Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
evolution equation; impulsive integro-differential equation; nonlocal condition; lower and upper solutions; monotone iterative technique; mild solution
evolution equation; impulsive integro-differential equation; nonlocal condition; lower and upper solutions; monotone iterative technique; mild solution
Summary:
In this paper we use a monotone iterative technique in the presence of the lower and upper solutions to discuss the existence of mild solutions for a class of semilinear impulsive integro-differential evolution equations of Volterra type with nonlocal conditions in a Banach space $E$ $$ \begin{cases} u'(t)+Au(t)= f(t,u(t),Gu(t)),\quad t\in J, t\neq t_k, \Delta u |_{t=t_k}=u(t_k^+)-u(t_k^-)=I_k(u(t_k)),\quad k=1,2,\dots ,m, u(0)=g(u)+x_0, \end{cases} $$ where $A\colon D(A)\subset E\to E$ is a closed linear operator and $-A$ generates a strongly continuous semigroup $T(t)$ $(t\geq 0)$ on $E$, $f\in C(J\times E\times E, E)$, $J=[0,a]$, $0<t_1<t_2<\nobreak \dots <t_m<\nobreak a$, $I_k\in C(E,E)$, $k=1,2,\dots ,m$, and $g$ constitutes a nonlocal condition. Under suitable monotonicity conditions and noncompactness measure conditions, we obtain the existence of the extremal mild solutions between the lower and upper solutions assuming that $-A$ generates a compact semigroup, a strongly continuous semigroup or an equicontinuous semigroup. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. Some concrete applications to partial differential equations are considered.
References:
[1] Ahmed, N. U.: Impulsive evolution equations in infinite dimensional spaces. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 10 (2003), 11-24. MR 1974227 | Zbl 1023.49025
[2] Ahmed, N. U.: Optimal feedback control for impulsive systems on the space of finitely additive measures. Publ. Math. 70 (2007), 371-393. MR 2310657 | Zbl 1164.34026
[3] Banas, J., Goebel, K.: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics 60 Marcel Dekker, New York (1980). MR 0591679 | Zbl 0441.47056
[4] Banasiak, J., Arlotti, L.: Perturbations of Positive Semigroups with Applications. Springer Monographs in Mathematics Springer, London (2006). MR 2178970 | Zbl 1097.47038
[5] Benchohra, M., Henderson, J., Ntouyas, S. K.: Impulsive Differential Equations and Inclusions. Contemporary Mathematics and Its Applications 2 Hindawi Publishing Corporation, New York (2006). MR 2322133 | Zbl 1130.34003
[6] Benchohra, M., Ntouyas, S. K.: Nonlocal Cauchy problems for neutral functional differential and integrodifferential inclusions in Banach spaces. J. Math. Anal. Appl. 258 (2001), 573-590. DOI 10.1006/jmaa.2000.7394 | MR 1835560 | Zbl 0982.45008
[7] Byszewski, L.: Existence, uniqueness and asymptotic stability of solutions of abstract nonlocal Cauchy problems. Dyn. Syst. Appl. 5 (1996), 595-605. MR 1424567 | Zbl 0869.47034
[8] Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162 (1991), 494-505. DOI 10.1016/0022-247X(91)90164-U | MR 1137634 | Zbl 0748.34040
[9] 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. 40 (1991), 11-19. DOI 10.1080/00036819008839989 | MR 1121321 | Zbl 0694.34001
