R} ^{4}:\gamma _{i}(t)\leq x_{i}\leq \Gamma _{i}(t),i=0,1,2\big\} . \] \begin{definition}\rm An $L^{1}$-Carath\'eodory function $f:E\to \mathbb{R}$ is said to satisfy the one-sided Nagumo-type growth condition in $E$ if it satisfies either \begin{equation} f(t,x,y,z,w)\leq \psi (t)h(| w| ),\quad \forall (t,x,y,z,w)\in E, \label{cn1} \end{equation} or \begin{equation} f(t,x,y,z,w)\geq -\psi (t)h(| w| ),\quad \forall (t,x,y,z,w)\in E, \label{cn1a} \end{equation} for some positive continuous functions $\psi ,h$, and some $\nu >1$, such that \begin{equation} \int_0^{+\infty }\psi (s)ds<+\infty ,\sup_{0\leq t<+\infty }\psi (t)(1+t)^{\nu }<+\infty ,\quad \int_0^{+\infty }\frac{s}{h(s)}ds=+\infty . \label{cn2} \end{equation} \end{definition} Next lemma provides an \textit{a priori} bound. \begin{lemma} \label{Lemma1} Let $f:[0,+\infty )\times \mathbb{R}^{4}\to \mathbb{R} $ be an $L^{1}$-Carath\'eodory function satisfying \eqref{cn1}, or \eqref{cn1a}, and \eqref{cn2} in $E$. Then for every $r>0$ there exists $R>0$ (not depending on $u$) such that every $u$ solution of \eqref{edo}, \eqref{cf} satisfying \begin{equation} \gamma _0(t)\leq u(t)\leq \Gamma _0(t),\gamma _{1}(t)\leq u'(t)\leq \Gamma _{1}(t),\gamma _{2}(t)\leq u''(t)\leq \Gamma _{2}(t), \label{lem-a} \end{equation} for $t\in [ 0,+\infty )$, satisfies $\| u'''\| _00$ such that \begin{equation} r>\max \Big\{ | \frac{C-\Gamma _{2}(0)}{a}| ,| \frac{C-\gamma _{2}(0)}{a}|, | D | \Big\} . \label{eq-r} \end{equation} By this inequality we cannot have $| u'''(t)| >r$ for all $t\in [ 0,+\infty )$, because \begin{equation} | u'''(0)| =| \frac{ C-u''(0)}{a}| \leq \max \Big\{ | \frac{ C-\Gamma _{2}(0)}{a}| ,| \frac{C-\gamma _{2}(0)}{a} | \Big\} r/2$ the proof is complete as \[ \| u'''\|_0 =\sup_{0\leq t<+\infty }| \frac{u'''(t)}{2}| \leq \frac{r}{2 }r$, then by \eqref{cn2} we can take $R>r$ such that \[ \int_{r}^{R}\frac{s}{h(s)}ds>M\max \Big\{ M_{1}+\sup_{0\leq t<+\infty } \frac{\Gamma _{2}(t)}{1+t}\frac{\nu }{\nu -1},M_{1}-\inf_{0\leq t<+\infty } \frac{\gamma _{2}(t)}{1+t}\frac{\nu }{\nu -1}\Big\} \] with $ M:=\sup_{0\leq t<+\infty }\psi (t)(1+t)^{\nu }$ and $ M_{1}:=\sup_{0\leq t<+\infty }\frac{\Gamma _{2}(t)}{(1+t)^{\nu }}-\inf_{0\leq t<+\infty }\frac{\gamma _{2}(t)}{(1+t)^{\nu }}$. Assume that growth condition \eqref{cn1} holds. By \eqref{eq-r}, suppose that there are $t_{\ast }$, $t_{+}\in (0,+\infty ) $ such that $u'''(t_{\ast })=r$, $u'''(t)>r$ for all $t\in (t_{\ast },t_{+}]$. Then \begin{align*} \int_{u'''(t_{\ast })}^{u'''(t_{+})}\frac{s}{h(s)}ds &= \int_{t_{\ast }}^{t_{+}}\frac{u'''(s)}{h(u'''(s))}u^{(4)}(s)ds \\ &= \int_{t_{\ast }}^{t_{+}}\frac{f(s,u(s),u'(s),u''(s),u'''(s))}{h(u'''(s))} u'''(s)ds \\ &\leq \int_{t_{\ast }}^{t_{+}}\psi (s)\text{ }u'''(s) \text{ }ds\leq M\int_{t_{\ast }}^{t_{+}}\frac{u'''(s)}{ (1+s)^{\nu }}\text{ }ds \\ &= M\int_{t_{\ast }}^{t_{+}}\Big(\frac{u''(s)}{(1+s)^{\nu }} \Big) '+\frac{\nu u''(s)}{(1+s)^{1+\nu }}ds \\ &= M\Big(\frac{u''(t_{+})}{(1+t_{+})^{\nu }}-\frac{ u''(t_{\ast })}{(1+t_{\ast })^{\nu }}+\int_{t_{\ast }}^{t_{+}} \frac{\nu u''(s)}{(1+s)^{1+\nu }}ds\Big) \\ &\leq M\Big(M_{1}+\sup_{0\leq t<+\infty }\frac{\Gamma _{2}(t)}{1+t} \int_0^{+\infty }\frac{\nu }{(1+s)^{\nu }}ds\Big) \\ &< \int_{r}^{R}\frac{s}{h(s)}ds. \end{align*} So $u'''(t_{+})-R$ for all $t\in [ 0,+\infty) $ and, therefore $\| u'''\| 0$, there exists a $t_{\epsilon }>0$ such that \[ \big| \frac{u^{(i)}(t)}{1+t^{3-i}}-\underset{t\to +\infty }{ \lim }\frac{u^{(i)}(t)}{1+t^{3-i}}\big| <\epsilon ,\text{ for all } t>t_{\epsilon }, x\in M\text{ and }i=0,1,2,3. \] \end{enumerate} \end{lemma} The functions considered as lower and upper solutions for the initial problem are defined as it follows. \begin{definition} \label{sub} \rm Given $a<0$ and $A,B,C,D\in\mathbb{R}$, a function $\alpha \in C^{4}[0,+\infty )\cap X$ is said to be a lower solution of problem \eqref{edo},\eqref{cf} if \[ \alpha ^{(4)}(t)\geq f(t,\overline{\alpha }(t),\alpha '(t),\alpha ''(t),\alpha '''(t)),\quad t\in [ 0,+\infty ), \] and \begin{equation} \alpha '(0)\leq B,\quad \alpha ''(0)+a\text{ } \alpha '''(0)\leq C,\quad \alpha '''(+\infty )0$ such that \begin{gather*} \overline{\alpha }(t) \leq u(t)\leq \overline{\beta }(t),\alpha '(t)\leq u'(t)\leq \beta '(t), \\ \alpha ''(t) \leq u''(t)\leq \beta ''(t),-R\overline{\beta }(t) \\ x, & \overline{\alpha }(t)\leq x\leq \overline{\beta }(t) \\ \overline{\alpha }(t),& x <\overline{\alpha }(t), \end{cases} \\ \delta _{i}(t,y_{i}) =\begin{cases} \beta ^{(i)}(t), & y_{i}>\beta ^{(i)}(t) \\ y_{i}, & \alpha ^{(i)}(t)\leq y_{i}\leq \beta ^{(i)}(t) \\ \alpha ^{(i)}(t), & y_{i}<\alpha ^{(i)}(t), \end{cases}\qquad i=1,2, \\ \delta _{3}(t,w) = \begin{cases} R, & w>R \\ w, & -R\leq w\leq R \\ -R, & w<-R. \end{cases} \end{gather*} For clearness, we do the proof in several steps. \smallskip \noindent\textbf{Step 1:} Every solution of \eqref{edo-m}, \eqref{cf} satisfies $\alpha ''(t)\leq u''(t)\leq \beta ''(t)$ for all $t\in [ 0,+\infty )$. Let $u$ be a solution of the modified problem \eqref{edo-m}, \eqref{cf} and suppose, by contradiction, that there exists $t\in (0,+\infty )$ such that $\alpha ''(t)>u''(t)$. Therefore \[ \inf_{0\leq t<+\infty }(u''(t)-\alpha ''(t))<0. \] By \eqref{BCsub} this infimum can not be attained at $+\infty $. If \[ \inf_{0\leq t<+\infty }(u''(t)-\alpha ''(t)):=u''(0^{+})-\alpha ''(0^{+})<0, \] then the following contradiction is achieved \begin{align*} 0 &\leq u'''(0^{+})-\alpha '''(0^{+}) =\frac{C-u''(0)}{a}+\frac{\alpha ''(0)-C}{a} \\ &= -\frac{1}{a}(u''(0)-\alpha ''(0)) <0. \end{align*} If there is $t_{\ast }\in (0,+\infty )$ then, we can