Doubly-measure pseudo S-asymptotically Bloch type periodicity and applications to some stochastic integrodifferential equations

Mamadou Moustapha Mbaye, Amadou Diop, Bipan Hazarika

Abstract


In the present work, for a separable complex Hilbert space V, we introduce the concept of doubly-measure pseudo $S$-asymptotically Bloch-type periodicity to define the space of $(\nu,\mu)-$pseudo $S$-asymptotically Bloch-type periodic (or $(\omega,k)-$periodic) stochastic process with values in the complex Banach space of all strongly-measurable, $p$-integrable V-valued random variables. We first looked into some completeness, composition and convolution theorems for such stochastic processes. Second, the existence and uniqueness in the $p^{th}$-mean $(\nu,\mu)-$pseudo $S$-asymptotically Bloch-type periodic ($(\nu,\mu)$-PSABP, in short) mild solutions of some stochastic integrodifferential equations is formally investigated. In conclusion, we provide examples to support our findings.


Full Text:

PDF

References


S. Abbas, M. Benchohra, G.M. N’Guérékata, Topics in Fractional Differential Equations,Springer, New York, 2012.

B. De Andrade, C. Cuevas, S-asymptotically ω-periodic and asymptotically ω-periodic solutions to semilinear Cauchy problems with non-dense domain, Nonl. Anal. 72 (2010), no. 6, 3190-3208.

E. Alvarez, C. Lizama, R. Ponce, Weighted pseudo antiperiodic solutions for fractional integrodifferential equations in Banach spaces, Appl. Math. Comput. 259 (2015), 164-172.

F. Bloch, Uberdie quanten mechanik der elektronen in kristall gittern, Z. Phys. 52 (1929), 555-600.

J. Blot, P. Cieutat, K. Ezzinbi, Measure theory and pseudo almost automorphic function: New developments and applications, Nonlinear Analysis 75 (2012), 2426-2447.

J. Blot, P. Cieutat, K. Ezzinbi, New approach for weighted pseudo-almost periodic functions under the light of measure theory, basic results and applications, Applicable Analysis 92 (2013), no. 3, 493-526.

N. Belmabrouk, M. Damak, M. Miraoui, Measure pseudo almost periodic solution for a class of nonlinear delayed stochastic evolution equations driven by Brownian motion, Filomat 35 (2021), no. 2, 515-534.

D. Brindle, G.M. N’Guérékata, S-asymptotically ω-periodic mild solutions to fractional differential equations, Electron. J. Differ. Equ. 30 (2020), 1-12.

D. Brindle, G.M. N’Guérékata, S-asymptotically ω-periodic integrodifferential equations, PanAmer. Math. J. 29 (2019), no. 2, 63-74.

Y.T. Bian, Y.K. Chang, J.J. Nieto, Weighted asymptotic behavior of solutions to a semilinear integro-differential equation in Banach spaces, Electron. J. Diff. Equ. 91 (2014), 1-16.

C. Cuevas, J.C. De Souza, Existence of S-asymptotically ω-periodic solutions for fractional order functional integro-differential equations with infinite delay, J. Nonl. Anal. 72 (2010), no. 3, 1683-1689.

A. Caicedo, C. Cuevas, H.R. Henr´ıquez, Asymptotic periodicity for a class of partial integrodifferential equations, Int. Sch. Res. Notices 2011 (2011), Article ID 537890. https://doi.org/10.5402/2011/537890

Y.K. Chang, M.J. Zhang, R. Ponce, Weighted pseudo almost automorphic solutions to a semilinear fractional differential equation with Stepanov-like weighted pseudo almost automorphic nonlinear term, Appl. Math. Comput. 257 (2015), 158-168.

Y.K. Chang, Y. Wei, S-asymptotically Bloch-type periodic solutions to some semi-linear evolution equations in Banach spaces, Acta Math. Sci. Ser. B 41 (2021), 413-425.

Y.K. Chang, Y. Wei, Pseudo S-asymptotically Bloch-type periodicity with applications to some evolution equations, Z. Anal. Anwend. 40 (2021), 33-50.

Y.K. Chang, J. Zhao, Weighted pseudo asymptotically Bloch periodic solutions to nonlocal Cauchy problems of integrodifferential equations in Banach spaces, Int. J. Nonlinear Sci. Numer. Simul. 24 (2021), no. 2, 581-598. https://doi.org/10.1515/ijnsns-2021-0251

Y.K. Chang, Y. Wei, Pseudo S-asymptotically Bloch-type periodicity with applications to some evolution equations, Zeitschrift f¨ur Analysis und ihre Anwendungen 40 (2021), no. 1, 33-50.

Y.K. Chang, R. Ponce, Uniform exponential stability and its applications to bounded solutions of integro-differential equations in Banach spaces, J. Integral Equ. Appl. 30 (2018), 347-369.

Y.K. Chang, X.Y. Wei, G.M. N’Guérékata, Some new results on bounded solutions to a semi-linear integro-differential equation in Banach spaces, J. Integral Equ. Appl. 27 (2015), 153-178.

G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge, 1992.

A. Diop, M.M. Mbaye, G.M. N’Guérékata, Y.K. Chang, On square-mean S-asymptotically Bloch-type periodicity of some stochastic evolution equations, Analele Universitatti Oradea Fasc. Matematica 30 (2023), no. 2, 83-112.

A. Diop, M.M. Mbaye, Y.K. Chang, G.M. N’Guérékata, Measure Pseudo-S-asymptotically Bloch-Type Periodicity of Some Semilinear Stochastic Integrodifferential Equations, Journal of Theoretical Probability 37 (2024), no. 3, 1-24.

