Some applications of the topological degree theory to multi-valued boundary value problems. (English) Zbl 0543.34008

A powerful tool in the qualitative theory of boundary value problems for (single-valued) ordinary differential equations is the Leray-Schauder topological degree technique for (single-valued) nonlinear operators. On the other hand, there exists a vast literature on both degree theory for multi-valued operators [see e.g. Yu. G. Borisovich, B. D. Gel’man, A. D. Myshkis and V. V. Obukhovskij, Itogi Nauki Tekh., Ser. Mat. Anal. 19, 127-230 (1982; Zbl 0514.54009)] and boundary value problems with multi-valued right-hand side [see e.g. A. F. Filippov, Dokl. Akad. Nauk SSSR 151, 65-68 (1963; Zbl 0141.277)]. When dealing with the interactions between these two topics, however, most authors confine themselves usually to applications of degree methods to first order equations of the form x’(t)\(\in F(t,x(t))\). In the present work, the author gives a detailed and self-contained systematic survey on the applicability of degree techniques to a general class of so-called admissible multi-valued boundary value problems (not necessarily of first order, see also the author’s paper in Nonlinear Anal., Theory Methods Appl. 5, 959-975 (1981; Zbl 0478.34017). As pointed out in the Introduction, the author indicates some applications as well, viz. to: 1) the Cauchy, the Nicoletti and the Floquet boundary value problem for first order equations; 2) problems with nonlinear boundary conditions for first order equations; 3) the Picard boundary value problem for second order equations; 4) the Darboux problem for hyperbolic equations; 5) problems with nonlinear boundary condition for hyperbolic equations; 6) the general boundary value problem for elliptic equations. In the first three points the multi-valued nonlinearities are not necessarily assumed to be convex-valued.
Reviewer: J.Appell


34B15 Nonlinear boundary value problems for ordinary differential equations
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34A60 Ordinary differential inclusions
55M25 Degree, winding number
47J05 Equations involving nonlinear operators (general)