Mark Alexandrovich Krasnosel'skii (April 27, 1920, Starokostiantyniv

Starokostiantyniv

Starokostiantyniv is a city in the Khmelnytskyi Oblast of western Ukraine. Serving as the administrative center of the Starokostiantynivsky Raion , the city itself is also designated as a separate raion within the oblast...

 – February 13, 1997, Moscow

Moscow

Moscow is the capital, the most populous city, and the most populous federal subject of Russia. The city is a major political, economic, cultural, scientific, religious, financial, educational, and transportation centre of Russia and the continent...

) was a Soviet

Soviet Union

The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....

, Russia

Russia

Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...

n and Ukrainian

Ukraine

Ukraine is a country in Eastern Europe. It has an area of 603,628 km², making it the second largest contiguous country on the European continent, after Russia...

 mathematician renowned for his work on nonlinear functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

 and its applications.

Early years

Mark Krasnosel'skii was born in the town of
Starokostiantyniv

Starokostiantyniv

Starokostiantyniv is a city in the Khmelnytskyi Oblast of western Ukraine. Serving as the administrative center of the Starokostiantynivsky Raion , the city itself is also designated as a separate raion within the oblast...

 in Ukraine

Ukrainian SSR

The Ukrainian Soviet Socialist Republic or in short, the Ukrainian SSR was a sovereign Soviet Socialist state and one of the fifteen constituent republics of the Soviet Union lasting from its inception in 1922 to the breakup in 1991...

 on the 27 April 1920 where his father worked as a
construction engineer and his mother taught in an elementary school.
In 1932 the Krasnosel'skii family moved to Berdyansk and in 1938 Mark
entered the physico-mathematical faculty of Kiev University

Kiev University

Taras Shevchenko University or officially the Taras Shevchenko National University of Kyiv , colloquially known in Ukrainian as KNU is located in Kiev, the capital of Ukraine. It is the third oldest university in Ukraine after the University of Lviv and Kharkiv University. Currently, its structure...

, which was
evacuated at the beginning of World War II

World War II

World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...

  to Kazakhstan

Kazakhstan

Kazakhstan , officially the Republic of Kazakhstan, is a transcontinental country in Central Asia and Eastern Europe. Ranked as the ninth largest country in the world, it is also the world's largest landlocked country; its territory of is greater than Western Europe...

  where it
became known as the Joint Ukrainian University.

He graduated in 1942, in the middle of the war, served
four years in the Soviet Army

Soviet Army

The Soviet Army is the name given to the main part of the Armed Forces of the Soviet Union between 1946 and 1992. Previously, it had been known as the Red Army. Informally, Армия referred to all the MOD armed forces, except, in some cases, the Soviet Navy.This article covers the Soviet Ground...

, became Candidate in Science in 1948, with a
dissertation on self-adjoint extensions of operators with nondense domains,
before getting the title of Doctor in Science in 1950, with a thesis on
investigations in Nonlinear Functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

.

Scientific career

From 1946 till 1952, Mark was a Research Fellow at the Mathematical
Institute of the Ukrainian Academy of Science

National Academy of Science of Ukraine

The National Academy of Sciences of Ukraine is the highest government research body in Ukraine and one of the six state academies. Its presidium is located at 57 Volodymyr Street, across the street from the Building of Pedagogical Museum where used to preside the Central Rada during the...

 in Kiev

Kiev

Kiev or Kyiv is the capital and the largest city of Ukraine, located in the north central part of the country on the Dnieper River. The population as of the 2001 census was 2,611,300. However, higher numbers have been cited in the press....

. From 1952 till 1967,
he was Professor at Voronezh

Voronezh

Voronezh is a city in southwestern Russia, the administrative center of Voronezh Oblast. It is located on both sides of the Voronezh River, away from where it flows into the Don. It is an operating center of the Southeastern Railway , as well as the center of the Don Highway...

 State University. He then moved to Moscow as a Senior Scientific Fellow
(1967-1974) and then a Head of a Laboratory (1974-1990) at the Institute of
Control Sciences of the USSR Academy of Science

Russian Academy of Sciences

The Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

 in Moscow. From 1990, he worked
at the Institute for Information Transmission Problems of the same Academy.

Family

When Mark was 18 he married Sarra Belotserkovskaya (10.09.1921-31.01.2009), they had 3 children (Veniamin, 1939; Alexandra (Alla), 1945; Alexander (Sasha), 1955). Now there are 7 grandchildren and 9 great-grandchildren.

