2018
Том 70
№ 11

All Issues

Piven’ A. L.

Articles: 2
Article (Ukrainian)

Entire solutions of one linear implicit differential-difference equation in Banach spaces

Gefter S. L., Piven’ A. L.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1044-1057

We establish the existence and uniqueness conditions for the solution for the initial problem $Bu\prime (z) = Au(z + h) + f(z),\; z \in C, u(0) = u_0$ in the classes of entire functions of exponential type. Closed linear operators $A$ and $B$ act on Banach spaces and can be degenerate. We also present an example of application of abstract results to partial differential equations.

Article (Russian)

Criteria for the Well-Posedness of the Cauchy Problem for Differential Operator Equations of Arbitrary Order

Piven’ A. L., Rutkas A. G., Vlasenko L. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1484-1500

In Banach spaces, we investigate the differential equation \(\mathop \sum \nolimits_{j = 0}^n \;A_j u^{(j)} (t) = 0\) with closed linear operators A j (generally speaking, the operator coefficient A n of the higher derivative is degenerate). We obtain well-posedness conditions that characterize the continuous dependence of solutions and their derivatives on initial data. Abstract results are applied to partial differential equations.