Point Theorem For Set-valued Mappings Research Articles

Nonempty intersection theorems in topological spaces with applications

In this paper, we establish some new nonempty intersection theorems for generalized L-KKM mappings and prove some new fixed point theorems for set-valued mappings under suitable conditions in topological spaces. As applications, an existence theorem for an equilibrium problem with lower and upper bounds and two existence theorems for a quasi-equilibrium problem with lower and upper bounds are obtained in topological spaces. Our results generalize some known results in the literature.

Generalized variational inequalities with weight-like vectors

A class of generalized vector variational inequalities with weight-like vectors over product of sets and system of generalized vector variational inequalities are considered. The relationships of solution sets among those variational inequality problems are presented. With the hemicontinuity and several kinds of generalized monotonicities with respect to a vector, several existence results for the solutions of problems mentioned above are established by fixed point theorems of set-valued mappings.

Best approximation, coincidence and fixed point theorems for set-valued maps in [formula omitted]-trees

Suppose X is a closed, convex and geodesically bounded subset of a complete R -tree M , and suppose F : X ⊸ M is an almost lower semicontinuous set-valued map whose values are nonempty closed convex. Suppose also G : X ⊸ X is a continuous, onto quasiconvex set-valued map with compact, convex values. Then there exists x 0 ∈ X such that d ( G ( x 0 ) , F ( x 0 ) ) = inf x ∈ X d ( x , F ( x 0 ) ) . As applications, we give some coincidence and fixed point results for weakly inward set-valued maps. Our results generalize some well-known results in literature.

New types of common fixed point theorems in 2-metric spaces

The purpose of this paper is to establish common fixed point theorems for set-valued mappings between 2-metric spaces. Generalizations of some definitions in 2-metric spaces and theorems by Ahmed [Ahmed MA. Common fixed point theorems for weakly compatible mappings. Rocky Mt J Math 2003;33 (4):1189–203.], generalized of Fisher [Fisher B. Common fixed points of mappings and set-valued mappings on a metric spaces. Kyungpook Math J 1985;25:35–42.] in 2-metric spaces.

Mizoguchi–Takahashi's fixed point theorem is a real generalization of Nadler's

We give an example which says that Mizoguchi–Takahashi's fixed point theorem for set-valued mappings is a real generalization of Nadler's. We also give a very simple proof of Mizoguchi–Takahashi's theorem.

Convergence analysis of a family of Steffensen-type methods for generalized equations

A class of Steffensen-type algorithms for solving generalized equations on Banach spaces is proposed. Using well-known fixed point theorem for set-valued maps [A.L. Dontchev, W.W. Hager, An inverse function theorem for set-valued maps, Proc. Amer. Math. Soc. 121 (1994) 481–489] and some conditions on the first-order divided difference, we provide a local convergence analysis. We also study the perturbed problem and we present a new regula-falsi-type method for set-valued mapping. This study follows the works on the Secant-type method presented in [S. Hilout, A uniparametric Secant-type methods for nonsmooth generalized equations, Positivity (2007), submitted for publication; S. Hilout, A. Piétrus, A semilocal convergence of a Secant-type method for solving generalized equations, Positivity 10 (2006) 673–700] and extends the results related to the resolution of nonlinear equations [M.A. Hernández, M.J. Rubio, The Secant method and divided differences Hölder continuous, Appl. Math. Comput. 124 (2001) 139–149; M.A. Hernández, M.J. Rubio, Semilocal convergence of the Secant method under mild convergence conditions of differentiability, Comput. Math. Appl. 44 (2002) 277–285; M.A. Hernández, M.J. Rubio, ω-Conditioned divided differences to solve nonlinear equations, in: Monogr. Semin. Mat. García Galdeano, vol. 27, 2003, pp. 323–330; M.A. Hernández, M.J. Rubio, A modification of Newton's method for nondifferentiable equations, J. Comput. Appl. Math. 164/165 (2004) 323–330].

