About a composite fractional relaxation equation via regularized families

Document Type : Article

Authors

1 Departamento de Matem´atica, Universidade Federal de Pernambuco, Recife-PE, CEP 50540-740, Brazil

2 Departamento de Matem´atica y Estad´ıstica, Universidad de La Frontera, Casilla 54-D, Temuco, Chile

Abstract

This work deals with asymptotic periodicity and compactness for a class of composite fractional relaxation
equation. Some difficulties arises when the effect of different kinds of nonhomogeneous terms are taken into
consideration. To overcome these we use methods coming from regularized families and fixed point techniques,
which are an important tool to study of nonlinear phenomena. We can cover a large class of nonlinearities.

Keywords

Main Subjects


References

1. Bazhlekova, E. \Fractional evolution equations in Banach spaces", PhD. Thesis, Eindhoven University of
Technology (2001).
2. Hilfer, R. and Seybold, H. \Computation of the generalized Mittag-Leer function and its inverse in
the complex plane", Integral Transforms and Special Functions, 17(9), pp. 637-652 (2006).
3. Kilbas, A., Srivastava, H. and Trujillo, J. Theory and Applications of Fractional Di erential Equations,
North Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam (2006).
4. Lizama, C. and N'Guerekata, G. \Mild solutions for
abstract fractional di erential equations", Appl. Anal.,
98(3), pp. 1731-1754 (2013).
5. Lizama, C. \Regularized solutions for abstract
Volterra equations", J. Math. Anal. Appl., 243, pp.
278-292 (2000).
6. Karczewska, A. and Lizama, C. \Solutions to stochastic
fractional relaxation equations", Physica Scripta T,
136, pp. 1-7 (2009).
7. Erdelyi, A., Magnus, W., Oberhettinger, F. and
Tricomi, F., Higher Transcendental Functions, 2,
McGraw-Hill, New York (1953).
8. Henrquez, H., Cuevas, C. and Caicedo, A. \Asymptotically
periodic solutions of neutral partial di erential
equations with in nite delay", Commun. Pure Appl.
Anal., 12(5), pp. 2031-2068 (2013).
9. Henrquez, H., Pierri, M. and Taboas, P. \On Sasymptotically
!-periodic functions on Banach spaces
and applications", J. Math. Anal. Appl., 343(2), pp.
1119-1130 (2008).
10. Pierri, M. and Rolnik, V. \On pseudo S-asymptotically
!-periodic functions", Bull. Aust. Math. Soc., 87(2),
pp. 238-254 (2013).
11. Cuevas, C., Henrquez, H. and Soto, H. \Asymptotically
periodic solutions of fractional di erential
equations", Appl. Math. Comput., 236, pp. 524-545
(2014).
338 A. Aparcana et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 329{338
12. De Andrade, B., Cuevas, C., Silva, C. and Soto, H.
\Asymptotic periodicity for
exible structural systems
and applications", Acta Appl. Math., 143, pp. 105-164
(2016).
13. Andrade, F., Cuevas, C., Silva, C. and Soto, H.
\Asymptotic periodicity for hyperbolic evolution equations
and applications", Appl. Math. Comput., 269,
pp. 169-195 (2015).
14. Cuevas, C. and Lizama, C. \S-asymptotically !-
periodic solutions for semilinear Volterra equations",
Math. Meth. Appl. Sci., 33, pp. 1628-1636 (2010).
15. Miller, R.K., Nonlinear Volterra Integral Equations,
W.A. Benjamin, Inc., California (1971).
16. De Andrade, B., Cuevas, C. and Henrquez, E.
\Asymptotic periodicity and almost automorphy for a
class of Volterra integro-di erential equations", Math.
Meth. Appl. Sci., 35, pp. 795-811 (2012).
17. Henrquez, H., Poblete, V. and Pozo, J. \Mild solutions
of non-autonomous second order problems with
nonlocal initial conditions", J. Math. Anal. Appl., 412,
pp. 1064-1083 (2014).