On the nonlocal Cauchy problem for semilinear fractional order evolution equations

JinRong Wang; Yong Zhou; Michal Fečkan

Open Mathematics (2014)

  • Volume: 12, Issue: 6, page 911-922
  • ISSN: 2391-5455

Abstract

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In this paper, we develop the approach and techniques of [Boucherif A., Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 2007, 16(3), 507–516], [Zhou Y., Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinar Anal. Real World Appl., 2010, 11(5), 4465–4475] to deal with nonlocal Cauchy problem for semilinear fractional order evolution equations. We present two new sufficient conditions on existence of mild solutions. The first result relies on a growth condition on the whole time interval via Schaefer fixed point theorem. The second result relies on a growth condition splitted into two parts, one for the subinterval containing the points associated with the nonlocal conditions, and the other for the rest of the interval via O’Regan fixed point theorem.

How to cite

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JinRong Wang, Yong Zhou, and Michal Fečkan. "On the nonlocal Cauchy problem for semilinear fractional order evolution equations." Open Mathematics 12.6 (2014): 911-922. <http://eudml.org/doc/269554>.

@article{JinRongWang2014,
abstract = {In this paper, we develop the approach and techniques of [Boucherif A., Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 2007, 16(3), 507–516], [Zhou Y., Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinar Anal. Real World Appl., 2010, 11(5), 4465–4475] to deal with nonlocal Cauchy problem for semilinear fractional order evolution equations. We present two new sufficient conditions on existence of mild solutions. The first result relies on a growth condition on the whole time interval via Schaefer fixed point theorem. The second result relies on a growth condition splitted into two parts, one for the subinterval containing the points associated with the nonlocal conditions, and the other for the rest of the interval via O’Regan fixed point theorem.},
author = {JinRong Wang, Yong Zhou, Michal Fečkan},
journal = {Open Mathematics},
keywords = {Fractional order evolution equations; Nonlocal Cauchy problem; Mild solution; Existence; fractional order evolution equations; nonlocal Cauchy problem; mild solution; existence},
language = {eng},
number = {6},
pages = {911-922},
title = {On the nonlocal Cauchy problem for semilinear fractional order evolution equations},
url = {http://eudml.org/doc/269554},
volume = {12},
year = {2014},
}

TY - JOUR
AU - JinRong Wang
AU - Yong Zhou
AU - Michal Fečkan
TI - On the nonlocal Cauchy problem for semilinear fractional order evolution equations
JO - Open Mathematics
PY - 2014
VL - 12
IS - 6
SP - 911
EP - 922
AB - In this paper, we develop the approach and techniques of [Boucherif A., Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 2007, 16(3), 507–516], [Zhou Y., Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinar Anal. Real World Appl., 2010, 11(5), 4465–4475] to deal with nonlocal Cauchy problem for semilinear fractional order evolution equations. We present two new sufficient conditions on existence of mild solutions. The first result relies on a growth condition on the whole time interval via Schaefer fixed point theorem. The second result relies on a growth condition splitted into two parts, one for the subinterval containing the points associated with the nonlocal conditions, and the other for the rest of the interval via O’Regan fixed point theorem.
LA - eng
KW - Fractional order evolution equations; Nonlocal Cauchy problem; Mild solution; Existence; fractional order evolution equations; nonlocal Cauchy problem; mild solution; existence
UR - http://eudml.org/doc/269554
ER -

