\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 95, pp. 1--9.\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 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/95\hfil Three positive solutions] {Three positive solutions for p-Laplacian functional dynamic equations on time scales} \author[D.-B. Wang\hfil EJDE-2007/95\hfilneg] {Da-Bin Wang} \address{Da-Bin Wang \newline Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, China} \email{wangdb@lut.cn} \thanks{Submitted May 17, 2007. Published June 29, 2007.} \subjclass[2000]{39A10, 34B15} \keywords{Time scale; $p$-Laplacian functional dynamic equation; \hfill\break\indent boundary value problem; positive solution; fixed point} \begin{abstract} In this paper, we establish the existence of three positive solutions to the following $p$-Laplacian functional dynamic equation on time scales, \begin{gather*} [ \Phi _p(u^{\Delta }(t))] ^{\nabla}+a(t)f(u(t),u(\mu (t)))=0,\quad t\in (0,T)_{\mathbf{T}}, \\ u_0(t)=\varphi (t),\quad t\in [-r,0] _{\mathbf{T}},\\ u(0)-B_0(u^{\Delta }(\eta ))=0,\quad u^{\Delta }(T)=0,. \end{gather*} using the fixed-point theorem due to Avery and Peterson \cite{a8}. An example is given to illustrate the main result. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Let $\mathbf{T}$ be a time scale; i.e., $\mathbf{T}$ is a nonempty closed subset of $R$. Let $0, T$ be points in $\mathbf{T}$, an interval $(0,T) _{\mathbf{T}}$ denotes time scales interval, that is, $(0,T) _{\mathbf{T}}:=(0,T) \cap \mathbf{T}$. Other types of intervals are defined similarly. The theory of dynamic equations on time scales has been a new important mathematical branch (see, for example, \cite{a1,a2,b1,b2,k1}) since it was initiated by Hilger \cite{h4}. At the same time, boundary value problems (BVPs) for dynamic equation on time scales have received considerable attention \cite{a3,a4,a5,a6,c1,e1,h1,h2,h3,k2,s1,s2,s3,w1}. However, to the best of our knowledge, few papers can be found in the literature on bvps of $p$-Laplacian dynamic equations on time scales \cite{a5,h2,h3,s1,s2,w1}, especially for $p$-Laplacian functional dynamic equations on time scales \cite{s1}. This paper concerns the existence of positive solutions for the $p$-Laplacian functional dynamic equation on time scale, \begin{aligned} \hspace{0.1cm}[\Phi_p(u^{\Delta }(t))]^{\nabla }+a(t)f(u(t),u(\mu (t)))=0,\quad t\in (0,T) _{\mathbf{T}}, \\ u_0(t)=\varphi (t),\quad t\in [-r,0] _{\mathbf{T}},\\ u(0)-B_0(u^{\Delta }(\eta ))=0,\quad u^{\Delta }(T)=0, \end{aligned} \label{e1.1} where $\Phi _p(s)$ is $p$-Laplacian operator, i.e., $\Phi _p(s)=|s| ^{p-2}s$, $p>1$, $(\Phi _p)^{-1}=\Phi _q$, $\frac 1p+\frac 1q=1$, $\eta \in (0,\rho (T))_{\mathbf{T}}$ and \begin{itemize} \item[(C1)] $f:(\mathbb{R}^+) ^2\to \mathbb{R}^+$ is continuous; \item[(C2)] $a:\mathbf{T}\to \mathbb{R}^+$ is left dense continuous (i.e., $a\in C_{\mathbf{ld}}(\mathbf{T},\mathbb{R}^+)$) and does not vanish identically on any closed subinterval of $[0,T]$, where $C_{\mathbf{ld}}(\mathbf{T},\mathbb{R}^+)$ denotes the set of all left dense continuous functions from $\mathbf{T}$ to $\mathbb{R}^+$; \item[(C3)] $\varphi :[-r,0] _{\mathbf{T}}\to \mathbb{R}^+$ is continuous and $r>0$; \item[(C4)] $\mu :[0,T] _{\mathbf{T}}\to [-r,T]_{\mathbf{T}}$ is continuous, $\mu (t)\leq t$ for all $t$; \item[(C5)] $B_0:R\to R$ is continuous and there exist constant $A\geq 1$, $B>0$ such that $Bv\leq B_0(v)\leq Av,\text{ for all }v\geq 0.