3. Links 4. My publications, preprints, etc.

# RESEARCH

### Topology and Applications

My publications, preprints, talks, etc.

Applications of Lefschetz numbers in control theory. Preprint. The goal of this paper is to develop some applications the Lefschetz theory techniques, already available in dynamics, in control theory. A dynamical system on a manifold M is determined by a map f:M→M. The current state, xÎ M, of the system determines the next state, f(x). The equilibria are fixed points of f, f(x)=x. More generally, one deals with coincidences of a pair of maps f,g:N→M, f(x)=g(x), between manifolds of the same dimension. The main tool is the Lefschetz number λfg defined in terms of the homology of M,N,f: if λfg≠0 then there is at least one coincidence. In the control situation, the next state f(x,u) in determined by the current one, xÎ M, as well as by the input, uÎU. It is described by a map f:N=U×M→M and its equilibria are coincidences of f and the projection g:N=U×M→M. In this paper the Lefschetz number and homomorphism are applied to detection of equilibria, controllability, and their robustness.

Higher order Nielsen numbers. To appear in Fixed Point Theory and Its Applications. For the Coincidence Problem: "If X and Y are a manifolds of dimension n+m and n respectively and f,g:X® Y are maps, what can be said about the set C=Coin(f,g) of coincidences, i.e., the set of x in X such that f(x)=g(x)?'' we suggest a generalization of the classical Nielsen theory (see Nielsen Fixed Point Theory by Robert F. Brown). More general is the Preimage Problem. For a map f:X® Y and a submanifold Y of Z, it studies the preimage set C={x:f(x)ÎY}, and the codimension is m=dimX+dimY-dimZ. In case of codimension 0, the classical Nielsen number N(f,Y) is a lower estimate of the number of points in C changing under homotopies of f, and for an arbitrary codimension, of the number of components of C. We extend this theory to take into account other topological characteristics of C. The goal is to find a "lower estimate" of the bordism group Wp(C) of C. The answer is the Nielsen group Sp(f,Y).

Removing coincidences of maps between manifolds of different dimensions. Topological Methods in Nonlinear Analysis, 22 (2003) 1, 105-114.  This is further development of the study in my previous paper Lefschetz coincidence theory for maps between spaces of different dimensions applied to manifolds. Consider the Coincidence Problem: "If X and Y are a manifolds of dimension n+m and n respectively and f,g:X® X are maps, what can be said about the set Coin(f,g) of coincidences, i.e., the set of x in X such that f(x)=g(x)?'' While the coincidence theory for codimension m=0 is well developed, very little is known about the case m>0. If the coincidence homomorphism (index) is not identically 0 then Coin(f,g) is nonempty. Here we consider the converse of this theorem. If the index is identically 0, does it mean that there are no coincidences? For m=0, the answer is yes under very mild restrictions. Under certain conditions, the answer is yes for m>0 as well.  More...

Lomonosov's invariant subspace theorem for multivalued linear operators. Proceedings of the American Mathematical Society, 131 (2003) 3, 825-834, Lomonosov's invariant subspace theorem states that a linear operator T that commutes with a compact operator C (i.e., TC=CT) has a nontrivial invariant subspace L (i.e., T(L)Í L). The proof makes use of the Schauder fixed point theorem. The idea of the present paper is to prove a similar theorem for multivalued linear operators by means of a fixed point theorem for multivalued maps, e.g., the Kakutani Theorem. Then a (single valued or multivalued) linear operator has an invariant subspace if it commutes with a multivalued "compact" operator. More...

Lefschetz coincidence theory for maps between spaces of different dimensions. Topology and Its Applications, 116 (2001) 1, 137-152. This is further development of the study in my previous paper A Lefschetz-type coincidence theorem. Consider the Coincidence Problem: "If X and Y are a topological spaces and f,g:X® X are maps, what can be said about the set Coin(f,g) of x in X such that f(x)=g(x)?'' (cf. the Fixed Point Problem). While the coincidence theory of maps between manifolds of the same dimension is well developed, very little is known if the dimensions are different or one of the spaces is not a manifold. For a given pair of maps f,g:X® Y from an arbitrary topological space (in particular, an m-manifold) to an n-manifold, we generalize the coincidence index and the Lefschetz coincidence number. The former is generalized by the coincidence homomorphism defined by R. Brooks and R. Brown. For the latter, we consider the Lefschetz homomorphism as a certain graded homomorphism of degree (-n) that depends only on the homomorphisms generated by f and g on homology groups. We prove that they coincide (the graded Lefschetz coincidence theorem). It follows that if the Lefschetz homomorphism is not identically 0 then there is an x in X such that f(x)=g(x) (a coincidence). The theorem contains previously known results concerning the following. (i) Coincidences for X,Y n-manifolds with f boundary-preserving. (ii) Fixed points of a multivalued map G:Y® Y with acyclic values, here X=Graph(G), f,g are projections (in particular, the Eilenberg-Montgomery Fixed Point Theorem). (iii) Fixed points of a parametrized map F:TxY® Y, here g=F, f is the projection and Fix(F)={(t,x):F(t,x)=x}More...

Fixed Points and Coincidences. Ph.D. Thesis. University of Illinois at Urbana-Champaign, 1999. It expands A Lefschetz-type coincidence theorem and Fixed points and selections of set valued maps on spaces with convexity. More...

