The Uniqueness of Strong Solutions for the Camassa-Holm Equation

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May 10, 2013 - 1 The School of Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China. 2 The School of Finance,Β ...
Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 409760, 7 pages http://dx.doi.org/10.1155/2013/409760

Research Article The Uniqueness of Strong Solutions for the Camassa-Holm Equation Meng Wu1 and Chong Lai2 1 2

The School of Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China The School of Finance, Southwestern University of Finance and Economics, Chengdu 610074, China

Correspondence should be addressed to Meng Wu; [email protected] Received 28 February 2013; Accepted 10 May 2013 Academic Editor: Janusz Matkowski Copyright Β© 2013 M. Wu and C. Lai. 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. Assume that there exists a strong solution of the Camassa-Holm equation and the initial value of the solution belongs to the Sobolev space 𝐻1 (𝑅). We provide a new proof of the uniqueness of the strong solution for the equation.

1. Introduction

We consider the equivalent form of the Cauchy problem for (1)

The integrable Camassa-Holm model [1]

𝑒𝑑 βˆ’ 𝑒𝑑π‘₯π‘₯ + 3𝑒𝑒π‘₯ = 2𝑒π‘₯ 𝑒π‘₯π‘₯ + 𝑒𝑒π‘₯π‘₯π‘₯

1 𝑒𝑑 + (𝑒2 )π‘₯ + πœ•π‘₯ 𝑃𝑒 (𝑑, π‘₯) = 0, 2

(2)

𝑒 (0, π‘₯) = 𝑒0 ,

(1)

∞

has been investigated by many scholars. Equation (1) has peaked solitary wave solutions, which takes the form π‘π‘’βˆ’|π‘₯βˆ’π‘π‘‘| , 𝑐 ∈ 𝑅. The existence and uniqueness of the global weak solutions for (1) have been given by Constantin and Escher [2] and Constantin and Molinet [3] in which the π‘š = 𝑒0 βˆ’ 𝑒0π‘₯π‘₯ is a positive (or negative) Radon measure. The local wellposedness of strong solutions for the Camassa-Holm model and its various generalized forms are provided in [4–8]. For the initial value 𝑒0 satisfying 𝑒0 βˆ’ 𝑒0π‘₯π‘₯ β‰₯ 0 or 𝑒0 βˆ’ 𝑒0π‘₯π‘₯ ≀ 0, it is shown in [9] that the Camassa-Holm equation has unique global strong solutions in the Sobolev space 𝐻𝑠 (𝑅) with 𝑠 > 3/2. If the initial data satisfy certain conditions, we know that the local strong solutions blow up in finite time [10, 11]. It means that the slope of the solution becomes unbounded while the solution itself remains bounded. For other techniques to obtain the dynamic properties for the Camassa-Holm equation and other related shallow water equations, the reader is referred to [12–16] and the references therein.

where 𝑃𝑒 (𝑑, π‘₯) = Ξ›βˆ’2 (𝑒2 + (1/2)𝑒π‘₯2 ) = (1/2) βˆ«βˆ’βˆž 𝑒|π‘₯βˆ’π‘¦| (𝑒2 (𝑑, 𝑦)+ (1/2)𝑒𝑦2 (𝑑, 𝑦))𝑑𝑦. If (1) has a suitable smooth strong solution, we have the conservation law ∫

∞

βˆ’βˆž

(𝑒2 (𝑑, π‘₯) + 𝑒π‘₯2 (𝑑, π‘₯)) 𝑑π‘₯ ∞

=∫

βˆ’βˆž

(3) 2

(𝑒 (0, π‘₯) +

𝑒π‘₯2

(0, π‘₯)) 𝑑π‘₯,

which derives σ΅„© σ΅„© β€–π‘’β€–πΏβˆž (𝑅) ≀ ‖𝑒‖𝐻1 (𝑅) = 󡄩󡄩󡄩𝑒0 󡄩󡄩󡄩𝐻1 (𝑅) .

(4)

The objective of this work is to give a new proof of the uniqueness for the solutions of the Camassa-Holm equation (1). Firstly, we establish the following inequality: ‖𝑒 (𝑑, β‹…) βˆ’ V (𝑑, β‹…)‖𝐿1 (𝑅) ≀ 𝑐𝑒𝑐𝑑 (∫

∞

βˆ’βˆž

∞

󡄨󡄨 2 2 󡄨󡄨 󡄨󡄨𝑒0π‘₯ βˆ’ V0π‘₯ 󡄨󡄨 𝑑π‘₯) , 󡄨 󡄨 βˆ’βˆž (5)

󡄨 󡄨󡄨 󡄨󡄨𝑒0 (π‘₯) βˆ’ V0 (π‘₯)󡄨󡄨󡄨 𝑑π‘₯ + ∫

2

Journal of Function Spaces and Applications

where 𝑑 ∈ [0, 𝑇0 ), functions 𝑒 and V are two local or global strong solutions of problem (2) with initial data 𝑒(0, β‹…) = 𝑒0 ∈ 𝐻1 (𝑅) and V(0, β‹…) = V0 ∈ 𝐻1 (𝑅), respectively. Constant 𝑐 depends on ‖𝑒0 ‖𝐻1 (𝑅) , β€–V0 ‖𝐻1 (𝑅) , and the maximum existence time 𝑇0 . Secondly, from (5), we immediately arrive at the goal of the uniqueness. Here we state that the approach to establish (5) is the device of doubling variables which was presented in Kruzkov’s paper [17]. This paper is organized as follows. Several lemmas are given in Section 2, while the proofs of the main results are established in Section 3.

