\documentclass[twoside]{article} \usepackage{amsfonts,amsmath} \pagestyle{myheadings} \markboth{\hfil Minimal and maximal solutions \hfil EJDE--2003/21} {EJDE--2003/21\hfil Myron K. Grammatikopoulos \& Petio S. Kelevedjiev \hfil} \begin{document} \title{\vspace{-1in}% \parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2003}(2003), No. 21, pp. 1--14. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Minimal and maximal solutions for two-point boundary-value problems % \thanks{\emph{Mathematics Subject Classifications:} 34B15. \hfil\break \indent {\em Key words:} Boundary-value problems, minimal and maximal solutions, \hfil\break\indent monotone method, barrier strips. \hfil\break\indent \copyright 2003 Southwest Texas State University. \hfil\break\indent Submitted December 12, 2002. Published February 28, 2003.} } \date{} \author{Myron K. Grammatikopoulos \& Petio S. Kelevedjiev} \maketitle \begin{abstract} In this article we consider a boundary-value problem for the equation ${f(t,x,x',x'')=0}$ with mixed boundary conditions. Assuming the existence of suitable barrier strips, and using the monotone iterative method, we obtain the minimal and maximal solutions. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \newtheorem{lemma}[theorem]{Lemma} \numberwithin{equation}{section} \section{Introduction} We apply the monotone iterative method to obtain minimal and maximal solutions to the nonlinear boundary-value problem (BVP) $$\begin{gathered} f(t,x,x',x'')=0,\quad 0\le a\le t\le b,\\ x(a)=A,\quad x'(b)=B, \end{gathered} \tag{1.1}$$ where the scalar function $f(t,x,p,q)$ is continuous and has continuous first derivatives on suitable subsets of $[a,b]\times \mathbb{R}^{3}$. For results, which guarantee the existence of $C^{2}[a,b]$-solutions to BVPs for the equation $x''=f(t,x,x',x'')-y(t)$ with various linear boundary conditions, see \cite{f1,f2,m1,m2,p1,p2,p3}. Concerning the uniqueness results, we refer to \cite{p1}. A result, concerning the existence and uniqueness of $C^{2}[a,b]$-solutions to the BVP for the equation $x''=f(t,x,x',x'')$, with general linear boundary conditions, can be found in \cite{t2}. The results of \cite{m3} guarantee the existence of $W^{2,\infty }[a,b]$-solutions or of $C^{2}[a,b]$-solutions to the Dirichlet BVP for the equation $% f(t,x,x',x'')=0$, where the function $f(t,x,p,q)$is defined on $[a,b]\times \mathbb{R}^{n}\times \mathbb{R}^{n}\times Y$, and $Y$ is a non-empty closed connected or locally connected subset of $\mathbb{R}^{n}$. Finally, the $C^{2}[a,b]$-solvability of BVPs for the equation ${% f(t,x,x',x'')=0}$ with fully nonlinear boundary conditions is studied in \cite{k2}. Note that, in the literature, the monotone iterative method is applied on BVPs for equations