\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage[compress]{cite} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 131, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/131\hfil Triple positive solutions] {Triple positive solutions for $m$-point boundary-value problems of dynamic equations on time scales with $p$-Laplacian} \author[A. Dogan \hfil EJDE-2015/131\hfilneg] {Abdulkadir Dogan} \address{Abdulkadir Dogan \newline Department of Applied Mathematics, Faculty of Computer Sciences, Abdullah Gul University, Kayseri 38039, Turkey \newline Phone +90 352 224 88 00, Fax +90 352 338 88 28} \email{abdulkadir.dogan@agu.edu.tr} \thanks{Submitted March 18, 2015. Published May 10, 2015.} \subjclass[2010]{34B15, 34B16, 34B18, 39A10} \keywords{Time scales; boundary value problem; $p$-Laplacian; \hfill\break\indent positive solutions; five functionals fixed-point theorem} \begin{abstract} In this article we study the existence of positive solutions for $m$-point dynamic equation on time scales with $p$-Laplacian. We prove that the boundary-value problem has at least three positive solutions by applying the five functionals fixed-point theorem. An example demonstrates the main results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In recent years, dynamic equations on time scales have found a considerable interest and attracted many researchers; see for example \cite{RPAgarwal,Agarwal, APeterson, Avery, AGraef, AKong,Dogan, ADogan,Liang,Peterson,Zhang}. The reasons seem to be two-fold. Theoretically, dynamic equations on time scales can not only unify differential and difference equations \cite{Hilger}, but also have displayed much more complicated dynamics \cite{Bohner,Peter, Laks}. Moreover, the study of time scales has led to several important applications in the study of insect population models, neural networks, stock market, heat transfer, wound healing and epidemic models; see for example \cite{Jones,Sped,Thomas}. In this paper, we study the existence of positive solutions of $m$-point $p$-Laplacian equation on time scales \begin{equation} (\phi_p( u^\Delta(t)))^\nabla+g(t)f(u(t))=0,\quad t\in[0,T]_{\mathbb{T} }, \label{e1.1} \end{equation} with the boundary conditions \begin{equation} u(0)=\sum_{i=1}^{m-2}a_i u(\xi_i), \quad u^\Delta(T)=0,\label{e1.2} \end{equation} or \begin{equation} u^\Delta(0)=0, \quad u(0)=\sum_{i=1}^{m-2}b_i u(\xi_i), \label{e1.3} \end{equation} where $\phi_p(s)$ is $p$-Laplacian operator; i.e., $\phi_p(s)=| s | ^{p-2} s$ for $p>1$, with $(\phi_p)^{-1}=\phi_q \text{and} \frac{1}{p}+ \frac{1}{q}=1 $ and $\xi_i \in (0,T)_{\mathbb{T}} $ with $ 0<\xi_1<\xi_2<\dots<\xi_{m-2}< T$ and $a_i,b_i \in [0,\infty)$ satisfy $1-\sum_{i=1}^{m-2}a_i \neq 0 $ and $ 1- \sum_{i=1}^{m-2}b_i \neq 0 $ $ (i=1,2,\dots,m-2)$. Some basic knowledge and definitions about time scales, which can be found in \cite{Bohner,Peter}, will be used here. By using the five functionals fixed-point theorem, we prove that the boundary-value problems \eqref{e1.1} \eqref{e1.2} and \eqref{e1.1} \eqref{e1.3} has