\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 113, pp. 1--8.\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 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/113\hfil Existence of solutions] {Existence of solutions for $p$-Laplacian functional dynamic equations on time scales} \author[C. Song\hfil EJDE-2006/113\hfilneg] {Changxiu Song} \address{Changxiu Song \newline School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China \newline School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, China} \email{scx168@sohu.com} \date{} \thanks{Submitted August 28, 2006. Published September 19, 2006.} \thanks{Supported by grants 10571064 from NNSF of China, and 011471 from NSF of Guangdong.} \subjclass[2000]{39K10, 34B15} \keywords{Time scale; boundary value problem; positive solutions; \hfill\break\indent $p$-Laplacian functional dynamic equations; fixed point theorem} \begin{abstract} In this paper, the author studies boundary-value problems for $p$-Laplacian functional dynamic equations on a time scale. By using the fixed point theorem, sufficient conditions are established for the existence of positive solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \section{Introduction} Let $\mathbb{T}$ be a closed nonempty subset of $\mathbb{R}$, and let subspace have the topology inherited from the Euclidean topology on $\mathbb{R}$. In some of the current literature, $\mathbb{T}$ is called a time scale (or measure chain). For notation, we shall use the convention that, for each interval of $J$ of $\mathbb{R}$, $J$ will denote time scales interval, that is, $J:=J\cap \mathbb{T}$. In this paper, let $\mathbb{T}$ be a time scale such that $-r$, $0$, $T\in \mathbb{T}$. We are concerned with the existence of positive solutions of the $p$-Laplacian dynamic equation, on a time scale, \begin{equation} \begin{gathered} {[\phi_p(x^\triangle(t))]}^\nabla+\lambda a(t)f(x(t),x(\mu(t)))=0 ,\quad t\in (0,T),\\ x_0(t)=\psi(t),\quad t\in[-r,0],\quad x(0)-B_0(x^\triangle(0))=0, \quad x^\triangle(T)=0, \end{gathered} \label{e1.1} \end{equation} where $\lambda>0$ and $\phi_p(u)$ is the $p$-Laplacian operator, i.e., $\phi_p(u)=|u|^{p-2}u$, $p>1$, $(\phi_p)^{-1}(u)=\phi_q(u)$, $\frac{1}{p}+\frac{1}{q}=1$. Also we assume the following: \begin{itemize} \item[(A)] The function $f:(\mathbb{R}^+)^2\to \mathbb{R}^+$ is continuous; \item[(B)] the function $a:\mathbb{T}\to \mathbb{R}^+$ is left dense continuous (i.e., $a\in C_{ld}(\mathbb{T},\mathbb{R}^+))$. Here $C_{ld}(\mathbb{T},\mathbb{R}^+)$ denotes the set of all left dense continuous functions from $\mathbb{T}$ to $\mathbb{R}^+$; \item[(C)] $ \psi:[-r,0]\to\mathbb{R}^+$ is continuous and $r>0$; \item[(D)] $\mu:[0,T]\to [-r,T]$ is continuous, $\mu(t)\leq t$ for all $t$; \item[(E)] $B_0:\mathbb{R}\to \mathbb{R}$ is continuous and satisfies that there are $\beta\geq\delta> 0$ such that $$ \delta s\leq B_0(s)\leq \beta s\mbox{ for }s\in \mathbb{R}^{+}. $$ \end{itemize} $p$-Laplacian problems with two-, three-, $m$-point boundary conditions for ordinary differential equations and finite difference equations have been studied extensively, for example see \cite{a3,c1,l1,w2} and references therein. However, there are not many concerning the $p$-Laplacian problems on time scales, especially for $p$-Laplacian functional dynamic equations on time scales. The motivations for the present work stems from many recent investigations in \cite{a2,k1,w1} and