Lemma 2 (see [17]). If the function |πœ•πΉ(𝑒)/πœ•π‘’| is bounded, then the function 𝐻(𝑒, V) = sign(𝑒 βˆ’ V)(𝐹(𝑒) βˆ’ 𝐹(V)) satisfies the Lipschitz condition in 𝑒 and V, respectively. Lemma 3. Let 𝑒0 ∈ 𝐻1 (𝑅). It holds that the function 𝑄𝑒 (𝑑, π‘₯) = πœ•π‘₯ 𝑃𝑒 (𝑑, π‘₯) = πœ•π‘₯ Ξ›βˆ’2 (𝑒2 + (1/2)𝑒π‘₯2 ) satisfies σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑃𝑒 (𝑑, π‘₯)σ΅„©σ΅„©σ΅„©πΏβˆž (𝑅) < ∞, σ΅„©σ΅„© σ΅„© 󡄩󡄩𝑃𝑒 (𝑑, π‘₯)󡄩󡄩󡄩𝐿1 (𝑅) < ∞, 󡄩󡄩󡄩𝑃𝑒 (𝑑, π‘₯)σ΅„©σ΅„©σ΅„© 2 < ∞, σ΅„© 󡄩𝐿 (𝑅) σ΅„©σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑄𝑒 (𝑑, π‘₯)σ΅„©σ΅„©πΏβˆž (𝑅) < ∞, σ΅„©σ΅„© σ΅„© 󡄩󡄩𝑄𝑒 (𝑑, π‘₯)󡄩󡄩󡄩𝐿1 (𝑅) < ∞,

2. Notations and Several Lemmas Set πœ‰π‘‡ = [0, 𝑇] Γ— 𝑅 for an arbitrary 𝑇 > 0. The space of all infinitely differentiable functions 𝑓(𝑑, π‘₯) with compact support in [0, 𝑇] Γ— 𝑅 is denoted by 𝐢0∞ (πœ‰π‘‡ ). We define 𝜌(𝜎) as a function which is infinitely differentiable on (βˆ’βˆž, +∞) such ∞ that 𝜌(𝜎) β‰₯ 0, 𝜌(𝜎) = 0 for |𝜎| β‰₯ 1 and βˆ«βˆ’βˆž 𝜌(𝜎)π‘‘πœŽ = 1. For any number β„Ž > 0, we let πœŒβ„Ž (𝜎) = 𝜌(β„Žβˆ’1 𝜎)/β„Ž. Then we have that πœŒβ„Ž (𝜎) is a function in 𝐢∞ (βˆ’βˆž, ∞) and πœŒβ„Ž (𝜎) β‰₯ 0,

πœŒβ„Ž (𝜎) = 0 if |𝜎| β‰₯ β„Ž,

󡄨 𝑐 󡄨󡄨 σ΅„¨σ΅„¨πœŒβ„Ž (𝜎)󡄨󡄨󡄨 ≀ , β„Ž

∫

∞

βˆ’βˆž

πœŒβ„Ž (𝜎) = 1.

(6)

σ΅„© σ΅„©σ΅„© 󡄩󡄩𝑄𝑒 (𝑑, π‘₯)󡄩󡄩󡄩𝐿2 (𝑅) < ∞. The proof of Lemma 3 can be found in [13, 15] (see [13, Lemma 5.1]). Lemma 4. Let 𝑒 be the strong solution of problem (2), 𝑓(𝑑, π‘₯) ∈ 𝐢0∞ (πœ‰π‘‡ ), and 𝑓(0, π‘₯) = 0. Then ∬ {|𝑒 βˆ’ π‘˜| 𝑓𝑑 + sign (𝑒 βˆ’ π‘˜)

Assume that the function V(π‘₯) is locally integrable in (βˆ’βˆž, ∞). We define the approximation of function V(π‘₯) as Vβ„Ž (π‘₯) =

π‘₯βˆ’π‘¦ 1 ∞ ) V (𝑦) 𝑑𝑦, ∫ 𝜌( β„Ž βˆ’βˆž β„Ž

β„Ž > 0.

lim

1

󡄨󡄨 󡄨 󡄨󡄨V (π‘₯) βˆ’ V (π‘₯0 )󡄨󡄨󡄨 𝑑π‘₯ = 0. |π‘₯βˆ’π‘₯ |β‰€β„Ž

∫

(8)

0

Lemma 1 (see [17]). Let the function V(𝑑, π‘₯) be bounded and measurable in cylinder Ξ© = [0, 𝑇] Γ— π»π‘Ÿ . If 𝛿 ∈ (0, min[π‘Ÿ, 𝑇]) and any number β„Ž ∈ (0, 𝛿), then the function 1 󡄨 󡄨 ∭∫ |(π‘‘βˆ’πœ)/2|β‰€β„Ž, 󡄨󡄨󡄨V (𝑑, π‘₯) βˆ’ V (𝜏, 𝑦)󡄨󡄨󡄨 𝑑π‘₯ 𝑑𝑑 𝑑𝑦 π‘‘πœ β„Ž2 𝛿≀(𝑑+𝜏)/2β‰€π‘‡βˆ’π›Ώ, |(π‘₯βˆ’π‘¦)/2|β‰€β„Ž, |(π‘₯+𝑦)/2|β‰€π‘Ÿβˆ’π›Ώ (9)

satisfies limβ„Ž β†’ 0 π‘‰β„Ž = 0.

1 2 [𝑒 βˆ’ π‘˜2 ] 𝑓π‘₯ 2

(11)

βˆ’ sign (𝑒 βˆ’ π‘˜) 𝑄𝑒 (𝑑, π‘₯) 𝑓} 𝑑π‘₯ 𝑑𝑑 = 0, where π‘˜ is an arbitrary constant.

At any Lebesgue point π‘₯0 of the function V(π‘₯), we have limβ„Ž β†’ 0 Vβ„Ž (π‘₯0 ) = V(π‘₯0 ). Since the set of points which are not Lebesgue points of V(π‘₯) has measured zero, we get Vβ„Ž (π‘₯) β†’ V(π‘₯) as β„Ž β†’ 0 almost everywhere. We introduce notation connected with the concept of a characteristic cone. For any 𝑀0 > 0, we define 𝑁 > supπ‘‘βˆˆ[0,∞) β€–π‘’β€–πΏβˆž (𝑅) < ∞. Let ℧ designate the cone {(𝑑, π‘₯) : |π‘₯| ≀ 𝑀0 βˆ’ 𝑁𝑑, 0 < 𝑑 < 𝑇0 = min(T, 𝑀0 π‘βˆ’1 )}. We let π‘†πœ designate the cross section of the cone ℧ by the plane 𝑑 = 𝜏, 𝜏 ∈ [0, 𝑇0 ]. Let π»π‘Ÿ = {π‘₯ : |π‘₯| ≀ π‘Ÿ}, where π‘Ÿ > 0.