define \[ \min_{0\leq t<+\infty }(u''(t)-\alpha ''(t)):=u''(t_{\ast })-\alpha ''(t_{\ast })<0, \] with $u'''(t_{\ast })=\alpha '''(t_{\ast })$ and $u^{(4)}(t_{\ast })-\alpha ^{(4)}(t_{\ast })\geq 0$. Therefore by \eqref{desi} and Definition \ref{sub} we get the contradiction \begin{align*} 0 &\leq u^{(4)}(t_{\ast })-\alpha ^{(4)}(t_{\ast }) \\ &= f(t_{\ast },\delta _0(t_{\ast },u(t_{\ast })),\delta _{1}(t_{\ast },u'(t_{\ast })),\delta _{2}(t_{\ast },u''(t_{\ast })),\delta _{3}(t_{\ast },u'''(t_{\ast }))) \\ &\quad +\frac{1}{1+t_{\ast }^{2}}\frac{u''(t_{\ast })-\delta _{2}(t_{\ast },u''(t_{\ast }))}{1+| u''(t_{\ast })-\delta _{2}(t_{\ast },u''(t_{\ast }))| }-\alpha ^{(4)}(t_{\ast }) \\ &= f(t_{\ast },\delta _0(t_{\ast },u(t_{\ast })),\delta _{1}(t_{\ast },u'(t_{\ast })),\alpha ''(t_{\ast }),\alpha '''(t_{\ast })) \\ &\quad +\frac{1}{1+t_{\ast }^{2}}\frac{u''(t_{\ast })-\alpha ''(t_{\ast })}{1+| u''(t_{\ast })-\alpha ''(t_{\ast })| }-\alpha ^{(4)}(t_{\ast }) \\ &\leq \frac{1}{1+t_{\ast }^{2}}\frac{u''(t_{\ast })-\alpha ''(t_{\ast })}{1+| u''(t_{\ast })-\alpha ''(t_{\ast })| }<0. \end{align*} So $u''(t)\geq \alpha ''(t),\forall t\in [ 0,+\infty )$. Analogously it can be shown that $u''(t)\leq \beta ''(t),\forall t\in [ 0,+\infty )$. As $\alpha '(0)\leq B\leq \beta '(0)$ and $u'(0)=B $, integrating on $[0,+\infty )$, \begin{gather*} \begin{aligned} \alpha '(t)-\alpha '(0) &= \int_0^{t}\alpha ''(s)ds\leq \int_0^{t}u''(s)ds=u'(t)-B \\ &\leq \int_0^{t}\beta ''(s)ds=\beta '(t)-\beta'(0), \end{aligned}\\ \alpha '(t)\leq \alpha '(t)-\alpha '(0)+B\leq u'(t)\leq \beta '(t)-\beta '(0)+B\leq \beta '(t), \\ \alpha (t)-\alpha (0)=\int_0^{t}\alpha '(s)ds\leq \int_0^{t}u'(s)ds=u(t)-A\leq \int_0^{t}\beta '(s)ds=\beta (t)-\beta (0), \\ \overline{\alpha }(t)\leq u(t)\leq \overline{\beta }(t). \end{gather*} \smallskip \noindent\textbf{Step 2:} Problem \eqref{edo-m}, \eqref{cf} has at least one solution. Let us define the operator $T:X\to X$ by \[ Tu(t)=A+Bt+\frac{C-aD}{2}t^{2}+\frac{D}{6}t^3-\int_0^{+\infty }G(t,s)F(u(s))ds, \] with \begin{align*} F(u(s)) &:=f(s,\delta _0(s,u),\delta _{1}(s,u'),\delta _{2}(s,u''),\delta _{3}(s,u''')) \\ &\quad +\frac{1}{1+s^{2}}\frac{u''(s)-\delta _{2}(s,u'')}{1+| u''(s) -\delta _{2}(s,u'')| }. \end{align*} By Lemma \ref{lemma-equi}, the fixed points of $T$ are solutions of problem \eqref{edo-m}, \eqref{cf}. So it is sufficient to prove that $T$ has a fixed point. (i) $T:X\to X$ is well defined. Let $u\in X$. As $f$ is an $L^{1}$-Carath\'eodory function, so, for \[ \rho >\max \{ {\| u\| _{X},\| \alpha \| _{X},\| \beta \| _{X}}\} , \] we obtain \begin{align*} \int_0^{+\infty }| F(u(s))| ds &\leq \int_0^{+\infty}\phi _{\rho }(s)+\frac{1}{1+s^{2}}\frac{u''(s)-\delta _{2}(s,u'')}{1+| u''(s)-\delta _{2}(s,u'')| }ds \\ &\leq \int_0^{+\infty }\phi _{\rho }(s)+\frac{1}{1+s^{2}}ds<+\infty , \end{align*} that is, $F$ is also an $L^{1}$-Carath\'eodory function. By the Lebesgue Dominated Convergence Theorem, \begin{align*} \lim_{t\to +\infty }\frac{(Tu)(t)}{1+t^3} &= \frac{D}{6} -\int_0^{+\infty }\lim_{t\to +\infty }\frac{G(t,s)}{1+t^3} F(u(s))ds \\ &= \frac{D}{6}-\frac{1}{6}\int_0^{+\infty }F(u(s))ds<+\infty , \end{align*} and analogously for \[ \lim_{t\to +\infty }\frac{(Tu)'(t)}{1+t^{2}},\text{ } \lim_{t\to +\infty }\frac{(Tu)''(t)}{1+t}\text{ and } \lim_{t\to +\infty }\frac{(Tu)'''(t)}{2}. \] Therefore $Tu\in X$. (ii) $T$ is continuous. For any convergent sequence $u_{n}\to u$ in $X$, there exists $\rho >0$ such that $\sup_{n}\| u_{n}\| _{X}<\rho $, we have \begin{align*} \| Tu_{n}-Tu\| _{X} &= \max \Big\{\| Tu_{n}-Tu\| ,\| (Tu_{n})'-(Tu)'\| , \\ &\quad \| (Tu_{n})''-(Tu)''\|,\| (Tu_{n})'''-(Tu)'''\|\Big\} \\ &\leq \int_0^{+\infty }M| F(u_{n}(s))-F(u(s))| ds\to 0,\\quad n\to +\infty , \end{align*} where \begin{align*} M&=\max \Big\{\sup_{0\leq t<+\infty }| \frac{G(t,s)}{1+t^3}|,\; \sup_{0\leq t<+\infty }| \frac{1}{1+t^{2}}\frac{\partial G(t,s)}{ \partial t}| , \\ &\quad \sup_{0\leq t<+\infty }| \frac{1}{1+t}\frac{\partial ^{2}G(t,s)}{ \partial t^{2}}| ,\; 1 \Big\}. \end{align*} (iii) $T$ is compact. Let $B\subset X$ be any bounded subset, therefore there is $R>0$ such that $ \| u \|_X< R$ for all $u\in B$. For each $u\in B$, one has \begin{align*} \| Tu\| &= \sup_{0\leq t<+\infty }\frac{| Tu(t)| }{1+t^3}\leq |A|+|B|+|C-aD|+|D| \\ &\quad +\int_0^{+\infty }\sup_{0\leq t<+\infty }\frac{| G(t,s)| }{1+t^3}| F(u(s))| ds \\ &\leq |A|+|B|+|C-aD|+|D|+\int_0^{+\infty }M(\phi _{R}(s)+\frac{1}{ 1+s^{2}}) <+\infty . \end{align*} By the same arguments it can be proved that $\| (Tu) ^{(i)}\| <+\infty $, for $i=1,2,3$, and, therefore, \[ \| Tu\| _{X}=\max \left\{ \| Tu\| ,\| (Tu)'\| ,\| (Tu)''\| ,\| (Tu)'''\| \right\} <+\infty , \] that is, $TB$ is uniformly bounded. $TB$ is equicontinuous, because, for $L>0$ and $t_{1},t_{2}\in [ 0,L]$, we have \begin{align*} \big| \frac{Tu(t_{1})}{1+t_{1}^3}-\frac{Tu(t_{2})}{1+t_{2}^3}\big| &\leq \big| \frac{A+Bt_{1}+\frac{C-aD}{2}t_{1}^{2}+\frac{D }{6}t_{1}^3}{1+t_{1}^3}-\frac{A+Bt_{2}+\frac{C-aD}{2}t_{2}^{2}+\frac{D}{6 }t_{2}^3}{1+t_{2}^3}\big| \\ &\quad +\int_0^{+\infty }\big| \frac{G(t_{1},s)}{1+t_{1}^3}-\frac{ G(t_{2},s)}{1+t_{2}^3}\big| | F(u(s))| ds \\ &\leq \big| \frac{A+Bt_{1}+\frac{C-aD}{2}t_{1}^{2}+\frac{D}{6}t_{1}^3 }{1+t_{1}^3}-\frac{A+Bt_{2}+\frac{C-aD}{2}t_{2}^{2}+\frac{D}{6}t_{2}^3}{ 1+t_{2}^3}\big| \\ &\quad +\int_0^{+\infty }\big| \frac{G(t_{1},s)}{1+t_{1}^3}-\frac{ G(t_{2},s)}{1+t_{2}^3}\big| \big(\phi _{R}(s)+\frac{1}{1+s^{2}} \big) ds\to 0, \end{align*} as $\ t_{1}\to t_{2}$. By the same technique one shows that \[ \big| \frac{(Tu) ^{(i)}(t_{1})}{1+t_{1}^{3-i}}-\frac{( Tu) ^{(i)}(t_{2})}{1+t_{2}^{3-i}}\big| \to 