Bregman common skew-attractive point theorems for semigroups of nonlinear mappings in Banach spaces

We extend the application of nearly contraction mapping principle introduced by Sahu (2005) for existence of fixed points of demicontinuous mappings to certain hemicontinuous nearly Lipschitzian nonlinear mappings in Banach spaces. We have applied certain results due to Sahu (2005) to obtain conditions for existence—and to introduce an asymptotic iterative process for construction—of fixed points of these hemicontractions with respect to a new auxiliary operator.

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Existence and Approximation of Attractive Points of the Widely More Generalized Hybrid Mappings in Hilbert Spaces

Abstract and Applied Analysis ◽

10.1155/2013/904164 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 2

Author(s):

Sy-Ming Guu ◽

Wataru Takahashi

Keyword(s):

Hilbert Space ◽

Hilbert Spaces ◽

Convergence Theorem ◽

Nonexpansive Mappings ◽

Strong Convergence Theorem ◽

Mean Convergence ◽

Nonspreading Mappings ◽

Weak Convergence Theorem ◽

Attractive Point ◽

Nonlinear Mappings

We study the widely more generalized hybrid mappings which have been proposed to unify several well-known nonlinear mappings including the nonexpansive mappings, nonspreading mappings, hybrid mappings, and generalized hybrid mappings. Without the convexity assumption, we will establish the existence theorem and mean convergence theorem for attractive point of the widely more generalized hybrid mappings in a Hilbert space. Moreover, we prove a weak convergence theorem of Mann’s type and a strong convergence theorem of Shimizu and Takahashi’s type for such a wide class of nonlinear mappings in a Hilbert space. Our results can be viewed as a generalization of Kocourek, Takahashi and Yao, and Hojo and Takahashi where they studied the generalized hybrid mappings.

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Convergence of approximants to fixed points of nonexpansive nonlinear mappings in banach spaces

Archive for Rational Mechanics and Analysis ◽

10.1007/bf00251595 ◽

1967 ◽

Vol 24 (1) ◽

pp. 82-90 ◽

Cited By ~ 267

Author(s):

Felix E. Browder

Keyword(s):

Fixed Points ◽

Banach Spaces ◽

Nonlinear Mappings

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On local convexity of nonlinear mappings between Banach spaces

Open Mathematics ◽

10.2478/s11533-012-0101-z ◽

2012 ◽

Vol 10 (6) ◽

Cited By ~ 3

Author(s):

Iryna Banakh ◽

Taras Banakh ◽

Anatolij Plichko ◽

Anatoliy Prykarpatsky

Keyword(s):

Banach Space ◽

Lower Bound ◽

Banach Spaces ◽

Lipschitz Constant ◽

Second Order ◽

Local Convexity ◽

Convexity Number ◽

Locally Convex ◽

Nonlinear Mappings

AbstractWe find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ɛ < c the image f(B ɛ(x)) of each ɛ-ball B ɛ(x) ⊂ U is convex. We give a lower bound on c via the second order Lipschitz constant Lip2(f), the Lipschitz-open constant Lipo(f) of f, and the 2-convexity number conv2(X) of the Banach space X.

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

Carpathian Journal of Mathematics ◽

10.37193/cjm.2020.01.03 ◽

2020 ◽

Vol 36 (1) ◽

pp. 27-34 ◽

Cited By ~ 2

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.

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