π‘‰β„Ž =

πœ‰π‘‡

(7)

We call π‘₯0 a Lebesgue point of function V(π‘₯) if β„Žβ†’0β„Ž

(10)

Proof. Let Ξ¦(𝑒) be an arbitrary twice smooth function on the line βˆ’βˆž < 𝑒 < ∞. We multiply (2) by the function Ξ¦σΈ€  (𝑒)𝑓(𝑑, π‘₯), where 𝑓(𝑑, π‘₯) ∈ 𝐢0∞ (πœ‰π‘‡ ). Integrating over πœ‰π‘‡ and transferring the derivatives with respect to 𝑑 and π‘₯ to the test function 𝑓, for any constant π‘˜, we obtain 𝑒

∬ {Ξ¦ (𝑒) 𝑓𝑑 + [∫ Ξ¦σΈ€  (𝑧) 𝑧 𝑑𝑧] 𝑓π‘₯ π‘˜

πœ‰π‘‡

(12)

σΈ€ 

βˆ’ Ξ¦ (𝑒) 𝑄𝑒 (𝑑, π‘₯) 𝑓} 𝑑π‘₯ 𝑑𝑑 = 0, ∞

𝑒

in which we have used βˆ«βˆ’βˆž [βˆ«π‘˜ Ξ¦σΈ€  (𝑧)𝑧 𝑑𝑧]𝑓π‘₯ 𝑑π‘₯ ∞

=

σΈ€ 

βˆ’ βˆ«βˆ’βˆž [𝑓Φ (𝑒)𝑒𝑒π‘₯ ]𝑑π‘₯. We have ∫

∞

βˆ’βˆž

𝑒

[∫ Ξ¦σΈ€  (𝑧) 𝑧 𝑑𝑧] 𝑓π‘₯ 𝑑π‘₯ π‘˜

∞

1 1 [Ξ¦σΈ€  (𝑒) ( 𝑒2 βˆ’ π‘˜2 ) 2 2 βˆ’βˆž

=∫

βˆ’

1 𝑒 2 ∫ (𝑧 βˆ’ π‘˜2 ) Ξ¦σΈ€ σΈ€  (𝑧) 𝑑𝑧] 𝑓π‘₯ 𝑑π‘₯. 2 π‘˜

(13)

Journal of Function Spaces and Applications

3

Let Ξ¦β„Ž (𝑒) be an approximation of the function |𝑒 βˆ’ π‘˜| and set Ξ¦(𝑒) = Ξ¦β„Ž (𝑒). Using the properties of the sign(𝑒 βˆ’ π‘˜), (12), and (13) and sending β„Ž β†’ 0, we have ∬ {|𝑒 βˆ’ π‘˜| 𝑓𝑑 + sign (𝑒 βˆ’ π‘˜) πœ‰π‘‡

1 2 [𝑒 βˆ’ π‘˜2 ] 𝑓π‘₯ 2

∞ 󡄨󡄨 2 󡄨 󡄨 2 2 󡄨󡄨 󡄨󡄨𝑒0 βˆ’ V02 󡄨󡄨󡄨 𝑑π‘₯ + ∫ 󡄨󡄨󡄨𝑒0π‘₯ 󡄨󡄨 𝑑π‘₯ βˆ’ V0π‘₯ 󡄨 󡄨 󡄨 󡄨 βˆ’βˆž βˆ’βˆž

≀ 𝑐 (∫

∞

+∫

∞

βˆ’βˆž

|𝑒 βˆ’ V| 𝑑π‘₯) , (16)

(14) in which we have used the Fubini theorem, β€–π‘’β€–πΏβˆž ≀ ‖𝑒0 ‖𝐻1 (𝑅) and β€–Vβ€–πΏβˆž ≀ β€–V0 ‖𝐻1 (𝑅) . The proof is completed.

βˆ’ sign (𝑒 βˆ’ π‘˜) 𝑄𝑒 (𝑑, π‘₯) 𝑓} 𝑑π‘₯ 𝑑𝑑 = 0, which completes the proof.

3. Main Results In fact, the proof of (11) can also be found in [17]. Lemma 5. Assume 𝑒 and V are two strong solutions of problem (2). It has 󡄨󡄨 ∞ 󡄨󡄨 󡄨󡄨 󡄨 σ΅„¨σ΅„¨βˆ« sign (𝑒 βˆ’ V) [𝑄𝑒 (𝑑, π‘₯) βˆ’ 𝑄V (𝑑, π‘₯)] 𝑓 𝑑π‘₯󡄨󡄨󡄨 󡄨󡄨 βˆ’βˆž 󡄨󡄨 ≀ π‘βˆ«

∞

βˆ’βˆž

󡄨 󡄨 2 󡄨 2 󡄨󡄨 󡄨󡄨 + |𝑒 βˆ’ V|) 𝑑π‘₯. (󡄨󡄨󡄨󡄨𝑒02 βˆ’ V02 󡄨󡄨󡄨󡄨 + 󡄨󡄨󡄨󡄨𝑒0π‘₯ βˆ’ V0π‘₯ 󡄨

(15)

∞

βˆ’βˆž

(𝑒π‘₯2 βˆ’ Vπ‘₯2 ) 𝑑π‘₯ = ∫

∞

βˆ’βˆž

2 2 [(𝑒02 + 𝑒0π‘₯ ) βˆ’ (V02 + V0π‘₯ )] 𝑑π‘₯ ∞

βˆ’βˆ«

βˆ’βˆž

(𝑒2 βˆ’ V2 ) 𝑑π‘₯,

󡄨󡄨 ∞ 1 2 1 2 󡄨󡄨 βˆ’2 2 2 σ΅„¨σ΅„¨βˆ« Ξ› (𝑒 + 𝑒π‘₯ βˆ’ V βˆ’ Vπ‘₯ ) 󡄨󡄨 βˆ’βˆž 2 2

󡄨󡄨 󡄨 Γ— πœ•π‘₯ [sign [𝑒 (𝑑, π‘₯) βˆ’ V (𝑑, π‘₯)] 𝑓 (𝑑, π‘₯)] 𝑑π‘₯󡄨󡄨󡄨 󡄨󡄨

󡄨󡄨 1 ∞ 1 1 󡄨 = 󡄨󡄨󡄨 ∬ π‘’βˆ’|π‘₯βˆ’π‘¦| (𝑒2 + 𝑒𝑦2 βˆ’ V2 βˆ’ V𝑦2 ) 𝑑𝑦 󡄨󡄨 2 βˆ’βˆž 2 2 󡄨󡄨 󡄨 Γ— πœ•π‘₯ [sign [𝑒 (𝑑, π‘₯) βˆ’ V (𝑑, π‘₯)] 𝑓 (𝑑, π‘₯)] 𝑑π‘₯󡄨󡄨󡄨 󡄨󡄨 ≀

