In the paper, we introduce a new class of noncyclic -contractions as a generalization of the class of noncyclic contractions which was first introduced in the paper [R. Esp´ınola, M. Gabeleh, On the structure of minimal sets of relatively nonexpansive mappings, Numerical Functional Analysis and Optimization 34 (8), 845-860, 2013] and study the existence, uniqueness and convergence of a fixed point for such class of noncyclic mapping in the framework of uniformly convex Banach spaces. We obtain existence results of the best proximity points for cyclic -contractions as a consequence of our main theorems.

Let $ \mathcal{A} $ be a Banach algebra, $ x,y\in  \mathcal{A}  $ and $ \varphi,\psi: \mathcal{A} \rightarrow  \mathcal{A}  $ be two additive mappings. In this paper, we solve    \begin{align*}        \begin{cases}            &2\varphi(x+y)=\psi(x-y),\\            &\psi(x+y)=2\varphi(y-x)+4\varphi(x).        \end{cases}    \end{align*}Also, we investigate the Hyers-Ulam stability of the system of additive functional equations in complex Banach spaces. Furthermore, we prove the Hyers-Ulam stability by fixed point method in Banach algebras.

This article consists of a new concept of generalized metric space, called $\phi$-metric space which is developed by making a suitable modification in the `triangle inequality. The notion of $\phi$-metric generalizes the concept of some existing metrizable generalized spaces like S-metric, b-metric, etc.  It is shown that one can easily construct a $\phi$-metric from those generalized metric functions and the notion of convergence of a sequence on those generalized metric spaces are identical with the respective induced $\phi$-metric spaces. Moreover, $\phi$-metric space is metrizable and its properties are pretty similar to the metric functions. So $\phi$-metric functions substantially may play the role of metric functions. Also, the structure of $\phi$-metric space is studied and some well-known fixed point theorems are established.

This paper introduces a new symmetric single-phase 13-Level Flying Capacitor Inverter (13-LFCI) structure, using a two dc sources and three capacitors. The proposed single-phase 13-level inverter has the ability to increase the output voltage with fewer semiconductor components compared to the state-of-the-art structures. The Phase Disposition Sinusoidal Pulse Width Modulation (PDSPWM) method is utilized to produce switching pulses. Using this modulation scheme facilitate the 13-LFCI with self-balancing capacitors' voltage capability. The optimal capacitors are designed for minimum voltage drop in the different loads. The 13-LFCI is also capable of transferring reactive power through R-L loads without any limitations. Furthermore, the inverter can properly generate 5-level, 7-Level, 9-level, 11-level, and 13-level output voltage for different applications with change of modulation index. Moreover, a comparison with state-of-the-art 13-level inverters is provided in terms of the number of active and passive components, boosting ability, reactive power support and voltage conversion ratio in order to show the advantages of the proposed structure. In addition, the theoretically losses are calculated to show the efficiency of the proposed topology in various powers. The performance of the converter is illustrated through different operating conditions simulations of resistive and resistive-inductive loads. Eventually, to confirm different aspects and implementation of the proposed structure, the proposed 13-LFCI has been approved in MATLAB/SIMULINK software, and the achieved simulation results are considered utilizing a laboratory prototype.

In this paper, by providing an example, we show that the condition which produced by Radenovi\'{c} and Kadelburg in [Generalized weak contractions in partially ordered metric spaces, Comput. Math. Appl. 60 (2010) pp. 1776-1783] is not sufficient for the uniqueness of the fixed point. Furthermore, a new sufficient condition is introduced for the uniqueness of the fixed point. Some suitable examples are furnished to demonstrate the validity of the hypotheses of our results.

In this paper we have developed C∗-class function and introduced (Φ-C∗)-contractions of Hardy-Rogers type on C ∗-algebra valued metric spaces. We have also established some unique common fixed point results for six maps in C ∗-algebra valued metric spaces using this type contractions. Some basic definitions, properties and lemmas are also discussed in the introduction and preliminaries parts. Some corollaries and examples are also given on the basis of the results.

In this paper, we choose an approach to generalize the idea  of usual norm keeping up  analogous structure and introduce a  generalized norm, called  $ \Phi $-norm using a  function $ \Phi $ which satisfies some special condition. We show that the induced metric of it is $ \phi $-metric. We also study some basic properties and obtain some fundamental results in such  spaces.