Abstract
In the present paper, the well-posedness of the initial value problem for the delay differential equation , ; () in an arbitrary Banach space E with the unbounded linear operators A and in E with dense domains is studied. Two main theorems on well-posedness of this problem in fractional spaces are established. In practice, the coercive stability estimates in Hölder norms for the solutions of the mixed problems for delay parabolic equations are obtained.
MSC:35G15.
Similar content being viewed by others
1 Introduction
The stability of delay ordinary differential and difference equations and delay partial differential and difference equations with bounded operators acting on delay terms has been studied extensively in a large cycle of works (see [1–13] and the references therein) and insight has developed over the last three decades. The theory of stability and coercive stability of delay partial differential and difference equations with unbounded operators acting on delay terms has received less attention than delay ordinary differential and difference equations (see [14–19]). It is well known that various initial-boundary value problems for linear evolutionary delay partial differential equations can be reduced to an initial value problem of the form
in an arbitrary Banach space E with the unbounded linear operators A and in E with dense domains . Let A be a strongly positive operator, i.e. −A is the generator of the analytic semigroup () of the linear bounded operators with exponentially decreasing norm when . That means the following estimates hold:
for some , . Let be closed operators.
A function is called a solution of the problem (1) if the following conditions are satisfied:
-
(i)
is continuously differentiable on the interval . The derivative at the endpoint is understood as the appropriate unilateral derivative.
-
(ii)
The element belongs to for all , and the function is continuous on the interval .
-
(iii)
satisfies the equation and the initial condition (1).
A solution of the initial value problem (1) is said to be coercive stable (well-posed) if
for every t, . We are interested in studying the coercive stability of solutions of the initial value problem under the assumption that
holds for every . We have not been able to obtain the estimate (3) in the arbitrary Banach space E. Nevertheless, we can establish the analog of estimates (3) where the space E is replaced by the fractional spaces () under an assumption stronger than (4). The coercive stability estimates in Hölder norms for the solutions of the mixed problem of the delay differential equations of the parabolic type are obtained.
The present paper is organized as follows. Section 1 is introduction. In Section 2, two main theorems on well-posedness of the initial value problem (1) are established. In Section 3, the coercive stability estimates in Hölder norms for the solutions of the initial-boundary value problem for delay parabolic equations are obtained. Finally, Section 4 is our conclusion.
2 Theorems on well-posedness
The strongly positive operator A defines the fractional spaces () consisting of all for which the following norms are finite:
We consider the initial value problem (1) for delay differential equations of parabolic type in the space of all continuous functions defined on the segment with values in a Banach space . First, we consider the problem (1) when and commute, i.e.
Theorem 2.1 Assume that the condition
holds for every , where M is the constant from (2). Then for every t, , , we have the following coercive stability estimate:
where does not depend on and . Here, we put when .
Proof It is clear that
where is the solution of the problem
and is the solution of the problem
First, we consider the problem (9). Using the formula
the semigroup property, condition (5), and the estimates (2), (6), we obtain
for every t, and λ, . This shows that
for every t, . Applying mathematical induction, one can easily show that it is true for every t. Namely, assume that the inequality
is true for t, , for some n. Letting , we have
Using the estimate (12), we obtain
for every t, , and λ, . This shows that
for every t, , . Therefore
is true for every . Applying (9), the triangle inequality, condition (5), and the estimates (6) and (13), we get
for every . Second, we consider the problem (10). To prove the theorem it suffices to establish the following stability inequality:
for the solution of the problem (10) for every t, , . Using the formula
the semigroup property, and the definition of the spaces , we obtain
for every t, and λ, . This shows that
for every t, . Applying mathematical induction, one can easily show that it is true for every t. Namely, assume that the inequality (15) is true for t, , , for some n. Using the formula
the semigroup property, the definition of the spaces , the estimate (2), and condition (6), we obtain
for every t, , and λ, . This shows that
for every t, , . Applying equation (9), the triangle inequality, and condition (5) and estimates (6) and (19), we get
for every . This result completes the proof of Theorem 2.1. □
Now, we consider the problem (1) when
for some . Note that A is a strongly positive operator in a Banach spaces E iff its spectrum lies in the interior of the sector of angle φ, , symmetric with respect to the real axis, and if on the edges of this sector, and and outside it the resolvent is the subject to the bound
for some . First of all let us give lemmas from the paper [18] that will be needed in the sequel.
