\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 245, pp. 1--16.\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-2013/245\hfil Existence of solutions] {Existence of solutions for two-point boundary-value problems with singular differential equations of variable order} \author[S. Zhang \hfil EJDE-2013/245\hfilneg] {Shuqin Zhang} % in alphabetical order \address{Shuqin Zhang \newline Department of Mathematics, China University of Mining and Technology, \newline Beijing 100083, China} \email{zsqjk@163.com, Tel +86 10 62331118, Fax +86 10 62331465} \thanks{Submitted May 22, 2013. Published November 12, 2013.} \subjclass[2000]{26A33, 34B15} \keywords{Derivatives and integrals of variable order; singular; \hfill\break\indent differential equations of variable order; Arzela-Ascoli theorem} \begin{abstract} In this work, we show the existence of a solution for a two-point boundary-value problem having a singular differential equation of variable order. We use some analysis techniques and the Arzela-Ascoli theorem, and then illustrate our results with examples. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Fractional calculus (fractional derivatives and integrals) refer to the differential and integral operators of arbitrary order, and fractional differential equations refer to those containing fractional derivatives. The former are the generalization of integer-order differential and integral operators and the latter, the generalization of differential equations of integer order. The derivatives and integrals of variable-order, which fall into a more complex category, are those whose orders are the functions of certain variables. Recently, derivatives and integrals and differential equations of variable-order have been considered, see the references in this article. In these works, authors consider the applications of variable-order derivatives in various topics, such as anomalous diffusion modeling, mechanical applications, multifractional Gaussian noises. Moreover, a physical experimental study of calculus of variable-order has been considered in \cite{r1}, a comparative study of constant-order and variable-order models has been considered in \cite{s5}. The nonlinear functional analysis methods (such as some fixed point theorems) have played a very important role in considering existence of solutions to differential equations of integer order and fractional order (constant order, such as 1/3). For such applications, because differential equations can be transformed into integral equations, by means of some fundamental properties of differential and integral calculus of integer order and fractional calculus (constant order). But, in general, we find that calculus of variable-order lacks these fundamental properties, thereby making it difficult to apply nonlinear functional analysis methods to consider existence of solution to problems for differential equations of variable-order. The following are several definitions of derivatives and integrals of variable-order for a