In this paper, we introduce a new class of expansive mappings called generalized (ξ , α)-expansive mappings and investigate the existence of a fixed point for the mappings in this class. We conclude that several fixed-point theorems can be considered as a consequence of main results. Moreover, some examples are given to illustrate the usability of the obtained results. MSC: 46T99; 54H25; 47H10; 54E50 Keywords: expansive mapping; complete metric space; fixed point IntroductionFixed-point theory has attracted many mathematicians since it provides a simple proof for the existence and uniqueness of the solutions to various mathematical models (integral and partial differential equations, variational inequalities etc.). After the celebrated results of Banach [], fixed-point theory became one of the most interesting topics in nonlinear analysis. Consequently, a number of the papers have appeared since then; see e.g. +∞ n= ψ n (t) < +∞ for each t > , where ψ n is the nth iterate of ψ .(ii) ψ is non-decreasing. for all x, y ∈ X.

The aim of our paper is to present a new class of functions and to define some new contractive mappings in b-metric spaces. We establish some fixed point results for these new contractive mappings in b-metric spaces. Furthermore, we extend our main result in the framework of b-metric-like spaces. Some consequences of main results are also deduced. We present some examples to illustrate and support our results. We provide an application to solve simultaneous linear equations. In addition, we present some open problems.

A new, simple and unified approach in the theory of contractive mappings was recently given by Samet et al. (Nonlinear Anal. 75, 2012, 2154-2165 by using the concepts of α-ψ-contractive type mappings and α-admissible mappings in metric spaces. The purpose of this paper is to present a new class of contractive pair of mappings called generalized α-ψ contractive pair of mappings and study various fixed point theorems for such mappings in complete metric spaces. For this, we introduce a new notion of α-admissible w.r.t g mapping which in turn generalizes the concept of g-monotone mapping recently introduced byĆirić et al. (Fixed Point Theory Appl. 2008, Article ID 131294, 11 pages). As an application of our main results, we further establish common fixed point theorems for metric spaces endowed with a partial order as well as in respect of cyclic contractive mappings. The presented theorems extend and subsumes various known comparable results from the current literature. Some illustrative examples are provided to demonstrate the main results and to show the genuineness of our results.

No abstract

Abstract. We introduce here new modified cosine and sine sums as a 0 2 + and study their integrability and L 1 -convergence. The L 1 -convergence of cosine and sine series have been obtained as corollary. In this paper, we have been able to remove the necessary and sufficient condition a k log k = o(1) as k → ∞ for the L 1 -convergence of cosine and sine series.