Lemma 2.1 For any z on the edges of the sector,
and
and outside it the estimate
holds for any . Here and in the future M and are the same constants of the estimates (2) and (20).
Lemma 2.2 Let for all the operator with domain which coincide with admit a closure bounded in E. Then for all the following estimate holds:
Here .
Suppose that
holds for every . Here and in the future ε is some constant, .
The application of Lemmas 2.1 and 2.2 enables us to establish the following fact.
Theorem 2.2 Assume that the condition
holds for every . Then for every the coercive stability estimate (7) holds.
Proof In a similar manner as in the proof of Theorem 2.1 we establish estimates for the solution of the problems (9) and (10), separately. First, we consider the problem (9). Let and λ, . Then using (11), we have
where
Using the estimates (2), (20), and condition (22), we obtain
for every t, and λ, . Now let us estimate . By Lemma 2.1 and using the estimate (21), we obtain
for every t, and λ, . Using the triangle inequality, we obtain
for every t, and λ, . This shows that
for every t, . In a similar manner as with Theorem 2.1 applying mathematical induction, one can easily show that it is true for every t. Therefore, to prove the theorem it suffices to establish the coercive stability inequality (15) for the solution of the problem (10). Now, we consider the problem (10). Exactly in the same manner, using (16), the semigroup property, and the definition of the spaces , we obtain (15) for every t, . Applying mathematical induction, one can easily show that it is true for every t. Namely, assume that the inequality (15) is true for t, , for some n. Using (18) and the semigroup property, we write
where
Using the estimate (2) and condition (22), we obtain
for every t, , , and λ, . Now let us estimate . By Lemma 2.2 and using the estimate (2) and condition (21), we obtain
for every t, , and λ, . Using the triangle inequality and estimates for all , , we obtain
for every t, , and λ, . This shows that
for every t, , . This result completes the proof of Theorem 2.2. □
Note that these abstract results are applicable to the study of stability of various delay parabolic equations with local and nonlocal boundary conditions with respect to the space variables. However, it is important to study the structure of for space operators in Banach spaces. The structure of for some space differential and difference operators in Banach spaces has been investigated (see [20–30]). In Section 3, applications of Theorem 2.1 to the study of the coercive stability of initial-boundary value problem for delay parabolic equations are given.
3 Applications
First, we consider the initial-boundary value problem for one dimensional delay differential equations of parabolic type
where , , , are given sufficiently smooth functions and is a sufficiently large number. We will assume that . The problem (23) has a unique smooth solution. This allows us to reduce the initial-boundary value problem (23) to the initial value problem (1) in Banach space with a differential operator defined by the formula
with domain . Let us give a number of corollaries of the abstract Theorem 2.1.
Theorem 3.1 Assume that
Then for all the solutions of the initial-boundary value problem (23) satisfy the following coercive stability estimates:
where is not dependent on and . Here is the space of functions satisfying a Hölder condition with the indicator .
The proof of Theorem 3.1 is based on the estimate
and on the abstract Theorem 2.1, on the strong positivity of the operator in (see [31, 32]) and on Theorem 3.2 on the structure of the fractional space for .
Theorem 3.2 For , the norms of the space and the Hölder space are equivalent [21].
Second, we consider the initial nonlocal boundary value problem for one dimensional delay differential equations of parabolic type,
where , , , are given sufficiently smooth functions and is a sufficiently large number. We will assume that . The problem (26) has a unique smooth solution. This allows us to reduce the initial-boundary value problem (26) to the initial value problem (1) in Banach space with a differential operator defined by the formula
with domain . Let us give a number of corollaries of the abstract Theorem 2.1.
Theorem 3.3 Assume that condition (25) holds. Then for all the solutions of the initial-boundary value problem (26) satisfy the following coercive stability estimates:
where is not dependent on and .