󡄨󡄨 1 ∞ 󡄨󡄨󡄨󡄨 ∞ 1 1 󡄨 ∫ σ΅„¨σ΅„¨βˆ« (𝑒2 + 𝑒𝑦2 βˆ’ V2 βˆ’ V𝑦2 ) 𝑑𝑦󡄨󡄨󡄨 󡄨󡄨 2 βˆ’βˆž 󡄨󡄨 βˆ’βˆž 2 2 󡄨 󡄨 Γ— π‘’βˆ’|π‘₯βˆ’π‘¦| σ΅„¨σ΅„¨σ΅„¨πœ•π‘₯ [sign [𝑒 (𝑑, π‘₯) βˆ’ V (𝑑, π‘₯)] 𝑓 (𝑑, π‘₯)]󡄨󡄨󡄨 𝑑π‘₯

≀

󡄨󡄨 1 󡄨󡄨󡄨󡄨 ∞ 1 2 1 2 󡄨 2 2 σ΅„¨σ΅„¨βˆ« (𝑒 + 𝑒𝑦 βˆ’ V βˆ’ V𝑦 ) 𝑑𝑦󡄨󡄨󡄨 󡄨󡄨 2 󡄨󡄨 βˆ’βˆž 2 2 Γ—βˆ«

∞

βˆ’βˆž

‖𝑒 (𝑑, β‹…) βˆ’ V (𝑑, β‹…)‖𝐿1 (𝑅) ≀ 𝑒𝑐𝑑 𝑐 (1 + 𝑇0 ) ∞

∞ 󡄨 2 󡄨 󡄨󡄨 2 󡄨 󡄨󡄨𝑒0 (π‘₯) βˆ’ V0 (π‘₯)󡄨󡄨󡄨 𝑑π‘₯ + ∫ 󡄨󡄨󡄨󡄨𝑒0π‘₯ βˆ’ V0π‘₯ 󡄨󡄨󡄨󡄨 𝑑π‘₯) , βˆ’βˆž βˆ’βˆž (17)

Γ— (∫

Proof. We have ∫

Theorem 6. Let 𝑒 and V be two local or global strong solutions of problem (2) with initial data 𝑒(0, β‹…) = 𝑒0 ∈ 𝐻1 (𝑅) and V(0, β‹…) = V0 ∈ 𝐻1 (𝑅), respectively. Let 𝑇0 be the maximum existence time of solutions 𝑒 and V. For any 𝑑 ∈ [0, 𝑇0 ), it holds that

󡄨󡄨 󡄨 σ΅„¨σ΅„¨πœ•π‘₯ [sign [𝑒 (𝑑, π‘₯) βˆ’ V (𝑑, π‘₯)] 𝑓 (𝑑, π‘₯)]󡄨󡄨󡄨 𝑑π‘₯

󡄨󡄨 ∞ 󡄨󡄨 1 1 󡄨 󡄨 ≀ 𝑐 σ΅„¨σ΅„¨σ΅„¨βˆ« (𝑒2 + 𝑒𝑦2 βˆ’ V2 βˆ’ V𝑦2 ) 𝑑𝑦󡄨󡄨󡄨 󡄨󡄨 βˆ’βˆž 󡄨󡄨 2 2

where 𝑐 depends on ‖𝑒0 ‖𝐻1 (𝑅) and β€–V0 ‖𝐻1 (𝑅) . From Theorem 6, we immediately obtain the uniqueness result. Theorem 7. Let 𝑒(𝑑, π‘₯) be a strong solution of (1) with 𝑒0 ∈ 𝐻1 (𝑅), and let 𝑇0 be the maximum existence time of solution 𝑒. Then any strong solution of (1) is unique. Proof of Theorem 6. For an arbitrary 𝑇 > 0, set πœ‰π‘‡ = [0, 𝑇]×𝑅. Let 𝑓(𝑑, π‘₯) ∈ 𝐢0∞ (πœ‰π‘‡ ). We assume that 𝑓(𝑑, π‘₯) = 0 outside some cylinder z = {(𝑑, π‘₯)} = [𝛿, 𝑇 βˆ’ 2𝛿] Γ— π»π‘Ÿβˆ’2𝛿 ,

0 < 2𝛿 ≀ min (𝑇, π‘Ÿ) . (18)

We define 𝑔 = 𝑓(

π‘₯βˆ’π‘¦ π‘‘βˆ’πœ 𝑑+𝜏 π‘₯+𝑦 , ) πœŒβ„Ž ( ) πœŒβ„Ž ( ) 2 2 2 2

(19)

= 𝑓 (β‹… β‹… β‹…) πœ† β„Ž (βˆ—) , where (β‹… β‹… β‹…) = ((𝑑 + 𝜏)/2, (π‘₯ + 𝑦)/2) and (βˆ—) = ((𝑑 βˆ’ 𝜏)/2, (π‘₯ βˆ’ 𝑦)/2). The function 𝜌(𝜎) is defined in (6). Note that 𝑔𝑑 + π‘”πœ = 𝑓𝑑 (β‹… β‹… β‹…) πœ† β„Ž (βˆ—) , 𝑔π‘₯ + 𝑔𝑦 = 𝑓π‘₯ (β‹… β‹… β‹…) πœ† β„Ž (βˆ—) .

(20)

4

Journal of Function Spaces and Applications

Taking π‘˜ = V(𝜏, 𝑦) and assuming 𝑓(𝑑, π‘₯) = 0 outside the cylinder z, from Lemma 4, we have ∭∫

πœ‰π‘‡ Γ—πœ‰π‘‡

󡄨󡄨 󡄨 + σ΅„¨σ΅„¨σ΅„¨σ΅„¨βˆ¬ sign (𝑒 (𝑑, π‘₯) βˆ’ V (𝑑, π‘₯)) 󡄨󡄨 πœ‰π‘‡

󡄨󡄨 󡄨 Γ— [𝑄𝑒 (𝑑, π‘₯) βˆ’ 𝑄V (𝑑, π‘₯)] 𝑓 𝑑π‘₯ 𝑑𝑑󡄨󡄨󡄨󡄨 . 󡄨󡄨

󡄨 󡄨 { 󡄨󡄨󡄨𝑒 (𝑑, π‘₯) βˆ’ V (𝜏, 𝑦)󡄨󡄨󡄨 𝑔𝑑 + sign (𝑒 (𝑑, π‘₯) βˆ’ V (𝜏, 𝑦))

(24)

2 𝑒2 (𝑑, π‘₯) V (𝜏, 𝑦) Γ—( βˆ’ ) 𝑔π‘₯ 2 2

We note that the first two terms in the integrand of (23) can be represented in the form

+ sign (𝑒 (𝑑, π‘₯) βˆ’ V (𝜏, 𝑦))

π‘Œβ„Ž = π‘Œ (𝑑, π‘₯, 𝜏, 𝑦, 𝑒 (𝑑, π‘₯) , V (𝜏, 𝑦)) πœ† β„Ž (βˆ—) .

