\documentclass[twoside]{article} \usepackage{amsfonts} % used for R in Real numbers \pagestyle{myheadings} \markboth{ An abstract existence result } {Sui Sun Cheng, Bin Liu, \& Jian-She Yu } \begin{document} \setcounter{page}{101} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent USA-Chile Workshop on Nonlinear Analysis, \newline Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 101--107. \newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % An abstract existence result and its applications % \thanks{ {\em Mathematics Subject Classifications:} 34K10, 34C20. \hfil\break\indent {\em Key words:} Borsuk's theorem, Fredholm mapping, perturbed differential equation, \hfil\break\indent periodic solution. \hfil\break\indent \copyright 2001 Southwest Texas State University. \hfil\break\indent Published January 8, 2001. } } \date{} \author{Sui Sun Cheng, Bin Liu, \& Jian-She Yu} \maketitle \begin{abstract} By means of Borsuk's theorem and continuation through an admissible homotopy, we establish an existence theorem for operator equation with homogeneous nonlinearity. We illustrate our theorem by considering a perturbed functional differential equation under periodic boundary conditions. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode@=11 \@addtoreset{equation}{section} \catcode@=12 \section{Introduction} Continuation theorems have been used to derive periodic solutions for differential systems with perturbations. In particular, in [1], existence criteria for $\omega$-periodic solutions are given for the equation $x'=g(x)+e(t,x)$ by means of `continuation' through an admissible homotopy carrying the given problem to the equation $x'=g(x),$ which admits only the trivial $\omega$-periodic solution (see [1, pp. 101-103]). In this note, we are interested in the study of a similar problem for the perturbed functional differential system $x'=g(t,x_{t})+h(t,x_{t}),\quad 0\leq t\leq \omega ,$ with solutions that satisfy the periodic boundary condition $x(0)=x(\omega )\,.$ This will be achieved by first proving an abstract existence theorem utilizing Borsuk's theorem and continuation through an admissible homotopy carrying our given problem to the equation $x'=g(t,x_{t}),$ which admits only the trivial periodic solution. \section{Main Results} Let $X,Y$ be real normed spaces with respective norms $\left\| \cdot \right\| _{X}$ and $\left\| \cdot \right\| _{Y}$. Let $L:\mathop{\rm dom}(L)\subseteq X\to Y$ be a linear Fredholm mapping of index zero, and let $\Omega$ be an open and bounded subset of $X$. It is well known [1, Section 2.2] that there exist projections $P:X\to X$ and $Q:Y\to Y$ such that $\mathop{\rm Im}P=\ker L,\ker Q= \mathop{\rm Im}L$ and $X=\ker L\oplus \ker P,Y=\mathop{\rm Im}L\oplus \mathop{\rm Im}Q$. Suppose $F:\mathop{\rm dom}(L)\cap \overline{\Omega }\to Y$ has the form $F=$ $L-N$ where $N:\overline{\Omega }\to Y$ is $L$-compact on $\overline{\Omega }$ and satisfies the condition $0\notin F(\mathop{\rm dom}(L)\cap \partial \Omega )$. Then