Some new results and generalizations in metric fixed point theory

The main purpose of this paper is the study of the generalization of some results given in [M. Berinde, V. Berinde, On a general class of multi-valued weakly Picard mappings, J. Math. Anal. Appl. 326 (2007) 772-782] and references therein. Some generalizations of the Mizoguchi-Takahashi fixed point theorem, Kannan’s fixed point theorems and Chatterjea’s fixed point theorems are established by using our new fixed point theorems.     

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.     

A GENERALIZATION OF CLARKE＇S FIXED POINT THEOREM

This paper proposes a formally stronger set-valued Claxke‘s fixed point theorem. By this-theorem we can improve a fixed point theorem for weakly inward contraction set-valued mapping of D. Dowing and W.A. Kirk.     

Common fixed point theorems under strict contractive conditions in Menger spaces

The main purpose of this paper is to establish some common fixed point theorems under strict contractive conditions for mappings satisfying the property (E.A) in Menger probabilistic metric spaces. As applications, we obtain the corresponding common fixed point theorems under strict contractive in metric spaces.     

Fixed points of products and the strong fixed point property

This problem motivates the present work: If ordered sets X and Y both have the fixed point property for order preserving maps has their product as well? Here we present a related condition — the so-called strong fixed point property — which arises from naive attempts to solve the problem. We are concerned with determining the nature and extent of this property. Several questions are raised concerning its relation to the fixed point property and other conditions such as dismantlability and contractibility.     

Note on “Coupled fixed point theorems for contractions in fuzzy metric spaces”

In the paper “Coupled fixed point theorems for contractions in fuzzy metric spaces” by Sedghi et al. [S. Sedghi, I. Altun, N. Shobec, Coupled fixed point theorems for contractions in fuzzy metric spaces, Nonlinear Analysis 72 (2010) 1298-1304], a coupled common fixed point result was presented. However, our purpose is to show that this result and its proof are false. We give a counterexample and also explain how to correct this result. As a modification, we state and prove a coupled fixed point theorem under some hypotheses of fuzzy metric and t-norm.     

Convergence theorems for inertial KM-type algorithms

This paper deals with the convergence analysis of a general fixed point method which unifies KM-type (Krasnoselskii–Mann) iteration and inertial type extrapolation. This strategy is intended to speed up the convergence of algorithms in signal processing and image reconstruction that can be formulated as KM iterations. The convergence theorems established in this new setting improve known ones and some applications are given regarding convex feasibility problems, subgradient methods, fixed point problems and monotone inclusions.     

Some generalizations of Caristi’s fixed point theorem with applications to the fixed point theory of weakly contractive set-valued maps and the minimization problem

In this paper, we first prove some generalizations of Caristi’s fixed point theorem. Then we give some applications to the fixed point theory of weakly contractive set-valued maps and the minimization problem.     

A note on cone metric fixed point theory and its equivalence

The main aim of this paper is to investigate the equivalence of vectorial versions of fixed point theorems in generalized cone metric spaces and scalar versions of fixed point theorems in (general) metric spaces (in usual sense). We show that the Banach contraction principles in general metric spaces and in TVS-cone metric spaces are equivalent. Our theorems also extend some results in Huang and Zhang (2007) [L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468-1476], Rezapour and Hamlbarani (2008) [Sh. Rezapour, R. Hamlbarani, Some notes on the paper Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 345 (2008) 719-724] and others.     

Fixed point results for set-valued contractions by altering distances in complete metric spaces

Nadler’s contraction principle has led to fixed point theory of set-valued contraction in non-linear analysis. Inspired by the results of Nadler, the fixed point theory of set-valued contraction has been further developed in different directions by many authors, in particular, by Reich, Mizoguchi–Takahashi, Feng–Liu and many others. In the present paper, the concept of generalized contractions for set-valued maps in metric spaces is introduced and the existence of fixed point for such a contraction are guaranteed by certain conditions. Our first result extends and generalizes the Nadler, Feng–Liu and Klim–Wardowski theorems and the second result is different from the Reich and Mizoguchi–Takahashi results. As a consequence, we derive some results related to fixed point of set-valued maps satisfying certain conditions of integral type.     

