×

On the nonlocal Cauchy problem for semilinear fractional order evolution equations. (English) Zbl 1296.26035

Summary: We develop the approach and techniques of [A. Boucherif and R. Precup, Dyn. Syst. Appl. 16, No. 3, 507–516 (2007; Zbl 1154.34027); Y. Zhou and F. Jiao, Nonlinear Anal., Real World Appl. 11, No. 5, 4465–4475 (2010; Zbl 1260.34017)] to deal with nonlocal Cauchy problem for semilinear fractional order evolution equations. We present two new sufficient conditions on the existence of mild solutions. The first result relies on a growth condition on the whole time interval via Schaefer’s 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’s fixed point theorem.

MSC:

26A33 Fractional derivatives and integrals
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
47D06 One-parameter semigroups and linear evolution equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baleanu D., Machado J.A.T., Luo A.C.J. (Eds.), Fractional Dynamics and Control, Springer, New York, 2012;
[2] Boucherif A., Precup R., On the nonlocal initial value problem for first order differential equations, Fixed Point Theory, 2003, 4(2), 205-212; · Zbl 1050.34001
[3] Boucherif A., Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 2007, 16(3), 507-516; · Zbl 1154.34027
[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; · Zbl 1105.34036
[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; · Zbl 0748.34040
[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; · Zbl 0694.34001
[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; · Zbl 1159.49042
[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; · Zbl 0798.35076
[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] 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; · Zbl 1228.26012
[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; · Zbl 1188.34073
[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; · Zbl 1193.35099
[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; · Zbl 1034.34096
[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; · Zbl 0814.35060
[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; · Zbl 1092.45003
[16] Lakshmikantham V., Leela S., Devi J.V., Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cottenham, 2009; · Zbl 1188.37002
[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; · Zbl 1083.34045
[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; · Zbl 1170.34346
[19] Michalski M.W., Derivatives of Noninteger Order and Their Applications, Dissertationes Math. (Rozprawy Mat.), 328, Polish Academy of Sciences, Warsaw, 1993; · Zbl 0880.26007
[20] Miller K.S., Ross B., An introduction to the fractional calculus and differential equations, Wiley-Intersci. Publ., John Wiley & Sons, New York, 1993; · Zbl 0789.26002
[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; · Zbl 1166.34320
[22] N’Guérékata G.M., Corrigendum: A Cauchy problem for some fractional differential equations, Commun. Math. Anal., 2009, 7(1), 11; · Zbl 1171.34332
[23] Nica O., Initial value problems for first-order differential systems with general nonlocal conditions, Electron. J. Differential Equations, 2012, #74; · Zbl 1261.34016
[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; · Zbl 1274.34041
[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; · Zbl 0884.34069
[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] Podlubny I., Fractional Differential Equations, Math. Sci. Engrg., 198, Academic Press, San Diego, 1999; · Zbl 0924.34008
[28] Smart D.R., Fixed Point Theorems, Cambridge Tracts in Math., 66, Cambridge University Press, London-New York, 1974; · Zbl 0297.47042
[29] Tarasov V.E., Fractional Dynamics, Nonlinear Phys. Sci., Springer, Heidelberg, 2010 http://dx.doi.org/10.1007/978-3-642-14003-7; · Zbl 1186.83027
[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; · Zbl 1207.34099
[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; · Zbl 1214.34010
[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; · Zbl 1223.93059
[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; · Zbl 1340.34207
[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; · Zbl 1268.34034
[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; · Zbl 1095.34040
[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; · Zbl 1260.34017
[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; · Zbl 1189.34154
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.