of the forms $x''=f(t,x,x')$ and $% (\phi (x'))'=f(t,x,x')$ with various boundary conditions (see, for example, \cite{c1,c2,c3,d1,h1,j1,k1,k3,l2,m4,t1,w1}). The sequences of iterates, considered in \cite{c1,c2,c3,d1,j1,k3,w1}, converge to the extremal solutions, while the sequences of iterates, considered in\cite{h1,l2,m3}, converge to the unique solution. The first elements $u_{0}(t)$ and $v_{0}(t)$ of such sequences of iterates usually are lower and upper solutions respectively of the problems under consideration (see, for example, \cite{c1,c2,c3,d1,j1,k3,w1}. To derive the needed monotone iterates, the authors of \cite{c1,c2,c3,d1,j1,k3,l2,w1} use suitable growth conditions. For more applications of the monotone iterative method, see cite{b1,l1,l3,s1,y1}. In this article, following cite{k1}, we obtain the extremal solutions to (1.1) under assumption of the existence of suitable barrier strips (see Remarks 2.1 and 2.2 below), which immediately imply the first iterates $% u_{0}(t)$ and $v_{0}(t)$. A version of \cite[Theorem 5.1]{k2} implies the existence of the next iterates, and a suitable comparison result guarantees the monotone properties for the sequences of iterates. Finally, the Arzela-Askoli's theorem ensures the existence of the extremal solutions of the problem (1.1) as limits of the sequences of iterates. \section{Basic hypotheses} The following four hypotheses will be a tool for obtaining our results. \begin{itemize} \item[(H1)] There are constants $K>0$, $F,F_{1},L,L_{1}$ such that $Fa\leq A\leq La,\quad F_{1}G_{1}^{+}\geq 2C,\quad G_{2}^{-}>G_{1}^{-}\geq 2C, \\ H_{2}^{+}0 is fixed and such that $$H_{1}^{+}>H_{2}^{+}+\varepsilon, \quad H_{1}^{-}>H_{2}^{-}+\varepsilon,\quad G_{2}^{+}>G_{1}^{+}+\varepsilon,\quad G_{2}^{-}>G_{1}^{-}+\varepsilon. \tag{2.1}$$ f_{t}(t,x,p,q), f_{x}(t,x,p,q) and f_{p}(t,x,p,q) are continuous for % (t,x,p,q) in [a,b]\times[m_1,M_1]\times[F,L]\times[m_2,M_2]; \[ f_{t}(t,x,p,q)+\ f_{x}(t,x,p,q)p+ f_{p}(t,x,p,q)q\geq 0$ for $(t,x,p,q)$ in $[a,b]\times[m_1,M_1]\times[F,L]\times\big([ H_{2}^{+},H_{1}^{+}]\cup [ G_{1}^{+},G_{2}^{+}]\big)$, and $f_{t}(t,x,p,q)+\ f_{x}(t,x,p,q)p+\ f_{p}(t,x,p,q)q\leq 0$ for $(t,x,p,q)$ in $[a,b]\times[m_1,M_1]\times[F,L]\times\bigl([ H_{2}^{-},H_{1}^{-}]\cup [ G_{1}^{-},G_{2}^{-}]\bigr)$, where $F$ and $L$ are the constants of {H1}. \end{itemize} \begin{remark} \label{rmk2.2} \rm Set $\Phi _{2}(t,x,p,q)\equiv f_{t}(t,x,p,q)+f_{x}(t,x,p,q)p+f_{p}(t,x,p,q)q$. Then, the pair of strips $\Omega _{1}=[a,b]\times ([H_{2}^{+},H_{1}^{+}]\cup [G_{1}^{+},G_{2}^{+}])$, where $\Phi _{2}(t,x,p,q)\geq 0$, and the pair of strips $\Omega _{2}=[a,b]\times ([H_{2}^{-},H_{1}^{-}]\cup [ G_{1}^{-},G_{2}^{-}])$, where $\Phi _{2}(t,x,p,q)\leq 0$, are such that the