We adopted a non decomposition method to study the existence and stability of Stepanov almost periodic solutions in the distribution sense of stochastic shunting inhibitory cellular neural networks (SICNNs) with mixed time delays. Due to the lack of linear structure in the set composed of Stepanov almost periodic stochastic processes in the distribution sense. Due to the lack of linear structure in the set composed of distributed Stepanov periodic stochastic processes, it poses difficulties for the existence of Stepanov almost periodic solutions in the distribution sense of SICNNs. To overcome this difficulty, we first proved that the network under consideration has a unique solution in a space composed of $ \mathcal{L}^p $ bounded and $ \mathcal{L}^p $ uniformly continuous stochastic processes. Then, using stochastic analysis, inequality techniques, and the definition of Stepanov almost periodic stochastic processes in the distribution sense, we proved that this solution is also a Stepanov almost periodic solution in the distribution sense. Moreover, the result of the global exponential stability of this almost periodic solution is given. It is worth noting that even if the network under consideration degenerated into a real-valued network, our results are novel. Finally, we provided a numerical example to validate our theoretical findings.
Citation: Qi Shao, Yongkun Li. Almost periodic solutions for Clifford-valued stochastic shunting inhibitory cellular neural networks with mixed delays[J]. AIMS Mathematics, 2024, 9(5): 13439-13461. doi: 10.3934/math.2024655
We adopted a non decomposition method to study the existence and stability of Stepanov almost periodic solutions in the distribution sense of stochastic shunting inhibitory cellular neural networks (SICNNs) with mixed time delays. Due to the lack of linear structure in the set composed of Stepanov almost periodic stochastic processes in the distribution sense. Due to the lack of linear structure in the set composed of distributed Stepanov periodic stochastic processes, it poses difficulties for the existence of Stepanov almost periodic solutions in the distribution sense of SICNNs. To overcome this difficulty, we first proved that the network under consideration has a unique solution in a space composed of $ \mathcal{L}^p $ bounded and $ \mathcal{L}^p $ uniformly continuous stochastic processes. Then, using stochastic analysis, inequality techniques, and the definition of Stepanov almost periodic stochastic processes in the distribution sense, we proved that this solution is also a Stepanov almost periodic solution in the distribution sense. Moreover, the result of the global exponential stability of this almost periodic solution is given. It is worth noting that even if the network under consideration degenerated into a real-valued network, our results are novel. Finally, we provided a numerical example to validate our theoretical findings.
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Qi Shao, Yongkun Li. Almost periodic solutions for Clifford-valued stochastic shunting inhibitory cellular neural networks with mixed delays[J]. AIMS Mathematics, 2024, 9(5): 13439-13461. doi: 10.3934/math.2024655
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