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Mircea Balaj

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Kunle Oladeji Babalola

University of Ilorin

Fadipe-Joseph Olubunmi

University of Ilorin

Kocaeli University

Işık University

Ritu (Goyal) Agarwal

Malaviya National Institute of Technology, Jaipur

University of East London

Gwen Robbins Schug

University of North Carolina at Greensboro

Gabriel Gutierrez-Alonso

University of Salamanca

Macquarie University

Swansea University

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Papers by Mircea Balaj

Existence results for quasi-equilibrium problems under a weaker equilibrium condition

Operations Research Letters, May 1, 2021

Abstract By a quasi-equilibrium problem is understood an equilibrium problem with a constraint se... more

Solving Equilibrium Problems with Bifunctions Having the Property of Negative Transitivity

Numerical Functional Analysis and Optimization, Jan 19, 2022

Matching Theorems and Simultaneous Relation Problems

Bulletin of The Korean Mathematical Society, Sep 30, 2011

In this paper we give two matching theorems of Ky Fan type concerning open or closed coverings of... more In this paper we give two matching theorems of Ky Fan type concerning open or closed coverings of nonempty convex sets in a topological vector space. One of them will permit us to put in evidence, when X and Y are convex sets in topological vector spaces, a new subclass of KKM(X, Y) different by any admissible class Ac(X, Y). For this class of set-valued mappings we establish a KKM-type theorem which will be then used for obtaining existence theorems for the solutions of two types of simultaneous relation problems.

Common fixed point theorems for set-valued mappings in normed spaces

Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, Oct 15, 2018

Some common fixed point theorems for a pair of multi-valued non-self mappings in complete convex ... more

Intersection theorems for weak KKM set-valued mappings in the finite dimensional setting

Topology and its Applications, Aug 1, 2019

Let X be a convex set in a vector space, Y be a nonempty set and S, T : X ⇒ Y two set-valued mapp... more Let X be a convex set in a vector space, Y be a nonempty set and S, T : X ⇒ Y two set-valued mappings. S is said to be a weak KKM mapping w.r.t. T if for each nonempty finite subset A of X and any x ∈ conv A, T (x) ∩ S(A) = ∅. Recently, the authors obtained two intersection theorems for a pair of such mappings, when X is a compact convex subset of a topological vector space. In this paper, we obtain open versions of the above mentioned theorems when X is a compact convex set in R n. As applications, we establish several minimax inequalities and existence criteria for the solutions of three types of set-valued equilibrium problems.

New criteria for existence of solutions for equilibrium problems

Computational Management Science, Feb 6, 2023

Equilibrium problems provide a mathematical framework which includes optimization, variational in... more Equilibrium problems provide a mathematical framework which includes optimization, variational inequalities, fixed point and saddle point problems, and noncooperative games as particular cases. In this paper sufficient conditions for the existence of solutions of an equilibrium problem are given by weakening the assumption of quasiconvexity of the involved equilibrium bifunction. The existence of solutions is established both in presence of compactness of the feasible set as well with a coercivity assumption. The results are obtained in an infinite dimensional setting, and they are based on the so called finite solvability property which is weaker than the recently introduced finite intersection property and in turn, weaker than most common cyclic and proper quasimonotonicity. Some examples are presented to illustrate the various cases in which other existence results for equilibrium problems do not apply. Finally, applications to the solution of quasiequilibrium problems, quasioptimization problems and generalized quasivariational inequalities are discussed.

Intersection theorems with applications in set-valued equilibrium problems and minimax theory

Carpathian Journal of Mathematics, 2019

In this paper, we obtain three intersection theorems that can be considered versions of Theorem 3... more

Fixed Point Theorems, Section Properties and Minimax Inequalities on K-G-Convex Spaces

Journal of The Korean Mathematical Society, May 1, 2002

Intersection theorems for generalized weak KKM set‐valued mappings with applications in optimization

Mathematische Nachrichten, May 20, 2021

In this paper, we introduce the concept of generalized weak KKM mapping that is more general than... more In this paper, we introduce the concept of generalized weak KKM mapping that is more general than many others encountered in the KKM theory. Then, two previous intersection theorems of the author are extended from weak KKM mappings to generalized weak KKM mappings. Applications of these results to set‐valued equilibrium problems and minimax inequalities are given in the last two sections.

