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 solutions for a class of abstract differential equations with nonlocal conditions. (English) Zbl 1221.47079

The authors consider the system

y ' (t)=Ay(t)+f(t,y(t))(0ta),y(0)=g(u)+y 0 (1)

in a Banach space E, where A is the infinitesimal generator of an analytic semigroup. The nonlinearity f maps [0,a]×E α into E, where E α is the domain of the fractional power (-A) α (0<α<1)· Finally, g maps C(I,E α ) into E, where I(0,a]· The results are on existence and uniqueness of solutions of (1), where “solution” is understood in various ways, two of them classical and mild (the latter means a solution of the integral equation version of (1)). The results are applied to partial differential equations, with nonlocal conditions involving partial derivatives or nonlinear expressions of the solution.

MSC:
47D06One-parameter semigroups and linear evolution equations
34K30Functional-differential equations in abstract spaces
34B10Nonlocal and multipoint boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
References:
[1]Aizicovici, S.; Lee, H.: Nonlinear nonlocal Cauchy problems in Banach spaces, Appl. math. Lett. 18, No. 4, 401-407 (2005) · Zbl 1084.34002 · doi:10.1016/j.aml.2004.01.010
[2]Anguraj, A.; Karthikeyan, K.: Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions, Nonlinear anal. 70, No. 7, 2717-2721 (2009) · Zbl 1165.34416 · doi:10.1016/j.na.2008.03.059
[3]Balachandran, K.; Park, J. Y.; Chandrasekaran, M.: Nonlocal Cauchy problem for delay integrodifferential equations of Sobolev type in Banach spaces, Appl. math. Lett. 15, No. 7, 845-854 (2002) · Zbl 1028.45006 · doi:10.1016/S0893-9659(02)00052-6
[4]Bahuguna, D.: Existence, uniqueness and regularity of solutions to semilinear nonlocal functional differential problems, Nonlinear anal. 57, No. 7–8, 1021-1028 (2004) · Zbl 1065.34078 · doi:10.1016/j.na.2004.03.026
[5]Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. math. Anal. appl. 162, No. 2, 494-505 (1991) · Zbl 0748.34040 · doi:10.1016/0022-247X(91)90164-U
[6]Byszewski, L.: Existence, uniqueness and asymptotic stability of solutions of abstract nonlocal Cauchy problems, Dynam. systems appl. 5, No. 4, 595-605 (1996) · Zbl 0869.47034
[7]Byszewski, L.; Akca, H.: Existence of solutions of a semilinear functional differential evolution nonlocal problem, Nonlinear anal. 34, No. 1, 65-72 (1998) · Zbl 0934.34068 · doi:10.1016/S0362-546X(97)00693-7
[8]Byszewski, L.; Akca, H.: On a mild solution of a semilinear functional-differential evolution nonlocal problem, J. appl. Math. stoch. Anal. 10, No. 3, 265-271 (1997) · Zbl 1043.34504 · doi:10.1155/S1048953397000336
[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, No. 1, 11-19 (1991) · Zbl 0694.34001 · doi:10.1080/00036819008839989
[10]Chang, Jung-Chan; Liu, Hsiang: Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the α-norm, Nonlinear anal. 71, No. 9, 3759-3768 (2009) · Zbl 1185.34112 · doi:10.1016/j.na.2009.02.035
[11]Chang, Yong-Kui; Anguraj, A.; Karthikeyan, K.: Existence for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators, Nonlinear anal. 71, No. 10, 4377-4386 (2009) · Zbl 1178.34071 · doi:10.1016/j.na.2009.02.121
[12]Ezzinbi, Khalil; Fu, Xianlong; Hilal, Khalid: Existence and regularity in the α-norm for some neutral partial differential equations with nonlocal conditions, Nonlinear anal. 67, No. 5, 1613-1622 (2007) · Zbl 1119.35105 · doi:10.1016/j.na.2006.08.003
[13]Fan, Zhenbin; Li, Gang: Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. funct. Anal. 258, No. 5, 1709-1727 (2010) · Zbl 1193.35099 · doi:10.1016/j.jfa.2009.10.023
[14]Hernández, E.: Existence results for partial neutral functional differential equations with nonlocal conditions, Dynam. systems appl. 11, No. 2, 241-252 (2002) · Zbl 1020.34072
[15]Karakostas, G. L.; Tsamatos, P. Ch.: Sufficient conditions for the existence of nonnegative solutions of a nonlocal boundary value problem, Appl. math. Lett. 15, No. 4, 401-407 (2002) · Zbl 1028.34023 · doi:10.1016/S0893-9659(01)00149-5
[16]Liu, James H.: A remark on the mild solutions of non-local evolution equations, Semigroup forum 66, No. 1, 63-67 (2003) · Zbl 1015.37045 · doi:10.1007/s002330010158
[17]Liu, Hsiang; Chang, Jung-Chan: Existence for a class of partial differential equations with nonlocal conditions, Nonlinear anal. 70, No. 9, 3076-3083 (2009) · Zbl 1170.34346 · doi:10.1016/j.na.2008.04.009
[18]Ntouyas, S. K.; Tsamatos, P.: Global existence for semilinear evolution equations with nonlocal conditions, J. math. Anal. appl. 210, 679-687 (1997) · Zbl 0884.34069 · doi:10.1006/jmaa.1997.5425
[19]Olmstead, W. E.; Roberts, C. A.: The one-dimensional heat equation with a nonlocal initial condition, Appl. math. Lett. 10, No. 3, 89-94 (1997) · Zbl 0888.35042 · doi:10.1016/S0893-9659(97)00041-4
[20]Paicu, Angela; Vrabie, Ioan I.: A class of nonlinear evolution equations subjected to nonlocal initial conditions, Nonlinear anal. 72, No. 11, 4091-4100 (2010) · Zbl 1200.34068 · doi:10.1016/j.na.2010.01.041
[21]Pazy, A.: Semigroups of linear operators and applications to partial differential equations, (1983)