A new fixed point theorem is obtained for the class of cyclic weak -contractions on partially metric spaces. It is proved that a self-mapping on a complete partial metric space has a fixed point if it satisfies the cyclic weak -contraction principle.

In this manuscript, we discuss the existence of a coupled coincidence point for mappings F : X × X → X and g : X → X, where F has the mixed g-monotone property, in the context of partially ordered metric spaces with an implicit relation. Our main theorem improves and extends various results in the literature. We also state some examples to illustrate our work. MSC: 47H10; 54H25; 46J10; 46J15

In this paper, it is shown that every point in the hyperbolic 3-space is moved at a distance at least 0.5 log 12 · 3 k−1 − 3 by one of the isometries of length at most k ≥ 2 in a 2-generator Klenian group Γ which is torsion-free, not co-compact and contains no parabolic. Also some lower bounds for the maximum of hyperbolic displacements given by symmetric subsets of isometries in purely loxodromic finitely generated free Kleinian groups are conjectured. 54C30,20E05; 26B25,26B35 IntroductionThe following is sequel to Yüce [21] in which the machinery developed by Culler and Shalen [8] that gives a lower bound for the maximum of the displacements under the generators of Γ is extended to calculate a lower bound for the maximum of the displacements under any finite set of isometries in Γ in connection with the solutions of certain minimax problems with a constraint. Here Γ is a Kleinian group generated by two non-commuting isometries ξ and η of H 3 that satisfies the hypothesis of the log 3 Theorem which can be stated as follows:Log 3 Theorem Suppose that Γ is torsion-free, not co-compact and contains no parabolic. Let Γ 1 be the set {ξ, η}. Then we have max γ∈Γ 1 {dist(z 0 , γ ·z 0 )} ≥ 0.5 log 9 for any z 0 ∈ H 3 .The use of this extension for the set of isometries Γ † = {ξ, η, ξη} ⊂ Γ implies, for instance, the fact that max γ∈Γ † {dist(z 0 , γ · z 0 )} ≥ 0.5 log(5 + 3 √ 2) for any z 0 ∈ H 3 [21, Theorem 5.1].Since it has implications on Margulis numbers and volume estimates for a large class of closed hyperbolic 3-manifolds, the log 3 theorem is the main tool or motivation behind many deep results that connect the topology of hyperbolic 3-manifolds to their geometry (see Agol-Culler-Shalen [2], Culler-Hersonski-Shalen [7], Culler-Shalen 2Ílker S. Yüce [8,9,10]). For example, if M is a closed hyperbolic 3-manifold whose first Betti number b 1 (M ) is at least 4 and the fundamental group π 1 (M ) of M has no subgroup isomorphic to the fundamental group of a genus two surface, then a generalization of the log 3 theorem due to Anderson-Canary-Culler-Shalen [3] implies that 0.5 log 5 is a Margulis number for M and, 3.08 is a lower bound for the volume of M . In [8], as well as proving the log 3 Theorem, Culler and Shalen show that 0.5 log 3 is a Margulis number and, 0.92 is a lower bound for the volume of M if b 1 (M ) ≥ 3 and π 1 (M ) has no subgroup of finite index. In [7], Culler, Hersonsky and Shalen increase the previous lower bound for M to 0.94. It must be noted that the lower volume estimates computed in [3] and [8] are improved by the works of Calegari-Meyerhoff-Milley [11] and Milley [15] in which a newer method called Mom technology was introduced.Aiming to set the ground work to investigate the further applications of the methods developed in [2,7,3,8,9,10] to improve on the Margulis numbers and volume estimates for the classes of closed hyperbolic 3-manifolds mentioned in the previous paragraph, in this paper we shall prove the following:Theorem 4.1 If Γ k is the set of isometries of length at most k ≥ 2...

We prove quadruple fixed point theorems in partially ordered metric spaces depending on another function. Also, we state some examples showing that our results are real generalization of known ones in quadruple fixed point theory.

The log 3 theorem, proved by Culler and Shalen, states that every point in the hyperbolic 3-space H 3 is moved a distance at least log 3 by one of the noncommuting isometries or Á of H 3 provided that and Á generate a torsion-free, discrete group which is not cocompact and contains no parabolic. This theorem lies in the foundations of many techniques that provide lower estimates for the volumes of orientable, closed hyperbolic 3-manifolds whose fundamental groups have no 2generator subgroup of finite index and, as a consequence, gives insights into the topological properties of these manifolds. Under the hypotheses of the log 3 theorem, the main result of this paper shows that every point in H 3 is moved a distance at least log p 5 C 3 p 2 by one of the isometries , Á or Á.