In this article, we are concerned with the existence of extremal solutions to the initial value problem of impulsive differential equations in ordered Banach spaces. The existence and uniqueness theorem for the solution of the associated linear impulsive differential equation is established. With the aid of this theorem, the existence of minimal and maximal solutions for the initial value problem of nonlinear impulsive differential equations is obtained under the situation that the nonlinear term and impulsive functions are not monotone increasing by using perturbation methods and monotone iterative technique. The results obtained in this paper improve and extend some related results in abstract differential equations. An example is also given to illustrate the feasibility of our abstract results.
References
H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620–709. 10.1137/1018114 MR415432 0345.47044 H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620–709. 10.1137/1018114 MR415432 0345.47044
M. Benchohra, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclutions, Hindawi, New York, 2006. 10.1155/9789775945501 1130.34003 M. Benchohra, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclutions, Hindawi, New York, 2006. 10.1155/9789775945501 1130.34003
P. Y. Chen and Y. X. Li, Mixed monotone iterative technique for a class of semilinear impulsive evolution equations in Banach spaces, Nonlinear Anal. 74 (2011), no. 11, 3578–3588. 10.1016/j.na.2011.02.041 MR2803085 1220.34018 P. Y. Chen and Y. X. Li, Mixed monotone iterative technique for a class of semilinear impulsive evolution equations in Banach spaces, Nonlinear Anal. 74 (2011), no. 11, 3578–3588. 10.1016/j.na.2011.02.041 MR2803085 1220.34018
––––, Monotone iterative method for abstract impulsive integro-differential equations with nonlocal initial conditions in Banach spaces, Appl. Math. 59 (2014), no. 1, 99–120. 10.1007/s10492-014-0044-8 MR3164579 1313.34230 ––––, Monotone iterative method for abstract impulsive integro-differential equations with nonlocal initial conditions in Banach spaces, Appl. Math. 59 (2014), no. 1, 99–120. 10.1007/s10492-014-0044-8 MR3164579 1313.34230
K. Deimling, Ordinary Differential Equations in Banach Spaces, Springer-Verlag, Berlin-New York, 1977. 10.1007/bfb0091636 0361.34050 K. Deimling, Ordinary Differential Equations in Banach Spaces, Springer-Verlag, Berlin-New York, 1977. 10.1007/bfb0091636 0361.34050
Y. H. Du, Fixed points of increasing operators in ordered Banach spaces and applications, Appl. Anal. 38 (1990), no. 1-2, 1–20. 10.1080/00036819008839957 MR1116172 0671.47054 Y. H. Du, Fixed points of increasing operators in ordered Banach spaces and applications, Appl. Anal. 38 (1990), no. 1-2, 1–20. 10.1080/00036819008839957 MR1116172 0671.47054
S. W. Du and V. Lakshmikantham, Monotone iterative technique for differential equtions in a Banach space, J. Math. Anal. Appl. 87 (1982), no. 2, 454–459. 10.1016/0022-247x(82)90134-2 0523.34057 S. W. Du and V. Lakshmikantham, Monotone iterative technique for differential equtions in a Banach space, J. Math. Anal. Appl. 87 (1982), no. 2, 454–459. 10.1016/0022-247x(82)90134-2 0523.34057
D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, San Diego, 1988. 10.1016/0378-4754(89)90065-7 0661.47045 D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, San Diego, 1988. 10.1016/0378-4754(89)90065-7 0661.47045
D. J. Guo and X. Z. Liu, Extremal solutions of nonlinear impulsive integrodifferential equations in Banach spaces, J. Math. Anal. Appl. 177 (1993), no. 2, 538–552. 10.1006/jmaa.1993.1276 MR1231500 0787.45008 D. J. Guo and X. Z. Liu, Extremal solutions of nonlinear impulsive integrodifferential equations in Banach spaces, J. Math. Anal. Appl. 177 (1993), no. 2, 538–552. 10.1006/jmaa.1993.1276 MR1231500 0787.45008
H.-P. Heinz, On the behaviour of measure of noncompactness with respect to differentiation and integration of rector-valued functions, Nonlinear Anal. 7 (1983), no. 12, 1351–1371. 10.1016/0362-546x(83)90006-8 MR726478 0528.47046 H.-P. Heinz, On the behaviour of measure of noncompactness with respect to differentiation and integration of rector-valued functions, Nonlinear Anal. 7 (1983), no. 12, 1351–1371. 10.1016/0362-546x(83)90006-8 MR726478 0528.47046
V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Teaneck, NJ, 1989. 10.1142/0906 0719.34002 V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Teaneck, NJ, 1989. 10.1142/0906 0719.34002
Y. X. Li and Z. Liu, Monotone iterative technique for addressing impulsive integro-differential equtions in Banach spaces, Nonlinear Anal. 66 (2007), no. 1, 83–92. 10.1016/j.na.2005.11.013 MR2271638 1109.34005 Y. X. Li and Z. Liu, Monotone iterative technique for addressing impulsive integro-differential equtions in Banach spaces, Nonlinear Anal. 66 (2007), no. 1, 83–92. 10.1016/j.na.2005.11.013 MR2271638 1109.34005
L. S. Liu, C. X. Wu and F. Guo, A unique solution of initial value problems for first order impulsive integro-differential equations of mixed type in Banach spaces, J. Math. Anal. Appl. 275 (2002), no. 1, 369–385. 10.1016/s0022-247x(02)00366-9 MR1941790 1014.45007 L. S. Liu, C. X. Wu and F. Guo, A unique solution of initial value problems for first order impulsive integro-differential equations of mixed type in Banach spaces, J. Math. Anal. Appl. 275 (2002), no. 1, 369–385. 10.1016/s0022-247x(02)00366-9 MR1941790 1014.45007
H. Q. Lu, Extremal solutions of nonlinear first order impulsive integro-differential equations in Banach spaces, Indian J. Pure Appl. Math. 30 (1999), no. 11, 1181–1197. MR1717753 0943.45004 H. Q. Lu, Extremal solutions of nonlinear first order impulsive integro-differential equations in Banach spaces, Indian J. Pure Appl. Math. 30 (1999), no. 11, 1181–1197. MR1717753 0943.45004
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. 0771.34059 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. 0771.34059
