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doi: 10.3934/dcdsb.2021248
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## Pseudo compact almost automorphy of neutral type Clifford-valued neural networks with mixed delays

 1 Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China 2 School of Mathematics and Computer Science, Yunnan Nationalities University, Kunming, Yunnan 650500, China

* Corresponding author: Yongkun Li

Received  April 2020 Revised  July 2021 Early access October 2021

Fund Project: The first author is supported by the National Natural Science Foundation of China under Grant No. 11861072 and the second author is supported by the Applied Basic Research Foundation of Yunnan Province under Grant No. 2019FB003

We consider a class of neutral type Clifford-valued cellular neural networks with discrete delays and infinitely distributed delays. Unlike most previous studies on Clifford-valued neural networks, we assume that the self feedback connection weights of the networks are Clifford numbers rather than real numbers. In order to study the existence of $(\mu, \nu)$-pseudo compact almost automorphic solutions of the networks, we prove a composition theorem of $(\mu, \nu)$-pseudo compact almost automorphic functions with varying deviating arguments. Based on this composition theorem and the fixed point theorem, we establish the existence and the uniqueness of $(\mu, \nu)$-pseudo compact almost automorphic solutions of the networks. Then, we investigate the global exponential stability of the solution by employing differential inequality techniques. Finally, we give an example to illustrate our theoretical finding. Our results obtained in this paper are completely new, even when the considered networks are degenerated into real-valued, complex-valued or quaternion-valued networks.

