Strong Convergence and Weak Convergence

1995 ◽  
pp. 119-145
Author(s):  
Kôsaku Yosida
Keyword(s):  
Weak Convergence ◽  
Strong Convergence

scholarly journals An iterative process for a hybrid pair of generalized I-asymptotically nonexpansive single-valued mappings and generalized nonexpansive multi-valued mappings in Banach spaces

2018 ◽  
Vol 34 (1) ◽  
pp. 31-45
Author(s):  
ALI FARAJZADEH ◽  
◽  
PREEYANUCH CHUASUK ◽  
ANCHALEE KAEWCHAROEN ◽  
MOHAMMAD MURSALEEN ◽  
...  
Keyword(s):  
Weak Convergence ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Iterative Process ◽  
Finite Family ◽  
Convergence Theorems ◽  
Asymptotically Nonexpansive ◽  
Hybrid Pair

In this paper, an iterative process for a hybrid pair of a finite family of generalized I-asymptotically nonexpansive single-valued mappings and a finite family of generalized nonexpansive multi-valued mappings is established. Moreover, the weak convergence theorems and strong convergence theorems of the proposed iterative process in Banach spaces are proven. The examples are established for supporting our main results. The obtained results can be viewed as an improvement and extension of the several results in the literature.

Weak Convergence Is Not Strong Convergence For Amenable Groups

2001 ◽  
Vol 44 (2) ◽  
pp. 231-241 ◽  
Author(s):  
Joseph M. Rosenblatt ◽  
George A. Willis
Keyword(s):  
Weak Convergence ◽  
Strong Convergence ◽  
Cayley Graphs ◽  
Linear Equations ◽  
Amenable Group ◽  
Amenable Groups

AbstractLet G be an infinite discrete amenable group or a non-discrete amenable group. It is shown how to construct a net (fα) of positive, normalized functions in L1(G) such that the net converges weak* to invariance but does not converge strongly to invariance. The solution of certain linear equations determined by colorings of the Cayley graphs of the group are central to this construction.

scholarly journals Common Attractive Points of Generalized Hybrid Multi-Valued Mappings and Applications

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1307
Author(s):  
Lili Chen ◽  
Ni Yang ◽  
Jing Zhou
Keyword(s):  
Variational Inequality ◽  
Weak Convergence ◽  
Strong Convergence ◽  
Approximation Method ◽  
Variational Inequality Problem ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Banach Limits ◽  
Weak Convergence Theorem ◽  
The Common

In this paper, we first propose the concepts of (ζ,η,λ,π)-generalized hybrid multi-valued mappings, the set of all the common attractive points (CAf,g) and the set of all the common strongly attractive points (CsAf,g), respectively for the multi-valued mappings f and g in a CAT(0) space. Moreover, we give some elementary properties in regard to the sets Af, Ff and CAf,g for the multi-valued mappings f and g in a complete CAT(0) space. Furthermore, we present a weak convergence theorem of common attractive points for two (ζ,η,λ,π)-generalized hybrid multi-valued mappings in the above space by virtue of Banach limits technique and Ishikawa iteration respectively. Finally, we prove strong convergence of a new viscosity approximation method for two (ζ,η,λ,π)-generalized hybrid multi-valued mappings in CAT(0) spaces, which also solves a kind of variational inequality problem.

scholarly journals Strong convergence of selections implied by weak

1989 ◽  
Vol 39 (2) ◽  
pp. 201-214 ◽  
Author(s):  
Tadeusz Rzezuchowski
Keyword(s):  
Weak Convergence ◽  
Strong Convergence ◽  
Differential Inclusions ◽  
Control Functions ◽  
Measurable Multifunctions ◽  
Derivatives Of ◽  
Convergent Sequences ◽  
Convex Subsets

In some situations weak convergence in L1, implies strong convergence. Let P, L: T → C∘(ℝd) be measurable multifunctions (C∘(ℝd) being the set of closed, convex subsets of ℝd) the values L(t) affine sets and W(t) = P(t) ∩ L(t) extremal faces of P(t). Let pk be integrable selections of P, the projection of pk,(t) on L(t) and pk(t) on W(t). We prove that if converges weakly to zero then pk − k converges to zero in measure. We give also some extensions of this theorem. As applications to differential inclusions we investigate convergence of derivatives of convergent sequences of solutions and we describe solutions which are in some sense isolated. Finally we discuss what can be said about control functions u when the corresponding trajectories of ẋ = f(t, x, u) are convergent to some trajectory.

Some remarks concerning quasiconvexity and strong convergence

1987 ◽  
Vol 106 (1-2) ◽  
pp. 53-61 ◽  
Author(s):  
L. C. Evans ◽  
R. F. Gariepy
Keyword(s):  
Sobolev Space ◽  
Weak Convergence ◽  
Strong Convergence ◽  
Calculus Of Variations ◽  
Heuristic Principle ◽  
Sequence Of Functions

SynopsisWe show that the weak convergence of a sequence of functions in a Sobolev space plus the convergence of appropriately quasiconvex “energies” imply, in fact, strong convergence. This assertion makes rigorous, for example, the heuristic principle that “quasiconvexity damps out oscillations in the gradients” of minimising sequences in the calculus of variations.

Weak and strong convergence theorems and a nonlinear ergodic theorem for N-generalized hybrid mappings

2017 ◽  
Vol 10 (01) ◽  
pp. 1750001
Author(s):  
Sattar Alizadeh ◽  
Fridoun Moradlou
Keyword(s):  
Weak Convergence ◽  
Strong Convergence ◽  
Ergodic Theorem ◽  
Hilbert Spaces ◽  
Convergence Theorem ◽  
Strong Convergence Theorem ◽  
Weak And Strong Convergence ◽  
Weak Convergence Theorem ◽  
Nonlinear Ergodic Theorem ◽  
Nonlinear Mappings

In this paper, assuming an appropriate condition, we prove that [Formula: see text]-generalized hybrid mappings are demiclosed in Hilbert spaces. Using this fact, we prove a weak convergence theorem of Ishikawa type for these nonlinear mappings. Also, a strong convergence theorem of Halpern–Ishikawa type and a nonlinear ergodic theorem for [Formula: see text]-generalized hybrid mappings have been proven in Hilbert spaces.

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