Coincidence Theorems Research Articles

New existence theorems for quasi-equilibrium problems and a minimax theorem on complete metric spaces

Motivated by the Suzuki's type fixed point theorems, we give several new existence theorems for scalar quasi-equilibrium problems, and vector quasi-equilibrium problem on complete metric spaces. We give important examples for our results. Note that the solution of quasi-equilibrium problem (resp. vector quasi-equilibrium problem) is unique under suitable conditions, and we can find the unique solution by the Picard iteration. Besides, we also give a new coincidence theorem on complete metric spaces. Finally, we give a new minimax theorem on complete metric spaces. Note that the solution of minimax theorem is unique under suitable conditions, and we can find the unique solution by the Picard iteration.

Weak contractive inequalities and compatible mixed monotone random operators in ordered metric spaces

Abstract The aim of this work is to define the notion of compatible random operators in a partially ordered metric space and prove some coupled random coincidence theorems for a pair of compatible mixed monotone random operators satisfying ( ϕ , φ ) -weak contractive conditions. These results present random versions and extensions of recent results of Ćirić and Lakshmikantham (Stoch. Anal. Appl. 27:1246-1259, 2009), Choudhury and Kundu (Nonlinear Anal. 73:2524-2531, 2010), Alotaibi and Alsulami (Fixed Point Theory Appl. 2011:44, 2011) and many others.

A Matching Theorem in FC-Spaces with the Application to Coincidence Questions

A Ky Fan type matching theorem for weakly transfer compactly open valued mappings is established in FC-spaces. As applications, Fan–Browder type fixed point theorems and Ky Fan type coincidence theorems are obtained in FC-spaces.

New representations of the Moore–Penrose inverse of [formula omitted] block matrices

This paper presents a full rank factorization of a 2×2 block matrix without any restriction. Applying this factorization, we obtain an explicit representation of the Moore–Penrose inverse in terms of four individual blocks of the partitioned matrix, which simplifies results of Hung and Markham (1975) [12], and Miao (1991) [16]. We also derive some important coincidence theorems, including in the expressions of the Moore–Penrose inverse of a sum of matrices. All these results extend earlier works by various authors.

On the Brézis Nirenberg Stampacchia-type theorems and their applications

In this paper, we obtain several new generalized KKM-type theorems under a new coercivity condition and the condition of intersectionally closedness which improves condition of transfer closedness. As applications, we obtain new versions of equilibrium problem, minimax inequality, coincidence theorem, fixed point theorem and an existence theorem for an 1-person game.

A constructive version of Ky Fan's coincidence theorem

In this paper we examine Ky Fan's coincidence theorem from the point of view of Bishop style constructive mathematics using constructive versions of Kakutani's fixed point theorem, Berge's maximum theorem and the separating hyperplane theorem.

On a Parameterized System of Nonlinear Equations with Economic Applications

We study a parameterized system of nonlinear equations. Given a nonempty, compact, and convex set, an affine function, and a point-to-set mapping from the set to the Euclidean space containing the set, we constructively prove that, under certain (boundary) conditions on the mapping, there exists a connected set of zero points of the mapping, i.e., the origin is an element of the image for every point in the connected set, such that the connected set has a nonempty intersection with both the face at which the affine function is minimized and the face at which that function is maximized. This result generalizes and unifies several well-known existence theorems including Browder’s fixed point theorem and Ky Fan’s coincidence theorem. An economic application with constrained equilibria is also discussed.

On the Weak Stability of Picard Iteration for Some Contractive Type Mappings and Coincidence Theorems

Starting from a weak concept of stability, introduced by Berinde [1] and called “weak stability”, in [27] we develop a weaker notion, named “w2-stability”. Therefore, in this paper we prove some results of this weaker stability concept for certain class of mappings and also we give some examples of w2-stable but not weak stable nor stable iterations. Because of the restriction of an “approximate” sequence, some fixed point iteration procedures are not weakly stable so if it is used a weaker type of sequence, the stability can be obtained in the meaning of a new concept. General Terms Fixed-point and coincidence theorems.

A section theorem with applications to coincidence theorems and minimax inequalities in [formula omitted]-spaces

In this paper, an improved version of section theorem is proved in FWC-spaces without any linear and convex structure under much weaker assumptions, and next as its applications, some new coincidence theorems and minimax inequalities are established in FWC-spaces. These results generalize many known theorems in the literature.

Tripled coincidence theorems for monotone mappings in partially ordered metric spaces

In this paper, we establish tripled coincidence point theorems for a pair of mappings F : X × X × X → X and g : X → X satisfying a nonlinear contractive condition ordered metric spaces. Presented theorems extend several existing results in the literature: [Borcut, M. and Berinde, V., Tripled coincidente point theorems for contractive type mappings in partially ordered metric spaces, Aplied Mathematics and Computation, 218 (2012), No. 10, 5929–5936], and Berinde, Borcut in article [Berinde, V., Borcut, M., Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal., 74 (2011), 4889-4897].

Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces

In this paper we introduce the concept of a tripled coincidence point for a pair of nonlinear contractive mappings F : X 3 → X and g : X → X. The obtained results extend recent coincidence theorems due to Ćirić and Lakshmikantham [V. Lakshmikantham, L. Ćirić, L., Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70 (2009) 4341–4349].

Coincidence theorems and minimax inequalities in abstract convex spaces

Abstract In this paper, we deal with the notion of abstract convex spaces via minimal spaces as an extended version of other forms of convexity and establish some well-known results such as coincidence theorems for the classes m-KKM and ms-KKM of multimaps and Ky Fan's type minimax inequality. Mathematics Subject Classification (2000) 26A51; 26B25; 54H25; 55M20; 47H10; 54A05.

Coincidence Theorem and the Inner Core

We prove a variant of Fan's coincidence theorem. Then, by applying it, we give another proof to a theorem on the nonemptiness of the inner core due to Aubin (1973) and Inoue (2011). Our coincidence theorem is a suitable mathematical tool for proving the nonemptiness of the inner core in the sense that one of the assumptions of our coincidence theorem is equivalent to the balancedness of a non-transferable utility (NTU) game when two correspondences are defined properly.

A coincidence theorem for commuting involutions

Let M m M^{m} be an m m -dimensional, closed and smooth manifold, and let S , T : M m → M m S, T:M^{m} \to M^{m} be two smooth and commuting diffeomorphisms of period 2 2 . Suppose that S ≠ T S \not = T on each component of M m M^{m} . Denote by F S F_{S} and F T F_{T} the respective sets of fixed points. In this paper we prove the following coincidence theorem: if F T F_{T} is empty and the number of points of F S F_{S} is of the form 2 p 2p , with p p odd, then C o i n c ( S , T ) = { x ∈ M m | S ( x ) = T ( x ) } Coinc(S,T)=\{x \in M^{m} \ \vert \ S(x)=T(x) \} has at least some component of dimension m − 1 m-1 . This generalizes the classic example given by M m = S m M^{m}=S^{m} , the m m -dimensional sphere, S ( x 0 , x 1 , . . . , x m ) = ( − x 0 , − x 1 , . . . , − x m − 1 , x m ) S(x_{0},x_{1},...,x_{m}) = (-x_{0},-x_{1},...,-x_{m-1},x_{m}) and T T the antipodal map.

Some Results for the Family KKM(X,Y) and the Φ-Mapping in FC-Spaces

We first establish a fixed point theorem for a k-set contraction map on the family KKM(X,X), which does not need to be a compact map. Next, we present the KKM type theorems, matching theorems, coincidence theorems, and minimax theorems on the family KKM(X,Y) and the Φ-mapping in FC-spaces. Our results improve and generalize some recent results.

Fixed Points and Existence Theorems of Maximal Elements with Applications in FC-Spaces

We present some fixed point theorems and existence theorems of maximal elements in FC-space from which we derive several coincidence theorems. Applications of these results to generalized equilibrium problems and minimax theory will be given. Our results improve and generalize some recent results.

Coincidence theorems and its applications to equilibrium problems

Given two maps \({h : X \times K \rightarrow \mathbb{R}}\) and g : X → K such that, for all \({x \in X, h(x, g(x)) = 0}\) , we consider the equilibrium problem of finding \({\tilde{x} \in X}\) such that \({h(\tilde{x}, g(x)) \geq 0}\) for every \({x \in X}\) . This question is related to a coincidence problem.

Coincidence and Maximal Element Theorems in Abstract Convex Spaces with Applications

In this paper, by using a nonempty intersection lemma due to the authors, we obtain two coincidence theorems involved $\mathfrak{RC}$-maps in abstract convex spaces, which are actually equivalent. We then derive some maximal element theorems for set-valued maps in abstract convex spaces. As an application, we study the existence of solutions for a system of generalized equilibrium problems in abstract convex spaces. We also give some examples to illustrate our results.

Coincidence and Maximal Element Theorems in Abstract Convex Spaces with Applications

In this paper, by using a nonempty intersection lemma due to the authors, we obtain two coincidence theorems involved $\mathfrak{RC}$-maps in abstract convex spaces, which are actually equivalent. We then derive some maximal element theorems for set-valued maps in abstract convex spaces. As an application, we study the existence of solutions for a system of generalized equilibrium problems in abstract convex spaces. We also give some examples to illustrate our results.

Random coincidence theorems and applications

In this article we state conditions under which the existence of deterministic coincidence points of two multivalued random functions implies the existence of random coincidence points. Moreover, two existence results of measurable selections are obtained for the intersection of the multivalued function evaluated at the random coincidence point. Our results extend or improve some theorems in the literature.