Fixed point theorems and their applications to theory of Nash equilibria

In this paper we prove fixed point theorems for set-valued mappings in products of posets. Applications to the theory of Nash equilibria are presented.

An Application of the Stiefel–Whitney Classes to the Proof of a Fixed Point Theorem for Set-Valued Mappings

An Application of the Stiefel–Whitney Classes to the Proof of a Fixed Point Theorem for Set-Valued Mappings

Fixed point theorems for fuzzy mappings in complete metric spaces

In this paper, we are concerned with mathematical properties of the previous contractive types of set-valued maps or fuzzy mappings on a metric space (see, for instance, Nadler, Pacific J. Math. 30 (1969) 475–488; Heilpern, J. Math. Anal. Appl. 83 (1981) 566–569; Papageorgiou, Nonlinear Anal. Theory Methods Appl. 7 (1983) 763–770; Bose and Sahani, Fuzzy Sets and Systems 21 (1987) 53–58; Som and Mukherjee, Fuzzy Sets and Systems 33 (1989) 213–219; Park and Jeong, Fuzzy Sets and Systems 59 (1993) 231–235; 87 (1997) 111–116; Lee et al., Fuzzy Sets and Systems 101 (1999) 143–152) and we prove some crucial theorems relating fixed points of fuzzy mappings on complete metric spaces. First, we prove a theorem by using the concept of w-distance (Kada et al., Math. Japonica 44 (1996) 381–391), which generalizes the results of Amemiya and Takahashi (Fuzzy Sets and Systems 114 (2000) 469–476), and Kada et al. (1996). Next, we prove two theorems for fuzzy mappings, which are connected with fixed point theorems for set-valued maps. Then we shall observe that our theorems can be used to obtain almost all results proved so far concerning fixed points of fuzzy mappings or set-valued maps on complete metric spaces and therefore, that our results imply the existence of fixed points of various contractive types of fuzzy mappings or set-valued maps on complete metric spaces.

A fixed point theorem for set-valued mappings

Fixed points for set-valued mappings from a metric space X (not necessarily complete) intoB(X), the collection of nonempty bounded subsets of X are obtained. The result generalizes some known results.

Fixed Point Theorems for Set-Valued Maps and Existence Principles for Integral Inclusions

New fixed point theorems of Mönch type are presented for set-valued maps. These theorems are then used to establish general existence principles for Hammerstein integral inclusions in Banach spaces.

The Characterization of Generalized Metric KKM Mappings with Open Values in Hyperconvex Metric Spaces and Some Applications

In this note, the characterization for a set-valued mapping with finitely metrically open values being a generalized metric KKM mapping in hyperconvex metric spaces is established. This result could be regarded as a dual form of corresponding results for the Fan–KKM principle in hyperconvex metric spaces obtained recently by M. A. Khamsi [J. Math. Anal. Appl.204 (1996) 298–306] and W. A. Kirk, B. Sims, and X. Z. Yuan [Nonlinear Analysis (1999) (in press)]. Then we show that the finite intersection property of generalized metric KKM mappings with finitely metrically open values indeed is equivalent to the finite intersection property of generalized metric KKM mappings with finitely metrically closed values in hyperconvex spaces. As applications, we first obtain Ky Fan type matching theorems for both closed and open covers in hyperconvex spaces, which, in turn, are used to establish fixed point theorems for set-valued mappings in hyperconvex metric spaces.

Common fixed point theorems and variational principle in generating spaces of quasi-metric family

In this paper, we introduce the concept of order relation in generating spaces of quasi-metric family and establish common fixed point theorems for set-valued mappings in this spaces. As consequences, we also give the variational principle, fixed point theorems for single-valued and set-valued mappings and equivalence between them in complete generating spaces of quasi-metric family. Some applications of these results to fuzzy metric spaces in sense of Kaleva and Seikkala (1984) and probabilistic metric spaces are presented.

