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Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. (English) Zbl 0748.34040
Theorem on existence and uniqueness for semilinear evolution equations in Banach spaces are given. The problem is a “nonlocal” one, i.e., a relation between the solution values at different time-points is given. The theory of semigroups of linear operators is the main tool used.

34G20Nonlinear ODE in abstract spaces
47D06One-parameter semigroups and linear evolution equations
Full Text: DOI
[1] Byszewski, L.: Strong maximum and minimum principles for parabolic problems with nonlocal inequalities. Z. angew. Math. mech. 70.3, 202-206 (1990) · Zbl 0709.35018
[2] Byszewski, L.: Strong maximum principles for parabolic nonlinear problems with nonlocal inequalities together with integrals. J. appl. Math. stochastic anal. 3.5, 65-79 (1990) · Zbl 0726.35023
[3] Byszewski, L.: Strong maximum principles for parabolic nonlinear problems with nonlocal inequalities together with arbitrary functionals. J. math. Anal. appl. 156, 457-470 (1991) · Zbl 0737.35135
[4] Byszewski, L.: Existence and uniqueness of solutions of nonlocal problems for hyperbolic equation uxt = F (x, t, u, ux). J. appl. Math. stochastic anal. 3.3, 163-168 (1990) · Zbl 0725.35059
[5] Byszewski, L.: Theorem about existence and uniqueness of continuous solution of nonlocal problem for nonlinear hyperbolic equation. Appl. anal. 40, 173-180 (1991) · Zbl 0725.35060
[6] 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
[7] Pazy, A.: Semigroups of linear operators and applications to partial differential equations. (1983) · Zbl 0516.47023