Homotopy of extremal problems (Q2746430)

The homotopic method for nonlinear equations (in the sense, that a ``difficult'' equation \(A_0(x)=0\) is getting connected with an ``easy'' one \(A_1(x)=0\) by a one-parametric family \(A_\lambda (x)=0\)) are known since 1904 (S. N. Bernstein, analytic variant). However, its variational analog was published first in 1980, 1983 (N. A. Bobylev, finite and infinite dimensions). In short it is as follows: ``If the extremal of the variational problem is isolated for every parameter value during the deformation of the problem, and if this extremal is a minimum for a certain parameter value, then it is a minimum for every parameter value as well.'' Like each simple and universal principle, this one not only solves new problems, but implies the elegant and more precise methods for some classical problems, too. In the book the authors develop this principle comprehensively and give numerous of its applications.NEWLINENEWLINE We turn to the contents, briefly concerning the basic parts, and in more detail about the applications. Chapter 1 contains preliminary information on linear and nonlinear functional analysis. Chapters 2 and 3 are devoted to the development of the above homotopic principle for finite and infinite dimensional problems. Chapter 4 may partially be attributed as an application: it deals with the Conley index. Its first section contains the original Conley theory, and the second one is devoted to the infinite dimensional generalization, in which some constructions originating to N. A. Bobylev and M. A. Krasnosel'skij have been used. The main applications form Chapter 5, and consist of: 1) Sharpenings of the classical inequalities, both well known and new (Cauchy, Friedrichs, Jensen, Minkowski, Young, etc). 2) Some problems from calculus of variations, optimal control, nonlinear and multiobjective programming. 3) Stability of ODE solutions. 4) Bifurcations of extremals.NEWLINENEWLINEThis book is self-contained and intelligible for students. Specialists will find the one-type approach to the various problems. The bibliography consists of 238 items.