\documentclass[twoside]{article} \usepackage{amssymb} % used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Quadratic Convergence of Approximate Solutions \hfil}% {\hfil V. Doddaballapur, P. W. Eloe \& Y. Zhang \hfil} \begin{document} \setcounter{page}{81} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Differential Equations and Computational Simulations III}\newline J. Graef, R. Shivaji, B. Soni J. \& Zhu (Editors)\newline Electronic Journal of Differential Equations, Conference~01, 1997, pp. 81--95. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp 147.26.103.110 or 129.120.3.113 (login: ftp)} \vspace{\bigskipamount} \\ Quadratic Convergence of Approximate Solutions of Two-Point Boundary Value Problems with Impulse \thanks{ {\em 1991 Mathematics Subject Classifications:} 34A37, 34B15. \hfil\break\indent {\em Key words and phrases:} Quasilinearization, boundary value problem with impulse, \hfil\break\indent quadratic convergence, Nagumo conditions. \hfil\break\indent \copyright 1998 Southwest Texas State University and University of North Texas. \hfil\break\indent Published November 12, 1998.} } \date{} \author{Vidya Doddaballapur, Paul W. Eloe, \& Yongzhi Zhang} \maketitle \begin{abstract} The method of quasilinearization, coupled with the method of upper and lower solutions, is applied to a boundary value problem for an ordinary differential equation with impulse that has a unique solution. The method generates sequences of approximate solutions which converge monotonically and quadratically to the unique solution. In this work, we allow nonlinear terms with respect to velocity; in particular, Nagumo conditions are employed. \end{abstract} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \section{Introduction} Let $0=t_{0}0,\ (t,x,y)\in [0,1]\times {\mathbb R}^2,& \label{e5}\\ &v_{k}\in C^{1}({\mathbb R}^2),&\label{e6} \end{eqnarray} and for$k=1,\dots ,m$, $$v_{kx}(x,y)>0,\ (x,y)\in {\mathbb R}^2,\quad v_{ky}(x,y)>0,\ (x,y)\in {\mathbb R}^2. \label{e7}$$ In order to obtain Theorem 2, we shall define an appropriate fixed point operator,$T$. For$x\in B$, define an operator$T$on$x$by $$\label{e8} Tx(t)=p(t)+I(t,x)+\int_{0}^{1}G(t,s)f(s,x(s),x'(s))\,ds,$$ where$p(t)=a(1-t)+bt$,$I(t,x)=\sum_{k=1}^{m}I_{k}(t,x)$. For$k=1,\dots ,m$, let $$I_{k}(t,x)=\left\{ \begin{array}{ll} t(-u_{k}-(1-t_{k})v_{k}(x(t_{k}),x'(t_{k})))&,0\le t\le t_{k},\\ (1-t)(u_{k}-t_{k}v_{k}(x(t_{k}),x'(t_{k})))&,t_{k}\le t\le 1\,. \end{array} \right.$$ Let $$G(t,s)=\left\{ \begin{array}{ll} t(s-1)&,0\le t0.$$ This provides a contradiction and so,$\tau \notin \cup_{k=0}^{m}(t_{k},t_{k+1})$. Now, assume that$\tau =t_{k}$for some$k\in \{ 1,\dots ,m\}$. By Taylor's theorem,$w'(t_{k}^{-})\ge 0$and$w'(t_{k}^{+})\le 0$, or$\Delta w'(t_{k})\le 0$and $$\alpha '(t_{k}^{-})=\alpha '(t_{k})\ge \beta '(t_{k})=\beta '(t_{k}^{-}).