graph of the function $x''(t),t\in [ a,b]$, can not cross the outer strips, of the four such ones, defined by $\Omega _{1}$ and $\Omega _{2}$. For this reason the outer strips of $\Omega _{1}$ and $\Omega _{2}$ are called barrier strips for $x''(t),t\in [ a,b]$. \end{remark} \begin{itemize} \item[(H3)] For $m_{3}=\min \{H_{1}^{+},H_{1}^{-}\}$ and $M_{3}=\max \{G_{1}^{+},G_{1}^{-}\}$ $h(\lambda ,t,x,p,m_{3}-\varepsilon)h(\lambda ,t,x,p,M_{3}+\varepsilon) \leq 0$ for $(\lambda,t,x,p)$ in $[0,1]\times[a,b]\times[ m_1-\varepsilon,M_1+% \varepsilon]\times[F-\varepsilon,L+\varepsilon]$, where $h(\lambda ,t,x,p,q)=(\lambda -1)Kq+\lambda f(t,x,p,q)$, $F,L,K$ are the constants of {% H1}, and $H_{1}^{+}$, $H_{1}^{-}$, $G_{1}^{+}$, $G_{1}^{-}$, $C$, $m_1$, $M_1$, and $\varepsilon$ are as in {H2}. \item[(H4)] For $(t,x,p,q)$ in $T\times[F,L]\times[\min% \{H_{1}^{+},H_{1}^{-}\},\max \{G_{1}^{+},G_{1}^{-}\}]$, $f_{x}(t,x,p,q)\ge 0$% , where the trapezoid $T$ and the constants $F$ and $L$ are as in H1, and $% H_{1}^{+}$, $H_{1}^{-}$, $G_{1}^{+}$ and $G_{1}^{-}$ are the constants in {H2% }, and $m_3$ and $M_3$ are as in {H3}. \end{itemize} \section{Main result} For a function $y(t)\in C[a,b]$ bounded on $[a,b]$, we define a mapping $\mathcal{A}y=x,$ where $x(t)\in C^{2}[a,b]$ is a solution to the BVP $$\begin{gathered} f(t,y(t),x',x'')=0,\quad t\in[a,b],\\ x(a)=A,\quad x'(b)=B. \end{gathered} \tag{3.1}$$ We will show that under the hypotheses H1, H2, and H3, the map $\mathcal{A}$ is uniquely determined. For this reason, we consider two sequences $\{u_{n}\}$ and $\{v_{n}\}$, $n=0,1,\dots$, defined by $u_{n+1}=\mathcal{A}u_{n}\quad \mbox{and}\quad v_{n+1}=\mathcal{A}v_{n},$ where $u_{0}=Ft$, $v_{0}=Lt$, $t\in [ a,b]$, and $F$ and $L$ are as in H1. Now we formulate our main result. \begin{theorem} \label{thm3.1} Under hypotheses H1--H4, there are sequences $\{u_{n}\}$ and $\{v_{n}\}$, $n=0,1,\dots$, such that for $n\to +\infty$: $u_{n}\to u^{m}$, $v_{n}\to v^{M}$ and $u_{0}\leq u_{1}\leq \dots\leq u_{n}\leq \dots\leq u^{m}\leq x\leq v^{M} \leq \dots \leq v_{n}\leq \dots\leq v_{1}\leq v_{0},$ where $u^{m}(t)$ and $v^{M}(t)$ are the minimal and maximal solutions of the BVP (1.1) respectively, and $x(t)\in C^{2}[a,b]$ is a solution of (1.1). \end{theorem} The proof of this theorem can be found at the end of this article and is based on the auxiliary results, which we present in the next section. \section{ Auxiliary results} %sec. 4 We begin this section with an existence result, which is a modification of \cite[Theorem 6.1, Chapter II]{g1}. Namely, we consider the family of BVPs $$\begin{gathered} Kx''=\lambda\big(Kx''+f(t,y(t),x',x'')\big),\quad t\in[a,b],\\ x(a)=A,\quad x'(b)=B, \end{gathered} \label{4.1l}$$ where $\lambda \in [ 0,1]$ and $K>0$. \begin{lemma} \label{lm4.1} Assume that there are constants $Q_{i}$, $i=0,1,\dots,5$, independent of $\lambda$ such that \begin{itemize} \item[(i)] For each solution $x(t)\in C^{2}[a,b]$ of \eqref{4.1l} it holds $Q_{0}Kx''(t_{0})=\lambda \left[ Kx^{\prime% \prime}(t_{0})+f(t_{0},x(t_{0}),x'(t_{0}),x''(t_{0}))% \right] \geq 0.