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Öğe Bipolar fuzzy soft filter and its application to multi-criteria group decision-making(Univ Nis, Fac Sci Math, 2025) Demir, Izzettin; Saldamli, Murat; Okurer, MerveThe convergence theory is not only a basic theory of topology but also has wide applications in other fields including information technology, economics and computer science. The convergence of filters is also one of the most important tools used in topology to characterize certain concepts such as the closure of a set, continuous mapping, Hausdorff space and so on. Besides, multi-criteria group decision making (for short MCGDM) aims to make unanimous decision based on different criterions to find the most accurate solution of real world problems and so that the MCGDM plays a very important role in our daily life problems. In this paper, taking into account all of these, we firstly introduce the notion of a bipolar fuzzy soft filter (for short BFS-filter) by using bipolar fuzzy soft sets (for short BFS-sets). Also, we define the idea of an ultra BFS-filter and establish some of its properties. Moreover, we investigate the convergence of BFS-filters in a bipolar fuzzy soft topological space (BFS-topological space) with related results. After introducing the concepts of a bipolar fuzzy soft continuity (BFS-continuity) and a bipolar fuzzy soft Hausdorfness (BFS-Hausdorffness), with the aid of the convergence of BFS-filters, we discuss the characterizations of these concepts. Next, we develop a multi-criteria group decision-making method based on the BFS-filters to deal with uncertainties in our daily life. Finally, we present a numerical example to make a decision for selection of best alternative.Öğe CONVERGENCE THEORY OF BIPOLAR FUZZY SOFT NETS AND ITS APPLICATIONS(Rocky Mt Math Consortium, 2024) Demir, Izzettin; Saldamli, MuratIn a different way than in the literature, we define the concept of a quasicoincident using the bipolar fuzzy soft points we previously proposed (2021) and investigate its basic properties. We introduce the notion of a bipolar fuzzy soft net (for short BFS-net) and give convergence of the BFS-nets in a bipolar fuzzy soft topological space with useful results. We show how a BFS-net is derived from a BFS-filter and obtain a characterization about bipolar fuzzy soft Hausdorff spaces. Based on the idea of quasicoincident, we give a new kind of bipolar fuzzy soft continuity and analyze its relationship with the BFS-nets. We put forward the idea of compactness in the setting of bipolar fuzzy soft sets and characterize it through the contribution of the BFS-subnets. Finally, we present some examples to illustrate the defined concepts.Öğe Error estimates for perturbed Milne-type inequalities by twice-differentiable functions using conformable fractional integrals(Springer, 2025) Unes, Esra; Demir, IzzettinMilne's inequality provides an upper bound for the error in definite integral approximations using Milne's rule, making it a useful tool for evaluating the rule's precision. For this reason, this inequality is widely applied in engineering, physics, and applied mathematics. Additionally, conformable fractional integral operators establish a stronger relationship between classical and fractional calculus, enhancing the modeling, analysis, and resolution of complex problems. Therefore, we focus on the study of conformable fractional integral operators and Milne-type inequalities, which have significant applications in various fields. In this study, we first obtain an integral identity involving conformable fractional integral operators and twice-differentiable functions. Building on this new identity, we develop various perturbed Milne-type integral inequalities for twice-differentiable convex functions. We also validate them numerically through examples, computational analysis, and visual representations. In conclusion, it is evident that our findings significantly enhance and expand upon prior findings regarding integral inequalities. In addition to improving the scope of previous discoveries, the obtained results offer meaningful approaches and methods for tackling mathematical and scientific issues.Öğe Fractional integral approaches to weighted corrected Euler-Maclaurin-type inequalities for different classes of functions(Pergamon-Elsevier Science Ltd, 2025) Demir, Izzettin; Unes, EsraIn recent years, a wide variety of integral inequalities, including Newton-type, Simpson-type, and corrected Euler-Maclaurin-type inequalities, have been extensively studied, particularly in the framework of fractional calculus using Riemann-Liouville or conformable fractional integrals. Among these, fractional corrected Euler-Maclaurin-type inequalities have emerged as a valuable tool due to their improved approximation capabilities. In this study, we focus on developing weighted corrected Euler-Maclaurin-type inequalities for different classes of functions using Riemann-Liouville fractional integrals. To achieve this, we first derive a key integral equality with the aid of a positive weighted function, providing the foundation for the primary outcomes. Through the use of this integral equality, we prove new inequalities for differentiable convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation. Also, for better explanation, we offer some examples together with their matching graphs. Moreover, these findings extend previous results. Consequently, the study clarifies the significance of corrected Euler-Maclaurin-type inequalities and suggests opportunities for further exploration.