\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011(2011), No. 21, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/21\hfil A priori estimates for solutions] {A priori estimates for solutions to a four point boundary value problem for singularly perturbed semilinear differential equations} \author[R. Vrabel\hfil EJDE-2011/21\hfilneg] {Robert Vrabel} % in alphabetical order \address{Robert Vrabel \newline Institute of Applied Informatics, Automation and Mathematics\\ Faculty of Materials Science and Technology, Hajdoczyho 1, 917 01 Trnava, Slovakia} \email{robert.vrabel@stuba.sk, epsilon.phi1@gmail.com} \thanks{Submitted January 5, 2010. Published February 7, 2011.} \subjclass[2000]{34K10, 34K26} \keywords{Singular perturbation; four point boundary value problem; \hfill\break\indent lower and upper solutions} \begin{abstract} This article concerns the existence and asymptotic behavior of solutions to a singularly perturbed second-order four-point boundary-value problem for nonlinear differential equations. Our analysis relies on the method of lower and upper solutions. We give accurate approximations of the solutions up to order $O(\epsilon)$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \section{Preliminaries} We consider the four (or three) point boundary value problem \begin{gather}\label{def_DE} \epsilon y''+ky=f(t,y),\quad t\in [a,b], \; k<0,\; 0<\epsilon\ll 1,\\ \label{def_BC} y(c)-y(a)=0,\quad y(b)-y(d)=0,\quad a0$,$0<\epsilon$such that$\tilde v_\epsilon(c)-\tilde v_\epsilon(a)=u(c)-u(a)>0$and$\tilde v_\epsilon(t)\to0^+$for$t\in (a,b]$and$\epsilon\to 0^+$, which could be used to solve this problem by the method of lower and upper solutions. Instead we compose barrier functions ($\alpha$,$\beta$) for two-endpoint boundary conditions to construct barrier functions for \eqref{def_DE}, \eqref{def_BC}, see e.g. \cite{ChHo}. In recent years multi-point boundary value problems have received a great deal of attention (see e.g. \cite{GG}, \cite{Kha} and the references therein). The reader is referred to \cite{Kha} where a four-point boundary value problem with boundary conditions$y(c)-\nu_1 y(a)=0, y(b)-\nu_2 y(d)=0$where the constants$\nu_1$,$\nu_2$are not simultaneously equal to$1$and$\epsilon=1$is studied. We apply the method of lower and upper solutions to prove the existence of a solution for problem \eqref{def_DE}, \eqref{def_BC} and by taking$\epsilon\to 0^+$, the corresponding solutions converge uniformly on compact subsets of$(a,b)$to$u$, the solution of the reduced problem. Moreover, we prove that these solutions converge to$u$up to order$O(\epsilon)$. As usual, we say that$\alpha_\epsilon \in C^2([a,b])$is a lower solution for problem \eqref{def_DE}, \eqref{def_BC} if for every$t\in(a,b)$we have$\epsilon \alpha''_\epsilon(t)+k\alpha_\epsilon (t) \geq f(t,\alpha_\epsilon (t) )$, and$\alpha_\epsilon (c)-\alpha_\epsilon (a)= 0$,$\alpha_\epsilon (b)-\alpha_\epsilon (d)\leq 0$. An upper solution$\beta_\epsilon \in C^2([ a,b])$satisfies$\epsilon \beta''_\epsilon(t)+k\beta_\epsilon (t) \leq f(t,\beta_\epsilon (t) )$and$\beta_\epsilon (c)-\beta_\epsilon (a)= 0$,$\beta_\epsilon (b)-\beta_\epsilon (d)\geq 0$for every$t\in(a,b)$. \begin{lemma}[\cite{Ma}]\label{mainlemma} If$\alpha_\epsilon ,\beta_\epsilon $are respectively lower and upper solutions for \eqref{def_DE}, \eqref{def_BC} such that$\alpha_\epsilon \leq \beta_\epsilon, $then there exists solution$y_\epsilon$of \eqref{def_DE}, \eqref{def_BC} with$\alpha_\epsilon\leq y_\epsilon\leq\beta_\epsilon$. \end{lemma} Consider the set$\mathcal{H}(u)=\{ (t,y): a\leq t\leq b, \vert y-u(t)\vert 0 with the boundary conditions \eqref{def_BC}. We apply the method of lower and upper solutions in order to obtain a solution. We define $$\alpha_\epsilon(t)=0$$ and $$\beta_\epsilon(t)=\frac{2w}{m}\max\{| v_\epsilon(t)|, t\in[ a,b]\} =\frac{2w}{m}| v_\epsilon(a)|.$$ Obviously, $| v_\epsilon(a)|=\vert u(c)-u(a)\vert \big(1+{O}\big(e^{\sqrt{\frac{m}{\epsilon}}(a-c)}\big)\big)$ and the constant functions $\alpha, \beta$ are, respectively, a lower and an upper solution for \eqref{def_LDE}, \eqref{def_BC}. Thus, in view of Lemma \ref{mainlemma}, for every $\epsilon>0$ there exists unique solution $v^{\rm (corr)}_{\epsilon}(t)$ of linear problem \eqref{def_LDE}, \eqref{def_BC} such that $$0\leq v^{\rm (corr)}_{\epsilon}(t)\leq \frac{2w}{m}| u(c)-u(a)| \Big(1+{O}\big(e^{\sqrt{\frac{m}{\epsilon}}(a-c)}\big)\Big)$$ on $[ a,b]$. We compute $v^{\rm (corr)}_{\epsilon}(t)$ exactly: $$v^{\rm (corr)}_{\epsilon}(t)=-\frac{\left(\psi_\epsilon (a) -\psi_\epsilon(c)\right)}{(u(c)-u(a))}v_\epsilon(t) +\frac{\left(\psi_\epsilon (d)-\psi_\epsilon(b)\right)}{\vert u(b) -u(d)\vert}\hat v_\epsilon(t)+\psi_\epsilon(t),$$ where $$\psi_\epsilon(t)=\frac{w\vert u(c)-u(a)\vert }{D\sqrt{m\epsilon}}t \left(e^{\sqrt{\frac m{\epsilon}}(b-t)}+e^{\sqrt{\frac m{\epsilon}} (t-b)}-e^{\sqrt{\frac m{\epsilon}}(d-t)}-e^{\sqrt{\frac m{\epsilon}} (t-d)}\right).$$ Hence \begin{align*} &\psi_\epsilon(a)-\psi_\epsilon(c)\\ &= \frac{w\vert u(c)-u(a)\vert }{D\sqrt{m\epsilon}}a \left(e^{\sqrt{\frac m{\epsilon}}(b-a)} +e^{\sqrt{\frac m{\epsilon}}(a-b)} -e^{\sqrt{\frac m{\epsilon}}(d-a)} -e^{\sqrt{\frac m{\epsilon}}(a-d)}\right)\\ &\quad - \frac{w\vert u(c)-u(a)\vert }{D\sqrt{m\epsilon}}c \left(e^{\sqrt{\frac m{\epsilon}}(b-c)} +e^{\sqrt{\frac m{\epsilon}}(c-b)} -e^{\sqrt{\frac m{\epsilon}}(d-c)} -e^{\sqrt{\frac m{\epsilon}}(c-d)}\right)\\ &= \frac{w\vert u(c)-u(a)\vert }{\sqrt{m\epsilon}}{O}(1), \end{align*} \begin{align*} \psi_\epsilon(d)-\psi_\epsilon(b) &=\frac{w\vert u(c)-u(a)\vert }{D\sqrt{m\epsilon}}d \left(e^{\sqrt{\frac m{\epsilon}}(b-d)} +e^{\sqrt{\frac m{\epsilon}}(d-b)}-2\right)\\ &\quad - \frac{w\vert u(c)-u(a)\vert }{D\sqrt{m\epsilon}}b \left(2-e^{\sqrt{\frac m{\epsilon}}(d-b)} -e^{\sqrt{\frac m{\epsilon}}(b-d)}\right)\\ &= \frac{w\vert u(c)-u(a)\vert }{\sqrt{m\epsilon}}{O} \left(e^{\sqrt{\frac{m}{\epsilon}}(a-d)}\right), \end{align*} $\psi_\epsilon(t)=\frac{w\vert u(c)-u(a)\vert }{\sqrt{m\epsilon}} {O}\left(e^{\sqrt{\frac{m}{\epsilon}}\chi (t)}\right).