\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 108, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/108\hfil Continuous solutions] {Continuous solutions of distributional Cauchy problems} \author[S. Heikkil\"a\hfil EJDE-2011/108\hfilneg] {Seppo Heikkil\"a} \address{Seppo Heikkil\"a \newline Department of Mathematical Sciences, University of Oulu, BOX 3000, FIN-90014 University of Oulu, Finland} \email{sheikki@cc.oulu.fi} \thanks{Submitted February 28, 2011. Published August 25, 2011.} \subjclass[2000]{26A24, 26A39, 34A12, 34A36, 47B38, 47H07, 58D25} \keywords{Distribution; primitive integral; continuous; Cauchy problem; \hfill\break\indent fixed point; smallest solution; greatest solution; minimal solution; maximal solution} \begin{abstract} Existence of the smallest, greatest, minimal, maximal and unique continuous solutions to distributional Cauchy problems, as well as their dependence on the data, are studied. The main tools are a continuous primitive integral and fixed point results in function spaces. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \section{Introduction}\label{S1} New existence results are derived for the smallest, greatest, minimal, maximal and unique continuous solutions of the distributional Cauchy problem $$\label{E1} y'=f(y), \quad y(a)=c.$$ Novel results for dependence of solutions on $f$ and on the initial value $c\in\mathbb{R}$ are also derived. The values of $f$ are distributions (generalized functions) on $[a,b]$, $-\infty< a x$, then $x\in C$ if and only if $x = \inf F[\{y\in D: y > x\}]$. \end{itemize} The greatest elements of $D$ are $n$-fold iterates $F^n(y_+)$, as long as they are defined and $F^{n}(y_+) 0$. Then $F$ has \begin{itemize} \item[(a)] minimal and maximal fixed points; \item[(b)] smallest and greatest fixed points $y_*$ and $y^*$ in the order interval $[\underline y,\overline y]$ of $P$, where $\underline y$ is the greatest solution of $y=-(-F(y))^+$, and $\overline y$ is the smallest solution of $y=F(y)^+$. \end{itemize} Moreover, $y^*$, $y_*$, $\underline y$ and $\overline y$ are all increasing with respect to $F$. \end{lemma} As an application of Lemma \ref{L31} we obtain the following result. \begin{proposition}\label{P31} Assume that the hypotheses (A0) and (B0) hold, and that the primitives $F(x)$ of $f(x)$ in $\mathcal B_c[a,b]$ satisfy the following hypothesis for some $R > 0$. \begin{itemize} \item[(C1)] $\|F(x)\|\le R$ for all $x\in L^1[a,b]$, $\|x\|_1\le R$. \end{itemize} Then the Cauchy problem \eqref{E36} has \begin{itemize} \item[(a)] minimal and maximal solutions in $B(\theta,R)$; \item[(b)] smallest and greatest solutions $y_*$ and $y^*$ in the order interval $[\underline y,\overline y]$ of $B(\theta,R)$, where $\underline y$ is the greatest solution of $y=-(-F(y))^+$, and $\overline y$ is the smallest solution of $y=F(y)^+$. \end{itemize} Moreover, $y^*$, $y_*$, $\underline y$ and $\overline y$ are all increasing with respect to $f$. \end{proposition} \begin{proof} For each $x\in L^1[a,b]$ the distribution $f(x)$ has by (A0) the primitive $F(x)$ in $\mathcal B_c[a,b]\subset L^1[a,b]$. The hypotheses (B0) and (C1) imply that $F$ satisfies the hypotheses of Lemma \ref{L31} when $P=B(\theta,R)$. Thus, by Lemma \ref{L31}(a), $F$ has in $B(\theta,R)$ minimal and maximal fixed points, which are by Lemma \ref{L20} also minimal and maximal solutions of \eqref{E36} in $B(\theta,R)$. The results of (b) and the last result of theorem follow from the corresponding results of Lemma \ref{L31} and from \eqref{E20}. \end{proof} As for the existence of minimal and maximal solutions of \eqref{E36} in the whole $\mathcal B_c[a,b]$, we have the following result. \begin{theorem}\label{T22} The distributional Cauchy problem \eqref{E36} has minimal and maximal solutions in $\mathcal B_c[a,b]$, and they are increasing with respect to $f$, if the hypotheses (A0) and (B0) hold, and if the primitives $F(x)$ of $f(x)$ in $\mathcal B_c[a,b]$ satisfy the following hypothesis. \begin{itemize} \item[(C2)] $\|F(x)\|_1\le Q(\|x\|_1)$ for all $x\in L^1[a,b]$, where $Q:\mathbb{R}_+\to\mathbb{R}_+$ is increasing, $R=Q(R)$ for some $R>0$, and $r\le Q(r)$ implies $r\le R$. \end{itemize} \end{theorem} \begin{proof} Hypothesis (C2) implies that $$\|F(x)\|_1\le Q(\|x\|_1)\le Q(R)=R\quad \text{for every } x\in B(\theta,R).$$ Thus hypothesis (C1) holds, whence \eqref{E36} has the by Proposition \ref{P31} minimal and maximal solutions in $B(\theta,R)$, and they are increasing with respect to $f$. If $y\in B(\theta,r)$ is a solution of \eqref{E1}, then $y$ is a fixed point of $F$ by Lemma \ref{L20}. Hypothesis (C2) with $r=\|y\|_1$ implies that $$\|y\|_1=\|F(y)\|_1\le Q(\|y\|_1)\le Q(R)=R.$$ Thus all the solutions of \eqref{E36} are in $B(\theta,R)$. The assertion follows from the above results. \end{proof} \section{Existence and uniqueness results}\label{S4} In this section, conditions are presented for distributions $f(x)$, $x\in C[a,b]$, which ensure that \eqref{E1} has for each $c\in\mathbb{R}$ a unique solution. Denoting $\lceil x\rceil =| x(\cdot)|$, $x\in C[a,b]$, we have the following fixed point result that is basis of our main existence and uniqueness theorem. \begin{proposition}[{\cite[Theorem 1.4.9]{HeiLak94}}]\label{P410.1} Let $F:C[a,b]\to C[a,b]$ satisfy the hypothesis: \begin{itemize} \item[(F0)] There exists a $v\in C_+[a,b]=\{u\in C[a,b]:\theta\le u\}$ and an increasing mapping $Q:[\theta,v]\to C_+[a,b]$ satisfying $Qv(t) < v(t)$ and $Q^nv(t)\to 0$ for each $t\in [a,b]$, such that $$\label{E410.5} \lceil F(x) - F(z)\rceil\le Q\lceil x - z\rceil$$ for all $x, z\in C[a,b]$, $\lceil x- z\rceil\le v$. \end{itemize} Then for each $y_0\in C[a,b]$ the sequence $(F^n(y_0))_{n=0}^\infty$ converges uniformly on $[a,b]$ to a unique fixed point of $F$. \end{proposition} In our main existence and uniqueness theorem we rewrite the inequality \eqref{E410.5} in terms of distributions. The modulus $|g|$ of a distribution $g$ on $(0,b)$ that is $DD$ integrable on $[a,b]$ is defined by $$\label{E0.01} |g|:=\sup\{g,-g\},$$ where the supremum is taken in the partial ordering $\preceq$ defined by \eqref{E20}. $|g|$ exists because $\preceq$ is a lattice ordering (cf. \cite[Sect. 9]{ET081}). Now we are able to prove an existence and uniqueness theorem for the solution of the Cauchy problem \eqref{E1}. \begin{theorem}\label{T411.1} Assume that distributions $f(x)$, $x\in C[a,b]$, and $h(w)$, $w\in[\theta,v]$, $v\in C_+[a,b]$, are $DD$ integrable on $[a,b]$, and that $$\label{E411.00} |f(x)-f(z)|\preceq h(\lceil x-z \rceil)$$ for all $x,\,z\in C[a,b]$ with $\lceil x-z\rceil\le v$, and that $Q:[\theta,v]\to C_+[a,b]$, defined by $$\label{E411.1} Q(w)(t)= \sideset{^c}{}{\!