Minimization theorems and fixed point theorems in generating spaces of quasi-metric family

By following the approaches of Kada et al. [13], we define a family of weak quasi-metrics in a generating space of quasi-metric family. By using a family of weak quasi-metrics, we prove a Takahashi-type minimization theorem, a generalized Ekeland variational principle and a general Caristi-type fixed point theorem for set-valued maps in complete generating spaces of quasi-metric family. Also, by following the approach of Aubin [11], we prove another fixed point theorem for set-valued maps in complete generating spaces of quasi-metric family without the assumption of lower semicontinuity. From our results in complete generating spaces of quasi-metric family, we obtain the corresponding theorems for set-valued maps in complete fuzzy metric spaces.

Some fixed point theorems for nonconvex spaces

We give a theorem for nonconvex topological vector spaces which yields the classical fixed point theorems of Ky Fan, Kim, Kaczynski, Kelly and Namioka as immediate consequences, and prove a new fixed point theorem for set‐valued maps on arbitrary topological vector spaces.

Caristi's fixed point theorem for fuzzy mappings and Ekeland's variational principle

The purpose of this paper is to generalize Caristi's fixed point theorem to fuzzy mappings and to obtain two formally stronger Caristi type fixed point and common fixed point theorems for set-valued mappings and sequences of set-valued mappings. The results presented in this paper generalize the corresponding results of Caristi [ Trans Amer. Math. Soc. 215 (1976) 241–251] and Kasahara [ Math. Seminar Notes 35 (2) (1975) 229–232]. In addition, by a simple method the equivalence between Caristi's fixed point theorem for fuzzy mappings and Ekeland's variational principle is shown. This result improves and extends some important results of Dancs et al., Kirk, Shi, and Sullivan.

Random common fixed point theorems of set-valued mappings

In this paper, we prove several random common fixed point theorems to general setvalued contractive mappings. These theorems improve and generalize the corresponding results in [1–18].

A monotone principle of fixed points

In this paper we formulate a new principle of fixed points, and we call it "monotone principle of fixed points". A fixed point theorem for set-valued mappings in a complete metric space and some theorems on fixed points in arbitrary topological spaces are presented in this paper. Also, we describe a class of conditions sufficient for the existence of a fixed point which generalize several known results. We introduce the concept of a contraction principle and CS-convergence. With such an extension, a very general fixed point theorem is obtained to include a recent result of the author, which contains, as special cases, some results of J. Dugundji and A. Granas, F. Browder, D. W. Boyd and J. S. Wong, J. Caristi, T. L. Hicks and B. E. Rhoades, B. Fisher, W. Kirk and M. Krasnoselskij.

Well-posedness, control and computation of a one-phase Stefan problem with Neumann condition*

SynopsisA one-phase Stefan problem can be reduced to an equivalent variational inequality by using the Baiocchi-Duvaut transformation. In this paper, we study the variational inequality by formulating it as a set-valued partial differential equation. The existence of solutions is proved by applying a generalized Schauder fixed fixed point theorem for set-valued mappings. Uniqueness and regularity of solutions are also obtained. In §3, we regard the boundary value Neumann data as boundary controls and combine both the variational inequality and the classical approaches to study the effects of controls on the free boundary and the state (i.e. temperature). In §4, we further use the theory to study an optimal “ice-melting” problem. Our results show that if the controls have fixed total input heat flux and are constrained in magnitude, then the optimal control is “bang-bang”. If the admissible controls are not constrained in magnitude, then the optimal control is a Dirac delta type distribution which is no longer admissible. In the last section, our existence theory is combined with the finite difference method and non-linear programming techniques to obtain numerical solutions.

Fixed-point theorems for set-valued mappings and existence of best approximants

The paper gives a proof of the following existence result in best approximation. Let M be a compact, convex, and non-empty subset of a normed space E and let g be a continuous almost affine mapping of M onto M. For each continuous mapping f from M into E there exists a point x in M such that g(x) is a best M- approximation to f(x). The proof uses Bohnenblust and Karlin's extension to normed spaces of Kakutani's Fixed Point Theorem for set-valued mappings on compact, convex, and non-empty subsets of Euclidean n-space.