**Volodymyr Petryshyn**'s name appears both as **Walter
Volodymyr Petryshyn** and as **Wolodymyr V Petryshyn**. His parents were
Vasyl and Maria Petryshyn. He was 10 years old when World War II broke out, and
following this his education was severely disrupted. He had commenced his
studies in Lviv during World War II, but he became a displaced person at the end
of the war and continued his schooling in Germany. In 1950 he emigrated from
Germany to the United States and completed his education there, living in
Paterson, New Jersey. He studied at Columbia University and was awarded a B.A.
in 1953, and an M.S. in 1954. In 1955, on the recommendation of the Department
of Mathematics, he was appointed Instructor in Mathematics at Columbia. He
studied for a doctorate, supervised by Francis Joseph Murray, while working as
an Instructor and, in 1961, he was awarded a Ph.D. from Columbia University for
his thesis *Linear Transformations Between Hilbert Spaces and the Application
of the Theory to Linear Partial Differential Equations*.

In 1956, while at Columbia University, Petryshyn married Arcadia Olenska (1934-1996). Arcadia was born in Roznoshentsi near Zbarazh in Ukraine, and emigrated with her parents to the United States in 1949, living in New York. She became a well-known artist, art critic and editor.

After completing his doctorate, Petryshyn was appointed as a
postdoctoral fellow at New York University. He held this position from 1961 to
1964 and during this time his first papers appeared. In 1962, *Direct and
iterative methods for the solution of linear operator equations in Hilbert
space* was published which does much toward developing a unified point of
view toward a number of important methods of solving linear equations. In the
same year, *The generalized overrelaxation method for the approximate solution
of operator equations in Hilbert space* appeared and in the following year
the two papers *On a general iterative method for the approximate solution of
linear operator equations* and *On the generalized overrelaxation method
for operation equations*.

From 1964 he taught at the University of Chicago where he was appointed as an associate professor. In 1967 he was appointed as a professor at Rutgers University, and he held this position until he retired in 1996.

Petryshyn's main work has been in iterative and projective methods, fixed point theorems, nonlinear Friedrichs extension, approximation-proper mapping theorem, and topological degree and index theories for multi-valued condensing maps. His mathematical achievements are described by Andrushkiw in [1]:-

Petryshyn's main achievements are in functional analysis. His major results include the development of the theory of iterative and projective methods for the constructive solution of linear and nonlinear abstract and differential equations.

The theory of *A*-proper maps was developed by Petryshyn
and this work is described in [1]:-

Petryshyn is a founder and principal developer of the theory of approximation-proper(A-proper)maps, a new class of maps which attracted considerable attention in the mathematical community. He has shown that the theory of A-proper type maps not only extends and unifies the classical theory of compact maps with some recent theories of condensing and monotone-accretive maps, but also provides a new approach to the constructive solution of nonlinear abstract and differential equations. ... The theory has been applied to ordinary and partial differential equations.

His main contributions are published in over 100 research
papers, but can be more easily appreciated from two important monographs which
he published in the 1990s. *Approximation-solvability of Nonlinear Functional
and Differential Equations* appeared in December 1992:-

This outstanding reference/text develops an essentially constructive theory of solvability on linear and nonlinear abstract and differential equations involving A-proper operator equations in separable Banach spaces, treats the problem of existence of a solution for equations involving pseudo-A-proper and weakly-A-proper mappings, and illustrates their applications. Facilitating the understanding of the solvability of equations in infinite dimensional Banach space through finite dimensional approximations, Approximation - solvability of Nonlinear Functional and Differential Equations: offers an important elementary introduction to the general theory of A-proper and pseudo-A-proper maps; develops the linear theory of A-proper maps; furnishes the best possible results for linear equations; establishes the existence of fixed points and eigenvalues for P-gamma-compact maps, including classical results; provides surjectivity theorems for pseudo-A-proper and weakly-A-proper mappings that unify and extend earlier results on monotone and accretive mappings; shows how Friedrichs' linear extension theory can be generalized to the extensions of densely defined nonlinear operators in a Hilbert space; presents the generalized topological degree theory for A-proper mappings; and applies abstract results to boundary value problems and to bifurcation and asymptotic bifurcation problems.

In 1995 his second monograph *Generalized Topological Degree
and Semilinear Equations* appeared in print, published by *Cambridge
University Press*. In the Preface he writes:-

In this monograph we develop the generalised degree theory for densely defined A-proper mappings, and then use it to study the solubility(sometimes constructive)and the structure of the solution set of[an]important class of semilinear abstract and differential equations ...

The publisher's publicity explains the contents more fully:-

This book describes the construction of the generalised topological degree for densely defined and not necessarily continuous A-proper operators, and presents important applications. A-proper mappings arise naturally in the solution to an equation in infinite dimensional space via the finite dimensional approximation.. The theory subsumes classical theory involving compact vector fields as well as more recent theories of condensing vector fields and strongly monotone and strongly accretive maps. The book begins with an outline of Brouwer degree theory and a description of some basic constructive results. Using these tools, the author defines the generalised topological degree for densely defined A-proper mappings, gives applications to the solubility of an important class of semilinear abstract and differential equations, and discusses global bifurcation results. These abstract results are then applied to boundary value problems of ODEs and PDEs with general nonlinearities, problems that are intractable under any other existing theory.

In addition he recently published *Development of mathematical
sciences in Ukraine* in Ukrainian in 2004.

Petryshyn received several significant honours for his excellent mathematical contributions. He was elected to the Shevchenko Scientific Society in 1980 and to the Ukrainian Academy of Sciences in 1992. He was also elected as an honorary member of the Kiev Mathematical Society in 1989. His greatest honour, however, was being awarded the M Krylov Award by the Ukrainian Academy of Sciences in 1992, this being the highest award of the Academy.

Finally we should mention the Vasyl and Maria Petryshyn Memorial Lectures in Ukrainian Studies established at the Harvard Ukrainian Research Institute in 1991 through a major gift donated by the estate of the late Vasyl and Maria Petryshyn, with additional funds provided by their son Wolodymyr Petryshyn, and other family members. The fund also supports the publication of the text of each lecture, the first of which was delivered on 24 April 1994.

**Article by:** *J J O'Connor* and *E F Robertson*

**December 2008**

http://www-history.mcs.st-andrews.ac.uk/Biographies/Petryshyn.html