Citation: Yongkun Li, Bing Li. Pseudo compact almost automorphy of neutral type Clifford-valued neural networks with mixed delays. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021248
##### References:
 [1] E. H. Ait Dads, F. Boudchich and B. Es-Sebbar, Compact almost automorphic solutions for some nonlinear integral equations with time-dependent and state-dependent delay, Adv. Diff. Equ., 2017 (2017), Paper No. 307, 21 pp. doi: 10.1186/s13662-017-1364-2. [2] E. H. Ait Dads, K. Ezzinbi and M. Miraoui, $(\mu, \nu)$-Pseudo almost automorphic solutions for some non-autonomous differential equations, Internat. J. Math., 26 (2015), 1550090, 21 pp. doi: 10.1142/S0129167X15500901. [3] C. Aouiti and F. Dridi, Weighted pseudo almost automorphic solutions for neutral type fuzzy cellular neural networks with mixed delays and $D$ operator in Clifford algebra, Int. J. Syst. Sci., 51 (2020), 1759-1781.  doi: 10.1080/00207721.2020.1777345. [4] C. Aouiti, F. Dridi, Q. Hui and E. Moulay, $(\mu, \nu)$-Pseudo almost automorphic solutions of neutral type Clifford-valued high-order hopfield neural networks with D operator, Neural Process. Lett., 53 (2021), 799-828. [5] J. Blot, P. Cieutat and K. Ezzinbi, New approach for weighted pseudo-almost periodic functions under the light of measure theory, basic results and applications, Applic. Anal., 92 (2013), 493-526.  doi: 10.1080/00036811.2011.628941. [6] S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. U.S.A., 48 (1962), 2039-2043.  doi: 10.1073/pnas.48.12.2039. [7] F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Pitman Books Limited, London, 1982. [8] S. Buchholz, A Theory of Neural Computation with Clifford Algebras, Ph. D. thesis, University of Kiel, Kiel, 2005. [9] S. Buchholz and G. Sommer, On Clifford neurons and Clifford multi-layer perceptrons, Neural Netw., 21 (2008), 925-935.  doi: 10.1016/j.neunet.2008.03.004. [10] B. de Andrade and C. Cuevas, Compact almost automorphic solutions to semilinear Cauchy problems with non-dense domain, Appl. Math. Comput., 215 (2009), 2843-2849.  doi: 10.1016/j.amc.2009.09.025. [11] T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, New York, 2003. doi: 10.1007/978-3-319-00849-3. [12] N. Drisi, B. Es-Sebbar and K. Ezzinbi, Compact almost automorphic solutions for some nonlinear dissipative differential equations in Banach spaces, Numer. Funct. Anal. Optim., 39 (2018), 825-841.  doi: 10.1080/01630563.2017.1423328. [13] B. Es-Sebbar, Almost automorphic evolution equations with compact almost automorphic solutions, C. R. Math. Acad. Sci. Paris, 354 (2016), 1071-1077.  doi: 10.1016/j.crma.2016.10.001. [14] E. Hernández and J. Wu, Existence, uniqueness and qualitative properties of global solutions of abstract differential equations with state-dependent delay, Proc. Edinb. Math. Soc. II, 62 (2019), 771-788.  doi: 10.1017/S001309151800069X. [15] Y. Hino and S. Murakami, Almost automorphic solutions for abstract functional differential equations, J. Math. Anal. Appl., 286 (2003), 741-752.  doi: 10.1016/S0022-247X(03)00531-6. [16] E. Hitzer, T. Nitta and Y. Kuroe, Applications of Clifford's geometric algebra, Adv. Appl. Clifford Algebras, 23 (2013), 377-404.  doi: 10.1007/s00006-013-0378-4. [17] B. Li and Y. Li, Existence and global exponential stability of pseudo almost periodic solution for Clifford-valued neutral high-order Hopfield neural networks with leakage delays, IEEE Access, 7 (2019), 150213-150225. [18] B. Li and Y. Li, Existence and global exponential stability of almost automorphic solution for Clifford-valued high-order Hopfield neural networks with leakage delays, Complexity, 2019 (2019), 6751806.  doi: 10.1155/2019/6751806. [19] Y. Li and N. Huo, $(\mu, \nu)$-pseudo almost periodic solutions of Clifford-valued high-order HNNs with multiple discrete delays, Neurocomputing, 414 (2020), 1-9. [20] Y. Li, N. Huo and B. Li, On $\mu$–pseudo almost periodic solution for Clifford-valued neutral type neural networks with leakage delays, IEEE Trans. Neural Netw. Learn. Syst., 32 (2021), 1365-1374.  doi: 10.1109/TNNLS.2020.2984655. [21] Y. Li and S. Shen, Almost automorphic solutions for Clifford-valued neutral-type fuzzy cellular neural networks with leakage delays on time scales, Neurocomputing, 417 (2020), 23-35. [22] Y. Li, Y. Wang and B. Li, The existence and global exponential stability of $\mu$-pseudo almost periodic solutions of Clifford-valued semi-linear delay differential equations and an application, Adv. Appl. Clifford Algebras, 29 (2019), Paper No. 105, 18 pp. doi: 10.1007/s00006-019-1025-5. [23] Y. Li and J. Xiang, Existence and global exponential stability of anti-periodic solution for Clifford-valued inertial Cohen-Grossberg neural networks with delays, Neurocomputing, 332 (2019), 259-269.  doi: 10.1016/j.neucom.2018.12.064. [24] J. Pearson and D. Bisset, Back propagation in a Clifford algebra, Artif. Neural Netw., 2 (1992), 413-416. [25] G. Rajchakit, R. Sriraman, P. Vignesh and C. P. Lim, Impulsive effects on Clifford-valued neural networks with time-varying delays: An asymptotic stability analysis, Appl. Math. Comput., 407 (2021), 126309, 18 pp. doi: 10.1016/j.amc.2021.126309. [26] S. Shen and Y. Li, $S^p$-Almost periodic solutions of Clifford-valued fuzzy cellular neural networks with time-varying delays, Neural Process. Lett., 51 (2020), 1749-1769. [27] W. A. Veech, Almost automorphic functions on groups, Amer. J. Math., 87 (1965), 719-751.  doi: 10.2307/2373071. [28] W. A. Veech, On a theorem of Bochner, Ann. of Math., 86 (1967), 117-137.  doi: 10.2307/1970363.

