Dr. Claudio H. Morales

Professor

Address Information

  • Office: 258J Shelby Center
  • Voice: (256) 824-2227
  • Fax: (256) 824-6173
  • E-Mail: morales@math.uah.edu

Research Interests

  • functional analysis
  • modern analysis
  • nonlinear operator theory
  • fixed point theory and its applications to partial differential equations
  • foundations of mathematics
  • symbolic logic
  • mathematics education

Current Project

The unification of the general theory of accretive operators defined on Banach spaces, basically motivated by the need of resolving general functional equations of the form

eq1

where A is an accretive operator and C is a compact perturbation under some kind of sign condition. This type of problem has been extensively studied due to its close connection to the theory of partial differential equations.

Accretive Operator in Banach Spaces

In 1967 Browder and Kato, independently, introduced the class of accretive operators, which arose as an extension of the well-known class of monotone mappings in Hilbert spaces. This latter family became an important source for the development of the theory of elliptical differential equations, variational problems, resonance problems, as well as network problems. Classical examples of accretive operators include the gradient of a convex functional and the negative of the Laplacian operator defined in an appropriate domain. This theory has been found to be intimately related to the class of nonexpansive mappings, which constitutes one of the first families of mappings for which fixed-point results can be proved under the absence of the compactness assumption while placing emphasis on the geometric structure.

As a matter of fact, an operator A, defined on a portion of a Banach space and mapping that portion into itself, is said to be accretive if, for each x and y in the domain of A and r > 0,

eq2

If, in addition, I + rA (where I is the identity mapping) is surjective, then A is called m-accretive. For this class of operators, the resolvent

Jr = (I + rA)−1.

is a nonexpansive mapping defined in the whole Banach space. Incidentally, this observation illustrates the close connection between these two theories.

Selected Works

  • Convergence of Paths for Pseudo-contractive Mappings in Banach Spaces (with J. S. Jung), Proc. Amer. Math. Soc., 128 (2000), 3411-34-19.
  • Convergence of the Steepest Descent Method for Accretive Operators (with C.E. Chidume), Proc. Amer. Math. Soc., 127 (1999), 3677-3683.
  • A Generalization of Bolzano's Theorem, The Lewis-Parker Lecture at the AACTM Annual Meeting, Alabama J. Math., 23 (1999), 3-13.
  • Approximations of Fixed Points for Locally Nonexpansive Mappings, Annales: Universitatis Mariae Curie-Sklodowska, Proceedings of Workshop on Fixed Point Theory, Kazimierz, Poland (1998), 203-212.
  • Locally Accretive Mappings in Banach Spaces, Bull. London Math. Soc., 28 (1996), 627-633.
  • On a Fixed Point Theorem of Kirk (with S.A. Mutangadura), Proc. Amer. Math. Soc., 123 (1995), 3397-3401.
  • On the Approximation of Fixed Points for Locally Pseudo-contractive Mappings (with S.A. Mutangadura), Proc. Amer. Math. Soc., 123 (1995), 417-423 .
  • On linear Ordinary Differential Equations with Functionally Commutative Coefficients Matrices (with J. Zhu), Linear Alg. Appl., 170 (1992), 81-105.
  • On the Range of Sums of Accretive and Continuous Operators in Banach Spaces, Nonlinear Analysis, Theory, Methods and Applications, 19 (1992), 1-9.
  • Multivalued Pseudo-contractive Mappings Defined on Unbounded Sets in Banach Spaces, Comment. Math. Univ. Carolinae, 33 (1992), 625-630.
  • Remarks on Compact Perturbations of M-Accretive Operators, Nonlinear Analysis, Theory, Methods and Applications, 16 (1991), 771-780.
  • Spatial Decomposition of Functionally Commutative Matrices (with J. Zhu), Linear Alg. Appl., 131 (1990), 71-92 .
  • Strong Convergence Theorems for Pseudo-contractive Mappings in Banach spaces, Houston J. Math., 16 (1990), 549-557.

Professional Background

As result of persistent work in the area of nonlinear operator theory, Professor Morales has attained a national and international recognition for his mathematical contribution in this area. He has been invited on several occasions to Europe, Africa and South America to present his work. In addition, he has directed a number of master theses in various areas of modern analysis, and, most recently, had his first Ph.D. student, M. Leigh Lunsford, complete her dissertation on Existence Results for Generalized Variational Inequalities.

Professor Morales has been at UAH since 1982.

Education

  • Ph.D. (math) University of Iowa.
  • M.S. (statistics) University of Iowa.
  • B.S. (math) University of Chile.