\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 45, 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 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/45\hfil Existence of positive solutions] {Existence of positive solutions for even-order $m$-point boundary-value problems on time scales} \author[\.I. Yaslan \hfil EJDE-2013/45\hfilneg] {\.Isma\.il Yaslan} % in alphabetical order \address{\.Isma\.il Yaslan \newline Pamukkale University\\ Department of Mathematics\\ 20070 Denizli, Turkey} \email{iyaslan@pau.edu.tr} \thanks{Submitted October 15, 2012. Published February 8, 2013} \subjclass[2000]{34B18, 34N05, 39A10} \keywords{Boundary value problems; cone; fixed point theorems; \hfill\break\indent positive solutions; time scales} \begin{abstract} In this article, we consider a nonlinear even-order $m$-point bound\-ary-value problems on time scales. We establish the criteria for the existence of at least one, two and three positive solutions for higher order nonlinear $m$-point boundary-value problems on time scales by using the four functionals fixed point theorem, Avery-Henderson fixed point theorem and the five functionals fixed point theorem, respectively. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} Higher order multi-point boundary value problems on time scales have attracted the attention of many researchers in recent years; see for example \cite{and04, and06, and08, do10, han08, hu11, hu112, sang07, su11, su08, wang09, yas12} and the references therein. In this article, we are concerned with the existence of single and multiple positive solutions to the following nonlinear higher order $m$-point boundary value problem (BVP) on time scales: $$\label{1.1} \begin{gathered} (-1)^{n}y^{\Delta ^{2n}}(t)= f(t,y(t)),\quad t\in [t_1,t_m]\subset \mathbb{T} ,\; n\in \mathbb{N} \\ y^{\Delta ^{2i+1} }(t_m)=0,\quad \alpha y^{\Delta^{2i}}(t_1)-\beta y^{\Delta^{2i+1} }(t_1)=\sum_{k=2}^{m-1}y^{\Delta^{2i+1} }(t_{k}), \end{gathered}$$ where $\alpha >0$ and $\beta > 0$ are given constants, $t_1 0$ and $\beta > 0$, then the Green's function $G(t,s)$ in \eqref{2.1} satisfies the inequality $G(t,s)\geq \frac{t-t_1}{t_m-t_1}G(t_m,s)$ for $(t,s)\in [t_1,t_m]\times[t_1,t_m]$. \end{lemma} \begin{proof} (i) Let $s\in [t_1,t_m]$ and $t\leq s$. Then we obtain $\frac{G(t,s)}{G(t_m,s)}=\frac{t+\frac{\beta+j-1}{\alpha}-t_1}{t_m+\frac{\beta+j-1}{\alpha}-t_1}> \frac{t-t_1}{t_m-t_1}.$ (ii) For $s\in [t_1,t_m]$ and $s\leq t$, we have $\frac{G(t,s)}{G(t_m,s)} = 1\geq\frac{t-t_1}{t_m-t_1}.