\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 156, 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} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/156\hfil Hyers-Ulam stability ] {Hyers-Ulam stability for Gegenbauer differential equations} \author[S.-M. Jung \hfil EJDE-2013/156\hfilneg] {Soon-Mo Jung} % in alphabetical order \address{Soon-Mo Jung \newline Mathematics Section, College of Science and Technology, Hongik University, 339-701 Sejong, South Korea} \email{smjung@hongik.ac.kr} \thanks{Submitted June 19, 2013. Published July 8, 2013.} \subjclass[2000]{39B82, 41A30, 34A30, 34A25, 34A05} \keywords{Gegenbauer differential equation; Hyers-Ulam stability; \hfill\break\indent power series method; second order differential equation} \begin{abstract} Using the power series method, we solve the non-homogeneous Gegenbauer differential equation $$( 1 - x^2 )y''(x) + n(n-1)y(x) = \sum_{m=0}^\infty a_m x^m.$$ Also we prove the Hyers-Ulam stability for the Gegenbauer differential equation. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} Let $Y$ be a normed linear space and $I$ be an open subinterval of $\mathbb{R}$. If for any function $f : I \to Y$ satisfying the differential inequality $$\big\| a_n(x)y^{(n)}(x) + a_{n-1}(x)y^{(n-1)}(x) + \cdots + a_1(x)y'(x) + a_0(x)y(x) + h(x) \big\| \leq \varepsilon$$ for all $x \in I$ and for some $\varepsilon \geq 0$, there exists a solution $f_0 : I \to Y$ of the differential equation $$a_n(x)y^{(n)}(x) + a_{n-1}(x)y^{(n-1)}(x) + \cdots + a_1(x)y'(x) + a_0(x)y(x) + h(x) = 0$$ such that $\| f(x) - f_0(x) \| \leq K(\varepsilon)$ for any $x \in I$, where $K(\varepsilon)$ depends on $\varepsilon$ only, then we say that the above differential equation satisfies the Hyers-Ulam stability (or the local Hyers-Ulam stability if the domain $I$ is not the whole space $\mathbb{R}$). We may apply these terminologies for other differential equations. For more detailed definition of the Hyers-Ulam stability, refer the reader to \cite{czerwik0,hir,jung2}. Apparently Ob\a{l}oza \cite{ob1,ob2} was the first author who investigated the Hyers-Ulam stability of linear differential equations. Here, we cite a result by Alsina and Ger \cite{ag}: If a differentiable function $f : I \to \mathbb{R}$ is a solution of the differential inequality $| y'(x) - y(x) | \leq \varepsilon$, where $I$ is an open subinterval of $\mathbb{R}$, then there exists a solution $f_0 : I \to \mathbb{R}$ of the differential equation $y'(x) = y(x)$ such that $| f(x) - f_0(x) | \leq 3\varepsilon$ for any $x \in I$. This result by Alsina and Ger was generalized by Takahasi, Miura and Miyajima \cite{tmm}. They proved that the Hyers-Ulam stability holds for the Banach space valued differential equation $y'(x) = \lambda y(x)$ (see also \cite{mjt,motn,popa}). Using the conventional power series method, the author investigated the general solution of the inhomogeneous linear first-order differential equation $$y'(x) - \lambda y(x) = \sum_{m=0}^\infty a_m (x-c)^m,$$ where $\lambda$ is a complex number and the convergence radius of the power series is positive. This result was applied for proving an approximation property of exponential functions in a neighborhood of $c$ (see \cite{115} and \cite{jung4,jung3}). Throughout this article, we assume that $\rho_1$ is a positive real number or infinity. In Section 2, using an idea from \cite{115}, we investigate the general solution of the inhomogeneous Gegenbauer differential equation $$\big( 1-x^2 \big) y''(x) + n(n-1)y(x) = \sum_{m=0}^\infty a_m x^m, \label{eq:1.1}$$ where the power series has a radius of convergence greater than or equal to $\rho_1$. Moreover, we prove the Hyers-Ulam stability of the Gegenbauer differential equation \eqref{eq:2.1} in a certain class of analytic functions. \section{General solution of \eqref{eq:1.1}} For an integer $n \geq 2$, the second-order ordinary differential equation $$\big( 1-x^2 \big) y''(x) + n(n-1)y(x) = 0 \label{eq:2.1}$$ is a kind of the ultraspherical or Gegenbauer differential equation and has a general solution of the form $y(x) = C_1 J_n(x) + C_2 H_n(x)$, where we denote by $J_n(x)$ and $H_n(x)$ the Gegenbauer functions which are expressed by using the Legendre functions of the first and second kind as follows: $$J_n(x) = \frac{P_{n-2}(x) - P_n(x)}{2n-1},\quad H_n(x) = \frac{Q_{n-2}(x) - Q_n(x)}{2n-1}.$$ The Gegenbauer differential equation \eqref{eq:2.1} is encountered in hydrodynamics when describing axially symmetric Stokes flows \cite{polyanin}. We recall that $\rho_1$ is a positive real number or infinity. \begin{theorem}\label{thm:2.1} Let $n$ be an integer greater than $1$ and let $\rho_1$ be the radius of convergence of power series $\sum_{m=0}^\infty a_m x^m$. Define $\rho := \min \{ 1, \rho_1 \}$. Then every solution $y : (-\rho, \rho) \to \mathbb{C}$ of the inhomogeneous Gegenbauer differential equation \eqref{eq:1.1} can be expressed as $$y(x) = y_h(x) + \sum_{m=2}^\infty c_m x^m, \label{eq:2.2}$$ where the coefficients $c_m$'s are given by \begin{gather*} c_{2m} = \sum_{k=0}^{m-1} \frac{(2k)! a_{2k}}{(2m)!} \prod_{i=k+1}^{m-1} (2i-n)(2i+n-1), \\ c_{2m+1} = \sum_{k=0}^{m-1} \frac{(2k+1)! a_{2k+1}}{(2m+1)!