Fixed Point Theorems for Generalized α-β-Weakly Contraction Mappings in Metric Spaces and Applications

Fixed point theorems for nonlinear weakly [formula omitted]-contractive mappings in metric spaces

Fixed point theorems for set-valued contraction mappings in metric spaces

Recently Suzuki (2008) and then Kikkawa and Suzuki (2008) gave a new generalization of the Banach contraction principle. Then, Mot and Petrusel (2009), Dhompapangasa and Yingtaweesttikulue (2009), Bose and Roychowd- hury (2011), Singh and Mishra (2010), and Doric and Lazovic (2011) further extended their work. Recently Bose (2012) obtained some Suzuki-type common fixed point theorems for generalized contractive multivalued mappings using a result of Bose and Mukherjee(1977) which extend the previously obtained re- sults. Recently Damjanovic and Doric(2011) obtained a multivalued generaliza- tion of theorems Kikkawa and Suzuki (2008) concerning Kannan mappings. Also Singh and Mishra (2010) considered coincidence and fixed point theorems for a class of hybrid pair of single-valued and multi-valued maps in metric space setting and Singh et al (2012) prsented a common fixed point theorem for a pair of multi-valued maps in a complete metric space extending a recent theorem of Doric and Lazovic. First we generalize the theorem of Singh et al which also extend the theorem of Singh and Mishra(2010). Then we extend the theorem of Damjanovic and Doric to a common fixed point theorem of a pair of mul- tivalued mappings. As an application, we consider the existence of a common solution for a class of functional equations arising in dynamic programming.

In this article, we investigate the relationship between freely quasiconformal mappings and locally weakly quasisymmetric mappings in quasiconvex and complete metric spaces. We first prove the equivalence of freely quasiconformal mappings, ring properties, and locally weakly quasisymmetric mappings. Finally, we prove that the composition of two locally weakly quasisymmetric mappings in metric spaces is locally weakly quasisymmetric mapping.

The objective of this paper is to present a comparative study of mapping in Metric Space and Controlled Metric Space. The study provides the structure, gap analysis and application of Metric Space and Controlled Metric Space. A Comparative Study of Mappings in Metric Space and in Controlled Metric Space is done with the help of studying the concept of metric space, its various types of mappings. Following this, a further conceptualization of Controlled Metric Space with its various types of mappings and its applications is done. A Controlled Metric Space is a specialized concept in Mathematics that is used to deal with issues related to the "control" of distances and the behavior of sequences or functions within a metric space. And finally we compare various mappings in metric space with various mappings in controlled metric spaces. In this way we find comparative results of various types of mappings. Thus the paper focuses on comparative study between metric spaces along its properties with controlled metric space and its properties.

Some fixed point theorems for (φ, ψ, p)‐contractive maps and (φ, k, p)‐contractive maps on a complete metric space are proved. Presented fixed point theorems generalize many results existing in the literature.

We provide a positive answer to a question raised by Al-Thagafi and Shahzad [Nonlinear Anal. 70 (2009), 3665–3671] about the existence of a best proximity point for a cyclic -contraction map in a reflexive Banach space by using appreciable generalized notions. In turn, this also gives a positive answer to a question raised by Eldred and Veeramani [J. Math. Anal. Appl. 323 (2006), 1001–1006]. To this end, we introduce a new concept of C-proximity point and a new class of maps, called cyclic -contractions, which contains cyclic contraction maps and cyclic -contraction maps as subclasses. Convergence and existence results of best C-proximity points for cyclic -contraction maps are also obtained.

Common fixed point theorems for some new generalized contractive type mappings

In topology, there are known results on preservation of the fixed point existence property under any homotopy of self-mappings on some spaces in the event that the initial mapping has a nonzero Lefschetz number. For contracting mappings of metric spaces and some of their generalizations, there are known results by M. Frigon on preservation of the contraction property and, consequently, the fixed point existence property under some special homotopies. In 1984, J.W.Walker introduced a discrete counterpart of homotopy for self-mappings of an ordered set, which he called an order isotone homotopy. The naturalness of this notion and its relation to the usual continuous homotopy follow from the work by R. E. Stong (1966). Recently, the author and D. A.Podoprikhin have extended Walker’s notion of an order isotone homotopy and suggested sufficient conditions for preservation of the fixed point (coincidence point) existence property under such discrete homotopy (a pair of homotopies) of a mapping (a pair of mappings) of ordered sets. This paper contains metric counterparts of the obtained results and some corollaries. The method of ordering a metric space proposed by A. Brondsted in 1974 is used.

General periodic and fixed point theorems are proved for a class of self maps of a quasi-metric space which satisfy the contractive definition (A) below. Two examples are presented to show that the class of mappings which satisfy (A) is indeed wider than a class of selfmaps which satisfy Caristi's contractive definition (C) below. Also a common fixed point theorem for a pair of maps which satisfy a contractive condition (D) below is established.

<abstract><p>In this paper, a novel implicit IH-multistep fixed point algorithm and convergence result for a general class of contractive maps is introduced without any imposition of the "sum conditions" on the countably finite family of the iteration parameters. Furthermore, it is shown that the convergence of the proposed iteration scheme is equivalent to some other implicit IH-type iterative schemes (e.g., implicit IH-Noor, implicit IH-Ishikawa and implicit IH-Mann) for the same class of maps. Also, some numerical examples are given to illustrate that the equivalence is true. Our results complement, improve and unify several equivalent results recently announced in literature.</p></abstract>

In this paper, we introduce a new class of mappings, called generalized - -contractive mappings, within the framework of complete fuzzy metric spaces. Our approach generalizes existing fixed point results and establishes conditions under which a unique fixed point exists. Some examples are provided to demonstrate the applicability of our theorem. This work contributes to the ongoing development of fixed point theory in fuzzy metric spaces, particularly in the context of non-classical contractions.

We study some minimization problems for noncyclic mappings in metric spaces. We then apply the solution to obtain some results in the theory of analytic functions.

For mappings in metric spaces satisfying one inequality with respect to modulus of families of curves, there is proved a lightness of the uniform limit of these mappings. It is proved that, the uniform limit of these mappings is light mapping, whenever a function which corresponds to distortion of families of curves, is of finite mean oscillation at every point. Besides that, for one class of homeomorphisms of metric spaces, there are obtained theorems about equicontinuity of inverse mappings.

This thesis contains two parts. In the first part, we provide a positive answer to a question raised by Al-Thagafi and Shahzad on the existence of a best proximity point for a cyclic ℓ -contraction map in a reflexive Banach space. Moreover, we extend the Al-Thagafi and Shahzad theorem to metric spaces with property UC. In the second part, we prove that the Picard projection iteration sequence converges to a fixed point and the rate of convergence is given. We also prove the generalize Collage Theorem for a multi-valued weak contraction map. In addition, we extend the cyclic condition to multi-valued maps and study the existence of a fixed point and the existence of a best proximity point for the map having been extended. Moreover, we study data dependence problem for a special class of multi-valued maps.