[10] Chang, Y.-K., Anguraj, A., Arjunan, M. M.: Existence results for impulsive neutral functional differential equations with infinite delay. Nonlinear Anal., Hybrid Syst. 2 (2008), 209-218. MR 2382006 | Zbl 1170.35467
[11] Chang, Y.-K., Anguraj, A., Karthikeyan, K.: Existence for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71 (2009), 4377-4386. DOI 10.1016/j.na.2009.02.121 | MR 2548667 | Zbl 1178.34071
[12] Deimling, K.: Nonlinear Functional Analysis. Springer Berlin (1985). MR 0787404 | Zbl 0559.47040
[13] Deng, K.: Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions. J. Math. Anal. Appl. 179 (1993), 630-637. DOI 10.1006/jmaa.1993.1373 | MR 1249842 | Zbl 0798.35076
[14] Du, Y.: Fixed points of increasing operators in ordered Banach spaces and applications. Appl. Anal. 38 (1990), 1-20. DOI 10.1080/00036819008839957 | MR 1116172 | Zbl 0671.47054
[15] Du, S. W., Lakshmikantham, V.: Monotone iterative technique for differential equations in a Banach space. J. Math. Anal. Appl. 87 (1982), 454-459. DOI 10.1016/0022-247X(82)90134-2 | MR 0658024 | Zbl 0523.34057
[16] Erbe, L. H., Liu, X.: Quasi-solutions of nonlinear impulsive equations in abstract cones. Appl. Anal. 34 (1989), 231-250. DOI 10.1080/00036818908839897 | MR 1387172 | Zbl 0662.34015
[17] Ezzinbi, K., Fu, X., Hilal, K.: Existence and regularity in the $\alpha$-norm for some neutral partial differential equations with nonlocal conditions. Nonlinear Anal., Theory Methods Appl. 67 (2007), 1613-1622. DOI 10.1016/j.na.2006.08.003 | MR 2323307 | Zbl 1119.35105
[18] Fan, Z.: Existence of nondensely defined evolution equations with nonlocal conditions. Nonlinear Anal., Theory Methods Appl. 70 (2009), 3829-3836. DOI 10.1016/j.na.2008.07.036 | MR 2515302 | Zbl 1170.34345
[19] Fan, Z.: Impulsive problems for semilinear differential equations with nonlocal conditions. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72 (2010), 1104-1109. DOI 10.1016/j.na.2009.07.049 | MR 2579372 | Zbl 1188.34073
[20] Fan, Z., Li, G.: Existence results for semilinear differential equations with nonlocal and impulsive conditions. J. Funct. Anal. 258 (2010), 1709-1727. DOI 10.1016/j.jfa.2009.10.023 | MR 2566317 | Zbl 1193.35099
[21] Fu, X., Ezzinbi, K.: Existence of solutions for neutral functional differential evolution equations with nonlocal conditions. Nonlinear Anal., Theory Methods Appl. 54 (2003), 215-227. DOI 10.1016/S0362-546X(03)00047-6 | MR 1979731 | Zbl 1034.34096
[22] Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Notes and Reports in Mathematics in Science and Engineering 5 Academic Press, Boston (1988). MR 0959889 | Zbl 0661.47045
[23] Guo, D., Liu, X.: Extremal solutions of nonlinear impulsive integrodifferential equations in Banach spaces. J. Math. Anal. Appl. 177 (1993), 538-552. DOI 10.1006/jmaa.1993.1276 | MR 1231500 | Zbl 0787.45008
[24] Heinz, H.-P.: On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions. Nonlinear Anal., Theory Methods Appl. 7 (1983), 1351-1371. MR 0726478 | Zbl 0528.47046
[25] Jackson, D.: Existence and uniqueness of solutions to semilinear nonlocal parabolic equations. J. Math. Anal. Appl. 172 (1993), 256-265. DOI 10.1006/jmaa.1993.1022 | MR 1199510 | Zbl 0814.35060
[26] Ji, S., Li, G., Wang, M.: Controllability of impulsive differential systems with nonlocal conditions. Appl. Math. Comput. 217 (2011), 6981-6989. DOI 10.1016/j.amc.2011.01.107 | MR 2775688 | Zbl 1219.93013
