$alpha

In this paper, we continue the investigation started in [1]. We obtain new results derived from novel concepts developed in analogy with others already established, e.g., the fact that leftoids (X, ∗) for φ are super-transitive if and only if φ(φ(x)) = φ(x) for all x ∈ X. In addition we apply fuzzy subsets in this context and we derive a number of results as consequences.
 

‎‎In his study‎, ‎we define $\alpha$-$\psi$-contractive type mapping‎, ‎twisted $(\alpha,\beta)$-admissible mapping‎, ‎and twisted $(\alpha,\beta)$-contractive mapping on orthogonally metric space‎. ‎After that‎, ‎we will examine the fixed point theorem for $\alpha$-$\psi$-contractive mappings with preserving orthogonality‎. ‎For instance‎, ‎we demonstrate that there is an orthogonally fixed point for orthogonally $\alpha$-$\psi$-contractive type mapping‎, ‎but under these conditions‎, ‎it does not exist in the metric space‎. ‎Next‎, ‎we set up a common fixed point with orthogonally preserving for new concepts defined under different conditions in an orthogonally metric space‎. ‎Moreover‎, ‎we demonstrate through an example that a fixed point with orthogonally preserving exists for new concepts defined under different conditions‎. ‎Lastly‎, ‎we show that the defined concepts can be stable‎

A hypertree is a special type of connected hypergraph that removes‎ ‎any‎, ‎its hyperedge then results in a disconnected hypergraph‎. ‎Relation between hypertrees (hypergraphs) and trees (graphs) can be helpful to solve real problems in hypernetworks and networks and it is the main tool in this regard‎. ‎The purpose of this paper is to introduce a positive relation (as $\alpha$-relation) on hypertrees that makes a connection between hypertrees and trees‎. ‎This relation is dependent on some parameters such as path‎, ‎length of a path‎, ‎and the intersection of hyperedges‎. ‎For any $q\in \mathbb{N}‎, ‎$ we introduce the concepts of a derivable tree‎, ‎$(\alpha‎, ‎q)$-hypergraph‎, ‎and fundamental $(\alpha‎, ‎q)$-hypertree for the first time in this study and analyze the structures of derivable trees from hypertrees via given positive relation‎. ‎In the final‎, ‎we apply the notions of derivable trees‎, ‎$(\alpha‎, ‎q)$-trees in real optimization problems by modeling hypernetworks and networks based on hypertrees and trees‎, ‎respectively.‎‎‎

In 1984, Khan et al. established some fixed point theorems in complete and compact metric spaces by altering distance functions. In 2020, Lotfy et al. introduced the $\alpha_{*}$-$\psi$-common rational type mappings on generalized metric spaces applied to fractional integral equations. In 2022, Roy et al. described the notion of Branciari  $ S_b$-metric space and related fixed point theorems with an application. In this paper, we introduce the notion of fixed point theorems for $\alpha_{*}$-$\psi$-$\beta_{i}$-contractive set-valued mappings on Branciari $S_{b}$-metric space with application to fractional integral equations.

In [24], Khan et al. established some fixed point theorems in complete and compact metric spaces by using altering distance functions. In [16] Gordji et al. described the notion of orthogonal set and orthogonal metric spaces. In [18] Gungor et al. established fixed point theorems on orthogonal metric spaces via altering distance functions. In [25] Lotfy et al introduced the notion of $\alpha_{*}$-$\psi$-common rational type mappings on generalized metric spaces with application to fractional integral equations. In [28] K. Royy et al. described the notion of Branciari $S_b$-metric space and related fixed point theorems with an application. In this paper, we introduce the notion of the common fixed point ($\alpha_*$-$\psi$-$\beta_{i}$)-contractive set-valued mappings on orthogonal Branciari $S_{b}$-metric space with the application of the existence of a unique solution to an initial value problem.

In this paper, we consider invariant (α, β)- metrics which are induced by invariant Riemannian metrics and invariant vector fields on homogeneous spaces. We first study geodesic vectors and investigates the set of all homogeneous geodesics of (α, β)- metrics. Then we study the geometry of simply connected two-step nilpotent Lie groups of dimension five equipped with a left invariant (α, β)- metrics and we examine Lie algebras with 1-dimensional center, 2-dimensional center and 3-dimensional center.

The relationship between derivations and algebraic structures of quotient near-rings has become a fascinating topic in modern algebra in recent decades. Assume $\mathcal{N}$ is a near-ring and $P$ is its prime ideal. In this paper we introduce the notion of $(\alpha, \tau )$-$P$-derivation in near-rings. Also, we study the structure of the quotient near-rings $N/P$ that satisfies certain algebraic identities involving $(\alpha, \tau )$-$P$-derivation.

In the present paper, the acceptance single sampling plan is developed by the generalized intuitionistic fuzzy numbers. The $\alpha_1$-cut and $\alpha_2$-cut sets of the generalized intuitionistic fuzzy numbers are applied to construct the acceptance single sampling plan. We also investigate the operating characteristic curve, where the parameter is considered the generalized intuitionistic fuzzy number. The $(\alpha_1,\alpha_2)$-cut set of generalized intuitionistic fuzzy operating characteristics is constructed. The bands are represented with upper and lower bounds instead of curves for operating characteristics and evaluated in detail. Finally, the numerical example is given to illustrate the proposed approach.