a coincidence degree $D_{L}(F,\Omega )$ can be defined which satisfies the properties listed in [1, Section 2.3]. As mentioned above, we will need the following Borsuk's Theorem: Suppose $\Omega$ is an open, bounded subset of $X$ which is symmetric with respect to the origin and suppose further that the function $% F$ mentioned above satisfies the additional condition that $F(-x)=-F(x)$ for every $x\in \mathop{\rm dom}(L)\cap \partial \Omega ,$ then the coincidence degree $D_{L}(F,\Omega )$ is odd. We remark that there are a number of studies which are concerned with the existence of periodic solutions of differential equations by means of coincidence theory, see for examples [2-6]. \begin{lemma} \label{lm1} Let $\overline{\Omega }=\left\{ x\in X|\;\left\| x\right\| _{X}\leq 1\right\}$. Let $N_{2}:X\to Y$ be a continuous mapping which maps bounded sets into bounded sets and satisfies $$\lim_{\left\| x\right\| _{X}\to \infty }\frac{\left\| N_{2}x\right\| _{Y}}{\left\| x\right\| _{X}^{\beta }}=0 \label{100}$$ for some $\beta \in (0,1]$. Suppose $H:\overline{\Omega }\times [0,1]\to Y$ is defined by $H(x,\mu )=\left\{ \begin{array}{ll} \mu ^{\beta }N_{2}(\mu ^{-\beta }x) &\mbox{if } \mu \in (0,1] \\[2pt] 0 & \mbox{if } \mu =0\,. \end{array} \right.$ Then $H$ is continuous and bounded on $\overline{\Omega }\times [0,1]$. \end{lemma} \paragraph{Proof.} To show that $H$ is continuous, it suffices to show that $H$ is continuous at $(x,0)$ where $x\in \overline{\Omega }$. For any $\varepsilon \in (0,1),$ in view of assumption (\ref{100}), we see that there exists a constant $\rho >0$ such that for arbitrary $x\in X$ which satisfies $\left\| x\right\| _{X}>\rho ,\left\| N_{2}x\right\| _{Y}\leq \varepsilon \left\| x\right\| _{X}^{\beta }$. Since $N_{2}$ maps bounded sets into bounded sets, hence $M=\sup \left\{ \left\| N_{2}x\right\| _{Y}:\;\left\| x\right\| _{X}\leq \rho <\infty \right\} >0.$ Let $\mu _{0}=\left( \frac{\varepsilon }{M+1}\right) ^{1/\beta }$. Clearly, $0<\mu _{0}<\left( \frac{1}{M+1}\right) ^{1/\beta }.$ For every positive $\mu \leq \mu _{0}$ and every $x\in \overline{\Omega },$ we assert that $\left\| H(x,\mu )\right\| _{Y}<\varepsilon$. In fact, if $% \mu ^{-\beta }\left\| x\right\| _{X}>\rho ,$ then \begin{eqnarray*} \left\| H(x,\mu )\right\| _{Y} &\leq &\mu ^{\beta }\left\| N_{2}(\mu ^{-\beta }x)\right\| _{Y} \\ &\leq& \mu ^{\beta }\varepsilon \left\| \mu ^{-\beta }x\right\| _{X}^{\beta } \\ &\leq& \mu ^{\beta }\varepsilon \mu ^{-\beta ^{2}} \left\| x\right\| _{X}^{\beta } \\ &\leq& \mu _{0}^{\beta (1-\beta )}\varepsilon \\ &<&\big( \frac{1}{M+1}\big) ^{1-\beta }\varepsilon <\varepsilon , \end{eqnarray*} and if $\mu ^{-\beta }\left\| x\right\| _{X}\leq \rho ,$ then $\left\| H(x,\mu )\right\| _{Y}\leq \mu ^{\beta }\left\| N_{2}(\mu ^{-\beta }x)\right\| _{Y}\leq \mu ^{\beta }M\leq \frac{\varepsilon }{M+1}% M<\varepsilon .$ Thus we have shown that $H$ is continuous at $(x,0)\in \overline{\Omega }% \times [0,1].