References

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  1. [1] Baleanu D., Machado J.A.T., Luo A.C.J. (Eds.), Fractional Dynamics and Control, Springer, New York, 2012 
  2. [2] Boucherif A., Precup R., On the nonlocal initial value problem for first order differential equations, Fixed Point Theory, 2003, 4(2), 205–212 Zbl1050.34001
  3. [3] Boucherif A., Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 2007, 16(3), 507–516 Zbl1154.34027
  4. [4] Boulite S., Idrissi A., Maniar L., Controllability of semilinear boundary problems with nonlocal initial conditions, J. Math. Anal. Appl., 2006, 316(2), 566–578 http://dx.doi.org/10.1016/j.jmaa.2005.05.006 Zbl1105.34036
  5. [5] Byszewski L., Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 1991, 162(2), 494–505 http://dx.doi.org/10.1016/0022-247X(91)90164-U Zbl0748.34040
  6. [6] Byszewski L., Lakshmikantham V., Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal., 1991, 40(1), 11–19 http://dx.doi.org/10.1080/00036819008839989 Zbl0694.34001
  7. [7] Chang Y.-K., Nieto J.J., Li W.-S., On impulsive hyperbolic differential inclusions with nonlocal initial conditions, J. Optim. Theory Appl., 2009, 140(3), 431–442 http://dx.doi.org/10.1007/s10957-008-9468-1 Zbl1159.49042
  8. [8] Deng K., Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 1993, 179(2), 630–637 http://dx.doi.org/10.1006/jmaa.1993.1373 Zbl0798.35076
  9. [9] Diethelm K., The Analysis of Fractional Differential Equations, Lecture Notes in Math., 2004, Springer, Berlin, 2010 http://dx.doi.org/10.1007/978-3-642-14574-2 
  10. [10] Dong X., Wang J., Zhou Y., On nonlocal problems for fractional differential equations in Banach spaces, Opuscula Math., 2011, 31(3), 341–357 http://dx.doi.org/10.7494/OpMath.2011.31.3.341 Zbl1228.26012
  11. [11] Fan Z., Impulsive problems for semilinear differential equations with nonlocal conditions, Nonlinear Anal., 2010, 72(2), 1104–1109 http://dx.doi.org/10.1016/j.na.2009.07.049 Zbl1188.34073
  12. [12] Fan Z., Li G., Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Funct. Anal., 2010, 258(5), 1709–1727 http://dx.doi.org/10.1016/j.jfa.2009.10.023 Zbl1193.35099
  13. [13] Fu X., Ezzinbi K., Existence of solutions for neutral functional differential evolution equations with nonlocal conditions, Nonlinear Anal., 2003, 54(2), 215–227 http://dx.doi.org/10.1016/S0362-546X(03)00047-6 Zbl1034.34096
  14. [14] Jackson D., Existence and uniqueness of solutions to semilinear nonlocal parabolic equations, J. Math. Anal. Appl., 1993, 172(1), 256–265 http://dx.doi.org/10.1006/jmaa.1993.1022 Zbl0814.35060
  15. [15] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud., 204, Elsevier, Amsterdam, 2006 http://dx.doi.org/10.1016/S0304-0208(06)80001-0 
  16. [16] Lakshmikantham V., Leela S., Devi J.V., Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cottenham, 2009 Zbl1188.37002
  17. [17] Liang J., Liu J., Xiao T.-J., Nonlocal Cauchy problems governed by compact operator families, Nonlinear Anal. TMA, 1994, 57(2), 183–189 http://dx.doi.org/10.1016/j.na.2004.02.007 Zbl1083.34045
  18. [18] Liu H., Chang J.-C., Existence for a class of partial differential equations with nonlocal conditions, Nonlinear Anal., 2009, 70(9), 3076–3083 http://dx.doi.org/10.1016/j.na.2008.04.009 Zbl1170.34346
  19. [19] Michalski M.W., Derivatives of Noninteger Order and Their Applications, Dissertationes Math. (Rozprawy Mat.), 328, Polish Academy of Sciences, Warsaw, 1993 Zbl0880.26007
  20. [20] Miller K.S., Ross B., An introduction to the fractional calculus and differential equations, Wiley-Intersci. Publ., John Wiley & Sons, New York, 1993 Zbl0789.26002
  21. [21] N’Guérékata G.M., A Cauchy problem for some fractional differential abstract differential equation with non local conditions, Nonlinear Anal., 2009, 70(5), 1873–1876 http://dx.doi.org/10.1016/j.na.2008.02.087 Zbl1166.34320
  22. [22] N’Guérékata G.M., Corrigendum: A Cauchy problem for some fractional differential equations, Commun. Math. Anal., 2009, 7(1), 11 
  23. [23] Nica O., Initial value problems for first-order differential systems with general nonlocal conditions, Electron. J. Differential Equations, 2012, #74 Zbl1261.34016
  24. [24] Nica O., Precup R., On the nonlocal initial value problem for first order differential systems, Stud. Univ. Babe?-Bolyai Math., 2001, 56(3), 113–125 
  25. [25] Ntouyas S.K., Tsamatos P.Ch., Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl., 1997, 210(2), 679–687 http://dx.doi.org/10.1006/jmaa.1997.5425 
  26. [26] O’Regan D., Fixed-point theory for the sum of two operators, Appl. Math. Lett., 1996, 9(1), 1–8 http://dx.doi.org/10.1016/0893-9659(95)00093-3 
  27. [27] Podlubny I., Fractional Differential Equations, Math. Sci. Engrg., 198, Academic Press, San Diego, 1999 
  28. [28] Smart D.R., Fixed Point Theorems, Cambridge Tracts in Math., 66, Cambridge University Press, London-New York, 1974 
  29. [29] Tarasov V.E., Fractional Dynamics, Nonlinear Phys. Sci., Springer, Heidelberg, 2010 http://dx.doi.org/10.1007/978-3-642-14003-7 
  30. [30] Tatar N., Existence results for an evolution problem with fractional nonlocal conditions, Comput. Math. Appl., 2010, 60(11), 2971–2982 http://dx.doi.org/10.1016/j.camwa.2010.09.057 Zbl1207.34099
  31. [31] Wang J., Zhou Y., A class of fractional evolution equations and optimal controls, Nonlinear Anal. Real World Appl., 2011, 12(1), 262–272 http://dx.doi.org/10.1016/j.nonrwa.2010.06.013 Zbl1214.34010
  32. [32] Wang J., Zhou Y., Analysis of nonlinear fractional control systems in Banach spaces, Nonlinear Anal., 2011, 74(17), 5929–5942 http://dx.doi.org/10.1016/j.na.2011.05.059 Zbl1223.93059
  33. [33] Wang J., Zhou Y., Fečkan M., Alternative results and robustness for fractional evolution equations with periodic boundary conditions, Electron. J. Qual. Theory Diff. Equ., 2011, #97 Zbl06528101
  34. [34] Wang J., Zhou Y., Fečkan M., Abstract Cauchy problem for fractional differential equations, Nonlinear Dynam., 2013, 71(4), 685–700 http://dx.doi.org/10.1007/s11071-012-0452-9 Zbl1268.34034
  35. [35] Xue X., Nonlinear differential equations with nonlocal conditions in Banach spaces, Nonlinear Anal., 2005, 63(4), 575–586 http://dx.doi.org/10.1016/j.na.2005.05.019 Zbl1095.34040
  36. [36] Zhou Y., Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinar Anal. Real World Appl., 2010, 11(5), 4465–4475 http://dx.doi.org/10.1016/j.nonrwa.2010.05.029 Zbl1260.34017
  37. [37] Zhou Y., Jiao F., Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 2010, 59(3), 1063–1077 http://dx.doi.org/10.1016/j.camwa.2009.06.026 Zbl1189.34154

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