zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Existence for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators. (English) Zbl 1178.34071
Summary: We study the existence of mild solutions for the following system in a general Banach space $X$ (with norm $\|\cdot\|$): $$\cases \frac{d}{dt}[x(t)-F(t,x(h_1(t)))]\in Ax(t)+\int^t_0K(t,s)G(s,x(h_2(s)))\,ds,\\ t\in J-\{t_1,\dots,t_m\},\quad \text{where }J=[0,a],\\ \Delta x|_{t_k}=I_k(x(t^-_k)),\quad k=1,\dots,m,\\ x(0)=g(x)\in X.\endcases\tag1.1$$ Here $A$ is the infinitesimal generator of a compact analytic semigroup $T(t($, $t>0$, $G$ is a multi-valued map and $\Delta x|_{t=t_k}=x(t^+_k)-x(t^-_k)$, where $x(t^-_k)$ and $x(t^+_k)$ represent the left and right limits of $x(t)$ and $t=t_k$, respectively. Let $K:D\to R$, $D=\{(t,s)\in J\times J:t\ge s\}$ and $F,G,g,I_k$ $(k=1,\dots,m)$ and $h_1, h_2$ are given functions. By using a fixed point theorem for multi-valued maps due to Dhage, a main existence theorem is established. Finally, we present an example to illustrate this main theorem.

34G25Evolution inclusions
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
[1] Bainov, D. D.; Simeonov, P. S.: Systems with impulsive effect. Ellis horwood series: mathematics and its applications (1989)
[2] Benchohra, M.; Henderson, J.; Ntouyas, S.: Impulsive differential equations and inclusions. (2006) · Zbl 1130.34003
[3] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations. (1989) · Zbl 0719.34002
[4] Samoilenko, A. M.; Perestyuk, N. A.: Impulsive differential equations. World scientific series on nonlinear science, series A 14 (1995) · Zbl 0837.34003
[5] Anguraj, A.; Arjunan, M. Mallika; Hernández, E. M.: Existence results for an impulsive neutral functional differential equation with state-dependent delay. Appl. anal. 86, 861-872 (2007) · Zbl 1131.34054
[6] A. Anguraj, K. Karthikeyan, Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions, Nonlinear Anal., doi:10.1016/j.na.2008.03.059 · Zbl 1165.34416
[7] Chang, Y. -K.; Anguraj, A.; Arjunan, M. Mallika: Existence results for non-densely defined neutral impulsive differential inclusions with nonlocal conditions. J. appl. Math. comput. 28, 79-91 (2008) · Zbl 1160.34072
[8] Chang, Y. -K.; Nieto, J. J.; Li, W. -S.: On impulsive hyperbolic differential inclusions with nonlocal initial conditions. J. optim. Theory appl. 140, 431-442 (2009) · Zbl 1159.49042
[9] Y.-K. Chang, J.J. Nieto, Existence of solutions for impulsive neutral integro-differential inclusions with nonlocal initial conditions via fractional operators, Numer. Funct. Anal. Optim. (in press) · Zbl 1176.34096
[10] Hernández, E.; Pierri, M.; Goncalves, G.: Existence results for an impulsive abstract partial differential equation with state-dependent delay. Comput. math. Appl. 52, 411-420 (2006) · Zbl 1153.35396
[11] Hernández, E. M.; Rabello, M.; Henríquez, H.: Existence of solutions for impulsive partial neutral functional differential equations. J. math. Anal. appl. 331, 1135-1158 (2007) · Zbl 1123.34062
[12] W.-S. Li, Y.-K. Chang, J.J. Nieto, Solvability of impulsive neutral evolution differential inclusions with state-dependent delay, Math. comput. Modelling, doi:10.1016/j.mcm.2008.12.010 · Zbl 1171.34304
[13] Li, J.; Nieto, J. J.; Shen, J.: Impulsive periodic boundary value problems of first-order differential equations. J. math. Anal. appl. 325, 226-236 (2007) · Zbl 1110.34019
[14] Nieto, J. J.: Basic theory for nonresonance impulsive periodic problems of first order. J. math. Anal. appl. 205, 423-433 (1997) · Zbl 0870.34009
[15] Nieto, J. J.: Impulsive resonance periodic problems of first order. Appl. math. Lett. 15, 489-493 (2002) · Zbl 1022.34025
[16] Zhang, H.; Chen, L.; Nieto, J. J.: A delayed epidemic model with stage-structure and pulses for pest management strategy. Nonlinear anal. RWA 9, 1714-1726 (2008) · Zbl 1154.34394
[17] Benchohra, M.; Ntouyas, S.: Existence and controllability results for multivalued semilinear differential equations with nonlocal conditions. Soochow J. Math. 29, 157-170 (2003) · Zbl 1033.34068
[18] Ezzinbi, K.; Fu, X.; Hilal, K.: Existence and regularity in the ${\alpha}$-norm for some neutral partial differential equations with nonlocal conditions. Nonlinear anal. 67, 1613-1622 (2007) · Zbl 1119.35105
[19] Erbe, L.; Kong, Q.; Zhang, B.: Oscillation theory for functional differential equations, pure and applied mathematics. (1994) · Zbl 0821.34067
[20] Hale, J.: Theory of functional differential equations. (1977) · Zbl 0352.34001
[21] Henderson, J.: Boundary value problems for functional differential equations. (1995) · Zbl 0834.00035
[22] Byszewski, L.: Theorems about the existence and uniqueness of a solution of a semilinear evolution nonlocal Cauchy problem. J. math. Anal. appl. 162, 496-505 (1991) · Zbl 0748.34040
[23] Deng, K.: Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions. J. math. Anal. appl. 179, 630-637 (1993) · Zbl 0798.35076
[24] Byszewski, L.; Lakshmikantham, V.: Theorem about existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. anal. 40, 11-19 (1990) · Zbl 0694.34001
[25] Byszewski, L.: Existence of solutions of semilinear functional-differential evolution nonlocal problem. Nonlinear anal. 34, 65-72 (1998) · Zbl 0934.34068
[26] Fu, X.; Ezzinbi, K.: Existence of solutions for neutral functional differential evolution equations with nonlocal conditions. Nonlinear anal. 54, 215-227 (2003) · Zbl 1034.34096
[27] Fu, X.: On solutions of neutral nonlocal evolution equations with nondense domain. J. math. Anal. appl. 299, 392-410 (2004) · Zbl 1064.34065
[28] Yosida, K.: Functional analysis. (1980) · Zbl 0435.46002
[29] Deimling, K.: Multivalued differential equations. (1992) · Zbl 0760.34002
[30] Pazy, A.: Semigroups of linear operators and applications to partial differential equations. (1983) · Zbl 0516.47023
[31] Lasota, A.; Opial, Z.: An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. acad. Pol sci. Ser. sci. Math. astronom. Phys. 13, 781-786 (1965) · Zbl 0151.10703
[32] Dhage, B. C.: Multi-valued mappings and fixed points II. Tamkang J. Math. 37, 27-46 (2006) · Zbl 1108.47046
[33] Travis, C. C.; Webb, G. F.: Partial functional differential equations with deviating arguments in the time variable. J. math. Anal. appl. 56, 397-409 (1976) · Zbl 0349.35071