\documentclass[reqno]{amsart} \usepackage{amssymb} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 111, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/111\hfil Nontrivial solution for a three-point BVP] {Nontrivial solution for a three-point boundary-value problem} \author[Y.-P. Sun \hfil EJDE-2004/111\hfilneg] {Yong-Ping Sun} \address{Yong-Ping Sun \hfill\break Department of Fundamental Courses \\ Hangzhou Radio and TV University \\ Hangzhou, Zhejiang 310012, China} \email{syp@mail.hzrtvu.edu.cn} \date{} \thanks{Submitted June 15, 2004. Published September 22, 2004.} \thanks{Supported by the Education Department of Zhejiang Province of China(20040495)} \subjclass[2000]{34B10, 34B15} \keywords{Three-point boundary value problem; Nontrivial solution; \hfill\break\indent Leray-Schauder nonlinear alternative} \begin{abstract} In this paper, we study the existence of nontrivial solutions for the second-order three-point boundary-value problem \begin{gather*} u''+f(t,u)=0,\quad 0 1$such that$T(x)=\lambda x$, or there exists a fixed point$x^\ast \in \overline{\Omega}$. \end{lemma} \section{Main Results} In this section, we present and prove our main results. \begin{theorem}\label{thm1} Suppose$f(t,0)\not\equiv 0$, and there exist nonnegative functions$k,h\in L^1[0,1]$such that \begin{gather*} |f(t,x)|\leq k(t)|x|+h(t),\ a.e.\ (t,x)\in [0,1]\times \mathbb{R},\\ 2\int_0^1(1-s)k(s)ds+ |\alpha|\int_0^\eta k(s)ds <1. \end{gather*} Then the BVP \eqref{e1.1} has at least one nontrivial solution$u^\ast \in C[0,1]$. \end{theorem} \begin{proof} Let \begin{gather*} A=2\int_0^1(1-s)k(s)ds+ |\alpha |\int_0^\eta k(s)ds,\\ B=2\int_0^1(1-s)h(s)ds+ |\alpha |\int_0^\eta h(s)ds. \end{gather*} Then$A<1$. Since$f(t,0)\not\equiv 0$, there exists an interval$[\sigma,\tau]\subset [0,1]$such that$\min_{\sigma \leq t\leq\tau}|f(t,0)|>0$. On the other hand, from$h(t)\geq |f(t,0)|$, a.e.$t\in [0,1]$, we know that$B>0$. Let$m=B(1-A)^{-1}$,$\Omega=\{u\in C[0,1]:\|u\|< m\}$. Suppose$u\in \partial \Omega$,$\lambda >1$such that$Tu=\lambda u, then \begin{align*} \lambda m&=\lambda \|u\|=\|Tu\|=\max_{0\leq t\leq 1}|(Tu)(t)|\\ &\leq\int_0^1(1-s)|f(s,u(s))|ds+ \max_{0\leq t\leq 1}\int_0^t(t-s)|f(s,u(s))|ds\\ &\quad +|\alpha |\int_0^\eta |f(s,u(s))|ds\\ &\leq 2\int_0^1(1-s)|f(s,u(s))|ds +|\alpha |\int_0^\eta |f(s,u(s))|ds\\ &\leq \Big[2\int_0^1(1-s)k(s)|u(s)|ds +|\alpha |\int_0^\eta k(s)|u(s)|ds\Big]\\ &\quad+\Big[2\int_0^1(1-s)h(s)ds +|\alpha |\int_0^\eta h(s)ds\Big]\\ &\leq A\|u\|+B=Am+B. \end{align*} Therefore, $$\lambda\leq A+\frac{B}{m}=A+\frac{B}{B(1-A)^{-1}}=A+(1-A)=1,$$ this contradicts\lambda>1$. By Lemma \ref{lem2},$T$has a fixed point$u^\ast\in \overline{\Omega}$. In view of$f(t,0)\not\equiv 0$, the BVP \eqref{e1.1} has a nontrivial solution$u^\ast\in C[0,1]$. This completes the proof. \end{proof} \begin{theorem}\label{thm2} Suppose$f(t,0)\not\equiv 0$, and there exist nonnegative functions$k,h\in L^1[0,1]$such that $$|f(t,x)|\leq k(t)|x|+h(t),\ a.e.