\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 149, 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} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/149\hfil Existence of three positive solutions] {Existence of three positive solutions for an $m$-point boundary-value problem on time scales} \author[A. Dogan \hfil EJDE-2013/149\hfilneg] {Abdulkadir Dogan} % in alphabetical order \address{Abdulkadir Dogan \newline Department of Applied Mathematics, Faculty of Computer Sciences, Abdullah Gul University, Kayseri, 38039 Turkey\newline Tel: +90 352 224 88 00\quad Fax:+90 352 338 88 28} \email{abdulkadir.dogan@agu.edu.tr} \thanks{Submitted March 29, 2013. Published June 27, 2013.} \subjclass[2000]{34B15, 34B16, 34B18, 39A10} \keywords{Time scales; boundary value problem; positive solutions; \hfill\break\indent fixed point theorem} \begin{abstract} We study an $m$-point boundary-value problem on time scales. By using a fixed point theorem, we prove the existence of at least three positive solutions, under suitable growth conditions imposed on the nonlinear term. An example is given to illustrate our results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} The theory of dynamic equation on time scales (or measure chains) was initiated by Stefan Hilger in his Ph. D. thesis in 1988 \cite{Hilger} (supervised by Bernd Aulbach) as a means of unifying structure for the study of differential equations in the continuous case and study of finite difference equations in the discrete case. In recent years, it has found a considerable amount of interest and attracted the attention of many researchers; see for example \cite{RPAgarwal,APeterson,Avery,AGraef,AKong,Sang,WLi,Sun,Zhang}. It is still a new area, and research in this area is rapidly growing. The study of time scales has led to several important applications, e.g., in the study of insect population models, heat transfer, neural networks, phytoremediation of metals, wound healing, and epidemic models \cite{Peter,Jones,Sped,Thomas}. Throughout the remainder of this article, let $\mathbb{T}$ be a closed nonempty subset of $R$, and let $\mathbb{T}$ have the subspace topology on $R$. In some of the current literature, $\mathbb{T}$ is called a time scale. For convenience, we make the blanket assumption that $0,T$ are points in $\mathbb{T}$. Sang and Xi \cite{YSang} considered the following $p$-Laplacian dynamic equation on time scales \begin{gather*} (\phi_p(u^\Delta(t)))^\nabla+a(t)f(t,u(t))=0,\quad t\in[0,T]_{\mathbb{T}}, \\ \phi_p(u^\Delta(0))=\sum_{i=1}^{m-2} a_i \phi_p(u^\Delta(\xi_i)), \quad u(T)= \sum_{i=1}^{m-2} b_i u(\xi_i), \end{gather*} where $\phi_p(s)=| s | ^{p-2} s$, $p>1$. He \cite{Zhe} studied the existence of at least two positive solutions by way of a new double fixed-point theorem for the equation \begin{gather*} [\varphi_p(u^\Delta(t))]^\nabla +a(t)f(u(t))=0,\quad t\in[0,T]_{\mathbb{T}}, \\ u(0)-B_0(u^{\Delta}(\eta))=0 \quad u^\Delta(T)=0,\text{or }\\ u^{\Delta}(0)=0, \quad u(T)+B_1(u^{\Delta}(\eta))=0, \end{gather*} where $\varphi_p(s)=| s | ^{p-2} s$, $p>1$, $\eta \in(0,\rho(t))_{\mathbb{T}}$. Anderson et al \cite{Avery} showed the existence of at least one solution for the corresponding boundary-value problem \begin{gather*} [g(u^\Delta(t))]^\nabla +c(t)f(u(t))=0,\quad t\in (a,b), \\ u(a)-B_0(u^{\Delta}(\upsilon))=0, \quad u^\Delta(b)=0, \end{gather*} where $g(z)=| z | ^{p-2} z$, $p>1$, and $\upsilon \in(a, b)\subset \mathbb{T}$. In recent years, much attention has been paid to the existence of positive solutions of boundary value problems (BVPs) on time scales for $p(t)\equiv 1$ and $\varphi(u)=| u | ^{p-2} u$, $p>1$; see \cite{Agarwal,Avery,Zhe, Erk,Wtli,HRSun,WLi} and the references therein. The key condition used in the above papers is the oddness of a $p$-Laplacian operator. Nevertheless, we define a new operator which improves and generalizes a $p$-Laplacian operator for some $p>1$, and $\varphi$ is not necessary odd. In addition, there are not many results concerning increasing homeomorphism and positive homomorphism on time scales; see \cite{SLiang,Liang}. Motivated by works mentioned above, in this paper, we study the existence of at least three positive solutions to the following $p$-Laplacian multipoint BVP on time scales \begin{gather} \label{e1.1} [\varphi( p(t)u^\Delta(t))]^\nabla+a(t)f(u(t))=0,\quad t\in[0,T]_{{\mathbb{T}}^K\cap{\mathbb{T}}_K }, \\ u(0)=\sum_{i=1}^{m-2}a_i u(\xi_i), \quad u^\Delta(T)=0, \quad \label{e1.2} \end{gather} where $\varphi:\mathbb{R}\to\mathbb{R}$ is an increasing homeomorphism and positive homomorphism and $\varphi(0)=0$, $p \in C([0,T]_{\mathbb{T}},(0,+\infty))$ and $\xi_i \in [0,T]_{\mathbb{T}}$ with $0<\xi_1<\xi_2<\dots <\xi_{m-2}0$ we define the set $$P(\gamma, d)=\{x \in P:\gamma(x)0 and M>0 such that$$ \gamma(x)\leq \beta(x)\leq \alpha(x), \|u \| \leq M \gamma(x) $$for all x\in \overline{P(\gamma, c)}. Suppose there exists a completely continuous operator T:\overline{P(\gamma, c)}\to P and 0b, for all x\in \partial P(\beta,b); \item[(S3)] P(\alpha,a)\ne \emptyset, and \alpha(Tx) \varphi (b/\lambda_2) for all u\in [b,Tb/\xi_1]; \item[(iii)]  f(u) < \varphi (a/\lambda_3) for all u\in [0,Ta/\xi_1]. \end{itemize} Then there exist at least three positive solutions u_1,u_2, u_3 of \eqref{e1.1} and\eqref{e1.2} such that$$ 0 \leq \alpha(u_1) < a <\alpha(u_2), \quad \beta(u_2)< b <\beta(u_3), \quad \gamma(u_3) < c. $$\end{theorem} \begin{proof} Define the completely continuous operator A by \eqref{e2.4}. Let u \in \partial P(\gamma,c), then (Au)(t)\geq 0 for t\in[0,T]_{\mathbb{T}}. By Lemma \ref{lem2.3} we know that A: \overline{ P(\gamma,c)}\to P. Now, we show that all the conditions of Theorem \ref{thm2.1} are satisfied. To verify (S1) of Theorem \ref{thm2.1} holds, we choose u \in \partial P(\gamma,c). Then \gamma (u)=\max_{t\in[0,\xi_{1}]_{\mathrm{T}} } u(t)=u(\xi_1)=c. If we recall that \|u\|\leq \frac{T}{\xi_1}\gamma(u)=\frac{T}{\xi_1}c. Therefore$$ 0 \leq u(t)\leq\frac{T}{\xi_1}c, \quad\text{for all } t\in[0,T]_{\mathbb{T}}. $$As a consequence of (i),$$ f(u(s))<\varphi (c/\lambda_1), \quad \text{for } s\in[0,T]_{\mathbb{T}}. Since Au \in P, we have \begin{align*} \gamma(Au) &= (Au)(\xi_1)\\ &= \int_0^{\xi_1} \frac{1}{p(s)}\varphi^{-1} \Big( \int_s^T a(\tau) f(u(\tau)) \nabla \tau \Big)\Delta s\\ &\quad +\frac {\sum_{i=1}^{m-2} a_i }{1-\sum_{i=1}^{m-2} a_i} \int_0^{\xi_i} \frac{1}{p(s)}\varphi^{-1} \Big( \int_s^T a(\tau) f(u(\tau) )\nabla \tau \Big)\Delta s\\ & \leq \frac{1}{p(0)}\int_0^{\xi_1} \varphi^{-1} \Big( \int_0^T a(\tau) f(u(\tau)) \nabla \tau \Big)\Delta s\\ &\quad + \frac {\frac{1}{p(0)} \sum_{i=1}^{m-2} a_i }{1-\sum_{i=1}^{m-2} a_i} \int_0^{\xi_i} \varphi^{-1} \Big( \int_0^T a(\tau) f(u(\tau)) \nabla \tau \Big)\Delta s\\ & < \frac{1}{p(0)} \Big(\xi_1+\frac {\sum_{i=1}^{m-2} a_i T}{1-\sum_{i=1}^{m-2} a_i}\Big) \varphi^{-1} \Big( \int_0^T a(\tau) \nabla \tau \Big)\frac{c}{\lambda_1}=c. \end{align*} Thus, (S1) of Theorem \ref{thm2.1} is satisfied. Secondly, we prove that (S2) of Theorem \ref{thm2.1} is fulfilled. For this, we choose u \in \partial P(\beta,b). Then \beta (u)=\min_{t\in[\xi_1,\xi_{m-2}]_{\mathbb{T}} } u(t)=u(\xi_1)=b. This means u(t)\geq b, t\in[\xi_i,T]_{\mathbb{T}} and since u \in P, we have  b \leq u(t) \leq \|u\|=u(T) for t\in[\xi_1,T]_{\mathbb{T}}. Note that \|u\| \leq \frac{T}{\xi_1}\gamma(u)= \frac{T}{\xi_1}\beta(u)=\frac{T}{\xi_1}b for all u \in P. Therefore, b \leq u(t)\leq\frac{T}{\xi_1}b, \quad \text{for all } t\in[\xi_1,T]_{\mathbb{T}}. $$From (ii), we have$$ f(u(s))>\varphi \left(\frac{b}{\lambda_2}\right ), \quad \text{for } s\in[\xi_1,T]_{\mathbb{T}}, and so \begin{align*} \beta(Au) &= (Au)(\xi_1)\\ &= \int_0^{\xi_1} \frac{1}{p(s)}\varphi^{-1} \Big( \int_s^T a(\tau) f(u(\tau)) \nabla \tau \Big)\Delta s\\ &\quad +\frac {\sum_{i=1}^{m-2} a_i }{1-\sum_{i=1}^{m-2} a_i} \int_0^{\xi_i} \frac{1}{p(s)}\varphi^{-1} \Big( \int_s^T a(\tau) f(u(\tau) )\nabla \tau \Big)\Delta s\\ & > \frac{1}{p(T)}\int_0^{\xi_1} \varphi^{-1} \Big( \int_{\xi_{m-2}}^T a(\tau) f(u(\tau)) \nabla \tau \Big)\Delta