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Jung, Convergence of paths for pseudo-contractive mappings in Banach spaces, Proc. Pras, Banach spaces with the uniform Opial property, Nonlinear Anal. Reich, Asymptotic behavior of contractions in Banach spaces, J. Reich, Extension problems for accretive sets in Banach spaces, J. Reich, Product formulas, nonlinear semigroups, and accretive operators, J. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J.

Reich, On the asymptotic behavior of nonlinear semigroups and the range of accretive operators, J. Reich and H. Xu, Nonlinear ergodic theory for semigroups of Lipschitizian mappings, Comm. Sahu, Z.

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Liu and S. Kang, Iterative approaches to common fixed points of asymptotically nonexpansive mappings. Rocky Mountain J. Sahu and D. O'Regan, Convergence theorems for semigroup-type families of non-self mappings. Palermo 2 57 , Schu, Approximation of fixed points of asymptotically nonexpansive mappings, Proc. Shahzad and A.

Udomeen, Fixed point solutions of variational inequalities for asymptotically nonexpansive mappings in Banach spaces, Nonlinear Anal. Shioji and W. Takahashi, Strong convergence of averaged approximants for asymptotically nonexpansive mappings in Banach spaces, J. Theory 97 , Wong, D. Sahu, and J.

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Yao, Solving variational inequalities involving nonexpansive type mappings. If you previously purchased this article, Log in to Readcube. Log out of Readcube.

Click on an option below to access. Log out of ReadCube. We present two types of iteration methods one is implicit, and the other is explicit. We establish strong convergence theorems for both methods. Some applications are given regarding convex optimization problems and split feasibility problems. These results improve some known results existing in the literatures.

Volume 37 , Issue 8. The full text of this article hosted at iucr. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account. If the address matches an existing account you will receive an email with instructions to retrieve your username. Tools Request permission Export citation Add to favorites Track citation. Share Give access Share full text access. It is worth noting that they are different from the Gromov hyperbolic space [ 25 ] or from other notions of hyperbolic space that can be found in the literature see, e.

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An important example of a hyperbolic space is a CAT 0 space. It is nonlinear in nature and its brief introduction is as follows. A metric space X , d is a length space if any two points of X are joined by a rectifiable path that is, a path of finite length and the distance between any two points of X is taken to be the infimum of the lengths of all rectifiable paths joining them.

In this case, d is known as a length metric otherwise an inner metric or intrinsic metric.

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Let X be a CAT 0 space. Then W satisfies the four properties of a hyperbolic space. Therefore, our hyperbolic space represents a unified approach for both linear and nonlinear structures simultaneously. To elaborate that there are hyperbolic spaces which are not imbedded in any Banach space, we give the following example. A metric space X , d is called a convex metric space introduced by Takahashi [ 29 ] if it satisfies only i. The hyperbolic space introduced by Kohlenbach [ 24 ] is slightly restrictive than the space of hyperbolic type [ 26 ] but general than hyperbolic space of [ 28 ].

Let C be a nonempty subset of hyperbolic space X. It is well known that uniformly convex Banach spaces and even CAT 0 spaces enjoy the property that bounded sequences have unique asymptotic centers with respect to closed convex subsets. The following lemma is due to Leustean [ 32 ] and ensures that this property also holds in a complete uniformly convex hyperbolic space.

Let X , d , W be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity. In this section, we approximate fixed point for nearly asymptotically nonexpansive mappings in a hyperbolic space. Lemma 3. Theorem 3.

Fixed Point Theory for Lipschitzian-type Mappings with Applications

Then by 3. By uniform continuity of T. Now we claim that v is a fixed point of T.


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  • By uniform continuity of T , we have. We now discuss the strong convergence for the S -iteration process defined by 3. Assume that F T is a closed set. From 3. In the next result, the closedness assumption on F T is not required. Proof By 3.


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    • By the uniformly continuous of T , we have. It follows that. Suppose that T satisfies the condition A. It also extends the corresponding results of Dhomponsga and Panyanak [ 22 ] from the class of nonexpansive mappings to a more general class of non-Lipschitzian mappings in the same space setting.