$ By arguments similar to those just described, we may show by means of the continuity of $H$ at $(x,0)\in \overline{\Omega }\times [0,1]$ that there exists a constant $\delta >0$ and a real number $M_{1}$ such that for $% (x,\mu )\in \overline{\Omega }\times [0,\delta ],$ $\left\| H(x,\mu )\right\| _{Y}\leq M_{1}$. Since $N_{2}$ maps bounded sets into bounded sets, there exists a number $M_{2}$ such that $\left\| H(x,\mu )\right\| _{Y}\leq M_{2}$ for $(x,\mu )\in \overline{\Omega }\times [\delta ,1]$. Thus $H$ is bounded on $\overline{\Omega }\times [0,1]$. The proof is complete. \smallskip Let us now consider the operator equation $$Lx=N_{1}x+N_{2}x,x\in X, \label{1}$$ where \begin{enumerate} \item[H1)] $L$ is a linear Fredholm mapping of index zero, \item[H2)] $N_{1}:X\to Y$ is a continuous mapping which satisfies $N_{1}(\lambda x)=\lambda N(x)$ for $\lambda \in (-\infty ,\infty )$ and $x\in X$, \item[H3)] $N_{2}:X\to Y$ is a continuous mapping which maps bounded sets into bounded sets and satisfies (\ref{100}) for some $\beta \in (0,1]$, \item[H4)] $N_{1},N_{2}$ are $L$-completely continuous. \end{enumerate} \begin{theorem} \label{thm1} Suppose the conditions H1-H4 hold. Suppose further that $$Lx=N_{1}x \label{2}$$ admits only the trivial solution. Then (\ref{1}) has a nontrivial solution in $\mathop{\rm dom}L\cap \overline{\Omega }$. \end{theorem} \paragraph{Proof.} Let $\Omega =\left\{ x\in X|\;\left\| x\right\| _{X}\leq 1\right\}$. Let $T:\overline{\Omega }\times [0,1]\to Y$ be defined by $$T(x,\mu )=\left\{ \begin{array}{ll} N_{1}x+\mu ^{\beta }N_{2}(\mu ^{-\beta }x) & \mbox{if }\mu \in (0,1] \$2pt] N_{1}x & \mbox{if }\mu =0\,. \end{array} \right. \label{3}$$ Then \[ T(x,1)=N_{1}x+N_{2}x,x\in \overline{\Omega },$ furthermore, in view of Lemma \ref{lm1}, $T$ is continuous and bounded on $\overline{% \Omega }\times [0,1]$. Since $N_{1}$ and $N_{2}$ are $L$-completely continuous, it is also easy to see that $T$ is $L$-compact on $\overline{% \Omega }\times [0,1].$ Note that, in view of the assumption that (\ref{2}) admits only the trivial solution, for any $x\in \partial \Omega ,(x,0)$ cannot be a solution of $$Lx=T(x,\mu ). \label{5}$$ Note further that if $(x,\mu )\in \partial \Omega \times (0,1]$ is a nontrivial solution of (\ref{5}), then in view of (\ref{3}) and (H2), $\mu ^{-\beta }x$ will be a nontrivial solution of (\ref{1}). Let $\tilde{F}=L-T$. Suppose to the contrary that the operator equation (\ref {1}) does not have any nontrivial solutions, then in view of the above discussions, $0\notin \tilde{F}((\mathop{\rm dom}(L)\cap \partial \Omega )\times [0,1])$. Thus the degree $D_{L}(\tilde{F}(\cdot ,\mu ),\Omega )$ can be defined for arbitrary $\mu \in [0,1],$ and it takes constant on $[0,1]$. But since \begin{eqnarray*} \tilde{F}(-x,0) &=&-Lx-T(-x,0)=-Lx-N_{1}(-x) \\ &=&-Lx+N_{1}x=-Lx+T(x,0)=-\tilde{F}(x,0) \end{eqnarray*} for all $x\in X,$ by Borsuk's Theorem stated above, we see that $D_{L}(% \tilde{F}(\cdot ,0),\Omega ),$ and (hence) $D_{L}(\tilde{F}(\cdot ,1),\Omega )$ are odd. But