Abstract
The wellposedness of difference schemes of the initial value problem for delay differential equations with unbounded operators acting on delay terms in an arbitrary Banach space is studied. Theorems on the wellposedness of these difference schemes in fractional spaces are proved. In practice, the coercive stability estimates in Hölder norms for the solutions of difference schemes of the mixed problems for delay parabolic equations are obtained.
Keywords:
wellposedness; delay parabolic equations; fractional spaces; coercive stability1 Introduction
Approximate solutions of the delay differential equations have been studied extensively in a series of works (see, for example, [16] and the references therein) and developed over the last three decades. In the literature mostly the sufficient condition
was considered for the stability of the following test delay differential equation:
with the initial condition
It is known that delay differential equations can be solved by applying standard numerical methods for ordinary differential equations without the presence of delay. However, it is difficult to generalize any numerical method to obtain a high order of accuracy algorithms, because highorder methods may not lead to efficient results. It is well known that even if , and are arbitrary differentiable functions, may not possess the higherorder derivatives for a sufficiently large t. Therefore, we have nonsmooth solution of delay differential equations for given smooth data. This is the main difficulty in the study of the convergence of numerical methods for delay differential equations.
Delay partial differential equations arise from various applications, like in climate models, biology, medicine, control theory, and many others (see, for example, [7] and the references therein).
The theory of approximate solutions of delay partial differential equations has received less attention than delay ordinary differential equations. A situation which occurs in delay partial differential equations when the delay term is an operator of lower order with respect to the other operator term is widely investigated (see, for example, [79] and the references therein). In the case where the delay term is an operator of the same order with respect to other operator term, this is studied mainly in a Hilbert space (see, for example, [10] and the references therein). In fact there are very few papers where the delay term is an operator of the same order with respect to the other operator term, this being investigated in a general Banach space (see [1114]) and in these works, the authors look only for partial differential equations under regular data. Additionally, approximate solutions of the delay parabolic equations in the case where the delay term is a simple operator of the same order with respect to the other operator term were studied recently in papers [1519].
It is known that various initialboundary 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.
The strongly positive operator A defines the fractional spaces () consisting of all for which the following norms are finite:
As noted in [19], it is important to study the stability of solutions of the initial value problem (4) for delay differential equations and of difference schemes for approximate solutions of problem (4) under the assumption that
holds for every . This assumption for the delay differential equation (2) follows from assumption (1) in the case when . Unfortunately, we have not been able to obtain the stability estimate for the solution of problem (4) in the arbitrary Banach space E. Nevertheless, in [20], the coercive stability estimate for the solution of problem (4) was established, when the space E is replaced by the fractional spaces () which is defined above under the condition
for every , where M is the constant from equation (5). However, the condition (7) is stronger than (6) and . Finally, in papers [21,22], theorems on the wellposedness in Hölder spaces in t of the initial value problem for the delay parabolic equation
in an arbitrary Banach space E with the small positive parameter ε in the high derivative and with the unbounded linear operators A and in E with dense domains were established.
Additionally, using the first and second order of the accuracy implicit difference schemes for differential equations without the presence of delay, the first and second order of the accuracy implicit difference schemes,
are presented for approximate solutions of the initial value problem (4). Here, we will put .
The main aim of present paper is to study the wellposedness of the difference schemes (9) and (10). We establish the coercive stability estimates in fractional spaces () under the assumption (7). In practice, the coercive stability estimates in Hölder norms for the solutions of difference schemes for the approximate solutions of the mixed problem of delay parabolic equations are obtained.
The paper is organized as follows. In Section 2, theorems on coercive stability of difference schemes (9) and (10) are established. In Section 3, the coercive stability estimates in Hölder norms for the solutions of difference schemes for the approximate solutions of delay parabolic equations are obtained. Finally, Section 4 is our conclusion.
2 The wellposedness of difference schemes (9) and (10)
First, we consider the difference scheme (9) when and commute, i.e.
Theorem 1Assume that the condition (7) holds for every, whereMis the constant from (5). Then for the solution of the difference scheme (9), the estimate
holds for any. Here and in future we putif.
