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The paper is devoted to obtaining the necessary and sufficient conditions of the solvability of weakly perturbed boundary-value problems for the nonlinear operator-differential Riccati equation in the Hilbert space on the interval and whole line with parameter ?. We find the solution of the given boundary value problem which for ε = 0 turns in one of the solutions of generating boundary value problem. Solution of the generating problem is constructed with the using generalized operator in analytical form. Iterative process for finding of solutions of weakly nonlinear equation with quadratic error is constructed.

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 [

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. [

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

Consider the case when the differential equation is defined on the finite interval

where

respectively. Obviously, that

Using the operator

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

is generalized Green’s operator and

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

where

where

is generalized Green’s operator.

3) There are exist the quasisolutions if and only if

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 [

1) If

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 and only if the following relation is hold

where

where

is generalized Green’s operator.

3) If the

with the same generalized invertible (see also the paper [

Now we consider the boundary value problem (1), (2). We find the solution

Theorem 2. (necessary condition) Let the boundary value problem (1), (2) has the solution

Proof. Suppose that the boundary value problem (1), (2) has solution _{0}. By virtue of the theorem 1 thefollowing condition of solvability is true

where

Such as condition (10) is true, then the condition of solvability (21) we can rewrite in the following form

Dividing by

or in the form

From this condition we obtain the theorem 2.

Now we obtain the sufficient condition of the solvability of boundary value problem (8), (9). We make the change of the variable

The family of solutions of the Equation (23) has the following form

under condition (21)

Substituting in this expression (25) and using the Equation (20), we have

Then we can rewrite this expression in the following form of the operator equation

where

and

If the following condition

is true then the equation (30) has the solution

Under condition (32), we can prove that boundary value problem (23), (24) have solutions. In a such way, we prove the following theorem.

Theorem 3. (sufficient condition) Under condition (32) boundary value problem (23), (24) is solvable. Solution of the given boundary problem can be found with using the following converging iterative process

with zero initial data.

Proof.

Proof of this theorem uses the modification of the fixed point theorem and is performed as well as the proof of the theorem 3 from the paper [

Example 1.

Considering the following boundary value problem with the matrix-valued in l_{2} functions,

nonhomogenous part has the following form

and conditions on infinity

In this case

Operator

with

In this case, the operator

From the condition (39) we obtain that

In the such way, we have

(46)

The unperturbed problem has the following form

Consider the following problem with the matrix,

nonhomogenous part has the following form

In this case

where

where

Here are

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