A result of suzuki type in partial G-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. Paesano and Vetro [D. Paesano and P. Vetro, Suzuki's type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces, Topology Appl., 159 (2012), 911-920] proved an analogous fixed point result for a self-mapping on a partial metric space that characterizes the partial metric 0-completeness. In this article, we introduce the notion of partial G-metric spaces and prove a result of Suzuki type in the setting of partial G-metric spaces. We deduce also a result of common fixed point.     

A RESULT OF SUZUKI TYPE IN PARTIAL G-METRIC SPACES

Recently, Suzuki [T. Suzuki characterizes metric completeness, Proc. A generalized Banach contractlon principle that Amer. Math. Soc. 136 （2008）, 1861-1869] proved a fixed point theorem that is a generalization of the Banach contraction principle and char- acterizes the metric completeness. Paesano and Vetro [D. Paesano and P. Vetro, Suzuki＇s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces, Topology Appl., 159 （2012）, 911-920] proved an analogous fixed point result for a self-mapping on a partial metric space that characterizes the partial metric O- completeness. In this article, we introduce the notion of partial G-metric spaces and prove a result of Suzuki type in the setting of partial G-metric spaces. We deduce also a result of common fixed point.     

On ?iri? maps with a generalized contractive iterate at a point and Fisher’s quasi-contractions in cone metric spaces

In this paper, we generalize and unify some results of Sehgal and Guseman, and ?iri?’s theorem for mappings with a generalized contractive iterate at a point to cone metric spaces, in which the cone does not need to be normal. As corollaries, we obtain recent results of Huang and Zhang, and Raja and Vaezpour. Furthermore, we introduce the definition of Fisher quasi-contractions on cone metric spaces and study their properties. Among other things, using new method of proof, we solve the open problem for the interval of contractive constant λ of (?iri?) quasi-contraction in non-normal cone metric spaces, and as sn immediate corollary, we recover the recent result of Rezapour and Hamlbarani.     

Best proximity point theorems for cyclic orbital Meir-Keeler contraction maps

We introduce a notion of cyclic orbital Meir-Keeler contraction and give sufficient conditions for the existence of fixed points and best proximity points of such a map. Our main result is a generalization of a best proximity point result due to Di Bari et al. [C. Di Bari, T. Suzuki, C. Vetro, Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Anal. 69 (2008) 3790-3794].     

Some fixed point results for a generalized ψ-weak contraction mappings in orbitally metric spaces

Samet and Vetro [Samet B, Vetro C. Berinde mappings in orbitally complete metric spaces. Chaos Solitons Fract 2011;44:1075–9.] studied a fixed point theorem for a self-mapping satisfying a general contractive condition of integral type in orbitally complete metric spaces. In this paper, we introduce the notion of a generalized ψ-weak contraction mapping and establish some results in orbitally complete metric spaces. Our results generalize several well-known comparable results in the literature. As an application of our results we deduce the result of Samet and Vetro. Some examples are given to illustrate the useability of our results.     

On cone metric spaces: A survey

Using an old M. Krein’s result and a result concerning symmetric spaces from [S. Radenovi?, Z. Kadelburg, Quasi-contractions on symmetric and cone symmetric spaces, Banach J. Math. Anal. 5 (1) (2011), 38-50], we show in a very short way that all fixed point results in cone metric spaces obtained recently, in which the assumption that the underlying cone is normal and solid is present, can be reduced to the corresponding results in metric spaces. On the other hand, when we deal with non-normal solid cones, this is not possible. In the recent paper [M.A. Khamsi, Remarks on cone metric spaces and fixed point theorems of contractive mappings, Fixed Point Theory Appl. 2010, 7 pages, Article ID 315398, doi:10.1115/2010/315398] the author claims that most of the cone fixed point results are merely copies of the classical ones and that any extension of known fixed point results to cone metric spaces is redundant; also that underlying Banach space and the associated cone subset are not necessary. In fact, Khamsi’s approach includes a small class of results and is very limited since it requires only normal cones, so that all results with non-normal cones (which are proper extensions of the corresponding results for metric spaces) cannot be dealt with by his approach.     

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 theory for generalized contractions in cone metric spaces

In this paper, we prove some fixed point theorems for generalized contractions in cone metric spaces. Our theorems extend some results of Suzuki (2008) [T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc Amer Math Soc 136(5) (2008), 1861-1869] and Kikkawa and Suzuki (2008) [M. Kikkawa and T. Suzuki, Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal 69(9) (2008), 2942-2949].     

Some fixed point theorems on ordered cone metric spaces

In the present work, two fixed point theorems for self maps on ordered cone metric spaces are proved motivated by [7, L. G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007) 1468–1476] and [15, A. C. M. Ran and M. C. B. Reuring, A fixed point theorem in partially ordered sets and some application to matrix equations, Proc. Amer. Math. Soc., 132, (2004), 1435–1443]      

Coincidence and fixed points in symmetric spaces under strict contractions

Some common fixed point theorems due to Aamri and El Moutawakil [M. Aamri, D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270 (2002) 181-188] and Pant and Pant [R.P. Pant, V. Pant, Common fixed points under strict contractive conditions, J. Math. Anal. Appl. 248 (2000) 327-332] proved for strict contractive mappings in metric spaces are extended to symmetric (semi-metric) spaces under tight conditions. Some related results are derived besides discussing illustrative examples which establish the utility of results proved in this note.     

