Fixed point theorems in "Equation missing"  spaces with applications

Lu, Haishu; Di, Lan; Hu, Qingwen; Yuan, George Xianzhi

doi:10.1186/1029-242x-2014-320

Cited by 6 publications

(3 citation statements)

References 27 publications

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“…Results depending on Hilbert space geometry have been especially amenable to this approach. A large number of recent papers have dealt with various approximation techniques, and also the recent paper [76] by Lu et al contains a nice review of various set-valued results in CAT(0) spaces, especially those related to the so-called KKM theory. A complete review of these results would extend well beyond the scope of these remarks.…”

Section: Cat(0) Spacesmentioning

confidence: 99%

Hyperbolic spaces and directional contractions

Kirk

Shahzad

2019

Bull. Math. Sci.

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The axiomatic approach to metric convexity goes back to the pioneering work of Karl Menger in 1928. This is an overview of this concept and the role it plays in metric fixed point theory especially in conjunction with spaces possessing a “hyperbolic” type structures. These include the CAT(0) spaces, hyperconvex metric spaces, and [Formula: see text]-trees. Much of the discussion involves the existence of “approximate” fixed point sequences for mappings satisfying weak contractive conditions. Applications of a well-known fixed point theorem due to Caristi are also included. These involve fixed and approximate fixed points for mappings satisfying local “directional” contractive and non-expansive conditions. Convexity plays a role in this part of the discussion as well. While the paper is semi-expository in nature, some detailed proofs appear here for the first time. Also the concept of a weak [Formula: see text]-directional contraction introduced in Sec. 8 appears to be new. Several suggestions for further research are also discussed.

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Section: Cat(0) Spacesmentioning

confidence: 99%

Hyperbolic spaces and directional contractions

Kirk

Shahzad

2019

Bull. Math. Sci.

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“…In fact, CAT(0) spaces constitute a suitable framework to obtain fixed points of nonexpansive mappings and its various generalizations [18][19][20]. Fixed point results of multivalued mappings in CAT(0) spaces have various applications in differential equations, optimization, control theory, graph theory, computer science, robotics [21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

Approximation of Fixed Points of Multivalued Generalized (α,β)-Nonexpansive Mappings in an Ordered CAT(0) Space

et al. 2021

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The purpose of this article is to initiate the notion of monotone multivalued generalized (α,β)-nonexpansive mappings and explore the iterative approximation of the fixed points for the mapping in an ordered CAT(0) space. In particular, we employ the S-iteration algorithm in CAT(0) space to prove some convergence results. Moreover, some examples and useful results related to the proposed mapping are provided. Numerical experiments are also provided to illustrate and compare the convergence of the iteration scheme. Finally, an application of the iterative scheme has been presented in finding the solutions of integral differential equation.

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“…Dhompongsa et al [] proved that a nonexpansive mapping from a nonempty bounded closed convex subset of a CAT() space to the family of nonempty compact subsets of the CAT() space has a fixed point under suitable conditions. Shahzad [] obtained fixed point theorems ©2014 Lu et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.…”

Section: Introductionmentioning

confidence: 99%

Fixed point theorems in "Equation missing" spaces with applications

et al. 2014

J Inequal Appl

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In this paper, noncompact CAT(0) versions of the Fan-Browder fixed point theorem are established. As applications, we obtain new minimax inequalities, a saddle point theorem, a fixed point theorem for single-valued mappings, best approximation theorems, and existence theorems of ϕ-equilibrium points for multiobjective noncooperative games in the setting of noncompact CAT(0) spaces. These results generalize many well-known theorems in the literature. MSC: 91A10; 47H04; 47H10; 54H25

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Fixed point theorems in "Equation missing" spaces with applications

Cited by 6 publications

References 27 publications

Hyperbolic spaces and directional contractions

Hyperbolic spaces and directional contractions

Approximation of Fixed Points of Multivalued Generalized (α,β)-Nonexpansive Mappings in an Ordered CAT(0) Space

Fixed point theorems in "Equation missing" spaces with applications

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