\ (t,x)\in [0,1]\times \mathbb{R}.$$ If one of the following conditions is fulfilled: \begin{itemize} \item[(1)] There exists constant$p>1$such that $$\int_0^1k^p(s)ds<\big[\frac{(1+q)^{1/q}}{2 +|\alpha|\ [\eta(1+q)]^{1/q} } \big]^p, \quad \big( \frac{1}{p} +\frac{1}{q}=1 \big).$$ \item[(2)] There exists a constant$\mu >-1$such that \begin{gather*} k(s)\leq \frac{(1+\mu)(2+\mu)} {2+|\alpha |(2+\mu)\eta^{1+\mu}}s^\mu ,\quad\mbox{a.e. }s\in [0,1],\\ \mathop{\rm meas} \big\{s\in [0,1]: k(s)< \frac{(1+\mu)(2+\mu)}{2 +|\alpha|(2+\mu)\eta^{1+\mu}}s^\mu \big\}>0. \end{gather*} \item[(3)] There exists a constant$\mu >-1$such that \begin{gather*} k(s)\leq \frac{(1+\mu)(2+\mu)}{2(1+\mu) +|\alpha |(2+\mu)}(1-s)^\mu ,\mbox{a.e. } s\in [0,1],\\ \mathop{\rm meas} \big\{s\in [0,1]: k(s)< \frac{(1+\mu)(2+\mu)}{2(1+\mu) +|\alpha |(2+\mu)}(1-s)^\mu \big\}>0. \end{gather*} \item[(4)]$k$satisfies \begin{gather*} k(s)\leq \frac{1}{1+|\alpha |\eta} ,\quad\mbox{a.e. } s\in [0,1],\\ \mathop{\rm meas} \big\{s\in [0,1]: k(s)< \frac{1}{1+|\alpha |\eta} \big\}>0. \end{gather*} \item[(5)]$f$satisfies $$\Lambda:=\limsup_{|x|\to\infty}\max_{t\in [0,1]}\big|\frac{f(t,x)}{x}\big|<\frac{1}{1+|\alpha |\eta}.$$ \end{itemize} Then the BVP \eqref{e1.1} has at least one nontrivial solution$u^\ast \in C[0,1]$. \end{theorem} \begin{proof} Let$A$be given in Theorem \ref{thm1}. In view of Theorem \ref{thm1}, we only need to prove$A<1. \noindent (1)\ By using the H\"{o}lder inequality, we have \begin{align*} A&\leq\Big[\int_0^1k^p(s)ds\Big]^{1/p} \Big\{2\Big[\int_0^1(1-s)^qds\Big]^{1/q}+|\alpha| \Big[\int_0^\eta 1^qds\Big]^{1/q}\Big\}\\ &\leq\Big[\int_0^1k^p(s)ds\Big]^{1/p} \Big[2\big(\frac{1}{1+q}\big)^{1/q}+|\alpha|\ \eta^{1/q}\Big]\\ &<\frac{(1+q)^{1/q}}{2 +|\alpha|\, [\eta(1+q)]^{1/q} } \cdot \frac{2 +|\alpha|\, [\eta(1+q)]^{1/q} }{(1+q)^{1/q}} =1. \end{align*} (2)\ In this case, we have \begin{align*} A&<\frac{(1+\mu)(2+\mu)} {2+|\alpha |(2+\mu)\eta^{1+\mu}} \Big[2\int_0^1(1-s)s^\mu ds+|\alpha| \int_0^\eta s^\mu ds\Big]\\ &\leq\frac{(1+\mu)(2+\mu)} {2+|\alpha |(2+\mu)\eta^{1+\mu}} \Big[\frac{2}{(1+\mu)(2+\mu)} +|\alpha|\cdot\frac{\eta^{1+\mu}}{1+\mu}\Big]\\ &=\frac{(1+\mu)(2+\mu)} {2+|\alpha |(2+\mu)\eta^{1+\mu}}\cdot \frac {2+|\alpha |(2+\mu)\eta^{1+\mu}}{(1+\mu)(2+\mu)}=1. \end{align*} (3)\ In this case, we have \begin{align*} A&<\frac{(1+\mu)(2+\mu)}{2(1+\mu) +|\alpha |(2+\mu)} \Big[2 \int_0^1(1-s)^{1+\mu} ds+|\alpha| \int_0^\eta(1-s)^\mu ds\Big]\\ &=\frac{(1+\mu)(2+\mu)}{2(1+\mu) +|\alpha |(2+\mu)} \Big[\frac{2}{2+\mu}+|\alpha| \cdot\frac{1-(1-\eta)^{1+\mu}}{1+\mu}\Big]\\ &\leq\frac{(1+\mu)(2+\mu)}{2(1+\mu) +|\alpha |(2+\mu)} \Big[\frac{2}{2+\mu}+ |\alpha|\cdot\frac{1}{1+\mu}\Big]\\ &=\frac{(1+\mu)(2+\mu)}{2(1+\mu) +|\alpha |(2+\mu)}\cdot \frac{2(1+\mu) +|\alpha |(2+\mu)}{(1+\mu)(2+\mu)}=1. \end{align*} (4)\ In this case, we have \begin{align*} A&<\frac{1}{1+|\alpha |\eta} \Big[2\int_0^1(1-s)ds+|\alpha|\int_0^\eta ds\Big]\\ &=\frac{1}{1+|\alpha |\eta}\left(1 +|\alpha|\eta\right)=1. \end{align*} (5)\ Let\varepsilon=\frac{1}{2}(\frac{1}{1+|\alpha |\eta}-\Lambda)$, then there exists$c>0$such that $$|f(t,x)|\leq\big(\frac{1}{1+|\alpha |\eta}-\varepsilon\big)|x|, \quad (t,x)\in [0,1]\times \mathbb{R}\setminus(-c,c).$$ Set$M=\max\{|f(t,x)|: (t,x)\in [0,1]\times [-c,c]\}$, then $$|f(t,x)|\leq\big(\frac{1}{1+|\alpha |\eta}-\varepsilon\big)|x|+M, \quad (t,x)\in [0,1]\times \mathbb{R}.$$ Set$k(s)=\frac{1}{1+|\alpha |\eta}-\varepsilon$,$h(s)=M$, then (4) holds. This completes the proof. \end{proof} \begin{corollary}\label{c1} Suppose$f(t,0)\not\equiv 0$, and there exist two nonnegative functions$k,h\in L^1[0,1]$such that $$|f(t,x)|\leq k(t)|x|+h(t),\ a.e.\ (t,x)\in [0,1]\times \mathbb{R}.$$ If one of the following conditions holds \begin{itemize} \item[(1)] There exists a constant$p>1$such that $$\int_0^1k^p(s)ds<\Big[\frac{(1+q)^{1/q}}{2 +|\alpha|\ (1+q)^{1/q} } \Big]^p,\quad \big( \frac{1}{p} +\frac{1}{q}=1 \big).$$ \item[(2)] There exists a constant$\mu >-1$such that \begin{gather*} k(s)\leq \frac{(1+\mu)(2+\mu)} {2+|\alpha |(2+\mu)}s^\mu ,\quad\mbox{a.e. } s\in [0,1],\\ \mathop{\rm meas}\big\{s\in [0,1]: k(s)< \frac{(1+\mu)(2+\mu)}{2 +|\alpha|(2+\mu)}s^\mu \big\}>0. \end{gather*} \item[(3)]$k$satisfies \begin{gather*} k(s)\leq \frac{1}{1+|\alpha |} ,\quad\mbox{a.e. } s\in [0,1],\\ \mathop{\rm meas} \big\{s\in [0,1]: k(s)< \frac{1}{1+|\alpha|} \big\}>0. \end{gather*} \item[(4)]$f$satisfies $$\Lambda=:\limsup_{|x|\to\infty}\max_{t\in [0,1]}\big|\frac{f(t,x)}{x}\big|<\frac{1}{1+|\alpha|}.$$ \end{itemize} Then the BVP \eqref{e1.1} has at least one nontrivial solution$u^\ast \in C[0,1]$. \end{corollary} \begin{proof} In this case, we have $A=\ 2\int_0^1(1-s)k(s)ds+|\alpha|\int_0^\eta k(s)ds \leq 2\int_0^1(1-s)k(s)ds+|\alpha|\int_0^1k(s)ds.$ The rest of the proof is the same as in Theorem \ref{thm2}. \end{proof} \begin{corollary}\label{c2} Suppose$f(t,0)\not\equiv 0$, and there exist two nonnegative functions$k,h\in L^1[0,1]$such that $$|f(t,x)|\leq k(t)|x|+h(t),\quad\mbox{a.e. } (t,x)\in [0,1]\times \mathbb{R}.