# Zhuravlev V. F.

### Conditions of solvability and representation of the solutions of equations with operator matrices

Fomin N. P., Zabrodskiy P. N., Zhuravlev V. F.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 4. - pp. 471-485

We propose new methods for the construction of generalized inverse operator matrices for the operator matrices in Banach spaces. The solvability criteria and the formulas for representations of the general solutions of operator equations with operator matrices are obtained. As an application, we consider the relationship between the obtained formulas and the well-known Frobenius formula for the construction of the matrix inverse to a nondegenerate block matrix.

### Bifurcation conditions for the solutions of weakly perturbed boundary-value problems for operator equations in Banach spaces

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 366-378

We obtain bifurcation conditions for the solutions of weakly perturbed boundary-value problems for operator equations in Banach spaces from the point $\varepsilon = 0$. A convergent iterative procedure is proposed for the construction of solutions as parts of series in powers of $\varepsilon$ with pole at the point $\varepsilon = 0$.

### Weakly perturbed operator equations in Banach spaces

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 6. - pp. 751-764

We obtain the conditions of bifurcation of the solutions of weakly perturbed operator equations in Banach spaces from the point $\varepsilon = 0$ and propose a convergent iterative procedure for finding the solutions in the form of parts of the series in powers of $\varepsilon$ with pole at the point $\varepsilon = 0$.

### Normally solvable operator equations in a Banach space

Boichuk A. A., Pokutnyi A. A., Zhuravlev V. F.

Ukr. Mat. Zh. - 2013. - 65, № 2. - pp. 163-174

On the basis of a generalization of the well-known Schmidt lemma to the case of linear, bounded, normally solvable operators in Banach spaces, we propose a procedure for the construction of a generalized inverse for a linear, bounded, normally solvable operator whose kernel and image are complementable in the indicated spaces. This construction allows one to obtain a solvability criterion for linear normally solvable operator equations and a formula for finding their general solutions.

### Solvability criterion and representation of solutions of $n$-normal and $d$-normal linear operator equations in a Banach space

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 167–182

On the basis of a generalization of the well-known Schmidt lemma to the case of n-normal and d-normal linear bounded operators in a Banach space, we propose constructions of generalized inverse operators. We obtain criteria for the solvability of linear equations with these operators and formulas for the representation of solutions of these equations.

### Weakly nonlinear boundary-value problems for operator equations with pulse influence

Boichuk A. A., Samoilenko A. M., Zhuravlev V. F.

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 272–288

We consider the problem of finding conditions of solvability and algorithms for construction of solutions of weakly nonlinear boundary-value problems for operator equations (with the Noetherian linear part) with pulse influence at fixed times. The method of investigation is based on passing by methods of the Lyapunov—Schmidt type from a pulse boundary-value problem to an equivalent operator system that can be solved by iteration procedures based on the fixed-point principle.

### Construction of the solutions of linear operator equations in Banach spaces

Boichuk A. A., Zhuravlev V. F.

Ukr. Mat. Zh. - 1991. - 43, № 10. - pp. 1343–1350