\!\!\int_a^t} h(w), \quad \theta\le w\le v, \ a\le t\le b,$$ is increasing, $Q(v)(t)) 0$, satisfies the condition. \begin{itemize} \item[(Q0)] $q(\cdot,x)$ is measurable for all $x\in[0,r]$, $q(\cdot,r)\in L^1([a,b], \mathbb{R}_+)$, $q(t,\cdot)$ is increasing and right-continuous for a.e. $t\in [a,b]$, and the zero-function is the only absolutely continuous (AC) solution with $u_0=0$ of the Cauchy problem $$\label{E411.2} u'(t) = q(t,u(t)) \text{ a.e. on } [a,b],\quad u(a) = u_0.$$ \end{itemize} Then there exists an $r_0>0$ such that the Cauchy problem \eqref{E411.2} has for every $u_0\in [0,r_0]$ the smallest AC solution $u=u(\cdot,u_0)$, which is increasing with respect to $u_0$. Moreover, $u(t,u_0)\to 0$ uniformly over $t\in [a,b]$ when $u_0\to 0$. \end{lemma} The next result is an application of Lemma \ref{L411.2} and Theorem \ref{T411.1}. \begin{proposition}\label{P411.0} The results of Theorem \ref{T411.1} are valid if the distributions $f(x)$, $x\in C[a,b]$, are $DD$ integrable on $[a,b]$ and satisfy the following hypothesis. \begin{itemize} \item[(F0)] There exists an $r > 0$ such that \eqref{E411.00} holds for all $x,z\in C[a,b]$ with $\|x-z\|_\infty \le r$ and for all $t\in [a,b]$, where $h$ is the Nemytskij operator defined by $$\label{E411.02} h(w) = q(\cdot,w(\cdot)), \quad w\in C_+[a,b], \ \|w\|_\infty\le r,$$ and $q:[a,b]\times[0,r]\to\mathbb{R}_+$ satisfies the hypothesis (q) of Lemma \ref{L411.2}. \end{itemize} \end{proposition} \begin{proof} According to Lemma \ref{L411.2} the Cauchy problem \eqref{E411.2} has for some $u_0=r_0> 0$ the smallest AC solution $v=u(\cdot,r_0)$, and $r_0\le v(t) \le r$ for each $t\in [a,b]$. Since $q(s,\cdot)$ is increasing and right-continuous in $[0,r]$ for a.e. $s\in [a,b]$, and because $q(\cdot, x)$ is measurable for all $x\in[0,r]$ and $q(\cdot,r)$ is Lebesgue integrable, it follows from \cite[Theorem 2.1.1 and Remarks 2.1.1]{CarHei00} that $q(\cdot,u(\cdot))$ is Lebesgue integrable whenever $u$ belongs to the order interval $[\theta,v]$ of $C_+[a,b]$. Thus the equation \eqref{E411.1}, where $h$ is the Nemytskij operator defined by \eqref{E411.02}, defines a mapping $Q:[\theta,v]\to C_+[a,b]$. Condition (Q0) ensures that $Q$ is increasing, and the choices of $r_0$ and $v$ imply that $$\label{E411.03} r_0 + Q(v) = v.$$ Thus $v(t)-Q(v)(t)=r_0>0$ for each $t\in[a,b]$. The sequence $(Q^n(v))_{n=0}^\infty$ is decreasing because $q(t,\cdot)$ is increasing for a.e. $t\in [a,b]$. Noticing that the functions $Q^n(v)$ are also continuous, the reasoning similar to that applied in the proof of Lemma \ref{L411.2} shows that $(Q^n(v))_{n=0}^\infty$ converges uniformly on $[a,b]$ to the zero function. The above proof shows that the hypotheses of Theorem \ref{T411.1} hold. \end{proof} \section{Dependence