show all references

##### References:
 [1] E. H. Ait Dads, F. Boudchich and B. Es-Sebbar, Compact almost automorphic solutions for some nonlinear integral equations with time-dependent and state-dependent delay, Adv. Diff. Equ., 2017 (2017), Paper No. 307, 21 pp. doi: 10.1186/s13662-017-1364-2. [2] E. H. Ait Dads, K. Ezzinbi and M. Miraoui, $(\mu, \nu)$-Pseudo almost automorphic solutions for some non-autonomous differential equations, Internat. J. Math., 26 (2015), 1550090, 21 pp. doi: 10.1142/S0129167X15500901. [3] C. Aouiti and F. Dridi, Weighted pseudo almost automorphic solutions for neutral type fuzzy cellular neural networks with mixed delays and $D$ operator in Clifford algebra, Int. J. Syst. Sci., 51 (2020), 1759-1781.  doi: 10.1080/00207721.2020.1777345. [4] C. Aouiti, F. Dridi, Q. Hui and E. Moulay, $(\mu, \nu)$-Pseudo almost automorphic solutions of neutral type Clifford-valued high-order hopfield neural networks with D operator, Neural Process. Lett., 53 (2021), 799-828. [5] J. Blot, P. Cieutat and K. Ezzinbi, New approach for weighted pseudo-almost periodic functions under the light of measure theory, basic results and applications, Applic. Anal., 92 (2013), 493-526.  doi: 10.1080/00036811.2011.628941. [6] S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. U.S.A., 48 (1962), 2039-2043.  doi: 10.1073/pnas.48.12.2039. [7] F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Pitman Books Limited, London, 1982. [8] S. Buchholz, A Theory of Neural Computation with Clifford Algebras, Ph. D. thesis, University of Kiel, Kiel, 2005. [9] S. Buchholz and G. Sommer, On Clifford neurons and Clifford multi-layer perceptrons, Neural Netw., 21 (2008), 925-935.  doi: 10.1016/j.neunet.2008.03.004. [10] B. de Andrade and C. Cuevas, Compact almost automorphic solutions to semilinear Cauchy problems with non-dense domain, Appl. Math. Comput., 215 (2009), 2843-2849.  doi: 10.1016/j.amc.2009.09.025. [11] T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, New York, 2003. doi: 10.1007/978-3-319-00849-3. [12] N. Drisi, B. Es-Sebbar and K. Ezzinbi, Compact almost automorphic solutions for some nonlinear dissipative differential equations in Banach spaces, Numer. Funct. Anal. Optim., 39 (2018), 825-841.  doi: 10.1080/01630563.2017.1423328. [13] B. Es-Sebbar, Almost automorphic evolution equations with compact almost automorphic solutions, C. R. Math. Acad. Sci. Paris, 354 (2016), 1071-1077.  doi: 10.1016/j.crma.2016.10.001. [14] E. Hernández and J. Wu, Existence, uniqueness and qualitative properties of global solutions of abstract differential equations with state-dependent delay, Proc. Edinb. Math. Soc. II, 62 (2019), 771-788.  doi: 10.1017/S001309151800069X. [15] Y. Hino and S. Murakami, Almost automorphic solutions for abstract functional differential equations, J. Math. Anal. Appl., 286 (2003), 741-752.  doi: 10.1016/S0022-247X(03)00531-6. [16] E. Hitzer, T. Nitta and Y. Kuroe, Applications of Clifford's geometric algebra, Adv. Appl. Clifford Algebras, 23 (2013), 377-404.  