$ This contradiction proves the assertion. \hfill$\diamondsuit$ \begin{lemma} \label{lm4.3} Let H1 hold, and $x(t)\in C^{2}[a,b]$ be a solution to \eqref{4.1l} with $y(t)\in V$. Then $Ft\leq x(t)\leq Lt,\quad F\leq x'(t)\leq L\quad \mbox{for}\quad t\in [ a,b].$ \end{lemma} \paragraph{Proof} Consider the sets $Y_{0}=\left\{ t\in [ a,b]:Lx'(\tau _{0})\quad \mbox{and}\quad x'(t_{1})x''(\tau _{0}).\tag{4.8}$ Since (4.6) holds for $t\in [ t_{0},\tau _{0}]$ and $\begin{gathered} G_{1}^{+}0 is as in H2. Thus, the condition (i) of Lemma \ref {lm4.1} holds for Q_{0}=m_1-\varepsilon , Q_{1}=M_1+\varepsilon , % Q_{2}=F-\varepsilon, Q_{3}=L+\varepsilon , Q_{4}=m_3-\varepsilon  and % Q_{5}=M_3+\varepsilon. Moreover, from (2.1) and H3 it follows that the conditions (ii) and (iii) of Lemma \ref{lm4.1} are satisfied. Also, \[ m_1-\varepsilon 0. \end{lemma} \begin{lemma} \label{lm4.8} Suppose that \phi \in C^{2}(a,b)\cap C^{1}[a,b] satisfies the inequality \[ \phi ''(t)+g(t)\phi '(t)\geq 0\quad \mbox{for }t\in (a,b),$ where $g(t)$ is bounded on $(a,b)$. If $\phi (a)\leq 0$ and $\phi '(b)\leq 0,\tag{4.12}$ then $\phi (t)\leq 0\quad\mbox{for }t\in [ a,b].\tag{4.13}$ \end{lemma} \paragraph{Proof} First, assume that $\phi (t)$achieves its maximum at $t_{0}\in (a,b)$. By Lemma \ref{lm4.6}, for $t\in[a,b]$ we obtain $\phi (t)\equiv \phi (t_{0})=\phi (a)\le 0$ and so (4.13) holds. Next, suppose that $\phi (t)$ achieves its maximum at the ends of the interval $[a,b]$. If we assume $\phi (t)\leq \phi(b)$, $t\in [ a,b]$, the application of Lemma \ref{lm4.7} shows that $\phi '(b)>0$, which contradicts (4.12). Thus, by our assumtions, $\phi (t)\leq \phi (a)\leq 0$, $% t\in [ a,b]$, and so (4.13) follows. The proof is complete. \hfill$% \diamondsuit$ In the last two lemmas we use the map $\mathcal{A}$ defined in the section 3. \begin{lemma} \label{lm4.9} Under assumptions H1, H2, and H3, for any $y\in V_{1}$, the image $x$ by the map $\mathcal{A}$ exists and it is unique. \end{lemma} \paragraph{Proof} The existence of the image of $x$ follows from Lemma \ref{lm4.5}. In order to see that $x$ is unique, fix $y$ and assume that$z$is an other image of $y$ by $\mathcal{A}$ and consider the function $\phi (t)=x(t)-z(t)$, $t\in [ a,b]$. Then, it is evident that $f\bigl(t,y(t),x'(t),x''(t)\bigr)-f\bigl(% t,y(t),z'(t),z''(t)\bigr)=0,\quad t\in [ a,b].