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We construct a class of minimal trees and use these trees to establish a number of coloring theorems on general trees. Among the applications of these trees and coloring theorems are quantification of the Bourgain $\ell_p$ and $c_0$ indices, dualization of the Bourgain $c_0$ index, establishing sharp positive and negative results for constant reduction, and estimating the Bourgain $\ell_p$ index of an arbitrary Banach space $X$ in terms of a subspace $Y$ and the quotient $X/Y$.

The best proximity point is a generalization of a fixed point that is beneficial when the contraction map is not a self-map. On other hand, best approximation theorems provide an approximate solution to the fixed-point equation Tҳ = ҳ. It is used to solve the problem to determine an approximate solution that is optimum. The main goal of this paper is to present new types of proximal contraction for nonself mappings in a fuzzy Banach space. At first, the notion of the best proximity point is presented. We introduce the notion of 𝛼̌–𝜂̌-𝛽̌ proximal contractive. After that, the best proximity point theorem for such type of mappings in a fuzzy Banach space is proved. In addition, the concept of 𝛼̌–𝜂̌-𝜑̌ proximal contractive mapping is presented in a fuzzy Banach space and under specific conditions, the best proximity point theorem for such type of mapping is proved. Additionally, some examples are supplied to show the results' applicability.

The best proximity point is a generalization of a fixed point that is beneficial when the contraction map is not a self-map. On other hand, best approximation theorems provide an approximate solution to the fixed-point equation Tҳ = ҳ. It is used to solve the problem to determine an approximate solution that is optimum. The main goal of this paper is to present new types of proximal contraction for nonself mappings in a fuzzy Banach space. At first, the notion of the best proximity point is presented. We introduce the notion of 𝛼̌–𝜂̌-𝛽̌ proximal contractive. After that, the best proximity point theorem for such type of mappings in a fuzzy Banach space is proved. In addition, the concept of 𝛼̌–𝜂̌-𝜑̌ proximal contractive mapping is presented in a fuzzy Banach space and under specific conditions, the best proximity point theorem for such type of mapping is proved. Additionally, some examples are supplied to show the results' applicability.

The best proximity point is a generalization of a fixed point that is beneficial when the contraction map is not a self-map. On other hand, best approximation theorems provide an approximate solution to the fixed-point equation Tҳ = ҳ. It is used to solve the problem to determine an approximate solution that is optimum. The main goal of this paper is to present new types of proximal contraction for nonself mappings in a fuzzy Banach space. At first, the notion of the best proximity point is presented. We introduce the notion of 𝛼̌–𝜂̌-𝛽̌ proximal contractive. After that, the best proximity point theorem for such type of mappings in a fuzzy Banach space is proved. In addition, the concept of 𝛼̌–𝜂̌-𝜑̌ proximal contractive mapping is presented in a fuzzy Banach space and under specific conditions, the best proximity point theorem for such type of mapping is proved. Additionally, some examples are supplied to show the results' applicability.

The best proximity point is a generalization of a fixed point that is beneficial when the contraction map is not a self-map. On other hand, best approximation theorems provide an approximate solution to the fixed-point equation Tҳ = ҳ. It is used to solve the problem to determine an approximate solution that is optimum. The main goal of this paper is to present new types of proximal contraction for nonself mappings in a fuzzy Banach space. At first, the notion of the best proximity point is presented. We introduce the notion of 𝛼̌–𝜂̌-𝛽̌ proximal contractive. After that, the best proximity point theorem for such type of mappings in a fuzzy Banach space is proved. In addition, the concept of 𝛼̌–𝜂̌-𝜑̌ proximal contractive mapping is presented in a fuzzy Banach space and under specific conditions, the best proximity point theorem for such type of mapping is proved. Additionally, some examples are supplied to show the results' applicability.

1970

In this note, we shall generalize the notion of a $P$-space to proximity spaces and investigate the basic properties of these proximities. We therefore define a $P_{\aleph_{1}}$-proximity to be a proximity where if $A_{n}\prec B$ for all $n\in\mathbb{N}$, then $\bigcup_{n}A_{n}\prec B$. It turns out that the class of $P_{\aleph_{1}}$-proximities is equivalent to the class of $\sigma$-algebras. Furthermore, the $P_{\aleph_{1}}$-proximity coreflection of a proximity space is the $\sigma$-algebra of proximally Baire sets.

For an asymptotic $\ell_1$ space $X$ with a basis $(x_i)$ certain asymptotic $\ell_1$ constants, $\delta_\alpha (X)$ are defined for $\alpha