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Metrical Common Fixed Points and Commuting Type Mappings
Published in Dhananjay Gopal, Poom Kumam, Mujahid Abbas, Background and Recent Developments of Metric Fixed Point Theory, 2017
Dhananjay Gopal, Ravindra K Bisht
Proving a common fixed point theorem for mappings satisfying the above contractive conditions involves the following steps. The first is the most crucial part of common fixed point theorems consisting of constructive procedures yielding a Cauchy sequence which converges to a point in X (where X is complete). The second step is to show the coincidence point by assuming suitable weaker forms of commutating and/or continuity type conditions; and the third step automatically gives rise to the coincidence point, which is simply a unique common fixed point due to contractive condition. Observing carefully the second step, one can find the existence of the coincidence point by suitable conditions including choice of weaker forms of commuting conditions. Many authors have considered generalizations of commuting mappings; see Table 2.1 in Section 2.3. Now, it has been shown that weak compatibility is the minimal noncommuting condition for the existence of common fixed points of contractive type mapping pairs.
Symmetric Spaces and Fixed Point Theory
Published in Dhananjay Gopal, Praveen Agarwal, Poom Kumam, Metric Structures and Fixed Point Theory, 2021
Pradip Ramesh Patle, Deepesh Kumar Patel
We now discuss coincidence and common fixed points in symmetric spaces. For us it is impossible to discuss all the results here, so we just give one of the basic result proved by Hicks and Rhodes [25]. Recall that, x ∈ X is said to be a coincidence point of mappings f and g on a symmetric space (X,s) if fx = gx holds and y ∈ X is called a common fixed point of f and g if y = fy = gy holds.
On Hölder calmness and Hölder well-posedness for optimal control problems
Published in Optimization, 2022
Lam Quoc Anh, Vo Thanh Tai, Tran Ngoc Tam
(EP) find such that Note further that the problem (EP) is not only a general form of (OCP) but also of many important problems in optimization, for instance, variational inequalities, fixed point and coincidence point problems, Nash equilibrium problems, traffic network problems, see [32]. The study of Hölder/Lipschitz conditions of solution maps to (EPs) has been an interesting and important topic in the stability for optimization theory. However, this topic is hard to discuss without taking account of conditions related to strong monotonicity [23,33,34] or strong convexity [22,35] of objective functions. In this study, we combine and adjust many techniques used in the literature, for instance [22,23,33–36], to investigate the stability in the sense of Hölder calmness for parametric optimal control problems with linear state equation and control constraints, and fortunately we have gained a little contribution on reducing the strongly convex condition of the objective function. In what follows, we will show that our results are valid for many situations, while the other ones are not (see Example 4.5 and Subsection 6.2).
Maximal element with applications to Nash equilibrium problems in Hadamard manifolds
Published in Optimization, 2019
Hai-Shu Lu, Rong Li, Zhi-Hua Wang
Maximal element theorem for set-valued mappings plays a crucial role in the study of nonlinear analysis and optimization theory. In 1983, Yannelis and Prabhakar [1] established a maximal element theorem, and then they used this theorem to prove the existence of equilibrium for an abstract economy in topological vector spaces. Since then, many authors have investigated generalizations of maximal element theorems in topological vector spaces and their applications to fixed point, coincidence point, variational inequalities, minimax inequalities, and equilibrium problems. For more details, the reader may consult Toussaint [2], Tulcea [3], Kim [4], Deguire et al. [5], Yuan [6], Balaj [7], Balaj and Lin [8], Duggan [9], Kassay et al. [10], Patriche [11], and the authors referenced by their work.
On the stability of approximate solutions to set-valued equilibrium problems
Published in Optimization, 2020
Lam Quoc Anh, Pham Thanh Duoc, Tran Ngoc Tam
It is well known that equilibrium problem [1] is a general mathematical model which includes as special cases many problems in optimization such as optimization problem, variational inequality, complementarity problem, Nash equilibrium problem, minimax problem, fixed-point and coincidence-point problems, traffic network problem, etc. This implies possibility of a wide application of results in equilibrium problem theory to several important fields, including physics (especially, mechanics), economics, engineering, transportation, sociology, chemistry, biology, etc. (see [2]).