# Stepanov-Like Pseudo Almost Periodic Solution of Quaternion-Valued for Fuzzy Recurrent Neural Networks with Mixed Delays

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## Abstract

Real-valued neural networks or complex-valued neural networks are sometimes inappropriate for some engineering and research problems for instance where the data is multi-dimensional, such as 4-D signals, color images and body images. Hence, researchers explored recently a more general and sophisticated model than the previous one, which is the quaternion-valued neural networks. The quaternions, which can also be defined as \(2\times 2\) matrix of complex numbers, have the capacity to analyze three or more dimensional signals and represent spatial transformations. In this paper, some sufficient conditions are given for the existence and various kinds of stability for the unique Stepanov-like pseudo almost periodic solution of quaternion-valued fuzzy recurrent neural networks. The results are established by employing Lyapunov functionals, Nemytskii’s operator and Banach fixed point theorem. Also, a new direct method is used to establish our theoretical results in order to avoid the decomposition of the considered model into real-valued or complex-valued system. Finally, a numerical example is given to illustrate the validity of the obtained results.

## Keywords

Stepanov-like pseudo almost periodic function Quaternion-valued fuzzy recurrent neural networks Existence and uniqueness Asymptotic and exponential stability## Notes

### Acknowledgements

The authors would like to thank the anonymous reviewers and the editor for their constructive comments, which greatly improved the quality of this paper.

