Existence of solutions for parabolic-type evolution differential inclusions and properties of the solution set. (English) Zbl 0940.34052
The purpose of this paper is to study the parabolic-type differential inclusion with the operator \(A\) depending on time \(t\) in Banach spaces
\[
x'(t)+A(t)x(t)\in F(t,x(t)),\quad \text{a.e.}\quad t\in I,\quad x(0)=x_0.\tag{1}
\]
An existence theorem to problem (1) with convex right-hand sides is proved and properties of the solution set are given. Some results by J.-P. Aubin and A. Cellina [Differential Inclusions. Set-valued maps and viability theory, Berlin, Springer-Verlag (1984; Zbl 0538.34007)] are used in the proofs.
Reviewer: Anatoly Martynyuk (Kyïv)
MSC:
| 34G25 | Evolution inclusions |
| 34A60 | Ordinary differential inclusions |
| 34K45 | Functional-differential equations with impulses |
Citations:
Zbl 0538.34007References:
| [1] | Aubin J P, Cellina A.Differential Inclusions[M]. Berlin: Springer-Verlag, 1984 |
| [2] | Frankowska H. Estimations a priori pour les inclusions differential operationelles[J].C R Acad Sci Paris, 1989,308(5):47–50 |
| [3] | Xue X, Song G. Existence results of mild-solutions to semilinear evolution inclusions in Banach spaces[J].Northeast Math J, 1995,11(7):151–156 |
| [4] | Vrabie V V. Some compactness methods in the theory of nonlinear evolution equations to P D E[J].Banach Centre Publ, 1987,19(7):351–360 |
| [5] | Wagner D. Survey of measurable selection theorems[J].SIAM J Control and Optim, 1977,15 (3):859–903 · Zbl 0407.28006 · doi:10.1137/0315056 |
| [6] | Zhou Hongxing, Wang Lianwen.Theory and Applications of Linear Operator Semigroup [M]. Jinan: Shandong Science Press, 1990 (in Chinese) |
| [7] | Aubin J P, Ekeland I.Applied Nonlinear Analysis[M]. New York: Wiley Sons, 1984 · Zbl 0641.47066 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