$ \end{itemize} In \cite{s1}, by using a double-fixed-point theorem due to Avery et al. \cite{a7} in a cone, Song and Xiao considered the problem \eqref{e1.1} and obtained the existence of two positive solutions. In paper \cite{h3}, Hong studied the problem \eqref{e1.1} when $\varphi (t)=0,t\in [-r,0] _{\mathbf{T}}$ and the nonlinear term is not involved $u(\mu (t))$. He imposed conditions on $f$ to yield at least three positive solutions to the problem \eqref{e1.1}, by applying the fixed-point theorem due to Avery and Peterson \cite{a8}. Motivated by \cite{h3,s1}, we shall show that the problem \eqref{e1.1}, has at least three positive solutions by means of the fixed point theorem due to Avery and Peterson. In the remainder of this section we list the following well known definitions which can be found in \cite{a2,a6,b1,b2}. \begin{definition} \label{def1.1} \rm For $t<\sup \mathbf{T}$ and $r>\inf\mathbf{T}$, define the forward jump operator $\sigma$ and the backward jump operator $\rho$, $\sigma (t)=\inf \{\tau \in \mathbf{T}|\tau >t\}\in \mathbf{T},\quad \rho (r)=\sup \{\tau \in \mathbf{T}|\tau t, t is said to be right scattered, and if \rho (r)0, there is a neighborhood U of t such that \[ | [x(\sigma (t))-x(s)] -x^{\Delta }(t)[\sigma (t)-s] | <\varepsilon | \sigma (t)-s| ,$ for all $s\in U$. For $x:\mathbf{T\to }R$ and $t\in \mathbf{T}_k$, we define the nabla derivative of $x(t)$, $x^\nabla (t)$, to be the number (when it exists), with the property that, for any $\varepsilon >0$, there is a neighborhood $V$ of $t$ such that $| [x(\rho (t))-x(s)] -x^\nabla (t)[\rho (t)-s]| <\varepsilon | \rho (t)-s| ,$ for all $s\in V$. If $\mathbf{T}=R$, then $x^{\Delta }(t)=x^\nabla (t)=x^{\prime }(t)$. If $\mathbf{T}=Z$, then $x^{\Delta }(t)=x(t+1)-x(t)$ is the forward difference operator while $x^\nabla (t)=x(t)-x(t-1)$ is the backward difference operator. \end{definition} \begin{definition} \label{def1.3}\rm If $F^{\Delta }(t)=f(t)$, then we define the delta integral by $\int_a^tf(s)\Delta s=F(t)-F(a).$ If $\Phi ^\nabla (t)=f(t)$, then we define the nabla integral by $\int_a^tf(s)\nabla s=\Phi (t)-\Phi (a).$ \end{definition} Throughout this papers, we assume $\mathbf{T}$ is closed subset of $\mathbb{R}$ with $0\in \mathbf{T}_k$ and $T\in \mathbf{T}^k$. \begin{lemma}[\cite{a6}] \label{lem1.1} The following formulas hold: \begin{itemize} \item[(i)] $\Big(\int_a^tf(s)\Delta s\Big) ^{\Delta }=f(t)$, \item[(ii)] $\Big(\int_a^tf(s)\Delta s\Big) ^{\nabla }=f(\rho (t))$, \item[(iii)] $\Big(\int_a^tf(s)\nabla s\Big) ^{\Delta }=f(\sigma(t))$, \item[(iv)] $\Big(\int_a^tf(s)\nabla s\Big) ^{\nabla }=f(t)$. \end{itemize} \end{lemma} \section{Preliminaries} In this section, we provide some background materials from the theory of cones in Banach spaces and we then state the fixed-point theorem due to Avery and Peterson. \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 $x\in P$ and $\lambda \geq 0$, then $\lambda x\in P$; \item[(ii)] if $x\in P$ and $-x\in P$, then $x=0$. \end{itemize} Every cone $P\subset E$ induces an ordering in $E$ given by $x\leq y \quad\text{if }y-x\in P.$ \end{definition} \begin{definition} \label{def2.2}\rm Given a cone $P$ in a real Banach space $E$, the map $\varsigma :P\to [0,\infty )$ is called a nonnegative continuous concave function on cone $P$ provided that $\varsigma$ is continuous and $\varsigma (tx+(1-t)y)\geq t\varsigma (x)+(1-t)\varsigma (y), \quad \text{for }x,\; y\in P\text{ and }0\leq t\leq 1.$ Dual to this, we call the map $\tau :P\to [0,\infty )$ is called a nonnegative continuous convex function on cone $P$ provided that $\tau$ is continuous and $\tau (tx+(1-t)y)\leq t\tau (x)+(1-t)\tau (y),\quad\text{for }x,\; y\in P \text{ and }0\leq t\leq 1.$ Let $\gamma$ and $\theta$ be nonnegative continuous convex functions on $P$, $\alpha$ be a nonnegative continuous concave function on $P$ and $\psi$ be a nonnegative continuous function on $P$. Then, for positive real numbers $a$, $b$, $c$ and $d$, we define the following convex sets \begin{gather*} P(\gamma ,d) =\{ x\in P:\gamma (x)b\} \neq \emptyset $and$\alpha (Fx)>b$for$x$in the set$P(\gamma ,\theta ,\alpha ,b,c,d)$; \item[(ii)]$\alpha (Fx)>b$for$x\in P(\gamma ,\alpha ,b,d)$with$\theta (Fx)>c$; \item[(iv)]$0\notin R(\gamma ,\psi ,a,d)$and$\psi (Fx)0\big\} ; \quad Y_3=Y_1\cap [l,T] _{\mathbf{T}}. \] Throughout this paper, we assume $Y_3\neq \phi$ and $\int_{Y_3}a(r)\nabla r>0$. Define the nonnegative continuous concave functionals $\alpha$, the nonnegative continuous convex functionals $\theta$, $\gamma$, and the nonnegative continuous functionals $\psi$ on the cone $P$ respectively as \begin{gather*} \gamma (u) =\|u\| ,\quad \theta (u)=\max_{t\in [l,T] _{\mathbf{T}^k}}u^{\Delta }(t), \\ \alpha (u)=\min_{t\in [\eta ,l] _{\mathbf{T}}}u(t),\quad \psi (u)=\min_{t\in [\eta ,T] _{\mathbf{T}}}u(t). \end{gather*} In addition, by Lemma \ref{lem3.1}, we have $\alpha (u)=\psi (u)=u(\eta )$, $\theta (u)=u^{\Delta }(l)$ for each $u\in P$. For convenience, we define \begin{gather*} \rho =(A+T)\Phi _q(\int_0^Ta(r)\nabla r) ,\quad \delta =(B+\eta )\Phi_q(\int_{Y_3}a(r)\nabla r) , \\ \lambda =(A+\eta )\Phi _q(\int_0^Ta(r)\nabla r). \end{gather*} We now state growth conditions on $f$ so that \eqref{e1.1} has at least three positive solutions. \begin{theorem} \label{thm3.1} Let $0<\frac T\eta a\Phi _p(\frac b\delta )$, if $b\leq u\leq d$, uniformly in $s\in [-r,0] _{\mathbf{T}}$, \item[(H3)] $f(u,\varphi (s))<\Phi _p(\frac a\lambda )$, if $0\leq u\leq \frac T\eta a$, uniformly in $s\in [-r,0] _{\mathbf{T}}$; $f(u_1,u_2)<\Phi _p(\frac a\lambda )$, if $0\leq u_i\leq \frac T\eta a$, $i=1,2$. \end{itemize} Then \eqref{e1.1} has at least three positive solutions of the form $u(t)=\begin{cases} u_i(t), & t\in [0,T] _{\mathbf{T}},\quad i=1,2,3, \\ \varphi (t), & t\in [-r,0] _{\mathbf{T}}, \end{cases}$ where $\gamma (u_i)\leq d$ for $i=1$, $2$, $3$, $b<\alpha (u_1)$, $a<\psi (u_2)$ with $\alpha (u_2)b\} \neq \phi$ and $\alpha (Fu)>b$ for $u\in P(\gamma ,\theta ,\alpha ,b,c,d)$. Let $u(t)=kb$ with $k=\frac \rho \delta >1$, then $u(t)=kb>b$ and $\theta (u)=0b\right\} \neq \emptyset. \] Moreover, for all$u\in P(\gamma ,\theta ,\alpha ,b,kb,d)$, we have$b\leq u(t)\leq d$,$t\in [\eta ,T] _{\mathbf{T}}. From (H2), we see that \begin{align*} &\alpha (Fu) \\ &= (Fu)(\eta ) \\ &= B_0(\Phi _q(\int_\eta ^Ta(r)f(u(r),u(\mu (r)))\nabla r) ) +\int_0^\eta \Phi _q(\int_s^Ta(r)f(u(r),u(\mu (r)))\nabla r) \Delta s \\ &\geq B\Phi _q(\int_\eta ^Ta(r)f(u(r),u(\mu (r)))\nabla r) +\eta \Phi _q(\int_\eta ^Ta(r)f(u(r),u(\mu (r)))\nabla r) \\ &\geq (B+\eta )\Phi _q(\int_l^Ta(r)f(u(r),u(\mu (r)))\nabla r) \\ &\geq (B+\eta )\Phi _q(\int_{Y_3}a(r)f(u(r),\varphi (\mu (r)))\nabla r) \\ &> (B+\eta )\Phi _q(\int_{Y_3}a(r)\nabla r) \frac b\delta =b, \end{align*} as required. Thirdly, we assert that\alpha (Fu)>b$for$u\in P(\gamma ,\alpha ,b,d) $with$\theta (Fu)>c$. For all$u\in P(\gamma ,\alpha ,b,d) $with$\theta (Fu)>kb, from Lemma \ref{lem3.1} we have $\theta (Fu)=(Fu)^{\Delta }(l)=\Phi _q\Big(\int_l^Ta(r)f(u(r),u(\mu (r)))\nabla r\Big) >kb.$ So, \begin{align*} &\alpha (Fu) \\ &=(Fu)(\eta ) \\ &=B_0\Big(\Phi _q\Big(\int_\eta ^Ta(r)f(u(r),u(\mu (r)))\nabla r\Big)\Big) +\int_0^\eta \Phi _q(\int_s^Ta(r)f(u(r),u(\mu (r)))\nabla r) \Delta s \\ &\geq B\Phi _q\Big(\int_\eta ^Ta(r)f(u(r),u(\mu (r)))\nabla r\Big) +\eta \Phi _q(\int_\eta ^Ta(r)f(u(r),u(\mu (r)))\nabla r) \\ &\geq (B+\eta ) \Phi _q\Big(\int_l^Ta(r)f(u(r),u(\mu (r)))\nabla r\Big) \\ &>(B+\eta ) kb=(B+\eta ) \frac \rho \delta b\\ &\geq (A+T) b>b. \end{align*} This implies that\alpha (Fu)>b$for$u\in P(\gamma ,\alpha,b,d) $with$\theta (Fu)>c$. Finally, we assert that$0\notin R(\gamma ,\psi ,a,d) $and$\psi (Fu)\Phi _p(\frac b\delta )=\sqrt{\frac{64}5}\approx 3.5777, \quad 1\leq u\leq 140, \\ f(u,\varphi (s))\leq \frac 8{1001}\approx 0.008<\Phi _p(\frac a\lambda )= \sqrt{\frac 1{50}}\approx 0.1414, \quad 0\leq u\leq \frac 1{10}; \\ f(u_1,u_2)\leq \frac 8{1002}\approx 0.008<\Phi _p(\frac a\lambda ) =\sqrt{\frac 1{50}}\approx 0.1414, \quad 0\leq u\leq \frac 1{10}, \end{gather*} Thus by Theorem \ref{thm3.1}, the \eqref{e4.1} has at least three positive solutions of the form $u(t)=\begin{cases} u_i(t), & t\in [0,1] _{\mathbf{T}},\quad i=1,2,3, \\ \varphi (t), & t\in [-\frac 34,0] _{\mathbf{T}}, \end{cases}$ where $\gamma (u_i)\leq 140$ for $i=1,2,3$, $1<\alpha (u_1)$, $\frac 1{40}<\psi (u_2)$ with $\alpha (u_2)<1$ and $\psi (u_3)<\frac 1{40}$. \subsection*{Acknowledgment} The author would like to thank the anonymous referees for their valuable suggestions which led to an improvement of this paper. \begin{thebibliography}{99} \bibitem{a1} R. 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