Infinite system of 2-systems of differential equations in Hilbert space Ibragimov G. I., Qushaqov H. Sh. Abstract

0.3 Main Part

We denote the space of continuous functions , , by . The following theorem is the main result of the present paper.

Theorem 3.1 If , and , , then there exists a unique solution of the infinite system of differential equations (0.1.4) in the space .

Proof. Obviously, each 2-system of the infinite system (0.1.4) has the unique solution which is given by the equation

(0.3.1)

where , . Therefore, system (0.1.4) can’t have more than one solution in the space .

Next, we show that . To prove the this we need to show that for each , , and that , , is continuous in the norm of the space . The proof that for each . We have from (0.3.1) that

(0.3.2)

By utilizing the relations from Property 2.1.

and in view of the Cauchy-Schwartz inequality we obtain

We have then

Hence,

and so, for each .

The proof that the function , , is continuious. We show that, for any positive , there exists such that whenever . Indeed, for , using Property 2.1. we have






Utilizing the inequality yields








Then,

where

where is a positive integer which will be chosen below.

Since and by item (iii) of Property 2.1. , we obtain

Since the series

are convergent, for any positive number , we can choose the positive integer such that .

Next, we evaluate . From , we obtain



(0.3.3)
By Property 2.1. as for each . Since the expression on the right hand side of (0.3.3) has finite number of terms, we can choose such that whenever .

To evaluate , we use the Cauchy-Schwartz inequality to obtain

Clearly, we can choose such that whenever . Therefore, can be done less than any given positive number by choosing .

Now, we consider , . Then

identical to the estimation of we can establish that for any positive number , there exists number such that whenever . Therefore, . This completes the proof of the theorem.