Some common fixed point theorems for Banach operator pairs with applications in best approximation

The ordered pair (T,I) of two self-maps of a metric space (X,d) is called a Banach operator pair if the set F(I) of fixed points of I is T-invariant i.e. T(F(I))⊆F(I). Some common fixed point theorems for a Banach operator pair and the existence of common fixed points of best approximation are presented in this paper. The results prove, generalize and extend some results of Al-Thagafi [M.A. Al-Thagafi, Common fixed points and best approximation, J. Approx. Theory 85 (1996) 318-323], Carbone [A. Carbone, Applications of fixed point theorems, Jnanabha 19 (1989) 149-155], Chen and Li [J. Chen, Z. Li, Common fixed points for Banach operator pairs in best approximations, J. Math. Anal. Appl. 336 (2007) 1466-1475], Habiniak [L. Habiniak, Fixed point theorems and invariant approximation, J. Approx. Theory 56 (1989) 241-244], Jungck and Sessa [G. Jungck, S. Sessa, Fixed point theorems in best approximation theory, Math. Japon. 42 (1995) 249-252], Sahab, Khan and Sessa [S.A. Sahab, M.S. Khan, S. Sessa, A result in best approximation theory, J. Approx. Theory 55 (1988) 349-351], Shahzad [N. Shahzad, Invariant approximations and R-subweakly commuting maps, J. Math. Anal. Appl. 257 (2001) 39-45] and of few others.     

Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces

Let X be a non-empty set and F:X×X→X be a given mapping. An element (x,y)∈X×X is said to be a coupled fixed point of the mapping F if F(x,y)=x and F(y,x)=y. In this paper, we consider the case when X is a complete metric space endowed with a partial order. We define generalized Meir-Keeler type functions and we prove some coupled fixed point theorems under a generalized Meir-Keeler contractive condition. Some applications of our obtained results are given. The presented theorems extend and complement the recent fixed point theorems due to Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379-1393].     

On the super fixed point property in product spaces

We prove that if F is a finite-dimensional Banach space and X has the super fixed point property for nonexpansive mappings, then F⊕X has the super fixed point property with respect to a large class of norms including all lp norms, 1?p<∞. This provides a solution to the “super-version” of the problem of Khamsi (1989).     

Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces

Let (X,?) be a partially ordered set and d be a complete metric on X. Let F,G be two set-valued mappings on X. We obtained sufficient conditions for the existence of common fixed point of F and G satisfying an implicit relation in partially ordered set X.     

Coefficient bounds for biholomorphic mappings which have a parametric representation

The main purpose of this paper is to prove a collection of new fixed point theorems for so-called weakly F-contractive mappings. By analogy, we introduce also a class of strongly F-expansive mappings and we prove fixed point theorems for such mappings. We provide a few examples, which illustrate these results and, as an application, we prove an existence and uniqueness theorem for the generalized Fredholm integral equation of the second kind. Finally, in Appendix A, we apply the Mönch fixed point theorem to prove two results on the existence of approximate fixed points of some continuous mappings.     

Elliptic problems with lack of compactness via a new fixed point theorem

This paper provides a new fixed point theorem for increasing self-mappings G:B→B of a closed ball B⊂X, where X is a Banach semilattice which is reflexive or has a weakly fully regular order cone X₊. By means of this fixed point theorem, we are able to establish existence results of elliptic problems with lack of compactness.     

On coincidence point and fixed point theorems for nonlinear multivalued maps

Several characterizations of MT-functions are first given in this paper. Applying the characterizations of MT-functions, we establish some existence theorems for coincidence point and fixed point in complete metric spaces. From these results, we can obtain new generalizations of Berinde-Berinde?s fixed point theorem and Mizoguchi-Takahashi?s fixed point theorem for nonlinear multivalued contractive maps. Our results generalize and improve some main results in the literature.     

Fixed point theorems for generalized set-contraction maps and their applications

A new generalized set-valued contraction on topological spaces with respect to a measure of noncompactness is introduced. Two fixed point theorems for the KKM type maps which are either generalized set-contraction or condensing ones are given. Furthermore, applications of these results for existence of coincidence points and maximal elements are deduced.     

Fixed point property on symmetric products

For a metric continuum X, let Fn(X)={A⊂X:A is nonempty and has at most n points}. In this paper we show a continuum X such that F₂(X) has the fixed point property while X does not have it.     

Convergence of a hybrid algorithm for a reversible semigroup of nonlinear operators in Banach spaces

The purpose of this paper is to study hybrid iterative schemes of Halpern types for a semigroup ℑ={T(s):s∈S} of relatively nonexpansive mappings on a closed and convex subset C of a Banach space with respect to a sequence {μ_n} of asymptotically left invariant means defined on an appropriate invariant subspace of l^∞(S). We prove that given a certain sequence {α_n} in [0,1], x∈C, we can generate an iterative sequence {x_n} which converges strongly to Π_F(ℑ)x where Π_F(ℑ)x is the generalized projection from C onto the fixed point set F(ℑ). Our main result is even new for the case of a Hilbert space.     

