Abstract
In this article, we first present the construction and basic properties of the Bochner integral for vector-valued functions on an arbitrary time scale. Using the properties of the Bochner integral, we develop an \(L^{p}\)-calculus for random processes on time scales, and present some results concerning the sample path and Lebesgue and \(L^{p}\)-integrability of a random process on time scales. Finally, we study random differential equations on time scales in the framework of the pth moment or \(L^{p}\)-calculus. An existence result is considered which gives sufficient conditions under which a sample path solution is also an \(L^{p}\)-solution.
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References
Arendt W, Batty CJK, Hieber M, Neubrander F (2011) Vector-valued laplace transforms and cauchy problems. Birkh äuser, Basel
Aulbach B, Neidhart L (2001) Integration on measure chains. In: Conference Proceedings of the sixth international conference on difference equations and applications, Augsburg
Bohner M, Peterson A (2001) Dynamic equations on time scales. An introduction with applications. Birkhäuser, Boston
Bohner M, Peterson A (2003) Advances in dynamic equations on time scales. Birkhäuser, Boston
Bohner M, Sanyal S (2010) The stochastic dynamic exponential and geometric Brownian motion on isolated time scales. Commun Math Anal 8(3):120–135
Bohner M, Guseinov Sh G (2006) Multiple Lebesgue integration on time scales. Adv Differ Equ 12:Art. ID 26391
Bohner M, Stanzhytskyi OM, Bratochkina AO (2013) Stochastic dynamic equations on general time scales. Electron J Differ Equ 57:1–15
Burgos C, Calatayud J, Cortés JC, Villafuerte L (2018) Solving a class of random non-autonomous linear fractional differential equations by means of a generalized mean square convergent power series. Appl Math Lett 78:95–104
Cabada A, Vivero DR (2005) Criterions for absolute continuity on time scales. J Differ Equ Appl 11(11):1013–1028
Cabada A, Vivero DR (2006) Expression of the Lebesgue \(\Delta \)-integral on time scales as a usual Lebesgue integral; application to the calculus of \(\Delta \)-antiderivatives. Math Comput Modell 43:194–207
Calatayud J, Cortés JC, Jornet M, Villafuerte L (2018) Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties. Adv Differ Equ 2018:392
Cichoń M (2011) On integrals of vector-valued functions on time scales. Commun Math Anal 11(1):94–110
Cohn DL (1980) Measure theory. Birkhäuser, Boston
Cortés JC, Villafuerte L, Burgos C (2017) A mean square chain rule and its application in solving the random Chebyshev differential equation. Mediterr J Math 14:35
Dhage BC (2009) On global existence and attractivity results for nonlinear random integral equations. Panamer Math J 19(1):97–111
Dhage BC (2011) On n\(^{\mathit{th}}\)-order nonlinear ordinary random differential equations. Nonlinear Oscill 13(4):535–549
Dhage BC, Ntouyas SK (2010) Existence and attractivity results for nonlinear first order random differential equations. Opuscula Math 30(4):411–429
Dhama S, Abbas S (2020) Square-mean almost automorphic solution of a stochastic cellular neural network on time scales. J Integral Equ Appl 32(2):151–170
Dhama S, Abbas S, Debbouche A (2020) Doubly-weighted pseudo almost automorphic solutions for stochastic dynamic equations with Stepanov-like coefficients on time scales. Chaos Solitons Fractals 137(8):109–119
Du NH, Tuan LA, Dieu NT (2020) Stability of stochastic dynamic equations with time-varying delay on time scales. Stoch Anal Appl 38(12):1–20
Edsinger R (1968) Random ordinary differential equations. PhD. Thesis, Univ. of California, Berkeley
Gulsen T, Jadlovská I, Yilmaz E (2021) On the number of eigenvalues for parameter-dependent diffusion problem on time scales. Math Methods Appl Sci 44(1):985–992
Heinonen J, Koskela P, Shanmugalingam N, Tyson JT (2015) Sobolev spaces on metric measure spaces. Cambridge University Press, Cambridge
Henríquez HR, Lizama C, Mesquita JG (2020) Semigroups on time scales and applications to abstract Cauchy problems. Topol Methods Nonlinear Anal 56(1):83–115
Hilger S (1988) Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. PhD Thesis, Universität Wü rzburg
Hille E, Phillips R (1937) Functional Analysis and semi-groups. Amer. Math. Soc. College Publ., Providence
Hytönen T, von Neerven J, Veraar M, Weis L (2016) Analysis in Banach spaces, volume I: martingales and Littlewood-Paley theory. Springer, Switzerland
Ladde GD, Lakshmikantham V (1980) Random differential inequalities. Academic Press, New York
Lungan C, Lupulescu V (2012) Random Differential equations on time scales. Electron J Differ Equ 86:1–14
Neidhart L (2001) Integration im Rahmen des Maßkettenkalkül. Diploma thesis, University of Augsburg
Pei Y, Bohner M, Pi D (2020) Impulsive synchronization of time-scales complex networks with time-varying topology. Commun Nonlinear Sci Numer Simulat 80:1–10
Qin G, Wang C (2021) Lebesgue-Stieltjes combined \( \Diamond _{\alpha }\)-measure and integral on time scales. Rev R Acad Cienc Exactas Fís Nat Ser A Mat 115:50
Rynne BP (2007) \(L^{2}\) spaces and boundary value problems on time scales. J Math Anal Appl 328:1217–1236
Sanyal S (2008) Stochastic dynamic equations. PhD dissertation, Missouri University of Science and Technology
Skrzypek E, Szymańska-Dębowska K (2019) On the Lebesgue and Sobolev spaces on a time-scale. Opuscula Math 39(5):705–731
Soong TT (1973) Random differential equations in science and engineering. Academic Press, New York
Strand JL (1968) Stochastic ordinary differential eqnations. PhD Thesis, Univ. of California, Berkeley
Strand JL (1970) Random ordinary differential equations. J Differ Equ 7:538–553
Wu W, Yang L (2020) Impulsive stochastic BAM neural networks on an invariant under a translation time scale. Acta Appl Math 169:647–665
Zhang LL, Yang XD (2021) Bochner definition of Stepanov-like almost automorphic functions on time scales and an application to cellular neural networks with delays. Adv Differ Equ 2021:83
Zhu Y, Jia G (2020) Linear feedback of mean-field stochastic linear quadratic optimal control problems on time scales. Math Probl Eng Article ID 8051918:11
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The authors would like to thank both referees for carefully reading this manuscript and making many important suggestions, leading to a better presentation of the results.
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Communicated by Juan Carlos Cortes.
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Bohner, M., Lupulescu, V., O’Regan, D. et al. Vector-valued functions on time scales and random differential equations. Comp. Appl. Math. 41, 153 (2022). https://doi.org/10.1007/s40314-022-01860-z
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DOI: https://doi.org/10.1007/s40314-022-01860-z