\documentclass[reqno]{amsart} \usepackage{amssymb} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 96, 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 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/96\hfil Existence, multiplicity and infinite solvability] {Existence, multiplicity and infinite solvability of positive solutions for $p$-Laplacian dynamic equations on time scales} \author[D.-B. Wang\hfil EJDE-2006/96\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} \date{} \thanks{Submitted April 14, 2006. Published August 22, 2006.} \subjclass[2000]{34B10, 34B18, 39A10} \keywords{Time scales; $p$-Laplacian; boundary value problem; \hfill\break\indent positive solution; existence; multiplicity; infinite solvability} \begin{abstract} In this paper, by using Guo-Krasnosel'skii fixed point theorem in cones, we study the existence, multiplicity and infinite solvability of positive solutions for the following three-point boundary value problems for $p$-Laplacian dynamic equations on time scales \begin{gather*} [ \Phi _p(u^{\triangle }(t))] ^{\triangledown}+a(t)f(t,u(t)) =0,\quad t\in [0,T]_{\mathbf{T}}, \\ u(0)-B_0(u^{\triangle }(\eta )) = 0,\quad u^{\triangle }(T)=0. \end{gather*} By multiplicity we mean the existence of arbitrary number of solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \section{Introduction} Let $\mathbb{T}$ be a closed nonempty subset of $\mathbb{R}$, and let $\mathbb{T}$ have the subspace 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 $J$ of $\mathbb{R}$, $J_{\mathbb{T}}=J\cap \mathbb{T}.$ The theory of dynamic equations on time scales has become a new important mathematical branch (see, for example \cite{a1,b1,k2}) since it was initiated by Hilger \cite{h2}. At the same time, boundary-value problems (BVPs) for scalar dynamic equations on time scales has received considerable attention \cite{a2,a3,a4,a5,a6,c1,e1,e2}. The purpose of this paper is to investigate the existence, multiplicity and infinite solvability of positive solutions for $p$-Laplacian dynamic equations on time scales $$\label{e1.1} [ \Phi _p(u^{\triangle }(t))]^{\triangledown }+a(t)f(t,u(t))=0, \quad t\in [0,T]_{\mathbb{T}},$$ satisfying the boundary conditions $$\label{e1.2} u(0)-B_0(u^{\triangle }(\eta ))=0,\quad u^{\triangle }(T)=0,$$ or $$\label{e1.3} u^{\triangle }(0)=0,\quad u(T)+B_1(u^{\triangle }(\eta ))=0,$$ where $\Phi _p(s)$ is $p$-Laplacian operator, i.e., $\Phi _p(s)=|s| ^{p-2}s$, $p>1,(\Phi _p)^{-1}=\Phi _{q}$, $\frac 1p+\frac 1q=1$, $\eta \in (0,\rho (T))_{\mathbb{T}}$. Here, by multiplicity we mean the existence of $m$ solutions, where $m$ is an arbitrary natural number. In this paper we assume the following hypotheses: \begin{itemize} \item[(H1)] $f:[0,T]_{\mathbb{T}}\times \mathbb{R}^{+}\to\mathbb{R}^{+}$ is continuous ($\mathbb{R}^{+}$ denotes the nonnegative real numbers) \item[(H2)] $a:\mathbb{T}\to\mathbb{R}^{+}$ is left dense continuous (i.e., $a\in C_{\rm ld}(\mathbb{T},\mathbb{R}^{+} )$) and does not vanish identically on any closed subinterval of $[0,T]_{\mathbb{T}}$, where $C_{\rm ld}(\mathbb{T},\mathbb{R}^{+} )$ denotes the set of all left dense continuous functions from $\mathbb{T}$ to $\mathbb{R}^{+}$. \item[(H3)] $B_0(v)$ and $B_1(v)$ are both continuous odd functions defined on $R$ and satisfy that there exist $C,D>0$ such that $Dv\leq B_j(v)\leq Cv,\quad \text{for all }v\geq 0,\quad j=0,1.