PPubMed24 Nov 2015

Almost automorphic solutions for shunting inhibitory cellular neural networks with time-varying delays

Changjin Xu, Maoxin Liao

Abstract

This paper is concerned with the shunting inhibitory cellular neural networks with time-varying delays. Under some suitable conditions, we establish some criteria on the existence and global exponential stability of the almost automorphic solutions of the networks. Numerical simulations are given to support the theoretical findings.

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The paper proves existence and global exponential stability of almost automorphic solutions for shunting inhibitory cellular neural networks with time‑varying delays, supported by simulations.

What’s new

i.Derived criteria guaranteeing the existence of almost automorphic solutions for shunting inhibitory cellular neural networks with time‑varying delays.
ii.Established global exponential stability of these almost automorphic solutions under suitable assumptions.
iii.Provided a rigorous fixed‑point operator framework and used contraction mapping arguments to prove uniqueness.
iv.Validated the theoretical results with numerical simulations illustrating the solution behavior.

How they did it

The authors formulated the network equations in the space of almost automorphic functions and constructed a nonlinear operator Φ. By verifying that Φ is a contraction on a closed convex subset, they applied the Banach fixed‑point theorem to obtain a unique almost automorphic solution. Exponential stability was then proved using Dini‑derivative estimates and Lyapunov‑type inequalities.

What they found

γ = 0.0015 < 1, satisfying the contraction condition.
‖L‖∞/(a⁻¹‑γ) = 10/17 < 1, confirming boundedness of the solution region.
max_{ij} Π_{ij}/a⁻ = 0.00188 < 1, ensuring Φ is a contraction.
a_{ij}⁻ – [...] = 1.000624 > 0, guaranteeing global exponential stability.

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