New existence criteria of the solutions for a variational relation problem

Optimization, Feb 28, 2017

Many quasivariational inclusions or quasiequilibrium problems, encountered in the literature, are... more Many quasivariational inclusions or quasiequilibrium problems, encountered in the literature, are special cases of a variational relation problem proposed in a recent paper by Agarwal et al. (J. Optim. Theory Appl. 2012;155:417-429). The purpose of this paper is to establish new existence results for the solutions of this problem. The main ingredients in the proofs are some continuous selection and fixed point theorems, and an interesting section result. In the last section, we prove that, applied for some concrete relations, our results are different from those obtained by other authors.

Alternative principles and their applications

Journal of Global Optimization, Sep 29, 2010

In this paper we establish two alternative principles of the following type: If X and Y are conve... more In this paper we establish two alternative principles of the following type: If X and Y are convex subsets of two locally convex Hausdorff topological vector spaces and F, S : X Y are two set-valued mappings satisfying certain conditions, then either there exists x 0 ∈ X such that F(x 0) = ∅ or x∈X S(x) = ∅. As first applications of the alternative principles we obtain two matching theorems of Ky Fan type. Next, are given several analytic alternatives and minimax inequalities. Finally we establish two very general alternative theorems concerning existence of solutions of a vector equilibrium problem.

Intersection Theorems with Applications in Optimization

Journal of Optimization Theory and Applications, Jun 14, 2018

In this paper, we establish two intersection theorems which are useful in considering some optimi... more In this paper, we establish two intersection theorems which are useful in considering some optimization problems (complementarity problems, variational inequalities, minimax inequalities, saddle point problems). Keywords Weak KKM map • Variational inequality • Minimax inequality Mathematics Subject Classification 47H10 • 49J53 1 Introduction An intersection theorem establishes sufficient conditions such that the values of a setvalued mapping between two sets, endowed with adequate topological and/or algebraic structures, have a common point. For example, problems in optimization and nonlinear functional analysis can be proved using an intersection theorem where an intersection point can be viewed as a fixed point, a coincidence point, an equilibrium point, a saddle

Stampacchia variational inequality with weak convex mappings

Optimization, Jun 14, 2018

In this paper, the concept of weak convex set-valued mapping is introduced and various conditions... more

Common Fixed Point Theorems in Topological Vector Spaces via Intersection Theorems

Journal of Optimization Theory and Applications, Feb 18, 2017

Our purpose in this paper is to present two methods for obtaining common fixed point theorems in ... more

Coincidence and maximal element theorems and their applications to generalized equilibrium problems and minimax inequalities

Nonlinear Analysis-theory Methods & Applications, Jun 1, 2008

In this paper we obtain first a very general coincidence theorem. From this we derive a new coinc... more In this paper we obtain first a very general coincidence theorem. From this we derive a new coincidence theorem and two alternative theorems concerning existence of maximal elements. Applications of these results to generalized equilibrium problems and minimax inequalities are given in the last sections.

A fixed point theorem for composed set-valued maps

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae, 1998

Families of sets in general position

Demonstratio Mathematica. Warsaw Technical University Institute of Mathematics, 1998

Fixed points and minimax inequalities

Acta Universitatis Apulensis. Mathematics - Informatics, 2004

Using Fan-Glicksberg fixed point theorem we obtain in this paper a fixed point theorem for the co... more

An Intersection Theorem and its Applications

CUBO, A Mathematical Journal, Dec 1, 2008

In this paper we obtain a very general intersection theorem for the values of a map. From this we... more In this paper we obtain a very general intersection theorem for the values of a map. From this we derive existence theorems for two types of vectorial equilibrium problems, an analytic alternative and a minimax inequality involving three real functions. En este artículo obtenemos un teorema general de intersección para los valores de una aplicación. A través de este resultado deducimos teoremas de existencia para dos tipos de problemas de equilibrio vectoriales, una alternativa analítica y una desigualdad minimax envolviendo tres funciones reales.

