Abstract
In the present article, the Schauder-type fixed point theorem for the class of fuzzy continuous, as well as fuzzy compact operators is established in a fuzzy normed linear space (fnls) whose underlying t-norm is left-continuous at (1,1). In the fuzzy setting, the concept of the measure of non-compactness is introduced, and some basic properties of the measure of non-compactness are investigated. Darbo’s generalization of the Schauder-type fixed point theorem is developed for the class of ψ-set contractions. This theorem is proven by using the idea of the measure of non-compactness.
Highlights
In 1930, Schauder established an important theorem in the field of fixed point theory
In 2007, Chu and Torres [1] proved the existence of positive solutions to the second order singular differential equations with the help of this fixed point theorem
Schauder’s fixed point theorem and its generalizations play a pivotal role in this context of nonlinear functional analysis
Summary
In 1930, Schauder established an important theorem in the field of fixed point theory. In 2007, Chu and Torres [1] proved the existence of positive solutions to the second order singular differential equations with the help of this fixed point theorem. Darbo extended the Schauder theorem to a more general class of mappings, the so-called α-set contractions, which contain compact, as well as continuous mappings. We establish Darbo’s generalization of the Schauder-type fixed point theorem in the fuzzy setting for the class of ψ-set contraction mappings using the properties of the measure of non-compactness.
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