$ \end{proof} \begin{lemma}\label{L2.2} If $\alpha >0$ and $\beta>0$, then the Green's function $G(t,s)$ in $\eqref{2.1}$ satisfies 00 and \beta>0, H_{j}(t,s)>0 for all j=1,2,\dots,m-1. Then we obtain G(t,s)>0 from \eqref{2.1}. Now, we will show that G(t,s)\leq G(s,s). (i) Let  s\in [t_1,t_m] and t\leq s. Since G(t,s) is nondecreasing in t, G(t,s)\leq G(s,s). (ii) For s\in [t_1,t_m] and s\leq t, it is clear that G(t,s)= G(s,s). \end{proof} \begin{lemma}\label{L2.3} If \alpha >0, \beta >0 and s\in [t_1,t_m], then the Green's function G(t,s) in \eqref{2.1} satisfies \begin{align*} \min_{t\in [t_{m-1},t_m]}G(t,s)\geq K\|G(.,s)\|, \end{align*} where $$\label{2.2} K=\frac{\beta+\alpha(t_{m-1}-t_1)}{\beta+m-2+\alpha(t_m-t_1)}$$ and \|x\|=\max_{t\in [t_1,t_m]}|x(t)|. \end{lemma} \begin{proof} Since the Green's function G(t,s) in \eqref{2.1} is nondecreasing in t, We have \min_{t\in [t_{m-1},t_m]}G(t,s)=G(t_{m-1},s) In addition, it is obvious that \|G(.,s)\|=G(s,s) for s\in [t_1,t_m] by Lemma \ref{L2.2}. Then we have \begin{align*} G(t_{m-1},s)\geq K G(s,s) \end{align*} from the branches of the Green's function G(t,s). \end{proof} If we let G_1(t,s):=G(t,s) for G as in \eqref{2.1}, then we can recursively define \begin{align*} G_{j}(t,s)=\int_{t_1}^{t_m}G_{j-1}(t,r)G(r,s)\Delta r \end{align*} for 2\leq j\leq n and G_{n}(t,s) is Green's function for the homogeneous problem \begin{gather*} (-1)^{n}y^{\Delta ^{2n}}(t)= 0,\quad t\in [t_1,t_m], \\ y^{\Delta ^{2i+1} }(t_m)=0,\quad \alpha y^{\Delta^{2i}}(t_1)-\beta y^{\Delta^{2i+1} }(t_1) =\sum_{k=2}^{m-1}y^{\Delta^{2i+1} }(t_{k}), \end{gather*} where m \geq 3 and 0\leq i\leq n-1. \begin{lemma}\label{L2.4} Let \alpha >0, \beta>0. The Green's function G_{n}(t,s) satisfies the following inequalities \begin{gather*} 0\leq G_{n}(t,s)\leq L^{n-1} \|G(.,s)\|,\quad (t,s)\in [t_1,t_m]\times [t_1,t_m], \\ G_{n}(t,s)\geq K^{n}M^{n-1}\|G(.,s)\|,\quad (t,s)\in [t_{m-1},t_m]\times [t_1,t_m] \end{gather*} where K is given in \eqref{2.2}, and \begin{gather}\label{2.3} L=\int_{t_1}^{t_m}\|G(.,s)\|\Delta s>0,\\ \label{2.4} M=\int_{t_{m-1}}^{t_m}\|G(.,s)\|\Delta s>0. \end{gather} \end{lemma} The proof of the above lemma is done using induction on n and Lemma \ref{L2.3}. Let \mathcal{B} denote the Banach space C[t_1,t_m] with the norm \|y\|=\max_{t\in [t_1,t_m]}|y(t)|. Define the cone P\subset \mathcal{B} by $$\label{2.5} P = \{y\in \mathcal{B}: y(t)\geq 0, \min_{t\in [t_{m-1},t_m]} y(t) \geq \frac{K^{n} M^{n-1}}{L^{n-1}} \|y\|\}$$ where K, L, M are given in \eqref{2.2}, \eqref{2.3}, \eqref{2.4}, respectively. Note that \eqref{1.1} is equivalent to the nonlinear integral equation $$\label{2.6} y(t)=\int_{t_1}^{t_m}G_{n}(t,s)f(s,y(s)) \Delta s.