We are concerned with the study of semilinear evolution equations with nonlocal initial conditions. We provide sufficient conditions on the nonlinearity which allow the use of variants of the nonlinear alternative to prove the existence of at least one solution. Our second result presents a novel growth condition splitted into two parts, one for the subinterval containing the points involved by the initial conditions, and another for the rest of the interval.
Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
Evolution equation; Nonlocal initial condition; Mild solution; Fixed point.
A. Boucherif, R. Precup, Semilinear evolution equations with nonlocal initial conditions, Dynamic Systems Appl. 16 (2007), 507-516.
Dynamic Systems and Applications
google scholar link
 A. Ashyralyev, H. Akca and L. Byszewski, On a semilinear evolution nonlocal Cauchy problem, in Some Problems of Applied Mathematics (A. Ashyralyev and H.Ali Yurtsever, Eds), Fatih University Publ., Istanbul, Turkey, 2000, 29–44.
 A. Boucherif and H. Akca, Nonlocal Cauchy problems for semilinear evolution equations, Dynam. Systems Appl., 11:415–420, 2002.
 A. Boucherif and R. Precup, On the nonlocal initial value problem for first order differential equations, Fixed Point Theory, 4:205–212, 2003.
 L. Byszewski, Theorems about the existence and uniqueness of a semilinar evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162:494–495, 1991.
 H.-K. Han and J.-Y. Park, Boundary controllability of differential equations with nonlocal conditions, J. Math. Anal. Appl., 230:242–250, 1999.
 D. Jackson, Existence and uniqueness of solutions to semilinear nonlocal parabolic equations, J. Math. Anal. Appl., 172:256–265, 1993.
 H.Chun Lee, Analysis of some nonlocal boundary value problems associated with feedback control, Bull. Korean Math. Soc., 35:325–338, 1998.
 Jin Liang, James Liu and Ti-Jun Xiao, Nonlocal Cauchy problems governed by compact operator families, Nonlinear Anal., 57:183–189, 2004.
 S.K. Ntouyas and P.Ch. Tsamatos, Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl., 210:679–687, 1997.
 D. O’Regan, Fixed-point theory for the sum of two operators, Applied Math. Letters, 9:1–8, 1996.
 D.R. Smart, Fixed Point Theorems, Cambridge University Press, 1974.