$ \end{itemize} We remark that by a solution $u$ of \eqref{e1.1}, \eqref{e1.2} (respectively \eqref{e1.1},\eqref{e1.3}), we mean $u:\mathbb {T}\to\mathbb {R}$ is delta differentiable, $u^{\triangle }:\mathbb{T}^\kappa \to \mathbb {R}$ is nabla differentiable on $\mathbb{T}^\kappa \cap \mathbb{T} _\kappa$ and $u^{\triangle \nabla }:\mathbb{T}^\kappa \cap \mathbb{T} _\kappa \to\mathbb {R}$ is continuous, and $u$ satisfies boundary conditions \eqref{e1.2} (respectively \eqref{e1.3}). If $u^{\triangle \nabla }(t)\leq 0$ on $[ 0,T]_{\mathbb{T}^\kappa \cap \mathbb{T}_\kappa }$, then we say $u$ is concave on $[0,T]_{\mathbb{T}}$. Anderson, Avery and Henderson \cite{a5} considered the problem \begin{gather*} [ \Phi _p(u^{\triangle }(t))]^{\triangledown }+c(t)f(u(t)) = 0, \quad t\in (a,b)_{\mathbb{T}}, \\ u(a)-B_0(u^{\triangle }(v)) = 0,\quad u^{\triangle }(b)=0, \end{gather*} where $v\in (a,b)_{\mathbb{T}}$, $f\in C( \mathbb{R}^{+},\mathbb{R}^{+})$, $c\in C_{\rm ld}((a,b)_{\mathbb{T}}, \mathbb{R}^{+})$ and $K_mx\leq B_0(x)\leq K_Mx$ for some positive constants $K_m$, $K_M$. They established the existence result of at least one positive solution by a fixed point theorem of cone expansion and compression of functional type. Very recently, in the case $f(t,l)=f(l)$, the existence of two positive solutions for the problem \eqref{e1.1}, \eqref{e1.2} and \eqref{e1.1}, \eqref{e1.3} has been established by He \cite{h1} by using a new double fixed-point theorem due to Avery, et al \cite{a7} in a cone. In this paper, we shall apply the method arising in papers \cite{y1,y2} to problem \eqref{e1.1}, \eqref{e1.2}. The main ingredient is the Guo-Krasnosel'skii fixed point theorem in cone. By considering the properties of $f$ on a bounded set of $[ 0,T]_{\mathbb{T}}\times \mathbb{R}^{+}$, we shall establish a basic existence criterion, that is theorem \ref{thm3.1}. Then, we shall prove the existence of $m$ positive solutions (in section 4) and the existence of infinitely many positive solutions (in section 5). In the remainder of this section we will provide without proof several foundational definitions and results from the calculus on time scales so that the paper is self-contained. For more details, one can see \cite{a1,a6,b1,h2,k2}. \begin{definition} \label{def1.1} \rm For $t<\sup \mathbb{T}$ and $r>\inf\mathbb{T}$, define the forward jump operator $\sigma$ and the backward jump operator $\rho$, respectively, $\sigma (t)=\inf \{\tau \in \mathbb{T}|\tau >t\}\in \mathbb{T},\quad \rho (r)=\sup \{\tau \in \mathbb{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^{\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 $s\in V$. \end{definition} 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 the forward difference operator while $x^\nabla (t)=x(t)-x(t-1)$ is the backward difference operator. \begin{definition} \label{def1.3}\rm If $F^{\triangle }(t)=f(t)$, then we define the delta integral by $\int_a^tf(s)\triangle s=F(t)-F(a).$ 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 paper, we assume $\mathbb{T}$ is a nonempty closed subset of $\mathbb{R}$ with $0\in \mathbb{T}_\kappa$, $T\in \mathbb{T}^\kappa$. \section{Preliminaries} Consider the Banach space $E=C_{\rm ld}([ 0,T]_{\mathbb{T}},\mathbb{R})$ with norm $\|u\|=\sup_{t\in [ 0,T]_{\mathbb{T}}}| u(t)|$. Then define the cone by $K=\left\{ u\in E|\quad u\text{ is concave and nonnegative valued on } [ 0,T]_{\mathbb{T}},\text{ and }u^{\triangle }(T)=0\right\}.$ From \cite{h1}, we know that if $u\in K$, then $\inf_{t\in [ \eta ,T]_{\mathbb{T} }}u(t)\geq (\eta /T)\|u\|.