$$ We can define the operator A:P \to \mathcal{B} by $$\label{2.7} Ay(t)=\int_{t_1}^{t_m}G_{n}(t,s)f(s,y(s)) \Delta s,$$ where y\in P. Therefore, solving \eqref{2.6} in P is equivalent to finding fixed points of the operator A. It is clear that AP\subset P and A:P\to P is a completely continuous operator by a standard application of the Arzela-Ascoli theorem. Now we state the fixed point theorems which will be applied to prove main theorems. We are now in a position to present the four functionals fixed point theorem. Let \varphi and \Psi be nonnegative continuous concave functionals on the cone P, and let \eta and \theta be nonnegative continuous convex functionals on the cone P. Then for positive numbers r, \tau, \mu and R, define the sets \begin{gather*} Q(\varphi, \eta, r, R)=\{x\in P: r\leq \varphi(x), \eta (x)\leq R\}, \\ U(\Psi, \tau )= \{x\in Q(\varphi, \eta, r, R): \tau\leq \Psi(x)\}, \\ V(\theta, \mu)= \{x\in Q(\varphi, \eta, r, R): \theta(x)\leq \mu\}. \end{gather*} The following theorem can be found in \cite{av08}. \begin{theorem}[Four Functionals Fixed Point Theorem] \label{T2.2} Suppose P is a cone in a real Banach space E, \varphi and \Psi are nonnegative continuous concave functionals on P, \eta and \theta are nonnegative continuous convex functionals on P, and there exist nonnegative positive numbers r, \tau, \mu and R, such that A:Q(\varphi, \eta, r, R)\to P is a completely continuous operator, and Q(\varphi, \eta, r, R) is a bounded set. If \begin{itemize} \item[(i)] \{x\in U(\Psi, \tau): \eta(x)r for all u\in \partial P(\phi ,r), \item[(ii)] \theta (Au)p for all u\in \partial P(\eta ,p), \end{itemize} then A has at least two fixed points u_1 and u_2 such that \[ p<\eta (u_1) \textmd{ with }\theta (u_1)b\}\neq \emptyset and \gamma (Ax)>b \\ for x\in P(\varphi, \theta, \gamma, b, k, c), \item[(ii)]  \{x\in Q(\varphi, \eta, \Psi, h, a, c): \eta(x)b, for x\in P(\varphi, \gamma, b, c), with \theta(Ax)>k, \item[(iv)] \eta (Ax)b,\quad \eta(x_{3})>a \quad\textmd{with } \gamma(x_{3})0 and \beta > 0. Suppose that there exist constants r, R, \mu, \tau  with 0r. \end{gather*} Then, we have \frac{\mu}{2}\in\{y\in U(\Psi,\tau):\eta(y)r\}, which means that (i) in Theorem \ref{T2.2} is fulfilled. Now, we shall verify that condition (ii) of Theorem \ref{T2.2} is satisfied. By Lemma \ref{L2.4}, we obtain \begin{align*} \theta (Ay) &=\int_{t_1}^{t_m}G_{n}(t_m,s)f(s,y(s)) \Delta s \\ &\leq L^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s. \end{align*} Since \theta(Ay)>\mu, we find that $$\label{3.2} \int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s > \frac{\mu}{L^{n-1}}.$$ Then, we obtain \begin{align*} \varphi (Ay) &=\int_{t_1}^{t_m}G_{n}(t_{m-1},s)f(s,y(s)) \Delta s \\ &\geq K^{n}M^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s >r, \end{align*} using Lemma \ref{L2.4} and \eqref{3.2}. Now, we shall show that condition (iii) of Theorem \ref{T2.2} holds. Since \varphi(y)=r and y\in V(\theta,\mu), we find that r\leq y(t)\leq \mu for t\in [t_{m-1},t_m]. By Lemma \ref{L2.4} and the hypothesis (i), we have \begin{align*} \varphi (Ay) &=\int_{t_1}^{t_m}G_{n}(t_{m-1},s)f(s,y(s)) \Delta