DOI QR코드

DOI QR Code

THE METHOD OF LOWER AND UPPER SOLUTIONS FOR IMPULSIVE FRACTIONAL EVOLUTION EQUATIONS IN BANACH SPACES

  • Gou, Haide (Department of Mathematics Northwest Normal University) ;
  • Li, Yongxiang (Department of Mathematics Northwest Normal University)
  • Received : 2018.08.02
  • Accepted : 2019.09.11
  • Published : 2019.12.30

Abstract

In this paper, we investigate the existence of mild solutions for a class of fractional impulsive evolution equation with periodic boundary condition by means of the method of upper and lower solutions and monotone iterative method. Using the theory of Kuratowski measure of noncompactness, a series of results about mild solutions are obtained. Finally, two examples are given to illustrate our results.

Acknowledgement

Supported by : National Natural Science Foundation of China

References

  1. 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 Appl. Math. 109 (2010), no. 3, 973-1033. https://doi.org/10.1007/s10440-008-9356-6 https://doi.org/10.1007/s10440-008-9356-6
  2. A. Aghajani, J. Banas, and N. Sabzali, Some generalizations of Darbo fixed point theorem and applications, Bull. Belg. Math. Soc. Simon Stevin 20 (2013), no. 2, 345-358. http://projecteuclid.org/euclid.bbms/1369316549 https://doi.org/10.36045/bbms/1369316549
  3. B. Ahmad and S. Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Anal. Hybrid Syst. 3 (2009), no. 3, 251-258. https://doi.org/10.1016/j.nahs.2009.01.008 https://doi.org/10.1016/j.nahs.2009.01.008
  4. Z. Bai, X. Dong, and C. Yin, Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions, Bound. Value Probl. 2016 (2016), Paper No. 63, 11 pp. https://doi.org/10.1186/s13661-016-0573-z https://doi.org/10.1186/s13661-015-0511-5
  5. K. Balachandran and S. Kiruthika, Existence of solutions of abstract fractional impulsive semilinear evolution equations, Electron. J. Qual. Theory Differ. Equ. 2010 (2010), No. 4, 12 pp. https://doi.org/10.14232/ejqtde.2010.1.4
  6. J. Banas and K. Goebel, Measures of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics, 60, Marcel Dekker, Inc., New York, 1980.
  7. M. Benchohra, J. Henderson, and S. Ntouyas, Impulsive differential equations and inclusions, Contemporary Mathematics and Its Applications, 2, Hindawi Publishing Corporation, New York, 2006. https://doi.org/10.1155/9789775945501
  8. M. Benchohra and D. Seba, Impulsive fractional differential equations in Banach spaces, Electron. J. Qual. Theory Differ. Equ. 2009 (2009), Special Edition I, No. 8, 14 pp. https://doi.org/10.14232/ejqtde.2009.4.8
  9. P. Chen and Y. Li, Mixed monotone iterative technique for a class of semilinear impulsive evolution equations in Banach spaces, Nonlinear Anal. 74 (2011), no. 11, 3578-3588. https://doi.org/10.1016/j.na.2011.02.041 https://doi.org/10.1016/j.na.2011.02.041
  10. P. Chen, Y. Li, Q. Y. Chen, and B. H. Feng, On the initial value problem of fractional evolution equations with noncompact semigroup, Comput. Math. Appl. 67 (2014), no. 5, 1108-1115. https://doi.org/10.1016/j.camwa.2014.01.002 https://doi.org/10.1016/j.camwa.2014.01.002
  11. S. W. Du and V. Lakshmikantham, Monotone iterative technique for differential equations in a Banach space, J. Math. Anal. Appl. 87 (1982), no. 2, 454-459. https://doi.org/10.1016/0022-247X(82)90134-2 https://doi.org/10.1016/0022-247X(82)90134-2
  12. M. Feckan, Y. Zhou, and J. Wang, On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 7, 3050-3060. https://doi.org/10.1016/j.cnsns.2011.11.017 https://doi.org/10.1016/j.cnsns.2011.11.017
  13. D. J. Guo and V. Lakshmikantham, Nonlinear problems in abstract cones, Notes and Reports in Mathematics in Science and Engineering, 5, Academic Press, Inc., Boston, MA, 1988.
  14. D. J. Guo and J. X. Sun, Ordinary Differential Equations in Abstract Spaces. Shandong Science and Technology, Ji'nan (1989) (in Chinese).
  15. H.-P. Heinz, On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal. 7 (1983), no. 12, 1351-1371. https://doi.org/10.1016/0362-546X(83)90006-8 https://doi.org/10.1016/0362-546X(83)90006-8
