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-general principles of the calculus of variations in the large; application to nonlinear integral equations; critical sets of geodesic curves; the functional space R of curves with common endpoints on the sphere; closed geodesics on manifolds homeomorphic to the n-dimensional sphere-

In the present paper we investigate the homological properties of func­tional spaces in connection with the estimation of the number of solutions of the corresponding variational problems. These properties are closely related, on the one hand, to previous work of Moscow mathematicians on topological methods in variational problems, and on the other hand to the investigations of the Moscow topological school on the homology theory of abstract spaces. In a certain sense this paper constitutes a synthesis of both directions of the Moscow topological school. Chapter I of this paper presents general methods in the calculus of variations in the large. It formulates general principles making it possible to determine when infinite families of extremals appear and to estimate the number of solutions of a variational problem in connection with the ring of products in the corresponding functional space. Chapter II gives applications of the general methods to the theory of eigenvalues of certain classes of nonlinear integral equations. Chapter III investigates “critical sets”, consisting of geodesic curves on manifolds. In addition, we reproduce the author’s theorem that each compact family of non-selfintersecting curves on the two-dimensional sphere can be reduced to a family of circumferences by a continuous deformation. Chapter IV investigates the functional space of all rectifiable curves on the sphere with common endpoints (admissible curves of a variational problem with fixed endpoints). Here we first give a complete investiga­ tion of the topological structure of an essentially nonlinear functional space. We construct upper and lower homology groups and the ring of intersections (for various coefficient groups). We also give investigations of certain homotopy properties of this space. As an application we present a complete investigation of the geodesic arcs joining two given points on a surface of genus 0, sharpening some of Morse’s results. In Chapter V general methods are used to study closed geodesics on manifolds homeomorphic to the n-dimensional sphere. For the case n = 2 we give a new proof of a theorem earlier proved by the author simul­ taneously with L. G. Snirel'man on the existence of three closed geo­ desics. For the case n > 2 under certain metric restrictions this theorem can be generalized: one proves the existence of at least (n -j- 1) closed geodesics. Translated from the Russian by J. M. Danskin