$ Thus, we obtain \label{corrfun} \begin{aligned} v^{\rm (corr)}_{\epsilon}(t) &= \frac{w\vert u(c)-u(a)\vert}{\sqrt{m\epsilon}}\cdot \Big[-{ O}(1)\frac{v_\epsilon(t)}{(u(c)-u(a))}\\ &\quad + O\left(e^{\sqrt{\frac{m}{\epsilon}}(a-d)}\right) \frac{\hat v_\epsilon(t)}{\vert u(b)-u(d)\vert}+t{ O} \left(e^{\sqrt{\frac{m}{\epsilon}}\chi (t)}\right)\Big]. \end{aligned} Hence, taking into consideration \eqref{corrfun} and the fact that $v^{\rm (corr)}_{\epsilon}(a)=v^{\rm (corr)}_{\epsilon}(c)$, the correction function $v^{\rm (corr)}_{\epsilon}$ converges uniformly to $0$ on $[ a,b]$ as $\epsilon\to 0^+$. \section{Proof of main theorem} First we analyze the case ${u(c)-u(a)\geq 0}$. Consider the lower solutions $$\alpha_\epsilon (t)=u(t)+v_\epsilon(t)-v^{\rm (corr)}_{\epsilon}(t) -\hat v_\epsilon(t)-\Gamma_\epsilon$$ and the upper solutions $$\beta_\epsilon (t)=u(t)+v_\epsilon(t) +\hat v_\epsilon(t)+\Gamma_\epsilon.$$ Here $\Gamma_\epsilon =\epsilon\Delta /m$, where $\Delta$ is a constant to be defined below, $\alpha_\epsilon\leq\beta_\epsilon$ on $[ a,b]$ and they satisfy the correspondent prescribed boundary conditions. Now we show that $\epsilon \alpha''_\epsilon(t)+k\alpha_\epsilon (t) \geq f(t,\alpha_\epsilon (t) )$ and $\epsilon \beta''_\epsilon(t)+k\beta_\epsilon (t) \leq f(t,\beta_\epsilon (t) )$. Denoting $h(t,y)=f(t,y)-ky$, by the Taylor we have \begin{align*} h(t,\alpha_\epsilon (t)) &= h(t,\alpha_\epsilon (t))-h(t,u(t))\\ &= \frac{\partial h(t,\theta_\epsilon (t))}{\partial y} (v_\epsilon(t)-v^{\rm (corr)}_{\epsilon}(t) -\hat v_\epsilon(t)-\Gamma_\epsilon), \end{align*} where $\alpha_\epsilon (t)<\theta_\epsilon (t)<\beta_\epsilon (t)$ and $(t,\theta_\epsilon (t))\in \mathcal{H}(u)$ for $\epsilon$ sufficiently small. Hence, from the inequalities $m\leq\frac{\partial h(t,\theta_\epsilon (t))}{\partial y}\leq m+2w$ in $\mathcal{H}(u)$ we have \begin{align*} &\epsilon \alpha''_\epsilon(t)-h(t,\alpha_\epsilon(t))\\ &\geq \epsilon u''(t)+\epsilon v''_\epsilon(t) -\epsilon v^{(corr)''}_{\epsilon}(t) -\epsilon \hat v''_\epsilon(t)\\ &\quad -(m+2w)v_\epsilon(t)+mv^{\rm (corr)}_{\epsilon}(t)+m\hat v_\epsilon(t) +m\Gamma_\epsilon. \end{align*} Since $v_\epsilon(t)=| v_\epsilon(t)|$, we have $-\epsilon v^{(corr)''}_{\epsilon}(t)-2wv_\epsilon(t)+mv^{\rm (corr)}_{\epsilon}(t)=0$ and using (\eqref {def_LDE}, we obtain $$\epsilon \alpha''_\epsilon(t)-h(t,\alpha_\epsilon(t)) \geq \epsilon u''(t)+m\Gamma_\epsilon \geq-\epsilon\vert u''(t)\vert+\epsilon\Delta.