Let F be an (α,β)-metric which is defined by a left invariant vector field and a left invariant Riemannian metric on a simply connected real Lie group G. We consider the automorphism and isometry groups of the Finsler manifold (G,F) and their intersection. We prove that for an arbitrary left invariant vector field X and any compact subgroup K of automorphisms which X is invariant under them, there exists an (α,β)-metric such that K is a subgroup of its isometry group.

In this paper a special class of R-quadratic generalized $(\alpha, \beta)$-metrics are considered. Some properties of this class are investigated. In special case, the Riemann curvature of this metrics is calculated. Moreover, it is proved that, in this class of metrics, there is not any (non-Riemannian) R-quadratic metrics of non-zero scalar curvature.

In this paper, first, we introduce $ (\alpha,\psi,\varphi) $-Geraghty and generalized $ (\alpha,\psi,\varphi) $-Geraghty contraction mappings in b-metric spaces and then we prove the existence and uniqueness of fixed point after exploring all the conditions which guarantee the existence of fixed point. Our results extend and generalize related fixed point results in the existing literature. We also provide examples in support of our main findings.

In this paper, we find necessary and sufficient conditions underwhich the infinite series metric and Randers metric on a manifold M of dimension n >3 be projectively related.

Recently, samet et al. introduced an interesting extension of the Banach contraction principle. In this paper, motivated by the main idea of Samet et al., we introduce the concept of α-admissible α-θ-generalized mappings in metric spaces and give and prove several theorems of the existence and uniqueness of a fixed point in complete metric spaces for such mappings. The results obtained in this study, generalize many of the results in this field, especially, the results presented by Jleli et al. and the work done by Geraghty. By presenting an example, we show that our results are real generalization of the previous results. Next, we get new results in ordered metric spaces and graphical metric spaces using the concept of α-admissible α-θ-generalized mappings. Finally, we present an application of our obtained results for the existence and uniqueness of the solution of nonlinear first-order ordinal differential equations and periodic boundary value problems.

The structure of an α(β,β) -topological ring is richer in comparison with the structure of an α(β,β) -topological group. The theory of α(β,β) -topological rings has many common features with the theory of α(β,β) -topological groups. Formally, the theory of α(β,β) -topological abelian groups is included in the theory of α(β,β) -topological rings. The purpose of this paper is to introduce and study the concepts of α(β,β) -topological rings and α(β,γ) -topological R-modules. we show how they may be introduced by specifying the neighborhoods of zero, and present some basic constructions. We provide fundamental concepts and basic results on α(β,β) -topological rings and α(β,γ) -topological Rmodules.

The theory of α- resolvent families was developed as a generalization of the classical theory of strongly continuous semigroups and cosine operator families, to study the fractional evolution equations with Caputo derivative of order α. An important problem in semigroup theory and also for cosine operator families is to discuss different type of boundedness (in terms of their generator) for these families. In this paper, we study polynomially bounded α- resolvent families. We impose conditions on the resolvent of a closed and densely defined linear operator to be the generator of an α-resolvent family. We also show that these conditions are necessary in the case of Hilbert spaces. This generalizes the result by T. Eisner on polynomially bounded semigroups. Moreover, since α- resolvent families describe the solutions to fractional evolution equations, with this generation result, we discuss the existence and stability of solutions to these problems at the same time.

‎In this paper‎, ‎we introduce more general contractions called $varphi $-fixed‎ ‎point point for $(F,varphi‎ ,‎alpha )_{s}$ and $(F,varphi‎ ,‎alpha )_{s}$-‎weak contractions‎. ‎We prove the existence and uniqueness of $varphi $-‎fixed point point for $(F,varphi‎ ,‎alpha )_{s}$ and $(F,varphi‎ ,‎alpha‎‎)_{s}$-weak contractions in complete $b$-metric spaces‎. ‎Some examples are‎ ‎supplied to support the usability of our results‎. ‎As applications‎, ‎necessary‎ ‎conditions to ensure the existence of a unique solution for a nonlinear‎ ‎inequality problem are also discussed‎. ‎Also‎, ‎some new fixed point results in‎ ‎partial metric spaces are proved.

Intuitionistic Fuzzy Modal Operator was defined by Atanassov in cite{at3} in 1999. In 2001, cite{at4}, he introduced the generalization of these modal operators. After this study, in 2004, Dencheva cite{dencheva} defined second extension of these operators. In 2006, the third extension of these was defined in cite{at6} by Atanassov. In 2007,cite{gc1}, the author introduced a new operator over Intuitionistic Fuzzy Sets which is a generalization of Atanassov's and Dencheva's operators. At the same year, Atanassov defined an operator which is an extension of all the operators defined until 2007. The diagram of One Type Modal Operators on Intuitionistic Fuzzy Sets was introduced first in 2007 by Atanassov cite{at10}. In 2008, Atanassov defined the most general operator and in 2010 the author expanded the diagram of One Type Modal Operators on Intuitionistic Fuzzy Sets with the operator $Z_{alpha, beta}^{omega}$. Some relationships among these operators were studied by several researchers% cite{at5}-cite{at8} cite{gc1}, cite{gc3}, cite{dencheva}- cite% {narayanan}. The aim of this paper is to expand the diagram of one type modal operators over intuitionistic fuzzy sets. For this purpose, we defined a new modal oparator $Z_{alpha, beta}^{omega, theta}$ over intuitionistic fuzzy sets. It is shown that this oparator is the generalization of the operators$Z_{alpha, beta}^{omega},E_{alpha, beta},boxplus _{alpha, beta},boxtimes _{alpha, beta}.$