} \prod_{i=k+1}^{m-1} (2i-n+1)(2i+n) \end{gather*} for each $m \in \mathbb{N}$ and $y_h(x)$ is a solution of the Gegenbauer differential equation \eqref{eq:2.1}. \end{theorem} \begin{proof} Since each solution of \eqref{eq:1.1} can be expressed as a power series in $x$, we put $y(x) = \sum_{m=0}^\infty c_m x^m$ in \eqref{eq:1.1} to obtain \begin{align*} &\big( 1-x^2 \big)y''(x) + n(n-1)y(x) \\ &= \sum_{m=0}^\infty \big[ (m+2)(m+1) c_{m+2} - (m-n)(m+n-1) c_m \big] x^m \\ &= \sum_{m=0}^\infty a_m x^m, \end{align*} from which we obtain the recurrence formula \begin{align} (m+2)(m+1) c_{m+2} - (m-n)(m+n-1) c_m = a_m \label{eq:rec} \end{align} for all $m \in \mathbb{N}_0$. Now we prove that the formula \begin{aligned} c_{2m} &= \sum_{k=0}^{m-1} \frac{(2k)! a_{2k}}{(2m)!} \prod_{i=k+1}^{m-1} (2i-n)(2i+n-1) \\ &\quad + \frac{c_0}{(2m)!} \prod_{i=0}^{m-1} (2i-n)(2i+n-1) \end{aligned}\label{eq:2.3a} holds for any $m \in \mathbb{N}$: If we set $m = 1$ in \eqref{eq:2.3a}, then we obtain $2c_2 + n(n-1)c_0 = a_0$ which coincides with \eqref{eq:rec} when $m = 0$. We assume that the formula \eqref{eq:2.3a} is true for some $m \in \mathbb{N}$. Then, it follows from \eqref{eq:rec} and the induction hypothesis that \begin{align*} c_{2m+2} & = \frac{a_{2m}}{(2m+2)(2m+1)} + \frac{(2m-n)(2m+n-1)}{(2m+2)(2m+1)} c_{2m} \\ & = \frac{a_{2m}}{(2m+2)(2m+1)} + \sum_{k=0}^{m-1} \frac{(2k)! a_{2k}}{(2m+2)!} \prod_{i=k+1}^m (2i-n)(2i+n-1) \\ &\quad + \frac{c_0}{(2m+2)!} \prod_{i=0}^m (2i-n)(2i+n-1) \\ &= \sum_{k=0}^m \frac{(2k)! a_{2k}}{(2m+2)!} \prod_{i=k+1}^m (2i-n)(2i+n-1) \\ &\quad + \frac{c_0}{(2m+2)!} \prod_{i=0}^m (2i-n)(2i+n-1), \end{align*} which can be obtained provided we replace $m$ in \eqref{eq:2.3a} with $m+1$. Hence, we conclude that the formula \eqref{eq:2.3a} is true for all $m \in \mathbb{N}$. Similarly, we can prove the validity of the formula \begin{aligned} c_{2m+1} &= \sum_{k=0}^{m-1} \frac{(2k+1)! a_{2k+1}}{(2m+1)!} \prod_{i=k+1}^{m-1} (2i-n+1)(2i+n) \\ &\quad + \frac{c_1}{(2m+1)!} \prod_{i=0}^{m-1} (2i-n+1)(2i+n) \end{aligned} \label{eq:2.3b} for all $m \in \mathbb{N}$. Indeed, we can set $c_0 = c_1 = 0$ in \eqref{eq:2.3a} and \eqref{eq:2.3b}. Under this assumption, we have \begin{align*} c_{2m} &= \sum_{k=0}^{m-1} \frac{(2k)! a_{2k}}{(2m)!} \prod_{i=k+1}^{m-1} (2i-n)(2i+n-1) \\ &= \sum_{k=0}^{[n/2]-1} \frac{(2k)! a_{2k}}{(2m)!} \prod_{i=k+1}^{m-1} (2i-n)(2i+n-1) \\ &\quad + \sum_{k=[n/2]}^{m-1} \frac{(2k)! a_{2k}}{(2m)!} \prod_{i=k+1}^{m-1} (2i-n)(2i+n-1) \\ &= \sum_{k=0}^{[n/2]-1} \frac{(2k)! a_{2k}}{(2m)!} \Big( \prod_{i=k+1}^{[n/2]} (2i-n)(2i+n-1) \Big)\! \Big( \prod_{i=[n/2]+1}^{m-1} (2i-n)(2i+n-1) \Big) \\ &+ \sum_{k=[n/2]}^{m-1} \frac{(2k)! a_{2k}}{(2m)!} \prod_{i=k+1}^{m-1} (2i-n)(2i+n-1). \end{align*} Hence, since $| 2i-n | | 2i+n-1 | < 2i(2i-1)$ for $i > [n/2]$, we obtain \begin{align*} | c_{2m} | &\leq \sum_{k=0}^{[n/2]-1} \frac{(2k)! | a_{2k} |}{(2m)!} \Big( \prod_{i=k+1}^{[n/2]} | 2i-n | | 2i+n-1 | \Big) \Big( \prod_{i=[n/2]+1}^{m-1} (2i)(2i-1) \Big) \\ &+ \sum_{k=[n/2]}^{m-1} \frac{(2k)! | a_{2k} |}{(2m)!} \prod_{i=k+1}^{m-1} (2i)(2i-1) \\ &= \sum_{k=0}^{[n/2]-1} \frac{(2k)! | a_{2k} |}{(2m)!