$$ For $\beta_\epsilon (t))$ we have the inequality \begin{align*} &h(t,\beta_\epsilon(t))-\epsilon \beta''_\epsilon(t)\\ &=\frac{\partial h(t,\tilde\theta_\epsilon (t))}{\partial y} (v_\epsilon(t)+\hat v_\epsilon(t)+\Gamma_\epsilon) -\epsilon \beta''_\epsilon(t)\\ &=m(v_\epsilon(t)+\hat v_\epsilon(t)+\Gamma_\epsilon) -\epsilon(u''(t)+ v''_\epsilon(t)+\hat v''_\epsilon(t))\\ &\geq \epsilon\Delta-\epsilon\vert u''(t)\vert, \end{align*} where $\alpha_\epsilon (t)<\tilde\theta_\epsilon (t) <\beta_\epsilon (t)$ and $(t,\tilde\theta_\epsilon (t))\in \mathcal{H}(u)$ for $\epsilon$ sufficiently small. Let us now analyse the case ${u(c)-u(a)\leq 0}$: The lower solutions $$\alpha_\epsilon (t)=u(t)+v_\epsilon(t) -\hat v_\epsilon(t)-\Gamma_\epsilon$$ and the upper solutions $$\beta_\epsilon (t)=u(t)+v_\epsilon(t)+v^{\rm (corr)}_{\epsilon}(t) +\hat v_\epsilon(t)+\Gamma_\epsilon$$ satisfy \begin{align*} \epsilon \alpha''_\epsilon-h(t,\alpha_\epsilon) &=\epsilon u''+\epsilon v''_\epsilon-\epsilon\hat v''_\epsilon -\frac{\partial h}{\partial y}(v_\epsilon-\hat v_\epsilon -\Gamma_\epsilon)\\ &=\epsilon u''+\epsilon v''_\epsilon-\epsilon\hat v''_\epsilon +\frac{\partial h}{\partial y}(-v_\epsilon +\hat v_\epsilon+\Gamma_\epsilon)\\ &\geq \epsilon u''+\epsilon v''_\epsilon-\epsilon\hat v''_\epsilon +m(-v_\epsilon+\hat v_\epsilon+\Gamma_\epsilon)\\ &=\epsilon u''+\epsilon\Delta\geq\epsilon\Delta-\epsilon\vert u''\vert \end{align*} and \begin{align*} h(t,\beta_\epsilon)-\epsilon \beta''_\epsilon &=\frac{\partial h}{\partial y} \Big(v_\epsilon+v^{\rm (corr)}_{\epsilon} +\hat v_\epsilon+\Gamma_\epsilon\Big)-\epsilon u'' -\epsilon v''_\epsilon-\epsilon v^{(corr)''}_{\epsilon} -\epsilon\hat v''_\epsilon\\ &\geq (m+2w)v_\epsilon+m\Big(v^{\rm (corr)}_{\epsilon} +\hat v_\epsilon+\Gamma_\epsilon\Big)-\epsilon u'' -\epsilon v''_\epsilon-\epsilon v^{(corr)''}_{\epsilon} -\epsilon\hat v''_\epsilon\\ &=-2w| v_\epsilon|+mv^{\rm (corr)}_{\epsilon} -\epsilon v^{(corr)''}_{\epsilon}+\epsilon\Delta-\epsilon u''\\ &=\epsilon\Delta-\epsilon u''\geq\epsilon\Delta-\epsilon\vert u''\vert. \end{align*} Now, if we choose a constant $\Delta$ such that $\Delta\geq\vert u''(t)\vert$, $t\in[ a,b]$ then $\epsilon \alpha''_\epsilon(t) \geq h(t,\alpha_\epsilon (t) )$ and $\epsilon \beta''_\epsilon(t) \leq h(t,\beta_\epsilon (t) )$ in $[ a,b]$. The existence of a solution for \eqref{def_DE}, \eqref{def_BC} satisfying the above inequality follows from Lemma \ref{mainlemma}. The uniqueness follows from the fact that the function $h(t,y)$ is increasing in the variable $y$ on the set $\mathcal{H}$. \begin{remark} \label{rmk4.1} \rm Theorem \ref{maintheorem} implies $y_\epsilon(t)=u(t)+O(\epsilon)$ on every compact subset of $(a,b)$ and $\lim_{\epsilon\to0^+ }y_\epsilon(a)=u(c)$, $\lim_{\epsilon\to 0^+ }y_\epsilon(b)=u(d)$. The boundary layer effect occurs