$ Next, we construct the equality \begin{align*} f\bigl(t,y(t),x'(t),x''(t)\bigr)-f\bigl(% t,y(t),z'(t),x''(t)\bigr)& \\ +f\bigl(t,y(t),z'(t),x''(t)\bigr)-f\bigl(% t,y(t),z'(t),z''(t)\bigr)&=0, \end{align*} which can be rewritten in the form $I_{1}(t)\phi '(t)+\ I_{2}(t)\phi ''(t)=0$, where \begin{gather*} I_{1}(t) =\int_{0}^{1} f_{p}\bigl(t,y(t),z'(t)+\theta (x'(t)-z'(t)),x''(t)\bigr)d\theta , \\ I_{2}(t) =\int_{0}^{1} f_{q}\bigl(t,y(t),z'(t),z''(t) +\theta (x''(t)-z''(t))\bigr)d\theta . \end{gather*} Hence, the function $\phi (t)$ is a solution to the BVP \begin{gather*} \phi ''(t)+ \frac{I_{1}(t)}{I_{2}(t)}\phi '(t)=0,\quad t\in [ a,b], \\ \phi (a)=0,\quad \phi '(b)=0. \end{gather*} Moreover, it is easy to conclude that $\phi (t)\equiv 0$, $t\in [ a,b]$, is the unique solution of the above BVP. Consequently, $x(t)\equiv z(t)$, $t\in [ a,b]$. The proof of the lemma is complete. \hfill $\diamondsuit$ \begin{lemma} \label{lm4.10} Under the hypotheses H1--H4, if $y_{1}(t),y_{2}(t)\in V_{1}$ are such that ${y_{1}(t)\leq y_{2}(t)}$ for $t\in [ a,b]$, then $x_{1}(t)\leq x_{2}(t)\;\;\mbox{for}\quad t\in [ a,b],$ where $x_{i}=\mathcal{A}y_{i},i=1,2$. \end{lemma} \paragraph{Proof} Observe that, by Lemma \ref{lm4.3}, we have $F\leq x_{1}'(t)\leq L$, $t\in [ a,b]$, and, by Lemma \ref{lm4.4}, $m_3\leq {x_1}''(t)\,\leq M_3,\quad t\in [ a,b].$ Moreover, $Ft\leq y_{1}(t)\leq y_{2}(t)\leq Lt,\quad t\in [ a,b].$ Thus, from $f_{x}(t,x,p,q)\geq 0$ for $(t,x,p,q)$ in $T\times [ F,L]\times[% m_3,M_3]$ it follows that $0=f\bigl(t,y_{1}(t),x_{1}'(t),x_{1}''(t)\bigr)\leq f% \bigl(t,y_{2}(t),x_{1}'(t),x_{1}''(t)\bigr),\quad t\in [ a,b].$ Hence, for $t\in [ a,b]$ we have $f\bigl(t,y_{2}(t),x_{2}'(t),x_{2}''(t)\bigr) -f\bigl(t,y_{2}(t),x_{1}'(t),x_{1}''(t)\bigr)\leq 0$ and then, as in Lemma \ref{lm4.9}, we construct the inequality \begin{align*} f\bigl(t,y_{2}(t),x_{1}'(t),x_{1}''(t)\bigr) -f\bigl(% t,y_{2}(t),x_{2}'(t),x_{1}''(t)\bigr)& \\ +f\bigl(t,y_{2}(t),x_{2}'(t),x_{1}''(t)\bigr) -f\bigl(% t,y_{2}(t),x_{2}'(t),x_{2}''(t)\bigr)&\geq 0 \end{align*} from which for $\phi (t)=x_{1}(t)-x_{2}(t)$, $t\in [ a,b]$, we find $\phi ''(t)+\frac{\ J_{1}(t)}{\ J_{2}(t)}\phi '(t)\geq 0, \quad t\in [ a,b],$ where \begin{gather*} J_{1}(t) =\int_{0}^{1}f_{p}\bigl(t,y_{2}(t),x_{2}'(t) +\theta (x_{1}'(t)-x_{2}'(t)),x_{1}''(t)\bigr)d\theta , \\ \ J_{2}(t) =\int_{0}^{1}f_{q}\bigl(t,y_{2}(t),x_{2}'(t),x_{2}^{% \prime\prime}(t) +\theta (x_{1}''(t)-x_{2}''(t))% \bigr)d\theta . \end{gather*} Furthermore, $\phi (a)=0$, $\phi '(b)=0$. Finally, applying Lemma \ref{lm4.8}, we see that $\phi (t)\leq 0$ for $t\in [ a,b]$, which completes the proof. \hfill $\diamondsuit$ \section{Proof of Theorem \ref{thm3.1}} %sec. 5 Consider the sequences $\{u_{n}\,\}$ and $\{v_{n}\,\}$, defined by $u_{n+1}=\mathcal{A}u_{n}\quad \mbox{and}\quad