$$ But $$\Delta w'(t_{k})=\Delta \alpha '(t_{k})-\Delta \beta '(t_{k}) \ge v_{k}(\alpha (t_{k}),\alpha '(t_{k}))-v_{k}(\beta (t_{k}), \beta '(t_{k}))>0$$ by (\ref{e7}). Thus,$\tau \notin \{ t_{1},\dots t_{m}\}$, and$w(t)\le 0$,$0\le t\le 1$. \begin{theorem} \label{t2} Assume$g\in C([0,1]\times {\mathbb R}^2)$,$z_{k}\in C({\mathbb R}^2)$,$k=1,\dots ,m$, and assume that each$z_{k}(x,y)$is monotone increasing in$y$for fixed$x$. Assume that each solution of$x''(t)=g(t,x(t),x'(t))$extends to$[0,1]$, or becomes unbounded on its maximal interval of convergence. Let$\alpha ,\beta $be lower and upper solutions of the BVP, \begin{eqnarray} &x''(t)=g(t,x(t),x'(t)),\quad t_{k}< t< t_{k+1},& \label{e9}\\ &\Delta x(t_{k})=u_{k} & \nonumber \\ &\Delta x'(t_{k})=z_{k}(x(t_{k}),x'(t_{k}))\,,& \label{e10} \end{eqnarray} with$k=1,\dots ,m$and boundary conditions given by (\ref{e2}), respectively, such that $$\alpha \le \beta.$$ Then, there exists a solution,$x$, of the BVP with impulse, (\ref{e9}), (\ref{e2}), (\ref{e10}), satisfying $$\alpha \le x\le \beta .$$ \end{theorem} \paragraph{Proof.} Define $$\hat f (t,x,y)=\left\{ \begin{array}{ll} g(t,\beta (t),y)+(x-\beta (t))/[1+(x-\beta (t))],&x>\beta (t),\\ g(t,x,y),&\alpha (t)\le x\le \beta (t),\\ g(t,\alpha (t),y)+(x-\alpha (t))/[1+|x-\alpha (t)|],&x<\alpha (t), \end{array}\right .$$ and for$k=1,\dots ,m$, define $$\hat v_{k} (x,y)= \left\{ \begin{array}{ll} z_{k}(\beta (t_{k}),y)+(x-\beta (t_{k}))/[1+(x-\beta (t_{k}))],&x>\beta (t_{k}),\\ z_{k}(x,y),&\alpha (t_{k})\le x\le \beta (t_{k}),\\ z_{k}(\alpha (t_{k}),y)+(x-\alpha (t_{k}))/[1+|x-\alpha (t_{k})|],&x<\alpha (t_{k}). \end{array}\right .$$ Let$N>0$be such that$|\alpha '(t)|\le N$,$|\beta '(t)|\le N$,$t\in [t_{k},t_{k+1}]$,$k=0,\dots ,m$. For each positive integer,$l$, define $$f_{l}(t,x,y) =\left\{ \begin{array}{ll} \hat f(t,x,N+l),&y>N+l,\\ \hat f(t,x,y),&|y|\le N+l,\\ \hat f(t,x, -(N+l)),&y<-(N+l), \end{array}\right .$$ and $$v_{kl}(t,x,y) =\left\{ \begin{array}{ll} \hat v_{k}(x,N+l),&y>N+l,\\ \hat v_{k}(x,y),&|y|\le N+l,\\ \hat v_{k}(x, -(N+l)),&y<-(N+l). \end{array}\right .$$ Notice that$f_{l}$and each$v_{kl}$are bounded and continuous. With a standard application of the Schauder fixed point theorem to the operator$T$, defined by (\ref{e8}), one obtains a solution,$x_{l}\in B$, to the BVP with impulse, (\ref{e1}), (\ref{e2}), (\ref{e3}), with$f=f_{l}$and each$v_{k}=v_{kl}$bounded and continuous. We now argue that each solution,$x_{l}$, satisfies$\alpha\le x_{l}\le \beta$. We shall show that$x_{l}\le \beta$. As in the proof of Theorem 1, assume for the sake of contradiction that$x_{l}-\beta$has a positive maximum at$\tau$. As in