\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Generalized quasilinearization method \hfil EJDE--2002/90} {EJDE--2002/90\hfil Bashir Ahmad, Rahmat Ali Khan, \& Paul W. Eloe \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2002}(2002), No. 90, pp. 1--12. \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} \\ % Generalized quasilinearization method for a second order three point boundary-value \\ problem with nonlinear boundary conditions % \thanks{ {\em Mathematics Subject Classifications:} 34B10, 34B15. \hfil\break\indent {\em Key words:} Generalized quasilinearization, boundary value problem, \hfil\break\indent nonlinear boundary conditions, quadratic and rapid convergence. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Submitted March 14, 2002. Published October 22, 2002.} } \date{} % \author{Bashir Ahmad, Rahmat Ali Khan, \& Paul W. Eloe} \maketitle \begin{abstract} The generalized quasilinearization technique is applied to obtain a monotone sequence of iterates converging uniformly and quadratically to a solution of three point boundary value problem for second order differential equations with nonlinear boundary conditions. Also, we improve the convergence of the sequence of iterates by establishing a convergence of order $k$. \end{abstract} \newtheorem{theorem}{Theorem}[section] \numberwithin{equation}{section} \section{Introduction} The method of quasilinearization pioneered by Bellman and Kalaba \cite{b1} and generalized by Lakshmikantham \cite{l1,l2} has been applied to a variety of problems \cite{c1, l3,l4,l5,l6,n1}. Multipoint boundary value problems for second order differential equations have also been receiving considerable attention recently. Kiguradze and Lomtatidze \cite{k1} and Lomtatidze \cite{l7,l8} have studied closely related problems. Gupta et.al. \cite{c2,c3,c4} have studied problems related to three point boundary value problems. More recently, Paul Eloe and Yang Gao \cite{e1} discussed the method of quasilinearization for a three point boundary value problem. In this paper, we develop the method of generalized quasilinearization for a three point boundary value problem involving nonlinear boundary conditions and obtain a monotone sequence of approximate solutions converging uniformly and quadratically to a solution of the problem. Also, we have discussed the convergence of order $k$. \subsection*{Basic Results} Consider the three point boundary value problem with nonlinear boundary conditions $$\begin{gathered} x'' =f(t,x(t)) ,\quad t\in [0,1]=J \\ x(0) =a,\quad x(1) =g(x(\frac{1}{2})), \end{gathered}\label{a}$$ where $f\in C[J\times \mathbb{R},\mathbb{R}]$ and $g:\mathbb{R}\to \mathbb{R}$ is continuous. Let $G(t,s)$ denote the Green's function for the conjugate or Dirichlet boundary value problem and is given by  G(t,s) =\begin{cases} t(s-1),& 0\leq t0$on$[0,1] \times \mathbb{R}$and$g$is continuous with$0\leq g'<1$on$\mathbb{R}$. Let$\beta $and$\alpha $be the upper and lower solutions of the BVP \eqref{a} respectively. Then$\alpha (t) \leq \beta (t)$,$t\in [0,1]$. \end{theorem} \begin{theorem}[Method of upper and lower solutions] \label{thm2} Assume that$f$is continuous on$[0,1] \times \mathbb{R}$and$g$is continuous on$\mathbb{R}$satisfying$0\leq g'<1$. Further, we assume that there exists an upper solution$\beta $and a lower solution$\alpha $of the BVP \eqref{a} such that$\alpha (t) \leq \beta (t)$,$t\in [0,1]$. Then there exists a solution$x$of the BVP \eqref{a} such that $\alpha (t) \leq x\leq \beta (t) ,\quad t\in [0,1] .