F-quadratic stochastic operators

In this paper, we introduce the notion of an F-quadratic stochastic operator. It is shown that each F-quadratic operator has a unique fixed point. Besides, it is proved that any trajectory of an F-quadratic stochastic operator exponentially rapidly converges to this fixed point.     

Remarks on Caristi’s fixed point theorem and Kirk’s problem

Without assumptions on the continuity and the subadditivity of η, by means of Caristi’s fixed point theorem, we investigated the existence of fixed points for a Caristi type mapping which partially answered Kirk’s problem and improved Caristi’s fixed point theorem, Jachymski’s fixed point theorem and Khamsi’s fixed point theorem since φ is not necessarily assumed to be bounded below on X.     

Topological fixed points in Boolean networks

We introduce the notion of a topological fixed point in Boolean Networks: a fixed point of Boolean network F is said to be topologic if it is a fixed point of every Boolean network with the same interaction graph as the one of F. Then, we characterize the number of topological fixed points of a Boolean network according to the structure of its interaction graph.     

Fixed point results for multimaps in CAT(0) spaces

Common fixed point results for families of single-valued nonexpansive or quasi-nonexpansive mappings and multivalued upper semicontinuous, almost lower semicontinuous or nonexpansive mappings are proved either in CAT(0) spaces or R-trees. It is also shown that the fixed point set of quasi-nonexpansive self-mapping of a nonempty closed convex subset of a CAT(0) space is always nonempty closed and convex.     

Towards Lim

The paper contains an elegant extension of the Nadler fixed point theorem for multivalued contractions (see Theorem 21). It is based on a new idea of the α-step mappings (see Definition 17) being more efficient than α-contractions. In the present paper this theorem is a tool in proving some fixed point theorems for “nonexpansive” mappings in the bead spaces (metric spaces that, roughly speaking, are modelled after convex sets in uniformly convex spaces). More precisely the mappings are nonexpansive on a set with respect to only one point - the centre of this set (see condition (4)). The results are pretty general. At first we assume that the value of the mapping under consideration at this central point looks “sharp” (see Definition 6). This idea leads to a group of theorems (based on Theorem 7). Their proofs are compact and the theorems, in particular, are natural extensions of the classical results for (usual) nonexpansive mappings. In the second part we apply the idea of Lim to investigate the regular sequences and here the proofs are based on our extension of Nadler's Theorem. In consequence we obtain some fixed point theorems that generalise the classical Lim Theorem for multivalued nonexpansive mappings (see e.g. Theorem 26).     

A general iterative algorithm for nonexpansive mappings in Hilbert spaces

Let H be a real Hilbert space. Suppose that T is a nonexpansive mapping on H with a fixed point, f is a contraction on H with coefficient 0<α<1, and F:H→H is a k-Lipschitzian and η-strongly monotone operator with k>0,η>0. Let . We proved that the sequence {x_n} generated by the iterative method x_n+1=α_nγf(x_n)+(I−μα_nF)Tx_n converges strongly to a fixed point , which solves the variational inequality , for x∈F_ix(T).     

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 : X³ → X and g : X → X. The obtained results extend recent coincidence theorems due to ?iri? and Lakshmikantham [V. Lakshmikantham, L. ?iri?, L., Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70 (2009) 4341-4349].     

Krasnosel’skii type fixed point theorems under weak topology features

In this paper we prove the following Krasnosel’skii type fixed point theorem: Let M be a nonempty bounded closed convex subset of a Banach space X. Suppose that A:M→X and B:X→X are two weakly sequentially continuous mappings satisfying:

(i)

AM is relatively weakly compact;

(ii)

B is a strict contraction;

(iii)

.

Then A+B has at least one fixed point in M.This result is then used to obtain some new fixed point theorems for the sum of a weakly compact and a nonexpansive mapping. The results presented in this paper encompass several earlier ones in the literature.     

Compositions of analytic functions of the form Fn(z) = Fn−1(fn(z)), fn(z) → f(z)

Frequently, in applications, a function is iterated in order to determine its fixed point, which represents the solution of some problem. In the variation of iteration presented in this paper fixed points serve a different purpose. The sequence {F_n(z)} is studied, where F₁(z) = f₁(z) and F_n(z) = F_n−1(f_n(z)), with f_n → f. Many infinite arithmetic expansions exhibit this form, and the fixed point, α, of f may be used as a modifying factor (z = α) to influence the convergence behaviour of these expansions. Thus one employs, rather than seeks the fixed point of the function f.