the proof of Theorem 1,$\tau\in (0,1)$. If$\tau \in \cup_{k=0}^{m}(t_{k},t_{k+1})$, then$x_{l}''(\tau )\le \beta ''(\tau )$, and$|x_{l}'(\tau )|=|\beta '(\tau )| \le N 0, $$which is a contradiction. If \tau =t_{k}, for some k\in\{ 1,\dots ,m\}, then x_{l}'(t_{k}) \ge \beta '(t_{k}). Since each z_{k}(x,y) is monotone increasing in y for fixed x, it follows that each v_{kl}(x,y) is monotone increasing in y for fixed x. Moreover, note that v_{kl}(\beta (t_{k}),\beta '(t_{k}))=$$z_{k}(\beta (t_{k}),\beta '(t_{k}))$. Thus, \begin{eqnarray*} \Delta (x_{l}-\beta )'(t_{k})&\geq& v_{kl}(\beta (t_{k}),x_{l}'(t_{k})) -v_{kl}(\beta (t_{k}),\beta '(t_{k})) \\ &&+ (x_{l}-\beta )(t_{k})/[1+(x_{l}-\beta (t_{k}))]\\ &\ge&(x_{l}-\beta )(t_{k})/[1+(x_{l}-\beta (t_{k}))]>0\, \end{eqnarray*} which is also a contradiction. Therefore,$x_{l}\le \beta$. To show that$\alpha\le x_{l}$we follow a similar procedure. For each$l$there exists$t_{l}\in [0,t_{1}]$such that $$t_{1}|x_{kl}^{'}(t_{l})|=|x_{kl}(t_{1})-a| \le \max \{ |\beta (0)-\alpha (t_{1})|, |\beta (t_{1})-\alpha (0)| \} .$$ Thus, each of the sequences$\{ x_{kl}(t_{l})\}$and$\{ x_{kl}^{'}(t_{l})\}$are bounded. One can now apply the Kamke convergence theorem (see \cite{l11}) for solutions of initial value problems and obtain a subsequence of$\{ x_{kl}\}$which converges to a solution of$x''(t)=\hat f(t,x(t),x'(t))$on a maximal subinterval of$[0,t_{1}]$. Clearly,$\alpha (t)\le x(t)\le \beta (t)$and solutions of$x''(t)=g(t,x(t),x'(t))$extend to all of$[0,1]$or become unbounded; thus,$x''(t)=\hat f(t,x(t),x'(t))$on$[0,t_{1}]$. Now, apply the impulse defined by (\ref{e10}) at$t_{1}$. Apply the Kamke theorem to the subsequence that was extracted in the preceding paragraph. Because of (\ref{e10}) one can employ$t_{1}=t_{l}$for each$l$. Thus, one obtains a further subsequence which converges to a solution,$x$, of$x''(t)=\hat f(t,x(t),x'(t))$on$(0,t_{1})\cup (t_{1},t_{2})$such that$x$satisfies (\ref{e10}) at$t_{1}$. Continue inductively, first applying (\ref{e10}) at each$t_{j}$and then applying the Kamke convergence theorem on that subinterval$(t_{j},t_{j+1})$. Finally, since$\alpha \le x\le \beta$,$\hat f(t,x(t),x'(t))= f(t,x(t),x'(t))$and the proof of Theorem 2 is complete. \paragraph{Remark.} For simplicity, we can assume that$g$satisfies a Nagumo condition in$x'$(\cite{l10}, \cite{l11}). That is, assume that for each$M>0$there exists a positive continuous function,$h_{M}(s)$, defined on$[0,\infty )$such that $$|g(t,x,x')|\le h_{M}(|x'|)$$ for all$(t,x,x')\in [0,1]\times [-M,M]\times {\mathbb R}$and such that $$\int_{0}^{\infty}(s/h_{M}(s))ds=+\infty .