Comparability invariance of the fixed point property

We show that the fixed point property is comparability invariant for finite ordered sets; that is, if P and Q are finite ordered sets with isomorphic comparability graphs, then P has the fixed point property if and only if Q does. In the process we give a characterization of comparability invariants which can also be used to give shorter proofs of some known results.     

Suzuki?s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces

Recently, Suzuki [T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc. 136 (2008) 1861-1869] proved a fixed point theorem that is a generalization of the Banach contraction principle and characterizes the metric completeness. In this paper we prove an analogous fixed point result for a self-mapping on a partial metric space or on a partially ordered metric space. Our results on partially ordered metric spaces generalize and extend some recent results of Ran and Reurings [A.C.M. Ran, M.C. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435-1443], Nieto and Rodríguez-López [J.J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223-239]. We deduce, also, common fixed point results for two self-mappings. Moreover, using our results, we obtain a characterization of partial metric 0-completeness in terms of fixed point theory. This result extends Suzuki?s characterization of metric completeness.     

Two abstract approaches in vectorial fixed point theory

In this paper a fixed point theory is established for operators defined on Cartesian product spaces. Two abstract approaches are presented in terms of closure operators and of general functionals called measures of deviations from zero resembling the measures of noncompactness. In particular, we give vectorial versions to Mönch’s fixed point theorems. An application is included to illustrate the theory.     

Fixed point property for nonexpansive mappings versus that for nonexpansive semigroups

The main result of this paper is that a closed convex subset of a Banach space has the fixed point property for nonexpansive mappings if and only if it has the fixed point property for nonexpansive semigroups.     

Generalized Levinson-Durbin sequences, binomial coefficients and autoregressive estimation

For a discrete time second-order stationary process, the Levinson-Durbin recursion is used to determine the coefficients of the best linear predictor of the observation at time k+1, given k previous observations, best in the sense of minimizing the mean square error. The coefficients determined by the recursion define a Levinson-Durbin sequence. We also define a generalized Levinson-Durbin sequence and note that binomial coefficients form a special case of a generalized Levinson-Durbin sequence. All generalized Levinson-Durbin sequences are shown to obey summation formulas which generalize formulas satisfied by binomial coefficients. Levinson-Durbin sequences arise in the construction of several autoregressive model coefficient estimators. The least squares autoregressive estimator does not give rise to a Levinson-Durbin sequence, but least squares fixed point processes, which yield least squares estimates of the coefficients unbiased to order 1/T, where T is the sample length, can be combined to construct a Levinson-Durbin sequence. By contrast, analogous fixed point processes arising from the Yule-Walker estimator do not combine to construct a Levinson-Durbin sequence, although the Yule-Walker estimator itself does determine a Levinson-Durbin sequence. The least squares and Yule-Walker fixed point processes are further studied when the mean of the process is a polynomial time trend that is estimated by least squares.     

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 weakly contractive mappings

We prove a fixed point theorem for a generalized weakly contractive mapping and a fixed point theorem for a pair of weakly contractive mappings. We also show that these mappings satisfy properties P and Q.     

Some notes on Bauschke’s condition

In this paper, we discuss Bauschke’s condition which requires that the fixed point set of the composition of nonexpansive mappings is equal to the intersection of the individual fixed point sets. A sufficient condition for Bauschke’s condition is provided. We also present an example rooted in the theory of one-parameter nonexpansive semigroups.     

Fixed point theorems for mixed monotone operators and applications to integral equations

The purpose of this paper is to present some coupled fixed point theorems for a mixed monotone operator in a complete metric space endowed with a partial order by using altering distance functions. We also present an application to integral equations.     

Extrapolation methods for some singular fixed point sequences

A fixed point sequence is singular if the Jacobian matrix at the limit has 1 as an eigenvalue. The asymptotic behaviour of some singular fixed point sequences in one dimension are extended toN dimensions. Three algorithms extrapolating singular fixed point sequences inN dimensions are given. Using numerical examples three algorithms are tested and compared.