KKM mappings in cone metric spaces and some fixed point theorems

In this paper, we define KKM mappings in cone metric spaces and define Nℜ-cone metric spaces to obtain some fixed point theorems and hence generalize the results obtained in [A. Amini, M. Fakhar, J. Zafarani, KKM mapping in metric spaces, Nonlinear Anal. 60 (2005) 1045-1052].     

锥b-度量空间上映射的公共不动点定理

本文研究了锥b-度量空间上四个自映射的公共不动点问题.利用序列逼近的方法,获得了锥b-度量空间上四个自映射的一些公共不动点结果,将锥度量空间中的几个相关结果推广到锥b-度量空间中,并且给出了一个例子以支撑我们的结果.     

Common fixed points in cone metric spaces

In this paper we consider a notion of g-weak contractive mappings in the setting of cone metric spaces and we give results of common fixed points. This results generalize some common fixed points results in metric spaces and some of the results of Huang and Zhang in cone metric spaces. Supported by Universitá degli Studi di Palermo, R. S. ex 60%.     

Equivalent conditions for generalized contractions on (ordered) metric spaces

We establish a geometric lemma giving a list of equivalent conditions for some subsets of the plane. As its application, we get that various contractive conditions using the so-called altering distance functions coincide with classical ones. We consider several classes of mappings both on metric spaces and ordered metric spaces. In particular, we show that unexpectedly, some very recent fixed point theorems for generalized contractions on ordered metric spaces obtained by Harjani and Sadarangani [J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal. 72 (2010) 1188-1197], and Amini-Harandi and Emami [A. Amini-Harandi, H. Emami A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal. 72 (2010) 2238-2242] do follow from an earlier result of O’Regan and Petru?el [D. O’Regan and A. Petru?el, Fixed point theorems for generalized contractions in ordered metric spaces, J. Math. Anal. Appl. 341 (2008) 1241-1252].     

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.     

Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces

Investigations concerning the existence of dynamic processes convergent to fixed points of set-valued nonlinear contractions in cone metric spaces are initiated. The conditions guaranteeing the existence and uniqueness of fixed points of such contractions are established. Our theorems generalize recent results obtained by Huang and Zhang [L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive maps, J. Math. Anal. Appl. 332 (2007) 1467–1475] for cone metric spaces and by Klim and Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (1) (2007) 132–139] for metric spaces. The examples and remarks provided show an essential difference between our results and those mentioned above.     

Some common fixed point theorems for a pair of tangential mappings in symmetric spaces

A metrical common fixed point theorem for a pair of self mappings due to Sastry and Murthy (K.P.R. Sastry, I.S.R. Krishna Murthy, A common fixed points of two partially commuting tangential selfmaps on a metric space, J. Math. Anal. Appl. 250 (2000) 731734.) [8] is extended to symmetric spaces which in turn generalises a fixed point theorem due to Pant (R.P. Pant, Common fixed points of Lipschitz type mapping pairs, J. Math. Anal. Appl. 248 (1999) 280283.) [11] besides deriving some related results. Some illustrative examples to highlight the realised improvements are also furnished.     

Meir–Keeler-type conditions in abstract metric spaces

Various Meir–Keeler-type conditions for mappings acting in abstract metric spaces are presented and their connections are discussed. Results about associated symmetric spaces, obtained in [S. Radenovi?, Z. Kadelburg, Quasi-contractions on symmetric and cone symmetric spaces, Banach J. Math. Anal. 5 (2011), 38–50] are used to show that the regularity condition for the underlying cone can be dropped in some fixed point results that have appeared recently.     

Conditions of regularity in cone metric spaces

In this paper we prove some fixed points results on cone metric spaces for maps satisfying general contractive type conditions. Among other things, we extend some results of Nguyen [11] from metric spaces to cone metric spaces. The example is included.     

COMMON FIXED POINTS WITH APPLICATIONS TO BEST SIMULTANEOUS APPROXIMATIONS

For a subset K of a metric space (X , d) and x ∈ X ,P K (x) ={y ∈ K : d(x, y) = d(x, K) ≡ inf { d(x, k) : k ∈ K }}is called the set of best K-approximant to x. An element g。∈ K is said to be a best simultaneous approximation of the pair y 1 , y 2 ∈ X if max{d(y 1 , g。), d(y 2 , g。) } = inf g ∈ K max { d(y 1 , g), d(y 2 , g)}.In this paper, some results on the existence of common fixed points for Banach operator pairs in the framework of convex metric spaces have been proved. For self mappings T and S on K, results are proved on both T - and S- invariant points for a set of best simultaneous approximation. Some results on best K-approximant are also deduced. The results proved generalize and extend some results of I. Beg and M. Abbas [1] , S. Chandok and T.D. Narang [2] , T.D. Narang and S. Chandok [11] , S.A. Sahab, M.S. Khan and S. Sessa [14] , P. Vijayaraju [20] and P. Vijayaraju and M. Marudai [21] .