doi: 10.1007/s00006-013-0378-4. [17] B. Li and Y. Li, Existence and global exponential stability of pseudo almost periodic solution for Clifford-valued neutral high-order Hopfield neural networks with leakage delays, IEEE Access, 7 (2019), 150213-150225. [18] B. Li and Y. Li, Existence and global exponential stability of almost automorphic solution for Clifford-valued high-order Hopfield neural networks with leakage delays, Complexity, 2019 (2019), 6751806.  doi: 10.1155/2019/6751806. [19] Y. Li and N. Huo, $(\mu, \nu)$-pseudo almost periodic solutions of Clifford-valued high-order HNNs with multiple discrete delays, Neurocomputing, 414 (2020), 1-9. [20] Y. Li, N. Huo and B. Li, On $\mu$–pseudo almost periodic solution for Clifford-valued neutral type neural networks with leakage delays, IEEE Trans. Neural Netw. Learn. Syst., 32 (2021), 1365-1374.  doi: 10.1109/TNNLS.2020.2984655. [21] Y. Li and S. Shen, Almost automorphic solutions for Clifford-valued neutral-type fuzzy cellular neural networks with leakage delays on time scales, Neurocomputing, 417 (2020), 23-35. [22] Y. Li, Y. Wang and B. Li, The existence and global exponential stability of $\mu$-pseudo almost periodic solutions of Clifford-valued semi-linear delay differential equations and an application, Adv. Appl. Clifford Algebras, 29 (2019), Paper No. 105, 18 pp. doi: 10.1007/s00006-019-1025-5. [23] Y. Li and J. Xiang, Existence and global exponential stability of anti-periodic solution for Clifford-valued inertial Cohen-Grossberg neural networks with delays, Neurocomputing, 332 (2019), 259-269.  doi: 10.1016/j.neucom.2018.12.064. [24] J. Pearson and D. Bisset, Back propagation in a Clifford algebra, Artif. Neural Netw., 2 (1992), 413-416. [25] G. Rajchakit, R. Sriraman, P. Vignesh and C. P. Lim, Impulsive effects on Clifford-valued neural networks with time-varying delays: An asymptotic stability analysis, Appl. Math. Comput., 407 (2021), 126309, 18 pp. doi: 10.1016/j.amc.2021.126309. [26] S. Shen and Y. Li, $S^p$-Almost periodic solutions of Clifford-valued fuzzy cellular neural networks with time-varying delays, Neural Process. Lett., 51 (2020), 1749-1769. [27] W. A. Veech, Almost automorphic functions on groups, Amer. J. Math., 87 (1965), 719-751.  doi: 10.2307/2373071. [28] W. A. Veech, On a theorem of Bochner, Ann. of Math., 86 (1967), 117-137.  doi: 10.2307/1970363.
Curves of $x_{p}^{0}(t)$ and $x_{p}^{1}(t)$ of system (1) with the initial values $(x_{1}^{0}(0), x_{2}^{0}(0))^{T} = (0.05, -0.1)^{T}, (-0.06, 0.09)^{T}$ and $(x_{1}^{1}(0), x_{2}^{1}(0))^{T} = (-0.1, 0.05)^{T}, (0.1, -0.04)^{T}$
Curves of $x_{p}^{2}(t)$ and $x_{p}^{3}(t)$ of system (1) with the initial values $(x_{1}^{2}(0), x_{2}^{2}(0))^{T} = (-0.03, 0.1)^{T}, (-0.1, 0.02)^{T}$ and $(x_{1}^{3}(0), x_{2}^{3}(0))^{T} = (0.1, -0.1)^{T}, (0.04, -0.02)^{T}$
Curves of $x_{p}^{12}(t)$ and $x_{p}^{13}(t)$ of system (1) with the initial values $(x_{1}^{12}(0), x_{2}^{12}(0))^{T} = (-0.04, 0.04)^{T}, (0.08, -0.07)^{T}$ and $(x_{1}^{13}(0), x_{2}^{13}(0))^{T} = (0.07, -0.06)^{T}, (-0.02, 0.03)^{T}$
Curves of $x_{p}^{23}(t)$ and $x_{p}^{123}(t)$ of system (1) with the initial values $(x_{1}^{23}(0), x_{2}^{23}(0))^{T} = (0.08, 0.02)^{T}, (-0.1, -0.04)^{T}$ and $(x_{1}^{123}(0), x_{2}^{123}(0))^{T} = (0.03, -0.02)^{T}, (-0.1, 0.08)^{T}$
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