Coincidence, averages and fractals

M.A. Navascués

Chaos, Solitons & Fractals, 2026, vol. 202, issue P1

Abstract: Countless problems of the applied and the social sciences can be formalized in terms of an equation of type Sx=Tx, where S and T are two mappings or operators on a common domain. Mathematically x is called a coincidence. This paper is devoted to the study of this kind of elements, in the context of b-metric and quasi-normed spaces, for a pair of maps satisfying a very general contractive condition. In the first place, the concept of Lyapunov and asymptotic stability of fixed points is extended to coincidences. Then a constructive theorem of existence of coincidence is given, which generalizes a classical result of Goebel. The theory is applied to the construction of a new type of fractal sets and fractal functions. Afterwards a recurrent algorithm to approximate coincidences is proposed, proving its convergence and stability. The particular case where the underlying space is Euclidean is considered, giving an iterative method to compute the sought points. Since the stability is sometimes related to a Jacobian matrix, an averaged method to find eigenvalues and singular values of matrices and infinite dimensional linear operators is developed. The algorithm proposed is used to compute the maximum singular value of the Volterra integral operator.

Keywords: Coincidence points; Jungck iteration; Fractals; Power method; Eigenvalues; Singular values (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:202:y:2026:i:p1:s0960077925015541

DOI: 10.1016/j.chaos.2025.117541

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