The proof of Theorem 3.3 is based on the estimate
and on the abstract Theorem 2.1, on the strong positivity of the operator in (see [6]) and on Theorem 3.4 on the structure of the fractional space for .
Theorem 3.4 For , the norms of the space and the Hölder space are equivalent [6].
Third, we consider the initial value problem on the range
for 2m th order multidimensional delay differential equations of parabolic type,
where , , , and are sufficiently smooth functions and is a sufficiently large number. We will assume that the symbol
of the differential operator of the form
acting on functions defined on the space , satisfies the inequalities
for , where . The problem (28) has a unique smooth solution. This allows us to reduce the initial value problem (28) to the initial value problem (1) in Banach space E with a strongly positive operator defined by (29). Let us give a number of corollaries of the abstract Theorem 2.1.
Theorem 3.5 Assume that condition (25) holds. Then for all the solutions of the initial-boundary value problem (28) satisfy the following coercive stability estimates:
where does not depend on and . Here is the space of functions satisfying a Hölder condition with the indicator .
The proof of Theorem 3.5 is based on the estimate
and on the abstract Theorem 2.1, on the strong positivity of the operator in , and on the equivalence of the norms in the spaces and when [20, 23].
4 Conclusion
In the present paper, two theorems on the well-posedness of the initial value problem for the delay parabolic differential equations with unbounded operators acting on delay terms in fractional spaces are established. In practice, the coercive stability estimates in Hölder norms for the solutions of the mixed problems for delay parabolic equations are obtained.
References
Al-Mutib AN: Stability properties of numerical methods for solving delay differential equations. J. Comput. Appl. Math. 1984, 10(1):71-79.
Bellen A: One-step collocation for delay differential equations. J. Comput. Appl. Math. 1984, 10(3):275-283. 10.1016/0377-0427(84)90039-6
Bellen A, Jackiewicz Z, Zennaro M: Stability analysis of one-step methods for neutral delay-differential equations. Numer. Math. 1988, 52(6):605-619. 10.1007/BF01395814
Cooke KL, Györi I: Numerical approximation of the solutions of delay differential equations on an infinite interval using piecewise constant arguments. Comput. Math. Appl. 1994, 28: 81-92.
Torelli L: Stability of numerical methods for delay differential equations. J. Comput. Appl. Math. 1989, 25: 15-26. 10.1016/0377-0427(89)90071-X
Yenicerioglu AF, Yalcinbas S: On the stability of the second-order delay differential equations with variable coefficients. Appl. Math. Comput. 2004, 152(3):667-673. 10.1016/S0096-3003(03)00584-8
Yenicerioglu AF: Stability properties of second order delay integro-differential equations. Comput. Math. Appl. 2008, 56(12):309-311.
Yenicerioglu AF: The behavior of solutions of second order delay differential equations. J. Math. Anal. Appl. 2007, 332(2):1278-1290. 10.1016/j.jmaa.2006.10.069
Ashyralyev A, Akca H, Guray U: Second order of accuracy difference scheme for approximate solutions of delay differential equations. Funct. Differ. Equ. 1999, 6(3-4):223-231.
Ashyralyev A, Akca H: Stability estimates of difference schemes for neutral delay differential equations. Nonlinear Anal., Theory Methods Appl. 2001, 44(4):443-452. 10.1016/S0362-546X(99)00270-9
Ashyralyev A, Akca H, Yenicerioglu AF: Stability properties of difference schemes for neutral differential equations. Differ. Equ. Appl. 2003, 3: 57-66.
Liu J, Dong P, Shang G: Sufficient conditions for inverse anticipating synchronization of unidirectional coupled chaotic systems with multiple time delays. Chinese Control and Decision Conference (CCDC 2010) 2010, 751-756.
Akca H, Shakhmurov VB, Arslan G: Differential-operator equations with bounded delay. Nonlinear Times Dig. 1989, 2: 179-190.
Sahmurova A, Shakhmurov VB: Parabolic problems with parameter occurring in environmental engineering. AIP Conference Proceedings 1470. In First International Conference on Analysis and Applied Mathematics (ICAAM 2012) Edited by: Ashyralyev A, Lukashov A. 2012, 39-41.
