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Weighted \(S^p\)-pseudo \(S\)-asymptotic periodicity and applications to Volterra integral equations. (English) Zbl 1460.43006

Summary: This paper is related to the function space formed by weighted \(S^p\)-pseudo \(S\)-asymptotic periodicity and their applications. Initially, the translation invariance and completeness of the function space are investigated. Additionally, the composition theorem and convolution operator generated by Lebesgue integrable functions are presented. Finally, existence and uniqueness of solutions with weighted \(S^p\)-pseudo \(S\)-asymptotic periodicity for two classes of Volterra equations are proved by using the results obtained above, and some concrete examples are given. The methods mainly include Minkowski’s inequality, convolution inequality, contraction mapping principle, and especially the generalized Minkowski’s inequality. Our results extend some known results on asymptotic periodicity.

MSC:

43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
45D05 Volterra integral equations
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[1] Agarwal, R. P.; Cuevas, C.; Soto, H.; El-Gebeily, M., Asymptotic periodicity for some evolution equations in banach spaces, Nonlinear Anal., 74, 5, 1769-1798 (2011) · Zbl 1221.34159
[2] Cushing, J. M., Forced asymptotically periodic solutions of predator-prey systems with or without hereditary effects, SIAM J. Appl. Math., 30, 4, 665-674 (1976) · Zbl 0331.93078
[3] Wei, F. Y.; Wang, K., Global stability and asymptotically periodic solutions for nonautonomous cooperative lotka-volterra diffusion system, Appl. Math. Comput., 182, 1, 161-165 (2006) · Zbl 1113.92062
[4] Zeng, Z. J., Asymptotically periodic solution and optimal harvesting policy for gompertz system, Nonlinear Anal., 12, 3, 1401-1409 (2011) · Zbl 1211.92046
[5] Henríquez, H. R.; Pierri, M.; Táboas, P., On s-asymptotically ω-periodic functions on banach spaces and applications, J. Math. Anal. Appl., 343, 2, 1119-1130 (2008) · Zbl 1146.43004
[6] Henríquez, H. R.; Pierri, M.; Táboas, P., Existence of s-asymptotically ω-periodic solutions for abstract neutral equations, Bull. Aust. Math. Soc., 78, 3, 365-382 (2008) · Zbl 1183.34122
[7] Cuevas, C.; Lizama, C., s-asymptotically ω-periodic solutions for semilinear volterra equations, Math. Methods Appl. Sci., 33, 1628-1636 (2010) · Zbl 1251.45007
[8] Pierri, M.; Rolnik, V., On pseudo s-asymptotically periodic functions, Bull. Aust. Math. Soc., 87, 2, 238-254 (2013) · Zbl 1263.35017
[9] Henríquez, H. R., Asymptotically periodic solutions of abstract differential equations, Nonlinear Anal., 80, 135-149 (2013) · Zbl 1266.34100
[10] Xia, Z. N., Pseudo asymptotically periodic solutions for volterra integro-differential equations, Math. Meth. Appl. Sci., 38, 5, 799-810 (2015) · Zbl 1315.45005
[11] Xia, Z. N., Weighted pseudo asymptotically periodic mild solutions of evolution equations. Acta Mathematica Sinica, English Series, 31, 8, 1215-1232 (2015) · Zbl 1320.47069
[12] Blot, J.; Cieutat, P.; N’Guérékata, G. M., s-asymptocially ω-periodic functins and applications to evolution equations, Afr Diaspora J Math, 12, 2, 113-121 (2011) · Zbl 1248.34093
[13] de Andrade, B.; Cuevas, C., s-asymptotically ω-periodic and asymptotically ω-periodic solutions to semilinear cauchy problems with non-dense domain, Nonlinear Anal., 72, 6, 3190-3208 (2010) · Zbl 1205.34074
[14] Dimbour, W.; Mophou, G.; N’Guérékata, G. M., s-asymptotically periodic solutions for partial differential equations with finite delay, Electron. J. Differ. Eq., 117, 966-967 (2011) · Zbl 1231.34136
[15] Pierri, M., On s-asymptotically ω-periodic functions and applications, Nonlinear Anal., 75, 2, 651-661 (2012) · Zbl 1232.45016
[16] Adivar, M.; Koyuncuoğlu, H. C.; Raffoul, Y. N., Periodic and asymptotically periodic solutions of systems of nonlinear difference equations with infinite delay, J. Differ. Equ. Appl., 19, 19, 1927-1939 (2013) · Zbl 1278.39020
[17] Agarwal, R. P.; Cuevas, C.; Frasson, M. V.S., Semilinear functional difference equations with infinite delay, Math. Comput. Modelling, 55, 1083-1105 (2012) · Zbl 1255.39012
[18] Caicedo, A.; Cuevas, C.; Henríquez, H. R., Asymptotic periodicity for a class of partial integrodifferential equations, ISRN Math. Anal., 2011, 1-18 (2011) · Zbl 1227.45007
[19] Cuevas, C.; de Souza, J. C., s-asymptocially ω-periodic solutions of semilinear fractional integro-differential equations, Appl. Math. Lett., 22, 6, 865-870 (2009) · Zbl 1176.47035
[20] Lizama, C.; N’Guérékata, G. M., Bounded mild solutions for semilinear integro differential equations in banach spaces, Integr. Equat. Oper. Th., 68, 2, 207-227 (2010) · Zbl 1209.45007
