International Journal of Differential Equations

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

Junfei Cao, Zaitang Huang, and Gaston M. N’Guérékata

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

This paper is concerned with the existence of asymptotically almost automorphic mild solutions to a class of abstract semilinear fractional differential equations Dtαxt=Axt+Dtα-1Ft,xt,Bxt,tR, where 1<α<2, A is a linear densely defined operator of sectorial type on a complex Banach space X and B is a bounded linear operator defined on X, F is an appropriate function defined on phase space, and the fractional derivative is understood in the Riemann-Liouville sense. Combining the fixed point theorem due to Krasnoselskii and a decomposition technique, we prove the existence of asymptotically almost automorphic mild solutions to such problems. Our results generalize and improve some previous results since the (locally) Lipschitz continuity on the nonlinearity F is not required. The results obtained are utilized to study the existence of asymptotically almost automorphic mild solutions to a fractional relaxation-oscillation equation.

Article information

Source
Int. J. Differ. Equ., Volume 2018 (2018), Article ID 8243180, 23 pages.

Dates
Received: 21 December 2017
Revised: 18 April 2018
Accepted: 10 May 2018
First available in Project Euclid: 19 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1537322438

Digital Object Identifier
doi:10.1155/2018/8243180

Mathematical Reviews number (MathSciNet)
MR3842713

Zentralblatt MATH identifier
1397.34101

Citation

Cao, Junfei; Huang, Zaitang; N’Guérékata, Gaston M. Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations. Int. J. Differ. Equ. 2018 (2018), Article ID 8243180, 23 pages. doi:10.1155/2018/8243180. https://projecteuclid.org/euclid.ijde/1537322438


