Existence and generic stability conditions of equilibrium points to controlled systems for n-player multiobjective generalized games using the Kakutani–Fan–Glicksberg fixed-point theorem

Abstract

The purpose of this article is to establish some results on the existence and generic stability of solution sets of the controlled system for multiobjective generalized games in infinite-dimensional spaces. First, we introduce a class of the controlled system for multiobjective generalized games. Afterward, we establish some conditions for existence of the solution set to this problem using the Kakutani-Fan-Glicksberg fixedpoint theorem. Finally, we study the generic stability of set-valued mappings where the set of essential points of a map is a dense residual subset of a (Hausdorff) metric space of the set-valued maps. The results presented in the paper are new and completely different from the main results given by some authors in the literature.

Controllability of Mild Solution to Hilfer Fuzzy Fractional Differential Inclusion with Infinite Continuous Delay

“…where ∆ ψ η i (t) is defined by ( 14)- (17). Due to the linearty of ∆, we can see the linearty of ∆ and by using Proposition 3 and the convexity of S u,(1,∞) Π,x , we have…”

“…In 2021, N. V. Hung [14] gave us strong and more worthwhile results on the generalization of Levitin-Polyak well-posedness for controlled systems of minty type-fuzzy mixed quasi-hemivariational inequalities (FMQHI). For more readings, it is worth looking into the engineering, mechanics, and economics literature as well, for example, [15][16][17][18][19][20].…”

Optimal control of generalized multiobjective games with application to traffic networks modeling

“…Remark Let us note that Lemmas 3.2 and 3.6 are different from results in Hung and Keller [17], because we established existence conditions to controlled systems for generalized multiobjective games under Browder‐type fixed point theorem in the noncompact case and the $$C_i$$‐quasi‐concavity, while Hung and Keller [17] studied on the existence of solutions using the Kakutani–Fan–Glicksberg fixed point theorem and $$C_i$$‐quasi‐convexity.…”

“…Let 𝑆 𝑛 𝑖 ∶ 𝐾 î ⇉ 𝐾 𝑖 be the sequences of the feasible strategy correspondence of player 𝑖, 𝑓 𝑛 𝑖 ∶ 𝐾 ×  𝑖 → 𝑌 𝑖 be the sequences of the payoff function of player 𝑖, for 𝑛 ∈ ℕ ⧵ {0}. Suppose that the following conditions hold: [17], because we established existence conditions to controlled systems for generalized multiobjective games under Browder-type fixed point theorem in the noncompact case and the 𝐶 𝑖 -quasi-concavity, while Hung and Keller [17] studied on the existence of solutions using the Kakutani-Fan-Glicksberg fixed point theorem and 𝐶 𝑖 -quasi-convexity.…”

“…Then, Shapley [32] established and studied the concept of equilibrium points in games with vector payoffs with mean games with incommensurable criteria in 1959. In recent years, there have been many authors studying models of multiobjective games on different topics such as existence conditions and stability of solutions, see, for example, [11,17,22,[39][40][41][42] and references therein.…”