## 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

### MSC:

 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)

### Citations:

Zbl 0514.54009; Zbl 0141.277; Zbl 0478.34017