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Differential equations on closed sets. (English) Zbl 1095.34002
Cañada, A.(ed.) et al., Ordinary differential equations. Vol. II. Amsterdam: Elsevier/North Holland (ISBN 0-444-52027-9/hbk). Handbook of Differential Equations, 147-238 (2005).
This piece of writing concerns the differential equation $$u'(t)=f(t,u(t))$$, the differential inclusion $$u'(t)\in F(t,u(t))$$, the evolution equation $$u'(t)=Au(t)+f(t,u(t))$$, and the evolution inclusion $$u'(t)\in Au(t)+F(t,u(t))$$. The starting point of the theory developed in this chapter is a result of M. Nagumo [Proc. Phys.-Math. Soc. Japan, III Ser. 24, 551–559 (1942; Zbl 0061.17204)], which concerns the differential equation in case that the domain $${\mathcal{D}}\subseteq \mathbb{R}\times \mathbb{R}^n$$ of the function $$f:{\mathcal{D}}\to \mathbb{R}^n$$ is not necessarily an open set. An all-inclusive definition of a solution to the differential equation originates from Carathéodory (1918). A function $$u:I\to \mathbb{R}^n$$ defined on an interval $$I\subseteq \mathbb{R}$$ is said to be a solution to the differential equation if $$(t,u(t))\in{\mathcal{D}}$$ for all $$t$$; the function $$t\to f(t,u(t))$$ is Lebesgue integrable; and $$u(t_2)-u(t_1)=\int_{t_1}^{t_2}f(s,u(s))ds$$ for all $$t_1\leq t_2$$. Therefore, $$u'(t)$$ exists and equals $$f(t,u(t))$$ for almost all $$t$$.
If $$u:[\tau,T)\to\mathbb{R}$$ is a solution to the differential equation and $$f$$ is continuous at $$(\tau,u(\tau))$$, then $$(1,f(\tau,u(\tau)))\in{\mathcal{T}}_{\mathcal{D}}(\tau,u(\tau))$$. Here, $${\mathcal{T}}$$ stands for the tangency concept devised by Bouligand (1931) and Severi (1931). Therefore, if $$f$$ is a continuous function on the set $${\mathcal{D}}$$, then the forward existence condition “for every $$(\tau,\xi)\in{\mathcal{D}}$$ there exists a solution $$u:[\tau,T)\to \mathbb{R}^n$$ to the differential equation such that $$u(\tau)=\xi$$” implies the tangency condition “$$(1,f(\tau,\xi))\in{\mathcal{T}}_{\mathcal{D}}(\tau,\xi)$$ for all $$(\tau,\xi)\in{\mathcal{D}}$$”. The result of Nagumo is surprisingly simple and essentially states that, if $$f$$ is a continuous function on the locally closed set $${\mathcal{D}}$$, then the forward existence condition and the tangency condition are equivalent. Since $${\mathcal{T}}_{\mathcal{D}}(\tau,\xi)=\mathbb{R}\times \mathbb{R}^n$$ whenever $$(\tau,\xi)$$ is an interior point of the set $${\mathcal{D}}$$, the Nagumo result extends to the result of Peano (1890), which essentially states that, if $$f$$ is a continuous function on the open set $${\mathcal{D}}$$, then both forward existence and backward existence hold. If $$g$$ stands for the restriction of the function $$f$$ to a subset $${\mathcal{K}}$$ of $${\mathcal{D}}$$, then the forward existence condition for the restricted differential equation $$u'(t)=g(t,u(t))$$ has been labelled by Nagumo as forward admissibility of the set $${\mathcal{K}}$$ with regard to the non restricted differential equation. But this property does not depend on any extension of $$g$$, so it is not surprising that the forward admissibility result of Nagumo is a pure forward existence result, for its elementary, one page proof does not essentially involve any extension of $$g$$.
If $$u:[\tau,T)\to \mathbb{R}^n$$ is a solution to the differential equation but $$f$$ is not necessarily continuous at $$(\tau,u(\tau))$$, then the relation $$(1,f(\tau,u(\tau)))\in{\mathcal{T}}_{\mathcal{D}}(\tau,u(\tau))$$ may fail. However, if $$f$$ is a Carathéodory function on the locally closed, cylindrical set $${\mathcal{D}}=[a,b)\times D$$ where $$[a,b)\subseteq \mathbb{R}$$ and $$D\subseteq \mathbb{R}^n$$, then forward existence is equivalent to the almost tangency condition “there exists a negligible set $$Z\subseteq \mathbb{R}$$ such that $$(1,f(\tau,\xi))\in{\mathcal{T}}_{\mathcal{D}}(\tau,\xi)$$ for all $$(\tau,\xi)\in{\mathcal{D}}\setminus(Z\times \mathbb{R}^n)$$” (see the reviewer’s paper [Math. Jap. 31, 483–491 (1986: Zbl 0603.34001)]). Moreover, the almost tangency condition is equivalent to the almost integral condition which roughly states that there exists a negligible set $$Z\subseteq \mathbb{R}$$ such that for every $$(\tau,\xi)\in{\mathcal{D}}\setminus(Z\times \mathbb{R}^n)$$ there exist functions $$u:[\tau,T)\to \mathbb{R}^n$$ such that: the range $$u([\tau,T))$$ is “sufficiently close” to $$\xi$$; $$(t,u(t)\in{\mathcal{D}}$$ for all $$t$$; the function $$t\to f(t,u(t))$$ is Lebesgue integrable; and there exist “sufficiently small” points $$(h,\eta)$$ such that $$(\tau+h,\xi+\int_\tau^{\tau+h}f(s,u(s))ds+h\eta)\in{\mathcal{D}}$$ (see the reviewer’s preprint [Carathéodory solutions of ordinary differential equations on locally compact sets in Fréchet spaces, “Al. I. Cuza” University of Iaşi, Preprint series in mathematics of “A. Myller” Mathematical Seminar. 18 (1982)]).