$$ The assumption that$g$satisfies a Nagumo condition implies that each solution of the differential equation,$x''(t)=g(t,x(t),x'(t))$, either extends to$[0,1]$or becomes unbounded on its maximal interval of existence (\cite{l10}, \cite{l11}). In our main result, Theorem 4,$g$will represent a modification of$f$. Thus, we shall assume in Theorem 4 that$f$satisfies a Nagumo condition in$x'$. \begin{theorem} \label{t3} Assume that (\ref{e4}), (\ref{e5}), (\ref{e6}), and (\ref{e7}) hold. Then, solutions of the BVP with impulse, (\ref{e1}), (\ref{e2}), (\ref{e3}), are unique. \end{theorem} \paragraph{Proof.} The uniqueness of solutions result follows immediately from Theorem 1 and the observation that solutions are respectively upper and lower solutions. \begin{theorem} \label{t4} Assume that (\ref{e4}), (\ref{e5}), (\ref{e6}), and (\ref{e7}) hold, and assume that $$(\partial ^2/\partial x^2)f\in C([0,1]\times {\mathbb R}^2),v_{k}'' \in C({\mathbb R}^2), k=1,\dots ,m\,.$$ Assume that$f$satisfies a Nagumo condition in$x'$. Assume that$\alpha _{0}$and$\beta_{0}$are lower and upper solutions of the BVP with impulse, (\ref{e1}), (\ref{e2}), (\ref{e3}), respectively. Then there exist monotone sequences,$\{ \alpha_{n}(t)\}$and$\{ \beta_{n}(t)\}$, which converge in$B$to the unique solution,$x(t)$, of the BVP with impulse, (\ref{e1}), (\ref{e2}), (\ref{e3}), and the convergence is quadratic. \end{theorem} \paragraph{Proof.} Let$F(t,x): [0,1]\times {\mathbb R}\rightarrow {\mathbb R}$be such that$F, F_{x}, F_{xx}$are continuous on$[0,1]\times {\mathbb R}$and $$\label{e11} F_{xx}(t,x)\ge 0, (t,x)\in [0,1]\times {\mathbb R} \,.$$ Set$\phi_{1} (t,x_{1},x_{2}) =F(t,x_{1})-f(t,x_{1},x_{2})$on$[0,1]\times {\mathbb R}^2$. From (\ref{e11}) it follows that, if$x_{1},y_{1}\in {\mathbb R}$, then$F(t,x_{1})\ge F(t,y_{1})+F_{x}(t,y_{1})(x_{1}-y_{1})$. In particular, for$x_{1},y_{1},x_{2},y_{2}\in {\mathbb R}$, $$\label{e12} f(t,x_{1},x_{2})\ge f(t,y_{1},y_{2})+ F_{x}(t,y_{1})(x_{1}-y_{1}) -\phi_{1} (t,x_{1},x_{2})+\phi_{1} (t,y_{1},y_{2}).$$ For each$k=1,\dots ,m$, let$V_{k}(x): {\mathbb R}\rightarrow {\mathbb R}$be such that$V_{k}, V_{k}', V_{k}''$are continuous on${\mathbb R}$and $$\label{e13} V_{k}''(x)\ge 0, \quad x\in {\mathbb R} \,.$$ Set$\phi_{2k} (x_{1},x_{2}) =V_{k}(x_{1})-v_{k}(x_{1},x_{2})$on${\mathbb R}^2$. From (\ref{e13}) it follows that, if$x_{1},y_{1}\in {\mathbb R}$, then$V_{k}(x_{1})\ge V_{k}(y_{1})+V_{k}'(y_{1})(x_{1}-y_{1})$. In particular, for$x_{1},y_{1},x_{2},y_{2}\in {\mathbb R}$, $$\label{e14} v_{k}(x_{1},x_{2})\geq v_{k}(y_{1},y_{2})+V_{k}'(y_{1})(x_{1}-y_{1}) -(\phi_{2k} (x_{1},x_{2})-\phi_{2k} (y_{1},y_{2})).