$$ If one of the following conditions is holds. \begin{itemize} \item[(1)] There exists a constant$p>1$such that $$\int_0^1k^p(s)ds<\Big[\frac{1}{2+|\alpha|}\cdot\big( \frac{1+q}{2^{1+q}-1}\big)^{1/q}\Big]^p,\quad \big( \frac{1}{p} +\frac{1}{q}=1 \big).$$ \item[(2)] There exists a constant$\mu >-1$such that \begin{gather*} k(s)\leq \frac{(1+\mu)(2+\mu)}{(2+|\alpha|)(\mu+3)}s^\mu ,\quad\mbox{a.e. } s\in [0,1],\\ \mathop{\rm meas} \big\{s\in [0,1]: k(s)< \frac{(1+\mu)(2+\mu)}{(2+|\alpha|)(\mu+3)}s^\mu \big\}>0. \end{gather*} \item[(3)] There exists a constant$\mu >-2$such that \begin{gather*} k(s)\leq \frac{(2+\mu)}{(2+|\alpha|)(2^{2+\mu}-1)}(2-s)^\mu ,\quad \mbox{a.e. } s\in [0,1],\\ \mathop{\rm meas} \big\{ s\in [0,1]: k(s)< \frac{(2+\mu)}{(2+|\alpha|)(2^{2+\mu}-1)} (2-s)^\mu \big\}>0. \end{gather*} \end{itemize} Then the BVP \eqref{e1.1} has at least one nontrivial solution$u^\ast \in C[0,1]. \end{corollary} \begin{proof} In this case, \begin{align*} A&= 2\int_0^1(1-s)k(s)ds+|\alpha|\int_0^\eta k(s)ds\\ &\leq 2\int_0^1(1-s)k(s)ds+|\alpha|\int_0^1k(s)ds\\ &\leq (2+|\alpha|)\int_0^1(2-s)k(s)ds. \end{align*} (1)\ Using the H\"{o}lder inequality, \begin{align*} A&\leq (2+|\alpha|)\int_0^1(2-s)k(s)ds\\ &\leq(2+|\alpha|)\Big[\int_0^1k^p(s)ds\Big]^{1/p} \Big[\int_0^1(2-s)^qds\Big]^{1/q}\\ &< (2+|\alpha|)\cdot\frac{1}{2+|\alpha|}\big( \frac{1+q}{2^{1+q}-1}\big)^{1/q}\cdot\big( \frac{2^{1+q}-1}{1+q}\big)^{1/q}=1. \end{align*} (2)\ In this case, we have \begin{align*} A&\leq (2+|\alpha|)\int_0^1(2-s)k(s)ds\\ &< (2+|\alpha|)\cdot \frac{(1+\mu)(2+\mu)}{(2+|\alpha|) (\mu+3)}\int_0^1(2-s)s^{\mu}ds\\ &=\frac{(1+\mu)(2+\mu)}{\mu+3}\cdot \frac{\mu+3}{(1+\mu)(2+\mu)}=1. \end{align*} (3)\ In this case, \begin{align*} A&\leq (2+|\alpha|)\int_0^1(2-s)k(s)ds\\ &<(2+|\alpha|)\cdot \frac{(2+\mu)}{(2+|\alpha|) (2^{2+\mu}-1)}\int_0^1(2-s)^{1+\mu}ds\\ &=\frac{2+\mu}{2^{2+\mu}-1}\cdot \frac{2^{2+\mu}-1}{2+\mu}=1. \end{align*} The proof is complete.\end{proof} \section{Examples} In this section, in order to illustrate our results, we consider some examples. \begin{example} \label{ex4.1} \rm Consider the three-point BVP \label{e4.1} \begin{gathered} u''+(t-t^2)|u|\sin u-t^2u+t^3-2\sin t=0,\quad 00.$Hence, by Theorem \ref{thm2} (2), the BVP \eqref{e4.3} has at least one nontrivial solution$u^*\in C[0, 1]$. \end{example} \begin{example} \label{ex4.4} \rm Consider the three-point BVP \label{e4.4} \begin{gathered} u''+\frac{u^2e^{-u^2}}{3(1+u^2)\sqrt[4]{1-t}}-3e^{-t}+\sqrt{\sin t}=0,\quad 00. \end{gather*} Hence, by Theorem \ref{thm2} (3), the BVP \eqref{e4.4} has at least one nontrivial solution$u^*\in C[0, 1]$. \end{example} \begin{example} \label{ex4.5} \rm Consider the three-point BVP \label{e4.5} \begin{gathered} u''+\frac{t^2u^2e^{-u^2}}{t^2+u^2}-\cos