Ashyralyev A, Agirseven D: Finite difference method for delay parabolic equations. AIP Conference Proceedings 1389. International Conference on Numerical Analysis and Applied Mathematics (ICNAAM 2011) 2011, 573-576.
Agirseven D: Approximate solutions of delay parabolic equations with the Dirichlet condition. Abstr. Appl. Anal. 2012., 2012: Article ID 682752
Ashyralyev A, Sobolevskii PE: On the stability of the delay differential and difference equations. Abstr. Appl. Anal. 2001, 6(5):267-297. 10.1155/S1085337501000616
Ashyralyev A, Sobolevskii PE: New Difference Schemes for Partial Differential Equations. Birkhäuser, Basel; 2004.
Di Blasio G: Delay differential equations with unbounded operators acting on delay terms. Nonlinear Anal., Theory Methods Appl. 2003, 52(1):1-18. 10.1016/S0362-546X(01)00868-9
Ashyralyev A, Sobolevskii PE: The theory of interpolation of linear operators and the stability of difference schemes. Dokl. Akad. Nauk SSSR 1984, 275(6):1289-1291. (in Russian)
Bazarov MA: On the structure of fractional spaces. In Proceedings of the XXVII All-Union Scientific Student Conference ‘The Student and Scientific-Technological Progress’. Gos. Univ., Novosibirsk; 1989:3-7. (in Russian)
Triebel H: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam; 1978.
Ashyralyev A, Sobolevskii PE: Well-Posedness of Parabolic Difference Equations. Operator Theory Advances and Applications. Birkhäuser, Basel; 1994.
Ashyralyev A: Fractional spaces generated by the positive differential and difference operator in a Banach space. In Proceedings of the Conference ‘Mathematical Methods and Engineering’. Edited by: Tas K, Tenreiro Machado JA, Baleanu D. Springer, Berlin; 2007:13-22.
Ashyralyev A, Akturk S, Sozen Y: Positivity of two-dimensional elliptic differential operators in Holder spaces. AIP Conference Proceedings 1470. First International Conference on Analysis and Applied Mathematics (ICAAM 2012) 2012, 77-79.
Ashyralyev A, Yaz N: On structure of fractional spaces generated by positive operators with the nonlocal boundary value conditions. In Proceedings of the Conference Differential and Difference Equations and Applications. Edited by: Agarwal RF, Perera K. Hindawi Publishing Corporation, New York; 2006:91-101.
Ashyralyev A, Tetikoğlu FS: The structure of fractional spaces generated by the positive operator with periodic conditions. AIP Conference Proceedings 1470. First International Conference on Analysis and Applied Mathematics (ICAAM 2012) 2012, 57-60.
Ashyralyev A, Agirseven D: Approximate solutions of delay parabolic equations with the Neumann condition. AIP Conference Proceedings 1479. International Conference on Numerical Analysis and Applied Mathematics (ICNAAM 2012) 2012, 555-558.
Ashyralyev A, Agirseven D: On convergence of difference schemes for delay parabolic equations. Comput. Math. Appl. 2013, 66(7):1232-1244. 10.1016/j.camwa.2013.07.018
Ashyralyev A, Agirseven D: Well-posedness of delay parabolic difference equations. Adv. Differ. Equ. 2014., 2014: Article ID 18 10.1186/1687-1847-2014-18
Solomyak MZ: Estimation of norm of the resolvent of elliptic operator in spaces . Usp. Mat. Nauk 1960, 15(6):141-148. (in Russian)
Stewart HB: Generation of analytic semigroups by strongly elliptic operators. Trans. Am. Math. Soc. 1974, 190: 141-162.
Acknowledgements
This work is supported by Trakya University Scientific Research Projects Unit (Project No: 2010-91).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Ashyralyev, A., Agirseven, D. Well-posedness of delay parabolic equations with unbounded operators acting on delay terms. Bound Value Probl 2014, 126 (2014). https://doi.org/10.1186/1687-2770-2014-126
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-2770-2014-126