Proof Let us consider . In this case
where
Let us estimate and for any . Using the formula
condition (11) and estimates (5) and (7), we obtain
where
Making the substitution and integrating by parts, we obtain
for every and . This shows that
for every . Using formula (13), and the estimate (5), we obtain
Applying the inequality
we get
for every and . This shows that
for every . Using the triangle inequality and the estimates (14) and (15), we get
Applying mathematical induction, one can easily show that it is true for every k. Actually, suppose that the estimate (16) is true for , . Letting , we have
Using the estimate (16), we obtain
for every , . Theorem 1 is proved. □
Now, we consider the difference scheme (9) when
Recall that (see, for example, [[23], Chapter 2, p.116]) A is a strongly positive operator in a Banach space E iff its spectrum lies in the interior of the sector of the angle φ, , symmetric with respect to the real axis, and if on the edges of this sector, and , and outside it the resolvent is subject to the bound
for some . First of all let us give lemmas from the paper [12] that will be needed in the sequel.
Lemma 1For anyzon the edges of the sector,
and outside it the estimate
holds for any. Here and in the futureMandare the same constants as of the estimates (5) and (17).
Lemma 2Let for allthe operatorwith domain which coincides withpermit the closurebounded inE. Then for allthe following estimate holds:
Suppose that
holds for every. Here and in the futureεis a constant, .
The use of Lemmas 1 and 2 enables us to establish the following statement.
Theorem 2Assume that the condition
holds for every. Then for the solution of the difference scheme (9), the coercive estimate (12) holds.
Proof Let us consider . Using formula (13), we can write
Using the estimates (5), (17), and condition (19), we obtain
for every and . Now let us estimate . By Lemma 2 and using the estimate (20), we can obtain
for every and . Using the triangle inequality, we obtain
for every and . This shows that
for every . Using the triangle inequality and the last estimate and (15), we get
In a similar manner as Theorem 1, applying mathematical induction, one can easily show that it is true for every k. Theorem 2 is proved. □
Now we consider the difference scheme (10). We have not been able to obtain the same result for the solution of the difference scheme (10) in spaces under assumption (7). Nevertheless, for the solution of the difference scheme (10) the coercive stability estimate in the norm of same fractional spaces () under the supplementary restriction of the operator A is established.
Theorem 3Suppose that the following estimates hold:
and
Then for the solution of the difference scheme (10), the coercive estimate (12) holds.
Proof Let us consider . In this case
where
Let us estimate and for any . Using formula (22), condition (11), and the estimates (5) and (7), we obtain
for every and . This shows that
for every . Using formula (22), the condition (21), and the estimate (5), we obtain
for every and . This shows that
for every . Using the triangle inequality and the estimates (23) and (24), we get
In a similar manner as Theorem 1, applying mathematical induction, one can easily show that it is true for every k. Theorem 3 is established. □
Now, we consider the difference scheme (10) when
for some . Suppose that the operator , with domain which coincides with , permits the closure bounded in E and that the estimate
Theorem 4Assume that all conditions of Theorem 2 and Theorem 3 are satisfied. Then for the solution of the difference scheme (10), the estimate (12) holds.
Proof Let us consider . Using formula (22), the estimates (5), (17), and the condition (19), we obtain
Using the estimates (5), (17), and the condition (19), we obtain
for every and . Now let us estimate . By Lemma 2 and using the estimate (20), we obtain
for every and . Using the triangle inequality, we obtain
for every and . This shows that
for every . Using the triangle inequality and the estimates (26) and (24), we get
In a similar manner as Theorem 1, applying mathematical induction, one can easily show that it is true for every k. Theorem 4 is established. □
Note that these abstract results are applicable to the study of the coercive stability of various delay parabolic equations with local and nonlocal boundary conditions with respect to the space variables. However, it is important to study structure of for space operators in Banach spaces. The structure of for some space differential and difference operators in Banach spaces has been investigated in some papers [2332]. In Section 3, applications of Theorem 1 to the study of the coercive stability of the difference schemes for delay parabolic equations are given.
3 Applications
First, the initialboundary value problem for onedimensional delay parabolic equations is considered:
where , , , are given sufficiently smooth functions and is a sufficiently large number. It will be assumed that . The discretization of problem (27) is carried out in two steps. In the first step, the uniform grid space
is defined. To formulate the result, one needs to introduce the Banach space () of the grid functions defined on satisfying the conditions , equipped with the norm
Here and in the future, is the space of the grid functions defined on , equipped with the norm
To the differential operator generated by problem (27), we assign the difference operator by the formula
acting in the space of grid functions satisfying the conditions . With the help of , we arrive at the initialboundary value problem
for the system of ordinary differential equations. In the second step, problem (28) is replaced by the firstorder accuracy in the difference scheme in t,
Theorem 5Assume that
Then for the solution of the difference scheme (29) the following coercive stability estimates hold:
for all, wheredoes not depend onand. Here and in the future we put
The proof of Theorem 5 is based on the estimate
and on the abstract Theorem 1, the positivity of the operator in , and on the following theorem on the structure of the fractional space .