references therein. Especially, Kaufmann and Raffoul \cite{k1} considered a nonlinear functional dynamic equation on a time scale and obtained sufficient conditions for the existence of positive solutions. In this paper, we apply the fixed point theorem to obtain at least one positive solution of boundary value problem (BVP for short) \eqref{e1.1}. We do not need the condition that $f(x_1,x_2)$ is increasing in each $x_i$, for $x_i>0,i=1,2$. And we claim the condition $\psi\equiv 0$ is not essential in our results. For convenience, we list the following well-known definitions which can be found in \cite{a1,b1,h1} and the references therein. \begin{definition} \rm For $t<\sup \mathbb{T}$ and $r>\inf \mathbb{T}$, define the forward jump operator $\sigma$ and the backward jump operator $\rho$: $$ \sigma(t)=\inf\{\tau\in \mathbb{T}|\tau>t\}\in \mathbb{T},\quad \rho(r)=\sup\{\tau\in \mathbb{T}|\taut$, $t$ is said to be right scattered, and if $\rho(r)0$, there is a neighborhood $U$ of $t$ such that $$ |[x(\sigma(t))-x(s)]-x^\triangle(t)[\sigma(t)-s]|<\varepsilon|\sigma(t)-s| $$ for all $s\in U$. For $x:\mathbb{T}\to\mathbb{R}$ and $t\in\mathbb{T}_\kappa$, we define the nabla derivative of $x(t)$, $x^\nabla(t)$, to be the number (when it exists), with the property that, for any $\varepsilon >0$, there is a neighborhood $V$ of $t$ such that $$ |[x(\rho(t))-x(s)]-x^\nabla(t)[\rho(t)-s]|<\varepsilon|\rho(t)-s| $$ for all $s\in V$. If $\mathbb{T}=\mathbb{R}$, then $x^\triangle(t)=x^\nabla(t)=x'(t)$. If $\mathbb{T}=\mathbb{Z}$, then $x^\triangle(t)=x(t+1)-x(t)$ is forward difference operator while $x^\nabla(t)=x(t)-x(t-1)$ is the backward difference operator. \end{definition} \begin{definition} \rm If $F^\triangle(t)=f(t)$, then we define the delta integral by $\int_a^t f(s)\triangle s=F(t)-F(a)$. If $\Phi^\nabla(t)=f(t)$, then we define the nabla integral by $\int_a^t f(s)\nabla s=\Phi(t)-\Phi(a)$. \end{definition} In the following, we provide the definition of cones in Banach spaces, and we then state the fixed-point theorem for a cone preserving operator. \begin{definition} \rm Let $X$ be a real Banach space. A nonempty, closed, convex set $K\in X$ is called a cone, if it satisfies the following two conditions: \begin{itemize} \item[(i)] $ x\in K$, $\lambda\geq 0$ implies $\lambda x\in K$; \item[(ii)] $ x$ and $-x$ in $K$ implies $x=0$. \end{itemize} \end{definition} Every cone $K\subset X$ induces an ordering in $X$ given by $x\leq y$ if and only if $y-x\in K$. \begin{lemma}[\cite{k2}] \label{lem1.1} Assume that $X$ is a $Banach$ space and $K\subset X$ is a cone in $X$; $\Omega_1$, $\Omega_2$ are open subsets of $X$, and $0 \in\overline\Omega_1\subset \Omega_2$. Furthermore, let $F:K\cap(\overline\Omega_2\setminus \Omega_1)\to K$ be a completely continuous operator satisfying one of the following conditions: \begin{itemize} \item[(i)] $ \|F(x)\|\leq\|x\|$ for all $x\in K\cap\partial \Omega_1$, $\|F(x)\|\geq\|x\|$ for all $x\in K\cap\partial \Omega_2$; \item[(i)] $ \|F(x)\|\leq\|x\|$ for all $x\in K\cap\partial \Omega_2$, $\|F(x)\|\geq\|x\|$ for all $x\in K\cap\partial \Omega_1$. \end{itemize} Then there is a fixed point of $F$ in $K\cap(\overline\Omega_2\setminus\Omega_1)$. \end{lemma} \section{Main results} We note that $x(t)$ is a solution of \eqref{e1.1} if and only if \begin{equation} x(t)=\begin{cases} B_0 \Big(\phi_q\Big(\int_{0}^{T}\lambda a(r)f(x(r),x(\mu(r)))\nabla r \Big)\Big) \\ + \int_{0}^{t}\phi_q\Big(\int_{s}^{T}\lambda a(r)f(x(r),x(\mu(r)))\nabla r \Big)\triangle s, & t\in [0,T],\\[4pt] \psi(t), & t\in[-r,0]. \end{cases} \label{e2.1} \end{equation} Let $X=C_{ld}([0,T],\mathbb{R})$ be endowed with the norm $\|x\|=\max_{t\in [0,T]}|x(t)|$ and $$ K=\{x\in X: x(t)\geq \frac{\delta}{T+\beta}\|x\|\mbox{ for }t\in [0,T]\}. $$ Clearly, $X$ is a Banach space with the norm $\|x\|$ and $K$ is a cone in $X$. For each $x\in X$, extend $x(t)$ to $[-r,T]$ with $x(t)=\psi (t)$ for $t\in [-r,0]$. For $t\in [0,T]$, define $F : P\to X$ as \begin{equation} \begin{aligned} F x(t)&= B_0\Big(\phi_q\Big(\int_{0}^{T}\lambda a(r)f(x(r),x(\mu(r))) \nabla r\Big)\Big)\\ &\quad + \int_{0}^{t}\phi_q\Big(\int_{s}^{T} \lambda a(r)f(x(r),x(\mu(r)))\nabla r\Big)\triangle s, \end{aligned} \label{e2.2} \end{equation} We seek a fixed point, $x_1$, of $F$ in the cone $P$. Define $$ x(t)=\begin{cases} x_1(t), & t\in [0,T],\\ \psi(t),& t\in[-r,0]. \end{cases} $$ Then $x(t)$ denotes a positive solution of \eqref{e1.1}. It follows from \eqref{e2.2} that \begin{equation} \begin{aligned} \|F x\|&=(F x)(T)\\ &= B_0 \Big(\phi_q\Big(\int_{0}^{T}\lambda a(r)f(x(r),x(\mu(r)))\nabla r\Big)\Big)\\ &\quad + \int_{0}^{T}\phi_q\Big(\int_{s}^{T} \lambda a(r)f(x(r),x(\mu(r)))\nabla r\Big)\triangle s\\ &\leq (T+\beta)\lambda^{q-1}\phi_q\Big(\int_{0}^{T} a(r)f(x(r),x(\mu(r)))\Big). \end{aligned}\label{e2.3} \end{equation} From \eqref{e2.2} and \eqref{e2.3}, we have the following lemma. \begin{lemma} \label{lem2.1} Let $F$ be defined by \eqref{e2.2}. If $x\in K$, then \begin{itemize} \item[(i)] $F (K) \subset K$. \item[(ii)] $F :K\to K$ is completely continuous. \item[(iii)] $ x(t)\geq \frac{\delta}{T+\beta}\|x\|$, $t\in[0,T]$. \end{itemize} \end{lemma} We need to define subsets of $[0,T]$ with respect to the delay $\mu$. Set $$ Y_1:=\{t\in[0,T]: \mu (t)<0\};\quad Y_2:=\{t\in[0,T]: \mu(t)\geq 0\}. $$ Throughout this paper, we assume $Y_1$ is nonempty and $\int_{Y_1}a(r)\nabla r>0$. Let \begin{gather*} l=:\frac{\lambda^{1-q}}{(T+\beta)\phi_q \big(\int_{0}^{T}a(r)\nabla r\big)}, \quad m=:\frac{(T+\beta)\lambda^{1-q}}{\delta^2\phi_q \big(\int_{Y_1}a(r)\nabla r\big)},\\ \widetilde{l}=:\frac{1}{(T+\beta)\phi_q \big(\int_{0}^{T}a(r)\nabla r\big)}. \end{gather*} In additions to Conditions (A)--(E), we shall also consider the following: \begin{itemize} \item[(H1)] $\lim_{x\to 0^+}\frac{f(x,\psi(s))}{x^{p-1}}m^{p-1}$, uniformly in $s\in [-r,0]$. \end{itemize} \begin{theorem} \label{thm2.1} Assume Conditions (A)--(E), (H1)--(H3) are satisfied. Then, for each $0<\lambda<\infty$, BVP \eqref{e1.1} has at least a positive solution. \end{theorem} \begin{proof} Apply Condition (H1) and set $\varepsilon_1>0$ such that if $00$ such that if $0\rho_1$ such that if $x\geq\frac{\rho_2}{T+1+\beta}$, then $$ f(x,\psi(s) )> (mx)^{p-1}, \quad \mbox {for each } s\in [-r,0]. $$ Define $\Omega_2=\{x\in K: \|x\|<\rho_2\}$. For $x\in K$ with $\|x\|=\rho_2$, we have $$ x(t)\geq \frac{\delta}{T+\beta}\|x\|,\quad t\in [0,T],\label{e2.6} $$ Thus, we have \begin{equation} \begin{aligned} \|F x\|&=(F x)(T)\\ &\geq \delta\phi_q\Big(\int_{0}^{T}\lambda a(r)f(x(r),x(\mu(r)))\nabla r\Big)\\ &\geq \delta\lambda^{q-1}\phi_q\Big(\int_{Y_1} a(r)f(x(r),\psi(\mu(r))) \nabla r\Big)\\ &\geq m\delta\lambda^{q-1}\min_{t\in Y_1}\{x(t)\}\phi_q \Big(\int_{Y_1} a(r)\nabla r\Big)\\ &\geq \frac{m\delta^2\lambda^{q-1}}{T+\beta}\|x\|\phi_q \Big( \int_{Y_1} a(r)\nabla r\Big)\\ &=\|x\| \quad \mbox{for } x\in K\cap\partial\Omega_2. \end{aligned}\label{e2.7} \end{equation} Applying Condition (i) of Lemma \ref{lem1.1}, the proof is complete. \end{proof} Note that Theorem \ref{thm2.1} is useful, but it does not apply if $$ f(x_1,x_2)=x_1^2+x_2^2, $$ and if $\mu(t)<0$ for some $t$ satisfying $\psi(\mu(t))>l$ and $p=2$, for example. That is, Condition (H1) is not satisfied in this case. We now provide a second theorem to address the above case. Firstly, we assume \begin{itemize} \item[(H2')] $$ \lim_{x_1\to 0^+;x_2\to 0^+} \frac{f(x_1,x_2)}{\max\{x_1^{p-1}, x_2^{p-1}\}}<\widetilde{l}^{p-1}. $$ \end{itemize} \begin{theorem} \label{thm2.2} Assume Conditions (A)--(E), (H2') and (H3) are satisfied. Then, there exists $L>0$ such that for each $0<\lambda0$ such that if $0m^{p-1}$, uniformly in $s\in [-r,0]$; \item[(H5)] $\lim_{x\to \infty}\frac{f(x,\psi(s))}{x^{p-1}}0$ such that if $0 (mx)^{p-1} . $$ Define $\Omega_1=\{x\in K: \|x\|<\rho_1\}$. For $x\in K$ with $\|x\|=\rho_1$, we have $$ x(t)\geq \frac{\delta}{T+\beta}\|x\|,\quad t\in [0,T], $$ Thus, \begin{align*} \|F x\|&=(F x)(T)\\ &\geq \delta\phi_q\Big(\int_{0}^{T}\lambda a(r)f(x(r),x(\mu(r)))\nabla r\Big)\\ &\geq \delta\lambda^{q-1}\phi_q\Big(\int_{Y_1} a(r)f(x(r),\psi(\mu(r))) \nabla r\Big)\\ &\geq m\delta\lambda^{q-1}\min_{t\in Y_1}\{x(t)\}\phi_q \Big(\int_{Y_1} a(r)\nabla r\Big)\\ &\geq \frac{m\delta^2\lambda^{q-1}}{T+\beta}\|x\|\phi_q \Big( \int_{Y_1} a(r)\nabla r\Big)\\ &=\|x\| \quad \mbox{for } x\in K\cap\partial\Omega_1. \end{align*} To construct $\Omega_2$, we consider two cases, $f$ bounded and $f$ unbounded: When $f$ is bounded, the construction is straightforward. If $f(x_1,x_2)$ is bounded by $N^{p-1}>0$, set $$ \rho_2=\max\big\{2\rho_1,N(T+\beta)\phi_q\big( \int_0^T a(r)\nabla r\big)\big\}. $$ Then define $\Omega_2=\{x\in K: \|x\|<\rho_2\}$. For $x\in K$ with $\|x\|=\rho_2$, we have $$ \|F x\|\leq N(T+\beta)\phi_q\big(\int_0^T a(r)\nabla r\big)\leq \rho_2. $$ Assume $f$ is unbounded. Apply Condition (H5) and set $\varepsilon_1>0$ such that if $x> \varepsilon_1$, then $$ f(x,\psi(s))< (lx)^{p-1}, \quad \mbox{for each } s\in [-r,0]. $$ Apply Condition (H6) and set $\varepsilon_2>0$ such that if $x_1\geq \varepsilon_2,x_2\geq \varepsilon_2$, then $$ f(x_1,x_2)<\max\big\{x_1^{p-1},x_2^{p-1}\big\}l^{p-1}. $$ Set $\rho_2=\max\{2\rho_1,\varepsilon_1,\varepsilon_2\}$. Then, for any $x\in K$ with $\|x\|=\rho_2$, from \eqref{e2.3}, we have \begin{align*} &\|F x\|\\ &\leq (T+\beta)\lambda^{q-1}\phi_q\Big(\int_{0}^{T} a(r)f(x(r),x(\mu(r)))\nabla r\Big)\\ &=(T+\beta)\lambda^{q-1}\Big[\phi_q\Big(\int_{Y_1} a(r)f(x(r),\psi(\mu(r)))\nabla r+\int_{Y_2} a(r)f(x(r),x(\mu(r)))\nabla r\Big)\Big]\\ &\leq l(T+\beta)\lambda^{q-1}\max_{t\in [0,T]}\{x(t)\} \phi_q\Big(\int_{0}^{T}a(r)\nabla r\Big)\\ &= l(T+\beta)\lambda^{q-1}\|x\|\phi_q\Big(\int_{0}^{T}a(r)\nabla r\Big)\\ &=\|x\| \quad \mbox{for } x\in K\cap\partial \Omega_2, \end{align*} where $\Omega_2=\{x\in K: \|x\|<\rho_2\}$. Apply Condition (ii) of Lemma \ref{lem1.1}, the proof is complete. \end{proof} Similarly, assuming \begin{itemize} \item[(H6')] $$ \lim_{x_1\to \infty;x_2\to \infty} \frac{f(x_1,x_2)}{\max\{x_1^{p-1},x_2^{p-1}\}}<\widetilde{l}^{p-1}, $$ \end{itemize} we have the following theorem which is analogous to Theorem \ref{thm2.2}. \begin{theorem} \label{thm2.4} Assume Conditions (A)--(E), (H5) and (H6') are satisfied. Then, there exists $L>0$ such that for each $0<\lambda