Generalization Of Fixed Point Theorem Research Articles

A general fixed point theorem for four weakly compatible mappings satisfying an implicit relation

In this paper, using a combination of methods used in [5], [14], [15] and [17] the results of [1, Theorems 2.2 and 2.3] and [15, Theorem 3] are improved by removing the assumptions of continuity and reciprocally continuity, relaxing compatibility and compatibility of type (A) to weakly compatibility and replacing the completeness of the space with a set of four alternative conditions for four mappings satisfying an implicit relation.

The KKM principle implies many fixed point theorems

We show that the KKM principle implies two new general fixed point theorems for the Kakutani maps or the Browder maps. Consequently, we give unified transparent proofs of many of well-known results.

PROBABILISTIC MULTI-VALUED CONTRACTIONS AND DECOMPOSABLE MEASURES

A general fixed point theorem for a multi-valued probabilistic q-contraction f : S → 2S is proved, where (S, ℱ, T) is a complete Menger space and ℱ satisfies a growth condition which is connected with the countable extension of a t-norm T. As a corollary a generalization of Tardiff's result27 is obtained. A random fixed point result is proved, where a measure space related to decomposable measure is used.

A general fixed point theorem for mappings in pseudocompact tichonoff spaces

In [2] Jain and Dixit have obtained some interesting results on fixed points in pseudocompact Tichonoff spaces. In this paper we present a general fixed point theorem in pseudocompact Tichonoff spaces for mappings satisfying an implicit relation which generalize Theorem 1 and 2 from [2] and some results from [3] and [4].

A Fixed Point Theorem of Leggett–Williams Type with Applications to Single- and Multivalued Equations

Abstract We establish a general fixed point theorem for multivalued maps defined on cones in Banach spaces. Applications to single and multivalued equations are presented.

Semicompatibility and fixed point theorems in an unbounded D‐metric space

Rhoades (1996) proved a fixed point theorem in a bounded D‐metric space for a contractive self‐map with applications. Here we establish a more general fixed point theorem in an unbounded D‐metric space, for two self‐maps satisfying a general contractive condition with a restricted domain of x and y. This has been done by using the notion of semicompatible maps in D‐metric space. These results generalize and improve the results of Rhoades (1996), Dhage et al. (2000), and Veerapandi and Rao (1996). These results also underline the necessity and importance of semicompatibility in fixed point theory of D‐metric spaces. All the results of this paper are new.

Stationary weak solutions for a class of non-Newtonian fluids with energy transfer

We prove the existence of weak solutions to the coupled system of stationary equations for a class of general non-Newtonian fluids with energy transfer. In particular, we may include Bingham flows that lead to classical free boundary problems of fluid dynamics. Using convex analysis and L 1-theory for elliptic mixed boundary value problems, we consider separately two auxiliary problems, obtained by prescribing the essential non-linearities in the equations for the velocity and for the energy. We use then a general fixed point theorem, due to Glicksberg, for the multivalued mapping in a product of Banach spaces endowed with the weak topologies.

Absolute retracts and a general fixed point theorem for fuzzy sets

The space of fuzzy sets on a compact metric space, with the sendograph metric for fuzzy sets, is shown to have the fixed point property. The relationship between absolute retracts and spaces of fuzzy sets is demonstrated.

Fixed point indexes and its applications to nonlinear integral equations modelling infectious diseases

In this paper the fixed point index problem for a class of positive operators with boundary control conditions is discussed, and some sufficient conditions for the fixed point index to be equal to 1 or 0 are given. Moreover, a general fixed point theorem of expansions and compressions for cone is obtained, which generalizes and improves the corresponding results of [3,8,9]. As an application, we utilize the results presented above to study the existence conditions of positive solutions of nonlinear integral equations modelling infectious diseases.

Equilibrium and quasi-equilibrium existence theorems for a general model without ordered preferences

In previous papers Gale and Mas-Colell (1975) and later Florenzano (1981) obtained a fixed point theorem. Gale and Mas-Colell proved also an existence equilibrium theorem. In this paper I offer a more general fixed point theorem that embodies as special cases the above mentionaed results. The new fixed point theorem is used as a main result in obtaining a general equilibrium theorem in weaker conditions than those of the previous proofs.

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.

