International Journal of Partial Differential EquationsVolume 2013 (2013), Article ID 796430, 20 pageshttp://dx.doi.org/10.1155/2013/796430

Research Article

Existence and Uniqueness of the Solutions for Some Initial-Boundary Value Problems with the Fractional Dynamic Boundary Condition

Mykola Krasnoschok and Nataliya Vasylyeva

Institute of Applied Mathematics and Mechanics of NAS of Ukraine, R.Luksemburg Street 74, Donetsk 83114, Ukraine

Received 22 April 2013; Accepted 28 July 2013

Academic Editor: Antonin Novotny

Copyright © 2013 Mykola Krasnoschok and Nataliya Vasylyeva. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we analyze some initial-boundary value problems for the subdiffusion equation with a fractional dynamic boundary condition in a one-dimensional bounded domain. First, we establish the unique solvability in the Hölder space of the initial-boundary value problems for the equation , , where L is a uniformly elliptic operator with smooth coefficients with the fractional dynamic boundary condition. Second, we apply the contraction theorem to prove the existence and uniqueness locally in time in the Hölder classes of the solution to the corresponding nonlinear problems.

1. Introduction

Let and be any numbers from and let , ; ; ; be a fixed value. In this paper, we consider a partial differential equation with the fractional derivative in time as follows: Here, denotes the Caputo fractional derivative with respect to and is defined by (see, e.g., in [1]), where is the gamma function, , , are the given functions, and is a positive. Note that if , then (1) represents a parabolic equation. As we are interested in the fractional cases, we restrict the order to the case .

We will solve (1) satisfying the following conditions: the fractional dynamic boundary condition on : and one of the following conditions on : the Dirichlet boundary condition: or the Neumann boundary condition: or the fractional dynamic boundary condition:

Here, and are given positive functions, and , , and , , are given functions.

Concerning problems (1)–(6), they have the following features. First of all, (4) is a fractional dynamic boundary condition; next, these problems are formulated for the subdiffusion linear equation.

Note that if , conditions (4) and (6) are called normal dynamic boundary conditions. These conditions are very natural in many mathematical models, including heat transfer in a solid in contact with a moving fluid, thermoelasticity, diffusion phenomena, and problems in fluid dynamics, and in the Stefan problem, (see [2–4] and the references therein). At the present moment, there are a lot of works concerning linear and nonlinear problems with dynamic boundary conditions. Here we make no pretence to provide a complete survey on the results related to problems of the type (1)-(6), if , and present only some of them. The initial-boundary value problems for the heat equation in the certain shape of domains with linear dynamic boundary condition have been solved with the separation variables method or with the Laplace transformation in [3]. In the case of smooth domains, these problems have been researched with the approaches of the general theory for evolution equations in Hilbert and Banach spaces, and the weak solutions of the above mentioned problems have been obtained in [5–7]. Using the Schauder method, Grigor’eva and Mogilevskii [8] have got the coercive estimates of the solution in the anisotropic Sobolev spaces. The one-to-one solvability in the case of the linear parabolic equation with variable coefficients has been proved by Bazaliy [4] in the Hölder spaces and by Bizhanova and Solonnikov [9] in the weighted Hölder classes. The global and local existence for the solution to initial-boundary value problem for linear and quasilinear equations with nonlinear dynamic boundary conditions has been discussed in [10–12] (see also references there).

Over the past few decades, an intensive effort has been put into developing theoretical models for systems with diffusive motion that cannot be modelled as the standard Brownian motion [13, 14]. The signature of this anomalous diffusion is that the mean square displacement of the diffusing species scales as a nonlinear power law in time, that is, . If , this is referred to as subdiffusion. In recent years, the additional motivation for these studies has been stimulated by experimental measurements of subdiffusion in porous media [15], glass forming materials [16], and biological media [17]. The review paper by Klafter et al. [18] provides numerous references to physical phenomena in which anomalous diffusion occurs.