at the point $a$ ($b$) whenever $u(a)\neq u(c)$ ($u(b)\neq u(d)$). \end{remark} \section{Approximation of the solutions for \eqref{def_DE}, \eqref{def_BC}} In this section we consider only the case $u(c)-u(a)\leq 0$ as the other case could be treated analogously. We define the approximate solution $\tilde y_\epsilon(t)$ of \eqref{def_DE}, \eqref{def_BC} by $$\label{approx} \tilde y_\epsilon(t)=\frac{1}{2}\left(\alpha_\epsilon (t) +\beta_\epsilon (t)\right)=u(t)+v_\epsilon(t)+\frac{v^{\rm (corr)}_{\epsilon}(t)}{2}.$$ Taking into consideration the conclusions of Theorem \ref{maintheorem}, in both cases we obtain the following estimate for the solution $y_\epsilon$ of problem \eqref{def_DE}, \eqref{def_BC} $| y_\epsilon(t)-\tilde y_\epsilon(t)| \leq\hat v_\epsilon(t)+\frac{v^{\rm (corr)}_{\epsilon}(t)}{2}+\frac{\epsilon}{m} \max\{| u''(t)|,t\in[ a,b]\}.$ \begin{example} \label{exa5.1} \rm Consider the nonlinear differential equation $$\label{example} \epsilon y''+ky=y^2+g(t),\quad k<0,\; g\in C([ a,b])$$ subject to the boundary conditions \eqref{def_BC}. The assumptions of Theorem \ref{maintheorem} are satisfied if and only if there exists $w>0$ such that \begin{gather} \frac14\big(k^2-(w-k)^2\big) < g(t) <\frac14\big(k^2-(w+k)^2\big)\quad\text{on }[ a,b],\label{c1}\\ | g(c)-g(a)|<\frac18\big(w-k-\zeta(a)\big) \big(\zeta(a)+\zeta(c)\big),\label{c2}\\ | g(b)-g(d)|<\frac18\big(w-k-\zeta(b)\big) \big(\zeta(b)+\zeta(d)\big),\label{c3}\\ | g(c)-g(a)|<\frac18\big(w+k+\zeta(a)\big) \big(\zeta(a)+\zeta(c)\big),\label{c4}\\ | g(b)-g(d)|<\frac18\big(w+k+\zeta(b)\big) \big(\zeta(b)+\zeta(d)\big),\label{c5} \end{gather} where $\zeta(t)=\sqrt{k^2-4g(t)}$. As an illustrative example we consider the problem \eqref{example}, \eqref{def_BC} with $k=-2$, $g(t)=t$, $a=0$, $b=1/2$, $c=d=1/4$. It is not difficult to verify that the solution $u(t)=-1+\sqrt{1-t}$ of the reduced problem satisfies conditions \eqref{c1}--\eqref{c5} for every $w\in\big(\frac2{\sqrt2+\sqrt3}+2-\sqrt2,2\big)$. Thus, on the basis of Theorem \ref{maintheorem}, there exists $\epsilon_0=\epsilon_0(w)$ such that for every $\epsilon \in (0,\epsilon_0]$ the problem $\epsilon y''-2y=y^2+t$, \eqref{def_BC} has a unique solution which is $O(\epsilon)$ close to approximate solution \eqref{approx}; i.e., $\tilde y_\epsilon(t)=-1+\sqrt{1-t}+v_\epsilon(t)+\frac{v^{\rm (corr)}_{\epsilon}(t)}{2}.$ \end{example} \subsection*{Acknowledgments} This research was supported by grant 1/0068/08 from the Slovak Grant Agency, Ministry of Education of Slovak Republic. \begin{thebibliography}{00} \bibitem{GG} Guo, Y., Ge, W.; \emph{Positive solutions for three-point boundary value problems with dependence on the first order derivative}, J. Math. Anal. Appl., 290(1) (2004), pp. 291–-301. \bibitem{ChHo} Chang, K. W., Howes, F. 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