$ We define a operator $F:K\to E$ by $(Fu)(t)=B_0(\Phi _q(\int_\eta ^Ta(r)f(r,u(r))\nabla r))+\int_0^t\Phi _q(\int_s^Ta(r)f(r,u(r))\nabla r)\triangle s,$ and from \cite{h1}, we also know $F:K\to K$ is completely continuous. We denote the constants $A=\Big[ C\Phi _q(\int_\eta ^Ta(r)\nabla r)+\int_0^T\Phi _q(\int_s^Ta(r)\nabla r)\triangle s\Big]^{-1}, \quad B=\Big[ D\Phi _q(\int_\eta ^Ta(r)\nabla r)\Big]^{-1}.$ Clearly, $0(TB^{p-1})/\eta$, then $\overline{ \beta }_0>B^{p-1}$. \item If $\min \underline{f}_\infty >(TB^{p-1})/\eta$, then $\overline{\beta }_\infty >B^{p-1}$. \end{enumerate} \end{lemma} \begin{proof} It is easy to show that the following inequalities hold: \begin{gather*} \limsup _{l\to0}\max_{t\in [ 0,T]_{\mathbb{T}}}f(t,l)/l^{p-1} \geq \liminf _{l\to0}\max \{f(t,c):(t,c)\in [ 0,T]_{\mathbb{T}}\times [ 0,l] \}/l^{p-1}, \\ \limsup _{l\to+\infty }\max_{t\in [ 0,T]_{ \mathbb{T}}}f(t,l)/l^{p-1} \geq \liminf _{l\to+\infty }\max \{f(t,c):(t,c)\in [ 0,T]_{\mathbb{T}} \times [ 0,l]\}/l^{p-1}, \\ \begin{aligned} &\liminf _{l\to0}\min_{t\in [ \eta ,T]_{\mathbb{T}}}f(t,l)/l^{p-1}\\ &\leq \limsup _{l\to0}\min \{f(t,c):(t,c)\in [ \eta ,T]_{\mathbb{T}}\times [ ( \eta /T)l,l]\}/((l^{p-1}\eta )/T), \end{aligned} \\ \begin{aligned} &\liminf _{l\to+\infty }\min_{t\in [ \eta ,T]_{ \mathbb{T}}}f(t,l)/l^{p-1}\\ &\leq \limsup _{l\to+\infty }\min \{f(t,c):(t,c)\in [ \eta ,T]_{\mathbb{T}}\times [ ( \eta /T)l,l]\}/((l^{p-1}\eta )/T). \end{aligned} \end{gather*} The statements of the lemma follow from these inequalities. \end{proof} The following Lemma is crucial in our argument, which is the well-known Guo-Krasnosel'skii fixed point theorem in cone. \begin{lemma}[\cite{g1,k1}] \label{lem2.2} Let $X$ be a Banach space and $K\subset E$ be a cone in $X$. Assume $\Omega _1$, $\Omega _2$ are bounded open subsets of $K$ with $0\in \Omega _1\subset \overline{\Omega }_1\subset \Omega _2$ and $T:K\to K$ is a completely continuous operator such that either: \begin{enumerate} \item $\|Tw\|\leq \|w\|$, $w\in \partial \Omega _1$, and $\|Tw\|\geq \|w\|$, $w\in \partial \Omega _2$; or \item $\|Tw\|\geq \|w\|$, $w\in \partial \Omega _1$, and $\|Tw\|\leq \|w\|$, $w\in \partial \Omega _2$. \end{enumerate} Then $T$ has a fixed point in $\overline{\Omega }_2\backslash \Omega _1$. \end{lemma} \section{Existence results} \begin{theorem} \label{thm3.1} Assume that there exist two positive numbers $a$, $b$ such that $\alpha (a)\leq (aA)^{p-1}$, $\beta (b)\geq (bB)^{p-1}$. Then problem \eqref{e1.1}, \eqref{e1.2} has at least one positive solution $u^{*}\in K$ satisfying $\min \{a,b\}\leq u^{*}\leq \max \{a,b\}.$ \end{theorem} \begin{proof} First of all, we claim $a\neq b$. If not, $a=b$. Noticing that $A0}\alpha (l)/l^{p-1}0}\beta (l)/l^{p-1}>B^{p-1}$. Then problem \eqref{e1.1}, \eqref{e1.2} has at least one positive solution. \end{corollary} By Theorem \ref{thm3.1} and Lemma \ref{lem2.1} we have the following results. \begin{corollary} \label{coro3.3} Assume that one of the following conditions holds: \begin{enumerate} \item $\underline{\alpha }_0B^{p-1}$ (in particular, $\underline{\alpha }_0=0$ and $\overline{\beta }_\infty =+\infty$, \item $\overline{\beta }_0>B^{p-1}$ and $\underline{\alpha }_\infty (TB^{p-1})/\eta$ (in particular, $\max f_0=0$ and $\min f_\infty =+\infty$) \item $\min \underline{f}_0>(TB^{p-1})/\eta$ and $\max \overline{f}_\infty 0}\alpha (l)/l^{p-1}0$ such that $\alpha (a)<(aA)^{p-1})$, \item $\max \{\min \underline{f}_0,\min \underline{f}_\infty \} >(TB^{p-1})/\eta$ (in particular when $\min f_0=+\infty$ or $\min f_\infty =+\infty$). \end{enumerate} Then problem \eqref{e1.1}, \eqref{e1.2} has at least one positive solution. \end{corollary} \begin{corollary} \label{coro3.6} Assume that: \begin{enumerate} \item $\sup_{l>0}\beta (l)/l^{p-1}>B^{p-1}$ (in particular, there exists $b>0$ such that $\beta (b)>(bB)^{p-1})$, \item $\min \{\max \overline{f}_0,\max \overline{f}_\infty \}(a_{2k}B)^{p-1}$, $k=1,\dots$, $[ \frac{ m+1}2]$ \item $\beta (a_{2k-1})>(a_{2k-1}B)^{p-1}$, $k=1,\dots$, $[ \frac{m+2} 2]$, $\alpha (a_{2k})<(a_{2k}A)^{p-1}$, $k=1,\dots$, $[ \frac{ m+1}2]$. \end{enumerate} Then problem \eqref{e1.1}, \eqref{e1.2} has at least $m$ positive solutions $u_1^{*}$, $u_2^{*}$, $\dots$, $u_m^{*}$ satisfying $a_k<\|u_k^{*}\|0}\alpha (l)/l^{p-1}0 such that \alpha (a)<(aA)^{p-1}) \item \min \{\min \underline{f}_0, \min \underline{f}_\infty \}>( TB^{p-1})/\eta  (in particular, \min f_0=\min f_\infty =+\infty ). \end{enumerate} Then problem \eqref{e1.1}, \eqref{e1.2} has at least two positive solutions. \end{corollary} \begin{corollary} \label{coro4.3} Assume that: \begin{enumerate} \item \sup_{l>0}\beta (l)/l^{p-1}>B^{p-1} (in particular, there exists b>0 such that \beta (b)>(bB)^{p-1}) \item \max \{\max \overline{f}_0, \max \overline{f}_\infty \}(TB^{p-1})/\eta  and \max \overline{ f}_\infty (a_2B)^{p-1}. \end{enumerate} Then problem \eqref{e1.1}, \eqref{e1.2} has at least three positive solutions. \end{corollary} \begin{corollary} \label{coro4.5} Let 0( TB^{p-1})/\eta  (in particular, \max f_0=0 and \min f_\infty =+\infty ) \item \beta (a_1)>(a_1B)^{p-1} and \alpha (a_2)<(a_2A)^{p-1}. \end{enumerate} Then problem \eqref{e1.1}, \eqref{e1.2} has at least three positive solutions. \end{corollary} Obviously, analogous results still hold for arbitrary number m. Also we have the following result. \begin{theorem} \label{thm4.6} Let 0B^{p-1} (in particular, \underline{\alpha }_0=0 and \overline{\beta }_0=+\infty ). Then problem \eqref{e1.1}, \eqref{e1.2} has a sequence of positive solutions \{u_k^{*}\} _{k=1}^\infty  satisfying \|u_k^{*}\| \to 0 as k\to\infty. \end{theorem} \begin{proof} Since \liminf _{l\to0}\alpha (l)/l^{p-1}B^{p-1}, there exist two sequences of positive numbers a_k\to 0 and b_k\to 0 such that \[ \alpha (a_k)\leq (a_kA)^{p-1},\quad \beta (b_k)\geq (b_kB)^{p-1},\quad k=1,\quad 2,\dots$ Without loss of generality, we may assume $a_1>b_1>a_2>b_2>\dots >a_k>b_k>\dots .$ Now applying Theorem \ref{thm3.1} to each pair of numbers $\{ b_k,a_k\}$, $k=1,2,\dots$, problem \eqref{e1.1}, \eqref{e1.2} has a sequence of positive solutions $\{ u_k^{*}t\} _{k=1}^\infty$ satisfying $b_k\leq \|u_k^{*}\|\leq a_k$. The proof is complete. \end{proof} \begin{corollary} \label{coro5.2} Assume that there exists an $l_0$ such that \[ \inf_{00,\quad \sup_{0(k_2TB^{p-1})/\eta $and$\min \underline{f}_0\eta /(Tk_2) (k_2TB^{p-1}/\eta )=B^{p-1}. \end{align*} Now the conclusion follows from Theorem \ref{thm5.1}. \end{proof} Similarly, we have the following statement. \begin{theorem} \label{thm5.3} Assume that $\underline{\alpha }_\infty B^{p-1}$ (in particular, $\underline{\alpha } _\infty =0$ and $\overline{\beta }_\infty =+\infty$). Then problem \eqref{e1.1}, \eqref{e1.2} has a sequence of positive solutions $\{u_k^{*}\} _{k=1}^\infty$ satisfying $\|u_k^{*}\|\to+\infty$ as $k\to\infty$. \end{theorem} \begin{corollary} \label{coro5.4} Assume that \begin{gather*} \inf_{00,\\ \sup_{0(k_2TB^{p-1})/\eta $and$\min \underline{f}_\infty \sqrt{6}=(a_2B)^{p-1},\\ \beta (a_4)=\min \{ f(u):u\in [ 12,24]\} =7>\sqrt{48}=(a_4B)^{p-1}. \end{gather*} Therefore, by Theorem \ref{thm4.6}, problem \eqref{e6.1}, \eqref{e6.2} has at least two positive solutions $u_1^{*}$, $u_2^{*}$ satisfying $1\leq \|u_1^{*}\| < \|u_2^{*}\|\leq 24$. The methods in this paper can also be used for studying problem \eqref{e1.1}, \eqref{e1.3}. \begin{thebibliography}{00} \bibitem{a1} R. P. Agarwal, M. Bohner; \emph{Basic calculus on time scales and some of its applications}, Result. Math. \textbf{35}(1999), 3-22. \bibitem{a2} R. P. Agarwal, M. Bohner and P. Wong; \emph{Sturm-Liouville eigenvalue problem on time scales}, Appl. Math. Comput. \textbf{99} (1999), 153-166. \bibitem{a3} R. P. Agarwal, D. O'Regan; \emph{Nonlinear boundary value problems on time scales}, Nonlinear Anal. \textbf{44}(2001), 527-535. \bibitem{a4} D. Anderson, \emph{Solutions to second-order three-point problem on time scales}, J. Difference Equations Appl. \textbf{8}(2002), 673-688. \bibitem{a5} D. Anderson, R. Avery and J. Henderson; \emph{Existence of solutions for a one-dimensional $p$-Laplacian on time scales}, J. Difference Equations Appl. \textbf{10}(2004), 889-896. \bibitem{a6} F. M. Atici, G. Sh. Guseinov; \emph{On Green's functions and positive solutions for boundary value problems on time scales}, J. Comput. Appl. Math. \textbf{141}(2002), 75-99. \bibitem{a7} R. I. Avery, C. J. Chyan and J. Henderson; \emph{Twin solutions of boundary value problems for ordinary differential equations and finite difference equations}, Comput. Math. Appl. \textbf{42}(2001), 695-704. \bibitem{b1} M. Bohner, A. Peterson, \emph{Dynamic Equations on Time Scales, An Introduction with Applications}, Birkh\"{a}user, Boston, 2001. \bibitem{c1} C. J. Chyan, J. Henderson; \emph{Eigenvalue problems for nonlinear differential equations on a measure chain}, J. Math. Anal. Appl. \textbf{245}(2000), 547-559. \bibitem{e1} L. Erbe, A. Peterson; \emph{Positive solutions for nonlinear differential equations on measure chain}, Math. Comput. Modelling. \textbf{32}(5-6)(2000), 571-585. \bibitem{e2} L. Erbe, S. Hilger, Sturmian theory on measure chains, \emph{Differential Equations Dyn. Systems}, \textbf{1}(1993), 223-246. \bibitem{g1} D. Guo, V. Lakshmikantham; \emph{Nonlinear Problems in Abstract Cones}, Academic press, Boston, 1988. \bibitem{h1} Z. M. He; \emph{Double positive solutions of three-point boundary value problems for p-Laplacian dynamic equations on time scales}, J. Comput. Appl. Math. \textbf{182}(2005), 304-315. \bibitem{h2} S. Hilger; \emph{Analysis on measure chains-a unified approach to continuous and discrete calculus}, Results Math., \textbf{18} (1990), 18-56. \bibitem{k1} M. Q. Krasnosel'skii, \emph{Positive solutions of operator equations}, P. Noordhoff Ltd., Groningen, The Netherlands, 1964. \bibitem{k2} B. Kaymakcalan, V. Lakshmikantham and S. Sivasundaram; \emph{Dynamic Systems on Measure Chains}, Kluwer Academic Publishers, Boston, 1996. \bibitem{y1} Q. Yao; \emph{Positive solutions for eigenvalue problems of fourth-order elastic beam equations}, Appl. Math. Letter. \textbf{17} (2004), 237-243. \bibitem{y2} Q. Yao; \emph{Existence, multiplicity and infinite solvability of positive solutions to a nonlinear fourth-order periodic boundary value problem}, Nonlinear Analysis, \textbf{63}(2005), 237-246. \end{thebibliography} \end{document}