on the Initial Value}\label{S5} We shall first prove that under the hypotheses of Proposition \ref{P411.0} the difference of solutions $y$ of \eqref{E1} belonging to initial values $c$ and $\hat c$, respectively, can be estimated by the \emph{smallest solution} of the comparison problem \eqref{E411.2} with initial value $u_0=|c-\hat c|$. This estimate implies by Lemma \ref{L411.2} the continuous dependence of $y$ on $c$. \begin{proposition}\label{P411.1} Let the distributions $f(x)$, $x\in C[a,b]$, satisfy the hypotheses of Proposition \ref{P411.0}. If $y = y(\cdot,c)$ denotes the solution of the Cauchy problem \eqref{E1} and $u=u(\cdot,u_0)$ the smallest solution of the Cauchy problem \eqref{E411.2}, then for all $c,\,\hat c\in \mathbb{R}$, with $| c-\hat c|$ small enough, $$\label{E411.5} | y(t,c) - y(t,\hat c))|\le u(t,| c-\hat c|), \quad t\in [a,b].$$ In particular, $y(\cdot,c)$ depends continuously on $c$ in the sense that $y(t,\hat c)\to y(t,c)$ uniformly over $t\in [a,b]$ as $\hat c\to c$. \end{proposition} \begin{proof} Assume that $c,\hat c\in \mathbb{R}$, and that $| c-\hat c|\le r_0$, where $r_0$ is chosen as in Lemma \ref{L411.2}. The solutions $y=y(\cdot,c)$ and $\hat y = y(\cdot,\hat c)$ exist by Proposition \ref{P411.0}, and they satisfy by Lemma \ref{L20} the fixed point equations $$y(t)=F(y)(t) = c + \sideset{^c}{}{\!\!\!\int_a^t}f(y),\ \hbox {and}\ \hat y(t)= \hat F(\hat y)(t) = \hat c + \sideset{^c}{}{\!\!\!\int_a^t}f(\hat y), \quad t\in [a,b].$$ Moreover, $F$ satisfies by the proof of Proposition \ref{P411.0} the hypotheses of Proposition \ref{P410.1} with $Q$ defined by \eqref{E411.1}, or equivalently, by $$Q(w)(t) = \int_a^t q(s,w(s))\,ds, \quad t\in [a,b],$$ and $u=u(\cdot,|c-\hat c|)$ is the smallest AC solution of $$u=|c-\hat c|+Q(u).$$ Denote $$V = \{y\in C[a,b]: \lceil y - \hat y\rceil\le u\}.$$ Since $Q$ is increasing, and since $$F(\hat y)(t)-\hat y(t)= F(\hat y)(t)-\hat F(\hat y)(t)=c-\hat c$$ for all $t\in [a,b]$, we have for every $y\in V$, \begin{align*} \lceil F(y)-\hat y\rceil &\le \lceil F(\hat y)-\hat y\rceil+\lceil F(y)-F(\hat y)\rceil\\ &\le \lceil F(\hat y)-\hat y\rceil +Q(\lceil y-\hat y\rceil)\\ &\le |c-\hat c|+Q(u)=u. \end{align*} Thus $F[V]\subseteq V$. Since $\hat y\in V$, then $(F^n(\hat y))\in V$ for every $n\in \mathbb{N}_0$. The uniform limit $y=\lim_nF^n(\hat y)$ exists by Theorem \ref{T411.1} and is the solution of \eqref{E1}. Because $V$ is closed, then $y\in V$, so that $\lceil y-\hat y\rceil\le u$. This proves \eqref{E411.5}. According to Lemma \ref{L411.2}, $u(t,|c-\hat c|)\to 0$ uniformly over $t\in [a,b]$ as $|c-\hat c|\to 0$. This result and \eqref{E411.5} imply that the last assertion of the proposition is true. \end{proof} \begin{remark}\label{R411.1}\rm If in condition {\rm (Q0)}, $r=\infty$ and $q(\cdot,z)\le \overline q\in L^1([a,b]$ for each $z\in\mathbb{R}_+$, then \eqref{E411.5} holds for all $c,\,\hat c\in \mathbb{R}$. The hypotheses imposed on $q:[a,b]\times [0,r]\to \mathbb{R}_+$ in (Q0) hold if $q(t,\cdot)$ is increasing for a.e. $t\in [a,b]$, and if $q$ is an $L^1$-bounded Carath\'eodory function such that the following local Kamke's condition holds. $$u\in C([a,b],[0,r])\text{ and u(t)\le \int_a^tq(s,u(s))\,ds for all t\in [a,b] imply } \ u(t)\equiv 0.