# Slyusarchuk V. Yu.

### Method of local linear approximation for nonlinear discrete equations

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 9. - pp. 1284-1296

UDC 517.988.6

We obtain new conditions for the existence of bounded solutions of nonlinear discrete equations with application of the local linear approximation of these equations.

### On the Favard theory without $H$ -classes for differential-functional equations in Banach spaces

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 7. - pp. 952-967

UDC 517.937

We obtain necessary and sufficient conditions for the existence and uniqueness of solutions bounded on the real axis with
precompact sets of values of linear almost periodic differential-functional equations in Banach spaces.

### Investigation of systems of differential equations with delays and constraints imposed on the delays and derivatives of the solutions

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 5. - pp. 677-691

UDC 517.929; 517.958:531–133

We establish conditions for the existence and uniqueness of the solutions of nonlinear systems of differential equations with delays and restrictions imposed on the delays and derivatives of the solutions.

### Necessary and sufficient conditions for the absolute instability of solutions of linear differential-difference equations with self-adjoint operator coefficients

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 5. - pp. 715-724

For linear differential-difference equations of retarded and neutral types with infinitely many deviations and self-adjoint operator coefficients, we present necessary and sufficient conditions for the absolute instability of the zero solutions.

### Favard – Amerio theory for almost periodic functional-differential equations without the use of $H$-classes of these equations

Ukr. Mat. Zh. - 2017. - 69, № 6. - pp. 788-802

The Favard – Amerio theory is constructed for almost periodic functional-differential equations in a Banach space without the use of $\scr H$ -classes of these equations. For linear equations, we present the first example of an almost periodic operator, which has no analogs in the classical Favard – Amerio theory.

### Conditions of existence of bounded and almost periodic solutions of nonlinear differential equation with perturbations of solutions

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1286-1296

We present the conditions of existence and uniqueness of bounded solutions of a nonlinear scalar differential equation $dx(t)/dt=f(x(t)+h(t)),\; t \in R$, in the case where a function $f$ is continuous on $R$ and a function $h$ is bounded and continuous. In addition, we study the case of an almost periodic function $h$.

### Necessary and sufficient conditions for the invertibility of nonlinear differentiable maps

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 563-576

We establish necessary and sufficient conditions for the invertibility of nonlinear differentiable maps in the case of arbitrary Banach spaces. We establish conditions for the existence and uniqueness of bounded and almost periodic solutions of nonlinear differential and difference equations.

### Mykola Oleksiiovych Perestyuk (on his 70th birthday)

Boichuk О. A., Gorbachuk M. L., Gorodnii M. F., Khruslov E. Ya., Lukovsky I. O., Makarov V. L., Parasyuk I. O., Samoilenko A. M., Samoilenko V. G., Sharkovsky O. M., Shevchuk I. A., Slyusarchuk V. Yu., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 142-144

### Almost periodic and Poisson stable solutions of difference equations in metric spaces

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1707-1714

We introduce a new class of almost periodic operators and establish the conditions of existence of almost periodic and Poisson stable solutions of difference equations in metric spaces that can be not almost periodic in Bochner’s sense.

### A Criterion for the Existence of Almost Periodic Solutions of Nonlinear Differential Equations with Impulsive Perturbation

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 838–848

We establish conditions for the existence of almost periodic solutions of nonlinear almost periodic differential equations with impulsive perturbation in Banach spaces without using the \( \mathcal{H} \)-classes of these equations.

### Almost Periodic Solutions of Nonlinear Equations that are not Necessarily Almost Periodic in Bochner’s Sense

Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 230-244

We introduce a new class of almost periodic operators and establish conditions for the existence of almost periodic solutions of nonlinear equations that are not necessarily almost periodic in Bochner’s sense.

### Conditions for Almost Periodicity of Bounded Solutions of Nonlinear Differential Equations Unsolved with Respect to the Derivative

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 384–393

We establish conditions for the existence of almost periodic solutions of nonlinear almost periodic differential equations in Banach spaces without using the *H*-classes of these equations.

### Differential Equations with Absolutely Unstable Trivial Solutions

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1266–1275

We obtain conditions for the absolute instability of trivial solutions of the nonlinear differential equations.

### Conditions for the existence of almost periodic solutions of nonlinear differential equations in Banach spaces

Ukr. Mat. Zh. - 2013. - 65, № 2. - pp. 307-312

We obtain conditions for the existence of almost periodic solutions of nonlinear almost periodic differential equations in a Banach space without using the $\mathcal{H}$-classes of these equations.