Theorem 6For anythe norms in the spacesandare equivalent uniformly inh[25].
Second, the initial nonlocal boundary value problem for onedimensional delay parabolic equations is considered:
where , , , are given sufficiently smooth functions and is a sufficiently large number. It will be assumed that . The discretization of problem (32) is carried out in two steps. In the first step, let us use the discretization in the space variable x. To formulate the result, one needs to introduce the Banach space () of the grid functions defined on satisfying the conditions , equipped with the norm
To the differential operator generated by problem (32) we assign the difference operator by the formula
acting in the space of grid functions satisfying the conditions , . With the help of , we arrive at the initial value problem
for the system of ordinary differential equations. In the second step, problem (34) is replaced by the firstorder accuracy of the difference scheme in t
Theorem 7Assume that all the conditions of Theorem 5 are satisfied. Then for the solution of the difference scheme (35) the coercive stability estimate (31) holds.
The proof of Theorem 7 is based on the estimate
and on the abstract Theorem 1, the positivity of the operator in , and on the following theorem on the structure of the fractional space .
Theorem 8For anythe norms in the spacesandare equivalent uniformly inh[27].
Third, the initial value problem on the range
for 2mthorder multidimensional delay differential equations of parabolic type is considered:
where , , , are given sufficiently smooth functions and is a sufficiently large number. We will assume that the symbol []
of the differential operator of the form
acting on the functions defined on the space , satisfies the inequalities
for , where . The discretization of problem (36) is carried out in two steps. In the first step the uniform grid space () is defined as the set of all points of the Euclidean space whose coordinates are given by
The difference operator is assigned to the differential operator , defined by equation (36). The operator
acts on functions defined on the entire space . Here is a vector with nonnegative integer coordinates,
where is the unit vector of the axis .
An infinitely differentiable function of the continuous argument that is continuous and bounded together with all its derivatives is said to be smooth. We say that the difference operator is a λthorder () approximation of the differential operator if the inequality
holds for any smooth function . The coefficients are chosen in such a way that the operator approximates in a specified way the operator . It will be assumed that the operator approximates the differential operator with any prescribed order [33,34].
The function is obtained by replacing the operator in the righthand side of the equality (38) with the expression , respectively, and it is called the symbol of the difference operator .
It will be assumed that for and fixed x the symbol of the operator satisfies the inequalities
Suppose that the coefficient of the operator is bounded and satisfies the inequalities
With the help of we arrive at the initial value problem
for an infinite system of ordinary differential equations. Now, problem (41) is replaced by the firstorder accuracy of the difference scheme in t
To formulate the result, one needs to introduce the spaces and of all bounded grid functions defined on , equipped with the norms
Theorem 9Assume that all the conditions of Theorem 7 are satisfied. Then for the solution of the difference scheme (42) the following coercive stability estimates hold:
for all, wheredoes not depend onand.
The proof of Theorem 9 is based on the estimate
and on the abstract Theorem 1, the positivity of the operator in , and on the fact that the norms are equivalent to the norms uniformly in h for [[23], Chapter 4, p.283].
4 Conclusion
In the present paper, the wellposedness of the difference schemes for the approximate solutions of the initial value problem for delay parabolic equations with unbounded operators acting on delay terms in an arbitrary Banach space is established. Theorems on the coercive stability of these difference schemes in fractional spaces are established. In practice, the coercive stability estimates in Hölder norms for the solutions of the difference schemes for the approximate solutions of the mixed problems for delay parabolic equations are obtained. Note that in the present paper is a time variable unbounded space operator acting on the delay term. The delay w is a positive constant. In general, it is interesting to consider the delay as a function , dependent on t. A wellknown parabolic problem with delay used in population dynamics is the socalled Hutchinson equation where is a time variable bounded nonlinear space operator acting on the delay term [8,9]. It would be interesting to consider the case when is a nonlinear unbounded space operator acting on the delay term. Actually, it will be possible after establishing theorems on the existence, uniqueness, and stability of the solutions, and the smoothness property of the solutions, and obtaining a suitable contractivity condition of the numerical solutions.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Each of the authors read and approved the final version of the manuscript.
Acknowledgements
This work is supported by Trakya University Scientific Research Projects Unit (Project No: 201091). We would like to thank to the reviewers, whose careful reading, helpful suggestions, and valuable comments helped us to improve the manuscript.
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