Sergei Kornev

Method of guiding functions in problems of nonlinear analysis

1 Background.- 2 MGF in Finite-Dimensional Spaces.- 3 Guiding Functions in Hilbert Spaces.- 4 Second-Order Differential Inclusions.- 5 Nonlinear Fredholm Inclusions.

Asymptotic behavior of solutions of differential inclusions and the method of guiding functions

To study the asymptotic behavior of solutions of differential inclusions, we suggest to use a generalization of the Krasnosel’skii-Perov method of guiding functions to the nonsmooth case.

Guiding functions and periodic solutions for inclusions with causal multioperators

In the present paper, the method of guiding functions is applied to study the periodic problem for a differential inclusion with a causal multioperator. At first we consider the case when the multioperator is closed and convex-valued. Then the case of a non-convex-valued and lower semicontinuous right-hand part is considered. Thereafter, the theory is extended to the case of non-smooth guiding functions.

Method of generalized integral guiding functions in the problem of the existence of periodic solutions for functional-differential inclusions

We suggest new methods for the solution of a periodic problem for a nonlinear object described by the differential inclusion x′(t) ∈ F(t, xt) under the assumption that the multimapping F has convex compact values and satisfies the upper Carathéodory conditions. We also study the case in which this multimapping is not convex-valued but is normal. The class of normal multimappings includes, for example, bounded almost lower semicontinuous multimappings with compact values and mappings satisfying the Carathéodory conditions. In both cases, a generalized integral guiding function is used to study the problem.

Second-Order Differential Inclusions

Various aspects of the theory of second-order differential inclusions attract the attention of many researchers (see., e.g., [1, 2, 6, 12, 18, 42, 46, 47, 68, 70, 97]). In this chapter we consider the boundary value problem of form

Method of Guiding Functions in Finite-Dimensional Spaces

\displaystyle{ {u}^{{\prime\prime}}\in Q(u),\;\;u(0) = u(1) = 0, }

On some developments of the method of integral guiding functions

(4.1) for second-order differential inclusions which arises naturally from some physical and control problems. Using the method of guiding functions we study the existence of solutions of problem (4.1) in an one-dimensional and in Hilbert spaces.

On asymptotics of solutions for a class of functional differential inclusions

The necessity of studying coincidence points of nonlinear Fredholm operators and nonlinear (compact and condensing) maps of various classes arises in the investigation of many problems in the theory of partial differential equations and optimal control theory.