at least three positive solutions. Throughout this paper, we assume that the following conditions are satisfied: \begin{itemize} \item[(H1)] $f:\mathbb{R}\to \mathbb{R}^{+} $ is continuous, and does not vanish identically on any closed subinterval of $[0,T]_{\mathbb{T}};$ \item[(H2)] $g:\mathbb{T} \to\mathbb{R}^{+} $ is left dense continuous ($ g \in C_{ld}(\mathbb{T},\mathbb{R}^{+})), $ and does not vanish identically on any closed subinterval of $[0,T]_{\mathbb{T}}$. \end{itemize} Recently, the boundary-value problems with $p$-Laplacian in the continuous case have been studied extensively in the literature; see for example \cite{ Bai,Feng,RLiang, Lu, YWang,WangGe,Zhao}. However, to the best of our knowledge, there are not many results concerning $p$-Laplacian dynamic equations on time scales, see \cite{ Avery,Su, DBWang}. Zhao et al \cite{Zhao} studied the existence of at least three positive solutions to the following $p$-Laplacian problem, \begin{gather*} (\phi_p(u'(t)))'+a(t)f(u,u')=0, \quad t\in[0,1], \\ u'(0)=u(1)=0. \end{gather*} To show their main results, they applied Leggett-Williams fixed-point theorem. Anderson et al \cite{Avery} considered the following BVP on time scales: \begin{gather*} [\phi_p(u^\Delta(t))]^\nabla +c(t)f(u(t))=0,\quad t\in (a,b)_{\mathbb{T}}, \\ u(a)-B_0(u^{\Delta}(\upsilon))=0, \quad u^\Delta(b)=0, \end{gather*} where $\upsilon \in (a,b)_{\mathbb{T}}$, $f\in C_{ld}([0,+\infty),[0,+\infty))$, $c\in C_{ld}([a,b],[0,+\infty))$, and $K_m x \leq B_0 (x) \leq K_M x$ for some positive constants $K_m, K_M$. By using a fixed-point theorem, they established the existence result for at least one positive solution. Wang \cite{DBWang} studied existence criteria of three positive solutions to the following boundary-value problems for $p$-Laplacian dynamic equations on time scales \begin{gather*} [\phi_p(u^\Delta(t))]^\nabla +a(t)f(u(t))=0,\quad t\in[0,T]_{\mathbb{T}}, \\ u^{\Delta}(0)=0, \quad u(T)+B_1(u^{\Delta}(\eta))=0, \quad \text{or } \\ u(0)-B_0(u^{\Delta}(\eta))=0, \quad u^\Delta(T)=0. \end{gather*} The main tool used in \cite{DBWang} is the Leggett-Williams fixed-point theorem. Motivated by the results mentioned above, we consider the existence of solutions to \eqref{e1.1} \eqref{e1.2} and \eqref{e1.1} \eqref{e1.3}. Our main results will depend on an application of the five functionals fixed-point theorem. \section{Preliminaries} In this section, we provide some background materials from theory of cones in Banach spaces, and we then state the five functionals fixed-point theorem for a cone preserving operator. \begin{definition} \label{def2.1} \rm Let $E$ be a real Banach space. A nonempty, closed, convex set $P \subset E$ is said to be a cone provided the following conditions are satisfied: \begin{itemize} \item[(i)] If $u \in P$ and $\lambda \geq 0$, then $\lambda u\in P$; \item[(ii)] If $u \in P$ and $-u\in P$, then $u=0$. \end{itemize} Every cone $P \subset E$ induces an ordering in $E$ given by $x\leq y$ if and only if $y-x \in P$. \end{definition} \begin{definition} \label{def2.2} \rm