Here we refer to several works on the mathematical treatments for linear equation (1). Kochubei [19, 20], and Pskhu [21, 22] constructed the fundamental solution in and proved the maximum principle for the Cauchy problem. Gejji and Jafari [23] solved a nonhomogeneous fractional diffusion-wave equation in a one-dimensional bounded domain. Metzler and Klafter [14], using the method of images and the Fourier-Laplace transformation technique, obtained the solutions of different boundary value problems for the homogenous fractional diffusion equation in a half-space and in a box. Agrawal [24] constructed a solution of a fractional diffusion equation using a finite transform technique and presented numerical results in a one-dimensional bounded domain. Mophou and N’Guérékata [25] and Sakamoto and Yamamoto [26] proved the one-valued solvability of the initial-boundary value problem for the fractional diffusion equation with variable coefficients which is -independent with the homogenous Dirichlet conditions in the Sobolev space. Note that, in [26], the authors obtained the certain regularities of the solution given by the eigenfunction expansions and established several results of uniqueness for related inverse problems.

As source books related with fractional derivatives, see the work of Samko et al. [27] which is an encyclopedic treatment of the fractional calculus and also Kilbas et al. [1], Mainardi [28], Podlubny [29], and Pskhu [21].

As for the quasilinear equation like (1), Clément et al. [30] analyzed the abstract fractional parabolic quasilinear equations. Via maximal regularity results in the corresponding linear equation, they arrived to results on existence (locally in time), uniqueness, and continuation on the quasilinear equation in the BUC classes with a weight. As for investigation of the problem with fractional dynamic boundary conditions, Kirane and Tatar [31] have analyzed the issue of nonexistence of local and global solutions for elliptic systems with nonlinear fractional dynamic boundary conditions.

To the authors’ best knowledge, there are no works published concerning the solvability of problems (1)–(6) in the Hölder classes. The first purpose of this paper is to prove the well-posedness and the regularity of the solutions to problems (1)–(6) in the smooth classes. Second, we obtain a local in time solvability in the smooth classes of the corresponding nonlinear problems. This paper is organized as follows. In the second section, we state the main results, Theorems 3–5, and define the functional spaces. In Section 3, we establish the one-valued solvability of certain model problems in . The principal results of this section are given in Theorems 9 and 13. In Section 4, we prove the main results of this paper. To this end, we will combine ideas from [32] with coercive estimates of the solutions to the corresponding model problems (Section 3). In Section 5, we address the corresponding nonlinear problems. We first reduce them to a form , where is a nonlinear function of and is the linear operator derived in Section 4; that is, is the solution of the model problem for data . Setting , we will then prove that the mapping , where , is a contraction, so that it has a unique fixed point. The principal results of this section are formulated in Theorem 18 and Remarks 19 and 20. The Appendix contains the proofs of some auxiliary assertions which are applied in Section 3.

2. The Functional Spaces and the Main Results

Let us introduce the functional spaces. Let be a bounded or an unbounded domain in , , ; ; . Denote

Definition 1. We will say that functions and that if and only if the functions and , , are continuous and the following norms are finite:

Note that if , the spaces , , coincide with the ordinary Hölder spaces (see (1.10)–(1.12) in Chapter 1 in [32]). Further, we also use the Banach spaces and of the functions and with the finite norms In a similar way, we introduce the spaces , .

Definition 2. We will say that functions and that if and only if and and and .If , , and , then .

We introduce the spaces of , , with the same way.

Let and be some positive numbers. We assume that the given functions , , and , , in (1), (4), and (7) are subject to the following conditions:  and one of the following:  or  or Note that requirements (15)–(18) are called the consistency conditions.

The main results of our paper are the following:

Theorem 3. Let , and conditions (13)–(16) hold, and , , , for any positive number . Then there exists a unique solution of problem (1)–(5): , , and where a positive constant depends only on the measure of and , , , .

Theorem 4. Let , and conditions (13)–(15), and (17) hold, and let , , , for any positive number . Then there exists a unique solution of problem (1)–(4), (6) as follows: , , and where is a positive constant and depends only on , , , , and the measure of .

Theorem 5. Let , and conditions (13)–(15) and (18) hold, and , , , and for any positive number . Then there exists a unique solution of problem (1)–(4), (7) as follows: , , , and where is a positive constant and depends only on ,, ,  , and the measure of .

3. Model Problems

Let , and , and and be some positive numbers. Here we will discuss the first initial-boundary value problem for the fractional diffusion equation in and the initial-boundary value problem with the fractional boundary condition in .

3.1. The Solvability of the First Initial-Boundary Value Problem for the Subdiffusion Equation

We look for the function by the following conditions: where and , are some given functions.

We assume that the following conditions hold: for some positive number .

