On the nonlocal Cauchy problem for semilinear fractional order evolution equations

详细信息    查看全文

• 作者：JinRong Wang (1) (2)
  Yong Zhou (3)
  Michal Fe?kan (4) (5)
  
• 关键词：26A33 ; 34A12 ; 47D06 ; 34G20 ; Fractional order evolution equations ; Nonlocal Cauchy problem ; Mild solution ; Existence
• 刊名：Central European Journal of Mathematics
• 出版年：2014
• 出版时间：June 2014
• 年：2014
• 卷：12
• 期：6
• 页码：911-922
• 全文大小：1,164 KB
• 参考文献：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), 205-12
  3. Boucherif A., Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 2007, 16(3), 507-16
  4. Boulite S., Idrissi A., Maniar L., Controllability of semilinear boundary problems with nonlocal initial conditions, J. Math. Anal. Appl., 2006, 316(2), 566-78 g/10.1016/j.jmaa.2005.05.006" target="_blank" title="It opens in new window">CrossRef
  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), 494-05 g/10.1016/0022-247X(91)90164-U" target="_blank" title="It opens in new window">CrossRef
  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), 11-9 g/10.1080/00036819008839989" target="_blank" title="It opens in new window">CrossRef
  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), 431-42 g/10.1007/s10957-008-9468-1" target="_blank" title="It opens in new window">CrossRef
  8. Deng K., Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 1993, 179(2), 630-37 g/10.1006/jmaa.1993.1373" target="_blank" title="It opens in new window">CrossRef
  9. Diethelm K., The Analysis of Fractional Differential Equations, Lecture Notes in Math., 2004, Springer, Berlin, 2010 g/10.1007/978-3-642-14574-2" target="_blank" title="It opens in new window">CrossRef
  10. Dong X., Wang J., Zhou Y., On nonlocal problems for fractional differential equations in Banach spaces, Opuscula Math., 2011, 31(3), 341-57 g/10.7494/OpMath.2011.31.3.341" target="_blank" title="It opens in new window">CrossRef
  11. Fan Z., Impulsive problems for semilinear differential equations with nonlocal conditions, Nonlinear Anal., 2010, 72(2), 1104-109 g/10.1016/j.na.2009.07.049" target="_blank" title="It opens in new window">CrossRef
  12. Fan Z., Li G., Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Funct. Anal., 2010, 258(5), 1709-727 g/10.1016/j.jfa.2009.10.023" target="_blank" title="It opens in new window">CrossRef
  13. Fu X., Ezzinbi K., Existence of solutions for neutral functional differential evolution equations with nonlocal conditions, Nonlinear Anal., 2003, 54(2), 215-27 g/10.1016/S0362-546X(03)00047-6" target="_blank" title="It opens in new window">CrossRef
  14. Jackson D., Existence and uniqueness of solutions to semilinear nonlocal parabolic equations, J. Math. Anal. Appl., 1993, 172(1), 256-65 g/10.1006/jmaa.1993.1022" target="_blank" title="It opens in new window">CrossRef
  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 g/10.1016/S0304-0208(06)80001-0" target="_blank" title="It opens in new window">CrossRef
  16. Lakshmikantham V., Leela S., Devi J.V., Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cottenham, 2009
  17. Liang J., Liu J., Xiao T.-J., Nonlocal Cauchy problems governed by compact operator families, Nonlinear Anal. TMA, 1994, 57(2), 183-89 g/10.1016/j.na.2004.02.007" target="_blank" title="It opens in new window">CrossRef
  18. Liu H., Chang J.-C., Existence for a class of partial differential equations with nonlocal conditions, Nonlinear Anal., 2009, 70(9), 3076-083 g/10.1016/j.na.2008.04.009" target="_blank" title="It opens in new window">CrossRef
  19. Michalski M.W., Derivatives of Noninteger Order and Their Applications, Dissertationes Math. (Rozprawy Mat.), 328, Polish Academy of Sciences, Warsaw, 1993
  20. Miller K.S., Ross B., An introduction to the fractional calculus and differential equations, Wiley-Intersci. Publ., John Wiley & Sons, New York, 1993