$ \end{theorem} \section{Main Result} \begin{theorem}[Generalized quasilinearization method] \label{thm3}\quad \begin{enumerate} \item[$(A_1)$]$f,f_x$are continuous on$[0,1]\times \mathbb{R}$and$f_{xx}$exists on$[0,1]\times \mathbb{R}$. Further,$f_x>0$and$f_{xx}+\phi _{xx}\leq 0$, where$\phi ,\phi _x$are continuous on$[0,1]\times \mathbb{R}$and$\phi _{xx}\leq 0$. \item[$(A_2)$]$g$,$g'$are continuous on$\mathbb{R}$and$g''$exists and$0\leq g'<1$,$g''(x) \geq 0$,$x\in \mathbb{R}$. \item[$(A_3) $]$\alpha $and$\beta $are lower and upper solutions of the BVP \eqref{a} respectively. \end{enumerate} Then there exists a monotone sequence$\left\{ w_n\right\} $of solutions converging quadratically to the unique solution$x$of the BVP \eqref{a}. \end{theorem} \paragraph{Proof.} Define$F:[0,1] \times \mathbb{R}\to \mathbb{R}$as $F(t,x) =f(t,x) +\phi (t,x) .$ Then, in view of$(A_1) $, we note that$F$,$F_x$are continuous on$[0,1] \times \mathbb{R}$, and$F_{xx}$exists such that $$F_{xx}(t,x) \leq 0. \label{2.1}$$ Using the mean value theorem and the assumptions$(A_1) $and$(A_2) $, we obtain \begin{gather} f(t,x) \leq F(t,y) +F_x(t,y) (x-y) -\phi (t,x) , \label{2.2} \\ g(x) \geq g(y) +g'(y) (x-y) , \label{2.3} \end{gather} where$x,y\in \mathbb{R}$such that$x\geq y$and$t\in [0,1]$. Here, we remark that \eqref{2.2} and \eqref{2.3} are also valid independent of the requirement$x\geq y$. Define the functions$\stackrel *F (t,x,y) $and$h(x,y) as \begin{gather*} \stackrel *F(t,x,y) =F(t,y) +F_x(t,y) (x-y) -\phi (t,x) , \\ h(x,y) =g(y) +g'(y) (x-y) . \end{gather*} We observe that $$f(t,x) =\min_yF^* (t,x,y) . \label{2.4}$$ Further \begin{aligned} \stackrel *F_x(t,x,y) =& F_x(t,y) -\phi_x(t,x) \geq F_x(t,x) -\phi _x(t,x) \\ =&f_x(t,x) >0, \end{aligned}\label{2.4a} implies that\stackrel *F (t,x,y) $is increasing in$x$for each fixed$(t,y) \in [0,1]\times \mathbb{R}$. Similarly \begin{gather} g(x) =\max_y h(x,y), \label{2.4b} \\ 0 \leq h'(x,y) <1. \label{2.4c} \end{gather} Now, set$\alpha =w_0$, and consider the three point BVP $$\begin{gathered} x'' =\stackrel *F (t,x(t) ,w_0(t)) ,\quad t\in [0,1]=J \\ x(0) =a,\quad x(1) =h(x(\frac12),w_0(\frac 12)) . \end{gathered} \label{2.6}$$ Using$(A_3) $together with \eqref{2.4} and \eqref{2.4b}, we have \begin{gather*} w_0'' \geq f(t,w_0) =\stackrel *F (t,w_0,w_0) ,\quad t\in [0,1] \\ w_0(0) \leq a,\quad w_0(1) \leq g(w_0(\frac 12)) =h(w_0(\frac 12), w_0(\frac 12)) , \end{gather*} and \begin{gather*} \beta '' \leq f(t,\beta) \leq \stackrel *F (t,\beta ,w_0) ,\quad t\in [0,1] \\ \beta (0) \geq a,\quad \beta (1) \geq g(\beta (\frac 12)) \geq h(\beta (\frac 12),w_0(\frac 12)) , \end{gather*} which imply that$w_0$and$\beta $are lower and upper solutions of the BVP \eqref{2.6} respectively. In view of \eqref{2.4a} \eqref{2.4c} and the fact that$w_0$and$\beta $are lower and upper solutions of the BVP \eqref{2.6} respectively, it follows by Theorems \ref{thm1} and \ref{thm2} that there exists a unique solution$w_1$of the BVP \eqref{2.6} such that $w_0(t) \leq w_1(t) \leq \beta (t) ,\quad t\in [0,1].