M.A. Diop, K. Ezzinbi, M.M. Mbaye, Existence and global attractiveness of a pseudo almost periodic solution in p-th mean sense for stochastic evolution equation driven by a fractional Brownian motion, Stochastics 87 (2015), no. 6, 1061-1093.

J.P.C. Dos Santos, H.R. Henr'iquez, Existence of S-asymptotically ω-periodic solutions to abstract integro-differential equations, Appl. Math. Comput. 256 (2015), 109-118.

T. Diagana, Weighted pseudo-almost periodic functions and applications, C. R. Math. Acad. Sci. Paris 343(10) (2006), no. 10, 643-646.

T. Diagana, Weighted pseudo-almost periodic solutions to some differential equations, Nonlinear Anal. Theory Methods Appl. 68 (2008), no. 8, 2250-2260.

W.S. Du, M. Kostić, M. Pinto, Almost Periodic Functions and Their Applications: A Survey of Results and Perspectives, J. Math. 2021 (2021), Article ID 5536018.

H. Henríquez, M. Pierri, P. T'aboas, Existence of S-asymptotically ω-periodic solutions for abstract neutral equations, Bull. Austral. Math. Soc. 78 (2008), no. 3, 365-382.

H. Henríquez, M. Pierri, P. Táboas, On S-asymptotically ω-periodic functions on Banach spaces and applications, J. Math. Anal. Appl. 343 (2008), no. 2, 1119-1130.

Y. Hino, S. Murakami, T. Naito, Functional-differential equations with infinite delay Lecture notes in mathematics 1473, Springer-Verlag, Berlin, 1991.

M.F. Hasler, G.M. N’Guérékata, Bloch-periodic functions and some applications, Nonlinear Stud. 21 (2014), 21-30.

M. Kostić, D. Velinov, Asymptotically Bloch-periodic solutions of abstract fractional nonlinear differential inclusions with piecewise constant argument, Funct. Anal. Approx. Comput. 9 (2017), 27-36.

Q. Li, L. Liu, M. Wei, Existence of positive S-asymptotically periodic solutions of the fractional evolution equations in ordered Banach spaces, Nonlinear Anal. Model. Control 26 (2021), no. 5, 928-946.

Q. Li, L. Liu, M. Wei, S-asymptotically Periodic Solutions for Time-Space Fractional Evolution Equation, Mediterr. J. Math. 18 (2021), no. 4, 1-21.

C. Lizama, G.M. N’Guérékata, Bounded mild solutions for semilinear integro-differential equations in Banach spaces, Integr. Equat. Oper. Th. 68 (2010), no. 2, 207-227.

C. Lizama, R. Ponce, Bounded solutions to a class of semilinear integrodifferential equations in Banach spaces, Nonlin. Anal. 74 (2011), 3397-3406.

M.A. Meyers, K.K. Chawla, Mechanical Behavior of Materials, Cambridge University Press, Cambridge, 2009.

X. Mao, Stochastic differential equations and their applications, Horwood Publishing Ltd., Chichester, 1997.

M.M. Mbaye, A. Diop, M. Dieye, Weighted pseudo S-asymptotically Bloch type periodic solutions for a class of mean field stochastic fractional evolution equations, Malaya Journal of Matematik 11 (2024), no. 4, 378-402.

S. Nicola, M. Pierri, A note on S-asymptotically ω-periodic functions, Nonl. Anal. 10 (2009), no. 5, 2937-2938.

E.R. Oueama-Guengai, G.M. N’Guérékata, S-asymptotically ω-periodic mild solutions to some fractional integro-differential equations with infinite delay, Lib. Math. (NS) 38 (2018), no. 2, 307 111-124.

M. Pierri, D. O’Regan, S-asymptotically ω-periodic solutions for abstract neutral differential equations, Electron. J. Diff. Equ. 2015 (2015), Article no. 210.

R. Ponce, Asymptotic behavior of mild solutions to fractional Cauchy problems in Banach spaces, Appl. Math. Lett. 105 (2020), 1-9.

R. Ponce, Bounded mild solutions to fractional integro-differential equations in Banach spaces, Semigroup Forum 87 (2013), 377-392.

L. Qiang, X. Wu, Existence and asymptotic behavior of square-mean S-asymptotically periodic solutions for fractional stochastic evolution equation with delay, Fract. Calc. Appl. Anal. 26 316 (2023), 718-750.

X. Shu, F. Xu, Y. Shi, S-asymptotically ω-positive periodic solutions for a class of neutral fractional differential equations, Appl. Math. Comput. 270 (2015), 768-776.

C. Zhang, Integration of vector-valued pseudo almost periodic functions, Proc. Amer. Math. Soc. 121 (1994), no. 1, 167--174.

C. Zhang, Pseudo almost-periodic solutions of some Differential Equations, J. Math. Anal. Appl. 181 (1994), no. 1, 62-76.

C. Zhang, Pseudo almost periodic solutions of some differential equations II, J. Math. Anal. Appl. 192 (1995), no. 2, 543-561.

S. Zhao, M. Song, S-asymptotically ω-periodic solutions in distribution for a class of Stochastic fractional functional differential equations, arXiv : 1609.01453v1 [math.DS](2016).

S. Zhao, M. Song, Square-mean S-asymptotically ω-periodic solutions for a Stochastic fractional evolution equation driven by Levy noise with piecewise constant argument, arXiv : 1609.01444v1 [math.DS] (2016).




DOI: https://doi.org/10.52846/ami.v52i2.2002