Öğe Hermite-Hadamard-Mercer type inequalities for fractional integrals: A study with h-convexity and ψ-Hilfer operators(Springer, 2025) Azzouz, Noureddine; Benaissa, Bouharket; Budak, Hueseyin; Demir, IzzettinIn this paper, we first prove a generalized fractional version of Hermite-Hadamard-Mercer type inequalities using h-convex functions by means of psi-Hilfer fractional integral operators. Then, we give new identities of this type with special functions depending on psi. Moreover, we establish some new fractional integral inequalities connected with the right- and left-hand sides of Hermite-Hadamard-Mercer inequalities involving differentiable mappings whose absolute values of the derivatives are h-convex. For the development of these novel integral inequalities, we utilize h-Mercer inequality and H & ouml;lder's integral inequality. These results offer the significant advantage of being convertible into classical integral inequalities and Riemann-Liouville fractional integral inequalities for convex functions, s-convex functions, and P-convex functions.Öğe A new approach to n-soft topological structures(Rocky Mt Math Consortium, 2023) Demir, Izzettin; Okurer, MerveWe study the topological structure of N-soft sets given by Riaz et al. (J. Intell. Fuzzy Syst. 36:6 (2019), 6521-6536). Firstly, we redefine N-soft closed sets using the complement operation of N-soft sets proposed in (Math. Methods Appl. Sci. 44:8 (2021), 7343-7358) and investigate their basic properties. Then, we introduce the concept of an N-soft continuous mapping and also obtain the initial N-soft topology determined by a family of N-soft mappings. Furthermore, we establish a new concept of N-soft topological subspace and analyze some related properties of this concept. Finally, we present some examples to better understand the defined concepts.Öğe A new approach to Simpson-type inequality with proportional Caputo-hybrid operator(Wiley, 2024) Demir, Izzettin; Tunc, TubaIn this article, we begin by deriving a new identity with the help of twice-differentiable convex functions for the proportional Caputo-hybrid operator. Then, using this newly uncovered identity, we obtain various integral inequalities associated with the Simpson's integral inequality for proportional Caputo-hybrid operator. Moreover, we indicate that the acquired results improve and refine certain existing discoveries in the realm of integral inequalities. Finally, for a better understanding of the newly obtained inequalities, we establish illustrative examples and visualize them through their corresponding graphs.Öğe New midpoint-type inequalities in the context of the proportional Caputo-hybrid operator(Springer, 2024) Demir, Izzettin; Tunc, TubaFractional calculus is a crucial foundation in mathematics and applied sciences, serving as an extremely valuable tool. Besides, the new hybrid fractional operator, which combines proportional and Caputo operators, offers better applications in numerous fields of mathematics and computer sciences. Due to its wide range of applications, we focus on the proportional Caputo-hybrid operator in this research article. Firstly, we begin by establishing a novel identity for this operator. Then, based on the newfound identity, we establish some integral inequalities that are relevant to the left-hand side of Hermite-Hadamard-type inequalities for the proportional Caputo-hybrid operator. Furthermore, we show how the results improve upon and refine many previous findings in the setting of integral inequalities. Later, we present specific examples together with their related graphs to offer a better understanding of the newly obtained inequalities. Our results not only extend previous studies but also provide valuable viewpoints and methods for tackling a wide range of mathematical and scientific problems.Öğe On a new version of Hermite-Hadamard-type inequality based on proportional Caputo-hybrid operator(Springer, 2024) Tunc, Tuba; Demir, IzzettinIn mathematics and the applied sciences, as a very useful tool, fractional calculus is a basic concept. Furthermore, in many areas of mathematics, it is better to use a new hybrid fractional operator, which combines the proportional and Caputo operators. So we concentrate on the proportional Caputo-hybrid operator because of its numerous applications. In this research, we introduce a novel extension of the Hermite-Hadamard-type inequalities for proportional Caputo-hybrid operator and establish an identity. Then, taking into account this novel generalized identity, we develop some integral inequalities associated with the left-side of Hermite-Hadamard-type inequalities for proportional Caputo-hybrid operator. Moreover, to illustrate the newly established inequalities, we give some examples with the help of graphs.Öğe Refinements of the Jensen Inequality and Estimates of the Jensen Gap Based on Interval-Valued Functions(Wiley, 2025) Demir, IzzettinThe significance of the Jensen inequality stems from its impactful and compelling outcomes. As a generalization of classical convexity, it plays a key role in deriving other well-known inequalities such as Hermite-Hadamard, H & ouml;lder, Minkowski, arithmetic-geometric, and Young's inequalities. So, this inequality has become an influential concept in a wide range of scientific fields. Besides, interval analysis provides methods for managing uncertainty in data, making it possible to build mathematical and computer models of various deterministic real-world phenomena. In this paper, taking into account all of these, we first present several refinements of the Jensen inequality for the left and right convex interval-valued functions. We also provide examples with corresponding graphs to demonstrate these refinements more clearly. Next, we adopt a novel approach to derive several bounds for the Jensen gap in integral form using the gH-differentiable interval valued functions as well as various related notions. Moreover, we obtain the proposed bounds by utilizing the renowned Ostrowski inequality. The fundamental benefit of the newly discovered inequalities is that they extend to many known inequalities in the literature, as discussed in this work.
