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Applicable Analysis
An International Journal
Volume 95, 2016 - Issue 6
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Articles

Almost automorphic mild solutions to fractional partial difference-differential equations

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Pages 1347-1369 | Received 02 Mar 2015, Accepted 13 Jun 2015, Published online: 07 Jul 2015

References

  • Atici FM, Eloe PW. Initial value problems in discrete fractional calculus. Proc. Amer. Math. Soc. 2009;137:981–989.
  • Ferreira R. Calculus of variations on time scales and discrete fractional calculus [PhD thesis]. Aveiro: Universidade de Aveiro; 2010.
  • Ferreira R. Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one. J. Differ. Equ. Appl. 2013;19:712–718.
  • Holm MT. The theory of discrete fractional calculus: development and applications [PhD thesis]. Lincoln: University of Nebraska; 2011.
  • Holm MT. The Laplace transform in discrete fractional calculus. Comput. Math. Appl. 2011;62:1591–1601.
  • Abdeljawad T, Atici FM. On the definition of nabla fractional operator. Abst. Appl. Anal. 2012;2012:1–13.
  • Ortigueira MD, Coito FJV, Trujillo JJ. Discrete-time differential systems. Signal Process. 2015;107:198–217.
  • Goodrich CS. Existence of a positive solution to a system of discrete fractional boundary value problems. Appl. Math. Comput. 2011;217:4740–4753.
  • Goodrich CS. A convexity result for fractional differences. Appl. Math. Lett. 2014;35:58–62.
  • Atici FM, Sengül S. Modeling with fractional difference equations. J. Math. Anal. Appl. 2010;369:1–9.
  • Baleanu D, Diethelm K, Scalas E. Fractional calculus: models and numerical methods. Singapore: World Scientific; 2012.
  • Chandler RE, Herman R, Montroll EW. Traffic dynamics: studies in car following. Oper. Res. 1958;6:165–184.
  • Li Y, Sun D. Microscopic car-following model for the traffic flow: the state of the art. J. Control Theory Appl. 2012;10:133–143.
  • Engel KJ, Nagel R. One-parameter semigroups for linear evolution equations. Vol. 194, Graduate texts in Mathematics. New York (NY): Springer; 2000.
  • Bateman H. Some simple differential difference equations and the related functions. Bull. Amer. Math. Soc. 1943;49:494–512.
  • N’Guérékata GM. Almost automorphic and almost periodic functions in abstract spaces. New York (NY): Kluwer Academic; 2001.
  • Liu JH, N’Guérékata GM, Van Minh N. Topics on stability and periodicity in abstract differential equations. Vol. 6, Series on Concrete and Applicable Mathematics. Singapore: World Scientific; 2008.
  • Diagana T. Almost automorphic type and almost periodic type functions in abstract spaces. Cham: Springer; 2013.
  • Liu Z, Sun K. Almost automorphic solutions for stochastic differential equations driven by Lévy noise. J. Funct. Anal. 2014;266:1115–1149.
  • Campos J, Tarallo M. Almost automorphic linear dynamics by Favard theory. J. Differ. Equ. 2014;256:1350–1367.
  • Lizama C, Mesquita JG. Almost automorphic solutions of dynamic equations on time scales. J. Funct. Anal. 2013;265:2267–2311.
  • Fu M, Chen F. Almost automorphic solutions for some stochastic differential equations. Nonlinear Anal. 2013;80:66–75.
  • Xia Z. Weighted pseudo almost automorphic solutions of hyperbolic semilinear integro-differential equations. Nonlinear Anal. 2014;95:50–65.
  • Corduneanu C. Almost periodic oscillations and waves. New York (NY): Springer; 2009.
  • Ding HS, Fu JD, N’Guérékata GM. Positive almost periodic type solutions to a class of nonlinear difference equations. Electron. J. Qual. Theory Differ. Equ. 2011;2011:1–16.
  • Long W, Pan WH. Asymptotically almost periodic solution to a class of Volterra difference equations. Adv. Differ. Equ. 2012;2012:1–12.
  • Minh NV, Naito T, N’Guérékata G. A spectral countability condition for almost automorphy of solutions of differential equations. Proc. Amer. Math. Soc. 2006;134:3257–3266.
  • Caraballo T, Cheban D. Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard’s separation condition. I. J. Differ. Equ. 2009;246:108–128.
  • Caraballo T, Cheban D. Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard’s separation condition. II. J. Differ. Equ. 2009;246:1164–1186.
  • Araya D, Castro R, Lizama C. Almost automorphic solutions of difference equations. Adv. Differ. Equ. 2009;2009:1–15.
  • Li X, Sun X. Existence and uniqueness of weighted pseudo almost automorphic sequence solutions to some semilinear difference equations. Ann. Differ. Equ. 2011;27:183–189.
  • Diagana T. Existence of globally attracting almost automorphic solutions to some nonautonomous higher-order difference equations. Appl. Math. Comput. 2013;219:6510–6519.
  • Lizama C, Mesquita JG. Almost automorphic solutions of non-autonomous difference equations. J. Math. Anal. Appl. 2013;407:339–349.
  • Abbas S, Chang Y-K, Hafayed M. Stepanov type weighted pseudo almost automorphic sequences and their applications to difference equations. Nonlinear Stud. 2014;21:99–111.
  • Yi Y. On almost automorphic oscillations. Vol. 42, Fields Institute Communications. Providence (RI): American Mathematical Society; 2004. p. 75–99.
  • Calzadillas S, Lizama C, Mesquita JG. A unified approach to discrete fractional calculus and applications. Submitted, 2014.
  • Cuesta E, Palencia C. A numerical method for an integro-differential equation with memory in Banach spaces: qualitative properties. SIAM J. Numer. Anal. 2003;41:1232–1241.
  • Lizama C. The Poisson distribution, abstract fractional difference equations, and stability. Proc. Amer. Math. Soc. Forthcoming.
  • Zygmund A. Trigonometric series. 2nd ed. Vol. I, II. New York (NY): Cambridge University Press; 1959.
  • Lizama C. lp-maximal regularity for fractional difference equations on UMD spaces. Math. Nachr. Forthcoming.
  • Agarwal RP, Cuevas C, Dantas F. Almost automorphy profile of solutions for difference equations of Volterra type. J. Appl. Math. Comput. 2013;42:1–18.
  • Yosida K. Functional analysis. Berlin: Springer Verlag; 1980.
  • Podlubny I. Fractional differential equations. Vol. 198, Mathematics in Science and Engineering. San Diego (CA): Academic Press; 1999.
  • Mainardi F. Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. London: Imperial College Press; 2010.
  • Ponce R. Bounded mild solutions to fractional integro-differential equations in Banach spaces. Semigroup Forum. 2013;79:377–392.
  • Gradshteyn IS, Ryzhik IM. Table of integrals, series and products. 7th ed. London: Academic Press; 2007.
  • Pazy A. Semigroups of linear operators and applications to partial differential equations. Vol. 44, Applied Mathematical Sciences. New York (NY): Springer; 1983.

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