## References

- 1.Amdouni M, Chérif F (2018) The pseudo almost periodic solutions of the new class of Lotka–Volterra recurrent neural networks with mixed delays. Chaos Solitons Fractals 113:79–88MathSciNetzbMATHCrossRefGoogle Scholar
- 2.Ammar B, Chérif F, Alimi Adel M (2012) Existence and uniqueness of pseudo almost-periodic solutions of recurrent neural networks with time-varying coefficients and mixed delays. IEEE Trans Neural Netw Learn Syst 23(1):109–118CrossRefGoogle Scholar
- 3.Blot J, Cieutat P, N’Guérékata GM, Pennequin D (2009) Superposition operators between various almost periodic function spaces and applications. Commun Math Anal 6(1):42–70MathSciNetzbMATHGoogle Scholar
- 4.Cai Z, Huang J, Huang L (2018) Periodic orbit analysis for the delayed Filippov system. Proc Am Math Soc 146(11):4667–4682MathSciNetzbMATHCrossRefGoogle Scholar
- 5.Chen X, Li Z, Song Q, Hu J, Tan Y (2017) Robust stability analysis of quaternion-valued neural networks with time delays and parameter uncertainties. Neural Netw 91:55–65CrossRefGoogle Scholar
- 6.Chérif F (2011) A various types of almost periodic functions on Banach spaces: part I. Int Math Forum 6(19):921–952MathSciNetzbMATHGoogle Scholar
- 7.Chérif F (2011) A various types of almost periodic functions on Banach spaces: part II. Int Math Forum 6(20):953–985MathSciNetzbMATHGoogle Scholar
- 8.Chérif F (2015) Pseudo almost periodic solution of Nicholson’s blowflies model with mixed delays. Appl Math Model 39:5152–5163MathSciNetzbMATHCrossRefGoogle Scholar
- 9.Chérif F (2013) Analysis of global asymptotic stability and pseudo almost periodic solution of a class of chaotic neural networks. Math Model Anal 18(4):489–504MathSciNetzbMATHCrossRefGoogle Scholar
- 10.Duan L, Huang C (2016) Existence and global attractivity of almost periodic solutions for a delayed differential neoclassical growth model. Math Methods Appl Sci 40(3):814–822MathSciNetzbMATHCrossRefGoogle Scholar
- 11.Duan L, Fang X, Huang C (2018) Global exponential convergence in a delayed almost periodic Nicholson’s blowflies model with discontinuous harvesting. Math Methods Appl Sci 41(5):1954–1965MathSciNetzbMATHCrossRefGoogle Scholar
- 12.Duan L, Huang L, Guo Z, Fang X (2017) Periodic attractor for reaction-diffusion high-order Hopfield neural networks with time-varying delays. Comput Math Appl 73(2):233–245MathSciNetzbMATHCrossRefGoogle Scholar
- 13.Huo N, Li Y (2018) Anti-periodic solutions for quaternion-valued shunting inhibitory cellular neural networks with distributed delays and impulses. Complexity 6420256:1–12zbMATHGoogle Scholar
- 14.Hu J, Wang J (2015) Global exponential periodicity and stability of discrete-time complex-valued recurrent neural networks with time-delays. Neural Netw 66:119–130zbMATHCrossRefGoogle Scholar
- 15.Huang C, Long X, Huang L, Fu S (2019) Stability of almost periodic Nicholson’s blowflies model involving patch structure and mortality terms. Can Math Bull. https://doi.org/10.4153/S0008439519000511 CrossRefGoogle Scholar
- 16.Huang C, Su R, Cao J, Xiao S (2019) Asymptotically stable of high-order neutral cellular neural networks with proportional delays and D operators. Comput Simul Math. https://doi.org/10.1016/j.matcom.2019.06.001 CrossRefGoogle Scholar
- 17.Haijun H, Zou X (2017) Existence of an extinction wave in the Fisher equation with a shifting habitat. Proc Am Math Soc 145(11):4763–4771MathSciNetzbMATHCrossRefGoogle Scholar
- 18.Huang C, Zhang H, Cao J, Hu H (2019) Stability and Hopf bifurcation of a delayed prey-predator model with disease in the predator. Int J Bifurc Chaos 29(7):1950091:1–23MathSciNetzbMATHCrossRefGoogle Scholar
- 19.Huang C, Yang Z, Yi T, Zou X (2014) On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities. J Differ Equ 256:2101–2114MathSciNetzbMATHCrossRefGoogle Scholar
- 20.Huang C, Liu B, Tian X, Yang L, Zhang X (2019) Global convergence on asymptotically almost periodic SICNNs with nonlinear decay functions. Neural Process Lett 49(2):625–641CrossRefGoogle Scholar
- 21.Huang C, Liu B (2019) New studies on dynamic analysis of inertial neural networks involving non-reduced order method. Neurocomputing 325:283–287CrossRefGoogle Scholar
- 22.Huang C, Qiao Y, Huang L, Agarwal R (2018) Dynamical behaviors of a food-chain model with stage structure and time delay. Adv Differ Equ 2018:186. https://doi.org/10.1186/s13662-018-1589-8 MathSciNetCrossRefzbMATHGoogle Scholar
- 23.Jian J, Jiang W (2015) Lagrange exponential stability for fuzzy Cohen–Grossberg neural networks with time-varying delays. Fuzzy Sets Syst 277:65–80MathSciNetzbMATHCrossRefGoogle Scholar
- 24.Kumari S, Chugh R, Cao J, Huang C (2019) Multi fractals of generalized multivalued iterated function systems in b-metric spaces with applications. Mathematics 7(10):967. https://doi.org/10.3390/math7100967 CrossRefGoogle Scholar
- 25.Li N, Cao J (2018) Global dissipativity analysis of quaternion-valued memristor-based neural networks with proportional delay. Neurocomputing 321:103–113CrossRefGoogle Scholar
- 26.Li Y, Qin J, Li B (2018) Existence and global exponential stability of anti-periodic solutions for delayed quaternion-valued cellular neural networks with impulsive effects. Math Methods Appl Sci 42(1):5–23MathSciNetzbMATHCrossRefGoogle Scholar
- 27.Li L, Jin Q, Yao B (2018) Regularity of fuzzy convergence spaces. Open Math 16:1455–1465MathSciNetzbMATHCrossRefGoogle Scholar
