×

zbMATH — the first resource for mathematics

Existence of mild solutions for abstract semilinear evolution equations in Banach spaces. (English) Zbl 1235.34174
Summary: The nonlocal initial value problem, \[ \begin{cases} x'(t)=Ax(t)+f(t,x(t)), \quad t\in I=[0,1],\\ x(0)=g(x),\end{cases} \] where \(A\) is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators (i.e. \(C_0\)-semigroup) \(T(t)\) in Banach space \(X\), and \(f:I\times X\to X\), \(g:C([0,1];X)\to X\) are given \(X\)-valued functions.

MSC:
34G20 Nonlinear differential equations in abstract spaces
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 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
[2] Byszewski, L., Existence and uniqueness of solutions of semilinear evolution nonlocal Cauchy problem, Zesz. nauk. Pol. rzes. mat. fiz., 18, 109-112, (1993) · Zbl 0858.34045
[3] Byszewski, L.; Lakshmikantham, V., Theorems about the existence and uniqueness of a solutions of nonlocal Cauchy problem in a Banach space, Appl. anal., 40, 11-19, (1990) · Zbl 0694.34001
[4] Ntouyas, S.; Tsamotas, P., Global existence for semilinear evolution equations with nonlocal conditions, J. math. anal. appl., 210, 679-687, (1997) · Zbl 0884.34069
[5] Ntouyas, S.; Tsamotas, P., Global existence for semilinear integrodifferential equations with delay and nonlocal conditions, Anal. appl., 64, 99-105, (1997) · Zbl 0874.35126
[6] Xue, X., Existence of solutions for semilinear nonlocal Cauchy problems in Banach spaces, Elec. J. differential equations, 64, 1-7, (2005)
[7] Fan, Z.; Dong, Q.; Li, G., Semilinear differential equations with nonlocal conditions in Banach spaces, Inter. J. nonlinear sci., 2, 131-139, (2006) · Zbl 1394.34117
[8] Banas, J.; Goebel, K., ()
[9] Mönch, H., Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear anal. TMA, 4, 985-999, (1980) · Zbl 0462.34041
[10] Xue, X., Semilinear nonlocal differential equations with measure of noncompactness in Banach spaces, J. Nanjing. univ. math. big., 24, 264-276, (2007) · Zbl 1174.34046
[11] Bothe, D., Multivalued perturbation of \(m\)-accretive differential inclusions, Israel. J. math., 108, 109-138, (1998) · Zbl 0922.47048
[12] Zhang, X.; Liu, L.S.; Wu, C.X., Global solutions of nonlinear second-order inpulsive integro-differential equations of mixed type in Banach spaces, Nonlinear anal., 67, 2335-2349, (2007) · Zbl 1121.45004
[13] Liu, L.S.; Guo, F.; Wu, C.X.; Wu, Y.H., Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. math. anal. appl., 309, 638-649, (2005) · Zbl 1080.45005
[14] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer-Verlag New York · 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.