Many authors used the concept of $\mathcal{F}-$contraction introduced by Wardowski in 2012 in order to define and prove new results on fixed points in complete metric spaces. In some later papers (for example Proinov P.D., J. Fixed Point Theory Appl. (2020)22:21, doi:10.1007/s11784-020-0756-1) it is shown that conditions (F2) and (F3) are not necessary to prove Wardowski’s results. In this article we use a new approach in proving that the Picard–Jungck sequence is a Cauchy one. It helps us obtain new Jungck–Fisher–Wardowski type results using Wardowski’s condition (F1) only, but in a way that differs from the previous approaches. Along with that, we came to several new contractive conditions not known in the fixed point theory so far. With the new results presented in the article, we generalize, extend, unify and enrich methods presented in the literature that we cite.

Zero-and-one inflated count time series have only recently become the subject of more extensive interest and research. One of the possible approaches is represented by first-order, non-negative, integer-valued autoregressive processes with zero-and-one inflated innovations, abbr. ZOINAR(1) processes, introduced recently, around the year 2020 to the present. This manuscript presents a generalization of ZOINAR processes, given by introducing the zero-and-one inflated power series (ZOIPS) distributions. Thus, the obtained process, named the ZOIPS-INAR(1) process, has been investigated in terms of its basic stochastic properties (e.g., moments, correlation structure and distributional properties). To estimate the parameters of the ZOIPS-INAR(1) model, in addition to the conditional least-squares (CLS) method, a recent estimation technique based on probability-generating functions (PGFs) is discussed. The asymptotic properties of the obtained estimators are also examined, as well as their Monte Carlo simulation study. Finally, as an application of the ZOIPS-INAR(1) model, a dynamic analysis of the number of deaths from the disease COVID-19 in Serbia is considered.

This manuscript deals with a novel, nonlinear, and non-stationary stochastic model with symmetric, Laplacian distributed innovations. The obtained model, named Laplacian Split-BREAK (LSB) process, is intended for dynamic analysis of time series with pronounced and permanent fluctuations. By using the method of characteristic functions (CFs), the basic stochastic properties of the LSB process are proven, with a special emphasis on its asymptotic behaviour. The different procedures for estimating its parameters are also given, along with numerical simulations of the obtained estimators. Finally, it has been shown that the LSB process, as an adequate stochastic model, can be applied in the analysis of dynamics in the world market of crude oil and natural gas.

One of the main goals of this paper is to obtain new contractive conditions using the method of a strictly increasing mapping $\mathbf{F}:(0,+\infty )\to (-\infty ,+\infty )$. According to the recently obtained results, this was possible (Wardowski’s method) only if two more properties $\left(F_2\right)$ and $\left(F_3\right)$ were used instead of the aforementioned strictly increasing $\left(F_1\right)$. Using only the fact that the function $\mathbf{F}$ is strictly increasing, we came to new families of contractive conditions that have not been found in the existing literature so far. Assuming that $\alpha (\mathfrak{u},\mathfrak{v})=1$ for every $\mathfrak{u}$ and $\mathfrak{v}$ from metric space $\Xi$, we obtain some contractive conditions that can be found in the research of Rhoades (Trans. Amer. Math. Soc. 1977, 222) and Collaco and Silva (Nonlinear Anal. TMA 1997). Results of the paper significantly improve, complement, unify, generalize and enrich several results known in the current literature. In addition, we give examples with results in line with the ones we obtained.