s \\ &\geq K^{n}M^{n-1}\int_{t_{m-1}}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s \geq r. \end{align*} Now, we shall verify that condition (iv) of Theorem \ref{T2.2} is fulfilled. We get \begin{align*} \Psi (Ay) &=\int_{t_1}^{t_m}G_{n}(t_{m-1},s)f(s,y(s)) \Delta s \\ &\geq K^{n}M^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s \end{align*} using Lemma \ref{L2.4}. Since \Psi(Ay)<\tau, $$\label{3.3} \int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s <\frac{\tau}{K^{n}M^{n-1}}.$$ Then, by Lemma \ref{L2.4} and \eqref{3.3} we obtain \begin{align*} \eta (Ay) &=\int_{t_1}^{t_m}G_{n}(t_m,s)f(s,y(s)) \Delta s \\ &\leq L^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s < R. \end{align*} Finally, we shall show that condition (v) of Theorem \ref{T2.2} is satisfied. Since \eta(y)=R, we find 0\leq y(t)\leq R for t\in [t_1,t_m]. Using Lemma \ref{L2.4} and the hypothesis (ii), we have \begin{align*} \eta (Ay) &=\int_{t_1}^{t_m}G_{n}(t_m,s)f(s,y(s)) \Delta s \\ &\leq L^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s \leq R. \end{align*} Hence, by Theorem \ref{T2.2}, the \eqref{1.1} has at least one positive solution y such that r\leq y(t)\leq R for t\in [t_1,t_m]. This completes the proof. \end{proof} Now we will use the Avery-Henderson fixed point theorem to prove the next theorem. \begin{theorem}\label{T4.1} Assume \alpha >0, \beta>0. Suppose there exist numbers 0\frac{r}{K^{n} M^{n}} for (t,y)\in [t_{m-1},t_m]\times[r,\frac{rL^{n-1}}{K^{n}M^{n-1}}]; \item[(ii)] f(t,y)<\frac{q}{L^{n}} for (t,y)\in [t_1,t_m]\times[0,\frac{qL^{n-1}}{K^{n}M^{n-1}}]; \item[(iii)] f(t,y)>\frac{p}{K^{n} M^{n}} for t\in [t_{m-1},t_m]\times[\frac{K^{n}M^{n-1}}{L^{n-1}}p,p], \end{itemize} where K, L, M, are defined in \eqref{2.2}, \eqref{2.3}, \eqref{2.4}, respectively. Then \eqref{1.1} has at least two positive solutions y_1 and y_2 such that \begin{gather*} p < \max_{t\in [t_1,t_m]}y_1(t)\quad \text{with }\max_{t\in [t_{m-1},t_m]}y_1(t) r: Since y\in \partial P(\phi ,r) and \|y\| \leq \frac{L^{n-1}}{K^{n}M^{n-1}}\phi (y), we have r\leq y(t)\leq \frac{rL^{n-1}}{K^{n}M^{n-1}} for t\in [t_{m-1},t_m]. Then, by hypothesis (i) and Lemma \ref{L2.4} we find that \begin{align*} \phi(Ay)&= \int_{t_1}^{t_m}G_{n}(t_{m-1},s)f(s,y(s)) \Delta s \\ &\geq K^{n} M^{n-1}\int_{t_{m-1}}^{t_m} \|G(.,s)\|f(s,y(s)) \Delta s >r. \end{align*} \noindent \textbf{Claim 2:} If y\in \partial P(\theta ,q), then \theta(Ay)p  for all y\in \partial P(\eta ,p): Since \frac{p}{2}\in P and p>0, \frac{p}{2}\in P(\eta,p). If y\in \partial P(\eta ,p) and \eta(y)\geq \frac{K^{n}M^{n-1}}{L^{n-1}}\|y\|, we obtain \frac{K^{n}M^{n-1}}{L^{n-1}}p\leq y(t)\leq \|y\|=p for t\in [t_{m-1},t_m]. Hence, by hypothesis (iii) and Lemma \ref{L2.4} we have \begin{align*} \eta(Ay)&= \int_{t_1}^{t_m}G_{n}(t_m,s)f(s,y(s)) \Delta