### Denseness of the set of Cauchy problems with nonunique solutions in the set of all Cauchy problems

Ukr. Mat. Zh. - 2012. - 64, № 7. - pp. 1001-1006

We prove the following theorem: Let $E$ be an arbitrary Banach space, $G$ be an open set in the space $R×E$, and $f : G → E$ be an arbitrary continuous mapping. Then, for an arbitrary point $(t_0, x_0) ∈ G$ and an arbitrary number $ε > 0$, there exists a continuous mapping $g : G → E$ such that $$\sup_{(t,x)∈G}||g(t, x) − f(t, x)|| \leq \varepsilon$$ and the Cauchy problem $$\frac{dz(t)}{dt} = g(t, z(t)), z(t0) = x_0$$ has more than one solution.

### Existence and exponential stability of periodic solution for fuzzy BAM neural networks with periodic coefficient

Ukr. Mat. Zh. - 2011. - 63, № 12. - pp. 1685-1698

We obtain conditions for the existence of solutions of nonlinear differential equations in the space of functions bounded on the axis by using a local linear approximation of these equations.

### Strengthening of the Kneser theorem on zeros of solutions of the equation $u″ + q(t)u = 0$ using one functional equation

Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1705–1714

We present conditions under which a linear homogeneous second-order equation is nonoscillatory on a semiaxis and conditions under which its solutions have infinitely many zeros.

### Conditions for the existence of bounded solutions of nonlinear differential and functional differential equations

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 837–846

Let $E$ be a finite-dimensional Banach space, let $C^0(R; E)$ be a Banach space of functions continuous and bounded on $R$ and taking values in $E$; let $K:\;C^0(R ,E) → C^0(R, E)$ be a $c$-continuous bounded mapping, let $A:\;E → E$ be a linear continuous mapping, and let $h ∈ C^0(R, E)$. We establish conditions for the existence of bounded solutions of the nonlinear equation $$\frac{dx(t)}{dt} + (Kx)(t)Ax(t) = h(t),\;t ∈ R.$$

### Method of local linear approximation in the theory of bounded solutions of nonlinear differential equations

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1541-1556

The conditions for the existence of solutions of nonlinear differential equations in a space of functions bounded on the axis are established by using local linear approximations of these equations.

### Green–Samoilenko operator in the theory of invariant sets of nonlinear differential equations

Perestyuk N. A., Slyusarchuk V. Yu.

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 948-957

We establish conditions for the existence of an invariant set of the system of differential equations $$\frac{dφ}{dt} = a(φ),\quad \frac{dx}{dt} = P(φ)x + F(φ,x),$$ where $a: Φ → Φ, P: Φ → L(X, X)$, and $F: Φ × X→X$ are continuous mappings and $Φ$ and $X$ are finite-dimensional Banach spaces.

### Conditions for the existence and uniqueness of bounded solutions of nonlinear differential equations

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 268-279

We establish conditions required for the existence and uniqueness of bounded solutions of the nonlinear differential equation $f_1\left(\frac{dx(t)}{dt} \right) = f_2(x(t)), t ∈ ℝ$.

### Generalization of the Mukhamadiev theorem on the invertibility of functional operators in the space of bounded functions

Ukr. Mat. Zh. - 2008. - 60, № 3. - pp. 398–412

We obtain necessary and sufficient conditions of reversibility of the linear bounded operator *d ^{m }* /

*d t*+

^{m }*A*in the space of functions bounded on

*R*.

### Generalization of the Kneser theorem on zeros of solutions of the equation $y" + p (t) y = 0$

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 571–576

We obtain conditions of the oscillation of solutions of the equation $y" + p(t)Ay = 0$ in the Banach space, where $A$ is a bounded linear operator and $p : R_+ \rightarrow R_+$ is a continuous function.

### Periodic solutions of nonlinear differential equations with pulse influence in a banach space

Perestyuk N. A., Slyusarchuk V. Yu.

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 370–380

Rank conditions for control of linear pulse systems are established. An example of control synthesis in a problem for linear pulse systems is given.

### Necessary and sufficient conditions for oscillation of solutions of nonlinear pulse systems with multiplicatively separated right-hand side

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 381–389

Necessary and sufficient conditions for oscillation of solutions of nonlinear differential equations with fixed times of pulse influence and multiplicatively separated right-hand sides are established.