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Asymptotic behavior of the time-dependent solution of an M/G/1 queueing model

Geni Gupur1* and Rena Ehmet2

Author affiliations

1 College of Mathematics and Systems Science, Xinjiang University, Urumqi, 830046, P.R. China

2 School of Mathematical Sciences, Xinjiang Normal University, Urumqi, 830054, P.R. China

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Citation and License

Boundary Value Problems 2013, 2013:17  doi:10.1186/1687-2770-2013-17

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/17


Received:6 October 2012
Accepted:22 January 2013
Published:11 February 2013

© 2013 Gupur and Ehmet; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the spectrum on the imaginary axis of the underlying operator which corresponds to the M/G/1 queueing model with exceptional service time for the first customer in each busy period that was described by infinitely many partial differential equations with integral boundary conditions and obtain that all points on the imaginary axis except 0 belong to the resolvent set of the operator and 0 is an eigenvalue of the operator and its adjoint operator. Thus, by combining these results with our previous results, we deduce that the time-dependent solution of the model converges strongly to its steady-state solution. Moreover, we show that our result on convergence is optimal.

MSC: 47A10, 47D99.

Keywords:
M/G/1 queueing model with exceptional service time for the first customer in each busy period; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M1">View MathML</a>-semigroup; eigenvalue; resolvent set

1 Introduction

According to Takagi [1], the M/G/1 queueing system with exceptional service time for the first customer in each busy period can be described by the following partial differential equations with integral boundary conditions:

(1.1)

(1.2)

(1.3)

(1.4)

(1.5)

(1.6)

(1.7)

(1.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M10">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M11">View MathML</a> represents the probability that there is no customer in the system and the server is idle at time t; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M12">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M13">View MathML</a>) represents the probability that at time t there are n customers in the system and the server is busy with remaining service time lying between in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M14">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M15">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M13">View MathML</a>) represents the probability that at time t there are n customers in the system and the server is busy with the elapsed service time of the first service lying between x and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M17">View MathML</a>; λ represents the arrival rate of customers; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M18">View MathML</a> is the service rate at x; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M19">View MathML</a> is the exceptional service rate at x.

Many papers have been published about queueing systems with server vacations. But most works on vacation models have been limited to the analysis of steady-states. There are few treatments of transient behavior, see Welch [2], Minh [3], Takagi [1], Gupur [4,5] for instance. In 1990, Takagi [1] first established the mathematical model of the M/G/1 queueing system with exceptional service time for the first customer in each busy period by using the supplementary variable technique, then studied the time-dependent solution of the model by using probability generating functions and got the Laplace transform of the probability generating function. Roughly speaking, he obtained the existence of a time-dependent solution of the model. In 2002, by using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M1">View MathML</a>-semigroup theory in functional analysis, Gupur [6] proved that the model has a unique positive time-dependent solution which satisfies the probability condition. In 2003, Gupur [4] considered the asymptotic behavior of the time-dependent solution of the model when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M19">View MathML</a> are constants. Firstly, he determined the resolvent set of the adjoint operator of the operator corresponding to the model; next he proved that 0 is an eigenvalue of the operator and its adjoint operator with geometric multiplicity one. Thus, by using Theorem 14 in Gupur, Li and Zhu [7] obtained that the time-dependent solution of the model converges strongly to its steady-state solution. In 2009, Zhang and Gupur [8] found that the operator has one eigenvalue on the left complex half-plane. In 2011, Lin and Gupur [9] proved that the operator has infinitely many eigenvalues on the left complex half-plane which converges to zero and therefore showed that the convergence of the time-dependent solution of the model obtained in Gupur [4] is the best result on the convergence, that is to say, it is impossible that the time-dependent solution exponentially (uniformly) converges to its steady-state solution. In the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M19">View MathML</a> are functions, any literature about asymptotic behavior of the above model has not been found. This paper is an effort on this subject.

