In 1956, while at Columbia University, Petryshyn married Arcadia Olenska (1934-1996). Arcadia was born in Roznoshentsi near Zbarazh in the 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 Andrushkov in :-
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 :-
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.Unfortunately, after this work was published Petryshin believed that he had found a serious error in what he had published. He thought that this would lead to his being ridiculed by his fellow mathematicians. While under the strain of this, he had a complete mental breakdown and on 6 May 1966 he killed his wife Arcadia. In August 1967 he was found not guilty of murder by reason of insanity. 
Petryshin recently published Development of mathematical sciences in the 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 and a fellow of the American Mathematical Society in 2012. 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 Volodymyr 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