scholarly journals Convergence Theorems for Fixed Points of Multivalued Mappings in Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
N. Djitte ◽  
M. Sene
Keyword(s):  
Hilbert Space ◽  
Fixed Points ◽  
Hilbert Spaces ◽  
Convex Subset ◽  
Real Hilbert Space ◽  
Nonempty Closed Convex Subset ◽  
Multivalued Mappings ◽  
Convergence Theorems ◽  
Lipschitz Pseudocontractive Mapping ◽  
Iteration Parameters

Let H be a real Hilbert space and K a nonempty closed convex subset of H. Suppose T:K→CB(K) is a multivalued Lipschitz pseudocontractive mapping such that F(T)≠∅. An Ishikawa-type iterative algorithm is constructed and it is shown that, for the corresponding sequence {xn}, under appropriate conditions on the iteration parameters, lim infn→∞⁡ d (xn,Txn)=0 holds. Finally, convergence theorems are proved under approximate additional conditions. Our theorems are significant improvement on important recent results of Panyanak (2007) and Sastry and Babu (2005).

scholarly journals Convergence Theorems for Fixed Points of Multivalued Strictly Pseudocontractive Mappings in Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
C. E. Chidume ◽  
C. O. Chidume ◽  
N. Djitté ◽  
M. S. Minjibir
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Hilbert Spaces ◽  
Convex Subset ◽  
Real Hilbert Space ◽  
Convergence Theorems ◽  
Strictly Pseudocontractive Mapping ◽  
Pseudocontractive Mappings ◽  
Iteration Sequence ◽  
Strictly Pseudocontractive Mappings

LetKbe a nonempty, closed, and convex subset of a real Hilbert spaceH. Suppose thatT:K→2Kis a multivalued strictly pseudocontractive mapping such thatF(T)≠∅. A Krasnoselskii-type iteration sequence{xn}is constructed and shown to be an approximate fixed point sequence ofT; that is,limn→∞d(xn,Txn)=0holds. Convergence theorems are also proved under appropriate additional conditions.

scholarly journals Strong Convergence Theorems for a Pair of Strictly Pseudononspreading Mappings

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Bin-Chao Deng ◽  
Tong Chen
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Fixed Points ◽  
Real Hilbert Space ◽  
Convergence Theorems ◽  
Mild Conditions ◽  
Strongly Monotone

LetHbe a real Hilbert space. LetT1,T2:H→Hbek1-,k2-strictly pseudononspreading mappings; letαnandβnbe two real sequences in (0,1). For givenx0∈H, the sequencexnis generated iteratively byxn+1=βnxn+1-βnTw1αnγfxn+I-μαnBTw2xn,∀n∈N, whereTwi=1−wiI+wiTiwithi=1,2andB:H→His strongly monotone and Lipschitzian. Under some mild conditions on parametersαnandβn, we prove that the sequencexnconverges strongly to the setFixT1∩FixT2of fixed points of a pair of strictly pseudononspreading mappingsT1andT2.

scholarly journals Approximating fixed points of enriched nonexpansive mappings in Banach spaces by using a retraction-displacement condition

2020 ◽  
Vol 36 (1) ◽  
pp. 27-34 ◽  
Author(s):  
VASILE BERINDE
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Convergence Theorems ◽  
Displacement Condition ◽  
Nonlinear Mappings

In this paper, we prove convergence theorems for a fixed point iterative algorithm of Krasnoselskij-Mann typeassociated to the class of enriched nonexpansive mappings in Banach spaces. The results are direct generaliza-tions of the corresponding ones in [Berinde, V.,Approximating fixed points of enriched nonexpansive mappings byKrasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35 (2019), No. 3, 293–304.], from the setting of Hilbertspaces to Banach spaces, and also of some results in [Senter, H. F. and Dotson, Jr., W. G.,Approximating fixed pointsof nonexpansive mappings, Proc. Amer. Math. Soc.,44(1974), No. 2, 375–380.], [Browder, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197–228.], byconsidering enriched nonexpansive mappings instead of nonexpansive mappings. Many other related resultsin literature can be obtained as particular instances of our results.

scholarly journals Implicit iteration process of nonexpansive non-self-mappings

2005 ◽  
Vol 2005 (19) ◽  
pp. 3103-3110
Author(s):  
Somyot Plubtieng ◽  
Rattanaporn Punpaeng
Keyword(s):  
Hilbert Space ◽  
Convex Subset ◽  
Real Hilbert Space ◽  
Iteration Process ◽  
Nonempty Closed Convex Subset ◽  
Implicit Iteration Process ◽  
Nearest Point ◽  
Nearest Point Projection ◽  
Point Projection ◽  
Implicit Iteration

SupposeCis a nonempty closed convex subset of real Hilbert spaceH. LetT:C→Hbe a nonexpansive non-self-mapping andPis the nearest point projection ofHontoC. In this paper, we study the convergence of the sequences{xn},{yn},{zn}satisfyingxn=(1−αn)u+αnT[(1−βn)xn+βnTxn],yn=(1−αn)u+αnPT[(1−βn)yn+βnPTyn], andzn=P[(1−αn)u+αnTP[(1−βn)zn+βnTzn]], where{αn}⊆(0,1),0≤βn≤β<1andαn→1asn→∞. Our results extend and improve the recent ones announced by Xu and Yin, and many others.

scholarly journals Strong Convergence Theorems for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-13
Author(s):  
Jian-Wen Peng ◽  
Yan Wang
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Fixed Points ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Common Element ◽  
Convergence Theorems