According to Theorem 14 in Gupur, Li and Zhu [7], to obtain the asymptotic behavior of the time-dependent solution of the above model, we need to know the spectrum of the underlying operator on the imaginary axis. By investigating the above model and comparing with Gupur [10], one may find that the main difficult points of the above equations (1.1)-(1.8) are that there are infinitely many equations and boundary conditions. When studying the population equation, Greiner [11] put forward an idea to perturb the boundary condition which states ‘one can introduce the maximal operator without the boundary condition and define a boundary operator, and by studying the spectrum of the boundary operator and the maximal operator can discuss the spectrum of the underlying operator which corresponds to the population equation.’ In 2007, Haji and Radl [12] successfully applied Greiner’s idea to the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M25">View MathML</a> queueing model, in which both the service rate and arrival rate are constants, and studied the asymptotic behavior of its time-dependent solution. Gupur [5,13] obtained the asymptotic behavior of the time-dependent solutions of two queueing models by using Greiner’s idea. In this paper, firstly, by using probability generating functions, we prove that 0 is an eigenvalue of the underlying operator; next, by using the idea in Gupur [5,13], the result in Haji and Radl [12] and Corollary 2.3 in Nagel [14], we deduce the resolvent set of the underlying operator; thirdly, we show that 0 is an eigenvalue of the adjoint operator of the underlying operator, and therefore, by using Theorem 14 in Gupur, Li and Zhu [7], we obtain that the time-dependent solution of the above model converges strongly to its steady-state solution. Finally, by Lin and Gupur [9] we show that our result on convergence is optimal, that is to say, it is impossible that the time-dependent solution of the model converges exponentially to its steady-state solution. Although the idea and method in Gupur [4] are quite different, the main result is a special case of our result.

In this paper, we use the notations in Gupur [5,6,13]. Take the state space as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M26">View MathML</a>

It is obvious that X is a Banach space. In addition, X is also a Banach lattice under the following order relation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M27">View MathML</a>

For convenience, we introduce

We define

We choose a boundary space as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M30">View MathML</a>

and define the boundary operators

Now we introduce the underlying operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M32">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M33">View MathML</a>

Then the system of the above equations (1.1)-(1.8) can be written as an abstract Cauchy problem in the Banach space X, which is just the form given in Gupur [6]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M34">View MathML</a>

(1.9)

Gupur [6] has proved the following result for the system (1.9).

Theorem 1.1The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M35">View MathML</a>generates a positive contraction<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M1">View MathML</a>-semigroup<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M37">View MathML</a>and the system (1.9) has a unique positive time-dependent solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M38">View MathML</a>which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M39">View MathML</a>

2 Main results

Lemma 2.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M40">View MathML</a>, then 0 is an eigenvalue ofAwith geometric multiplicity one.

Proof We consider the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M41">View MathML</a>, i.e.,

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

(2.6)

(2.7)

By solving (2.2)-(2.5), we have

(2.8)

(2.9)

(2.10)

(2.11)

Through using (2.8)-(2.11) repeatedly, we deduce

(2.12)

(2.13)

By combining (2.10) and (2.11) with (2.7) and using (2.13), we deduce

(2.14)

(2.15)

It is difficult to determine directly all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M57">View MathML</a> and to verify <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M58">View MathML</a>. In the following, we use another method. We introduce the probability generating function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M59">View MathML</a> for all complex variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M60">View MathML</a>. Theorem 1.1 ensures that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M61">View MathML</a> is well defined. (2.2) and (2.3) give

(2.16)

By applying (2.6), (2.16), (2.14), (2.1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M64">View MathML</a> and the L’Hospital rule it follows that

(2.17)

(2.16) and (2.17) give

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M66">View MathML</a>

(2.18)

(2.18) and (2.15) show that 0 is an eigenvalue of A. Moreover, from (2.12), (2.14), (2.1) and (2.6), it is easy to see that the eigenvectors corresponding to zero span one dimensional linear space, that is, the geometric multiplicity of 0 is one. □

According to Theorem 14 in Gupur, Li and Zhu [7], we know that in order to obtain the asymptotic behavior of the time-dependent solution of the system (1.9), we need the spectrum of A on the imaginary axis. Through investigating the system (1.9), we find that the infinite number of equations and the boundary conditions are the difficult points. Greiner [11] put forward an idea to study the spectrum of A by perturbing boundary conditions. And by using the Greiner idea, Haji and Radl [12] gave a result which was described by the Dirichlet operator. In the following, by applying the result, we deduce the resolvent set of A on the imaginary axis. To do this, define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M67">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M68">View MathML</a>

and discuss the inverse of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M69">View MathML</a>. For any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M70">View MathML</a>, consider the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M71">View MathML</a>, that is,

(2.19)

(2.20)

(2.21)

(2.22)

(2.23)

(2.24)

By (2.19)-(2.24) it is easy to calculate

(2.25)

(2.26)

(2.27)

(2.28)

(2.29)

If we set

then the above equations (2.25)-(2.29) give, if the resolvent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M69">View MathML</a> exists,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M85">View MathML</a>

From which together with the definition of the resolvent set we have the following result.