Γ— 𝑄𝑒 (𝑑, π‘₯) 𝑔} 𝑑π‘₯ 𝑑𝑑 𝑑𝑦 π‘‘πœ = 0. (21) Similarly, it has ∭∫

πœ‰π‘‡ Γ—πœ‰π‘‡

Since β€–π‘’β€–πΏβˆž ≀ ‖𝑒0 ‖𝐻1 (𝑅) and β€–Vβ€–πΏβˆž ≀ β€–V0 ‖𝐻1 (𝑅) , from Lemma 2, we know π‘Œβ„Ž satisfies the Lipschitz condition in 𝑒 and V, respectively. By the choice of 𝑔, we have π‘Œβ„Ž = 0 outside the region {(𝑑, π‘₯; 𝜏, 𝑦)} = {𝛿 ≀

󡄨 󡄨 { 󡄨󡄨󡄨V (𝜏, 𝑦) βˆ’ 𝑒 (𝑑, π‘₯) π‘”πœ 󡄨󡄨󡄨 + sign (V (𝜏, 𝑦) βˆ’ 𝑒 (𝑑, π‘₯)) Γ—(

V2 (𝜏, 𝑦) 𝑒2 (𝑑, π‘₯) βˆ’ ) 𝑔𝑦 2 2

𝑑+𝜏 |𝑑 βˆ’ 𝜏| ≀ 𝑇 βˆ’ 2𝛿, ≀ β„Ž, 2 2

󡄨 󡄨 󡄨󡄨 󡄨󡄨 󡄨󡄨π‘₯ + 𝑦󡄨󡄨󡄨 󡄨π‘₯ βˆ’ 𝑦󡄨󡄨󡄨 ≀ π‘Ÿ βˆ’ 2𝛿, 󡄨 ≀ β„Ž} , 2 2 ∭∫

πœ‰π‘‡ Γ—πœ‰π‘‡

+ sign (V (𝜏, 𝑦) βˆ’ 𝑒 (𝑑, π‘₯))

(25)

π‘Œβ„Ž 𝑑π‘₯ 𝑑𝑑 𝑑𝑦 π‘‘πœ

= ∭∫

[π‘Œ (𝑑, π‘₯, 𝜏, 𝑦, 𝑒 (𝑑, π‘₯) , V (𝜏, 𝑦))

πœ‰π‘‡ Γ—πœ‰π‘‡

Γ— 𝑄V (𝜏, 𝑦) 𝑔} 𝑑π‘₯ 𝑑𝑑 𝑑𝑦 π‘‘πœ = 0,

(26)

βˆ’π‘Œ (𝑑, π‘₯, 𝑑, π‘₯, 𝑒 (𝑑, π‘₯) , V (𝑑, π‘₯)) ]

(22)

Γ— πœ† β„Ž (βˆ—) 𝑑π‘₯ 𝑑𝑑 𝑑𝑦 π‘‘πœ

from which we obtain 0 ≀ ∭∫

πœ‰π‘‡ Γ—πœ‰π‘‡

+ ∭∫

πœ‰π‘‡ Γ—πœ‰π‘‡

󡄨 󡄨 { 󡄨󡄨󡄨𝑒 (𝑑, π‘₯) βˆ’ V (𝜏, 𝑦)󡄨󡄨󡄨 (𝑔𝑑 + π‘”πœ )

Γ— πœ† β„Ž (βˆ—) 𝑑π‘₯ 𝑑𝑑 𝑑𝑦 π‘‘πœ

+ sign (𝑒 (𝑑, π‘₯) βˆ’ V (𝜏, 𝑦)) Γ—(

= 𝐽1 (β„Ž) + 𝐽2 .

2

Considering the estimate |πœ†(βˆ—)| ≀ 𝑐/β„Ž2 and the expression of function π‘Œβ„Ž , we have 󡄨󡄨󡄨𝐽1 (β„Ž)󡄨󡄨󡄨 󡄨 󡄨

𝑒2 (𝑑, π‘₯) V (𝜏, 𝑦) βˆ’ ) 2 2

Γ— (𝑔π‘₯ + 𝑔𝑦 ) } 𝑑π‘₯ 𝑑𝑑 𝑑𝑦 π‘‘πœ

(23)

󡄨󡄨 󡄨 sign (𝑒 (𝑑, π‘₯) βˆ’ V (𝑑, π‘₯)) + σ΅„¨σ΅„¨σ΅„¨σ΅„¨βˆ­βˆ« 󡄨󡄨 πœ‰π‘‡ Γ—πœ‰π‘‡

󡄨󡄨 󡄨 Γ— (𝑄𝑒 (𝑑, π‘₯) βˆ’ 𝑄V (𝜏, 𝑦)) 𝑔 𝑑π‘₯ 𝑑𝑑 𝑑𝑦 π‘‘πœσ΅„¨σ΅„¨σ΅„¨σ΅„¨ 󡄨󡄨

= ∭∫

πœ‰π‘‡ Γ—πœ‰π‘‡

(𝐼1 + 𝐼2 + 𝐼3 ) 𝑑π‘₯ 𝑑𝑑 𝑑𝑦 π‘‘πœ.

