... conditions of strongconvergence theorem are different from the
strong convergence theorems for the implicit iterative processes in the literatures.
Finally, we discuss a strongconvergence theorem ... Takahashi, W: Weak and strongconvergence theorems for relatively nonexpansive mappings in Banach
spaces. Fixed Point Theory Appl 37–47 (2004). 2004
21. Matsushita, S, Takahashi, W: A strongconvergence ... we first consider weakconvergence theorems of implicit iterative
processes for two nonexpansive mappings and a mapping which satisfies condition (C).
Next, we consider strongconvergence theorem...
... conditions of strongconvergence theorem are different from the
strong convergence theorems for the implicit iterative processes in the literatures.
Finally, we discuss a strongconvergence theorem ... Takahashi, W: Weak and strongconvergence theorems for relatively nonexpansive mappings in Banach
spaces. Fixed Point Theory Appl 37–47 (2004). 2004
21. Matsushita, S, Takahashi, W: A strongconvergence ... we first consider weakconvergence theorems of implicit iterative
processes for two nonexpansive mappings and a mapping which satisfies condition (C).
Next, we consider strongconvergence theorem...
... amendment
process.
I call the new model of judicial review weak- form judicial review, in con-
trast with the strong form of judicial review in the United States. Strong- form
review itself has numerous variants. ... provide a measure of the degree to which the HRA
creates a weak or strong form of judicial review. The HRA system would be
indistinguishable from strong- form review if statutes that courts declared ... States as weak- form. The contemporary Supreme Court certainly re-
gards judicial review as having the strong form, and much of the public ap-
pears to agree.
40
Exactly when we got strong- form...
... W: Weak and strongconvergence theorems for families of nonexpansive
mappings and their applications. Ann. Univ. Mariae Curie-Sklodowska 51, 277–292
(1997)
[21] Takahashi, W, Shimoji, K: Convergence ... sufficient condition for the convergence of
Ishikawa iteration on an arbitrary interval. J. Math. Anal. Appl. 323, 1383–1386
(2006)
[8] Atsushiba, S, Takahashi, W: Strongconvergence theorems for ... sufficient condition
for the strongconvergence of the proposed iterative methods for continuous map-
pings on an arbitrary interval is given. We also compare the rate of convergence of
those iterations....
... ε-approximate fixed points of
T, i.e., F
ε
(T; D)={x Î D :||x - T
x
|| ≤ ε }. Weakconvergence is denoted by ⇀ and
strong convergence is denoted by ®.
3 Main results
Theorem 3.1 Let S be a semigroup, ... solutions of variational inequality for b-inverse strongly
monotone mapping in a real Hilbert space. Under suitable conditions, we prove the
strong convergence theorem for approximating a common ... implica-
tion holds:
x
n
→ u ∈ C, Tx
n
→ vim
p
l
y
Tu = v
,
where ® (resp. ⇀) denotes strong (resp. weak) convergence.
Lemma 2.2 [17]LetCbeanonemptyclosedconvexsubsetofaHilbertspaceHand
suppose...
... subset of H.Wewritex
n
⇀ x to indicate that the sequence {x
n
}con-
verges weakly to x,andx
n
® x implies that {x
n
} converges strongly to x. For every
point x Î H, there exists a unique nearest poi ... A strong convergence
theorem of the proposed iterative scheme to a common fixed point of a countable
family of nonexpansive mappings and a solution of variational inequality of an
inverse strongly ... C, f is a contraction, A is a strongly positive lin-
ear bounded operator, T
n
is a countable family of nonexpansive mappings of C into
itself, B is a b-inverse strongly monotone mapping, {a
n
}...
... Strongconvergence theorem for a generalized equilibrium problem and a k-strict
pseudocontraction in Hilbert spaces. Appl Math Mech , English 2009, 30(6):685-694.