A Lefschetz-type coincidence theorem. Fundamenta Mathematicae, 162 (1999) 1-2, 65-89. Consider the Coincidence Problem: "If X and Y are a topological spaces and f,g:X® X are maps, what can be said about the set Coin(f,g) of x in X such that f(x)=g(x)?'' (cf. the Fixed Point Problem). While the coincidence theory of maps between manifolds of the same dimension is well developed, very little is known if the dimensions are different or one of the spaces is not a manifold. In this paper a Lefschetz-type coincidence theorem for two maps f,g:X® Y from an arbitrary topological space X to a manifold Y is given: I(f,g)=L(f,g), the coincidence index is equal to the Lefschetz number. It follows that if L(f,g) is not equal to zero then there is an x in X such that f(x)=g(x) (i.e., Coin(f,g) is nonempty). The theorem contains some well known coincidence results for (i) X,Y n-manifolds, f boundary-preserving and (ii) Y Euclidean, f with acyclic fibers (in particular, the Eilenberg-Montgomery Fixed Point Theorem). We also provide examples of how to use our results to detect coincidences that would not fit into the classical theory.  More...

Fixed points and selections of set valued maps on spaces with convexity. International Journal of Mathematics and Mathematical Sciences, 24 (2000) 9, 595-612. We provide two results that unite the following two pairs of theorems respectively.
Kakutani Fixed Point Theorem. Let X be a nonempty convex compact subset of a locally convex Hausdorff topological vector space, and let F:X® Y be an upper semicontinuous multifunction with nonempty closed convex images. Then F has a fixed point.
Browder Fixed Point Theorem. Let X be a nonempty convex compact subset of a Hausdorff topological vector space, and let F:X® X be a multifunction with nonempty convex images and fibers relatively open in X. Then F has a fixed point.
Michael Selection Theorem. Let X be a paracompact Hausdorff topological space, and let Y be a Banach space. Let T:X® Y be a lower semicontinuous multifunction with nonempty closed convex images. Then T has a continuous selection, i.e. a map g:X® Y such that g(x)Î T(x) for all x.
Browder Selection Theorem. Let X be a paracompact Hausdorff topological space, and let Z be any topological vector space. Let T:X® Z be a multifunction having nonempty convex images and open fibers. Then T has a continuous selection.
For this purpose we introduce convex structures on topological spaces ones of more general than those of topological vector spaces, or topological convexity structures due to Michael, Van de Vel, Horvath, and others. We are able to construct a convexity structure for a wide class of topological spaces, which makes it possible to prove a generalization of the following purely topological fixed point theorem.
Eilenberg-Montgomery Fixed Point Theorem. Let X be an acyclic compact ANR, and let F:X® X be an upper semicontinuous multifunction with nonempty closed acyclic values. Then F has a fixed point. More...

Dissipativity in the plane. Russian Academy of Sciences. Sbornik. Mathematics, 75 (1993) 1, 125-135. A classical method of studying dissipativity (ultimate boundedness of solutions) of systems of ordinary differential equations on the plane is considered. The method consists in the construction of a system of compact sets covering the plane, whose boundaries are given by trajectories of a certain auxiliary system, and the trajectories of the given system intersect them from outside in. In this connection the problem of coincidence and intersection of trajectories of two differential inclusions (multivalued differential equations) is solved. In addition, this method is used to prove a theorem that generalizes certain well-known results on the existence of a periodic solution and the dissipativity of the Lienard x''+f(x,x')x'+g(x)=e(t,x,x') and Rayleigh x"+f(x')+g(x)=e(t,x,x') equations. These results follow from the dissipativity and the Brouwer Fixed Point Theorem, but for differential inclusions one needs to apply (in a nontrivial fashion) the Eilenberg-Montgomery Fixed Point Theorem. More...

The dissipativity of the generalized Lienard equation. Differential Equations, 28 (1992) 6, 794-800. In this paper we consider a generalization of a result of a Cartwright and Swinnerton-Dyer on the dissipativity (ultimate boundedness of solutions) of the autonomous Lienard equation x''+f(x,x')x'+g(x)=0. We extend this result to include nonautonomous spaces of solutions. This allows us to consider the generalized Lienard equation x''+f(x,x')x'+g(x)=e(t,x,x') with relaxed restrictions on f and g, and f, g, and e with arbitrary singularities. We also prove the existence of periodic solutions through the Brouwer Fixed Point Theorem. More...

On Poincare-Bendixson theorem and dissipativity on a plane. Moscow University Mathematics Bulletin, 46 (1991) 3, 54-55. The famous Poincare-Bendixson Theorem states that under suitable conditions the limit set of a trajectory of a differential equation with uniqueness in the plane is a cycle. It was generalized by V. V. Filippov to the limit set of a sequence of trajectories without uniqueness provided there are no intersections or self-intersections of trajectories. The proof is purely topological. In this note we extend it further to include the following result: the limit set of a sequence of cycles is a cycle. More...

Accuracy estimation of the multilateral scheme (with N. Chmutin). Raketno-Kosmicheskaya Tekhnika (Rocket and Space Technology), 6 (1989) 3, 62-65 (in Russian). The paper studies a certain technique of approximation of multidimensional integrals and its applications.

The dissipativity of systems of ordinary differential equations in the plane. Moscow University Mathematics Bulletin, 43 (1988) 4, 93-96. The results deal with the existence of a periodic solution and the ultimate boundedness of the trajectories of systems of ordinary differential equations, in particular the Lienard x''+f(x,x')x'+g(x)=e(t,x,x') and Rayleigh x"+f(x')+g(x)=e(t,x,x') equations. The results have been developed further in The dissipativity of the generalized Lienard equation and Dissipativity in the plane. More...

 3. Links 4. My publications, preprints, etc.