We will show that 0 ≀ ∬ { |𝑒 (𝑑, π‘₯) βˆ’ V (𝑑, π‘₯)| 𝑓𝑑 + sign (𝑒 (𝑑, π‘₯) βˆ’ V (𝑑, π‘₯)) πœ‰π‘‡

Γ—(

𝑒2 (𝑑, π‘₯) V2 (𝑑, π‘₯) βˆ’ ) 𝑓π‘₯ } 𝑑π‘₯ 𝑑𝑑 2 2

π‘Œ (𝑑, π‘₯, 𝑑, π‘₯, 𝑒 (𝑑, π‘₯) , V (𝑑, π‘₯))

[ [ 1 [ ≀ 𝑐 [β„Ž + 2 [ β„Ž [ [ Γ— ∭∫

|(π‘‘βˆ’πœ)/2|β‰€β„Ž, 𝛿≀(𝑑+𝜏)/2β‰€π‘‡βˆ’π›Ώ, |(π‘₯βˆ’π‘¦)/2|β‰€β„Ž, |(π‘₯+𝑦)/2|β‰€π‘Ÿβˆ’π›Ώ

󡄨󡄨 󡄨󡄨V (𝑑, π‘₯)

] ] ] 󡄨󡄨 βˆ’V (𝜏, 𝑦)󡄨󡄨 𝑑π‘₯ 𝑑𝑑 𝑑𝑦 π‘‘πœ, ] , ] ] ] (27)

Journal of Function Spaces and Applications

5

where the constant 𝑐 does not depend on β„Ž. Using Lemma 1, we obtain 𝐽1 (β„Ž) β†’ 0 as β„Ž β†’ 0. The integral 𝐽2 does not depend on β„Ž. In fact, substituting 𝑑 = 𝛼, (𝑑 βˆ’ 𝜏)/2 = 𝛽, π‘₯ = πœ‚, (π‘₯ βˆ’ 𝑦)/2 = πœ‰ and noting that β„Ž

By Lemmas 1 and 3, we have 𝐾1 (β„Ž) β†’ 0 as β„Ž β†’ 0. Using (28), we have 𝐾2 = 22 ∬ 𝐼3 (𝛼, πœ‚, 𝛼, πœ‚, 𝑒 (𝛼, πœ‚) , V (𝛼, πœ‚)) πœ‰π‘‡

∞

∫ ∫

βˆ’β„Ž βˆ’βˆž

πœ† β„Ž (𝛽, πœ‰) π‘‘πœ‰ 𝑑𝛽 = 1,

(28)

β„Ž

Γ— {∫ ∫

∞

βˆ’β„Ž βˆ’βˆž

we have 𝐽2 = 22 ∬ π‘Œβ„Ž (𝛼, πœ‚, 𝛼, πœ‚, 𝑒 (𝛼, πœ‚) , V (𝛼, πœ‚))

= 4∬ 𝐼3 (𝑑, π‘₯, 𝑑, π‘₯, 𝑒 (𝑑, π‘₯) , V (𝑑, π‘₯)) 𝑑π‘₯ 𝑑𝑑

πœ‰π‘‡

β„Ž

Γ— {∫ ∫

∞

βˆ’β„Ž βˆ’βˆž

πœ† β„Ž (𝛽, πœ‰) π‘‘πœ‰ 𝑑𝛽} π‘‘πœ‚ 𝑑𝛼

= 4∬ sign (𝑒 (𝑑, π‘₯) βˆ’ V (𝑑, π‘₯))

(29)

πœ‰π‘‡

Γ— (𝑄𝑒 (𝑑, π‘₯) βˆ’ 𝑄V (𝑑, π‘₯)) 𝑓 (𝑑, π‘₯) 𝑑π‘₯ 𝑑𝑑.

πœ‰π‘‡

From (30) and (33), we prove that inequality (24) holds. Let

Hence lim ∭∫

πœ‰π‘‡ Γ—πœ‰π‘‡

π‘Œβ„Ž 𝑑π‘₯ 𝑑𝑑 𝑑𝑦 π‘‘πœ

πœ” (𝑑) = ∫

(30)

= 4∬ π‘Œβ„Ž (𝑑, π‘₯, 𝑑, π‘₯, 𝑒 (𝑑, π‘₯) , V (𝑑, π‘₯)) 𝑑π‘₯ 𝑑𝑑.

βˆ’βˆž

Since 𝐼3 = sign (𝑒 (𝑑, π‘₯) βˆ’ V (𝜏, 𝑦)) (𝑄𝑒 (𝑑, π‘₯) βˆ’ 𝑄V (𝜏, 𝑦)) π‘“πœ† β„Ž (βˆ—) , πœ‰π‘‡ Γ—πœ‰π‘‡

πœ‰π‘‡ Γ—πœ‰π‘‡

πœƒβ„Ž = ∫

πœ‰π‘‡ Γ—πœ‰π‘‡

𝜎

βˆ’βˆž

πœŒβ„Ž (𝜎) π‘‘πœŽ

(πœƒβ„ŽσΈ€  (𝜎) = πœŒβ„Ž (𝜎) β‰₯ 0)

𝑓 = [πœƒβ„Ž (𝑑 βˆ’ 𝜏1 ) βˆ’ πœƒβ„Ž (𝑑 βˆ’ 𝜏2 )] πœ’ (𝑑, π‘₯) ,

[𝐼3 (𝑑, π‘₯, 𝜏, 𝑦) βˆ’ 𝐼3 (𝑑, π‘₯, 𝑑, π‘₯)]

β„Ž < min (𝜏1 , 𝑇0 βˆ’ 𝜏2 ) ,

Γ— πœ† β„Ž (βˆ—) 𝑑π‘₯ 𝑑𝑑 𝑑𝑦 π‘‘πœ + ∭∫

|𝑒 (𝑑, π‘₯) βˆ’ V (𝑑, π‘₯)| 𝑑π‘₯.

(35)

(36)

where

𝐼3 (𝑑, π‘₯, 𝑑, π‘₯) πœ† β„Ž (βˆ—) 𝑑π‘₯ 𝑑𝑑 𝑑𝑦 π‘‘πœ

πœ’ (𝑑, π‘₯) = πœ’πœ€ (𝑑, π‘₯) = 1 βˆ’ πœƒπœ€ (|π‘₯| + 𝑁𝑑 βˆ’ 𝑀0 + πœ€) ,

= 𝐾1 (β„Ž) + 𝐾2 , (31) we obtain 󡄨󡄨 󡄨 󡄨󡄨𝐾1 (β„Ž)󡄨󡄨󡄨

πœ€ > 0.