First, we will study problem (22)–(24) under restriction We will search a solution of (22)–(24) under restriction (27) in the class , if for .

Note that conditions (25) and (26) together with restriction (27) allow us to apply the Laplace transformation in to the right hand sides of (22)–(24). Indeed, conditions (25)–(27) mean that the right hand sides in (22)–(24) except equal zero and . Thus, we can extend the right hand sides in (22)–(24) by for and save, for simplicity, the same notation for the extension of the function . Therefore, we can apply, at least formally, the Laplace transformation in to (22)–(24) in the case of (25)–(27) hold.

Denote by the Laplace transformation of the function ; that is, The Laplace transformation in (22)–(24) leads to the problem Here we used the following formula from [33]: One can easily check that the following function solves the equations in (29): Due to formula (2.30) in [34] and the inverse Laplace transformation, we get the integral representations of as follows: where Here is the Wright function, which is defined for as (see formula () in v.3 [35]) The main properties of the Wright functions are described in Chapters 4.1, v.1 and 18.1 v3 in [35], Chapter 1.11 in [1], Chapter 1.3 in [34], and Chapter 2 in [21, 36].

In Lemma 6, we describe the properties of the kernel which will be necessary to estimate the function . Its proof is represented in Appendix A.

Lemma 6. Let , , and be some positive constants, , , , . Then one has the following.  where is the Riemann-Liouville fractional derivative, and its definition is in (2.1.8) in [1].

Lemma 7. Let , and conditions (25)–(27) hold, and . Then the function represented by formula (32) satisfies the following inequalities:

Proof. First, we obtain the representation of . To this end, we need the following properties of the fractional derivative (see Lemma 2.10 and formula () in [1]): (i)          where (see () in [1]) (ii) If the functions , , and are bounded in , then  where is the Riemann-Liouville fractional integral of order (see, e.g. () in [1]) One can easily see that and (see (25) and (27)). Then, using properties (46) and (48) and equality (36), we represent the function as Namely, this representation will be useful below. To prove inequality (43), we will use this representation, statement from Lemma 6 and the first estimate in (37) as follows: In view of , if , we can rewrite representation (50) as Let , and . Denote Then, Using inequality (38) with , , we obtain As for the term , we apply the mean-value theorem to the difference together with inequality (39) (where , ) and deduce Thus, representation (54) together with inequalities (55) and (56) prove the correctness of (44).To complete the proof of Lemma 7, we need to obtain inequality (45). Let and . Denote We analyze the difference As for the last term in this sum, it is estimated by . We change the variable in the term and apply estimate (38) with , . Thus, we have In the same way, we evaluate the function . The estimate of the term follows from the properties of the function and inequality (36). At last, the mean-value theorem together with estimate (40), where , , lead to Therefore, inequality (45) is deduced from (58)–(60).

Now, based on the results of Lemmas 6 and 7, we can infer the next assertion.

Lemma 8. Let conditions of Lemma 7 hold. Then there exists a unique solution of problem (22)–(24), which is represented with (32) and

Proof. First of all we obtain estimate (61). One can get the following inequality using the results of Lemma 7 and (22), where Next, we use formula () from [1] as follows: to evaluate the maximum of . Hence, (43) and (63) lead to inequality here we use the fact that .After that, the minor seminorms of the function are estimated with the interpolation inequalities from Section 8.8 [37] and (43)–(45), (62), and (64). Therefore, the arguments above prove inequality (61) and the embedding .Next, we show that the function given by (32) satisfies (22). To this end, we use equality (46) and (48) and represent as Then (41) leads to Next, due to property (42) of the function , one can check that As it follows from the first inequality in (37), the function represented by (32) satisfies the following conditions: Finally, it is necessary to show that the function meets boundary condition (24). To this end, we observe the next difference (here we will essentially use statement (35)) Applying inequality (38) with , to the term , we get To estimate the term in (69), we use the Wright formula (see [38] or () in [21]): and rewrite the function as After that, due to Lemmas and in [21] and representation (72), we have Thus, Then, we joint estimates (69)–(74) and obtain which means that .Therefore, as it was written above, the function given by (32) is a solution of (22)–(24) in the case of (27). The uniqueness of this solution is proved like the arguments of Theorem 3.2 from [39].

