scholarly journals A further necessary and sufficient condition for strong convergence of nonlinear contraction semigroups and of iterative methods for accretive operators in Banach spaces

1995 ◽  
Vol 38 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Zong-Ben Xu ◽  
Yao-Lin Jiang ◽  
G. F. Roach
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Iterative Methods ◽  
Accretive Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accretive Operators ◽  
Uniformly Smooth ◽  
Necessary And Sufficient

Let A be a quasi-accretive operator defined in a uniformly smooth Banach space. We present a necessary and sufficient condition for the strong convergence of the semigroups generated by – A and of the steepest descent methods to a zero of A.

scholarly journals EQUILIBRIUM POINTS FOR A SYSTEM INVOLVING M-ACCRETIVE OPERATORS

2001 ◽  
Vol 44 (1) ◽  
pp. 187-199 ◽  
Author(s):  
C. E. Chidume ◽  
M. O. Osilike
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Iterative Methods ◽  
Equilibrium Points ◽  
Accretive Operator ◽  
Evolution System ◽  
Open Domain ◽  
Accretive Operators ◽  
Uniformly Smooth ◽  
Error Terms

AbstractLet $E$ be a real uniformly smooth Banach space and let $A$ be a nonlinear $\phi$-strongly quasi-accretive operator with range $R(A)$ and open domain $D(A)$ in $E$. For a given $f\in E$, let $A$ satisfy the evolution system $\rd u(t)/\rd t+Au(t)=f$, $u(0)=u_0$. We establish the strong convergence of the Ishikawa and Mann iterative methods with appropriate error terms recently introduced by Xu to the equilibrium points of this system. Related results deal with the strong convergence of the iterative methods to the fixed points of $\phi$-strong pseudocontractions defined on open subsets of $E$.AMS 2000 Mathematics subject classification: Primary 47H06; 47H15; 47H17

ON BI-BANACH FRAMES IN BANACH SPACES

2014 ◽  
Vol 12 (02) ◽  
pp. 1450015 ◽  
Author(s):  
SHALU SHARMA
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Banach Frames ◽  
Banach Frame ◽  
Necessary And Sufficient

Bi-Banach frames in Banach spaces have been defined and studied. A necessary and sufficient condition under which a Banach space has a Bi-Banach frame has been given. Finally, Pseudo exact retro Banach frames have been defined and studied.

ON FRAME SYSTEMS IN BANACH SPACES

2009 ◽  
Vol 07 (01) ◽  
pp. 1-7 ◽  
Author(s):  
P. K. JAIN ◽  
S. K. KAUSHIK ◽  
NISHA GUPTA
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bessel Sequence ◽  
Banach Frame ◽  
Frame System ◽  
Frame Systems ◽  
Necessary And Sufficient

Banach frame systems in Banach spaces have been defined and studied. A sufficient condition under which a Banach space, having a Banach frame, has a Banach frame system has been given. Also, it has been proved that a Banach space E is separable if and only if E* has a Banach frame ({φn},T) with each φn weak*-continuous. Finally, a necessary and sufficient condition for a Banach Bessel sequence to be a Banach frame has been given.

FRAMES OF SUBSPACES FOR BANACH SPACES

2010 ◽  
Vol 08 (02) ◽  
pp. 243-252 ◽  
Author(s):  
P. K. JAIN ◽  
S. K. KAUSHIK ◽  
VARINDER KUMAR
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Banach Frame ◽  
Necessary And Sufficient ◽  
Compactly Generated

Frames of subspaces for Banach spaces have been introduced and studied. Examples and counter-examples to distinguish various types of frames of subspaces have been given. It has been proved that if a Banach space has a Banach frame, then it also has a frame of subspaces. Also, a necessary and sufficient condition for a sequence of projections, associated with a frame of subspaces, to be unique has been given. Finally, we consider complete frame of subspaces and prove that every weakly compactly generated Banach space has a complete frame of subspaces.

scholarly journals Sequences and bases in p-Banach spaces

1986 ◽  
Vol 34 (1) ◽  
pp. 87-92
Author(s):  
M. A. Ariño
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dimensional Subspace ◽  
Basic Sequence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Necessary And Sufficient ◽  
Infinite Dimensional Subspace

Necessary and sufficient condition are given for an infinite dimensional subspace of a p-Banach space X with basis to contain a basic sequence which can be extended to a basis of X.

Atomic systems for operators

2019 ◽  
Vol 17 (01) ◽  
pp. 1850066 ◽  
Author(s):  
Khole Timothy Poumai ◽  
Shah Jahan
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Atomic System ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bessel Sequence ◽  
Atomic Systems ◽  
The Family ◽  
Necessary And Sufficient

Gavruta [L. Gavruta, Frames for operators, Appl. Comput. Harmon. Anal. 32 (2012) 139–144] introduced the notion of [Formula: see text]-frame and atomic system for an operator [Formula: see text] in Hilbert spaces. We extend these notions to Banach spaces and obtain various new results. A necessary and sufficient condition for the existence of an atomic system for an operator [Formula: see text] in a Banach space is given. Also, a characterization for the family of local atoms of subspaces of Banach spaces has been given. Further, we give methods to construct an atomic system for an operator [Formula: see text] from a given Bessel sequence and an [Formula: see text]-Bessel sequence. Finally, a result related to stability of atomic system for an operator [Formula: see text] in a Banach space has been given.

Convergence Theorems for Accretive Operators in Banach Spaces

2011 ◽  
Vol 50-51 ◽  
pp. 224-228
Author(s):  
Yan Guo ◽  
Ji Wei He ◽  
Yan Mei Yang
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Nonexpansive Mapping ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Accretive Operator ◽  
Common Element ◽  
Accretive Operators ◽  
Uniformly Smooth ◽  
Strongly Accretive Operator

In this paper, a new iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of the variational inequality for an inverse-strongly accretive operator in a 2-uniformly smooth Banach space was introduced . It was verified that the sequence converged to a common element of two sets.

scholarly journals Convergence of an Iterative Algorithm for Common Solutions for Zeros of Maximal Accretive Operator with Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Uamporn Witthayarat ◽  
Yeol Je Cho ◽  
Poom Kumam
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Accretive Operator ◽  
Uniformly Convex ◽  
Common Solution ◽  
Uniformly Smooth ◽  
Convex Feasibility

The aim of this paper is to introduce an iterative algorithm for finding a common solution of the sets(A+M2)−1(0) and(B+M1)−1(0), where M is a maximal accretive operator in a Banach space and, by using the proposed algorithm, to establish some strong convergence theorems for common solutions of the two sets above in a uniformly convex and 2-uniformly smooth Banach space. The results obtained in this paper extend and improve the corresponding results of Qin et al. 2011 from Hilbert spaces to Banach spaces and Petrot et al. 2011. Moreover, we also apply our results to some applications for solving convex feasibility problems.

Sign in / Sign up

Export Citation Format

Share Document