Export citation

References

  • S. Bochner, “Continuous mappings of almost automorphic and almost periodic functions,” Proceedings of the National Acadamy of Sciences of the United States of America, vol. 52, pp. 907–910, 1964.MR0168997
  • S. Bochner, “Uniform convergence of monotone sequences of functions,” Proceedings of the National Acadamy of Sciences of the United States of America, vol. 47, pp. 582–585, 1961.MR0126094
  • S. Bochner, “A new approach to almost periodicity,” Proceedings of the National Acadamy of Sciences of the United States of America, vol. 48, pp. 2039–2043, 1962.MR0145283
  • S. Bochner and J. Von Neumann, “On compact solutions of operational-differential equations,” I. Annals of Mathematics: Second Series, vol. 36, no. 1, pp. 255–291, 1935.MR1503222
  • G. M. N'Guérékata, Almost Automorphic Functions and Almost Periodic Functions in Abstract Spaces, Kluwer Academic/Plenum Publishers, New York, London, Moscow, 2001.MR1880351
  • G. M. N'Guerekata, Topics in Almost Automorphy, Springer, New York, NY, USA, 2005.MR2107829
  • W. A. Veech, “Almost automorphic functions,” Proceedings of the National Acadamy of Sciences of the United States of America, vol. 49, pp. 462–464, 1963.MR0152830
  • J. Campos and M. Tarallo, “Almost automorphic linear dynamics by Favard theory,” Journal of Differential Equations, vol. 256, no. 4, pp. 1350–1367, 2014.MR3145760
  • T. Caraballo and D. Cheban, “Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition,” I. Journal of Differential Equations, vol. 246, no. 1, pp. 108–128, 2009.MR2467017
  • L. Mahto and S. Abbas, “PC-almost automorphic solution of impulsive fractional differential equations,” Mediterranean Journal of Mathematics, vol. 12, no. 3, pp. 771–790, 2015.MR3376811
  • D. Araya and C. Lizama, “Almost automorphic mild solutions to fractional differential equationsčommentComment on ref. [11?]: We deleted reference [51] in the original manuscript, which was a repetition of [11]. Consequently we will replace all the citations of [51] within text with those of [11]. Please check.,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 69, no. 11, pp. 3692–3705, 2008.MR2463325
  • G. M. Mophou and G. M. N'Guérékata, “On some classes of almost automorphic functions and applications to fractional differential equations,” Computers & Mathematics with Applications. An International Journal, vol. 59, no. 3, pp. 1310–1317, 2010.MR2579493
  • L. Abadias and C. Lizama, “Almost automorphic mild solutions to fractional partial difference-differential equations,” Applicable Analysis: An International Journal, vol. 95, no. 6, pp. 1347–1369, 2016.MR3479007
  • M. Fu and Z. Liu, “Square-mean almost automorphic solutions for some stochastic differential equations,” Proceedings of the American Mathematical Society, vol. 138, no. 10, pp. 3689–3701, 2010.MR2661567
  • J. Cao, Q. Yang, and Z. Huang, “Existence and exponential stability of almost automorphic mild solutions for stochastic functional differential equations,” Stochastics. An International Journal of Probability and Stochastic Processes, vol. 83, no. 3, pp. 259–275, 2011.MR2810592
  • Z. Liu and K. Sun, “Almost automorphic solutions for stochastic differential equations driven by Lévy noise,” Journal of Functional Analysis, vol. 266, no. 3, pp. 1115–1149, 2014.MR3146813
  • G. M. N'guérékata, “Comments on almost automorphic and almost periodic functions in Banach spaces,” Far East Journal of Mathematical Sciences (FJMS), vol. 17, no. 3, pp. 337–344, 2005.MR2183191
  • G. M. N'Guérékata, “Sur les solutions presqu automorphes d'équations différentielles abstraites,” Annales des Sciences Mathématiques du Québec, no. 1, pp. 69–79, 1981.MR626577
  • D. Bugajewski and G. M. N'Guérékata, “On the topological structure of almost automorphic and asymptotically almost automorphic solutions of differential and integral equations in abstract spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 59, no. 8, pp. 1333–1345, 2004.
  • T. Diagana, E. M. Hernández, and J. dos Santos, “Existence of asymptotically almost automorphic solutions to some abstract partial neutral integro-differential equations,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 71, no. 1-2, pp. 248–257, 2009.MR2518032
  • H.-S. Ding, T.-J. Xiao, and J. Liang, “Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions,” Journal of Mathematical Analysis and Applications, vol. 338, no. 1, pp. 141–151, 2008.MR2386405
  • J.-Q. Zhao, Y.-K. Chang, and G. M. N'guérékata, “Existence of asymptotically almost automorphic solutions to nonlinear delay integral equations,” Dynamic Systems and Applications, vol. 21, no. 2-3, pp. 339–349, 2012.MR2918384
  • Y.-K. Chang and C. Tang, “Asymptotically almost automorphic solutions to stochastic differential equations driven by a Lévy process,” Stochastics. An International Journal of Probability and Stochastic Processes, vol. 88, no. 7, pp. 980–1011, 2016.MR3529857