Given a cone $P$ in a real Banach space $E$, a functional $\psi: P\to \mathbb{R}$ is said to be increasing on $P$, provided $\psi(x) \leq \psi(y)$ for all $x,y \in P$ with $x\leq y$. \end{definition} \begin{definition} \label{def2.3} \rm A map $\alpha$ is said to be a nonnegative continuous concave functional on a cone $P$ of a real Banach space $E$ if $\alpha: \to [0,\infty)$ is continuous and $$ \alpha(tx+(1-t)y)\geq t\alpha(x)+(1-t)\alpha(y) $$ for all $x,y\in P$ and $0 \leq t \leq 1$. Similarly, we say the map $\beta$ is a nonnegative continuous convex functional on a cone $P$ of a real Banach space $E$ if $\beta: P \to [0,\infty)$ is continuous and $$ \beta (tx+(1-t)y) \leq t\beta(x)+(1-t)\beta(y) $$ for all $x,y \in P$ and $0 \leq t \leq 1$. \end{definition} Let $\gamma, \beta,\theta $ be nonnegative continuous convex functionals on $P$, and $\alpha,\psi$ be nonnegative continuous concave functionals on $P$. Then for nonnegative real numbers $h,a,b,d$ and $c$, we define the following convex sets, \begin{gather*} P (\gamma,c) = \{u\in P : \gamma(u)b \}\ne \emptyset$ and $\alpha(Fu)>b$ for \\ $u\in P( \gamma,\theta,\alpha,b,k,c)$; \item[(ii)] $\{ u\in Q( \gamma,\beta,\psi,h,a,c): \beta(u)b$ for $ u\in P( \gamma,\alpha,b,c)$ with $ \theta(Fu) >k$; \item[(iv)] $\beta(Fu)< a$ for $u\in Q( \gamma,\beta,a,c)$ with $\psi(Fu) < h$. \end{itemize} Then $F$ has at least three fixed points $u_1,u_2, u_3 \in \overline{P(\gamma, c)}$ such that $\beta(u_1) < a$, $b<\alpha(u_2)$ and $ a <\beta(u_3)$, with $\alpha(u_3)0$ hold, then we will use the following lemma. \begin{lemma} \label{lem3.1} If $u \in P$, then \begin{itemize} \item[(i)] $u(t)\geq\frac{t}{T} \|u\|$ for $t\in [0,T]_{\mathbb{T}}$; \item[(ii)] $s u(t) \geq t u(s)$ for $ t,s \in [0,T]_{\mathbb{T}}$, with $t \leq s$. \end{itemize} \end{lemma} \begin{proof} (i) Since $u^{\Delta \nabla}(t)\leq 0$, it follows that $u^{\Delta}(t)$ is nonincreasing. Thus, for $ 0< t < T$, \begin{gather*} u(t)-u(0)=\int_0^t u^{\Delta}(s) \Delta s\geq t u^{\Delta}(t),\\ u(T)-u(t)=\int_t^T u^{\Delta}(s) \Delta s\leq (T-t) u^{\Delta}(t) \end{gather*} from which we have $$ u(t)\geq \frac{t u(T)+(T-t)u(0)}{T} \geq \frac{t}{T} u(T) = \frac{t}{T}\|u\|. $$ (ii) If $t=s$, then the conclusion of (ii) holds. If $t t u(s)$, for $t < s$ with $t,s \in [0,T]_{\mathbb{T}}$. This completes the proof. \end{proof} We define the nonnegative, continuous concave functionals $\alpha,\psi$ and nonnegative continuous convex functionals $\beta,\theta,\gamma$ on the cone $P$ by \begin{gather*} \gamma (u)=\theta(u):=\max_{t\in[0,\xi_{m-2}]_{\mathbb{T}} } u(t)=u(\xi_{m-2}),\\ \alpha (u):=\min_{t\in[l,T]_{\mathbb{T}} } u(t)=u(l),\\ \beta (u):=\max_{t\in[0,l]_{\mathbb{T}} } u(t)=u(l),\\ \psi (u):=\min_{t\in[\xi_{m-2},T]_{\mathbb{T}} } u(t)=u(\xi_{m-2}). \end{gather*} We see that, for all $u \in P$, $$ \alpha(u)=u(l)=\beta (u). $$ For notational convenience, we define \begin{gather*} M = \Big(\frac { \sum_{i=1}^{m-2} a_i \xi_i} {1-\sum_{i=1}^{m-2} a_i} +\xi_{m-2} \Big) \phi_q\Big( \int_{0}^T