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    A new approach to n-soft topological structures
    (Rocky Mt Math Consortium, 2023) Demir, Izzettin; Okurer, Merve
    We 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.
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    A new approach to Simpson-type inequality with proportional Caputo-hybrid operator
    (Wiley, 2024) Demir, Izzettin; Tunc, Tuba
    In 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.
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    New midpoint-type inequalities in the context of the proportional Caputo-hybrid operator
    (Springer, 2024) Demir, Izzettin; Tunc, Tuba
    Fractional 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.
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    On a new version of Hermite-Hadamard-type inequality based on proportional Caputo-hybrid operator
    (Springer, 2024) Tunc, Tuba; Demir, Izzettin
    In 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.