  • Z.-H. Zhao, Y.-K. Chang, and J. J. Nieto, “Square-mean asymptotically almost automorphic process and its application to stochastic integro-differential equations,” Dynamic Systems and Applications, vol. 22, no. 2-3, pp. 269–284, 2013.MR3100203
  • G. M. N'Guérékata, Spectral Theory for Bounded Functions and Applications to Evolution Equations, Nova Science Publishers, NY, USA, 2017.MR3676744
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 24 of North-Holland Mathematics Studies, Elsevier Science B.V., Am-sterdam, 2006.MR2218073
  • S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integral and derivatives: Theory and applications, Gordon and Breach Science Publishers, Switzerland, 1993.MR1347689
  • K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer Verlag, Berlin, Heidelberg, Germany, 2010.
  • R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.MR1890104
  • I. Podlubny, Fractional Differential Equations, Academic Press, CA, USA, 1999.MR1658022
  • K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley Interscience Publication, John Wiley & Sons, NY, USA, 1993.MR1219954
  • Y. Zhou, Basicheory of Fractional Diferential Equations, World Scientiic, Singapore, 2014.
  • R. P. Agarwal, M. Belmekki, and M. Benchohra, “A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative,” Advances in Difference Equations, vol. 2009, Article ID 981728, 47 pages, 2009.MR2505633
  • R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, “On the concept of solution for fractional differential equations with uncertainty,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 72, no. 6, pp. 2859–2862, 2010.MR2580143
  • M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, “Existence results for fractional order functional differential equations with infinite delay,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1340–1350, 2008.MR2386501
  • R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973–1033, 2010.MR2596185
  • M. M. El-Borai, “Some probability densities and fundamental solutions of fractional evolution equations,” Chaos, Solitons & Fractals, vol. 14, no. 3, pp. 433–440, 2002.MR1903295
  • V. Lakshmikantham, “Theory of fractional functional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 60, pp. 3337–3343, 2008.MR2450543
  • V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 69, no. 8, pp. 2677–2682, 2008.MR2446361
  • V. Lakshmikantham and A. S. Vatsala, “Theory of fractional differential inequalities and applications,” Communications in Applied Analysis, vol. 11, no. 3-4, pp. 395–402, 2007.MR2368191
  • V. Lakshmikantham and J. V. Devi, “Theory of fractional differential equations in a Banach space,” European Journal of Pure and Applied Mathematics, vol. 1, no. 1, pp. 38–45, 2008.MR2379647
  • G. Mophou, O. Nakoulima, and G. M. N'Guérékata, “Existence results for some fractional differential equations with nonlocal conditions,” Nonlinear Studies. The International Journal, vol. 17, no. 1, pp. 15–21, 2010.MR2647799
  • G. M. Mophou and G. M. N'Guérékata, “Existence of the mild solution for some fractional differential equations with nonlocal conditions,” Semigroup Forum, vol. 79, no. 2, pp. 315–322, 2009.MR2538728
  • G. M. Mophou and G. M. N'Guérékata, “On integral solutions of some nonlocal fractional differential equations with nondense domain,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 71, no. 10, pp. 4668–4675, 2009.MR2548700
  • G. M. Mophou, “Existence and uniqueness of mild solutions to impulsive fractional differential equations,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 72, no. 3-4, pp. 1604–1615, 2010.MR2577561
  • G. M. N'Guérékata, “A Cauchy problem for some fractional abstract differential equation with non local conditions,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 70, no. 5, pp. 1873–1876, 2009.MR2492125
  • Y. Zhou and L. Peng, “On the time-fractional Navier-Stokes equations,” Computers & Mathematics with Applications. An International Journal, vol. 73, no. 6, pp. 874–891, 2017.MR3623092
  • Y. Zhou and L. Peng, “Weak solutions of the time-fractional Navier-Stokes equations and optimal control,” Computers & Mathematics with Applications. An International Journal, vol. 73, no. 6, pp. 1016–1027, 2017.MR3623103
  • Y. Zhou and L. Zhang, “Existence and multiplicity results of homoclinic solutions for fractional Hamiltonian systems,” Computers & Mathematics with Applications. An International Journal, vol. 73, no. 6, pp. 1325–1345, 2017.MR3623125
  • Y. Zhou, V. Vijayakumar, and R. Murugesu, “Controllability for fractional evolution inclusions without compactness,” Evolution Equations and Control Theory, vol. 4, no. 4, pp. 507–524, 2015.MR3461697
  • C. Cuevas and C. Lizama, “Almost automorphic solutions to a class of semilinear fractional differential equations,” Applied Mathematics Letters, vol. 21, no. 12, pp. 1315–1319, 2008.MR2464387
