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Existence results of semilinear differential equations with nonlocal initial conditions in Banach spaces. (English) Zbl 1227.34060
The authors study the existence of solutions to the following problem $$u'(t)=Au(t)+f(t,u(t)),\quad t\in (0,T],\quad u(0) = g(u),$$ where $A$ is the generator of a linear semigroup, $f,g$ satisfy Lipschitz conditions. When $g$ is a constant function this is a Cauchy problem associated with a semilinear evolution equation. If $g$ is an arbitrary Lipschitz function this problem is called “nonlocal” Cauchy problem. The authors use the measure of noncompactness and a fixed point theorem to prove an existence result.

34G10Linear ODE in abstract spaces
47D06One-parameter semigroups and linear evolution equations
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
[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 · doi:10.1016/0022-247X(91)90164-U
[2] 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 · doi:10.1080/00036819008839989
[3] Ntouyas, S.; 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
[4] Byszewski, L.; Akca, H.: Existence of solutions of a semilinear functional-differential evolution nonlocal problem, Nonlinear anal. 34, 65-72 (1998) · Zbl 0934.34068 · doi:10.1016/S0362-546X(97)00693-7
[5] Fu, X.; Ezzinbi, K.: Existence of solutions for neutral functional differential evolution equations with nonlocal conditions, Nonlinear anal. 54, 215-227 (2003) · Zbl 1034.34096 · doi:10.1016/S0362-546X(03)00047-6
[6] 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
[7] Aizicovici, S.; Mckibben, M.: Existence results for a class of abstract nonlocal Cauchy problems, Nonlinear anal. 39, 649-668 (2000) · Zbl 0954.34055 · doi:10.1016/S0362-546X(98)00227-2
[8] Aizicovici, S.; Lee, H.: Existence results for nonautonomous evolution equations with nonlocal initial conditions, Commun. appl. Anal. 11, 285-297 (2007) · Zbl 1147.34043
[9] Aizicovici, S.; Staicu, V.: Multivalued evolution equations with nonlocal initial conditions in Banach spaces, Nonlinear differential equations appl. 14, 361-376 (2007) · Zbl 1145.35076 · doi:10.1007/s00030-007-5049-5
[10] Cardinali, T.; Rubbioni, P.: J. math. Anal. appl., J. math. Anal. appl. 308, 620-635 (2005)
[11] Garcia-Falset, J.: Existence results and asymptotic behavior for nonlocal abstract Cauchy problems, J. math. Anal. appl. 338, 639-652 (2008) · Zbl 1140.34026 · doi:10.1016/j.jmaa.2007.05.045
[12] Kamenskii, M.; Obukhovskii, V.; Zecca, P.: Condensing multivalued maps and semilinear differential inclusions in Banach spaces, De gruyter series in nonlinear anal. Appl. 7 (2001) · Zbl 0988.34001
[13] Liang, J.; Liu, J.; Xiao, T.: Nonlocal Cauchy problems governed by compact operator families, Nonlinear anal. 57, 183-189 (2004) · Zbl 1083.34045 · doi:10.1016/j.na.2004.02.007
[14] Liang, J.; Liu, J.; Xiao, T.: Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Math. comput. Modelling 49, 798-804 (2009) · Zbl 1173.34048 · doi:10.1016/j.mcm.2008.05.046
[15] Xue, X.: Semilinear nonlocal problems without the assumptions of compactness in Banach spaces, Anal. appl. 8, 211-225 (2010) · Zbl 1202.34107 · doi:10.1142/S021953051000159X
[16] Xue, X.: Lp theory for semilinear nonlocal problems with measure of noncompactness in separable Banach spaces, J. fixed point theory appl. 5, 129-144 (2009) · Zbl 1182.34081 · doi:10.1007/s11784-008-0090-5
[17] Xue, X.: Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces, Nonlinear anal. 70, 2593-2601 (2009) · Zbl 1176.34071 · doi:10.1016/j.na.2008.03.046
[18] Zhu, L.; Li, G.: On a nonlocal problem for semilinear differential equations with upper semicontinuous nonlinearities in general Banach spaces, J. math. Anal. appl. 341, 660-675 (2008) · Zbl 1145.34034 · doi:10.1016/j.jmaa.2007.10.041
[19] Zhu, L.; Li, G.: Nonlocal differential equations with multivalued perturbations in Banach spaces, Nonlinear anal. 69, 2843-2850 (2008) · Zbl 1163.34041 · doi:10.1016/j.na.2007.08.057
[20] Bothe, D.: Multivalued perturbations of m-accretive differential inclusions, Israel J. Math. 108, 109-138 (1998) · Zbl 0922.47048 · doi:10.1007/BF02783044
[21] Obukhovski, V.; Zecca, P.: Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup, Nonlinear anal. 70, 3424-3436 (2009) · Zbl 1157.93006 · doi:10.1016/j.na.2008.05.009
[22] Sun, J.; Zhang, X.: The fixed point theorem of convex-power condensing operator and applications to abstract semilinear evolution equations, Acta math. Sinica (Chin. Ser.) 48, No. 3, 439-446 (2005) · Zbl 1124.34342
[23] Banas, J.; Goebel, K.: Measure of noncompactness in Banach spaces, Lecture notes in pure and applied mathematics 60 (1980) · Zbl 0441.47056
[24] Agarwal, R.; Meehan, M.; O’regan, D.: Fixed point theory and applications, Cambridge tracts in mathematics, (2001)
[25] Pazy, A.: Semigroups of linear operators and applications to partial differential equations, (1983) · Zbl 0516.47023
[26] Martin, R. H.: Nonlinear operators and differential equations in Banach spaces, (1976) · Zbl 0333.47023