Accretive operators and nonlinear evolution equations in Banach spaces.

1970 ◽  
pp. 138-161 ◽  
Author(s):  
Tosio Kato
Keyword(s):  
Banach Spaces ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Accretive Operators

scholarly journals NONLINEAR EVOLUTION GOVERNED BY ACCRETIVE OPERATORS IN BANACH SPACES: ERROR CONTROL AND APPLICATIONS

2006 ◽  
Vol 16 (03) ◽  
pp. 439-477 ◽  
Author(s):  
RICARDO H. NOCHETTO ◽  
GIUSEPPE SAVARÉ
Keyword(s):  
Error Analysis ◽  
Banach Spaces ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
A Posteriori Error Estimates ◽  
Nonlinear Evolution Equations ◽  
Degenerate Parabolic Equations ◽  
Optimal Order ◽  
Accretive Operators ◽  
Unified Framework

Nonlinear evolution equations governed by m-accretive operators in Banach spaces are discretized via the backward or forward Euler methods with variable stepsize. Computable a posteriori error estimates are derived in terms of the discrete solution and data, and shown to converge with optimal order [Formula: see text]. Applications to scalar conservation laws and degenerate parabolic equations (with or without hysteresis) in L1, as well as to Hamilton–Jacobi equations in C0are given. The error analysis relies on a comparison principle, for the novel notion of relaxed solutions, which combines and simplifies techniques of Benilan and Kružkov. Our results provide a unified framework for existence, uniqueness and error analysis, and yield a new proof of the celebrated Crandall–Liggett error estimate.

scholarly journals Convergence Theorems for Generalized Viscosity Explicit Methods for Nonexpansive Mappings in Banach Spaces and Some Applications

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 161
Author(s):  
Pongsakorn Sunthrayuth ◽  
Nuttapol Pakkaranang ◽  
Poom Kumam ◽  
Phatiphat Thounthong ◽  
Prasit Cholamjiak
Keyword(s):  
Banach Spaces ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonexpansive Mappings ◽  
Nonlinear Evolution Equations ◽  
Fredholm Integral Equations ◽  
Explicit Method ◽  
Convergence Theorems ◽  
New Techniques ◽  
The Given

In this paper, we introduce a generalized viscosity explicit method (GVEM) for nonexpansive mappings in the setting of Banach spaces and, under some new techniques and mild assumptions on the control conditions, prove some strong convergence theorems for the proposed method, which converge to a fixed point of the given mapping and a solution of the variational inequality. As applications, we apply our main results to show the existence of fixed points of strict pseudo-contractions and periodic solutions of nonlinear evolution equations and Fredholm integral equations. Finally, we give some numerical examples to illustrate the efficiency and implementation of our method.

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