[21] He, B.; Wang, Q. R., On completeness of the space of weighted stepanov-like pseudo almost automorphic (periodic) functions, J. Math. Anal. Appl., 465, 1176-1185 (2018) · Zbl 1403.46024
[22] Wang, Q. R.; Zhu, Z. Q., Almost periodic solutions of neutral functional dynamic systems in the sense of stepanov, Difference equations, discrete dynamical systems and applications, 133-143 (2015), Springer International Publishing: Springer International Publishing Switzerland · Zbl 1338.34175
[23] Xia, Z.; Fan, M., Weighted stepanov-like pseudo almost automorphy and applications, Nonlinear Anal., 75, 2378-2397 (2012) · Zbl 1306.35140
[24] Corduneanu, C., Integral equations and applications (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0714.45002
[25] Prüs, J., Evolutionary integral equations and applications (2012), Birkhäuser: Birkhäuser Basel · Zbl 1258.45008
[26] Razdolsky, L., Integral Volterra equations, Probability based high temperature engineering, 55-100 (2016), Springer International Publishing: Springer International Publishing Switzerland
[27] Islam, M. N., Asymptotically periodic solutions of volterra integral equations, Electron. J. Differ. Eq., 83, 1-9 (2016) · Zbl 1342.45002
[28] Ji, D. S.; Zhang, C. Y., Some properties of weighted stepanov-like pseudo almost automorphic functions and applications to volterra integral equations (in chinese), Sci. Sin. Math., 44, 4, 349-368 (2014) · Zbl 1488.32023
[29] de Andrade, B.; Cuevas, C.; Henríquez, E., Asymptotic periodicity and almost automorphy for a class of volterra integro-differential equations, Math. Meth. Appl. Sci., 35, 7, 795-811 (2012) · Zbl 1243.45015
[30] Diagana, T., Existence results for some damped second-order volterra integro-differential equations, Appl. Math. Comput., 237, 7, 304-317 (2014) · Zbl 1334.45013
[31] Wei, F. Y.; Wang, K., Asymptotically periodic logistic equation, J. Biomath., 20, 4, 399-405 (2005) · Zbl 1127.46308
[32] Pankov, A., Bounded and almost periodic solutions of nonlinear operator differential equations (1990), Springer: Springer Dordrecht · Zbl 0712.34001
[33] Nicola, S. H.J.; Pierri, M., A note on s-asymptotically periodic functions, Nonlinear Anal., 10, 5, 2937-2938 (2009) · Zbl 1163.42305
[34] Hardy, G. H.; Littlewood, J. E.; Póya, G., Inequalities (1952), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0047.05302
[35] Bruno, G.; Pankov, A., On convolution operators in the spaces of almost periodic functions and l^p spaces, J. Anal. Appl., 19, 359-367 (2000) · Zbl 0972.47036
[36] Alvarez, E.; Lizama, C., Weighted pseudo almost automorphic and s-asymptotically omega-periodic solutions to fractional difference-differential equations, Electron. J. Differ. Eq., 270, 1-12 (2016) · Zbl 1355.35194
[37] Chang, J. C., Asymptotically periodic solutions of a partial differential equation with memory, J. Fix Point Theory A, 2016, 1-26 (2016)
[38] Yang, M.; Wang, Q. R., Pseudo asymptotically periodic solutions for fractional integro-differential neutral equations, Sci. China Math., 62, 1705-1718 (2019) · Zbl 1425.34088
[39] Li, Y., Existence and asymptotic stability of periodic solution for evolution equations with delays, J. Funct. Anal., 261, 5, 1309-1324 (2011) · Zbl 1233.34028
[40] Burton, T. A.; Furumochi, T., Periodic and asymptotically periodic solutions of volterra integral equations, Funkc. Ekvacioj, 39, 87-107 (1996) · Zbl 0861.45003
[41] Cuevas, C.; Henríquez, H. R.; Soto, H., Asymptotically periodic solutions of fractional differential equations, Appl. Math. Comput., 236, 524-545 (2014) · Zbl 1334.34173
[42] Alvarez-Pardo, E.; Lizama, C., Pseudo asymptotic solutions of fractional order semilinear equations, Banach J. Math. Anal., 7, 42-52 (2013) · Zbl 1275.47092
[43] Shu, X. B.; Xu, F.; Shi, Y., s-asymptotically ω-positive periodic solutions for a class of neutral fractional differential equations, Appl. Math. Comput., 270, 768-776 (2015) · Zbl 1410.34203
[44] Wang, H.; Li, F., s-asymptotically t-periodic solutions for delay fractional differential equations with almost sectorial operator, Adv. Differ. Equ., 2016, 1, 315 (2016) · Zbl 1419.34040
[45] Diblík, J.; Fečkan, M.; Pospíšil, M., Nonexistence of periodic solutions and s-asymptotically periodic solutions in fractional difference equations, Appl. Math. Comput., 257, 230-240 (2015) · Zbl 1338.39026
[46] Henríquez, H. R.; Pierri, M.; Rolnik, V., Pseudo s-asymptotically periodic solutions of second-order abstract cauchy problems, Appl. Math. Comput., 274, 590-603 (2016) · Zbl 1410.34128
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