## 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.

### MSC:

 34G20 Nonlinear differential equations in abstract spaces 47D06 One-parameter semigroups and linear evolution equations
Full Text:

### References:

 [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 $$u_{ xt } = F (x, t, u, u_x )$$, 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), Springer-Verlag: Springer-Verlag New York/Berlin/Heidelberg/Tokyo · 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.