zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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 of mild solutions for semilinear evolution equations with non-local initial conditions. (English) Zbl 1179.34063
The authors study the existence of mild solutions for a class of semilinear evolution equations with nonlocal initial conditions. They mainly suppose that the nonlocal term $g$ is continuous or depends only on the value of $u$ in the interval $[\epsilon, T]$, where $\epsilon$ is small enough.

MSC:
34G20Nonlinear ODE in abstract spaces
47D06One-parameter semigroups and linear evolution equations
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
WorldCat.org
Full Text: DOI
References:
[1] Cannon, J.: The one-dimensional heat equation. Encyclopedia of mathematics and its applications 23 (1984)
[2] Chadam, J.; Yin, H. -M.: Determination of an unknown function in a parabolic equation with an overspecified condition. Math. meth. Appl. sci. 13, 421-430 (1990) · Zbl 0738.35097
[3] Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. math. Anal. appl. 162, 494-505 (1991) · Zbl 0748.34040
[4] Byszewski, L.: Uniqueness of solutions of parabolic semilinear nonlocal-boundary problems. J. math. Anal. appl. 165, 472-478 (1992) · Zbl 0774.35038
[5] Byszewski, L.: Application of properties of the right-hand sides of evolution equations to an investigation of nonlocal evolution problems. Nonlinear anal. 33, 413-426 (1998) · Zbl 0933.34064
[6] Byszewski, L.; Akca, H.: Existence of solutions of a semilinear functional-differential evolution nonlocal problem. Nonlinear anal. 34, 65-72 (1998) · Zbl 0934.34068
[7] 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, 11-19 (1990) · Zbl 0694.34001
[8] Chabrowski, J.: On non-local problems for parabolic equations. Nagoya math. J. 93, 109-131 (1984) · Zbl 0506.35048
[9] Chabrowski, J.: On the non-local problem with a functional for parabolic equation. Funkcial. ekvac. 27, 101-123 (1984) · Zbl 0568.35046
[10] Deng, K.: Exponential decay of solutions of semilinear parabolic equations with non-local initial conditions. J. math. Anal. appl. 179, 630-637 (1993) · Zbl 0798.35076
[11] Jackson, D.: Existence and uniqueness of solutions of a semilinear nonlocal parabolic equations. J. math. Anal. appl. 172, 256-265 (1993) · Zbl 0814.35060
[12] Liang, J.; Van Casteren, J.; Xiao, T. J.: Nonlocal Cauchy problems for semilinear evolution equations. Nonlinear anal. 50, 173-189 (2002) · Zbl 1009.34052
[13] Liang, J.; Liu, J. H.; Xiao, T. J.: Nonlocal Cauchy problems governed by compact operator families. Nonlinear anal. 57, 183-189 (2004) · Zbl 1083.34045
[14] Liu, J. H.: A remark on the mild solutions of non-local evolution equations. Semigroup forum 26, 63-67 (2003) · Zbl 1015.37045
[15] Ntouyas, S. K.; Tsamatos, P. C.: Global existence for semilinear evolution equations with nonlocal conditions. J. math. Anal. appl. 210, 679-687 (1997) · Zbl 0884.34069
[16] Xue, X. M.: Existence of solutions for semilinear nonlocal Cauchy problems in Banach spaces. Electron. J. Differential equations 26, 1-7 (2005) · Zbl 1075.34051
[17] Xue, X. M.: Existence of semilinear differential equations with nonlocal initial conditions. Acta math. Sin. (Engl. Ser.) 23, 983-988 (2007) · Zbl 1129.34041
[18] Zhu, L. P.; Li, G.: Nonlocal differential equations with multivalued perturbations in Banach spaces. Nonlinear anal. 69, 2843-2850 (2008) · Zbl 1163.34041
[19] Aizicovici, S.; Gao, Y.: Functional differential equations with nonlocal initial conditions. J. appl. Math. stochastic anal. 10, 145-156 (1997) · Zbl 0883.34065
[20] Aizicovici, S.; Mckibben, M.: Existence results for a class of abstract nonlocal Cauchy problems. Nonlinear anal. 39, 649-668 (2000) · Zbl 0954.34055
[21] Aizicovici, S.; Lee, H.: Nonlinear nonlocal Cauchy problems in Banach spaces. Appl. math. Lett. 18, 401-407 (2005) · Zbl 1084.34002
[22] Aizicovici, S.; Lee, H.: Existence results for nonautonomous evolution equations with nonlocal initial conditions. Commun. appl. Anal. 11, 285-297 (2007) · Zbl 1147.34043
[23] Aizicovici, S.; Staicu, V.: Multivalued evolution equations with nonlocal initial conditions in Banach spaces. Nodea nonlinear differential equations appl. 14, 361-376 (2007) · Zbl 1145.35076
[24] Boucherif, A.: First-order differential inclusions with nonlocal initial conditions. Appl. math. Lett. 15, 409-414 (2002) · Zbl 1025.34009
[25] Liang, J.; Xiao, T. J.: Semilinear integrodifferential equations with nonlocal initial conditions. Comput. math. Appl. 47, 863-875 (2004) · Zbl 1068.45014
[26] Xue, X. M.: Nonlinear differential equations with nonlocal conditions in Banach spaces. Nonlinear anal. 63, 575-586 (2005) · Zbl 1095.34040
[27] Lin, Y.: Analytical and numerical solutions for a class of nonlocal nonlinear parabolic differential equations. SIAM J. Math. anal. 25, 1577-1594 (1994) · Zbl 0807.35069
[28] Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Applied math. Sci. 44 (1983) · Zbl 0516.47023
[29] Baras, P.; Hassan, J. -C.; Véron, L.: Compacité de l’opérateur définissant la solution d’une équation d’évolution non homogène. C. R. Acad. sci. Paris 284, 799-802 (1977) · Zbl 0348.47026
[30] Byszewski, L.; Lakshmikantham, V.: Monotone iterative technique for nonlocal hyperbolic differential problem. J. math. Phys. sci. 26, No. 4, 345-359 (1992) · Zbl 0811.35083
[31] Garcia-Falset, J.: Existence results and asymptotic behavior for nonlocal abstract Cauchy problems. J. math. Anal. appl. 338, 639-652 (2008) · Zbl 1140.34026
[32] Lin, Y.; Liu, J. H.: Semilinear integrodifferential equations with nonlocal Cauchy problem. Nonlinear anal. 26, 1023-1033 (1996) · Zbl 0916.45014