We continue our investigation of situations in which the fixed point sets for maps and their iterates are the same. In a previous paper [21] we demonstrated a number of situations in which the fixed point set of a map is the same as the fixed point set of each iterate of the map. We also investigated the same phenomenon for pairs of maps, with respect to common fixed points. In this paper we continue that study. In Section 1 we examine some contractive conditions involving pairs of commuting maps. In section 2 we examine some contractive conditions involving multivalued maps for which the ¿-metric is used. In section 3 we examine some contractive conditions on 2-metric spaces. Let F(T) denote the fixed point set of a map T. As in [21] we shall say that a map T has property P if F(T) = F(Tn) for each n G N. We shall say that a pair of maps S and T have property Q if F(S)nF(T) = F(Sn)DF(Tn) for each n G N. If Ti is a collections of maps, then, if flF(Tj) = nF(Tf ) we shall say that this collection has property R. An important consequence of this study is that none of these maps has any nontrivial periodic points.

More maps for which F(T)=F(Tn)

In this paper, we discuss the unique common fixed point of two pairs of weak commutative mappings on a complete multiplicative metric space. They satisfy the following inequality: d(Sx, Ty) ≤{ max{d(Ax, By), d(Ax, Sx), d(By, Ty), d(Sx, By), d(Ax, Ty)}} λ , where A and S are weak commutative, B and T also are weak commutative. Our results substantially generalize and extend the results of Ozavsar and Cevikel (Fixed point of multiplicative contraction mappings on multiplicative metric space). MSC: Primary 46B20; 47A12

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Common fixed points for weak commutative mappings on a multiplicative metric space

\"{O}zavsar and Cevikel(Fixed point of multiplicative contraction mappings on multiplicative metric space.arXiv:1205.5131v1 [math.GN] 23 may 2012)initiated the concept of the multiplicative metric space in such a way that the usual triangular inequality is replaced by "multiplicative triangle inequality $d(x,y)\leq d(x,z).d(z,y)$ for all $x,y,z\in X$". In this manuscript, we discussed some unique fixed point theorems in the context of multiplicative metric spaces. The established results carry some well known results from the literature to multiplicative metric space. We note that some fixed point theorems can be deduced in multiplicative metric space by using the established results. Appropriate examples are also given.

Some Unique Fixed Point Theorems in Multiplicative Metric Spaces

The purpose of this paper is to prove some common xed point theorems in metric and 2-metric spaces for two pairs of weakly compatible mappings satisfying an implicit relation which unify and generalize most of the existing relevant xed point theorems. As an application of our main result, a Bryant type xed point theorem is also derived besides furnishing two illustra- tive examples.

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S. L. Singh

Papers

Common Fixed Points of Commuting Mappings in 2‐Metric Spaces and an Application