Existence–uniqueness and stability of reaction–diffusion stochastic Hopfield neural networks with S-type distributed time delays

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Abstract The aim of this paper is to develop some theories of mild solutions to reaction–diffusion stochastic Hopfield neural networks with S-type distributed time delays driven by infinite dimensional Wiener processes. First, we transform the networks with infinite time delays defined on ( − ∞ , 0 ] into those with finite time delays defined on [ − τ , 0 ] by truncation method. Then, the existence–uniqueness theorem of the networks with finite time delays is established in terms of the generalized Halanay inequality. Next, the existence of equilibrium is demonstrated with homotopy invariance theorem and topological degree theory. And the stability is considered using M-function. Finally, with the help of approximation method, the well-posedness and stability of the original networks are obtained. An interesting example shows the effectiveness of our results.