The famous Newton–Kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of Newton's method to a solution of an equation in connection with the Lipschitz continuity of the Fréchet-derivative of the operator involved. Here using Lipschitz and center-Lipschitz conditions we show that the Newton–Kantorovich hypothesis can be weakened. The error bounds obtained under our semilocal convergence result are more precise than the corresponding ones given by the dominating Newton–Kantorovich theorem.

Pythagorean fuzzy set is a useful expansion of intuitionistic fuzzy set for dealing with ambiguities, which mostly occur in real-life problems. Hamacher t-norm also has important and compatible norms that incorporate a parameter that offers various options to decision-makers during the information fusion process, thereby enhancing their ability to model decision-making problems effectively compared to alternative methods. In this study, Hamacher operators are used to introduce several Pythagorean fuzzy Hamacher interactive weighted averaging (PFHIWA), Pythagorean fuzzy Hamacher interactive ordered weighted averaging (PFHIOWA), Pythagorean fuzzy Hamacher interactive weighted geometric (PFHIWG), and Pythagorean fuzzy Hamacher interactive ordered weighted geometric (PFHIOWG) operators. The properties of these operators are examined in detail. The benefit of using progressive operators is that they deliver more understanding of the scenario to the decision-makers. The proposed operators are utilized to elaborate multi-attribute decision-making (MADM). By showing the sensitivity analysis, our proposed operator has high stability related to multi-attribute decision-making (MADM) under the Pythagorean fuzzy data set.

A semilocal convergence analysis for directional Newton methods in $n$ -variables is provided in this study. Using weaker hypotheses than in the elegant related work by Y. Levin and A. Ben-Israel and introducing the center-Lipschitz condition we provide under the same computational cost as in Levin and Ben-Israel a semilocal convergence analysis with the following advantages: weaker convergence conditions; larger convergence domain; finer error estimates on the distances involved, and an at least as precise information on the location of the zero of the function. A numerical example where our results apply to solve an equation but not the ones in Levin and Ben-Israel is also provided in this study.

A plethora of sufficient convergence criteria has been provided for single-step iterative methods to solve Banach space valued operator equations. However, an interesting question remains unanswered: is it possible to provide unified convergence criteria for single-step iterative methods, which are weaker than earlier ones without additional hypotheses? The answer is yes. In particular, we provide only one sufficient convergence criterion suitable for single-step methods. Moreover, we also give a finer convergence analysis. Numerical experiments involving boundary value problems and Hammerstein-like integral equations complete this paper.

The design of the techniques and algorithms used by the static, dynamic and interactive security testing tools differ. Therefore, each tool detects to a greater or lesser extent each type of vulnerability for which they are designed for. In addition, their different designs mean that they have different percentages of false positives. In order to take advantage of the possible synergies that different analysis tools types may have, this paper combines several static, dynamic and interactive analysis security testing tools—static white box security analysis (SAST), dynamic black box security analysis (DAST) and interactive white box security analysis (IAST), respectively. The aim is to investigate how to improve the effectiveness of security vulnerability detection while reducing the number of false positives. Specifically, two static, two dynamic and two interactive security analysis tools will be combined to study their behavior using a specific benchmark for OWASP Top Ten security vulnerabilities and taking into account various scenarios of different criticality in terms of the applications analyzed. Finally, this study analyzes and discuss the values of the selected metrics applied to the results for each n-tools combination.