Haar spaces and polynomial selections

Mathematica Pannonica, 2003

Existence results for quasi-equilibrium problems under a weaker equilibrium condition

Operations Research Letters, May 1, 2021

Abstract By a quasi-equilibrium problem is understood an equilibrium problem with a constraint se... more

Solving Equilibrium Problems with Bifunctions Having the Property of Negative Transitivity

Numerical Functional Analysis and Optimization, Jan 19, 2022

Matching Theorems and Simultaneous Relation Problems

Bulletin of The Korean Mathematical Society, Sep 30, 2011

In this paper we give two matching theorems of Ky Fan type concerning open or closed coverings of... more In this paper we give two matching theorems of Ky Fan type concerning open or closed coverings of nonempty convex sets in a topological vector space. One of them will permit us to put in evidence, when X and Y are convex sets in topological vector spaces, a new subclass of KKM(X, Y) different by any admissible class Ac(X, Y). For this class of set-valued mappings we establish a KKM-type theorem which will be then used for obtaining existence theorems for the solutions of two types of simultaneous relation problems.

Common fixed point theorems for set-valued mappings in normed spaces

Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, Oct 15, 2018

Some common fixed point theorems for a pair of multi-valued non-self mappings in complete convex ... more

Intersection theorems for weak KKM set-valued mappings in the finite dimensional setting

Topology and its Applications, Aug 1, 2019

Let X be a convex set in a vector space, Y be a nonempty set and S, T : X ⇒ Y two set-valued mapp... more Let X be a convex set in a vector space, Y be a nonempty set and S, T : X ⇒ Y two set-valued mappings. S is said to be a weak KKM mapping w.r.t. T if for each nonempty finite subset A of X and any x ∈ conv A, T (x) ∩ S(A) = ∅. Recently, the authors obtained two intersection theorems for a pair of such mappings, when X is a compact convex subset of a topological vector space. In this paper, we obtain open versions of the above mentioned theorems when X is a compact convex set in R n. As applications, we establish several minimax inequalities and existence criteria for the solutions of three types of set-valued equilibrium problems.

New criteria for existence of solutions for equilibrium problems

Computational Management Science, Feb 6, 2023

Equilibrium problems provide a mathematical framework which includes optimization, variational in... more Equilibrium problems provide a mathematical framework which includes optimization, variational inequalities, fixed point and saddle point problems, and noncooperative games as particular cases. In this paper sufficient conditions for the existence of solutions of an equilibrium problem are given by weakening the assumption of quasiconvexity of the involved equilibrium bifunction. The existence of solutions is established both in presence of compactness of the feasible set as well with a coercivity assumption. The results are obtained in an infinite dimensional setting, and they are based on the so called finite solvability property which is weaker than the recently introduced finite intersection property and in turn, weaker than most common cyclic and proper quasimonotonicity. Some examples are presented to illustrate the various cases in which other existence results for equilibrium problems do not apply. Finally, applications to the solution of quasiequilibrium problems, quasioptimization problems and generalized quasivariational inequalities are discussed.

Intersection theorems with applications in set-valued equilibrium problems and minimax theory

Carpathian Journal of Mathematics, 2019

In this paper, we obtain three intersection theorems that can be considered versions of Theorem 3... more

Fixed Point Theorems, Section Properties and Minimax Inequalities on K-G-Convex Spaces

Journal of The Korean Mathematical Society, May 1, 2002

Intersection theorems for generalized weak KKM set‐valued mappings with applications in optimization

Mathematische Nachrichten, May 20, 2021

In this paper, we introduce the concept of generalized weak KKM mapping that is more general than... more In this paper, we introduce the concept of generalized weak KKM mapping that is more general than many others encountered in the KKM theory. Then, two previous intersection theorems of the author are extended from weak KKM mappings to generalized weak KKM mappings. Applications of these results to set‐valued equilibrium problems and minimax inequalities are given in the last two sections.