s\\ &\quad + \frac {\frac{1}{p(T)} \sum_{i=1}^{m-2} a_i }{1-\sum_{i=1}^{m-2} a_i} \int_0^{\xi_i} \varphi^{-1} \Big( \int_ {\xi_{m-2}}^T a(\tau) f(u(\tau)) \nabla \tau \Big)\Delta s\\ & > \frac{1}{p(T)} \Big(\xi_1+\frac {\sum_{i=1}^{m-2} a_i T}{1-\sum_{i=1}^{m-2} a_i}\Big) \varphi^{-1} \Big( \int_ {\xi_{m-2}}^T a(\tau) \nabla \tau \Big) \frac{b}{\lambda_2}=b. \end{align*} Thus, (S2) of Theorem \ref{thm2.1} is satisfied. Finally we prove that (S3) of Theorem \ref{thm2.1} is also satisfied. We note that u(t)=a/2, t\in[0,T]_{\mathbb{T}} is a member of P(\alpha,a) and \alpha (u)=\frac{a}{2}0, \end{cases} \] and $f(u)= \begin{cases} 0.1, & 0\leq u \leq 3,\\ 0.1+\frac{90(u-3)}{4\sqrt{3}-3}, & 3\leq u \leq4\sqrt{3},\\ 90.1, & 4\sqrt{3}\leq u. \end{cases}$ We take a=1, b=4\sqrt{3}, c=75. By simple calculations, we have \begin{gather*} \lambda_1 = \frac{1}{p(0)} \Big(\xi_1+\frac { \sum_{i=1}^{m-2} a_i T} {1-\sum_{i=1}^{m-2} a_i}\Big) \varphi ^{-1} \Big( \int_{0}^T a(\tau) \nabla \tau \Big)=\frac{4}{3}, \\ \lambda_2 = \frac{1}{p(T)} \Big(\xi_1+\frac { \sum_{i=1}^{m-2} a_i T} {1-\sum_{i=1}^{m-2} a_i}\Big) \varphi ^{-1}\Big( \int_{\xi_{m-2}}^T a(\tau) \nabla \tau \Big) = \frac{4\sqrt{3}}{9}, \\ \lambda_3 = \frac{1}{p(0)} \Big(\xi_{m-2}+\frac { \sum_{i=1}^{m-2} a_i T} {1-\sum_{i=1}^{m-2} a_i}\Big) \varphi ^{-1}\Big( \int_{0}^T a(\tau) \nabla \tau \Big)=\frac{5}{3}. \end{gather*} It is easy to see that 0 < a < \frac{\xi_1}{T} b < b < \frac{\lambda_2}{\lambda_1} c,  and that $f$ satisfies \begin{gather*} f(u) < \varphi \Big(\frac{c}{\lambda_1}\Big) = \Big(\frac{75}{\frac{4}{3}}\Big)^2\thickapprox 3164.0625, \quad u\in [0,225];\\ f(u) > \varphi \Big(\frac{b}{\lambda_2}\Big) = \Big(\frac{4\sqrt{3}} {\frac{4\sqrt{3}}{9}}\Big)^2=81, \quad u\in [4\sqrt{3},12\sqrt{3}];\\ f(u) < \varphi \Big(\frac{a}{\lambda_3}\Big) =\Big(\frac{1}{\frac{5}{3}}\Big)^2=\frac{9}{25}, \quad u\in [0,3]. \end{gather*} By Theorem \ref{thm3.1}, we see that BVP \eqref{e4.1} and \eqref{e4.2} has at least three positive solutions $u_1, u_2, u_3$ such that \begin{gather*} \max_{t\in [0,\frac{2}{3}]_{\mathbb{T}} } \{ u_i(t) \} \leq 75, \quad \text{for } i=1,2,3; \\ 0 \leq \max_{t\in [0,2/3]_{\mathbb{T}} } \{ u_1(t) \} < 1 < \max_{t\in [0,\frac{2}{3}]_{\mathbb{T}} } \{ u_2(t) \}; \\ \min_{t\in [ \frac{1}{3},\frac{2}{3}]_{\mathbb{T}} } \{ u_2(t) \} < 4\sqrt{3} < \min_{t\in [ \frac{1}{3},\frac{2}{3}]_{\mathbb{T}} } \{ u_3(t) \}, \quad \max_{t\in [0,\frac{1}{3}]_{\mathbb{T}} } \{ u_3(t)\} \leq 75. \end{gather*} \subsection*{Acknowledgments} The author would like to thank the anoymous referees and the editor for their helpful comments and suggestions. 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