Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
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NONLOCAL FRACTIONAL DIFFERENTIAL INCLUSIONS WITH IMPULSE EFFECTS AND DELAY

  • ALSARORI, NAWAL A. (DEPARTMENT OF MATHEMATICAL SCIENCES, DR. BABASAHEB UNIVERSITY) ;
  • GHADLE, KIRTIWANT P. (DEPARTMENT OF MATHEMATICAL SCIENCES, DR. BABASAHEB UNIVERSITY)
  • Received : 2019.12.20
  • Accepted : 2020.06.17
  • Published : 2020.06.25
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Abstract

Functional fractional differential inclusions with impulse effects in general Banach spaces are studied. We discuss the situation when the semigroup generated by the linear part is equicontinuous and the multifunction is Caratheodory. First, we define the PC-mild solutions for functional fractional semilinear impulsive differential inclusions. We then prove the existence of PC-mild solutions for such inclusions by using the fixed point theorem, multivalued properties and applications of NCHM (noncompactness Hausdorff measure). Eventually, we enhance the acquired results by giving an example.

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References

  1. Z. Agur, L. Cojocaru, G. Mazaur, R. M. Anderson, Y. L. Danon, Pulse mass measles vaccination across age shorts, Proc. Natl. Acad. Sci. USA, 90 (1993) 11698-11702. https://doi.org/10.1073/pnas.90.24.11698
  2. G. Ballinger, X. Liu, Boundedness for impulsive delay differential equations and applications in populations growth models, Nonlinear Anal., 53 (2003) 1041-1062. https://doi.org/10.1016/S0362-546X(03)00041-5
  3. A. D. Onofrio, On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Appl. Lett., 18 (2005) 729-732.
  4. M. Benchohra, J. Henderson, S. Ntouyas, Impulsive Differential Equations and Inclusions Hindawi, Philadelphia (2007).
  5. Z. Fan, Impulsive problems for semilinear differential equations with nonlocal conditions, Nonlinear Anal., 72 (2010) 1104-1109. https://doi.org/10.1016/j.na.2009.07.049
  6. T. Cardinali, P. Rubbioni, Impulsive mild solution for semilinear differential inclusions with nonlocal conditions in Banach spaces, Nonlinear Anal., 75 (2012) 871-879. https://doi.org/10.1016/j.na.2011.09.023
  7. J. Henderson, A. Ouahab, Impulsive differential inclusions with fractional order, Compu. Math. with Appl., 59 (2010) 1191-1226. https://doi.org/10.1016/j.camwa.2009.05.011
  8. A. G. Ibrahim, N. A. Alsarori, Mild solutions for nonlocal impulsive fractional semilinear differential inclusions with delay in Banach spaces, Applied Mathematics, 4 (2013) 40-56.
  9. O. K. Jaradat, A. Al-Omari, S. Momani, Existence of the mild solution for fractional semi-linear initial value problems, Nonlinear Anal. TMA, 69 (2008) 3153-3159. https://doi.org/10.1016/j.na.2007.09.008
  10. K. Li, J. Peng, J. Gao, Nonlocal fractional semilinear differential equations in separable Banach spaces. Electron. J. Differ. Equ., 7 (2013).
  11. G. M. Mophou, Existence and uniquness of mild solution to impulsive fractional differential equations, Nonlinear Anal.TMA, 72 (2010) 1604-1615. https://doi.org/10.1016/j.na.2009.08.046
  12. J. M. Ball, Initial boundary value problems for an extensible beam, J. Math. Anal. Appl., 42 (1973) 16-90.
  13. L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem J. Math. Anal. Appl., 162 (1991) 494-505. https://doi.org/10.1016/0022-247x(91)90164-u
  14. K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, Journal of Mathematical Analysis and Applications, 179 (1993) 630-637. https://doi.org/10.1006/jmaa.1993.1373
  15. E. A. Ddas, M. benchohra, S. hamani, Impulsive fractional differential inclusions involving The Caputo fractional derivative, Fractional Calculus and Applied Analysis, 12 (2009) 15-36.
  16. J. Wang, M. Feckan, Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolutions, Dynamics of PDE, Vol 8, No.4 (2011) 345-361.
  17. J. Wang, Y. Zhou, Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Anal., Real World Appl., 12 (2011) 3642-3653. https://doi.org/10.1016/j.nonrwa.2011.06.021
  18. Y. Zhou, F. Jiao, Nonlocal Cauchy problem for fractional netural evolution equations, Compu. Math. Appl., 59 (2010) 1063-1077. https://doi.org/10.1016/j.camwa.2009.06.026
  19. T. Lian, C. Xue, S. Deng, Mild solution to fractional differential inclusions with nonlocal conditions, Boundary Value problems, (2016) 2016:219. https://doi.org/10.1186/s13661-016-0724-2
  20. M. Kamenskii, V. Obukhowskii , P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Saur. Nonlinear Anal. Appl., Walter Berlin-New 7 (2001).
  21. J. Banas, K. Goebel, Measure of Noncompactness in Banach Spaces. Lect. Notes Pure Appl. Math., vol. 60. Dekker, New York (1980).
  22. H. R. Heinz, On the Behavior of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 7 (1983) 1351-1371. https://doi.org/10.1016/0362-546X(83)90006-8
  23. D. Bothe, Multivalued perturbation of m-accerative differential inclusions, Isreal J.Math., 108 (1998) 109-138. https://doi.org/10.1007/BF02783044
  24. J. P. Aubin, H. Frankoeska, Set-valued Analysis, Birkhauser, Boston, Basel, Berlin (1990).
  25. R. P. Agarwal, M. Meehan, D. O'regan, Fixed Point Theory and Applications, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge (2001).
  26. N. A. Alsarori, K. P. Ghadle, On the mild solution for nonlocal impulsive fractional semilinear differential inclusion in Banach spaces, J. Math. Modeling, Vol. 6, No. 2, 2018, pp. 239-258.
  27. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, Springer-Verlag, New York, (1983).