A Suzuki-type fixed point theorem for nonlinear contractions

We introduce the notion of admissible functions and show that the family of L-functions introduced by Lim in [Nonlinear Anal. 46(2001), 113--120] and the family of test functions introduced by Geraghty in [Proc. Amer. Math. Soc., 40(1973), 604--608] are admissible. Then we prove that if $\phi$ is an admissible function, $(X,d)$ is a complete metric space, and $T$ is a mapping on $X$ such that, for $\alpha(s)=\phi(s)/s$, the condition $1/(1+\alpha(d(x,Tx))) d(x,Tx) < d(x,y)$ implies $d(Tx,Ty) < \phi(d(x,y))$, for all $x,y\in X$, then $T$ has a unique fixed point. We also show that our fixed point theorem characterizes the metric completeness of $X$

Using an implicit function to prove common fixed point theorems

In this paper, we prove common fixed point results for a self-mappings satisfying an implicit function which is general enough to cover a multitude of known as well as unknown contractions. Our results modify, unify, extend and generalize many relevant results of the existing literature. Interestingly, unlike several other cases, our main results deduce a nonlinear order-theoretic version of a well-known fixed point theorem (proved for quasi-contraction) due to \'{C}iri\'{c} (Proc. Amer. Math. Soc. (54) 267-273, 1974). Finally, in the setting of metric spaces, we drive a sharpened version of Theorem 1 due to Berinde and Vetro (Fixed Point Theory Appl. 2012:105).Comment: 17 pages, Communicate

Fixed points and completeness in metric and in generalized metric spaces

The famous Banach Contraction Principle holds in complete metric spaces, but completeness is not a necessary condition -- there are incomplete metric spaces on which every contraction has a fixed point. The aim of this paper is to present various circumstances in which fixed point results imply completeness. For metric spaces this is the case of Ekeland variational principle and of its equivalent - Caristi fixed point theorem. Other fixed point results having this property will be also presented in metric spaces, in quasi-metric spaces and in partial metric spaces. A discussion on topology and order and on fixed points in ordered structures and their completeness properties is included as well.Comment: 89 pages, Additions in v5: paper reorganized, new subsections: Takahashi min. princ, strong EkVP, Aryutunov princ, weak sharp minima, Bao-Cobzas-Soubeyran. Published in Fundamental'naya i Prikladnaya Matematica vol. 22 (2018), no. 1, 127--215 (in Russian

Fixed point on partial metric type spaces

In this paper, we study some new fixed point results for self maps defined on partial metric type spaces. In particular, we give common fixed point theorems in the same setting. Some examples are given which illustrate the results.Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:1803.1151

Common fixed points for C∗C^{*}-algebra-valued modular metric spaces via C∗C_{*}-class functions with application

Based on the concept and properties of $C^{*}$-algebras, the paper introduces a concept of $C_{*}$-class functions. Then by using these functions in $C^{*}$-algebra- valued modular metric spaces of moeini et al. [14], some common fixed point theorems for self-mappings are established. Also, to support of our results an application is provided for existence and uniqueness of solution for a system of integral equations.Comment: 18 page

B-metric spaces, fixed points and Lipschitz functions

The paper is concerned with b-metric and generalized b-metric spaces. One proves the existence of the completion of a generalized b-metric space and some fixed point results. The behavior of Lipschitz functions on b-metric spaces of homogeneous type, as well as of Lipschitz functions defined on, or with values in quasi-Banach spaces, is studied.Comment: 32 pages; added some results of Aimar on balls in b-metric spaces A new proof of Czerwik result is included. The paper is revise

A ϕ\phi - contraction Principle in Partial Metric Spaces with Self-distance Terms

We prove a generalized contraction principle with control function in complete partial metric spaces. The contractive type condition used allows the appearance of self distance terms. The obtained result generalizes some previously obtained results such as the very recent " D. Ili\'{c}, V. Pavlovi\'{c} and V. Rako\u{c}evi\'{c}, Some new extensions of Banach's contraction principle to partial metric spaces, Appl. Math. Lett. 24 (2011), 1326--1330". An example is given to illustrate the generalization and its properness. Our presented example does not verify the contractive type conditions of the main results proved recently by S. Romaguera in " Fixed point theorems for generalized contractions on partial metric spaces, Topology Appl. 159 (2012), 194-199" and by I. Altun, F. Sola and H. Simsek in "Generalized contractions on partial metric spaces, Topology and Its Applications 157 (18) (2010), 2778--2785". Therefore, our results have an advantage over the previously obtained

On the Existence of Fixed Points of Contraction Mappings Depending of Two Functions on Cone Metric Spaces

In this paper, we study the existence of fixed points for mappings defined on complete, (sequentially compact) cone metric spaces, satisfying a general contractive inequality depending of two additional mappings.Comment: 9 pages, submitte

A new survey: Cone metric spaces

The purpose of this new survey paper is, among other things, to collect in one place most of the articles on cone (abstract, K-metric) spaces, published after 2007. This list can be useful to young researchers trying to work in this part of functional and nonlinear analysis. On the other hand, the existing review papers on cone metric spaces are updated. The main contribution is the observation that it is usually redundant to treat the case when the underlying cone is solid and non-normal. Namely, using simple properties of cones and Minkowski functionals, it is shown that the problems can be usually reduced to the case when the cone is normal, even with the respective norm being monotone. Thus, we offer a synthesis of the respective fixed point problems arriving at the conclusion that they can be reduced to their standard metric counterparts. However, this does not mean that the whole theory of cone metric spaces is redundant, since some of the problems remain which cannot be treated in this way, which is also shown in the present article.Comment: 27 page

Fixed point theorems of soft contractive mappings

The first aim of this paper is to examine some important properties of soft metric spaces. Second is to introduce soft continuous mappings and investigate properties of soft continuous mappings. Third is to prove some fixed point theorems of soft contractive mappings on soft metric spaces.Comment: arXiv admin note: text overlap with arXiv:1308.3390; and text overlap with arXiv:1305.4545 by other author