Lemma 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M86">View MathML</a>be measurable, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M87">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M88">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M89">View MathML</a>

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M90">View MathML</a>, by using integration by parts, we estimate

(2.30)

Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M92">View MathML</a>

(2.31)

From (2.30), (2.31), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M94">View MathML</a> we deduce, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M70">View MathML</a>,

which means that the result of this lemma is right. □

Lemma 2.3For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M97">View MathML</a>we have

(2.32)

(2.33)

(2.34)

(2.35)

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M102">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M103">View MathML</a>, which is equivalent to

(2.36)

(2.37)

(2.38)

(2.39)

(2.40)

By solving (2.37)-(2.40) we have

(2.41)

(2.42)

(2.43)

(2.44)

Through inserting (2.41) and (2.43) into (2.36), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M113">View MathML</a>

(2.45)

By using (2.41), (2.42), (2.43) and (2.44) repeatedly, we deduce

(2.46)

(2.47)

(2.48)

(2.49)

(2.50)

(2.51)

(2.52)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M121">View MathML</a>, by the imbedding theorem in Adams [15],

(2.53)

(2.54)

(2.45)-(2.54) show that (2.32)-(2.35) are true.

Conversely, if (2.32)-(2.35) hold, then by using the formulas

integration by parts and the Fubini theorem, we estimate

(2.55)

(2.56)

(2.33) and (2.34) give

(2.57)

(2.58)

(2.59)

(2.60)

By combining (2.57), (2.58), (2.59) and (2.60) with (2.55) and (2.56), we derive

(2.61)

(2.62)

(2.55)-(2.62) mean that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M133">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M103">View MathML</a>. □

It is not difficult to see that L is surjective. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M135">View MathML</a>

is invertible for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M97">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M137">View MathML</a> we define the Dirichlet operator as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M138">View MathML</a>

Lemma 2.3 gives the explicit form of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M139">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M97">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M141">View MathML</a>

(2.63)

where

From (2.63) and the definition of Φ, it is easy to determine the expression of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M143">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M97">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M145">View MathML</a>

Haji and Radl [12] gave the following result through which we deduce the resolvent set of A on the imaginary axis.

Lemma 2.4If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M97">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M147">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M148">View MathML</a>

By using Lemma 2.4 and Nagel [14], page 297, we derive the following result.

Lemma 2.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M86">View MathML</a>be measurable, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M87">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M88">View MathML</a>. Then all points on the imaginary axis except zero belong to the resolvent set ofA.

Proof Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M154">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M155">View MathML</a>. Then by the Riemann-Lebesgue lemma,

we know there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M157">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M158">View MathML</a>

(2.64)

By replacing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M160">View MathML</a> in (2.64) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M162">View MathML</a> and using the fact

we derive, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M158">View MathML</a>,

(2.65)

(2.65) means that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M158">View MathML</a>, the spectral radius <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M167">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M168">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M158">View MathML</a>, and therefore by Lemma 2.4, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M170">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M158">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M172">View MathML</a>

(2.66)

On the other hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M37">View MathML</a> is positive uniformly bounded by Theorem 1.1, by Corollary 2.3 in Nagel [14], page 297, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M174">View MathML</a> is imaginary additively cyclic, which states that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M175">View MathML</a> for all integer k, from which together with (2.66) and Lemma 2.1 we conclude <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M176">View MathML</a>. □

It is not difficult to prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M177">View MathML</a>, dual space of X, is as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M178">View MathML</a>

It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M177">View MathML</a> is a Banach space. Gupur [4] gave the expression of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M180">View MathML</a>, the adjoint operator of A as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M181">View MathML</a>

where

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M37">View MathML</a> is uniformly bounded, by Arendt and Batty [16] and Lemma 2.1, we know that 0 is an eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M180">View MathML</a>. Furthermore, by replacing μ and η in Lemma 3 in Gupur [4] with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M19">View MathML</a>, respectively, we deduce the following result.

Lemma 2.6If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M40">View MathML</a>, then 0 is an eigenvalue of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M180">View MathML</a>with geometric multiplicity one.

Since Theorem 1.1, Lemma 2.1, Lemma 2.5 and Lemma 2.6 satisfy the conditions of Theorem 14 in Gupur, Li and Zhu [7], the following conclusion is the direct result of Theorem 14 in Gupur, Li and Zhu [7].

Theorem 2.7Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M86">View MathML</a>be measurable, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M87">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M88">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M192">View MathML</a>, then the time-dependent solution of the system (1.9) converges strongly to its steady-state solution, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M193">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M194">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M195">View MathML</a>are the eigenvectors in Lemma 2.6 and Lemma 2.1, respectively.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M196">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M197">View MathML</a>, Lin and Gupur [9] proved that if , then are eigenvalues of A with geometric multiplicity one for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/17/mathml/M200">View MathML</a>. Which means that the result in Theorem 2.7 is optimal, that is to say, it is impossible that the time-dependent solution of the system (1.9) exponentially converges to its steady-state solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

This work was supported by the Natural Science Foundation of Xinjiang (No: 2012211A023).

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