# Volume 64, № 7, 2012

### Impulsive differential inclusions involving evolution operators in separable Banach spaces

Benchohra M., Nieto J. J., Ouahab A.

Ukr. Mat. Zh. - 2012. - 64, № 7. - pp. 867-891

We present some results on the existence of mild solutions and study the topological structure of the sets of solutions for the following first-order impulsive semilinear differential inclusions with initial and boundary conditions: $$y'(t) − A(t)y(t) \in F(t, y(t)) \text{for a.e.} t \in J\ \{t1,..., tm,...\},$$ $$y(t^+_k) − y(t^−_k) = I_k(y(t^−_k)),\quad k = 1,...,$$ $$y(0) = a$$ and $$y'(t) − A(t)y(t) \in F(t, y(t)) \text{for a.e.} t \in J\ \{t1,..., tm,...\},$$ $$y(t^+_k) − y(t^−_k) = I_k(y(t^−_k)),\quad k = 1,...,$$ $$Ly = a,$$ where $J = IR_+,\; 0 = t_0 < t_1 <...< t_m < ...;\; (m \in N), \lim_{k→∞} t_k = ∞,\; A(t)$ is the infinitesimal generator of a family of evolution operator $U(t, s)$ on a separable Banach space $E$, and $F$ is a set-valued mapping. The functions $I_k$ characterize the jump of solutions at the impulse points $t_k,\; k = 1,... .$ The mapping $L: P C_b → E$ is a bounded linear operator. We also investigate the compactness of the set of solutions, some regularity properties of the operator solutions, and the absolute retractness.

### Local deformations of positive-definite quadratic forms

Bondarenko V. M., Bondarenko V. V., Pereguda Yu. N.

Ukr. Mat. Zh. - 2012. - 64, № 7. - pp. 892-907

We give a complete description of real numbers that are $P$-limit numbers for integer-valued positive-definite quadratic forms with unit coefficients of the squares. It is shown that each of these $P$-limit numbers is realized in the Tits quadratic form of a Dynkin diagram.

### $C_\lambda$-semiconservative $FK$-spaces

Ukr. Mat. Zh. - 2012. - 64, № 7. - pp. 908-918

We study $C_\lambda$-semiconservative $FK$-spaces for $C_\lambda$-methods defined by deleting a set of rows from the Cesaro matrix $C_1$ and give some characterizations.

### The block by block method with Romberg quadrature for solving nonlinear Volterra integral equations on the large intervals

Ukr. Mat. Zh. - 2012. - 64, № 7. - pp. 919-931

We investigate the numerical solution of nonlinear Volterra integral equations by block by block method, which is useful specially for solving integral equations on large-size intervals. A convergence theorem is proved that shows that the method has at least sixth order of convergence. Finally, the performance of the method is illustrated by some numerical examples.

### On the Dirichlet problem for the Beltrami equations in finitely connected domains

Kovtonyuk D. A., Petkov I. V., Ryazanov V. I.

Ukr. Mat. Zh. - 2012. - 64, № 7. - pp. 932-944

We establish a series of criteria for the existence of regular solutions of the Dirichlet problem for degenerate Beltrami equations in arbitrary Jordan domains. We also formulate the corresponding criteria for the existence of pseudoregular and multivalued solutions of the Dirichlet problem in the case of finitely connected domains.

### Problem with pulse action for systems with Bessel?Kolmogorov operators

Ukr. Mat. Zh. - 2012. - 64, № 7. - pp. 945-953

We construct the fundamental matrix of solutions of the Cauchy problem and a problem with impulse action for systems with Bessel - Kolmogorov operators degenerate in all space variables. Estimates for the fundamental matrix are obtained, and its properties are established.

### Exact order of approximation of periodic functions by one nonclassical method of summation of Fourier series

Ukr. Mat. Zh. - 2012. - 64, № 7. - pp. 954-969

By using an exact estimate for approximation by known trigonometric polynomials, we strengthen a Jackson-type theorem. Moreover, we determine the exact order of approximation of some periodic functions by these polynomials. For this purpose, we introduce a special modulus of smoothness.

### Asymptotic *m*-phase soliton-type solutions of a singularly perturbed Korteweg?de Vries equation with variable coefficients

Samoilenko V. G., Samoilenko Yu. I.

Ukr. Mat. Zh. - 2012. - 64, № 7. - pp. 970-87

We propose an algorithm for the construction of asymptotic *m*-phase soliton-type solutions of a singularly perturbed
Korteweg – de Vries equation with varying coefficients and establish the accuracy with which the main term asymptotically satisfies the considered equation.

### Conditions for balance between survival and ruin

Ukr. Mat. Zh. - 2012. - 64, № 7. - pp. 988-993

Let $\xi_t$ be a classic risk process or a risk process with stochastic premiums. We establish conditions for balance between ruin and survival in the case of zero initial capital $u = 0$ (ruin probability $q_{+} = \psi(0) = 1/2$, survival probability $p_{+} = 1 — q_{+} = 1/2$) and determine premium estimates under these conditions.

### On transformation formulae for theta hypergeometric functions

Denis R. Y., Singh S. N., Singh S. P.

Ukr. Mat. Zh. - 2012. - 64, № 7. - pp. 994-1000

Using an identity and certain summation formulas for truncated theta hypergeometric series, we establish transformation formulas for finite bilateral theta hypergeometric series.

### 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.