# Mel'nik V. S.

### Construction of intermediate differentiable functions

Maslyuchenko V. K., Mel'nik V. S.

Ukr. Mat. Zh. - 2018. - 70, № 5. - pp. 672-681

For given upper and lower semicontinuous real-valued functions $g$ and $h$, respectively, defined on a closed parallelepiped $X$ in $R^n$ and such that $g(x) < h(x)$ on $X$ and points $x_0 \in X$ and $y_0 \in (g(x_0), h(x_0))$, we construct a smooth function $f : X \rightarrow R$ such that $f(x_0) = y_0$ and $g(x) < f(x) < h(x)$ on $X$. We also present similar constructions for functions defined on separable Hilbert spaces and Asplund spaces.

### Evolution inequalities with noncoercive *w *_{λ 0 }-pseudomonotone volterra-type mappings

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1499 – 1519

We consider a class of differential-operator inequalities with noncoercive *w *_{λ 0 }-pseudomonotone operators.
The problem of the existence of solutions of the Cauchy problem for these inequalities is investigated by using the Dubinsky method.
A priori estimates for these solutions and their derivatives are obtained. We give a model example that illustrates the results and generalizations obtained.

### Topological methods in the theory of operator inclusions in Banach spaces. II

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 505–521

We develop topological methods for the investigation of operator inclusions in Banach spaces, prove the generalized Ky Fan inequality, and study the critical points of many-valued mappings in topological spaces.

### Topological methods in the theory of operator inclusions in Banach spaces. I

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 184–194

We develop topological methods for the investigation of operator inclusions in Banach spaces, prove the generalized Ky Fan inequality, and study the critical points of many-valued mappings in topological spaces.

### On properties of subdifferential mappings in Fréchet spaces

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1385–1394

We present conditions under which the subdifferential of a proper convex lower-semicontinuous functional in a Fréchet space is a bounded upper-semicontinuous mapping. The theorem on the boundedness of a subdifferential is also new for Banach spaces. We prove a generalized Weierstrass theorem in Fréchet spaces and study a variational inequality with a set-valued mapping.

### Academician V. Ya. Bunyakovs'kyi (on 200-th anniversary of his birthday)

Mel'nik V. S., Mel'nyk O. M., Samoilenko A. M.

Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1675-1683

### Multivariational Inequalities and Operator Inclusions in Banach Spaces with Mappings of the Class $(S)_{+}$

Ukr. Mat. Zh. - 2000. - 52, № 11. - pp. 1513-1523

We prove theorems on the existence of solutions of variational inequalities and operator inclusions in Banach spaces with multivalued mappings of the class (*S*)_{+}. We justify the method of penalty operators for variational inequalities.

### On weak compactness of bounded sets in Banach and locally convex spaces

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 731-739

We investigate the compactness of one class of bounded subsets in Banach and locally convex spaces. We obtain a generalization of the Banach-Alaoglu theorem to a class of subsets that are not polars of convex balanced neighborhoods of zero.

### $T$-differentiable functionals and ther critical points

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 720–728

The critical points of the functionals $F:\; D \subset X \rightarrow \mathbb{R}$ defined on "nonlinear" sets $D$ in the topological vector spaces $X$ are studied. A construction of a $T$-derivative is suggested for these functionals and compared with to known constructions. The concept of a weak critical point is introduced and Coleman's principle is justified for $T$-differentiable functionals.

### Extremal solutions of certain operator-differential systems

Ukr. Mat. Zh. - 1989. - 41, № 11. - pp. 1494–1501