<p>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 α ̌–η ̌-β ̌ 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 α ̌–η ̌-φ ̌ 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.</p>
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 offer an approximate solution to the fixed point equation . It is used to solve the problem in order to come up with a good approximation. This paper's main purpose is to introduce new types of proximal contraction for nonself mappings in fuzzy normed space and then proved the best proximity point theorem for these mappings. At first, the definition of fuzzy normed space is given. Then the notions of the best proximity point and - proximal admissible in the context of fuzzy normed space are presented. The notion of α ̃–ψ ̃- proximal contractive mapping is introduced. After that, the best proximity point theorem for such type of mapping in a fuzzy normed space is state and prove. In addition, the idea of α ̃–ϕ ̃-proximal contractive mapping is presented in a fuzzy normed space and under specific conditions, the best proximity point theorem for such type of mappings is proved. Furthermore, some examples are offered to show the results' usefulness.
Soft operators defined on soft normed spaces are relatively modern concepts. Many properties of these operators have not been thoroughly studied. We have introduced in this paper some new concepts related to the soft compact operators such as spectrum of soft compact operator, null space of soft compact operator and rang of soft compact operator. Many important theorems related to these concepts were introduced.
The formal balls in fuzzy normed space (characterized by closed balls in) are ordered by reverse inclusion depending on the concept of level sets. The set of formal balls in a fuzzy normed space is called a fuzzy domain normed space denoted by. This set is directed complete partially ordered set (dcpo), its maximal elements are the suprema. A contraction mapping principle is defined on. Banach fixed point theorem is studied and proved on .
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