M-accretive Operators In Banach Spaces Research Articles

On Compact Perturbations and Compact Resolvents of Nonlinear m-Accretive Operators in Banach Spaces

On Compact Perturbations and Compact Resolvents of Nonlinear m-Accretive Operators in Banach Spaces

On compact perturbations and compact resolvents of nonlinear 𝑚-accretive operators in Banach spaces

Several mapping results are given involving compact perturbations and compact resolvents of accretive and m-accretive operators. A simple and straightforward proof is given to an important special case of a result of Morales who has recently improved and/or extended various results by the author and Hirano. Improved versions of results of Browder and Morales are shown to be possible by studying various homotopies of compact transformations.

An approximation theory for the identification of nonlinear volterra equations

An abstract approximation framework and convergence theory for Galerkin approximations to inverse problems involving nonlinear Volterra integral equations is developed. The approach relies on the theory of m-accretive operators in Banach spaces. An application to heat flow in materials with memory is also discussed.

Perturbation of m-accretive operators in banach spaces

THERE ARE not many results on multiplicative perturbation theory dealing with development of criteria for an m-accretive operator to remain m-accretive after being composed with other operators. In this connection we mention the work of Gustafson and Lumer [l 11, who gave a criterion for m-accretivity of the composition KA on a Banach space X, where K is assumed to be an everywhere defined strongly accretive bounded linear operator and KA is accretive. The main purpose of this note is to weaken the requirement of strong accretivity of K to simple accretivity to conclude that the closure of densely defined operator KA is m-accretive. An example is given to illustrate applicability of this result even though the Gustafson and Lumer result is not applicable. Further a nonlinear analogue of Lumer’s result is proved. We suggest two unsolved problems in this direction. It is well known that the class of m-accretive operators arise in initial boundary value problems of differential equations of evolution type. Multiplicative perturbations of such operators have been studied previously in [l, 5, 6, 7, 8, 9, 10, 11, 131. For a description of m-accretive operators and their properties one can refer to [2, 3, 6, 12, 151. We state the above mentioned result of Gustafson and Lumer [l l] in the following form which can be compared more easily with our results in this work.