e^t+3\sin^2 t=0,\quad 00. \end{gather*} Hence, by Theorem \ref{thm2} (4), the BVP \eqref{e4.5} has at least one nontrivial solution$u^*\in C[0, 1]\$. \end{example} \subsection*{Acknowledgment} The author wishes to express his sincere gratitude to the anonymous referees for their helpful suggestions in improving the paper. \begin{thebibliography}{99} \bibitem{d1} K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. \bibitem{f1} W. Feng, J. R. L. Webb, \emph{Solvability of a three-point boundary value problems at resonance}, Nonlinear Anal. 30(1997), 3227--3238. \bibitem{f2} W. Feng, J. R. L. Webb, \emph{Solvability of a m-point boundary value problems with nonlinear growth}, J.Math.Anal.Appl. 212(1997), 467--480. \bibitem{g1} C. P. Gupta, \emph{Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equations}, J. Math. Anal. Appl. 168(1992), 540--551. \bibitem{g2} C. P. Gupta, \emph{A shaper condition for the solvability of a three-point second order boundary value problem}, J.Math. Anal.Appl. 205(1997), 586--597. \bibitem{g3} C. P. Gupta, \emph{Solvability of an m-point nonlinear boundary value problem for second order ordinary differential equations}, J.Math.Anal.Appl. 189(1995)575--584. \bibitem{h1} X. He, W. Ge, \emph{Triple solutions for second-order three-point boundary value problems}, J. Math. Anal. Appl. 268 (2002), 256--265. \bibitem{i1} G. Infante, J.R.L.Webb, \emph{Nonzero solutions of Hammerstein integral equations with discontinuous kernels}, J.Math.Anal.Appl. 272(2002), 30--42. \bibitem{i2} G. Infante, \emph{Eigenvalues of some non-local boundary value problems}, Proc. Edinburgh Math. Soc. 46(2003), 75--86. \bibitem{l1} B. Liu, \emph{Positive solutions of a nonlinear three-point boundary value problem}, Comput. Math.Appl. 44(2002), 201--211. \bibitem{l2} B. Liu, \emph{Positive solutions of a nonlinear three-point boundary value problem}, Appl. Math. Comput. 132(2002), 11--28. \bibitem{m1} R. Ma, \emph{Existence theorems for second order three-point boundary value problems}, J.Math.Anal.Appl. 212 (1997), 545--555. \bibitem{m2} R. Ma, \emph{Positive solutions of a nonlinear three-point boundary value problems}, Electronic J.Differential Equations 34(1999), 1--8. \bibitem{m3} R. Ma, \emph{Positive solutions for second-order three-point boundary value problems}, Appl. Math. Letters 14(2001), 1--5. \bibitem{m4} R. Ma, \emph{Positive solutions of a nonlinear m-point boundary value problem}, Comput. Math.Appl. 42(2001), 755--765. \bibitem{w1} J. R. L. Webb, \emph{Positive solutions of some three point boundary value problems via fixed point index theory}, Nonlinear Anal. 47 (2001), 4319--4332. \end{thebibliography} \end{document}