We introduce an Ishikawa iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert space. Then, we prove some strong convergence theorems which extend and generalize S. Takahashi and W. Takahashi's results (2007).

scholarly journals Strong Convergence Theorems for Equilibrium Problems andk-Strict Pseudocontractions in Hilbert Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Dao-Jun Wen
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Real Hilbert Space ◽  
Finite Family ◽  
Common Element ◽  
Convergence Theorems

We introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed point of a finite family ofk-strictly pseudo-contractive nonself-mappings. Strong convergence theorems are established in a real Hilbert space under some suitable conditions. Our theorems presented in this paper improve and extend the corresponding results announced by many others.

scholarly journals Weak and strong convergence theorems for the Krasnoselskij iterative algorithm in the class of enriched strictly pseudocontractive operators

2018 ◽  
Vol 56 (2) ◽  
pp. 13-27 ◽  
Author(s):  
Vasile Berinde
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Fixed Points ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Weak And Strong Convergence ◽  
Convergence Theorems ◽  
Pseudocontractive Mappings ◽  
Strictly Pseudocontractive Mappings ◽  
Nonlinear Mappings

AbstractIn this paper, we introduce and study the class of enriched strictly pseudocontractive mappings in Hilbert spaces and extend some convergence theorems, i.e., Theorem 12 in [Brow-der, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197–228] and Theorem 3.1 in [Marino, G., Xu, H.-K., Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007), no. 1, 336–346], from the class of strictly pseudocontractive mappings to that of enriched strictly pseudocontractive mappings and thus include many other important related results from literature as particular cases.

scholarly journals An approximation method for monotone Lipschitzian operators in Hilbert spaces

1986 ◽  
Vol 41 (1) ◽  
pp. 59-63 ◽  
Author(s):  
C. E. Chidume
Keyword(s):  
Hilbert Space ◽  
Error Estimates ◽  
Hilbert Spaces ◽  
Approximation Method ◽  
Convex Subset ◽  
Nonempty Closed Convex Subset ◽  
Complex Hilbert Space ◽  
Real Sequence ◽  
Lipschitzian Mapping

AbstractSuppose H is a complex Hilbert space and K is a nonempty closed convex subset of H. Suppose T: K → H is a monotomc Lipschitzian mapping with constant L ≧ 1 such that, for x in K and h in H, the equation x + Tx Tx = h has a solution q in K. Given x0 in K, let {Cn}∞n=0 be a real sequence satisfying: (i) C0 = 1, (ii) 0 ≦ Cn < L-2 for all n ≧ 1, (iii) ΣnCn(1 − Cn) diverges. Then the sequence {Pn}∞n=0 in H defined by pn = (1 − Cn)xn + CnSxn, n ≧ 0, where {xn}∞n=0 in K is such that, for each n ≧ 1, ∥ xn – Pn−1 ∥ = infx ∈ k ∥ Pn−1 − x ∥, converges strongly to a solution q of x + Tx = h. Explicit error estimates are given. A similar result is also proved for the case when the operator T is locally Lipschitzian and monotone.

scholarly journals Dirac structures on Hilbert spaces

1999 ◽  
Vol 22 (1) ◽  
pp. 97-108 ◽  
Author(s):  
A. Parsian ◽  
A. Shafei Deh Abad
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Hilbert Spaces ◽  
Smooth Manifold ◽  
Real Hilbert Space ◽  
Dirac Structure ◽  
Dirac Structures ◽  
Basic Properties ◽  
Hilbert Subspace ◽  
Densely Defined

For a real Hilbert space(H,〈,〉), a subspaceL⊂H⊕His said to be a Dirac structure onHif it is maximally isotropic with respect to the pairing〈(x,y),(x′,y′)〉+=(1/2)(〈x,y′〉+〈x′,y〉). By investigating some basic properties of these structures, it is shown that Dirac structures onHare in one-to-one correspondence with isometries onH, and, any two Dirac structures are isometric. It is, also, proved that any Dirac structure on a smooth manifold in the sense of [1] yields a Dirac structure on some Hilbert space. The graph of any densely defined skew symmetric linear operator on a Hilbert space is, also, shown to be a Dirac structure. For a Dirac structureLonH, everyz∈His uniquely decomposed asz=p1(l)+p2(l)for somel∈L, wherep1andp2are projections. Whenp1(L)is closed, for any Hilbert subspaceW⊂H, an induced Dirac structure onWis introduced. The latter concept has also been generalized.

Counterexamples Concerning Support Theorems for Convex Sets in Hilbert Space

1988 ◽  
Vol 31 (1) ◽  
pp. 121-128 ◽  
Author(s):  
R. R. Phelps
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Convex Function ◽  
Convex Subset ◽  
Convex Sets ◽  
Lower Semicontinuous ◽  
Nonempty Closed Convex Subset ◽  
Common Point ◽  
Support Points ◽  
Support Theorems

AbstractThe Bishop-Phelps theorem guarantees the existence of support points and support functionals for a nonempty closed convex subset of a Banach space; equivalently, it guarantees the existence of subdifferentials and points of subdifferentiability of a proper lower semicontinuous convex function on a Banach space. In this note we show that most of these results cannot be extended to pairs of convex sets or functions, even in Hilbert space. For instance, two proper lower semicontinuous convex functions need not have a common point of subdifferentiability nor need they have a subdifferential in common. Negative answers are also obtained to certain questions concerning density of support points for the closed sum of two convex subsets of Hilbert space.

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