# Pavlenko V. N.

### Periodic solutions of a parabolic equation with homogeneous Dirichlet boundary condition and linearly increasing discontinuous nonlinearity

Fedyashev M. S., Pavlenko V. N.

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1080-1088

We consider a resonance problem of the existence of periodic solutions of parabolic equations with discontinuous nonli-nearities and a homogeneous Dirichlet boundary condition. It is assumed that the coefficients of the differential operator do not depend on time, and the growth of the nonlinearity at infinity is linear. The operator formulation of the problem reduces it to the problem of the existence of a fixed point of a convex compact mapping. A theorem on the existence of generalized and strong periodic solutions is proved.

### Resonance elliptic variational inequalities with discontinuous nonlinearities of linear growth

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 513-522

We consider resonance elliptic variational inequalities with second-order differential operators and discontinuous nonlinearity of linear grows. The theorem on the existence of a strong solution is obtained. The initial problem is reduced to the problem of the existence of a fixed point in a compact multivalued mapping and then, with the use of the Leray - Schauder method, the existence of the fixed point is established.

### Existence Theorem for One Class of Strongly Resonance Boundary-Value Problems of Elliptic Type with Discontinuous Nonlinearities

Ukr. Mat. Zh. - 2005. - 57, № 1. - pp. 102–110

We consider the Dirichlet problem for an equation of the elliptic type with a nonlinearity discontinuous with respect to the phase variable in the resonance case; it is not required that the nonlinearity satisfy the Landesman-Lazer condition. Using the regularization of the original equation, we establish the existence of a generalized solution of the problem indicated.

### Existence Theorems for Equations with Noncoercive Discontinuous Operators

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 349-364

In a Hilbert space, we consider equations with a coercive operator equal to the sum of a linear Fredholm operator of index zero and a compact operator (generally speaking, discontinuous). By using regularization and the theory of topological degree, we establish the existence of solutions that are continuity points of the operator of the equation. We apply general results to the proof of the existence of semiregular solutions of resonance elliptic boundary-value problems with discontinuous nonlinearities.

### Continuous approximations of discontinuous nonlinearities of semilinear elliptic-type equations

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 224–233

We obtain new variational principles of the existence of strong and semiregular solutions of principal boundary-value problems for elliptic-type second-order equations with discontinuous nonlinearity. We study a problem of proximity between the sets of solutions of an approximating problem with nonlinearity continuous in phase variable and solutions of the initial boundary-value problem with discontinuous nonlinearity.

### Control of singular distributed parabolic systems with discontinuous nonlinearities

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 729–736

We present the statement of control problems for singular distributed parabolic systems with discontinuous nonlinearties. Sufficient conditions for the existence of the optimal “control-state” pair are established under the assumption that the set of admissible “control-state” pairs is nonempty. The problem of existence of serniregular solutions is studied for the equation of state of a distributed system. It is not assumed that the nonlinearity of this equation increases sublinearly in the phase variable or possesses a bounded variation on any segment of the straight line.

### On solvability of variational inequalities with discontinuous semimonotone operators

Ukr. Mat. Zh. - 1993. - 45, № 3. - pp. 443–447

By using the method of monotone operators, a theorem on the existence of the solution with a special property is obtained for an elliptic variational inequality with discontinuous semimonotone operator; this theorem is then used to prove the existence of a semicorrect solution of a variational inequality with a differential semilinear high-order operator of elliptic type with a nonsymmetric linear part and discontinuous nonlinearity.

### Semiregular solutions of elliptic variational inequalities with discontinuous nonlinearities

Ukr. Mat. Zh. - 1991. - 43, № 2. - pp. 230–235

### Existence of semiregular solutions of the dirichlet problem for quasilinear equations of elliptic type with discontinuous nonlinearities

Ukr. Mat. Zh. - 1989. - 41, № 12. - pp. 1659–1664

### Existence of solutions of nonlinear equations with discontinuous semimonotonic operators

Ukr. Mat. Zh. - 1981. - 33, № 4. - pp. 547-552

### Nonlinear equations with discontinuous operators in Banach spaces

Ukr. Mat. Zh. - 1979. - 31, № 5. - pp. 569–572