Now we remove restriction (27). To this end, it is enough to consider the Cauchy problem: Here, and are extensions of the functions and , correspondingly, onto . These functions together with their corresponding derivatives have finite supports and The results of Theorem 3.2 from [39] give the one-valued solvability of (76) and or, due to inequalities (77), After that, we will look for the solution of problem (22)–(24) as where satisfies conditions (22)–(24) with the new right-hand sides which meet requirements of Lemma 8. Hence, we can apply the results of Lemma 8 to the function . This fact and the properties (see (77)) of the function allow us to obtain the next results.

Theorem 9. Let , and conditions (25) and (26) hold, and , , for any positive number . Then there exists a unique solution of problem (22)–(24) and

3.2. The Model Initial-Boundary Value Problem with a Fractional Dynamic Boundary Condition

Here we study the following problem: it is necessary to find the function by the following conditions: where are the given functions. Let the following conditions hold: for some positive number .

At the beginning, we assume that and search a solution of (82) under this restriction in the class , if , .

Note that conditions (83) and (84) allow us to extend the right-hand sides in (82) by for . We save, for simplicity, the same notation for the extended function .

After the application of the Laplace transformation in time to problem (82), we have the following: Here, we used again formula (30). Some simple calculations lead to the function which is the solution of problem (85). Due to formulas (2.30) in [34] and (1.80) in [29], we obtain, after applying of the inverse Laplace transformation to (86), that where the kernel is given by (33) and Here, is the function of the Mittag-Leffler type, which is defined by the series expansion (see, e.g., (1.56) in [29] or () in [1]) Note that this two-parameter function of the Mittag-Leffler type was in fact introduced by Agarwal [40].

The function has been studied in Section 3.1 (see Lemma 6). Thus, to describe the properties of the function , we have to observe the function . To this end, we will use the following properties of the kernel , which are proved in Appendix B.

Lemma 10. Let , , and be some positive constants, ; and be the fractional Riemann-Liouville integral and derivative, correspondingly (their definitions are given in (49) and (47)). Then the following is true.

Lemma 11. Let meet requirement (83) and let . Then the function represented by (88) belongs to , , and the following estimate holds:

Proof. First of all, we evaluate the value of . To this end, we use the first inequality in (90). Thus, one has This inequality gives that Next, we obtain the representation of . Due to equality (99) and properties (46) and (48), we conclude that Since , we can rewrite the last equality as or, applying (91) and (92), we have To estimate , we use representation (102) and get After that, we apply inequality (93) with to the first term in (103) and (94) to the second and get Finally, to complete the proof of Lemma 11 is necessary to estimate of . Let and . Denote Using formula (102), we represent the difference as Changing the variable in and in , we get Then, the property of the function together with inequality (93) (where ) lead to To get the same estimate for the term , we apply the mean-value theorem to the difference and inequality (96) and have after some simple calculations As for the estimate of , one follows immediately from (95) where and properties of the function are Hence, inequalities (106)–(110) lead to estimate which completes the proof of Lemma 11.

Due to results of this lemma and arguments like (70)–(74), we can infer that Based on the results of Lemma 11, properties of the function (see Lemmas 7 and 8), and (112), we can get the next assertion.

Lemma 12. Let conditions (83) and (84) hold and let meet the requirements of Lemma 11. Then the function represented by (87) satisfies (112) and

Now, we can prove the solvability of problem (82).

Theorem 13. Let , and conditions (83) hold, and , , for any positive number . Then there exists a unique solution of problem (82) and

Proof. To prove this theorem in the case of (84) it is enough to consider the following Dirichlet problem: and to apply the results of Theorem 9 and Lemma 12.To remove restriction (84), we will look for the solution of problem (82) in the following form: where is the solution of the following Dirichlet problem: and the function solves the following problem: The one-valued solvability of problem (117) follows from Theorem 9, which gives These properties of the function allow us to conclude that and ; that is, satisfies the conditions of Lemma 12, and the right-hand sides of (118) meet requirement (84). It means that there exists a unique solution of (118) which satisfies to inequality (119) with .Finally, returning to representation (116) for and using the described properties of , , we complete the proof of Theorem 13.

4. The Proofs of Theorems 3–5

Note that the proofs of Theorems 4 and 5 are analogous to the one of Theorem 3 and use the technique from Chapter 4 [32] together with the results of the solvability to the model problems from Section 3. That is why, we represent here only the proof of Theorem 3. In this route, we will need the solvability in of the next initial-boundary value problem: Reaching this goal is enough to repeat the arguments of Section 3.3 from [39] and apply the results of Theorem 9. Thus, we can assert the following.

