Editorial
TOPOLOGICAL AND VARIATIONAL METHODS OF NONLINEAR ANALYSIS AND THEIR APPLICATIONS
V. G. ZVYAGIN, YU. E. GLIKLIKH, AND V. V. OBUKHOVSKII Received 3 July 2006; Accepted 3 July 2006
Copyright © 2006 V. G. Zvyagin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This issue contains the papers selected from the talks of the international conference
“Topological and Variational Methods of Nonlinear Analysis and their Applications” ded- icated to the 85th Jubilee of Professor A. D. Myshkis and the 75th Jubilee of Professor Yu.
G. Borisovich. The conference took place in Voronezh, Russia, 27 June–2 July, 2005.
Professor Anatoly Dmitrievich Myshkis is the founder of a number of new scientific directions in the theory of functional differential equations (1949–1951), partial differ- ential equations, differential inclusions and multivalued dynamical systems, and many others. In particular, he was one of the first researchers who studied retarded-type equa- tions, he introduced the notion of a generalized solution for a differential equation with set-valued discontinuous right-hand side, his studies of set-valued maps with aspheric values (1954) found in the recent decades very important and interesting applications in the theory of differential equations, inclusions and control systems, and so forth. For the series of papers in the theory of set-valued differential equations, he was awarded a prize by the Moscow Mathematical Society.
In his activities, A. D. Myshkis pays special attention to the problems of applications.
It is also worth pointing out his contribution to approximate and numerical methods, difference equations and inequalities, turbulent systems, impulse impact systems, spectral problems with variable boundary, and his analysis of the influence of velocity forces on oscillatory stability. A. D. Myshkis pays much attention to the methodology of applied mathematics and in his works he expressed his original views about how engineers and other specialists should be taught mathematics.
The perennial educational work of A. D. Myshkis prompted him to write several text- books in mathematics and mathematical physics. These textbooks became very popular and were translated into many languages.
Hindawi Publishing Corporation Abstract and Applied Analysis
Volume 2006, Article ID 93926, Pages1–2 DOI 10.1155/AAA/2006/93926
2 Editorial
A. D. Myshkis is a member of editorial boards of well-known international journals such as “Nonlinear Analysis: Theory, Methods and Applications,” “Functional Differen- tial Equations,” and “Journal of Difference Equations and Applications.”
Professor Yuri Grigorievich Borisovich is worldwide known by his works on topologi- cal methods in nonlinear analysis and their applications to mathematical physics, control theory, geometry, and many other branches of mathematics. First we should mention a series of his works in the 50th, and 60th, of twentieth century on relative topological de- gree and relative rotation of vector fields that yielded the development of the degree the- ory for weakly continuous mappings in Banach spaces, for condensing operators (k-set contractions), and applications to functional-differential equations, partial differential equations, and so forth.
In the series of works with his collaborators in the 70th, a new version of degree the- ory for nonlinear Fredholm mappings was suggested that allowed one to cover Fredholm mappings with compact and condensing perturbations. This theory got plenty of appli- cations, first of all to partial differential equations.
Another topic, where Professor Borisovich’s influence is very well known, is the the- ory of set-valued mappings and differential inclusions. A lot of research and survey pa- pers (in particular, in Russian Mathematical Surveys) and two monographs (joint with B. Gel’man, A. D. Myshkis, and V. Obukhovskii; the last one in 2005) were published by him and his collaborators on this subject.
Borisovich’s text book “Introduction to Topology” (joint with N. Bliznyakov, T.
Fomenko, and Y. Izrailevich) has been translated into many foreign languages and is one of the best introductory books on this subject.
In this issue we include papers on the themes where the ideas of variational and topo- logical methods are applied. Here A. D. Myshkis and Yu. G. Borisovich made significant input. It is some parts of topology, topological index theory, functional analysis, global analysis and analysis on manifolds, stochastic analysis, hydrodynamics, and so forth.
V. G. Zvyagin Yu. E. Gliklikh V. V. Obukhovskii
Mathematical Problems in Engineering
Special Issue on
Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios
Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from “Qualitative Theory of Differential Equations,”
allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Prob- lems in Engineering aims to provide a picture of the impor- tance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.
Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophis- ticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System athttp://
mts.hindawi.com/according to the following timetable:
Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009
Guest Editors
José Roberto Castilho Piqueira,Telecommunication and Control Engineering Department, Polytechnic School, The University of São Paulo, 05508-970 São Paulo, Brazil;
Elbert E. Neher Macau,Laboratório Associado de Matemática Aplicada e Computação (LAC), Instituto Nacional de Pesquisas Espaciais (INPE), São Josè dos Campos, 12227-010 São Paulo, Brazil ; [email protected] Celso Grebogi,Center for Applied Dynamics Research, King’s College, University of Aberdeen, Aberdeen AB24 3UE, UK; [email protected]
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