} \alpha_n(k) \prod_{i=[n/2]+1}^{m-1} (2i)(2i-1) \\ &\quad + \sum_{k=[n/2]}^{m-1} \frac{(2k)! | a_{2k} |}{(2m)!} \prod_{i=k+1}^{m-1} (2i)(2i-1), \end{align*} where $\alpha_n(k) := \prod_{i=k+1}^{[n/2]} | 2i-n | | 2i+n-1 |$ for $k \in \{ 0, 1, \ldots, [n/2]-1 \}$. Moreover, taking into account that $\prod_{i=k+1}^{m-1} (2i)(2i-1) = (2m-2)!/(2k)!$, we have $$| c_{2m} | \leq \sum_{k=0}^{[n/2]-1} \frac{\alpha_n(k) | a_{2k} |}{2m(2m-1)} + \sum_{k=[n/2]}^{m-1} \frac{| a_{2k} |}{2m(2m-1)} \leq \frac{1}{m} \sum_{k=0}^{m-1} \frac{\alpha_n | a_{2k} |}{2(2m-1)}, \label{eq:20130507-1}$$ for all $m \in \mathbb{N}$, where $\alpha_n := \max \{ \alpha_n(0), \alpha_n(1), \ldots, \alpha_n([n/2]-1), 1 \}$. Similarly, we obtain $$| c_{2m+1} | \leq \frac{1}{m} \sum_{k=0}^{m-1} \frac{\beta_n | a_{2k+1} |}{2(2m+1)} \label{eq:20130507-2}$$ for any $m \in \mathbb{N}$, where $\beta_n := \max \{ \beta_n(0), \beta_n(1), \ldots, \beta_n([n/2]-1), 1 \}$ and $\beta_n(k) := \prod_{i=k+1}^{[n/2]} | 2i-n+1 | | 2i+n |$ for $k \in \{ 0, 1, \ldots, [n/2]-1 \}$. It follows from \eqref{eq:20130507-1}, \eqref{eq:20130507-2}, and \cite[Problem 8.8.1 (p)]{kosmala} that $\limsup_{m \to \infty} | c_{2m} | \leq \limsup_{m \to \infty} \frac{1}{m} \sum_{k=0}^{m-1} \frac{\alpha_n | a_{2k} |}{2(2m-1)} \leq \limsup_{m \to \infty} \frac{\alpha_n | a_{2m-2} |}{2(2m-1)} \leq \limsup_{m \to \infty} | a_{2m-2} |$ and $\limsup_{m \to \infty} | c_{2m+1} | \leq \limsup_{m \to \infty} \frac{1}{m} \sum_{k=0}^{m-1} \frac{\beta_n | a_{2k+1} |}{2(2m+1)} \leq \limsup_{m \to \infty} \frac{\beta_n | a_{2m-1} |}{2(2m+1)} \leq \limsup_{m \to \infty} | a_{2m-1} |$ which imply that the radius $\rho_2$ of convergence of the power series $\sum_{m=2}^\infty c_m x^m$ is not less than the radius $\rho_1$ of the power series $\sum_{m=0}^\infty a_m x^m$. If we define $\rho_3 := \min \{ \rho_0, \rho_1, \rho_2 \}$, where $\rho_0 = 1$ is the radius of convergence of the general solution to \eqref{eq:2.1}, then $\rho = \rho_3$. According to \cite[Theorem 2.1]{jungsevli} and our assumption that $c_0 = c_1 = 0$, every solution $y : (-\rho_3, \rho_3) \to \mathbb{C}$ of the inhomogeneous Gegenbauer differential equation \eqref{eq:1.1} can be expressed by \eqref{eq:2.2}. \end{proof} \section{Hyers-Ulam stability for \eqref{eq:2.1}} Let $n$ be an integer larger than $1$ and let $\rho_1$ be a positive real number larger than $1$ or infinity. We denote by $\tilde{C}$ the set of all functions $f : (-1, 1) \to \mathbb{C}$ with the following properties: \begin{itemize} \item[(a)] $f(x)$ is expressible by a power series $\sum_{m=0}^\infty b_m x^m$ whose radius of convergence is at least $\rho_1$; \item[(b)] There exists a constant $K \geq 0$ such that $\sum_{m=0}^\infty | a_m x^m | \leq K | \sum_{m=0}^\infty a_m x^m |$ for all $x \in (-\rho_1, \rho_1)$, where $a_m = (m+2)(m+1)b_{m+2} - (m-n)(m+n-1)b_m$ for all $m \in \mathbb{N}_0$. \end{itemize} If we define $$( y_1 + y_2 )(x) = y_1(x) + y_2(x) \quad\text{and}\quad ( \lambda y_1 )(x) = \lambda y_1(x)$$ for all $y_1, y_2 \in \tilde{C}$ and $\lambda \in \mathbb{C}$, then $\tilde{C}$ is a vector space over the complex numbers. We remark that the set $\tilde{C}$ is a vector space. In the following theorem, we investigate the Hyers-Ulam stability of the Gegenbauer differential equation \eqref{eq:2.1} for functions in $\tilde{C}$. \begin{theorem}\label{thm:3.1} If a function $y \in \tilde{C}$ satisfies the differential inequality $$\big| \big( 1 - x^2 \big) y''(x) + n(n-1) y(x) \big| \leq \varepsilon \label{eq:3.1}$$ for all $x \in (-1, 1)$ and for some $\varepsilon \geq 0$, then there exist constants $C_1, C_2 > 0$ and a solution $y_h : (-1, 1) \to \mathbb{C}$ of the Gegenbauer differential equation \eqref{eq:2.1} such that $$| y(x) - y_h(x) | \leq C_1 | x | \ln \frac{1 + | x |}{1 - | x |} + C_2 \Big( \ln \frac{1 + | x |}{1 - | x |} - 2 | x | \Big)$$ for any $x \in (-1, 1)$. \end{theorem} \begin{proof} According to (a), $y(x)$ can be expressed as $y(x) = \sum_{m=0}^\infty b_m x^m$ and it follows from (a) and (b) that \begin{aligned} &\big( 1 - x^2 \big) y''(x) + n(n-1) y(x) \\ &= \sum_{m=0}^\infty \big[ (m+2)(m+1) b_{m+2} - (m-n)(m+n-1) b_m \big] x^m \\ &= \sum_{m=0}^\infty a_m x^m \end{aligned}\label{eq:3.3} for all $x \in (-1, 1)$. By considering \eqref{eq:3.1} and \eqref{eq:3.3}, we have $$\Big| \sum_{m=0}^\infty a_m x^m \Big| \leq \varepsilon$$ for any $x \in (-1, 1)$. This inequality, together with (b), yields that $$\sum_{m=0}^\infty \big| a_m x^m \big| \leq K \Big| \sum_{m=0}^\infty a_m x^m \Big| \leq K \varepsilon \label{eq:condition1}$$ for all $x \in (-1, 1)$. Now, it follows from Theorem \ref{thm:2.1}, \eqref{eq:3.3}, and \eqref{eq:condition1} that there exists a solution $y_h : (-1, 1) \to \mathbb{C}$ of the Gegenbauer differential equation \eqref{eq:2.1} such that \begin{align*} | y(x) - y_h(x) | \leq \Big| \sum_{m=2}^\infty c_m x^m \Big| \leq \sum_{m=1}^\infty | c_{2m} | | x |^{2m} + \sum_{m=1}^\infty | c_{2m+1} | | x |^{2m+1} \end{align*} for all $x \in (-1, 1)$. By \eqref{eq:20130507-1} and \eqref{eq:20130507-2}, we moreover have \begin{aligned} &| y(x) - y_h(x) | \\ & \leq \alpha_n \sum_{m=1}^\infty \frac{|x|^{2m}}{2(2m-1)} \frac{1}{m} \sum_{k=0}^{m-1} | a_{2k} | + \beta_n \sum_{m=1}^\infty \frac{|x|^{2m+1}}{2(2m+1)} \frac{1}{m} \sum_{k=0}^{m-1} | a_{2k+1} | \end{aligned}\label{eq:20130604-1} for all $x \in (-1, 1)$. (See the proof of Theorem \ref{thm:2.1} for the definitions of $\alpha_n$ and $\beta_n$). In view of (a) and (b), the radius of convergence of the power series $\sum_{m=0}^\infty a_m x^m$ is $\rho_1$ which is larger than $1$. This fact implies that $$\sum_{m=0}^\infty | a_m | = \sum_{k=0}^\infty | a_{2k} | + \sum_{k=0}^\infty | a_{2k+1} | < \infty,$$ which again implies that $$\lim_{k \to \infty} | a_{2k} | = 0, \quad \lim_{k \to \infty} | a_{2k+1} | = 0.