this is contrary to the existence property of the coincidence degree. The proof is complete. \smallskip Let us now turn back to the perturbed functional differential equation $$x'=g(t,x_{t})+h(t,x_{t}),\quad 0\leq t\leq \omega , \label{6}$$ under the periodic boundary condition $$x(0)=x(\omega ), \label{7}$$ where $x(t)\in C(R,R^{n}),$ $x_{t}\in BC\left( R,R^{n}\right)$ are given by $x_{t}(s)=x(t+s),$ and $g,h:[0,\omega ]\times BC(R,R^{n})\to R^{n}$ are continuous mappings that take bounded sets into bounded sets. Here $% BC(R,R^{n})$ is the linear normed space of all continuous and bounded functions from $R$ into $R^{n}$ endowed with the usual supremum norm. \begin{theorem} \label{thm2} Assume that $$g(t,\lambda x)=\lambda g(t,x),\lambda ,t\in R;x\in BC(R,R^{n}), \label{8}$$ and there exists $\beta \in (0,1]$ such that $$\lim_{\left\| x\right\| \to \infty }\frac{\left| h(t,x)\right| }{% \left\| x\right\| ^{\beta }}=0\mbox{ uniformly in }t\in [0,\omega ]. \label{9}$$ Suppose further that the boundary value problem \begin{eqnarray} &x'=g(t,x_{t}) \quad t\in [0,\omega ]&\nonumber \\ & x(0)=x(\omega )& \label{10} \\ &x(t)=x(0) \quad t\in (-\infty ,0]\cup [\omega ,\infty )& \nonumber \end{eqnarray} admits only the trivial solution. Then (\ref{6}) has a nontrivial solution $% x$ that satisfies (\ref{7}). \end{theorem} \paragraph{Proof.} Let $X=\left\{ x\in C(R,R^{n})|\;x(0)=x(\omega ),x(t)=x(0),t\in (-\infty ,0]\cup [\omega ,\infty )\right\} ,$ and $Y=C\left( [0,\omega ],R^{n}\right)$. Then $X$ is a closed subset in $BC\left( R,R^{n}\right) ,$ and therefore it is a Banach space. Let $\mathop{\rm dom}(L)=\left\{ x\in X|\;x'\mbox{ is continuous on }[0,\omega ]\right\} ,$ let $L:\mathop{\rm dom}(L)\cap X\to Y$ be defined by $(Lx)(t)=x'(t)$ for $t\in R,$ and let $N:X\to Y$ be defined by $(Nx)(t)=(N_{1}x)(t)+(N_{2}x)(t),t\in R,$ where $(N_{1}x)(t)=g(t,x_{t}),(N_{2}x)(t)=h(t,x_{t})$ for $t\in R$. Then it is easy to show that the kernel of $L$ is $\ker L=\left\{ x\in X|\;x=c\in R^{n}\right\} ,$ the image of $L$ is $\mathop{\rm Im}L=\left\{ y\in Y|\;\frac{1}{\omega }\int_{0}^{\omega }y(s)ds=0\right\} ,$ and $\dim \ker L=\mbox{codim}\mathop{\rm Im}L=n$. Furthermore, if we define the projections $P:X\to X$ and $% Q:Y\to Y$ by $(Px)(t)=x(0),t\in R,$ and $(Qy)(t)=\frac{1}{\omega }\int_{0}^{\omega }y(s)ds,t\in R,$ respectively, then $\ker L=\mathop{\rm Im}P$ and $\ker Q=\mathop{\rm Im}% L$. Thus, $L$ is a Fredholm operator with index zero, and the generalized inverse $K_{P}:\mathop{\rm Im}L\to \ker P\cap \mathop{\rm dom}(L)$ of $L$ is given by $(K_{P}y)(t)=\left\{ \begin{array}{ll} \int_{0}^{t}y(s)ds & \mbox{if } 0\leq t\leq \omega \\[2pt] 0 & \mbox{if } t\in (-\infty ,0]\cup [\omega ,\infty) \,, \end{array} \right.$ and is compact. Since $(QN)(x)=\frac{1}{\omega }\int_{0}^{\omega }(g(s,x_{s})+h(s,x_{s}))ds,$ we easily see that $QN(\overline{\Omega })$ is bounded, furthermore, by the Arzela-Ascoli theorem, it is also easily seen that $K_{P}(I-Q)N:\overline{% \Omega }\to X$ is compact. As a consequence, $N$ is $L$-compact on $% \overline{\Omega }.