Distinctions

• Andronov Prize of the Soviet Academy of Science,
• Humboldt Prize
  Humboldt Prize
  The Humboldt Prize, also known as the Humboldt Research Award, is an award given by the Alexander von Humboldt Foundation to internationally renowned scientists and scholars, and is currently valued at € 60,000 with the possibility of further support during the prize winner's life. Up to one...
  
  
• Docteur {honoris causa} of the University of Rouen in France, 1996.

Scientific achievements

Mark Krasnosel'skii's has authored or co-authored some three hundred papers and fourteen
monographs.
Nonlinear techniques are roughly classified into analytical,
topological and variational methods. Mark Krasnosel'skii
has contributed to all three aspects in a
significant way, as well as to their application to many types of integral

Integral equation

In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...

,
differential

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

 and functional equation

Functional equation

In mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...

s coming from mechanics

Mechanics

Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....

, engineering

Engineering

Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

,
and control theory

Control theory

Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...

.

Mark Krasnosel'skii was the first to investigate the functional analytical
properties of fractional powers

Fractional calculus

Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation operator.and the integration operator J...

  of operators, at first for self-adjoint operators and then for more general situations. His theorem on the interpolation of complete continuity

Continuity

Continuity may refer to:In mathematics:* The opposing concept to discreteness; common examples include:** Continuous probability distribution or random variable in probability and statistics...

 of such fractional power operators has been a basic tool in the theory of partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s. Of comparable importance in applications is his extensive collection of works on the theory of positive operators, in particular results in which spectral gaps were estimated. His work on integral operators and superposition operators has also found many theoretical and practical applications. A major reason for this was his desire to always find readily verifiable conditions and estimates for whatever functional properties were under consideration. This is perhaps best seen in his work on topological

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 methods in nonlinear analysis which he developed into a universal method for finding answers to such qualitative problems such as evaluating the number of solutions, describing the structure of a solution set and conditions for the connectedness of this set, convergence of Galerkin type approximations

Galerkin method

In mathematics, in the area of numerical analysis, Galerkin methods are a class of methods for converting a continuous operator problem to a discrete problem. In principle, it is the equivalent of applying the method of variation of parameters to a function space, by converting the equation to a...

, the bifurcation

Bifurcation

Bifurcation means the splitting of a main body into two parts.Bifurcation or Bifurcated may refer to:*Bifurcation , the division of issues in a trial for example the division of a page into two parts....

 of solutions in nonlinear systems, and so on.

Krasnosel'skii also presented many new general principles on solvability of a large variety of nonlinear equations, including one-sided estimates, cone stretching and contractions, fixed-point theorem

Fixed-point theorem

In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point , under some conditions on F that can be stated in general terms...

s for monotone operators
and a combination of the Schauder fixed point

Schauder fixed point theorem

The Schauder fixed point theorem is an extension of the Brouwer fixed point theorem to topological vector spaces, which may be of infinite dimension...

 and contraction mapping theorems that was the genesis of condensing operators. He suggested a new general method for investigating degenerate extremals in variational problems and developed qualitative methods for studying critical and bifurcation parameter values based on restricted information of nonlinear equations. such as the properties of equations linearized at
zero or at infinity, which have been very useful in determining the existence of bounded or periodic solutions.

After he moved to Moscow he turned his attention increasingly to discontinuous processes and operators, in
connection firstly with nonlinear control systems and then with a mathematically rigorous formulation of hysteresis

Hysteresis

Hysteresis is the dependence of a system not just on its current environment but also on its past. This dependence arises because the system can be in more than one internal state. To predict its future evolution, either its internal state or its history must be known. If a given input alternately...

 which encompasses most classical models of hysteresis and is now standard. He also became actively involved with the analysis of desynchronized systems and the justification of the harmonic balance method commonly used by engineers.

Monographs

1. , 395p.
   1. , 249p.
      1. 1. , 242p.
            1. , 379p.
               1. , Translation of Mathematical Monographs, 19, 294p.
                  1. , 520 p.
                     1. , 443p.
                        1. , 484p.
                           1. , 366p.
                              1. , Grundlehren Der Mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics, 263, 409p.
                                 1. , 410p.
                                    1. 1. , 408p., [Russian].
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