Theorem 14. Let and the following conditions hold. Let and , , , , for any positive number . Then there exists a unique solution of problem (120): , and where a positive constant depends only on the measure of and , .

At the beginning, we prove Theorem 9 in the case of for some small . Then, we will show how the solution is extended from into , where , for all .

Lemma 15. Let the conditions of Theorem 3 hold. Then there exists a unique solution of problem (1)–(5), for any , and where the positive constant depends on , , , , and the measure of .

Proof. If the right-hand sides of (1)–(5) meet the following requirements: then by repeating the arguments of 4–7 from Chapter 4 [32] and using the results of Theorems 9 and 13 and Theorem 3.2 from [39], we have proved the assertion of Lemma 15.To remove restriction (124), we look for the solution of problem (1)–(5) as where the function is the solution of problem (120) with and the unknown function is the solution of the problem Our further arguments are divided into two parts. In the first step, we will show that the right-hand sides of (126) meet the requirements of Theorem 14, which will ensure the existence of the unique solution . The next step deals with the proving of the following equalities: We obtain after simple calculations the following properties for the functions , for all : After that, using (129), one can easily check that the right-hand sides in (126) satisfy conditions of Theorem 14. That is why, there exists a unique solution as follows as follows: where the positive constant depends on , , and the measure of .Then we return to problem (127). Using properties (129) and (131), we get that After that, to get statement (128) is enough to apply condition (15) to the right-hand sides of equalities (132).Finally, taking into account (129)–(131), we obtain the following statements: where the positive constant depends on , , , , and the measure of .Therefore, the right-hand sides of (127) meet requirements (124). It means that there exists a unique solution of problem (127) and that satisfies to inequality (123) with the corresponding right-hand sides. This fact together with the properties of the function (see (130)) and representation (125) completes the proof of Lemma 15.

Now we will extend the obtained solution , in Lemma 15, into . In other words, we construct the function , as follows: where the function is constructed in Lemma 15; that is, the function : , , is the unique solution of (1)–(5) for , .

First, we assume that and consider problem (120) with the following right-hand sides: where After some calculations, we can infer that Due to the fact the functions ,, and satisfy the consistency conditions, properties (137) allow us to conclude that the right-hand sides (135) meet requirements of Theorem 14. This means that there exists a unique solution , , and that After that, we search the solution of problem (134) in the following form: where the new unknown function satisfies the following conditions: As it follows from (136) and (138), if then

In problem (140), we introduce the new variable: . Let where and , satisfy the conditions of Theorem 3. Due to (141), we have and therefore

To rewrite problem (140) in the new variable , we use formula (3.110) from [39] as follows: Thus, problem (140) in the new variable can be rewritten as Then we apply Lemma 15 to problem (147) and get the one-to-one solvability in , and the estimate like (123) holds. Returning to the old variable , we obtain the unique solution of problem (140), , and satisfies inequality (123), where .

Thus, representation (139) together with the properties (138) for the functions and the corresponding properties of allow us to construct the solution of problem (1)–(5) for , if there is the solution for . Repeating this procedure, we prove the existence of the solution for and obtain estimate (19). As for the uniqueness of the solution, it follows from inequality (19). This ends the proof of Theorem 3.

5. The Local Solvability of the Nonlinear Problem with a Fractional Dynamic Boundary Condition

In this section, we indicate how our results may be applied to the nonlinear problem: We require the following conditions on the functions ,  , , ; : (i) There exist some positive constants such that (ii) Consistency conditions hold: (iii) There exist , , , , , such that for some positive constant  for any bounded functions together with their derivatives. (iv) There are positive constants ,   as follows: (v)We introduce the functional spaces and as follows: and for any elements , and , we denote

Let and , , be balls of radius in the space , centered at the origin, for some positive to be determined later on.

The simple calculations lead to the following assertion.

Proposition 16. Let conditions (149)–(153) hold and the function ; then , , ; , ; , , and Moreover, for the functions and , for all , , the following inequalities hold:

First, we linearize problem (148) on the initial data and represent one as a system , where is a linear operator and is a nonlinear perturbation. To this end, we introduce the new unknown function as follows: where the function is a solution of the following problem: By Theorem 5 and Proposition 16, there exists a unique solution of the problem (159) and where the positive constant depends on , , , , .