1 β„Ž2

(37)

We note that function πœ’(𝑑, π‘₯) = 0 outside the cone ℧ and 𝑓(𝑑, π‘₯) = 0 outside the set z. For (𝑑, π‘₯) ∈ ℧, we have the relations 󡄨 󡄨 0 = πœ’π‘‘ + 𝑁 σ΅„¨σ΅„¨σ΅„¨πœ’π‘₯ 󡄨󡄨󡄨 β‰₯ πœ’π‘‘ + π‘πœ’π‘₯ .

≀ 𝑐 (β„Ž +

(34)

and choose two numbers 𝜏1 and 𝜏2 ∈ (0, 𝑇0 ), 𝜏1 < 𝜏2 . In (24), we choose

𝐼3 𝑑π‘₯ 𝑑𝑑 𝑑𝑦 π‘‘πœ

= ∭∫

∞

We define

πœ‰π‘‡

∭∫

(33)

πœ‰π‘‡

= 4∬ π‘Œβ„Ž (𝑑, π‘₯, 𝑑, π‘₯, 𝑒 (𝑑, π‘₯) , V (𝑑, π‘₯)) 𝑑π‘₯ 𝑑𝑑.

β„Žβ†’0

πœ† β„Ž (𝛽, πœ‰) π‘‘πœ‰ 𝑑𝛽} π‘‘πœ‚ 𝑑𝛼

(38)

Applying (24) and (35)–(38), we have the inequality 0β‰€βˆ¬

Γ— ∭∫

|(π‘‘βˆ’πœ)/2|β‰€β„Ž, 𝛿≀(𝑑+𝜏)/2β‰€π‘‡βˆ’π›Ώ, |(π‘₯βˆ’π‘¦)/2|β‰€β„Ž, |(π‘₯+𝑦)/2|β‰€π‘Ÿβˆ’π›Ώ

󡄨󡄨 󡄨󡄨𝑄V (𝑑, π‘₯)

πœ‰π‘‡0

{[πœŒβ„Ž (𝑑 βˆ’ 𝜏1 ) βˆ’ πœŒβ„Ž (𝑑 βˆ’ 𝜏2 )] Γ— πœ’πœ€ |𝑒 (𝑑, π‘₯) βˆ’ V (𝑑, π‘₯)|} 𝑑π‘₯ 𝑑𝑑

󡄨󡄨 󡄨 + σ΅„¨σ΅„¨σ΅„¨σ΅„¨βˆ¬ [πœƒβ„Ž (𝑑 βˆ’ 𝜏1 ) βˆ’ πœƒβ„Ž (𝑑 βˆ’ 𝜏2 )] 󡄨󡄨 πœ‰π‘‡0 󡄨 βˆ’π‘„V (𝜏, 𝑦)󡄨󡄨󡄨 𝑑π‘₯ 𝑑𝑑 𝑑𝑦 π‘‘πœ) . (32)

󡄨󡄨 󡄨 Γ— [𝑄𝑒 (𝑑, π‘₯) βˆ’ 𝑄V (𝑑, π‘₯)] 𝐡 (𝑑, π‘₯) πœ’ (𝑑, π‘₯) 𝑑π‘₯ 𝑑𝑑󡄨󡄨󡄨󡄨 , 󡄨󡄨 (39) where 𝐡(𝑑, π‘₯) = sign[𝑒(𝑑, π‘₯) βˆ’ V(𝑑, π‘₯)].

6

Journal of Function Spaces and Applications From (39), we obtain

0β‰€βˆ¬

πœ‰π‘‡0

Using the similar proof of (43), we get 𝑇0

{[πœŒβ„Ž (𝑑 βˆ’ 𝜏1 ) βˆ’ πœŒβ„Ž (𝑑 βˆ’ 𝜏2 )]

𝐿󸀠 (𝜏1 ) = βˆ’ ∫ πœŒβ„Ž (𝑑 βˆ’ 𝜏1 ) πœ” (𝑑) 𝑑𝑑 󳨀→ βˆ’πœ” (𝜏1 ) 0

Γ— πœ’πœ€ |𝑒 (𝑑, π‘₯) βˆ’ V (𝑑, π‘₯)|} 𝑑π‘₯ 𝑑𝑑 + ∫ (πœƒβ„Ž (𝑑 βˆ’ 𝜏1 ) βˆ’ πœƒβ„Ž (𝑑 βˆ’ 𝜏2 ))

𝜏1

𝐿 (𝜏1 ) 󳨀→ 𝐿 (0) βˆ’ ∫ πœ” (𝜎) π‘‘πœŽ

0

󡄨󡄨 ∞ 󡄨󡄨 󡄨 󡄨 Γ— σ΅„¨σ΅„¨σ΅„¨βˆ« [𝑄𝑒 (𝑑, π‘₯) βˆ’ 𝑄V (𝑑, π‘₯)] 𝐡 (𝑑, π‘₯) πœ’ (𝑑, π‘₯) 𝑑π‘₯󡄨󡄨󡄨 𝑑𝑑. 󡄨󡄨 βˆ’βˆž 󡄨󡄨 (40)

0

𝜏2

𝐿 (𝜏2 ) 󳨀→ 𝐿 (0) βˆ’ ∫ πœ” (𝜎) π‘‘πœŽ 0

𝜏2

𝐿 (𝜏1 ) βˆ’ 𝐿 (𝜏2 ) 󳨀→ ∫ πœ” (𝜎) π‘‘πœŽ 𝜏1

Γ— πœ’πœ€ |𝑒 (𝑑, π‘₯) βˆ’ V (𝑑, π‘₯)|} 𝑑π‘₯ 𝑑𝑑 𝑇0

(41)

0

Γ— (𝐺 + ∫

∞

βˆ’βˆž

∞

|𝑒 βˆ’ V| 𝑑π‘₯) 𝑑𝑑,

0

𝜏1

∞ 󡄨󡄨 2 󡄨 󡄨 2 2 󡄨󡄨 󡄨󡄨𝑒0 βˆ’ V02 󡄨󡄨󡄨 𝑑π‘₯ + ∫ 󡄨󡄨󡄨𝑒0π‘₯ 󡄨󡄨 𝑑π‘₯) (𝜏2 βˆ’ 𝜏1 ) . βˆ’ V0π‘₯ 󡄨 󡄨 󡄨 󡄨 βˆ’βˆž βˆ’βˆž (49)

󡄨 󡄨󡄨 󡄨 󡄨 󡄨󡄨𝑒 (𝜏1 , π‘₯) βˆ’ V (𝜏1 , π‘₯)󡄨󡄨󡄨 ≀ 󡄨󡄨󡄨𝑒 (𝜏1 , π‘₯) βˆ’ 𝑒0 (π‘₯)󡄨󡄨󡄨 󡄨 󡄨 + 󡄨󡄨󡄨V (𝜏1 , π‘₯) βˆ’ V0 (π‘₯)󡄨󡄨󡄨 󡄨 󡄨 + 󡄨󡄨󡄨𝑒0 (π‘₯) βˆ’ V0 (π‘₯)󡄨󡄨󡄨 .