  • R. P. Agarwal, B. de Andrade, and C. Cuevas, “On type of periodicity and ergodicity to a class of fractional order differential equations,” Advances in Difference Equations, Article ID 179750, 2010.MR2595646
  • R. P. Agarwal, C. Cuevas, and H. Soto, “Pseudo-almost periodic solutions of a class of semilinear fractional differential equations,” Applied Mathematics and Computation, vol. 37, no. 1-2, pp. 625–634, 2011.MR2831559
  • R. P. Agarwal, B. de Andrade, and C. Cuevas, “Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 3532–3554, 2010.MR2683811
  • H.-S. Ding, J. Liang, and T.-J. Xiao, “Almost automorphic solutions to abstract fractional differential equations,” Advances in Difference Equations, Article ID 508374, 2010.MR2595645
  • C. Cuevas, A. Sepúlveda, and H. Soto, “Almost periodic and pseudo-almost periodic solutions to fractional differential and integro-differential equations,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1735–1745, 2011.MR2831397
  • Y.-K. Chang, R. Zhang, and G. M. N'Guérékata, “Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations,” Computers & Mathematics with Applications. An International Journal, vol. 64, no. 10, pp. 3160–3170, 2012.MR2989344
  • B. He, J. Cao, and B. Yang, “Weighted Stepanov-like pseudo-almost automorphic mild solutions for semilinear fractional differential equations,” Advances in Difference Equations, vol. 2015, 74 pages, 2015.MR3317932
  • J. Cao, Q. Yang, and Z. Huang, “Existence of anti-periodic mild solutions for a class of semilinear fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 277–283, 2012.MR2826006
  • C. Lizama and F. Poblete, “Regularity of mild solutions for a class of fractional order differential equations,” Applied Mathematics and Computation, vol. 224, pp. 803–816, 2013.MR3127665
  • Z. Xia, M. Fan, and R. P. Agarwal, “Pseudo almost automorphy of semilinear fractional differential equations in Banach spaces,” Fractional Calculus and Applied Analysis, vol. 19, no. 3, pp. 741–764, 2016.MR3563608
  • S. Abbas, V. Kavitha, and R. Murugesu, “Stepanov-like weighted pseudo almost automorphic solutions to fractional order abstract integro-differential equations,” Journal of Fractional Calculus and Applications, vol. 4, pp. 1–19, 2013.
  • G. M. Mophou, “Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations,” Applied Mathematics and Computation, vol. 217, no. 19, pp. 7579–7587, 2011.MR2799772
  • Y.-K. Chang and X.-X. Luo, “Pseudo almost automorphic behavior of solutions to a semi-linear fractional differential equation,” Mathematical Communications, vol. 20, no. 1, pp. 53–68, 2015.MR3352474
  • V. Kavitha, S. Abbas, and R. Murugesu, “($\mu $1,$\mu $2)-pseudo almost automorphic solutions of fractional order neutral integro-differential equations,” Nonlinear Stud, vol. 24, pp. 669–685, 2017.MR3701881
  • V. Kavitha, S. Abbas, and R. Murugesu, “Asymptotically almost automorphic solutions of fractional order neutral integro-differential equations,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 39, no. 3, pp. 1075–1088, 2016.MR3515067
  • S. Abbas, “Weighted pseudo almost automorphic solutions of fractional functional differential equations,” Cubo (Temuco), vol. 16, pp. 21–35, 2014.MR3185785
  • E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Eindhoven University of Technology, 2001.MR1868564
  • S. D. Eidelman and A. N. Kochubei, “Cauchy problem for fractional diffusion equations,” Journal of Differential Equations, vol. 199, no. 2, pp. 211–255, 2004.MR2047909
  • R. Gorenflo and F. Mainardi, “Fractional calculus: Integral and differential equations of fractional order,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., pp. 223–276, Springer-Verlag, NY, USA, Vienna, Austria, 1997.MR1611585
  • V. V. Anh and R. Mcvinish, “Fractional differential equations driven by Lévy noise,” Journal of Applied Mathematics and Stochastic Analysis, vol. 16, no. 2, pp. 97–119, 2003.MR1989577
  • H.-S. Ding, J. Liang, and T.-J. Xiao, “Some properties of Stepanov-like almost automorphic functions and applications to abstract evolution equations,” Applicable Analysis: An International Journal, vol. 88, no. 7, pp. 1079–1091, 2009.MR2561476
  • J. Liang, J. Zhang, and T.-J. Xiao, “Composition of pseudo almost automorphic and asymptotically almost automorphic functions,” Journal of Mathematical Analysis and Applications, vol. 340, no. 2, pp. 1493–1499, 2008.MR2390946
  • M. Haase, “The functional calculus for sectorial operators,” in Operator Theory: Advances and Applications, vol. 169, Birkhuser Verlag, Basel, Switzerland, 2006.
  • E. Cuesta, “Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations,” Discrete and Continuous Dynamical Systems - Series A, pp. 277–285, 2007.MR2409222
  • W. M. Ruess and W. H. Summers, “Compactness in spaces of vector valued continuous functions and asymptotic almost periodicity,” Mathematische Nachrichten, vol. 135, pp. 7–33, 1988.MR944213
  • D. R. Smart, Fixed Point Theorems, Cambridge University Press, London, UK, 1980.MR0467717 \endinput