Riccati equation plays an important role in the theory of optimal control, physics and many others applications. It should be emphasized that such equations are very often used in the games theory and calculus of variations. It should be noted here that in general, many papers are devoted to obtaining the conditions of the solvability in the regular case. We noted such papers as [1] -[12] where considering equation was investigated in the finite- dimensional and in the infinite-dimensional cases. It should be noted that this equation was investigated as in the operator and matrix form or as in the operator-differential and differential form (see the bibliography).

There are many papers where the matrix Riccati equations and operator-differential Riccati equations were investigated. As a rule, such equations were investigated in the regular case where the given problem had a unique solution. In the nonregular case, such equation was investigated (in the periodic case) in the work of Boichuk O. A. and Krivosheya S. A. [12] . In the paper [4] , the discrete Riccati equation was investigated. In the paper of Pronkin [8] , the question about quasiperiodic solutions of the matrix Riccati equation with coefficients which are Arnold functions is investigated. Dissertation of Christian Wyss [1] is also devoted to the perturbation theory for Hamiltonian operator matrices and Riccati equations in the Hilbert space.

In the present paper, using the technique of generalized inverse operators, we derive a criterion for the solvability of the given problem, generating problem and analyzing the structure of the solution set. We construct the iterative process for finding the solutions of weakly nonlinear problem which is the modification of Newton method and converges with quadratic error.

The article consists of three parts.

The first part of the given paper is devoted to the statement of the problem and denotations.

The second part is devoted to obtaining the necessary and sufficient conditions of the existence of bounded solutions of generating boundary value problem.

The last part is devoted to obtaining the necessary and sufficient conditions of the existence of solutions of weakly nonlinear Riccati operator-differential equation.

In this paper, Riccati equation was investigated in the critical case in the Hilbert space. And we obtain the full theorem of the solvability in linear case.

We consider the following boundary value problem

where is an unknown operator function from the space, and are bounded operator-valued functions; or in another words these operators are the ways in the space of linear and bounded operators, is a small parameter. We find the solution of boundary value problem (1), (2) which for turns in one of the solutions of the generating boundary value problem

Consider the case when the differential equation is defined on the finite interval or on the infinite inter- val. Let us the operator whose action on the operator-valued function is given by the formula

where, are evolution operators [13] , of the following operator-differential equations

respectively. Obviously, that satisfies the following operator-differential equation

Using the operator, we write out the general solution of the nonperturbed problem

in the form

where M is an arbitrary operator and

The main statement of this part is the following theorem.

Theorem 1. Consider the boundary value problem (8), (9).

1) There are exist solutions of the boundary value problem (8), (9) if and only if the following condition is true

where, and; under this condition the family of solutions have the following form

is generalized Green’s operator and is Moore-Penrose pseudoinvertible;

2) There are exist generalized solutions of boundary value problem (8), (9) if and only if

where is strong generalized invertible operator. Then the family of solutions of the Equation (8) has the form

where

is generalized Green’s operator.

3) There are exist the quasisolutions if and only if and in this case we have the solutions in the following form

with the same generalized invertible.

Sketch of the proof: Substituting in the boundary condition we will have

and then we obtain the following operator equation

Using the notion of generalized invertible operator [14] [15] , we have the following variants for the equation:

1) If then Equation (15) has the solution if and only if the following condition is true

If the condition (16) is satisfied then the set of the solutions of the Equation (15) has the following form:

for any linear and bounded operator C. Then the family of solutions of the boundary value problem (8), (9) has the form

where

is generalized Green’s operator.

2) If. If the we have the generalized solutions in the following form

if and only if the following relation is hold

where is strong generalized invertible operator [15] . Then the family of solutions of the boundary value problems (8), (9) has the form

where

is generalized Green’s operator.

3) If the we have the quasisolutions in the form

with the same generalized invertible (see also the paper [16] ).

O. O.Pokutnyi, (2015) Boundary Value Problem for an Operator-Differential Riccati Equation in the Hilbert Space on the Interval. Advances in Pure Mathematics,05,865-873. doi: 10.4236/apm.2015.514081