4. Peng JW, Yao JC: Strongconvergence ... be a real Banach space, and {x
n
}beasequenceinE.Wedenotebyx
n
® x and
x
n
⇀ x the strongconvergence and weakconvergence of {x
n
}, respectively. The normal-
ized duality mapping J : E ® 2
E*
is ... Fixed Point Theory Appl 2008, 2008:11, (Article ID 528476).
9. Takahashi W, Zembayashi K: Strong and weakconvergence theorems for equilibrium problems and relatively
nonexpansive mappings in Banach...
... doi:10.1016/j.na.2008.02.042
4. Iiduka, H, Takahashi, W: Weakconvergence theorem by Ces’aro means for nonexpansive mappings and inverse-strongly
monotone mappings. J Nonlinear Convex Anal. 7, ... Math Anal Appl. 20,
197–228 (1967). doi:10.1016/0022-247X(67)90085-6
9. Marino, G, Xu, HK: Weak and strongconvergence theorems for strict pseudo-contractions in Hilbert spaces. J Math Anal
Appl. ... strict pseudo-
contractions. Strongconvergence theorem of the purposed iteration is established in
the framework of Hilbert spaces.
Keywords: nonexpansive mappings, strongly positive operator,...
... paper, the set of fixed points of T is denoted by F(T). We use symbols ® and
⇀ to denote strong and weak convergence, respectively.
For each point x Î H, there exists a unique nearest point in ... Nonlinear Anal.
69, 2709–2719 (2008). doi:10.1016/j.na.2007.08.045
9. Tada, A, Takahashi, W: Weak and strongconvergence theorems for a nonexpansive mapping and an equilibrium
problem. J Optim Theory ... expansive.
(2) v-strongly monotone if there exists a constant v>0 such that
Tx − T
y
, x −
y
≥v||x −
y
||
2
, ∀x,
y
∈ C
.
Clearly, any v-strongly monotone mapping is v-expansive.
(3) u-inverse strongly...
... obtain strong conver-
gence theorems that extend and improve the corresponding results [3,4,13,14].
We will adopt the following notations:
1. ⇀ for the weakconvergence and ® for the strong convergence.
2.
ω
w
(x
n
)={x ... Nonlinear Anal. 70, 1902–1911 (2009). doi:10.1016/j.na.2008.02.090
14. Tada, A, Takahashi, W: Weak and strongconvergence theorems for a nonexpansive mappingand a equilibrium
problem. J Optim Theory ... HK: Weak and srong convergence theorems for strict pseudo-contractions in Hlibert spaces. J Math Anal
Appl. 329, 336–346 (2007). doi:10.1016/j.jmaa.2006.06.055
11. Takahashi, S, Takahashi, W: Strong...
... Takahashi, Strongconvergence of a proximal-type algorithm in a Banach
space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002.
24 F. Kohsaka and W. Takahashi, Strongconvergence ... 14 Fixed Point Theory and Applications
19 W. Takahashi and Kei Zembayashi, Strong and weakconvergence theorems for equilibrium
problems and relatively nonexpansive mappings i n Banach ... pp. 15–50, Marcel Dekker, New York, NY, USA,
1996.
2 S Y. Matsushita and W. Takahashi, Weak and strongconvergence theorems for relatively
nonexpansive mappings in Banach spaces,” Fixed Point...
... 4.
We will use the following notations:
1 for weakconvergence and → for strong convergence;
2 ω
w
x
n
{x : ∃x
n
j
x} denotes the weak ω-limit of {x
n
}.
We need some facts and tools ... Universities Grant ZXH2009D021.
References
1 H. H. Bauschke and P. L. Combettes, “A weak- to -strong convergence principle for Fej
´
er-monotone
methods in Hilbert spaces,” Mathematics of Operations ... Advanced
Mathematics, Cambridge University Press, Cambridge, UK, 1990.
6 C. Martinez-Yanes and H K. Xu, Strongconvergence of the CQ method for fixed point iteration
processes,” Nonlinear Analysis. Theory,...