function $f$, which can be founded in for example in \cite{r1,v1}, $$I_{a+}^{p(t)}f(t)=\int_{a}^{t}\frac{(t-s)^{p(t)-1}}{\Gamma(p(t))}f(s)ds,\quad p(t)>0,\; t>a,\label{e1.1}$$ where $\Gamma(\cdot)$ denotes the Gamma function, $-\infty0,\; t>a,\label{e1.2} provided that the right-hand side is pointwise defined. $$I_{a^{+}}^{p(t)}f(t)=\int_{a}^{t}\frac{(t-s)^{p(t-s)-1}}{\Gamma(p(t-s))}f(s)ds, \quad p(t)>0,\; t>a,\label{e1.3}$$ provided that the right-hand side is pointwise defined. $$D_{a+}^{p(t)}f(t)=\frac{ d^n}{dt^n}I_{a+}^{n-p(t)}f(t) =\frac{d^n}{dt^n}\int_{a}^{t}\frac{(t-s)^{n-1-p(t)}}{\Gamma(n-p(t))}f(s)ds, \quad t>a,\label{e1.4}$$ where$n-1a, n\in \mathbb{N}$, provided that the right-hand side is pointwise defined. $$D_{a+}^{p(t)}f(t)=\frac {d^n}{dt^n}I_{a+}^{n-p(t)}f(t) =\frac{ d^n}{dt^n}\int_{a}^{t}\frac{(t-s)^{n-1-p(s)}}{\Gamma(n-p(s))}f(s)ds, \quad t>a,\label{e1.5}$$ where$n-1a, n\in \mathbb{N}$, provided that the right-hand side is pointwise defined. $$D_{a^{+}}^{p(t)}f(t)=\frac{ d^n}{dt^n}I_{a+}^{n-p(t)}f(t) =\frac {d^n}{dt^n}\int_{a}^{t}\frac{(t-s)^{n-1-p(t-s)}}{\Gamma(n-p(t-s))}f(s)ds, \quad t>a,\label{e1.6}$$ where$n-1a$,$n\in \mathbb{N}$, provided that the right-hand side is pointwise defined. In particular, when$p(t)$is a constant function,$p(t)\equiv q$, where$q$is a finite positive constant, then$I_{a+}^{p(t)}, D_{a+}^{p(t)}$are usual Riemann-Liouville fractional integral$I_{a+}^q$and derivative$D_{a+}^q$, see \cite{k1}. It is well known that fractional calculus$I_{a+}^q, D_{a+}^q$have the following very important properties, which play a very important role in considering existence of solutions of fractional differential equation denoted by$D_{a+}^q$, by means of some fixed point theorems. \begin{proposition}[\cite{k1}] \label{prop1.1} The equality$I_{a+}^\gamma I_{a+}^\delta f(t)=I_{a+}^{\gamma+\delta}f(t)$,$\gamma>0$,$\delta>0$holds for$f\in L(a,b)$. \end{proposition} \begin{proposition}[\cite{k1}] \label{prop1.2} The equality$D_{a+}^\gamma I_{a+}^\gamma f(t)=f(t)$,$\gamma>0$holds for$f\in L(a,b)$. \end{proposition} \begin{proposition}[\cite{k1}] \label{prop1.3} Let$\alpha>0$. Then the differential equation$D_{a+}^\alpha u=0$has unique solution $$u(t)=c_1(t-a)^{\alpha-1}+c_2(t-a)^{\alpha-2}+\dots+c_n(t-a)^{\alpha-n},$$ where$c_i\in\mathbb{R}$,$i=1,2,\dots,n$, and$n-1<\alpha\leq n$. \end{proposition} \begin{proposition}[\cite{k1}] \label{prop1.4} Let$\alpha>0$,$u\in L(a,b)$,$D_{a+}^\alpha u\in L(a,b)$. Then the following equality holds $$I_{a+}^\alpha D_{a+}^\alpha u(t)=u(t)+c_1(t-a)^{\alpha-1}+c_2(t-a)^{\alpha-2}+\dots+c_n(t-a)^{\alpha-n},$$ where$c_i\in\mathbb{R}$,$i=1,2,\dots,n$, and$n-1<\alpha\leq n$. \end{proposition} In general, these properties do not hold for derivatives and integrals of variable-order$D_{a+}^{p(t)}, I_{a+}^{p(t)}$defined by \eqref{e1.1}--\eqref{e1.6}. For example, when$p(t),q(t)$are not constant functions, we have that $$I_{a+}^{p(t)}I_{a+}^{q(t)}f(t)\neq I_{a+}^{p(t)+q(t)}f(t), p(t)>0, q(t)>0,\quad f\in L(a,b).