$$ Define \begin{eqnarray*} g(t,x_{1},x_{2};\alpha_{0},\beta_{0},\alpha '_{0})&=& f(t,\alpha_{0}(t),\alpha '_{0}(t))+ F_{x}(t,\beta_{0}(t))(x_{1}-\alpha_{0}(t))\\ &&-\phi_{1}(t,x_{1},x_{2})+\phi_{1}(t, \alpha_{0}(t),\alpha '_{0}(t))\,, \\ G(t,x_{1},x_{2};\beta_{0},\beta '_{0})&=& f(t,\beta_{0}(t),\beta '_{0}(t))+ F_{x}(t,\beta_{0}(t))(x_{1}-\beta_{0}(t)) \\ &&-\phi_{1}(t,x_{1},x_{2})+\phi_{1}(t, \beta_{0}(t),\beta '_{0}(t))\,, \\ h_{k}(x_{1},x_{2};\alpha_{0},\beta_{0},\alpha '_{0})&=& v_{k}(\alpha_{0}(t_{k}),\alpha '_{0}(t_{k}))+ V'_{k}(\beta_{0}(t_{k}))(x_{1}-\alpha_{0}(t_{k}))\\ &&-(\phi_{2k}(x_{1},x_{2})-\phi_{2k}(\alpha_{0}(t_{k}),\alpha '_{0}(t_{k}))) \,, \\ H_{k}(x_{1},x_{2};\beta_{0},\beta '_{0})&=& v_{k}(\beta_{0}(t_{k}),\beta '_{0}(t_{k}))+ V'_{k}(\beta_{0}(t_{k}))(x_{1}-\beta_{0}(t_{k}))\\ &&-(\phi_{2k}(x_{1},x_{2})-\phi_{2k}(\beta_{0}(t_{k}),\beta '_{0}(t_{k})))\,. \end{eqnarray*} First consider the BVP with impulse, \label{e15} x''(t)=g(t,x(t),x'(t);\alpha_{0},\beta_{0},\alpha_{0}'),\ t_{k}0$, such that $$\label{e19} q''_{n+1}(t)-f_{x'}(t,c_{2}(t),c_{3}(t))q'_{n+1}(t)\ge -Me_{n}^2,$$ where $M>\max_{i}\max_{(t,x)\in D_{i}}F_{xx}(t,x)$, and for $i=0,\dots m$, $$D_{i}=\{ (t,x): t_{i}\le t\le t_{i+1}, \alpha_{0}(t)\le x\le \beta_{0}(t)\}\,.$$ Similarly, there exist appropriate $c_{4}$ and $c_{5}$ such that for $k=1,\dots ,m$, $$\label{e20} \Delta q_{n+1}'(t_{k})-v_{ky}(c_{4},c_{5})q'_{n+1}(t_{k})\ge -Me_{n}^2\,.$$ Let $m(t)=\exp\bigg(-\int_{0}^{t}f_{x'}(s,c_{2}(s),c_{3}(s))ds\bigg)$ denote the integrating factor associated with (\ref{e19}). Then $$\label{e21} (q'_{n+1}(t)m(t))'\ge -Mm(t)e_{n}^2\,.$$ Thus, for $t_{m}\le t\le 1$, $$q'_{n+1}(1)m(1)-q'_{n+1}(t)m(t)\ge -Me_{n}^2\int_{t}^{1}m(s)ds\,.$$ Since, $q'_{n+1}(1)\le 0$, it follows that $$q'_{n+1}(t)\le Me_{n}^2\int_{t}^{1}m(s)ds/m(t)\,.$$ Since $q_{n+1}$ converges to $0$ in $B$, eventually $(s,c_{2}(s),c_{3}(s))$ belongs to $$\hat D=\{ (s,x_{1},x_{2}):t_{m}\le s\le 1, \alpha_{0}(s)\le x_{1}\le \beta_{0}(s), x'(s)-1\le x_{2}\le x'(s)+1\}.$$ Thus, we can bound $m(t)$ away from both $0$ and $\infty$ for $n$ sufficiently large; in particular, there exists $N_{1}>0$ such that for $t_{m}\le t\le 1$ and $n$ sufficiently large, $$\label{e22} q'_{n+1}(t)\le N_{1}e_{n}^2\,.$$ Apply (\ref{e20}) at $t_{m}$. Then $$q'_{n+1}(t_{m}^{+})-q'_{n+1}(t_{m}) -v_{my}(c_{4},c_{5})q'_{n+1}(t_{m})\ge -Me_{n}^2\,.$$ Employ (\ref{e7}) and also bound $v_{my}$ away from both $0$ and $\infty$ to obtain some $\hat M >0$ such that $$\label{e23} q'_{n+1}(t_{m}^{-})\ge -\hat M e_{n}^2\,.$$ Now, employ (\ref{e21}) and (\ref{e23}) to obtain (\ref{e22}) for $t_{m-1} \le t\le t_{m}$ for some $N_{2}>0$. Again, apply (\ref{e20}) to obtain a suitable (\ref{e23}) at $t_{m-1}$. Proceed inductively and obtain that there exists $N>0$ such that for $t \in \cup_{k=0}^{m}[t_{k},t_{k+1}]$ and $n$ sufficiently large, $$\label{e24} q'_{n+1}(t)\le Ne_{n}^2.