×

Existence of semilinear differential equations with nonlocal initial conditions. (English) Zbl 1129.34041

The author considers the existence of mild solutions for semilinear Cauchy problems
\[ u'(t) =Au(t) +f(t,u(t)), \quad t\in [0,b]\text{ a.e., }u(0) =g(u) +u_0, \]
where \(A\) is an infinitesimal generator of a strongly continuous semigroup \(T(t)\) of bounded linear operators in a Banach space \(X\), \(f: [0,b] \times X\to X\), \(g\in C([0,b] ;X)\) are given \(X\)-valued functionals. Certain assumptions are imposed on the nonlinear terms which allow for using a suitable fixed point theorem in proving the existence result. The map \(g\) does not need to be compact in order to reach the existence conclusion. Instead some other weaker condition is assumed.

MSC:

34G20 Nonlinear differential equations in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchuy problem. J. Math. Anal. Appl., 162, 494–505 (1991) · Zbl 0748.34040 · doi:10.1016/0022-247X(91)90164-U
[2] 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 · doi:10.1006/jmaa.1993.1373
[3] Byszewski, L., Lakshmikantham, V.: Theorems about the existence and uniqueness of a solutions of a nonlocal Cauchuy problem in a Banach space. Appl. Anal., 40, 11–19 (1990) · Zbl 0694.34001 · doi:10.1080/00036819008839989
[4] Byszewski, L.: Existence and uniqueness of solutions of semilinear evolution nonlocal Cauchuy problem. Zesz. Nauk. Pol. Rzes. Mat. Fiz., 18, 109–112 (1993) · Zbl 0858.34045
[5] Jackson, D.: Existence and uniqueness of solutions of semilinear evolution nonlocal parabolic equations. J. Math. Anal. Appl., 172, 256–265 (1993) · Zbl 0814.35060 · doi:10.1006/jmaa.1993.1022
[6] Lin, Y., Liu, J.: Semilinear integrodi.erential equations with nonlocal Cauchy Problems. Nonlinear Analysis, 26, 1023–1033 (1996) · Zbl 0916.45014 · doi:10.1016/0362-546X(94)00141-0
[7] Ntouyas, S., Tsamotas, 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
[8] Ntouyas, S., Tsamotas, P.: Global existence for semilinear integrodifferential equations with delay and nonlocal conditions. Anal. Appl., 64, 99–105 (1997) · Zbl 0874.35126 · doi:10.1080/00036819708840525
[9] Byszewski, L., Akca, H.: Existence of solutions of a semilinear functional-differential evolution nonlocal problem. Nonlinear Analysis, 34, 65–72 (1998) · Zbl 0934.34068 · doi:10.1016/S0362-546X(97)00693-7
[10] Aizicovici, S., Mckibben, M.: Existence results for a class of abstract nonlocal Cauchy problems. Nonlinear Analysis, 39, 649–668 (2000) · Zbl 0954.34055 · doi:10.1016/S0362-546X(98)00227-2
[11] Benchohra, M., Ntouyas, S.: Nonlocal Cauchy problems for neutral functional differential and integrodifferential inclusions in Banach spaces. J. Math. Anal. Appl., 258, 573–590 (2001) · Zbl 0982.45008 · doi:10.1006/jmaa.2000.7394
[12] Balachandran, K., Park, J., Chanderasekran: Nonlocal Cauchy problems for delay integrodifferential equations of Sobolev type in Banach spaces. Appl. Math. Letters, 15, 845–854 (2002) · Zbl 1028.45006 · doi:10.1016/S0893-9659(02)00052-6
[13] Fu, X., Ezzinbi, K.: Existence of solutions for neutral functional di.erential evolution equations with nonlocal conditions. Nonlinear Analysis, 54, 215–227 (2003) · Zbl 1034.34096 · doi:10.1016/S0362-546X(03)00047-6
[14] Liu, J.: A remark on the mild solutions of nonlocal evolution equations. Semigroup Forum, 66, 63–67 (2003) · Zbl 1015.37045
[15] Kisielewicz, M.: Multivalued di.erential equations in separable Banach spaces. J. Optim. Theory Appl., 37, 231–249 (1982) · Zbl 0458.34008 · doi:10.1007/BF00934769
[16] Pazy, A.: Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983 · Zbl 0516.47023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.