DOI: 10.1371/journal.pone.0358480 ISSN: 1932-6203

A unified fixed point approach for enriched bivariate mappings with applications to volterra systems and fractional-order chaos

Khaleel Ahmad, Sahib Yar, Abdul Rahim Khan, Nida Najeeb, Adil Jhangeer, Walid Abdelfattah
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We develop a unified fixed point theory for enriched bivariate contractions in ordered Banach spaces and in ordered convex metric spaces. Two classes of mappings are introduced and analysed: enriched bivariate Ciric-Reich-Rus contractions (

๐’ž B C R R C e
) and enriched bivariate interpolative Ciric-Reich-Rus contractions (
๐’ž B I C R R C e
). For each class we prove the existence and uniqueness of coupled fixed points and establish the geometric convergence of the coupled Krasnoselskij iteration. The methodological contribution is a transparent product space reduction: the coupled averaged operator associated with an enriched bivariate map is shown to be an ordinary (single variable) Ciric-Reich-Rus operator on the product space, so that the classical theory applies directly and every algebraic step is explicit. This reduction yields a sharp sufficient condition for the existence of a coupled fixed point, namely
( 2 a + k 2 ) / ( k 1 + 1 ) + 2 b < 1
, where a , b are the contraction constants and
k 1 , k 2
are the enrichment constants. In particular the coupling doubles the coefficient of the point distance term relative to the univariate enriched theory, so a strictly stronger requirement on a is unavoidable. We further show that the condition 2 a โ€‰+โ€‰3 b โ€‰<โ€‰1 stated in earlier formulations is sufficient but not necessary, and we exhibit an explicit map for which a coupled fixed point exists while 2 a โ€‰+โ€‰3 b โ€‰<โ€‰1 fails. The abstract results are applied first to a Volterra type coupled integral system, and then to the fractional order Thomas cyclically symmetric attractor formulated with the Caputo-Fabrizio derivative, where an averaged mapping yields local in time existence and uniqueness even though the long time dynamics are chaotic. All numerical results are produced by direct computation. A bifurcation diagram and a largest Lyapunov exponent spectrum, computed independently, agree window for window and confirm a period doubling route to chaos with intermediate periodic windows, while the measured convergence factor of the coupled iteration is geometric.

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