s \\ &\geq K^{n} M^{n-1}\int_{t_{m-1}}^{t_m} \|G(.,s)\|f(s,y(s)) \Delta s >p. \end{align*} Since the conditions of Theorem \ref{T2.3} are satisfied, BVP \eqref{1.1} has at least two positive solutions y_1 and y_2 such that \begin{gather*} p < \max_{t\in [t_1,t_m]}y_1(t)\quad \textmd{with } \max_{t\in [t_{m-1},t_m]}y_1(t)0 and \beta >0. Suppose that there exist constants a, b, c with 0\frac{b}{K^{n}M^{n}} for (t,y)\in[t_{m-1},t_m]\times [b,\frac{bL^{n-1}}{K^{n}M^{n-1}}], \item[(iii)] f(t,y)<\frac{a}{L^{n}} for (t,y)\in[t_1,t_m]\times[0,a], \end{itemize} where K, L, M are as defined in \eqref{2.2}, \eqref{2.3}, \eqref{2.4}, respectively. Then \eqref{1.1} has at least three positive solutions y_1, y_2 and y_{3} such that \begin{gather*} \max_{t\in [t_1,t_m]}y_1(t)b,\\ \theta(y_1)=b+\epsilon_1<\frac{bL^{n-1}}{K^{n}M^{n-1}},\\ \varphi(y_1)=b+\epsilon_1<\frac{bL^{n-1}}{K^{n}M^{n-1}}b\}\neq \emptyset. If $y\in P(\varphi, \theta, \gamma, b, \frac{bL^{n-1}}{K^{n}M^{n-1}}, c)$, then we have $b\leq y(t)\leq \frac{bL^{n-1}}{K^{n}M^{n-1}}$ for all $t\in [t_{m-1},t_m]$. By using Lemma \ref{L2.4} and the hypothesis (ii), we obtain \begin{align*} \gamma (Ay) &=\int_{t_1}^{t_m}G_{n}(t_{m-1},s)f(s,y(s)) \Delta s \\ &\geq K^{n}M^{n-1}\int_{t_{m-1}}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s > b. \end{align*} Thus, the condition (i) of Theorem \ref{T2.4} holds. Let $y_2=\frac{a-\epsilon_2}{2}$ such that $0<\epsilon_2<(1-\frac{K^{n}M^{n-1}}{L^{n-1}})a$. Since \begin{gather*} \eta(y_2)=a-\epsilon_2\frac{K^{n}M^{n-1}}{L^{n-1}}a,\\ \varphi(y_2)=a-\epsilon_2\frac{bL^{n-1}}{K^{n}M^{n-1}}, we obtain $$\label{4.1} \int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s >\frac{b}{K^{n}M^{n-1}}.$$ Then, by Lemma \ref{L2.4} and \eqref{4.1} we find that \begin{align*} \gamma (Ay) &=\int_{t_1}^{t_m}G_{n}(t_{m-1},s)f(s,y(s)) \Delta s \\ &\geq K^{n}M^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s >b. \end{align*} Finally, we shall verify that the condition (iv) of Theorem \ref{T2.4} holds. By Lemma \ref{L2.4}, we obtain \begin{align*} \Psi (Ay) &=\int_{t_1}^{t_m}G_{n}(t_{m-1},s)f(s,y(s)) \Delta s \\ &\geq K^{n}M^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s. \end{align*} Since\Psi (Ay)<\frac{K^{n}M^{n-1}}{L^{n-1}}a, we have $$\label{4.2} \int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s <\frac{a}{L^{n-1}}.$$ Then, we find that \begin{align*} \eta (Ay) &= \int_{t_1}^{t_m}G_{n}(t_m,s)f(s,y(s)) \Delta s \\ &\leq L^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s < a. \end{align*} using Lemma \ref{L2.4} and \eqref{4.2}. Since the conditions of Theorem \ref{T2.4} are satisfied, \eqref{1.1} has at least three positive solutionsy_1, y_2, y_{3}\in \overline{P(\varphi,c)}\$ such that \begin{gather*} \max_{t\in [t_1,t_m]}y_1(t)