\label{e1.7}$$ \begin{example} \label{examp1.1}\rm Let$p(t)=t$,$0\leq t\leq 6, q(t)=\begin{cases} 2,& 0\leq t\leq 2\\ 1, & 2a, \label{e1.8} \\ D_{a+}^{p(t,f(t))}f(t)=\frac{d^2}{dt^2}I_{a+}^{2-p(t,f(t))}f(t) =\frac{d^2}{dt^2}\int_a^t\frac{(t-s)^{1-p(s,f(s))}}{\Gamma(2-p(s,f(s)))}f(s)ds, \quad t>a, \label{e1.9} \end{gather} provided that the right-hand side is pointwise defined. Of course, Propositions \ref{prop1.1}--\ref{prop1.4} do not usually hold for integral and derivative of variable-order defined by \eqref{e1.8}, \eqref{e1.9}. Therefore, without those properties, a variable-order differential equation cannot be transformed into an equivalent integral equation, so that one can consider existence of solutions of a differential equation of variable-order, by means of some fixed point theorems. In this paper, we will consider the existence of solutions to the following singular two-point boundary-value problem for differential equation of variable order \begin{gather} D_{0+}^{q(t,x(t))}x(t)=f(t,x),\quad 00 to be an arbitrary small number, which is important for the next step in the analysis. \begin{lemma} \label{lem2.1} Let {\rm (H1)} hold. And let x_n, x\in C[0,T], assume that x_n(t)\to x(t), t\in [0,T] as n\to \infty, then $$\int_0^{t-\delta}\frac{(t-s)^{1-q(s,x_n(s))}}{\Gamma(2-q(s,x_n(s)))}x_n(s)ds \to \int_0^{t-\delta}\frac{(t-s)^{1-q(s,x(s))}}{\Gamma(2-q(s,x(s)))}x(s)ds, \label{e2.1}$$ for t\in [\delta,T], as n\to \infty. \end{lemma} \begin{proof} For x_n, x\in C[0,T], we see that \begin{gather} \text{if 00, when n>N_0, we have that \begin{align*} &\big|\int_0^{t-\delta}\frac{(t-s)^{1-q(s,x_n(s))}}{\Gamma(2-q(s,x_n(s)))}x_n(s)ds -\int_0^{t-\delta}\frac{(t-s)^{1-q(s,x(s))}}{\Gamma(2-q(s,x(s)))}x(s)ds\big| \\ &\leq \int_0^{t-\delta}|\frac{(t-s)^{1-q(s,x_n(s))}}{\Gamma(2-q(s,x_n(s)))}| |x_n(s)-x(s)|ds\\ &\quad +\int_0^{t-\delta}|\frac{(t-s)^{1-q(s,x_n(s))}- (t-s)^{1-q(s,x(s))}}{\Gamma(2-q(s,x_n(s)))}||x(s)|ds \\ &\quad +\int_0^{t-\delta}|(t-s)^{1-q(s,x(s))}||\frac 1{\Gamma(2-q(s,x_n(s)))} -\frac 1{\Gamma(2-q(s,x(s)))}||x(s)|ds \\ &\leq \frac{L(2-q^{*})\varepsilon}{3LT^{*}T} \int_0^{t-\delta}(t-s)^{1-q(s,x_n(s))}ds + \frac{ML\varepsilon}{3MLT}\int_0^{t-\delta}ds \\ &\quad +\frac{M(2-q^{*})\varepsilon}{3MT^{*}T}\int_0^{t-\delta}(t-s)^{1-q(s,x(s))}ds \\ &= \frac{(2-q^{*})\varepsilon}{3T^{*}T}\int_0^{t-\delta}T^{1-q(s,x_n(s))} (\frac{t-s}T)^{1-q(s,x_n(s))}ds+\frac{\varepsilon}{3T}\int_0^{t-\delta}ds \\ &\quad +\frac{(2-q^{*})\varepsilon}{3T^{*}T}\int_0^{t-\delta}T^{1-q(s,x(s))} (\frac{t-s}T)^{1-q(s,x(s))}ds \\ &\leq \frac{(2-q^{*})\varepsilon}{3T^{*}T} \int_0^{t-\delta}T^{*}(\frac{t-s}T)^{1-q^{*}}ds +\frac{\varepsilon}{3T}\int_0^{t-\delta}ds \\ &\quad + \frac{(2-q^{*})\varepsilon}{3T^{*}T} \int_0^{t-\delta}T^{*}(\frac{t-s}T)^{1-q^{*}}ds \\ &= \frac{(2-q^{*})\varepsilon}{3T^{2-q^{*}}} \int_0^{t-\delta}(t-s)^{1-q^{*}}ds+\frac{\varepsilon}{3T}\int_0^{t-\delta}ds+ \frac{(2-q^{*})\varepsilon}{3T^{2-q^{*}}} \int_0^{t-\delta}(t-s)^{1-q^{*}}ds \\ &= \frac{\varepsilon}{3T^{2-q^{*}}}(t^{2-q^{*}}-\delta^{2-q^{*}}) +\frac{\varepsilon}{3T}(t-\delta)+ \frac{\varepsilon }{3T^{2-q^{*}}}(t^{2-q^{*}}-\delta^{2-q^{*}}) \\ &< \frac{\varepsilon T^{2-q^{*}}}{3T^{2-q^{*}}}+\frac{T\varepsilon}{3T} +\frac{\varepsilon T^{2-q^{*}}}{3T^{2-q^{*}}} \\ &= \frac\varepsilon 3+\frac\varepsilon 3+\frac\varepsilon 3=\varepsilon, \end{align*} which implies that \eqref{e2.1} holds. \end{proof} By a similar argument, we can show the following result. \begin{lemma} \label{lem2.2} Let {\rm (H1), (H2)} hold. And let x_n, x\in C[0,T], assume that x_n(t)\to x(t), t\in [0,T] as n\to \infty, then $$\int_0^{t-\delta}(t-s)f(s,x_n(s))ds\to \int_0^{t-\delta}(t-s)f(s,x(s))ds, \quad t\in [\delta,T], \label{e2.7}$$ as n\to \infty. \end{lemma} \begin{proof} By the convergence of x_n, for \zeta>0, there exists N_0\in \mathbb{N} such that |x_n(t)-x(t)|<\zeta, \quad t\in [0,T],\; n\geq N_0, by the continuity of tf, for \frac {2\varepsilon}{T^2} (where \varepsilon is arbitrary small number), when n\geq N_0, it holds s^r|f(s,x_n(s))-f(s,x(s))|<\frac {\Gamma(3-r)\varepsilon}{T^{2-r}\Gamma(1-r)}, \quad s\in [0,T]. Thus, we have \begin{align*} &|\int_0^{t-\delta}(t-s)(f(s,x_n(s))-f(s,x(s)))ds| \\ &\leq \int_0^{t-\delta}(t-s)s^{-r}s^r|f(s,x_n(s))-f(s,x(s))|ds \\ &<\frac{\Gamma(3-r)\varepsilon}{T^{2-r}\Gamma(1-r)}\int_0^{t-\delta}(t-s)s^{-r}ds \\ &\leq \frac{\Gamma(3-r)\varepsilon}{T^{2-r}\Gamma(1-r)}\int_0^t(t-s)s^{-r}ds \\ &= \frac{\Gamma(3-r)\Gamma(1-r)\varepsilon}{T^{2-r}\Gamma(1-r)\Gamma(3-r)}t^{2-r} \leq\varepsilon, \end{align*} which implies that \eqref{e2.7} holds. \end{proof} \begin{lemma}[\cite{k1}] \label{lem2.3} Let [a, b] be a finite interval and let AC[a, b] be the space of functions which are absolutely continuous on [a, b]. It is known that AC[a,b] coincides with the space of primitives of Lebesgue summable functions: f(t)\in AC[a,b]\Leftrightarrow f(t) =c+ \int_0^t\varphi(s)ds, \quad\varphi\in L(a,b),\;c\in\mathbb{R}, and therefore the absolutely continuous function f(t) has a summable derivative f'(t) = \varphi(t) almost everywhere on [a,b]. \end{lemma} \section{Existence result} By the definition of derivative of variable order, defined by \eqref{e1.9}, we see that problem \eqref{e1.10}-\eqref{e1.11} is equivalent to the equation $$\label{e3.1} \int_0^t\frac{(t-s)^{1-q(s,x(s))}}{\Gamma(2-q(s,x(s)))}x(s)ds =c_1+c_2t+\int_0^t(t-s)f(s,x(s))ds,$$ for t\in [0,T], where c_1,c_2\in\mathbb{R} such that x(0)=x(T)=0 holds. \begin{theorem} \label{thm3.1} Assume that {\rm (H1), (H2)} hold. Then problem \eqref{e1.10}-\eqref{e1.11} exists one solution x^{*}\in C[0,T]. \end{theorem} \begin{proof} To obtain the existence result for \eqref{e1.10}-\eqref{e1.11}, we firstly verify the following sequence has convergent subsequence, x_k(t)= \begin{cases} 0, & 0\leq t\leq \delta, \\ x_{k-1}(t)+\int_0^{t-\delta} \frac{(t-s)^{1-q(s,x_{k-1}(s))}}{\Gamma(2-q(s,x_{k-1}(s)))}x_{k-1}(s)ds \\ -c_{2,k-1}(t-\delta) -\int_0^{t-\delta}(t-s)f(s,x_{k-1}(s))ds, & \deltab^t>0, we have that k'(t)=a^t\ln a-b^t\ln b0, and all t_1, t_2\in [0,T], t_10, and all t_1, t_2\in [0,T], t_10, take \[ \delta_0=\min\{(\frac{\varepsilon(2-q^{*})} {MLT^{*}T^{q^{*}-1}})^{\frac 1{2-q^{*}}}, (\frac{\varepsilon(1-r)}{M_fT})^{\frac 1{1-r}}\} where $M=\max_{0\leq t\leq T}|x^{*}(t)|+1, \quad L=\max_{0\leq t\leq T, \|x^{*}\|\leq M} |\frac 1{\Gamma(2-q(t,x^{*}(t)))}|+1,$ when\delta<\delta_0, by \eqref{e2.2}, \eqref{e2.3}, \eqref{e2.4}, \eqref{e3.6}, we have \begin{align*} &\big|\int_0^{t-\delta}\frac{(t-s)^{1-q(s,x^{*}(s))}} {\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds -\int_0^t\frac{(t-s)^{1-q(s,x^{*}(s))}}{\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds\big| \\ &= \big|\int_{t-\delta}^t\frac{(t-s)^{1-q(s,x^{*}(s))}} {\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds\big| \\ &= |\int_{t-\delta}^t\frac{T^{1-q(s,x^{*}(s))}} {\Gamma(2-q(s,x^{*}(s)))}(\frac{t-s}T)^{1-q(s,x^{*}(s))}x^{*}(s)ds| \\ &\leq ML\int_{t-\delta}^tT^{*}(\frac{t-s}T)^{1-q^{*}}ds \\ &= \frac{MLT^{*}T^{q^{*}-1}}{2-q^{*}}\delta^{2-q^{*}} \\ &< \frac{MLT^{*}T^{q^{*}-1}}{2-q^{*}}\delta_0^{2-q^{*}} = \varepsilon, \end{align*} which implies that $$\lim_{\delta\to 0}\int_0^{t-\delta}\frac{(t-s)^{1-q(s,x^{*}(s))}} {\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds =\int_0^t\frac{(t-s)^{1-q(s,x^{*}(s))}}{\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds. \label{e3.14}$$ By the same arguments, we have that $$\lim_{\delta\to 0}\int_0^{T-\delta} \frac{(T-s)^{1-q(s,x^{*}(s))}}{\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds =\int_0^T\frac{(T-s)^{1-q(s,x^{*}(s))}}{\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds. \label{e3.15}$$ Similarly, we have \begin{align*} &\big|\int_0^{t-\delta}(t-s)f(s,x^{*}(s))-\int_0^t(t-s)f(s,x^{*}(s))ds\big| \\ &= |\int_{t-\delta}^t(t-s)f(s,x^{*}(s))ds| \\ &\leq M_f\int_{t-\delta}^t(t-s)s^{-r}ds \\ &\leq M_fT\int_{t-\delta}^ts^{-r}ds \\ &= \frac{M_fT}{1-r}(t^{1-r}-(t-\delta)^{1-r}) \\ &= \frac{M_fT}{1-r}((t-\delta+\delta)^{1-r}-(t-\delta)^{1-r}) \\ &\leq \frac{M_fT}{1-r}((t-\delta)^{1-r}+\delta^{1-r}-(t-\delta)^{1-r}) \\ &= \frac{M_fT}{1-r}\delta^{1-r} \\ &< \frac{M_fT}{1-r}\delta_0^{1-r}<\varepsilon, \end{align*} which implies $$\lim_{\delta\to 0}\int_0^{t-\delta}(t-s)f(s,x^{*}(s))ds =\int_0^t(t-s)f(s,x^{*}(s))ds.\label{e3.16}$$ By the same arguments, we also have $$\lim_{\delta\to 0}\int_0^{T-\delta}(T-s)f(s,x^{*}(s))ds =\int_0^T(T-s)f(s,x^{*}(s))ds.\label{e3.17}$$ Now, we let\delta\to 0\$ in \eqref{e3.11}, \eqref{e3.12}, \eqref{e3.13}, by \eqref{e3.14}, \eqref{e3.15}, \eqref{e3.16} and \eqref{e3.17}, we obtain \begin{gather} x^{*}(0)=x^{*}(T)=0,\label{e3.18}\\ \int_0^t\frac{(t-s)^{1-q(s,x^{*}(s))}}{\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds =\widetilde{c}t+\int_0^t(t-s)f(s,x^{*}(s))ds, \quad 0\leq t\leq T.\label{e3.19} \end{gather} where $\widetilde{c}=\frac{\int_0^T\frac{(T-s)^{1-q(s,x^{*}(s))}} {\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds-\int_0^T(T-s)f(s,x^{*}(s))ds}T.$ Differentiating on both sides of \eqref{e3.19}, we obtain \frac d{dt}I_{0+}^{2-q(t,x^{*}(t))}x^{*}(t)=\widetilde{c}+\int_0^tf(t,x^{*}), \quad 0