Perov type results in gauge spaces and their applications to integral systems on semi-axis

Abstract

ABSTRACT. In this paper we present Perov type fixed point theorems for contractive mappings in Gheorghiu's sense on spaces endowed with a family of vectorvalued pseudo-metrics. Applications to systems of integral equations are given to illustrate the theory. The examples also prove the advantage of using vectorvalued pseudo-metrics and matrices that are convergent to zero, for the study of systems of equations.

Nonlinear Dynamics, Fixed Points and Coupled Fixed Points in Generalized Gauge Spaces with Applications to a System of Integral Equations

“…Remark 9. The above results extend (to the case of nonselfoperators on a set endowed with two separating gauge structures) some fixed point theorems given in [3,18,22,24] and so forth.…”

“…Now, in a classical manner (see, e.g., Theorem 2.1 in Novac and Precup [24]), we get that, for any ∈̃( 0 ; 0 ) , the sequence ( ( )) ∈N is Cauchy in ( , Q). By assumption (ii), the sequence is also Cauchy in ( , P).…”

“…If map takes the values in R + (i.e., : × → R + and satisfies the axioms of Definition 1), then it is called a vectorvalued gauge (or a generalized gauge) on . In this case, the pair ( , P) (where P = { } ∈Λ is a family of separating vector-valued gauges on ) is called a generalized gauge space; see [24]. The properties of the generalized gauge spaces (i.e., the notions of convergent sequences, Cauchy sequences and completeness, open and closed sets, etc.)…”

"Iterates of multidimensional approximation operators via Perov theorem"

“…Perov [13] used the notion of vector-valued metric space and obtained a Banach type fixed theorem on such a complete generalized metric space by using matrices instead of Lipschitz constants. Perov's result have been exploited in various works, see, e.g., [3], [7], [8], [12], [15].…”