v_{n+1}=\mathcal{A} v_{n},\quad n=0,1,\dots$ In view of Lemma \ref{lm4.5}, from Lemma \ref{lm4.3} it follows that $% Ft=u_{0}\leq u_{1}$ and $v_{1}\leq v_{0}=Lt$. Moreover, Lemma \ref{lm4.10} and induction arguments imply that $u_{n-1}\leq u_{n},\quad v_{n}\leq v_{n-1},\quad n=1,2,\dots$ On the other hand, since $u_{0}\leq v_{0}$, by Lemma \ref{lm4.10} and induction arguments, we conclude that $u_{n}\leq v_{n}$, $n=0,1,\dots$. >From this observation it follows that $u_{0}\leq u_{n}\leq v_{0},\;\;n=0,1,\dots.$ Therefore, $\{u_{n}\}$ is uniformly bounded. Furthermore, since, by Lemma \ref{lm4.3}, $\{u_{n}'\;\}$ is uniformly bounded, we see that $% \{u_{n}\;\}$ is equicontinuous. Finally, since, by Lemma \ref{lm4.4}, $% \{u_{n}^{\prime\prime\prime}\}$ is uniformly bounded, it follows that the sequence $\{u_{n}''\}$ is uniformly bounded and equicontinuous. Thus, we can apply the Arzela-Ascoli theorem to conclude that there are a subsequence $\{u_{n_{i}}\}$ and a function $u\in C^{2}[a,b]$ such that $% \{u_{n_{i}}\}$, $\{u_{n_{i}}'\}$ and $\{u_{n_{i}}''\}$ are uniformly convergent on $[a,b]$ to $u,u'$ and $u''$ respectively. Now, using the fact that $u_{n_{i}}=\mathcal{A}u_{n_{i-1}}$can be rewritten equivalently in the form \begin{align*} u_{n_{i}}(t)&=\frac{1}{K}\int_{a}^{t}\Big( \int_{b}^{r}\bigl(% Ku_{n_{i}}''(s)+f(s,u_{n_{i-1}}(s),u_{n_{i}}'(s), u_{n_{i}}''(s))\bigr)ds\Big) dr \\ &\quad+B(t-a)+A, \end{align*} letting $i\to +\infty$, we obtain $\ u(t)=\frac{1}{K}\int_{a}^{t}\Big( \int_{b}^{r}\bigl(Ku''(s)+f(s,u(s),u'(s),u^{\prime% \prime}(s))\bigr)ds\Big) dr +B(t-a)+A,$ from which it follows that $u(t)$ is a solution to the BVP (1.1). Remark that, if $x(t)$ is a solution of (1.1), then, by Lemma \ref{lm4.3}, we have $u_{0}(t)\leq x(t)$ for $t\in [ a,b]$. Applying Lemma \ref{lm4.10} (it is possible, because $x=\mathcal{A}x$), by induction we obtain $u_{n}(t)\leq x(t),\quad t\in [ a,b],\quad n=0,1,\dots,$ and then $u(t)\leq x(t)$, $t\in [ a,b]$, which holds for each solution $% x(t)\in C^{2}[a,b]$ of the problem (1.1). Consequently, it follows that $u(t)\equiv u^{m}(t),\quad t\in [ a,b].$ By similar arguments, we conclude that $\lim v_{n}=v^{M}(t)$, $t\in [ a,b]$. Thus, the proof is complete. \hfill$\diamondsuit$ \paragraph{Acknowledgement} P. S. Kelevedjiev would like to thank the Ministry of National Economy of Helenic Republic for providing the NATO Science Fellowship (No. DOO 850/02) and the University of Ioannina for its hospitality. \begin{thebibliography}{99} \bibitem{b1} S. Bernfeld, V. 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Kelevedjiev} \newline Department of Mathematics, Technical University of Sliven \newline 8800 Sliven, Bulgaria \newline e-mail: keleved@mailcity.com \end{document}