In case that $${\mathcal{D}}$$ is not a cylindrical set, an elementary counterexample shows that both almost tangency and almost integral conditions may fail to imply forward existence even if $$f$$ is the null function on the graph $${\mathcal{D}}$$ of a continuous function $$c:[0,1)\to \mathbb{R}$$. Let $$c$$ be the restriction to $$[0,1)$$ of the Cantor function, then $$(1,0)\in{\mathcal{T}}_{\mathcal{D}}(\tau,c(\tau))$$ for almost all $$\tau\in[0,1)$$, but the differential system $$u'(t)=0$$, $$u(0)=0$$, does not have any solution $$u:[0,T)\to \mathbb{R}$$ such that $$(t,u(t))\in{\mathcal{D}}$$ for all $$t\in[0,T)$$. However, if the multifunction from $$\mathbb{R}$$ to $$\mathbb{R}^n$$ generated by the subset $${\mathcal{D}}$$ of $$\mathbb{R}\times \mathbb{R}^n$$ enjoys an appropriate property of left absolute continuity, then the forward existence and almost integral conditions are equivalent. Moreover, the almost integral and almost tangency conditions are equivalent if $$f$$ has a Carathéodory extension on a cylindrical set. These results are the differential equation transcription of some differential inclusion results in H. Frankowska, S. Plaskacz, and T. Rzezuchowski [J. Differ. Equations 116, No. 2, 265–305 (1995; Zbl 0836.34016)], where there are also defined the appropriate properties of left, right, and mere absolute continuity.
In case of the differential inclusion $$F:\mathbb{R}\times \mathbb{R}^n\to \mathbb{R}^n$$ is a multifunction, the domain of which is denoted by $${\mathcal{D}}$$. A function $$u:I\to \mathbb{R}^n$$ defined on an interval $$I\subseteq \mathbb{R}$$ is said to be a solution to the differential inclusion if: $$(t,u(t))\in{\mathcal{D}}$$ for all $$t$$; there exists a Lebesgue integrable function $$f:I\to \mathbb{R}^n$$ such that $$f(t)\in F(t,u(t))$$ for all $$t$$; and $$u(t_2)-u(t_1)=\int_{t_1}^{t_2}f(s)ds$$ for all $$t_1\leq t_2$$. Therefore $$u'(t)$$ exists and belongs to $$F(t,u(t))$$ for almost all $$t$$. Further, two tangency conditions are considered at a point $$(\tau,\xi)\in{\mathcal{D}}$$. The first one states that $$(\{1\}\times F(\tau,\xi))\cap{\mathcal{T}}_{\mathcal{D}}(\tau,\xi)\neq\emptyset$$ and is productive if $$F$$ is upper semi-continuous at $$(\tau,\xi)$$. The second one states that $$(\{1\}\times F(\tau,\xi))\subseteq{\mathcal{T}}_{\mathcal{D}}(\tau,\xi)$$ and is productive if $$F$$ is lower semi-continuous at $$(\tau,\xi)$$.
Throughout the chapter, the authors discuss the fundamental problem of forward existence as well as further problems related to the differential equation or to the differential inclusion: the equivalence of the Bouligand–Severi tangency condition to the tangency conditions generated by the tangency concepts of Clarke or Bony; the behavior of the funnel $${\mathcal{S}}(\tau,\xi)$$ of the non continuable forward solutions $$u$$ which satisfy the equality $$u(\tau)=\xi$$; the local or global invariance of a subset $${\mathcal{K}}$$ of $${\mathcal{D}}$$ with regard to the forward solutions. Applications are given to differential inequalities, Lyapunov functions and Bellman equations.
At the end of the chapter, the authors mention some problems concerning the evolution equation and the evolution inclusion in case that the autonomous evolution equation $$u'(t)=Au(t)$$ generates a semigroup of functions $$\{S(t):\overline{D(A)}\subseteq X\to X;t\geq0\}$$. Here, $$A:D(A)\subseteq X\to X$$ is a function and $$X$$ is a Banach space. In each case, a tangency condition is constructed from the corresponding definition of a solution. For example, in case of a linear, densely defined function $$A$$, a function $$u:I\to X$$ defined on an interval $$I\subseteq \mathbb{R}$$ is said to be a solution to the evolution equation if: $$(t,u(t))\in{\mathcal{D}}$$ for all $$t$$; the function $$t\to f(t,u(t))$$ is Lebesgue integrable; and $$u(t_2)-S(t_2-t_1)u(t_1)=\int_{t_1}^{t_2}S(t_2-s)f(s,u(s))ds$$ for all $$t_1\leq t_2$$. Now, in case of a continuous function $$f$$, the tangency condition at a point $$(\tau,\xi)\in{\mathcal{D}}$$ roughly states that there exist “sufficiently small” points $$(h,\eta)$$ such that $$(\tau+h,S(h)\xi+h(f(\tau,\xi)+\eta))\in{\mathcal{D}}$$.
For the entire collection see [Zbl 1074.34003].

##### MSC:
 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations 34A60 Ordinary differential inclusions 34G25 Evolution inclusions 34B20 Weyl theory and its generalizations for ordinary differential equations 34A26 Geometric methods in ordinary differential equations 34A99 General theory for ordinary differential equations
##### Keywords:
differential equations; differential inclusions