$$ According to \cite[Theorem 2.8.6]{kosmala}, the sequences $\big\{ | a_{2k} | \big\}$ and $\big\{ | a_{2k+1} | \big\}$ are $(C, 1)$ summable to $0$; i.e., $$\lim_{m \to \infty} \frac{1}{m} \sum_{k=0}^{m-1} | a_{2k} | = 0, \quad \lim_{m \to \infty} \frac{1}{m} \sum_{k=0}^{m-1} | a_{2k+1} | = 0.$$ Thus, there exists a constant $C > 0$ such that $$\frac{1}{m} \sum_{k=0}^{m-1} | a_{2k} | \leq C, \quad \frac{1}{m} \sum_{k=0}^{m-1} | a_{2k+1} | \leq C$$ for any $m \in \mathbb{N}$. Hence, from \eqref{eq:20130604-1} it follows that $$| y(x) - y_h(x) | \leq \frac{\alpha_n C}{2} \sum_{m=1}^\infty \frac{|x|^{2m}}{2m-1} + \frac{\beta_n C}{2} \sum_{m=1}^\infty \frac{|x|^{2m+1}}{2m+1} \label{eq:20130606-1}$$ for all $x \in (-1, 1)$. Since $$\frac{1}{2} \ln \frac{1 + | x |}{1 - | x |} = \sum_{m=1}^\infty \frac{| x |^{2m-1}}{2m-1} = \sum_{m=0}^\infty \frac{| x |^{2m+1}}{2m+1}$$ for $x \in (-1, 1)$, it holds that $$| y(x) - y_h(x) | \leq C_1 | x | \ln \frac{1 + | x |}{1 - | x |} + C_2 \Big( \ln \frac{1 + | x |}{1 - | x |} - 2 | x | \Big)$$ for any $x \in (-1, 1)$, where we set $$C_1 = \frac{\alpha_n C}{4}, \quad C_2 = \frac{\beta_n C}{4},$$ which completes the proof. \end{proof} According to the previous theorem, each approximate solution of the Gegenbauer differential equation \eqref{eq:2.1} can be well approximated by an exact solution of the Gegenbauer differential equation in a (small) neighborhood of $0$. \begin{corollary}\label{cor:3.2} If a function $y \in \tilde{C}$ satisfies the differential inequality \eqref{eq:3.1} for all $x \in (-1, 1)$ and for some $\varepsilon \geq 0$, then there exists a solution $y_h : (-1, 1) \to \mathbb{C}$ of the Gegenbauer differential equation \eqref{eq:2.1} such that $$| y(x) - y_h(x) | = O\big( x^2 \big)$$ as $x \to 0$, where $O(\cdot)$ denotes the Landau symbol $($big-$O)$. \end{corollary} \begin{proof} According to Theorem \ref{thm:3.1} and \eqref{eq:20130606-1}, there exists a solution $y_h : (-1, 1) \to \mathbb{C}$ of the Gegenbauer differential equation \eqref{eq:2.1} such that $$| y(x) - y_h(x) | \leq \frac{\alpha_n C}{2} | x |^2 \sum_{m=1}^\infty \frac{|x|^{2m-2}}{2m-1} + \frac{\beta_n C}{2} | x |^3 \sum_{m=1}^\infty \frac{|x|^{2m-2}}{2m+1}$$ for any $x \in (-1, 1)$, where we see the proof of Theorem \ref{thm:3.1} for the definition of $C$, which completes our proof. \end{proof} \subsection*{Acknowledgments} This work was supported by the 2013 Hongik University Research Fund. \begin{thebibliography}{00} \bibitem{ag} C. 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