$$ The hypotheses of (Q0) are valid also for the function $q:[a,b]\times [0,r]\to \mathbb{R}_+$, defined by $$\label{E411.05} q(t,s)= p(t)\,\phi(s)\quad t\in[a,b], \ s\in [0,r],$$ where $p\in L^1([a,b],\mathbb{R}_+)$, $\phi:[0,r]\to\mathbb{R}_+$ is increasing and right-continuous, and $\int_0^r\frac{dv}{\phi(v)} =\infty$. Let $\ln_n$ and $\exp_n$ denote $n$-fold iterated logarithm and exponential functions, respectively. The functions $\phi_n$, $n\in\mathbb{N}$, defined by $\phi_n(0)=0$, and $$\phi_n(s) = s\prod_{j=1}^n\ln_j\frac 1s, \quad 0< s\le \exp_n(1)^{-1},$$ have properties assumed above for the function $\phi$ when $r=\exp_n(1)^{-1}$. These properties hold also for the function $\phi(s)=s$, $s \ge 0$. Thus the following result is a special case of Theorem \ref{T411.1} and Proposition \ref{P411.1}. \end{remark} \begin{corollary}\label{C41} The Cauchy problem \eqref{E1} has for each $c\in \mathbb{R}$ a unique solution $y=y(\cdot,c)$, if $f(x)$ is $DD$ integrable on $[a,b]$ for all $x\in C[a,b]$, and if there exists a Lebesgue integrable function $p:[a,b]\to \mathbb{R}_+$ such that $$\big| \sideset{^c}{}{\!\!\!\int_a^t} f(y)- \sideset{^c}{}{\!\!\!\int_a^t}f(z)\big|\le \int_a^tp(s)|y(s)-z(s)|\,ds$$ for all $y,z\in C[a,b]$ and for all $t\in [a,b]$. Moreover, $$| y(t,c) - y(t,\hat c))|\le e^{\int_a^tp(s)ds}| c-\hat c|, \quad t\in [a,b], \; c,\,\hat c\in\mathbb{R}.$$ \end{corollary} In linear case we obtain the following consequence from Corollary \ref{C41}. \begin{corollary}\label{C42} For each $c\in \mathbb{R}$, the linear Cauchy problem $$y'=h+py, \quad y(0)=c,$$ has a unique solution in $C[a,b]$ whenever the distribution $h$ is $DD$ integrable on $[a,b]$, and $p:[a,b]\to \mathbb{R}_+$ is Lebesgue integrable. \end{corollary} \begin{remark}\label{R61.1} \rm The Cauchy problem $$\label{E611.2} y'(t) = g(t,y(t)) \text{ a.e. on } [a,b],\quad y(a) = c,$$ is a special case of problem \eqref{E1} when $f$ is the Nemytskij operator associated with the function $g:[a,b]\times\mathbb{R}\to \mathbb{R}$ by $$f(x):=g(\cdot,x(\cdot)), \quad x\in L^1[a,b].$$ \end{remark} For instance, Theorem \ref{T31} implies the following result. \begin{corollary}\label{C600} The Cauchy problem \eqref{E611.2} has the smallest and greatest continuous solutions that are increasing with respect to $g$ and $c$, if the following hypotheses are valid. \begin{itemize} \item[(G0)] $g(\cdot,x(\cdot))$ is Henstock-Kurzweil integrable on $[a,b]$ for every $x\in L^1[a,b]$. \item[(G1)] $\sideset{^K}{}{\!\!\!\int_a^t} g(s,x(s))\,ds\le \sideset{^K}{}{\!\!\!\int_a^t} g(s,y(s))\,ds$ for all $t\in[a,b]$ whenever $x\le y$ in $L^1[a,b]$. \item[(G2)] There exist Henstock-Kurzweil integrable functions $g_\pm:[a,b]\to\mathbb{R}$ such that\\ $\sideset{^K}{}{\!\!\!\int_a^t} g_-(s)\,ds\le \sideset{^K}{}{\!\!\!\int_a^t} g(s,x(s))\,ds \le \sideset{^K}{}{\!\!\!\int_a^t} g_+(s)\,ds$ for all $x\in L^1[a,b]$ and $t\in[a,b]$. \end{itemize} \end{corollary} \begin{thebibliography}{9} \bibitem{CarHei00} S. 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