$ Now, consider the BVP $$\begin{gathered} x'' =\stackrel *F (t,x(t),w_1(t)) ,\quad t\in [0,1]=J \\ x(0) =a,~~x(1) =h(x(\frac12),w_1(\frac 12)) . \end{gathered} \label{2.8}$$ Again, using$(A_3) $, \eqref{2.4} and \eqref{2.4b}, we find that$w_1$and$\beta $are lower and upper solutions of \eqref{2.8} respectively, that is, \begin{gather*} w_1'' =\stackrel *F (t,w_1,w_0) \geq \stackrel *F (t,w_1,w_1) ,\quad t\in [0,1] \\ w_1(0) =a,\quad w_1(1) =h(w_1(\frac12),w_0(\frac 12)) \leq h(w_1(\frac 12),w_1(\frac 12)) , \end{gather*} and \begin{gather*} \beta '' \leq f(t,\beta) \leq \stackrel *F (t,\beta ,w_1) ,\quad t\in [0,1] \\ \beta (0) \geq a,\quad \beta (1) \geq g(\beta (\frac 12)) \geq h(\beta (\frac 12),w_1(\frac 12)) . \end{gather*} Hence, by Theorems \ref{thm1} and \ref{thm2}, there exists a unique solution$w_2$of \eqref{2.8} such that $w_1(t) \leq w_2(t) \leq \beta (t) ,\quad t\in [0,1].$ Continuing this process successively, we obtain a monotone sequence$\{w_n\} $of solutions satisfying $w_0(t) \leq w_1(t) \leq \dots \leq w_n(t) \leq \beta (t) ,\quad t\in [0,1],$ where each element$w_n$of the sequence is a solution of the BVP \begin{gather*} x'' =\stackrel *F (t,x(t) ,w_{n-1}(t)) ,\quad t\in [0,1]=J \\ x(0) =a,\quad x(1) =h(x(\frac12),w_{n-1}(\frac 12)) , \end{gather*} and $$w_n(t) =a(1-t) +h(w_n(\frac 12),w_{n-1}(\frac 12)) t+\int_0^1G(t,s) \stackrel *F (s,w_n,w_{n-1}) ds. \label{2.9}$$ Employing the fact that$[0,1]$is compact and the monotone convergence is pointwise, it follows that the convergence of the sequence is uniform. If$x(t) $is the limit point of the sequence, then passing onto the limit$n\to \infty $, \eqref{2.9} gives \begin{eqnarray*} x(t) &=&a(1-t) +h(x(\frac 12),x(\frac12)) t+\int_0^1G(t,s) \stackrel *F (s,x(s) ,x(s)) ds \\ &=&a(1-t) +g(x(\frac 12)) t+\int_0^1G(t,s) f(s,x(s)) ds. \end{eqnarray*} Thus,$x(t) $is the solution of the BVP \eqref{a}. Now, we show that the convergence of the sequence is quadratic. For that, set $e_n(t) =x(t) -w_n(t) ,\quad t\in [0,1].$ Observe that \begin{gather*} e_n(t) \geq 0,\quad e_n(0) =0, \\ e_n(1) =g(x(\frac 12)) -h(w_n(\frac 12),w_{n-1}(\frac 12)) . \end{gather*} Using the mean value theorem repeatedly,$(A_1) $and the nonincreasing property of$F_x$, we have \begin{eqnarray*} e_{n+1}''(t) &=&x''(t) -w_{n+1}''(t) \\ &=&f(t,x) -[F(t,w_n) +F_x(t,w_n) (w_{n+1}-w_n) -\phi (t,w_{n+1}) ] \\ &=&F_x(t,c_1) (x-w_n) -F_x(t,w_n) (x-w_n)+F_x(t,w_n) (x-w_{n+1}) \\ &&-\phi _x(t,c_2) (x-w_{n+1}) \\ &=&(F_{xx}(t,c_3) (c_1-w_n)(x-w_n) +(F_x(t,w_n) -\phi _x(t,c_2) )(x-w_{n+1}) \\ &\geq &F_{xx}(t,c_3) (x-w_n) ^2+(F_x(t,c_2) -\phi _x(t,c_2) )(x-w_{n+1}) \\ &=&F_{xx}(t,c_3) (e_n) ^2+f_x(t,c_2) e_{n+1} \\ &\geq &F_{xx}(t,c_2) (e_n) ^2\geq -M\parallel e_n\parallel ^2, \end{eqnarray*} where$M$is a bound on$F_{xx}(t,x) $for$t\in [0,1]$,$w_n\sum_{i=0}^{k-1}\frac{\partial ^i% }{\partial x^i}f(t,y) \frac{(x-y) ^{i-1}}{(i-1)!}% \geq 0, \label{3.4} which implies that $\stackrel{**}{F}(t,x,y)$ is increasing in $x$ for each $(t,y) \in [0,1] \times \mathbb{R}$. Similarly, differentiation of \eqref{3.2}, in view of $(B_3)$, yields $\stackrel{*}{h}{}'(x,y) =\sum_{i=0}^{k-1}\frac{d^i}{% dx^i}g(y)\frac{(x-y) ^{i-1}}{(i-1)!}.$ Clearly $\stackrel *h{} '(x,y) \geq 0$ and \begin{eqnarray*} \stackrel *h{}'(x,y) &=&\sum_{i=0}^{k-1}\frac{d^i}{dx^i}g(y) \frac{(x-y) ^{i-1}}{(i-1)!}\leq \sum_{i=0}^{k-1} \frac{d^i}{dx^i}g(y)\frac{(\beta -\alpha) ^{i-1}}{(i-1)!} \\ &\leq &\sum_{i=0}^{k-1}\frac M{(i-1)!}