$ Note that the conditions (H2) and (H3) follow (\ref{8}) and (\ref{9}) respectively, and that $Lx=N_{1}x$ admits only the trivial solution. By Theorem \ref{thm1}, (\ref{6}) will have a nontrivial solution which satisfies (\ref{7}% ). The proof is complete. \smallskip As an example, consider the boundary value problem $$\displaylines{ x'=p(t)x(t-\tau )+p(t)\left( -x^{1/2}(t-\tau )+a\right) ,0\leq t\leq \omega , \cr x(0)=x(\omega ), }$$ where $a,\tau ,\omega$ are real numbers which satisfy $0<\omega <\tau$ and $a\leq 1/4$. The function $p\in C(R,R)$ is bounded and $\int_{0}^{\omega }p(s)ds\neq 0.$ Let $\beta =3/4$. Then $\lim_{\left| x\right| \to \infty }\frac{\left| p(t)\left( -x^{1/2}+a\right) \right| }{\left| x\right| ^{\beta }}\leq \lim_{\left| x\right| \to \infty }\frac{\max \left| p(t)\right| \left( \left| x\right| ^{1/2}+\left| a\right| \right) }{\left| x\right| ^{3/4}}=0.$ Furthermore, since $x(t-\tau )=x(0)$ for $0\leq t\leq \omega ,$ $x\equiv 0$ is the unique solution of the periodic boundary problem $$\displaylines{ x'=p(t)x(t-\tau ) \quad t\in [0,\omega ] \cr x(0)=x(\omega ) \cr x(t)=x(0) \quad -\tau \leq t\leq 0 }$$ By Theorem \ref{thm2}, there will be a nontrivial solution of our boundary value problem. In fact, $x(t)=\big( \frac{1+\sqrt{1-4a}}{2}\big) ^{1/2}, \quad -\tau \leq t\leq \omega ,$ is one of its nontrivial solutions. We remark that similar results can be obtained for boundary-value problems involving infinite delay, or problems of the form $$\displaylines{ x^{(m)}(t) = g\left( t,x_{t}',...,x_{t}^{(m-1)}\right) +h\left( t,x_{t}',...,x_{t}^{(m-1)}\right) , \quad 0\leq t\leq T, \cr x^{(i)}(0) = x^{(i)}(T),i=0,1,...,m-1. }$$ \begin{thebibliography}{0} \bibitem{r1} J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations, pp. 74-142 in Lecture Notes in Mathematics, No. 1537, edited by Fitzpatrick et al., Springer, Berlin, 1993. \bibitem{r2} F. Zanolin, Periodic solutions for differential systems of Rayleigh type, Tend. Istit. Mat. Univ. Trieste, 12(1980), no. 1-2, 69-77. \bibitem{r3} S. Invernizzi and F. Zanolin, Periodic solutions of a differential delay equation of Rayleigh type, Rend. Sem. Mat. Univ. Padova, 61(1979), 115-124. \bibitem{r4} G. Q. Wang and S. S. Cheng, A priori bounds for periodic solutions of a delay Rayleigh equation, Applied Math, Lett., 12(1999), 41-44. \bibitem{r5} S. W. Ma, J. S. Yu and Z. C. Wang, The periodic solutions of functional differential equations with perturbation, Chinese Sci. Bull., 43(1998), 1386-1388. \end{thebibliography} \noindent{\sc Sui Sun Cheng }\\ Department of Mathematics, Tsing Hua University \\ Taiwan 30043, R. O. China \\ e-mail: sscheng@math.nthu.edu.tw \smallskip \noindent{\sc Bin Liu }\\ Department of Mathematics, Hubei Normal Univeristy\\ Huangshi, Hubei 435002, P. R. China \smallskip \noindent{\sc Jian-She Yu }\\ Department of Applied Mathematics, Hunan University \\ Changsha, Hunan 410082, P. R. China \end{document}