  21. N’Guérékata G.M., A Cauchy problem for some fractional differential abstract differential equation with non local conditions, Nonlinear Anal., 2009, 70(5), 1873-876 g/10.1016/j.na.2008.02.087" target="_blank" title="It opens in new window">CrossRef
  22. N’Guérékata G.M., Corrigendum: A Cauchy problem for some fractional differential equations, Commun. Math. Anal., 2009, 7(1), 11
  23. Nica O., Initial value problems for first-order differential systems with general nonlocal conditions, Electron. J. Differential Equations, 2012, #74
  24. Nica O., Precup R., On the nonlocal initial value problem for first order differential systems, Stud. Univ. Babe?-Bolyai Math., 2001, 56(3), 113-25
  25. Ntouyas S.K., Tsamatos P.Ch., Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl., 1997, 210(2), 679-87 g/10.1006/jmaa.1997.5425" target="_blank" title="It opens in new window">CrossRef
  26. O’Regan D., Fixed-point theory for the sum of two operators, Appl. Math. Lett., 1996, 9(1), 1- g/10.1016/0893-9659(95)00093-3" target="_blank" title="It opens in new window">CrossRef
  27. Podlubny I., Fractional Differential Equations, Math. Sci. Engrg., 198, Academic Press, San Diego, 1999
  28. Smart D.R., Fixed Point Theorems, Cambridge Tracts in Math., 66, Cambridge University Press, London-New York, 1974
  29. Tarasov V.E., Fractional Dynamics, Nonlinear Phys. Sci., Springer, Heidelberg, 2010 g/10.1007/978-3-642-14003-7" target="_blank" title="It opens in new window">CrossRef
  30. Tatar N., Existence results for an evolution problem with fractional nonlocal conditions, Comput. Math. Appl., 2010, 60(11), 2971-982 g/10.1016/j.camwa.2010.09.057" target="_blank" title="It opens in new window">CrossRef
  31. Wang J., Zhou Y., A class of fractional evolution equations and optimal controls, Nonlinear Anal. Real World Appl., 2011, 12(1), 262-72 g/10.1016/j.nonrwa.2010.06.013" target="_blank" title="It opens in new window">CrossRef
  32. Wang J., Zhou Y., Analysis of nonlinear fractional control systems in Banach spaces, Nonlinear Anal., 2011, 74(17), 5929-942 g/10.1016/j.na.2011.05.059" target="_blank" title="It opens in new window">CrossRef
  33. Wang J., Zhou Y., Fe?kan M., Alternative results and robustness for fractional evolution equations with periodic boundary conditions, Electron. J. Qual. Theory Diff. Equ., 2011, #97
  34. Wang J., Zhou Y., Fe?kan M., Abstract Cauchy problem for fractional differential equations, Nonlinear Dynam., 2013, 71(4), 685-00 g/10.1007/s11071-012-0452-9" target="_blank" title="It opens in new window">CrossRef
  35. Xue X., Nonlinear differential equations with nonlocal conditions in Banach spaces, Nonlinear Anal., 2005, 63(4), 575-86 g/10.1016/j.na.2005.05.019" target="_blank" title="It opens in new window">CrossRef
  36. Zhou Y., Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinar Anal. Real World Appl., 2010, 11(5), 4465-475 g/10.1016/j.nonrwa.2010.05.029" target="_blank" title="It opens in new window">CrossRef
  37. Zhou Y., Jiao F., Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 2010, 59(3), 1063-077 g/10.1016/j.camwa.2009.06.026" target="_blank" title="It opens in new window">CrossRef
  
• 作者单位：JinRong Wang (1) (2)
  Yong Zhou (3)
  Michal Fe?kan (4) (5)
  
  1. School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, Guizhou, 550018, China
  2. Department of Mathematics, Guizhou University, Guiyang, Guizhou, 550025, China
  3. Department of Mathematics, Xiangtan University, Xiangtan, Hunan, 411105, China
  4. Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina, Bratislava, 842 48, Slovakia
  5. Mathematical Institute, Slovak Academy of Sciences, ?tefánikova 49, Bratislava, 814 73, Slovakia
  
• ISSN：1644-3616

文摘

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), 507-16], [Zhou Y., Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinar Anal. Real World Appl., 2010, 11(5), 4465-475] 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.