New existence criteria of the solutions for a variational relation problem

Optimization, Feb 28, 2017

Many quasivariational inclusions or quasiequilibrium problems, encountered in the literature, are... more Many quasivariational inclusions or quasiequilibrium problems, encountered in the literature, are special cases of a variational relation problem proposed in a recent paper by Agarwal et al. (J. Optim. Theory Appl. 2012;155:417-429). The purpose of this paper is to establish new existence results for the solutions of this problem. The main ingredients in the proofs are some continuous selection and fixed point theorems, and an interesting section result. In the last section, we prove that, applied for some concrete relations, our results are different from those obtained by other authors.

Alternative principles and their applications

Journal of Global Optimization, Sep 29, 2010

In this paper we establish two alternative principles of the following type: If X and Y are conve... more In this paper we establish two alternative principles of the following type: If X and Y are convex subsets of two locally convex Hausdorff topological vector spaces and F, S : X Y are two set-valued mappings satisfying certain conditions, then either there exists x 0 ∈ X such that F(x 0) = ∅ or x∈X S(x) = ∅. As first applications of the alternative principles we obtain two matching theorems of Ky Fan type. Next, are given several analytic alternatives and minimax inequalities. Finally we establish two very general alternative theorems concerning existence of solutions of a vector equilibrium problem.

Intersection Theorems with Applications in Optimization

Journal of Optimization Theory and Applications, Jun 14, 2018

In this paper, we establish two intersection theorems which are useful in considering some optimi... more In this paper, we establish two intersection theorems which are useful in considering some optimization problems (complementarity problems, variational inequalities, minimax inequalities, saddle point problems). Keywords Weak KKM map • Variational inequality • Minimax inequality Mathematics Subject Classification 47H10 • 49J53 1 Introduction An intersection theorem establishes sufficient conditions such that the values of a setvalued mapping between two sets, endowed with adequate topological and/or algebraic structures, have a common point. For example, problems in optimization and nonlinear functional analysis can be proved using an intersection theorem where an intersection point can be viewed as a fixed point, a coincidence point, an equilibrium point, a saddle

Stampacchia variational inequality with weak convex mappings

Optimization, Jun 14, 2018

In this paper, the concept of weak convex set-valued mapping is introduced and various conditions... more

Common Fixed Point Theorems in Topological Vector Spaces via Intersection Theorems

Journal of Optimization Theory and Applications, Feb 18, 2017

Our purpose in this paper is to present two methods for obtaining common fixed point theorems in ... more

Coincidence and maximal element theorems and their applications to generalized equilibrium problems and minimax inequalities

Nonlinear Analysis-theory Methods & Applications, Jun 1, 2008

In this paper we obtain first a very general coincidence theorem. From this we derive a new coinc... more In this paper we obtain first a very general coincidence theorem. From this we derive a new coincidence theorem and two alternative theorems concerning existence of maximal elements. Applications of these results to generalized equilibrium problems and minimax inequalities are given in the last sections.

A fixed point theorem for composed set-valued maps

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae, 1998

Families of sets in general position

Demonstratio Mathematica. Warsaw Technical University Institute of Mathematics, 1998

Fixed points and minimax inequalities

Acta Universitatis Apulensis. Mathematics - Informatics, 2004

Using Fan-Glicksberg fixed point theorem we obtain in this paper a fixed point theorem for the co... more

An Intersection Theorem and its Applications

CUBO, A Mathematical Journal, Dec 1, 2008

In this paper we obtain a very general intersection theorem for the values of a map. From this we... more In this paper we obtain a very general intersection theorem for the values of a map. From this we derive existence theorems for two types of vectorial equilibrium problems, an analytic alternative and a minimax inequality involving three real functions. En este artículo obtenemos un teorema general de intersección para los valores de una aplicación. A través de este resultado deducimos teoremas de existencia para dos tipos de problemas de equilibrio vectoriales, una alternativa analítica y una desigualdad minimax envolviendo tres funciones reales.

Haar spaces and polynomial selections

Mathematica Pannonica, 2003