  16. H. Lakzian, D. Gopal, and W. Sintunavarat, New fixed point results for mappings of contractive type with an application to nonlinear fractional differential equations, J. Fixed Point Theory Appl. 18 (2016), no. 2, 251-266. https://doi.org/10.1007/s11784-015-0275-7 https://doi.org/10.1007/s11784-015-0275-7
  17. B. Li and H. Gou, Monotone iterative method for the periodic boundary value problems of impulsive evolution equations in Banach spaces, Chaos Solitons Fractals 110 (2018), 209-215. https://doi.org/10.1016/j.chaos.2018.03.027 https://doi.org/10.1016/j.chaos.2018.03.027
  18. Y. Li, Positive solutions of abstract semilinear evolution equations and their applications, Acta Math. Sinica (Chin. Ser.) 39 (1996), no. 5, 666-672.
  19. J. Mu, Extremal mild solutions for impulsive fractional evolution equations with nonlocal initial conditions, Bound. Value Probl. 2012 (2012), 71, 12 pp. https://doi.org/10.1186/1687-2770-2012-71 https://doi.org/10.1186/1687-2770-2012-12
  20. J. Mu and Y. Li, Monotone iterative technique for impulsive fractional evolution equations, J. Inequal. Appl. 2011 (2011), 125, 12 pp. https://doi.org/10.1186/1029-242X-2011-125 https://doi.org/10.1186/1029-242X-2011-12
  21. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. https://doi.org/10.1007/978-1-4612-5561-1
  22. X.-B. Shu and Q. Wang, The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order 1 < ${\alpha}$ < 2, Comput. Math. Appl. 64 (2012), no. 6, 2100-2110. https://doi.org/10.1016/j.camwa.2012.04.006 https://doi.org/10.1016/j.camwa.2012.04.006
  23. J. X. Sun and Z. Q. Zhao, Extremal solutions of initial value problem for integro-differential equations of mixed type in Banach spaces, Ann. Differential Equations 8 (1992), no. 4, 469-475.
  24. G.Wang, B. Ahmad, L. Zhang, and J. J. Nieto, Comments on the concept of existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), no. 3, 401-403. https://doi.org/10.1016/j.cnsns.2013.04.003 https://doi.org/10.1016/j.cnsns.2013.04.003
  25. G. Wang, L. Zhang, and G. Song, Systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions, Nonlinear Anal. 74 (2011), no. 3, 974-982. https://doi.org/10.1016/j.na.2010.09.054 https://doi.org/10.1016/j.na.2010.09.054
  26. J. Wang, M. Feckan, and Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dyn. Partial Differ. Equ. 8 (2011), no. 4, 345-361. https://doi.org/10.4310/DPDE.2011.v8.n4.a3 https://doi.org/10.4310/DPDE.2011.v8.n4.a3
  27. J. Wang, M. Feckan, and Y. Zhou, Ulam's type stability of impulsive ordinary differential equations, J. Math. Anal. Appl. 395 (2012), no. 1, 258-264. https://doi.org/10.1016/j.jmaa.2012.05.040 https://doi.org/10.1016/j.jmaa.2012.05.040
  28. J. Wang, X. Li, and W. Wei, On the natural solution of an impulsive fractional differential equation of order q ${\in}$ (1; 2), Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 11, 4384-4394. https://doi.org/10.1016/j.cnsns.2012.03.011 https://doi.org/10.1016/j.cnsns.2012.03.011
  29. J. Wang, Y. Zhou, and M. Feckan, Alternative results and robustness for fractional evolution equations with periodic boundary conditions, Electron. J. Qual. Theory Differ. Equ. 2011 (2011), No. 97, 15 pp. https://doi.org/10.14232/ejqtde.2011.1.97 https://doi.org/10.1186/1687-1847-2011-15
  30. J. Wang, Y. Zhou, and M. Feckan, On recent developments in the theory of boundary value problems for impulsive fractional differential equations, Comput. Math. Appl. 64 (2012), no. 10, 3008-3020. https://doi.org/10.1016/j.camwa.2011.12.064 https://doi.org/10.1016/j.camwa.2011.12.064
  31. J. Wang, Y. Zhou, and M. Feckan, Abstract Cauchy problem for fractional differential equations, Nonlinear Dynam. 71 (2013), no. 4, 685-700. https://doi.org/10.1007/s11071-012-0452-9 https://doi.org/10.1007/s11071-012-0452-9
  32. H. Ye, J. Gao, and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl. 328 (2007), no. 2, 1075-1081. https://doi.org/10.1016/j.jmaa.2006.05.061 https://doi.org/10.1016/j.jmaa.2006.05.061
  33. W.-X. Zhou and Y.-D. Chu, Existence of solutions for fractional differential equations with multi-point boundary conditions, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 3, 1142-1148. https://doi.org/10.1016/j.cnsns.2011.07.019 https://doi.org/10.1016/j.cnsns.2011.07.019