0, \] uniformly, for $i=1,2,3$, as $t_{1}\to t_{2}$. Moreover $TB$ is equiconvergent at infinity, because \begin{align*} &\big| \frac{Tu(t)}{1+t^3}-\lim_{t\to +\infty }\frac{Tu(t)}{1+t^3}\big| \\ &\leq | \frac{A+Bt+\frac{C-aD}{2}t^{2}+\frac{D}{6}t^3}{1+t^3}-\frac{D}{6}| +\int_0^{+\infty }| \frac{G(t,s)}{1+t^3}-\frac{1}{6}| | F(u(s))| ds \\ &\leq \big| \frac{A+Bt+\frac{C-aD}{2}t^{2}+\frac{D}{6}t^3}{1+t^3} -\frac{D}{6}\big| +\int_0^{+\infty }\big| \frac{G(t,s)}{1+t^3}-\frac{1}{6} \big| \big(\phi _{\rho _{1}}+\frac{1}{1+s^{2}}\big) ds\to 0, \end{align*} as $t\to +\infty $, and analogously for \[ \big| \frac{(Tu) ^{(i)}(t)}{1+t^{3-i}}-\lim_{t\to +\infty }\frac{(Tu) ^{(i)}(t)}{1+t^{3-i}} \big| \to 0,\quad \text{as } t\to +\infty . \] So, by Lemma \ref{chin}, the set $TB$ is relatively compact. As $T$ is completely continuous then by Schauder Fixed Point Theorem, $T$ has at least one fixed point $u\in X$. \end{proof} \section{Example} Consider the fourth-order differential equation \begin{equation} (1+t^{2})u^{(4)}(t)=-u(t)| u'''(t)-6| e^{u'''(t)}-e^{-t}(6t+2-u''(t)),\quad t>0, \label{ex1} \end{equation} with the boundary conditions \begin{equation} u(0)=A,\ u'(0)=0,\quad u''(0)+au'''(0)=0,\quad u'''(+\infty )=D, \label{ex1-cf} \end{equation} where $A\geq 0$, $-\frac{1}{3}\leq a<0$ and $00$, such that \begin{gather*} A\leq u(t)\leq t^3+t^{2}+A,\quad 0\leq u'(t)\leq 3t^{2}+2t,\\ 0\leq u''(t)\leq 6t+2, \quad \| u'''\| _0\leq R,\quad \forall t\in [ 0,+\infty ). \end{gather*} We remark that, this solution is unbounded and, from the location part, we notice that $u$ is nondecreasing and convex. It is important to stress that the nonlinearity \eqref{f} does not satisfy the usual two-sided Nagumo-type condition. In fact, if there exist $\psi _0,h_0\in C(\mathbb{R}_0^{+},\mathbb{R} ^{+})$ satisfying \[ | f(t,x,y,z,w)| \leq \psi _0(t)h_0(|w| ),\quad \forall (t,x,y,z,w)\in E_0, \] with $\int_0^{+\infty }\frac{s}{h_0(s)}ds=+\infty $, then, in particular, \[ -f(t,x,y,z,w)\leq \psi _0(t)h_0(| w| ), \] and, for $t\in [ 0,+\infty )$, $x=1$, $0\leq y\leq 3t^{2}+2t$, $z=6t+2$, and $w\in \mathbb{R}$, \[ -f(t,1,y,6t+2,w)=\frac{| w-6| e^{w}}{1+t^{2}}\leq \psi _0(t)h_0(| w| ), \] For $\psi _0(t)=1/(1+t^{2})$ we have $| w-6| e^{w}\leq h_0(| w| )$ and the following contradiction holds \[ +\infty >\int_0^{+\infty }\frac{s}{(s-6)e^{s}}ds\geq \int_0^{+\infty } \frac{s}{h_0(s)}ds=+\infty . \] \subsection*{Acknowledgments} This work was supported by National Founds through FCT-Funda\c{c}\~{a}o para a Ci\^{e}ncia e a Tecnologia as part of project : SFRH/BSAB/114246/2016. The authors are grateful to the anonymous referee for his/her comments and suggestions. \begin{thebibliography}{99} \bibitem{agarwal1} R. P. 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