$$ Recall that $q_{n+1}(t)\ge 0$, and that $q_{n+1}\in C[0,1]$. Integrate (\ref{e24}) from $0$ to $t$; then for $n$ sufficiently large, $$\label{e25} 0\le q_{n+1}\le Ne_{n}^2\,.$$ Beginning again at (\ref{e21}), integrate from $0$ to $t\le t_{1}$ to obtain $$q'_{n+1}(t)m(t)-q'_{n+1}(0)\ge -Me_{n}^2\int_{0}^{t}m(s)\,ds\,.$$ Since, $q'_{n+1}(0)\ge 0$, it follows that for $0\le t\le t_{1}$, there exists $N_{1}>0$, such that $$q'_{n+1}(t)\ge -Me_{n}^2\int_{0}^{t}m(s)\,ds/m(t)\ge -N_{1}e_{n}^2\,,$$ for $n$ sufficiently large. This is analogous to (\ref{e22}). Proceed analogously, then, and choose $N$ large enough such that for $t \in \cup_{k=0}^{m}[t_{k},t_{k+1}]$ for $n$ sufficiently large, $$\label{e26} q'_{n+1}(t)\ge -Ne_{n}^2\,.$$ It now follows from (\ref{e24}), (\ref{e25}), and (\ref{e26}) that $\beta_{n}$ converges to $x$ quadratically in $B$. The argument that $\{ \alpha _{n}\}$ converges quadratically to $x$ in $B$ is similar and we provide some details. \begin{eqnarray*} \lefteqn{ p''_{n+1}(t)} \\ &=& F(t,x(t))-\phi_{1}(t,x(t),x'(t))\\ &&-(F(t,\alpha_{n}(t))+F_{x}(t,\beta_{n}(t))(\alpha_{n+1}-\alpha_{n})(t) -\phi_{1}(t, \alpha_{n+1}(t),\alpha '_{n+1}(t))) \\ &=& F_{x}(t,c_{1}(t))p_{n}(t)-F_{x}(t,\beta_{n}(t))p_{n}(t) +F_{x}(t,\beta_{n}(t))p_{n+1}(t) \\ &&-\phi_{1x}(t,c_{2}(t),c_{3}(t))p_{n+1}(t) -\phi_{1x'}(t,c_{2}(t),c_{3}(t))p'_{n+1}(t)\\ &=& F_{xx}(t,c_{4}(t))p_{n}(t)(c_{1}(t)-\beta_{n}(t))\\ &&+(F_{x}(t,\beta_{n}(t))-\phi_{1x}(t,c_{2}(t),c_{3}(t)))p_{n+1}(t) -\phi_{1x'}(t,c_{2}(t),c_{3}(t))p'_{n+1}(t) \\ &\ge& -F_{xx}(t,c_{4}(t))p_{n}(t)(p_{n}(t)+q_{n}(t))+ f_{x'}(t,c_{2}(t),c_{3}(t))p_{n+1}'(t)\,. \end{eqnarray*} In particular, $$p''_{n+1}(t)-f_{x'}(t,c_{2}(t),c_{3}(t))p '_{n+1}(t)\ge -2Me_{n}^2$$ on an appropriate set and for sufficiently large $n$. A similar inequality is obtained with respect to the impulse and the details for quadratic convergence follow as above. \begin{corollary} \label{t5} The sequence $\{ \beta ''_{n}(t)-f(t,\beta_{n}(t), \beta '_{n}(t))\}$ converges quadratically to $0$ in $B$. \end{corollary} \paragraph{Proof:} There exist $\beta_{n}\ge c_{2}\ge c_{1}\ge \beta_{n+1}$ such that \begin{eqnarray*} f(t,\beta_{n+1}(t),\beta_{n+1}'(t))&\ge& \beta_{n+1}''(t)\\ &=&f(t,\beta_{n}(t),\beta_{n}'(t))+F_{x}(t,\beta_{n}(t))(\beta_{n+1}(t) -\beta_{n}(t))\\ &&-(\phi_{1}(t,\beta_{n+1}(t),\beta_{n+1}'(t))-\phi_{1}(t,\beta_{n}(t),\beta_{n}'(t))) \\ &=&f(t,\beta_{n+1}(t),\beta_{n+1}'(t))\\ &&+F_{xx}(t,c_{2}(t))(\beta_{n+1}(t) -\beta_{n}(t))(\beta_{n}(t)-c_{1}(t))\,. \end{eqnarray*} Thus, \begin{eqnarray*} 0&\le& f(t,\beta_{n+1}(t),\beta_{n+1}'(t))-\beta_{n+1}''(t)\\ &\le& F_{xx}(t,c_{2}(t))(\beta_{n+1}(t)-\beta_{n}(t))^2\\ &\le & F_{xx}(t,c_{2}(t))e_{n}^2\,. \end{eqnarray*} Similar inequalities are obtained for the impulse. 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