g(r) \nabla r \Big), \\ m = \Big(\frac { \sum_{i=1}^{m-2} a_i \xi_i} {1-\sum_{i=1}^{m-2} a_i} +l \Big) \phi_q\Big( \int_{l}^T g(r) \nabla r \Big), \\ \lambda_l = \Big(\frac { \sum_{i=1}^{m-2} a_i \xi_i} {1-\sum_{i=1}^{m-2} a_i} +l \Big) \phi_q\Big( \int_{0}^T g(r) \nabla r \Big). \end{gather*} We note that $u(t)$ is a solution of \eqref{e1.1} and \eqref{e1.2}, if and only if \begin{align*} u(t)&=\frac {\sum_{i=1}^{m-2} a_i \Big( \int_0^{\xi_i} \phi_q \Big( \int_s^T g(r) f(u(r)) \nabla r \Big) \Delta s \Big)} {1-\sum_{i=1}^{m-2} a_i} \\ &\quad +\int_0^t \phi_q\Big( \int_s^T g(r) f(u(r) )\nabla r \Big)\Delta s, \quad t \in [0,T]_{\mathbb{T}}. \end{align*} \begin{theorem} \label{thm3.1} Let $0< a < lb/T <(l \xi_{m-2} c)/T^2$, $Mb \phi_p (\frac{b}{m})$, for all $b \leq w \leq T^2 b/\xi_{m-2}^2$; \item[(C3)] $ f(w) < \phi_p (\frac{a}{\lambda_l})$, for all $0 \leq w \leq Ta/l$. \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 $$ \max_{t \in [0,l]_{\mathbb{T}}} u_1 (t) < a, \quad b < \min_{t \in [l,T]_{\mathbb{T}}} u_2 (t) \quad\text{and}\quad a < \max_{t \in [0,l]_{\mathbb{T}}} u_3 (t) \quad \text{with } \min_{t \in [l,T]_{\mathbb{T}}} u_3 (t)< b. $$ \end{theorem} \begin{proof} Defining a completely continuous integral operator $F: P\to E$ by \begin{equation} \begin{aligned} (Fu)(t)&=\frac {\sum_{i=1}^{m-2} a_i \big( \int_0^{\xi_i} \phi_q \big( \int_s^T g(r) f(u(r)) \nabla r \big) \Delta s \big)} {1-\sum_{i=1}^{m-2} a_i} \\ &\quad +\int_0^t \phi_q\Big( \int_s^T g(r) f(u(r) )\nabla r \Big)\Delta s, \quad u\in P, \end{aligned} \label{e3.1} \end{equation} for $t\in [0,T]_{\mathbb{T}}$, we will search for fixed points of $F$ in the cone $P$. We note that, if $u\in P$, then $(Fu)(t)\geq 0$ for $t\in [0,T]_{\mathbb{T}}$, and $$ (Fu)^{\Delta}(t)=\phi_q\Big( \int_t^T g(r) f(u(r) )\nabla r \Big),\quad u\in P,\; t\in [0,T]_{\mathbb{T^\kappa}}. $$ We see that $(Fu)^{\Delta}(t)$ is continuous and nonincreasing on $[0,T]_{\mathbb{T^\kappa}}$ and, $(Fu)^{\Delta\nabla}(t)\leq 0$ for $[0,T]_{\mathbb{T^\kappa}\cap \mathbb{T_\kappa}}$. In addition, $(Fu)^{\Delta}(T)=0$. This implies that $Fu\in P$, and therefore $F:P\to P$. If $u \in \overline{ P(\gamma,c)}$, then $$ \gamma (u)=\max_{t \in[0,\xi_{m-2}]_{\mathbb{T}} } u(t)=u(\xi_{m-2})=c. $$ Consequently, $0 \leq u(t) \leq c$ for $t \in [0,\xi_{m-2}]_{\mathbb{T}}$. By Lemma \ref{lem3.1}, we have $$ \| u \| \leq \frac{ T u(\xi_{m-2})} {\xi_{m-2}} \leq \frac{Tc}{\xi_{m-2}}. $$ This implies $0\leq u(t) \leq \frac{Tc}{\xi_{m-2}}$ for $t \in[0,T]_{\mathbb{T}}$. It follows from (C1) of Theorem \ref{thm3.1} that \begin{align*} \gamma(F(u)) & = (Fu)(\xi_{m-2})\\ & = \frac {\sum_{i=1}^{m-2} a_i \left( \int_0^{\xi_i} \phi_q \left ( \int_s^T g(r) f(u(r) )\nabla r \right)\Delta s\right)} {1-\sum_{i=1}^{m-2} a_i}\\ &\quad + \int_0^{\xi_{m-2}} \phi_q\Big( \int_s^T g(r) f(u(r)) \nabla r \Big)\Delta s\\ & < \frac {\sum_{i=1}^{m-2} a_i \left( \int_0^{\xi_i} \phi_q \left ( \int_s^T g(r) f(u(r) )\nabla r \right)\Delta s\right)}{1-\sum_{i=1}^{m-2} a_i}\\ & \quad + \xi_{m-2} \phi_q\Big( \int_0^T g(r) f(u(r)) \nabla r \Big)\\ & < \frac {\sum_{i=1}^{m-2} a_i \left( \int_0^{\xi_i} \phi_q \left ( \int_0^T g(r) f(u(r) )\nabla r \right)\Delta s\right)}{1-\sum_{i=1}^{m-2} a_i}\\ &\quad + \xi_{m-2} \phi_q\Big( \int_0^T g(r) f(u(r)) \nabla r \Big)\\ & < \frac{c}{M}\Big( \frac{\sum_{i=1}^{m-2} a_i\xi_i}{1-\sum_{i=1}^{m-2} a_i} +\xi_{m-2} \Big) \phi_q\Big( \int_0^T g(r) \nabla r \Big)=c. \end{align*} So $F(u) \in \overline{ P(\gamma,c)}$. By Lemma \ref{lem3.1}, we obtain $\gamma(u) =u(\xi_{m-2}) \geq \frac{\xi_{m-2}}{T} \| u \| $, hence $$ \| u \| \leq \frac{ T u(\xi_{m-2})} {\xi_{m-2}} = \frac{T \gamma(u)}{\xi_{m-2}} \text{for all } u \in P.$$ Now we prove that (i)-(iv) of Theorem \ref{thm2.1} are satisfied. Firstly, if $u\equiv \frac{Tb}{\xi_{m-2}}$, $k= \frac{Tb}{\xi_{m-2}}$, then $$ \alpha(u)=u(l)= \frac{Tb}{\xi_{m-2}}>b, \quad \theta(u)= u(\xi_{m-2})= \frac{Tb}{\xi_{m-2}}=k, \quad \gamma(u)= \frac{Tb}{\xi_{m-2}}b \}\ne \emptyset. $$ For $u \in P( \gamma,\theta,\alpha,b,\frac{Tb}{\xi_{m-2}},c)$, we obtain $$ \theta(u)=\max_{t\in[0,\xi_{m-2}]_{\mathbb{T}} } u(t)=u(\xi_{m-2})\leq \frac{Tb}{\xi_{m-2}} , \quad \alpha (u)=\min_{t\in[l,T]_{\mathbb{T}} } u(t)=u(l)\geq b, $$ which imply $$ 0 \leq u(t) \leq \frac{Tb}{\xi_{m-2}} \quad \text{for all } t\in[0,\xi_{m-2}]_{\mathbb{T}}, $$ and $b \leq u(t)$ for all $t\in[l,T]_{\mathbb{T}}$. By Lemma \ref{lem3.1}, we obtain $$ \| u \| \leq \frac{ T u(\xi_{m-2})} {\xi_{m-2}} \leq \frac{ T^2 b} {\xi^2_{m-2}}, $$ as a result, $$b \leq u(t) \leq \frac{ T^2 b} {\xi^2_{m-2}} \quad \text{for all } t\in[l,T]_{\mathbb{T}}. $$ By (C2) of Theorem \ref{thm3.1}, we find \begin{align*} \alpha(F(u)) & = (Fu)(l)\\ & = \frac {\sum_{i=1}^{m-2} a_i \left( \int_0^{\xi_i} \phi_q \left ( \int_s^T g(r) f(u(r) )\nabla r \right)\Delta s\right)}{1-\sum_{i=1}^{m-2} a_i}\\ & \quad + \int_0^{l} \phi_q\Big( \int_s^T g(r) f(u(r)) \nabla r \Big)\Delta s\\ & > \frac {\sum_{i=1}^{m-2} a_i \left( \int_0^{\xi_i} \phi_q \left ( \int_l^T g(r) f(u(r) )\nabla r \right)\Delta s\right)}{1-\sum_{i=1}^{m-2} a_i}\\ & \quad + l \phi_q\Big( \int_l^T g(r) f(u(r)) \nabla r \Big)\\ & > \frac{b}{m}\Big( \frac{\sum_{i=1}^{m-2} a_i\xi_i}{1-\sum_{i=1}^{m-2} a_i} +l \Big) \phi_q\Big( \int_l^T g(r) \nabla r \Big)=b. \end{align*} Therefore, (i) of Theorem \ref{thm2.1} is satisfied. Secondly, we show that (ii) of Theorem \ref{thm2.1} is satisfied. Let $ u= \frac{a \xi_{m-2}}{T}$ and $ h= \frac{a \xi_{m-2}}{T}$, then \begin{gather*} \gamma (u)= u(\xi_{m-2})=\frac{a \xi_{m-2}}{T}< c, \quad \beta (u)=u(l)=\frac{a \xi_{m-2}}{T}< a, \\ \psi (u)= u(\xi_{m-2})= \frac{a \xi_{m-2}}{T}=h. \end{gather*} Thus $$ \{ u\in Q( \gamma,\beta,\psi,h,a,c): \beta(u) k= \frac{Tb}{\xi_{m-2}}, $$ then $$\alpha(F(u))=(Fu)( l) \geq \frac{l}{T}F(u(l)) \geq \frac{l}{T}F( u(\xi_{m-2}) ) > \frac{lb}{\xi_{m-2}}>b. $$ Lastly, if $$ u\in Q( \gamma,\beta,a,c) \quad \text{and} \quad \psi(F(u))= F( u(\xi_{m-2}) ) < h= \frac{a \xi_{m-2}}{T}, $$ then by Lemma \ref{lem3.1} we find $$ \beta(F(u))=(Fu)( l) \leq \frac{T}{l}F(u(l)) \leq \frac{T}{\xi_{m-2}}F( u(\xi_{m-2}) ) < a $$ which shows that condition (iv) of Theorem \ref{thm2.1} is satisfied. Hence, all the conditions in Theorem \ref{thm2.1} are fulfilled, therefore the boundary-value problems \eqref{e1.1} and \eqref{e1.2} has at least three positive solutions $u_1,u_2, u_3$ such that $$ \max_{t \in [0,l]_{\mathbb{T}}} u_1 (t) < a, \quad b < \min_{t \in [l,T]_{\mathbb{T}}} u_2 (t) , \quad a < \max_{t \in [0,l]_{\mathbb{T}}} u_3 (t) \quad \text{with } \min_{t \in [l,T]_{\mathbb{T}}} u_3 (t)< b. $$ The proof of Theorem \ref{thm3.1} is complete. \end{proof} Now, we apply the five functionals fixed-point theorem to establish the existence of at least three positive solutions of \eqref{e1.1} and \eqref{e1.3}. We define the cone, $P_1\subset E$, by $$ P_1=\{ u\in E: u^{\Delta}(0)=0, u \text{ is concave and nonnegative on } [0,T]_{\mathbb{T}}\}. $$ Suppose that there exists $l_1\in \mathbb{T}$ such that $0 < l_1<\xi_1 < T$ and $\int_0^{l_1} g(r)\nabla r >0$ hold. \begin{lemma} \label{lem3.2} If $u \in P_1$, then \begin{itemize} \item[(i)] $u(t)\geq\frac{T-t}{T} \|u\|$ for $t\in [0,T]_{\mathbb{T}}$; \item[(ii)] $(T-s) u(t) \geq(T-t) u(s)$ for $ t,s \in [0,T]_{\mathbb{T}}$, with $s \leq t$. \end{itemize} \end{lemma} \begin{proof} (i) Since $u^{\Delta \nabla}(t)\leq 0$, it follows that $u^{\Delta}(t)$ is nonincreasing. Thus, for $ 0< t < T$, \begin{gather*} u(t)-u(0)=\int_0^t u^{\Delta}(s) \Delta s\geq t u^{\Delta}(t), \\ u(T)-u(t)=\int_t^T u^{\Delta}(s) \Delta s\leq (T-t) u^{\Delta}(t) \end{gather*} from which we have $$ u(t)\geq \frac{t u(T)+(T-t)u(0)}{T} \geq \frac{T-t}{T} u(0) = \frac{T-t}{T}\|u\|. $$ (ii) If $t=s$, then the conclusion of (ii) holds. If $t>s$, $t,s\in [0,T]_{\mathbb{T}}$, setting $x(t)= u(t)-\frac{T-t}{T-s}u(s)$, for $u\in P_1$, we have $$ x^{\Delta\nabla}(t)=u^{\Delta\nabla}(t)\leq 0, x(T)=u(T)\geq 0, x(s)=0. $$ Therefore, the concavity of $x$ implies that $x(t)\geq 0, t\in (s,T]_{\mathbb{T}}$, i.e., $(T-s) u(t) > (T-t) u(s)$, for $t > s$, $t,s \in [0,T]_{\mathbb{T}}$. This completes the proof. \end{proof} We define the nonnegative continuous concave functionals $\alpha_1,\psi_1$ and the nonnegative continuous convex functionals $\gamma_1,\beta_1,\theta_1 $ on the cone $P_1$ by \begin{gather*} \gamma_1 (u)=\theta_1 (u):=\max_{t\in[\xi_{1},T]_{\mathbb{T}} } u(t)=u(\xi_{1}),\\ \alpha_1 (u):=\min_{t\in[0,l_1]_{\mathbb{T}} } u(t)=u(l_1),\\ \beta_1 (u):=\max_{t\in[l_1,T]_{\mathbb{T}} } u(t)=u(l_1),\\ \psi_1 (u):=\min_{t\in[0,\xi_1]_{\mathbb{T}} } u(t)=u(\xi_{1}). \end{gather*} We see that, for all $u \in P_1$, $\alpha_1(u)=u(l_1)=\beta_1 (u)$. \begin{gather*} M_1 = \Big(\frac { \sum_{i=1}^{m-2} b_i(T- \xi_i)} {1-\sum_{i=1}^{m-2} b_i} + T-\xi_{1} \Big) \phi_q\Big( \int_{0}^T g(r) \nabla r \Big), \\ m_1 = \Big(\frac { \sum_{i=1}^{m-2} b_i(T- \xi_i)} {1-\sum_{i=1}^{m-2} b_i} +T-l_1 \Big) \phi_q\Big( \int_{0}^{l_1} g(r) \nabla r \Big), \\ \lambda_{l_1} = \Big(\frac { \sum_{i=1}^{m-2} b_i (T-\xi_i)} {1-\sum_{i=1}^{m-2} b_i} +T-l_1 \Big) \phi_q\Big( \int_{0}^T g(r) \nabla r \Big). \end{gather*} We note that $u(t)$ is a solution of \eqref{e1.1} and \eqref{e1.2}, if and only if \begin{align*} u(t) &= \frac {\sum_{i=1}^{m-2} b_i \left( \int_{\xi_i}^T \phi_q \big( \int_0^s g(r) f(u(r)) \nabla r \big) \Delta s \right)} {1-\sum_{i=1}^{m-2} b_i} \\ &\quad + \int_t^T \phi_q\Big( \int_0^s g(r) f(u(r) )\nabla r \Big)\Delta s, \quad t\in[0,T]_{\mathbb{T}}. \end{align*} \begin{theorem} \label{thm3.2} Let $0< a < \frac{(T-l_1) b}{T} <\frac{(T-l_1)(T-\xi_{1}) c}{T^2}$, $M_1 b < m_1 c$, and assume that $f$ satisfies the following conditions: \begin{itemize} \item[(D1)] $ f(w) < \phi_p \big(\frac{c}{M_1}\big)$ for all $0\leq w \leq\frac{Tc}{T-\xi_{1}}$; \item[(D2)] $ f(w) > \phi_p \big(\frac{b}{m_1}\big)$ for all $b \leq w \leq\frac{T^2 b}{(T-\xi_{1})^2}$; \item[(D3)] $ f(w) < \phi_p \big(\frac{a}{\lambda_{l_1} }\big)$ for all $0 \leq w \leq\frac{Ta}{T-l_1}$. \end{itemize} Then, there exist at least three positive solutions $u_1,u_2, u_3$ of \eqref{e1.1} and \eqref{e1.3} such that $$ \max_{t \in [l_1,T]_{\mathbb{T}}} u_1 (t) < a, \quad b < \min_{t \in [0,l_1]_{\mathbb{T}}} u_2 (t) , \quad a < \max_{t \in [l_1,T]_{\mathbb{T}}} u_3 (t) \quad \text{with } \min_{t \in [0,l_1]_{\mathbb{T}}} u_3 (t)< b. $$ \end{theorem} \begin{proof} Defining a completely continuous integral operator $F_1: P_1\to E$ by \begin{equation} \begin{aligned} (F_1u)(t) &=\frac {\sum_{i=1}^{m-2} b_i \left( \int_{\xi_i}^T \phi_q \left ( \int_0^s g(r) f(u(r) ) \nabla r \right) \Delta s \right)} {1-\sum_{i=1}^{m-2} b_i} \\ &\quad +\int_t^T \phi_q\Big( \int_0^s g(r) f(u(r) )\nabla r \Big)\Delta s, \quad u\in P_1, \end{aligned} \label{e3.2} \end{equation} for $t\in [0,T]_{\mathbb{T}}$, each fixed point of $F_1$ in the cone $P_1$ is a positive solution of \eqref{e1.1} and \eqref{e1.3}. We note that, if $u\in P_1$, then $(F_1u)(t)\geq 0$ for $t\in [0,T]_{\mathbb{T}}$, and $$ (F_1u)^{\Delta}(t)=-\phi_q\Big( \int_0^t g(r) f(u(r) )\nabla r \Big),\quad u\in P_1,\; t\in [0,T]_{\mathbb{T^\kappa}}. $$ Note that $(F_1u)^{\Delta}(t)$ is continuous and nonincreasing on $[0,T]_{\mathbb{T^\kappa}}$, and $(F_1u)^{\Delta\nabla}(t)\leq 0$ for $ t\in [0,T]_{\mathbb{T^\kappa}\cap \mathbb{T_\kappa}}$. In addition, $(F_1u)^{\Delta}(0)=0$. This implies $F_1u\in P_1$, and therefore $F:P_1\to P_1$. In likeness to the proof of Theorem \ref{thm3.1}, we arrive at the conclusion. \end{proof} \section{An example} Let $\mathbb{T}=\{2-(\frac{1}{2})^{\mathbb{N}_0} \} \cup \{0,\frac{1}{8},\frac{1}{4},\frac{1}{3},\frac{1}{2},1,\frac{3}{2},2 \} \cup[\frac{1}{10},\frac{1}{9}]$. We consider the $p$-Laplacian dynamic equation with $k \in\mathbb{N}_0$, \begin{equation} (\phi_p( u^\Delta(t)))^\nabla+ \Big\{ \sum_{k=0}^6 t^k(\rho(t))^{6-k} \Big\} t^{\nabla} f(u(t))=0,\quad t\in[0,2]_{\mathbb{T}}, \label{e4.1} \end{equation} satisfying the boundary conditions \begin{equation} u(0)=\frac{1}{2}u\Big (\frac{1}{4} \Big )+\frac{1}{6}u \Big (\frac{1}{2} \Big),\quad u^{\Delta}(2)=0, \label{e4.2} \end{equation} where $p=4/3$, $\xi_1=1/4$, $\xi_2=1/2$, $a_1=1/2$, $a_2=1/6$, $T=2$ and \[ f(u)= \begin{cases} 1\times 10^{-7}, & 0\leq u \leq 4,\\ p(u), & 4\leq u \leq 10,\\ 7\times 10^{-6}, & 10\leq u \leq 800,\\ s(u), & u \geq 800, \end{cases} \] here $p(u)$ and $s(u)$ satisfy $p(4)= 1\times 10^{-7}$, $p(10)= 7\times 10^{-6}$, $s(800)= 7\times 10^{-6}$, $(p^{\nabla}(u))^{\nabla}=0$ for $u\in (4,10)$, and $s(u):\mathbb{R} \to \mathbb{R^{+} }$ is continuous. If \[ g(t)= \Big\{ \sum_{k=0}^6 t^k(\rho(t))^{6-k} \Big\} t^{\nabla}, \] then we obtain $(t^7)^{\nabla}= \Big\{ \sum_{k=0}^6 t^k(\rho(t))^{6-k} \Big\} t^{\nabla}$. Choose $a=2$, $b=10$, $c=200$, $l=1$. Then \begin{gather*} \begin{aligned} M & = \Big(\frac { \sum_{i=1}^{m-2} a_i \xi_i} {1-\sum_{i=1}^{m-2} a_i} +\xi_{m-2} \Big) \phi_q\Big( \int_{0}^T g(r) \nabla r \Big)\\ & = \big(\frac{5}{8} + \frac{1}{2}\big ) \Big(\int_0^2 \Big\{ \sum_{k=0}^6 t^k(\rho(t))^{6-k} \Big\} t^{\nabla}\nabla t \Big)^3\\ &= \frac{9}{8}\times 2^{21}=2.3593\times10^6, \end{aligned}\\ \begin{aligned} m &= \Big(\frac { \sum_{i=1}^{m-2} a_i \xi_i} {1-\sum_{i=1}^{m-2} a_i} +l \Big) \phi_q\Big( \int_{l}^T g(r) \nabla r \Big)\\ & = \Big(\frac{5}{8} + 1 \Big ) \Big(\int_1^2 \Big\{ \sum_{k=0}^6 t^k(\rho(t))^{6-k} \Big\} t^{\nabla}\nabla t \Big)^3\\ & = \frac{13}{8}\times( 2^{7}-1)^3=3.3286\times10^6, \end{aligned}\\ \begin{aligned} \lambda_l & = \Big(\frac { \sum_{i=1}^{m-2} a_i \xi_i} {1-\sum_{i=1}^{m-2} a_i} +l \Big) \phi_q\Big( \int_{0}^T g(r) \nabla r \Big) \\ & = \frac{13}{8} \Big(\int_0^2 \Big\{ \sum_{k=0}^6 t^k(\rho(t))^{6-k} \Big\} t^{\nabla}\nabla t \Big)^3 \\ &= \frac{13}{8}\times 2^{21}=3.4079\times10^6. \end{aligned} \end{gather*} It is easy to see that $$ 0 < a < \frac{lb}{T} < \frac{\xi_{2} lc}{T^2} , \quad Mb \phi_p \left(\frac{b}{m}\right)=3.0043\times 10^{-6}, \quad \text{for } 10 \leq w \leq\frac{T^2 b}{\xi_{m-2}^2}=160,\\ f(w) < \phi_p \left(\frac{c}{M}\right)= 8.4771\times 10^{-5},\quad \text{for } 0\leq w \leq\frac{Tc}{\xi_{m-2}}=800. \end{gather*} So, all the conditions of Theorem \ref{thm3.1} are satisfied. By Theorem \ref{thm3.1}, the problem \eqref{e4.1}, \eqref{e4.2} has at least three positive solutions $u_1$, $u_2$ and $u_3$ satisfying $$ \max_{t \in [0,1]_{\mathbb{T}}} u_1 (t) < 2, \quad 10 < \min_{t \in [1,2]_{\mathbb{T}}} u_2 (t) , \quad 2 < \max_{t \in [0,1]_{\mathbb{T}}} u_3 (t) \quad \text{with} \min_{t \in [1,2]_{\mathbb{T}}} u_3 (t)< 10. $$ \subsection*{Acknowledgments} The author would like to thank the anonymous referees and editor for their helpful comments and suggestions. The project is supported by Abdullah Gul University Foundation of Turkey. \begin{thebibliography}{99} \bibitem{RPAgarwal} R. P. Agarwal, M. Bohner; \emph{Basic calculus on time scales and its applications}, Results Math. 35 (1999), 3-22. \bibitem{Agarwal} R. P. Agarwal, D. O'Regan; \emph{Nonlinear boundary value problems on time scales}, Nonlinear Anal. 44 (2001), 527-535. \bibitem{APeterson} R. P. Agarwal, M. Bohner, D. O'Regan, A. Peterson; \emph {Dynamic equations on time scales: a survey}, J. 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