Explore the vast range of titles available.

Kaplan et al. (J. Inequal. Appl. 2025:54, 2025 ) introduced the notion of multiplicative modular metric spaces, motivated by the framework of multiplicative metric spaces (Özavşar and Cevikel in J. Eng. Technol. Appl. Sci. 2(2):65–79, 2017 ) and the classical modular metric spaces established by Chistyakov (Nonlinear Anal. 72(1):1–14, 2010 ). However, upon closer analysis, it turns out that, under a domain for the modular parameter, their multiplicative modular metric is independent of the modular parameters. In particular, every multiplicative modular metric space of Kaplan et al. can be decomposed into a disjoint union of multiplicative metric spaces. In this paper, we propose a refinement of their definition by restricting the choice of the multiplicative modular parameter, and we demonstrate the resulting connection with modular metric spaces of Chistyakov. In addition, we analyze two fixed point theorems stated by Kaplan et al., construct explicit counterexamples to illustrate their failure, and provide corrected formulations of these results.

In this paper, we provide a simplified proof of the existence of solutions for the system of nonlinear quasi-mixed equilibrium problems studied by Suantai and Petrot (Appl. Math. Lett. 24:308–313, 2011 ), utilizing Banach’s fixed point theorem. This result remains valid under weaker assumptions through the product space approach. Moreover, we show that the stability analysis of the iterative algorithm proposed in their work (and in (J. Inequal. Appl. 2010:437976, 2010 )) contains a flaw, which we address using Ostrowski’s classical result from 1967 (Z. Angew. Math. Mech. 47:77–81, 1967 ). Finally, we examine and improve the convergence theorem for solving a system of variational inequalities as established by Chang et al. (Appl. Math. Lett. 20:329–334, 2007 ).

We explore the intermixed method for finding a common element of the intersection of the solution set of a mixed variational inequality and the fixed point set of a nonexpansive mapping. We point out that Khuangsatung and Kangtunyakarn’s statement [J. Inequal. Appl. 2023:1, 2023 ] regarding the resolvent utilized in their paper is not correct. To resolve this gap, we employ the epigraphical projection and the product space approach. In particular, we obtain a strong convergence theorem with a weaker assumption.

Building upon the subgradient extragradient method proposed by Censor et al., we prove the strong convergence of the iterative sequence generated by a modification of this method by means of the Halpern method. We also consider the problem of finding a common element of the solution set of a variational inequality and the fixed-point set of a quasi-nonexpansive mapping with a demiclosedness property.

We discuss two functional inequalities proposed by Lu and Park (J. Inequal. Appl. 2012:294, 2012 ). Using the concept of γ -additive mappings and the results of Forti (J. Math. Anal. Appl. 295:127–133, 2004 ) we give an improvement of the stability results of these functional inequalities and obtain some hyperstability results. In some situation, we show that the estimate in our result is sharp.

In this paper, we study convergence theorems for finding a fixed point of a mapping with a directed graph. We give a counterexample to the result recently proved by Tripak (Fixed Point Theory Appl. 2016:87, 2016 ) and propose a better version of the theorem. In the presence of the transitivity of a directed graph, we prove convergence theorems without the uniform convexity of the space. Moreover, we obtain a sufficient condition for the existence of a fixed point of G -nonexpansive mappings. We also discuss a convergence theorem without assuming the transitivity on the graph G . In this setting, we obtain a result for a wider class of mappings including all G -nonexpansive mappings with a fixed point. Our results not only correct the original theorem, but also improve it by removing some of its assumptions.

In this paper we give a simple proof of three fixed point theorems of Secelean and Wardowski by using the fixed point result of Jachymski et al. Our result is established with weaker assumptions than the three theorems. Furthermore, the recent result of Secelean et al. in the setting of a complete metric space can be also deduced by our theorem.

We prove two key inequalities for metric and generalized projections in a certain Banach space. We then obtain some asymptotic behavior of a sequence generated by the shrinking projection method introduced by Takahashi et al. (J. Math. Anal. Appl. 341:276–286, 2008) where the computation allows some nonsummable errors. We follow the idea proposed by Kimura (Banach and Function Spaces IV (ISBFS 2012), pp. 303–311, 2014). The mappings studied in this paper are more general than the ones in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018). In particular, the results in (Ibaraki and Kimura in Linear Nonlinear Anal. 2:301–310, 2016; Ibaraki and Kajiba in Josai Math. Monogr. 11:105–120, 2018) are both extended and supplemented. Finally, we discuss our results for finding a zero of maximal monotone operator and a minimizer of convex functions on a Banach space.

We use the notion of Halpern-type sequence recently introduced by the present authors to conclude two strong convergence theorems for solving the bilevel equilibrium problems proposed by Yuying et al. and some authors. Our result excludes some assumptions as were the cases in their results.

We discuss the approximation method of finding a common fixed point of two certain mappings defined on a Banach space. This paper deals with three modes of convergence, namely, weak, strong, and △‐convergences in the presence of the uniform convexity of the norm. It is worth mentioning that the △‐convergence which is closely connected with the weak convergence is discussed without assuming the Opial’s condition. Our results not only generalize the results recently announced by Garodia et al. but also rectify a gap in their paper. Finally, some concrete numerical examples are shown that our results are genuine generalizations of a number of results in the literature.