Nonlocal Cauchy problems governed by compact operator families. (English) Zbl 1083.34045

Let \(A\) be the infinitesimal generator of a compact semigroup of linear operators on a Banach space \(X\). The authors establish the existence of mild solutions to the nonlocal Cauchy problem \[ u'(t)=Au(t)+f(t,u(t)), \quad t\in[t_0,t_0+T], \quad u(t_0)+g(u)=u_0, \] under some conditions on \(f\) and \(g\), where \(f:[t_0,t_0+T]\times X\to X\) and \(g:C([t_0,t_0+T];X)\to X\) are given functions. They assume a Lipschitz condition on \(f\) with respect to \(u\), but they do not require any compactness assumption on \(g\), opposed to S. Aizicovici and M. McKibben [Nonlinear Anal., Theory Methods Appl. 39, No. 5(A), 649–668 (2000; Zbl 0954.34055)] and L. Byszewski and H. Akca [Nonlinear Anal., Theory Methods Appl. 34, No. 1, 65–72 (1998; Zbl 0934.34068)], where the authors assume a compactness property for \(g\), but do not require any Lipschitz condition on \(f\).


34G20 Nonlinear differential equations in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
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[1] Aizicovici, S.; McKibben, M., Existence results for a class of abstract nonlocal Cauchy problems, Nonlinear Anal. Ser. A: Theory Methods, 39, 649-668 (2000) · Zbl 0954.34055
[2] Boucherif, A., First-order differential inclusions with nonlocal initial conditions, Appl. Math. Lett, 15, 409-414 (2002) · Zbl 1025.34009
[3] 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
[4] Byszewski, L., Uniqueness of solutions of parabolic semilinear nonlocal-boundary problems, J. Math. Anal. Appl, 165, 472-478 (1992) · Zbl 0774.35038
[5] Byszewski, L., Application of properties of the right-hand sides of evolution equations to an investigation of nonlocal evolution problems, Nonlinear Anal. TMA, 33, 413-426 (1998) · Zbl 0933.34064
[6] Byszewski, L.; Akca, H., Existence of solutions of a semilinear functional-differential evolution nonlocal problem, Nonlinear Anal. TMA, 34, 65-72 (1998) · Zbl 0934.34068
[7] Byszewski, L.; Lakshmikantham, V., Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Applicable Anal, 40, 11-19 (1990) · Zbl 0694.34001
[8] Byszewski, L.; Lakshmikantham, V., Monotone iterative technique for nonlocal hyperbolic differential problem, J. Math. Phys. Sci, 26, 4, 345-359 (1992) · Zbl 0811.35083
[9] Deng, K., Exponential decay of solutions of semilinear parabolic equations with non-local initial conditions, J. Math. Anal. Appl, 179, 630-637 (1993) · Zbl 0798.35076
[10] Jackson, D., Existence and uniqueness of solutions of a semilinear nonlocal parabolic equations, J. Math. Anal. Appl, 172, 256-265 (1993) · Zbl 0814.35060
[11] Liang, J.; van Casteren, J.; Xiao, T. J., Nonlocal Cauchy problems for semilinear evolution equations, Nonlinear Anal. Ser. A: Theory Methods, 50, 173-189 (2002) · Zbl 1009.34052
[13] Lin, Y.; Liu, J. H., Semilinear integrodifferential equations with nonlocal Cauchy problem, Nonlinear Anal. TMA, 26, 1023-1033 (1996) · Zbl 0916.45014
[14] Liu, J. H., A remark on the mild solutions of non-local evolution equations, Semigroup Forum, 26, 1023-1033 (2003)
[15] Ntouyas, S. K.; Tsamatos, P. Ch., Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl, 210, 679-687 (1997) · Zbl 0884.34069
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