Open Access Library Journal
Vol.05 No.06(2018), Article ID:85547,5 pages
10.4236/oalib.1104631
Solvability of a Class of Operator-Differential Equations of Third Order with Complicated Characteristic on the Whole Real Axis
Abdel Baset I. Ahmed, Mohamed A. Labeeb
Egyptian Russian University, Badr City, Egypt
Copyright © 2018 by authors and Open Access Library Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: May 2, 2018; Accepted: June 23, 2018; Published: June 26, 2018
ABSTRACT
On the whole real axis, we demonstrate sufficient conditions of regular solvability of third order operator-differential equations with complicated characteristics. These conditions were formulated only by the operator coefficients of the equation. In addition, by the principal part of the equation, the norms of the operators of intermediate derivative were estimated.
Subject Areas:
Ordinary Differential Equation
Keywords:
Operator-Differential Equation, Hilbert Space, Self-Adjoint Operator, Intermediate Derivative Operator
1. Introduction
In a separable Hilbert space H, we have the following equation:
(1)
where
A is a self-adjoint positive-definite operator, and are generally linear unbounded operators. All derivatives are understood in the sense of distributions theory.
We consider , where
(see [1] [2] ), and , which are determined as follows:
With the norm
See [2] .
Notice that the principal part of the investigated equation possesses complicated characteristic, not multiple characteristics as in [3] .
Definition 1. If for any there exists a vector function that satisfies Equation (1) almost everywhere in R, then it is called a regular solution of Equation (1)
Definition 2. If for any there exists a regular solution of Equation (1), and satisfies the inequality
(2)
then Equation (1) is called regularly solvable.
It is known that if , then , .
And the following inequalities are valid (see [2] ).
(3)
Definition 3. Parseval’s equality
where,
2. Main Results
Theorem 1. The operator is an isomorphism from the space to the space .
Proof. From (2), it is easy to prove that the operator acts from to be bounded. Using Fourier transforms for the equation , we obtain
(4)
(E is the unit operator), where are Fourier transform for the functions , respectively. The operator pencil is invertible and moreover
(5)
Hence,
(6)
We show that . By using the Parseval equality and (3), we obtain:
(7)
If is a spectrum of the operator A, then we consider
(8)
(9)
Taking into account (5) and (6) into (4) we obtain:
(10)
Consequently, .
Applying Banach theorem on the inverse operator, we get that the operator is an isomorphism from to .
Now, we estimate the norms of intermediate derivative operators participating in the main part of the Equation (1) for finding exact conditions on regular solvability of the given equation, expressed only by its operator coefficients.
From theorem 1, we have that the norms and are equivalent in the space . Therefore by the norm the theorem on intermediate derivatives is valid as well.
Theorem 2. Let . Then there hold the following inequalities:
(11)
where .
Proof. To establish the validity of inequality (11) we make change and apply the Fourier transformation. We get
(12)
For we estimate the following norms:
(13)
Finally, from (12), we have
(14)
Lemma. The operator continuously acts from to provided that the operators are bounded in H.
Taking into account the results found up [4] to now we get possibility to establish regular solvability conditions of Equation (1).
Theorem 3. Let the operators be bounded in H and it holds the inequality , where the numbers are determined in theorem 2. Then the Equation (1) is regularly solvable.
Proof. By theorem 1, provided that the operator has a bounded inverse operator acting from to , then after replacing in Equation (1) can be written as .
Now we prove under the theorem conditions (see [5] ), that the norm
By theorem (2), we have:
Consequently,
Thus, the operator is invertible in and hence can be determined by , moreover
(15)
The theorem is proved.
3. Conclusion
We formulated exact conditions on regular solvability of Equation (1), expressed only by its operator coefficients. We estimated the norms of intermediate derivative operators participating in the principle part of the given equation. In the case when in the perturbed part of Equation (1), the participant variable operator coefficients, i.e. are linear operators, which determined for all , are investigated in a similar way.
Cite this paper
Ahmed, A.B.I. and Labeeb, M.A. (2018) Solvability of a Class of Operator-Differential Equations of Third Order with Complicated Characteristic on the Whole Real Axis. Open Access Library Journal, 5: e4631. https://doi.org/10.4236/oalib.1104631
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