Generalized iteration methods for bounds of the solution of fixed point operator-equations

Some general fixed point theorems are obtained describing methods for preservation of a relation between fixed points or between fixed points and iteratives of related iteration operators. If the relation is especially an order relation or an inclusion relation some useful iteration methods to compute bounds for fixed points are stated. Applications and some practical results are discussed.

Fixed Point Theorems for Mappings Satisfying Inwardness Conditions

Let X be a normed linear space and let K be a convex subset of X. The inward set, ${I_K}(x)$, of x relative to K is defined as follows: ${I_K}(x) = \{ x + c(u - x):c \geqslant 1,u \in K\}$. A mapping $T:K \to X$ is said to be inward if $Tx \in {I_K}(x)$ for each $x \in K$, and weakly inward if Tx belongs to the closure of ${I_K}(x)$ for each $x \in K$. In this paper a characterization of weakly inward mappings is given in terms of a condition arising in the study of ordinary differential equations. A general fixed point theorem is proved and applied to derive a generalization of the Contraction Mapping Principle in a complete metric space, and then applied together with the characterization of weakly inward mappings to obtain some fixed point theorems in Banach spaces.

Fixed point theorems for mappings satisfying inwardness conditions

Let X be a normed linear space and let K be a convex subset of X. The inward set, I K ( x ) {I_K}(x) , of x relative to K is defined as follows: I K ( x ) = { x + c ( u − x ) : c ⩾ 1 , u ∈ K } {I_K}(x) = \{ x + c(u - x):c \geqslant 1,u \in K\} . A mapping T : K → X T:K \to X is said to be inward if T x ∈ I K ( x ) Tx \in {I_K}(x) for each x ∈ K x \in K , and weakly inward if Tx belongs to the closure of I K ( x ) {I_K}(x) for each x ∈ K x \in K . In this paper a characterization of weakly inward mappings is given in terms of a condition arising in the study of ordinary differential equations. A general fixed point theorem is proved and applied to derive a generalization of the Contraction Mapping Principle in a complete metric space, and then applied together with the characterization of weakly inward mappings to obtain some fixed point theorems in Banach spaces.

On Dugundji’s notion of positive definiteness

Dugundji’s notion of positive definiteness is generalized to nonnegative real-valued functions on a uniform space. Its relations with completeness and various notions of compactness are investigated. For an arbitrary uniform space X X , there may be lack of the right kind of lower semicontinuous real-valued functions on X X and so a further generalization of Dugundji’s notion of positive definiteness is needed for the development of the fixed point (or coincidence) theory. 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 S. Banach, F.E. Browder, D. W. Boyd and J. S. W. Wong, M. Edelstein and R. Kannan.

Fixed point theorems for various classes of 1-set-contractive and 1-ball-contractive mappings in Banach spaces

Let X be a real Banach space, D a bounded open subset of X, and D ¯ \bar D the closure of D. In §1 of this paper we establish a general fixed point theorem (see Theorem 1 below) for 1-set-contractions and 1-ball-contractions T : D ¯ → X T:\bar D \to X under very mild conditions on T. In addition to classical fixed point theorems of Schauder, Leray and Schauder, Rothe, Kransnoselsky, Altman, and others for T compact, Theorem 1 includes as special cases the earlier theorem of Darbo as well as the more recent theorems of Sadovsky, Nussbaum, Petryshyn, and others (see §1 for further contributions and details) for T k-set-contractive with k > 1 k > 1 , condensing, and 1-set-contractive. In §§2, 3, 4, and 5 of this paper Theorem 1 is used to deduce a number of known, as well as some new, fixed point theorems for various special classes of mappings (e.g. mappings of contractive type with compact or completely continuous perturbations, mappings of semicontractive type introduced by Browder, mappings of pseudo-contractive type, etc.) which have been recently extensively studied by a number of authors and, in particular, by Browder, Krasnoselsky, Kirk, and others (see §1 for details),

A general fixed point theorem

A general fixed point theorem

Some Remarks on Transformations in Metric Spaces

Recently A. Haimovici [1] has proved a general fixed point theorem of transformations in metric spaces from which he obtained existence theorems for certain types of ordinary and partial differential equations. However, both the result and the proof are given for a rather special case. One of the purposes of this present note is to put his result on a more concrete basis and give a stronger characterization of the kind of transformations used in [l]. (Theorem 3).