- 28.Liu Y, Wu J (2014) Fixed point theorems in piecewise continuous function spaces and applications to some nonlinear problems. Math Methods Appl Sci 37(4):508–517MathSciNetzbMATHCrossRefGoogle Scholar
- 29.Liu F, Feng L, Anh V, Li J (2019) Unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time-space fractional Bloch–Torrey equations on irregular convex domains. Comput Math Appl. https://doi.org/10.1016/j.camwa.2019.01.007 MathSciNetCrossRefGoogle Scholar
- 30.Long X, Gong S (2019) New results on stability of Nicholson’s blowflies equation with multiple pairs of time-varying delays. Appl Math Lett. https://doi.org/10.1016/j.aml.2019.106027 CrossRefzbMATHGoogle Scholar
- 31.Song Q, Chen X (2018) Multistability analysis of quaternion-valued neural networks with time delays. IEEE Trans Neural Netw Learn Syst 29(11):5430–5440MathSciNetCrossRefGoogle Scholar
- 32.Sudbery A (1979) Quaternionic analysis. Math Proc Camb 85(2):199–225MathSciNetzbMATHCrossRefGoogle Scholar
- 33.Shen S, Li B, Li Y (2018) Anti-periodic dynamics of quaternion-valued fuzzy cellular neural networks with time-varying delays on time scales. Discret Dyn Nat Soc 5290786:1–14MathSciNetzbMATHCrossRefGoogle Scholar
- 34.Tan Y, Huang C, Sun B, Wang T (2018) Dynamics of a class of delayed reaction-diffusion systems with Neumann boundary condition. J Math Anal Appl 458(2):1115–1130MathSciNetzbMATHCrossRefGoogle Scholar
- 35.Tang W, Zhang J (2019) Symmetric integrators based on continuous-stage Runge–Kutta–Nyström methods for reversible systems. Appl Math Comput 361:1–12MathSciNetzbMATHCrossRefGoogle Scholar
- 36.Wu J (2014) Some fixed-point theorems for mixed monotone operators in partially ordered probabilistic metric spaces. Fixed Point Theory Appl 2014:49MathSciNetzbMATHCrossRefGoogle Scholar
- 37.Wang J, Huang C, Huang L (2019) Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type. Nonlinear Anal Hybrid Syst 33:162–178MathSciNetzbMATHCrossRefGoogle Scholar
- 38.Wang J, Chen X, Huang L (2019) The number and stability of limit cycles for planar piecewise linear systems of node-saddle type. J Math Anal Appl 469(1):405–427MathSciNetzbMATHCrossRefGoogle Scholar
- 39.Wang W (2017) Uniform ultimate boundedness of numerical solutions to nonlinear neutral delay differential equations. J Comput Appl Math 309:132–144MathSciNetzbMATHCrossRefGoogle Scholar
- 40.Wang W, Chen Y (2017) Fast numerical valuation of options with jump under Merton’s model. J Comput Appl Math 318:79–92MathSciNetzbMATHCrossRefGoogle Scholar
- 41.Wang F, Yao Z (2016) Approximate controllability of fractional neutral differential systems with bounded delay. Fixed Point Theory 17(2):495–508MathSciNetzbMATHGoogle Scholar
- 42.Wei Y, Yin L, Long X (2019) The coupling integrable couplings of the generalized coupled Burgers equation hierarchy and its Hamiltonian structure. Adv Differ Equ 2019:58. https://doi.org/10.1186/s13662-019-2004-9 MathSciNetCrossRefzbMATHGoogle Scholar
- 43.Xing Z, Peng J (2012) Exponential lag synchronization of fuzzy cellular neural networks with time-varying delays. J Frankl Inst 349:1074–1086MathSciNetzbMATHCrossRefGoogle Scholar
- 44.Xu X, Zhang J, Shi J (2014) Exponential stability of complex-valued neural networks with mixed delays. Neurocomputing 128:483–490CrossRefGoogle Scholar
- 45.Xu Q, Xu X, Zhuang S, Xiao J, Song C, Che C (2018) New complex projective synchronization strategies for drive-response networks with fractional complex-variable dynamics. Appl Math Comput 338:552–566MathSciNetzbMATHGoogle Scholar
- 46.Xu C, Li P, Xiao Q, Yuan S (2019) New results on competition and cooperation model of two enterprises with multiple delays and feedback controls. Bound Value Probl 2019:36MathSciNetCrossRefGoogle Scholar
- 47.Xiao Q, Liu W (2016) On derivations of quantales. Open Math 14:338–346MathSciNetzbMATHCrossRefGoogle Scholar
- 48.Yang HZ, Sheng L (2009) Robust stability of uncertain stochastic fuzzy cellular neural networks. Neurocomputing 73:133–138CrossRefGoogle Scholar
- 49.Yang T, Yang L (1996) The global stability of fuzzy cellular neural networks. IEEE Trans Cricuits Syst 43(10):880–883MathSciNetCrossRefGoogle Scholar
- 50.Yoshizawa T (1975) Stability theory and the existence of periodic solutions and almost periodic solutions. Springer, New YorkzbMATHCrossRefGoogle Scholar
- 51.Yang C, Huang L, Li F (2018) Exponential synchronization control of discontinuous nonautonomous networks and autonomous coupled networks. Complexity 2018:1–10zbMATHGoogle Scholar
- 52.Yang X, Wen S, Liu Z, Li C, Huang C (2019) Dynamic properties of foreign exchange complex network. Mathematics 7:832. https://doi.org/10.3390/math7090832 CrossRefGoogle Scholar
- 53.You X, Song Q, Liang J, Liu Y, E Alsaadi F (2018) Global \(\mu \)-stability of quaternion-valued neural networks with mixed time-varying delays. Neurocomputing 90:12–25CrossRefGoogle Scholar
- 54.Zhang C (1992) Pseudo almost periodic functions and their applications. University of Western Ontario, LondonGoogle Scholar
- 55.Zhu J, Sun J (2018) Stability of quaternion-valued impulsive delay difference systems and its application to neural networks. Neurocomputing 284:63–69CrossRefGoogle Scholar
- 56.Zhou X (2015) Weighted sharp function estimate and boundedness for commutator associated with singular integral operator satisfying a variant of Hörmander’s condition. J Math Inequal 9(2):587–596MathSciNetzbMATHCrossRefGoogle Scholar