[27] Lakshmikantham, V., Bainov, D. D., Simeonov, P. S.: Theory of Impulsive Differential Equations. Series in Modern Applied Mathematics 6 World Scientific, Singapore (1989). MR 1082551 | Zbl 0719.34002
[28] Li, Y.: Existence of solutions of initial value problems for abstract semilinear evolution equations. Acta Math. Sin. 48 (2005), 1089-1094 Chinese. MR 2205049 | Zbl 1124.34341
[29] Li, Y.: The positive solutions of abstract semilinear evolution equations and their applications. Acta Math. Sin. 39 (1996), 666-672 Chinese. MR 1436036 | Zbl 0870.47040
[30] Li, Y., Liu, Z.: Monotone iterative technique for addressing impulsive integro-differential equations in Banach spaces. Nonlinear Anal., Theory Methods Appl. 66 (2007), 83-92. DOI 10.1016/j.na.2005.11.013 | MR 2271638 | Zbl 1109.34005
[31] Liang, J., Liu, J. H., Xiao, T.-J.: Nonlocal Cauchy problems governed by compact operator families. Nonlinear Anal., Theory Methods Appl. 57 (2004), 183-189. DOI 10.1016/j.na.2004.02.007 | MR 2056425 | Zbl 1083.34045
[32] Liang, J., Liu, J. H., Xiao, T.-J.: Nonlocal impulsive problems for nonlinear differential equations in Banach spaces. Math. Comput. Modelling 49 (2009), 798-804. DOI 10.1016/j.mcm.2008.05.046 | MR 2483682 | Zbl 1173.34048
[33] Liang, J., Casteren, J. van, Xiao, T.-J.: Nonlocal Cauchy problems for semilinear evolution equations. Nonlinear Anal., Theory Methods Appl. 50 (2002), 173-189. DOI 10.1016/S0362-546X(01)00743-X | MR 1904939
[34] Lin, Y., Liu, J. H.: Semilinear integrodifferential equations with nonlocal Cauchy problem. Nonlinear Anal., Theory Methods Appl. 26 (1996), 1023-1033. DOI 10.1016/0362-546X(94)00141-0 | MR 1362770 | Zbl 0916.45014
[35] Liu, J. H.: Nonlinear impulsive evolution equations. Dyn. Contin. Discrete Impulsive Syst. 6 (1999), 77-85. MR 1679758 | Zbl 0932.34067
[36] Ntouyas, S. K., Tsamatos, P. C.: Global existence for semilinear evolution integrodifferential equations with delay and nonlocal conditions. Appl. Anal. 64 (1997), 99-105. DOI 10.1080/00036819708840525 | MR 1460074 | Zbl 0874.35126
[37] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44 Springer, New York (1983). MR 0710486 | Zbl 0516.47023
[38] Rogovchenko, Y. V.: Impulsive evolution systems: Main results and new trends. Dyn. Contin. Discrete Impulsive Syst. 3 (1997), 57-88. MR 1435816 | Zbl 0879.34014
[39] Sun, J., Zhao, Z.: Extremal solutions of initial value problem for integro-differential equations of mixed type in Banach spaces. Ann. Differ. Equations 8 (1992), 469-475. MR 1215993
[40] Xiao, T.-J., Liang, J.: Existence of classical solutions to nonautonomous nonlocal parabolic problems. Nonlinear Anal., Theory Methods Appl. (electronic only) 63 (2005), e225--e232. DOI 10.1016/j.na.2005.02.067 | Zbl 1159.35383
[41] Xue, X.: Nonlinear differential equations with nonlocal conditions in Banach spaces. Nonlinear Anal., Theory Methods Appl. 63 (2005), 575-586. DOI 10.1016/j.na.2005.05.019 | MR 2175816 | Zbl 1095.34040
[42] Xue, X.: Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces. Nonlinear Anal., Theory Methods Appl. 70 (2009), 2593-2601. DOI 10.1016/j.na.2008.03.046 | MR 2499726 | Zbl 1176.34071
Search
Partner of