0

|𝑒 (𝑑, π‘₯) βˆ’ V (𝑑, π‘₯)| 𝑑π‘₯} 𝑑𝑑 (42)

+ 𝑐 ∫ (πœƒβ„Ž (𝑑 βˆ’ 𝜏1 ) βˆ’ πœƒβ„Ž (𝑑 βˆ’ 𝜏2 )) 0

Γ— (𝐺 + ∫

βˆ’βˆž

󡄨󡄨 󡄨󡄨 𝑇0 󡄨 󡄨󡄨 σ΅„¨σ΅„¨βˆ« πœŒβ„Ž (𝑑 βˆ’ 𝜏1 ) πœ” (𝑑) 𝑑𝑑 βˆ’ πœ” (𝜏1 )󡄨󡄨󡄨 󡄨󡄨 󡄨󡄨 0 󡄨 󡄨 󡄨󡄨 𝑇0 󡄨󡄨 󡄨 󡄨 = σ΅„¨σ΅„¨σ΅„¨σ΅„¨βˆ« πœŒβ„Ž (𝑑 βˆ’ 𝜏1 ) [πœ” (𝑑) βˆ’ πœ” (𝜏1 )] 𝑑𝑑󡄨󡄨󡄨󡄨 󡄨󡄨 0 󡄨󡄨

∞

βˆ’βˆž

By the properties of the function πœŒβ„Ž (𝜎) for β„Ž ≀ min(𝜏1 , 𝑇0 βˆ’ 𝜏1 ), we have

(43)

(50)

Thus, from (42), (43), (48), (49), and (50), for any 𝑑 ∈ [0, 𝑇0 ], we have ∫

|𝑒 βˆ’ V| 𝑑π‘₯) 𝑑𝑑.

∞

Let 𝜏1 β†’ 0 and 𝜏2 β†’ 𝑑, and note that

𝑇0

∞

(48)

𝜏2

= (∫

𝑇0

as β„Ž 󳨀→ 0.

𝑇0

0 ≀ ∫ { [πœŒβ„Ž (𝑑 βˆ’ 𝜏1 ) βˆ’ πœŒβ„Ž (𝑑 βˆ’ 𝜏2 )]

βˆ’βˆž

(47)

Furthermore, if β„Ž β†’ 0, we have

∞

Γ—βˆ«

as β„Ž 󳨀→ 0.

∫ (πœƒβ„Ž (𝑑 βˆ’ 𝜏1 ) βˆ’ πœƒβ„Ž (𝑑 βˆ’ 𝜏2 )) 𝐺 𝑑𝑑 󳨀→ 𝐺 ∫ 𝑑𝑑

2 2 where 𝐺 = βˆ«βˆ’βˆž |𝑒02 βˆ’V02 |𝑑π‘₯+βˆ«βˆ’βˆž |𝑒0π‘₯ βˆ’V0π‘₯ |𝑑π‘₯ and 𝑐 is defined in Lemma 5. Letting πœ€ β†’ 0 in (41) and sending 𝑀0 β†’ ∞, we have

∞

(46)

Then, we get

{[πœŒβ„Ž (𝑑 βˆ’ 𝜏1 ) βˆ’ πœŒβ„Ž (𝑑 βˆ’ 𝜏2 )]

+ 𝑐 ∫ (πœƒβ„Ž (𝑑 βˆ’ 𝜏1 ) βˆ’ πœƒβ„Ž (𝑑 βˆ’ 𝜏2 ))

as β„Ž 󳨀→ 0.

Similarly, we have

Using Lemma 5, we have

πœ‰π‘‡0

(45)

from which we obtain

𝑇0

0β‰€βˆ¬

as β„Ž 󳨀→ 0,

|𝑒 (𝑑, π‘₯) βˆ’ V (𝑑, π‘₯)| 𝑑π‘₯ ∞

β‰€βˆ«

βˆ’βˆž

|𝑒 (0, π‘₯) βˆ’ V (0, π‘₯)| 𝑑π‘₯ ∞

∞ 󡄨 󡄨 󡄨 2 2 󡄨󡄨 󡄨󡄨 𝑑π‘₯) + 𝑐𝑇0 (∫ 󡄨󡄨󡄨󡄨𝑒02 βˆ’ V02 󡄨󡄨󡄨󡄨 𝑑π‘₯ + ∫ 󡄨󡄨󡄨󡄨𝑒0π‘₯ βˆ’ V0π‘₯ 󡄨 βˆ’βˆž βˆ’βˆž 𝑑

∞

0

βˆ’βˆž

+∫ ∫

(51)

|𝑒 βˆ’ V| 𝑑π‘₯,

from which we complete the proof of Theorem 6 by using the Gronwall inequality.

1 𝜏1 +β„Ž 󡄨󡄨 󡄨 ≀𝑐 ∫ σ΅„¨πœ” (𝑑) βˆ’ πœ” (𝜏1 )󡄨󡄨󡄨 𝑑𝑑 󳨀→ 0 as β„Ž 󳨀→ 0, β„Ž 𝜏1 βˆ’β„Ž 󡄨

Acknowledgments

where 𝑐 is independent of β„Ž. Set 𝑇0

𝑇0

π‘‘βˆ’πœ1

0

0

βˆ’βˆž

𝐿 (𝜏1 ) = ∫ πœƒβ„Ž (𝑑 βˆ’ 𝜏1 ) πœ‡ (𝑑) 𝑑𝑑 = ∫ ∫

πœŒβ„Ž (𝜎) π‘‘πœŽπœ” (𝑑) 𝑑𝑑. (44)

Thanks are given to referees whose suggestions are very helpful to the paper. This work is supported by both the Fundamental Research Funds for the Central Universities (JBK120504) and the Applied and Basic Project of Sichuan Province (2012JY0020).

Journal of Function Spaces and Applications

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