New Contribution to the Advancement of Fixed Point Theory, Equilibrium Problems, and Optimization Problems

Abstract The fixed point theory is essential to various theoretical and applied fields, such as variational and linear inequalities, the approximation theory, nonlinear analysis, integral and differential equations and inclusions, the dynamic systems theory, mathematics of fractals, mathematical economics (game theory, equilibrium problems, and optimisation problems) and mathematical modelling. This paper presents a few benchmarks regarding the applications of the fixed point theory. This paper also debates if the results of the fixed point theory can be applied to the mathematical modelling of quality.

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Equilibrium problems and fixed point theory

Fixed Point Theory and Applications ◽

10.1186/1687-1812-2012-25 ◽

2012 ◽

Vol 2012 (1) ◽

Cited By ~ 1

Author(s):

Qamrul Hasan Ansari ◽

Suliman Al-Homidan ◽

Jen-Chih Yao

Keyword(s):

Fixed Point ◽

Equilibrium Problems ◽

Fixed Point Theory ◽

Point Theory

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Recent Developments on Fixed Point Theory in Function Spaces and Applications to Control and Optimization Problems

Journal of Function Spaces ◽

10.1155/2015/512564 ◽

2015 ◽

Vol 2015 ◽

pp. 1-2

Author(s):

Hemant Kumar Nashine ◽

Mujahid Abbas ◽

Marlène Frigon ◽

Calogero Vetro

Keyword(s):

Fixed Point ◽

Optimization Problems ◽

Fixed Point Theory ◽

Point Theory ◽

Function Spaces ◽

Recent Developments

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Probabilistic p-cyclic contractions using different types of t-norms

Random Operators and Stochastic Equations ◽

10.1515/rose-2018-0004 ◽

2018 ◽

Vol 26 (1) ◽

pp. 39-52 ◽

Cited By ~ 1

Author(s):

Binayak S. Choudhury ◽

Samir Kumar Bhandari ◽

Parbati Saha

Keyword(s):

Fixed Point ◽

Metric Spaces ◽

Optimization Problems ◽

Fixed Point Theory ◽

Point Theory ◽

Third Order ◽

Menger Space ◽

Different Types ◽

Cyclic Contractions ◽

Probabilistic Metric

Abstract Cyclic mappings have appeared prominently in fixed point theory during the last decade. They have also their applications in global optimization problems. Note that p-cyclic mappings are extensions of cyclic mappings over p number of sets. In this paper we introduce two p-cyclic contractions in probabilistic spaces. We have two corresponding fixed point theorems using third-order Hadzic-type t-norm and minimum t-norm, respectively. One of the probabilistic contractions is of Ciric type while the other is a general contraction. One illustrative example is given. The space we consider here is a 2-Menger space which is an extension of a probabilistic metric space in the same vein as the 2-metric spaces are extensions of metric spaces.

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MONOTONE AND PSEUDO-MONOTONE EQUILIBRIUM PROBLEMS IN HADAMARD SPACES

Journal of the Australian Mathematical Society ◽

10.1017/s1446788719000041 ◽

2019 ◽

pp. 1-23 ◽

Cited By ~ 5

Author(s):

HADI KHATIBZADEH ◽

VAHID MOHEBBI

Keyword(s):

Fixed Point ◽

Variational Inequalities ◽

Equilibrium Point ◽

Proximal Point Algorithm ◽

Equilibrium Problems ◽

Fixed Point Theory ◽

Point Theory ◽

Proximal Point ◽

Monotone Bifunction ◽

Monotone Bifunctions

As a continuation of previous work of the first author with Ranjbar [‘A variational inequality in complete CAT(0) spaces’, J. Fixed Point Theory Appl. 17 (2015), 557–574] on a special form of variational inequalities in Hadamard spaces, in this paper we study equilibrium problems in Hadamard spaces, which extend variational inequalities and many other problems in nonlinear analysis. In this paper, first we study the existence of solutions of equilibrium problems associated with pseudo-monotone bifunctions with suitable conditions on the bifunctions in Hadamard spaces. Then, to approximate an equilibrium point, we consider the proximal point algorithm for pseudo-monotone bifunctions. We prove existence of the sequence generated by the algorithm in several cases in Hadamard spaces. Next, we introduce the resolvent of a bifunction in Hadamard spaces. We prove convergence of the resolvent to an equilibrium point. We also prove $\triangle$ -convergence of the sequence generated by the proximal point algorithm to an equilibrium point of the pseudo-monotone bifunction and also the strong convergence under additional assumptions on the bifunction. Finally, we study a regularization of Halpern type and prove the strong convergence of the generated sequence to an equilibrium point without any additional assumption on the pseudo-monotone bifunction. Some examples in fixed point theory and convex minimization are also presented.

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