Next, we rewrite problem (148) in terms of the function and after some calculations get the problem in the form: where

Thus, we represent nonlinear problem (148) as Note that if we froze the functional arguments in the functions and , , then problem (161) will be a linear problem with variable coefficients, which has been studied in detail in Sections 2–4. By Theorem 5, has a bounded inverse , so that and is a nonlinear operator. We will show that is a contraction operator.

Lemma 17. The following inequalities hold for the right-hand sides of problem (167): with as .

Proof. First we prove (168). Note that Let Using definitions (163), (165) and notations (171) with and , we have It is easy to examine that the following are true: Hence, using inequalities (173), representation (172), and results of Proposition 16, we get after some tedious calculations where the constant is from (160).Let It is easy to see that , as . Thus inequalities (174) together with representation (170) lead to estimate (168), where is given by (175).Next, we will obtain inequality (169). Note that, as it follows from definitions (162) and (164), one has Denote Then, we have the following: After that, the simple calculations together with the results of Proposition 16 allow us to get the following for any function . Moreover, due to , we have the following: Thus, applying inequalities from Proposition 16 and (179)–(182) to the right-hand sides in (178), we get (169), where It is obviously as .

As it follows from Lemma 17, for sufficiently small and , the nonlinear operator satisfies the conditions of the fixed point theorem for a contraction operator. Hence, we have proved the following theorem.

Theorem 18. Let conditions (149)–(153) hold. Then there exists a unique solution of problem (148) for small interval , such that and that , .

Using the analogous arguments, it is possible to assert the following results.

Remark 19. If we exchange the dynamic boundary condition on by either (5) or (6) and assume that the corresponding consistency conditions hold, then the results of Theorem 18 save.

Remark 20. The results of Theorem 18 hold if conditions (152) are changed by and constants and in requirements (149) are replaced by and , correspondingly.

Appendix

Here we give the proof of some estimates used in the arguments above.

A. The proof of Lemma 6

The statement and the first inequality in of Lemma 6 follow immediately from representation (72) of the function and the next properties of the Wright functions: where the constant will be positive if and either or . The left inequality (A.1) and the right one if have been proved in Lemmas and in [21], where and .

Let us get the right estimate in the case of . To this end, we use the well-known Wright formula (see e.g., () in [21] or [38]) and represent the function as As we consider the case of and , which means we can apply inequality (A.1) to the terms in the right hand side of (A.2) and get It is easy to check that , if and that Thus, the last inequalities together with (A.4) allow us to get (A.1) with and if . Note that to prove and the first inequality in of Lemma 6, we need (A.1) in the case of .

To prove equality (36), we use formula (2.2.5) from [21] as follows: and represent the function as After that, equality (A.7) and estimates (A.1) lead to Then, using definition (34) of the function , we can deduce equality (36). To get inequalities (37), we need the well-known formula to a differentiation of the Wright function (see, e.g. () in [21] or [38]) as follows: Due to this formula and (A.6), we can represent as Furthermore, using inequalities (16) and (17) from Lemma 3 [22], we infer the estimate of the Wright function Thus, representations (A.10) and inequality (A.11) lead to (37). Note that statements (iv)–(vii) of Lemma 6 follow, after some simple calculations, from representations (A.10) and estimate (A.11).

In virtue of formulas in [21] or and in [41]:

and equality (A.9), we state (42).

B. The proof of Lemma 10

Using estimate of Lemma 3.1 in [26] or Theorem 1.6 in [29]: we have got Then Due to formula () in [42], we can conclude that is positive for . Thus, the statement of Lemma 10 follows immediately from (B.3) and the arguments above.

Due to formula (1.82) in [29]:

we can calculate value of and get Here we used the definition of (see (89)). Then, representation (B.6) leads to (91). Note that . Hence or due to (B.6) and (91),

To obtain (92), we apply estimate like (B.1) to the first term in the right-hand side (B.8). In the same way, we infer inequality (95).

Inequalities (93)–(95) follow from (B.5)–(B.8) and (B.1).

At last, proving inequality (96) is enough to use formula (1.83) in [29] as follows: and to estimate (B.1).

Acknowledgments

The authors are very grateful to the referee for the careful reading of the paper and the valuable comments. This work is partially supported by Grant PIRSES-GA-2011-295164.

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