On Coupled Best Proximity Points in Reflexive Banach Spaces

AJETI, LAURA; Ilchev, Atanas; Zlatanov, Boyan

doi:10.3390/math10081304

Cited by 8 publications

(7 citation statements)

References 34 publications

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“…The authors of [10] called these new type of maps cyclic again. A more natural name is introduced in [28], where the authors called them an ordered pair of semi-cyclic maps.…”

Section: Definition 3 ([26]mentioning

confidence: 99%

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Long-Run Equilibrium in the Market of Mobile Services in the USA

Badev,

Kabaivanov,

Kopanov

et al. 2024

Mathematics

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We develop an empirical model of the market for mobile services in the USA based on providers’ response functions. Guided by a duopoly model, we obtain our empirical response functions from an approximation of quarterly response data on smartphone subscriptions by sigmoid functions of time. The robustness analysis suggests that our model fits the data well and outperforms the regression model. Further, we demonstrate that our empirical response functions satisfy the conditions for semi-cyclic contractions which guarantee the existence, uniqueness and stability of long-run equilibrium.

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“…The authors of [10] called these new type of maps cyclic again. A more natural name is introduced in [28], where the authors called them an ordered pair of semi-cyclic maps.…”

Section: Definition 3 ([26]mentioning

confidence: 99%

“…There is a sequence of results that guarantees the existence and uniqueness of coupled fixed points for semi-cyclic kinds of maps and thus the existence and uniqueness of market equilibrium in duopoly markets [1,10,28].…”

Section: Definition 3 ([26]mentioning

confidence: 99%

Long-Run Equilibrium in the Market of Mobile Services in the USA

Badev,

Kabaivanov,

Kopanov

et al. 2024

Mathematics

Self Cite

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“…The authors have termed these new types of maps cyclic once again in Dzhabarova et al (2020). A more natural name a semi-cyclic map is introduced in Ajeti et al (2022).…”

Section: Coupled Fixed Pointsmentioning

confidence: 99%

“…We obtain the concept of coupled fixed points from Definition 2 anytime X 1 = X 2 = A and F 2 (x, y) = F 1 (y, x). Definition 6 ((Ajeti et al 2022;Dzhabarova et al 2020)). Let A, B ̸ = ∅ be sets in a metric space (X, ρ) and F : A × B → A, G : A × B → B.…”

Section: Coupled Fixed Pointsmentioning

confidence: 99%

Investigation of Equilibrium in Oligopoly Markets with the Help of Tripled Fixed Points in Banach Spaces

Ilchev,

Ivanova,

Kulina

et al. 2024

Econometrics

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In the study we explore an oligopoly market for equilibrium and stability based on statistical data with the help of response functions rather than payoff maximization. To achieve this, we extend the concept of coupled fixed points to triple fixed points. We propose a new model that leads to generalized triple fixed points. We present a possible application of the generalized tripled fixed point model to the study of market equilibrium in an oligopolistic market dominated by three major competitors. The task of maximizing the payout functions of the three players is modified by the concept of generalized tripled fixed points of response functions. The presented model for generalized tripled fixed points of response functions is equivalent to Cournot payoff maximization, provided that the market price function and the three players’ cost functions are differentiable. Furthermore, we demonstrate that the contractive condition corresponds to the second-order constraints in payoff maximization. Moreover, the model under consideration is stable in the sense that it ensures the stability of the consecutive production process, as opposed to the payoff maximization model with which the market equilibrium may not be stable. A possible gap in the applications of the classical technique for maximization of the payoff functions is that the price function in the market may not be known, and any approximation of it may lead to the solution of a task different from the one generated by the market. We use empirical data from Bulgaria’s beer market to illustrate the created model. The statistical data gives fair information on how the players react without knowing the price function, their cost function, or their aims towards a specific market. We present two models based on the real data and their approximations, respectively. The two models, although different, show similar behavior in terms of time and the stability of the market equilibrium. Thus, the notion of response functions and tripled fixed points seems to present a justified way of modeling market processes in oligopoly markets when searching whether the market has reached equilibrium and if this equilibrium is unique and stable in time

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“…It is interesting to mention that the presented technique in [10,17] enables to find exact solutions in cases where the classical fixed point methods can find only approximations. Best proximity point results have been used in searching of market equilibrium in duopoly markets, where the cyclic maps have been replaced by semicyclic maps [1,6]. The natural underlying space in the market equilibrium theory is close to non-convex spaces, rather than convex spaces as pointed in [1], where results about coupled best proximity points have been obtained in reflexive Banach spaces.…”

Section: Introductionmentioning

confidence: 99%

On the UC and UC$^*$ properties and the existence of best proximity points in metric spaces

Zhelinski,

Zlatanov

2022

Ann. Sofia Univ. Fac. Math. Informat.

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We investigate the connections between UC and UC$^*$ properties for ordered pairs of subsets $(A,B)$ in metric spaces, which are involved in the study of existence and uniqueness of best proximity points. We show that the UC property and the UC$^*$ property lead to one and the same corollaries, when iterated sequences, generated by cyclic maps, are investigated. We introduce some new notions: bounded UC (BUC) property and uniformly convex set about a function $\phi$. We prove that these new notions are generalizations of the UC property and that both of them are sufficient to ensure existence and uniqueness of best proximity points. We show that these two new notions are different from a uniform convexity and even from a strict convexity. If we consider the underlying space to be a Banach space, we find a sufficient condition which ensures that from the UC property follows the uniform convexity of the underlying Banach space. We illustrate the new notions with examples. We present an example of a cyclic contraction $T$ in a space, which is not even strictly convex and the ordered pair $(A,B)$ does not have the UC property, but has the BUC property and thus there is a unique best proximity point of $T$ in $A$.

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On Coupled Best Proximity Points in Reflexive Banach Spaces

Cited by 8 publications

References 34 publications

Long-Run Equilibrium in the Market of Mobile Services in the USA

Long-Run Equilibrium in the Market of Mobile Services in the USA

Investigation of Equilibrium in Oligopoly Markets with the Help of Tripled Fixed Points in Banach Spaces

On the UC and UC$^*$ properties and the existence of best proximity points in metric spaces

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