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International Journal of Differential Equations
VolumeΒ 2011Β (2011), Article IDΒ 679528, 31 pages
http://dx.doi.org/10.1155/2011/679528
Research Article

A High Order Iterative Scheme for a Nonlinear Kirchhoff Wave Equation in the Unit Membrane

1Nhatrang Educational College, 01 Nguyen Chanh Street, Nhatrang City, Vietnam
2Department of Mathematics and Computer Science, University of Natural Science, Vietnam National University Ho Chi Minh City, 227 Nguyen Van Cu Street, District 5, Ho Chi Minh City, Vietnam

Received 5 May 2011; Accepted 16 October 2011

Academic Editor: BashirΒ Ahmad

Copyright Β© 2011 Le Thi Phuong Ngoc and Nguyen Thanh Long. 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

A high-order iterative scheme is established in order to get a convergent sequence at a rate of order 𝑁 (𝑁β‰₯1) to a local unique weak solution of a nonlinear Kirchhoff wave equation in the unit membrane. This extends a recent result in (EJDE, 2005, No. 138) where a recurrent sequence converges at a rate of order 2.

1. Introduction

In this paper we consider the initial and boundary value problemπ‘’π‘‘π‘‘ξ‚€β€–β€–π‘’βˆ’π΅π‘Ÿβ€–β€–20π‘’ξ‚ξ‚€π‘Ÿπ‘Ÿ+1π‘Ÿπ‘’π‘Ÿξ‚||||=𝑓(π‘Ÿ,𝑑,𝑒),0<π‘Ÿ<1,0<𝑑<𝑇,limπ‘Ÿβ†’0+βˆšπ‘Ÿπ‘’π‘Ÿ||||𝑒(π‘Ÿ,𝑑)<∞,π‘Ÿ(1,𝑑)+β„Žπ‘’(1,𝑑)=0,𝑒(π‘Ÿ,0)=̃𝑒0(π‘Ÿ),𝑒𝑑(π‘Ÿ,0)=̃𝑒1(π‘Ÿ),(1.1) where 𝐡,𝑓,̃𝑒0,̃𝑒1 are given functions satisfying conditions specified later, β€–π‘’π‘Ÿβ€–20=∫10π‘Ÿ|π‘’π‘Ÿ(π‘Ÿ,𝑑)|2π‘‘π‘Ÿ, and β„Ž>0 is a given constant.

Equation (1.1)1 herein is the bidimensional nonlinear wave equation describing nonlinear vibrations of the unit membrane Ξ©1={(π‘₯,𝑦)∢π‘₯2+𝑦2<1}. In the vibration process, the area of the unit membrane and the tension at various points change in time. The condition on the boundary πœ•Ξ©1 describes elastic constraints, where the constant β„Ž1 has a mechanical signification. The boundary condition |limπ‘Ÿβ†’0+βˆšπ‘Ÿπ‘’π‘Ÿ(π‘Ÿ,𝑑)|<∞ is satisfied automatically if 𝑒 is a classical solution of the problem (1.1), for example, with π‘’βˆˆC1([0,1]Γ—(0,𝑇))∩𝐢2((0,1)Γ—(0,𝑇)). This condition is also used in connection with Sobolev spaces with weight π‘Ÿ (see [1–3]).

Equation (1.1)1 is related to the Kirchhoff equationπœŒβ„Žπ‘’π‘‘π‘‘=𝑃0+πΈβ„Žξ€œ2𝐿𝐿0||||πœ•π‘’||||πœ•π‘¦(𝑦,𝑑)2ξƒͺ𝑒𝑑𝑦π‘₯π‘₯(1.2) presented by Kirchhoff in 1876 (see [4]). This equation is an extension of the classical D’Alembert wave equation which considers the effects of the changes in the length of the string during the vibrations. The parameters in (1.2) have the following meanings: 𝑒 is the lateral deflection, 𝐿 is the length of the string, β„Ž is the area of the cross-section, 𝐸 is the Young modulus of the material, 𝜌 is the mass density, and 𝑃0 is the initial tension.

The Kirchhoff wave equation of the form (1.1)1 received much attention. Many interesting results about the existence, stability, regularity in time variable, asymptotic behavior, and asymptotic expansion of solutions can be found, for example, in [2, 3, 5–14] and references therein.

In [2], in a special case, sufficient conditions were established for a quadratic convergence to the solution of (1.1) with 𝑓(π‘Ÿ,𝑑,𝑒)=𝑓(π‘Ÿ,𝑒) and 𝐡(β€–π‘’π‘Ÿβ€–20)=𝑏0+β€–π‘’π‘Ÿβ€–20,𝑏0>0. Based on the ideas about recurrence relations for a third-order method for solving the nonlinear operator equation 𝐹(𝑒)=0 in [15], we extend the above result by the construction of a high-order iterative scheme for (1.1)1, where 𝑓 and 𝐡 are more generalized.

In this paper, we associate with (1.1)1 a recurrent sequence {π‘’π‘š} defined byπœ•2π‘’π‘šπœ•π‘‘2ξ‚€β€–β€–π‘’βˆ’π΅π‘šπ‘Ÿβ€–β€–20ξ‚ξ‚΅πœ•2π‘’π‘šπœ•π‘Ÿ2+1π‘Ÿπœ•π‘’π‘šξ‚Ά=πœ•π‘Ÿπ‘βˆ’1𝑖=01πœ•π‘–!π‘–π‘“πœ•π‘’π‘–ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1π‘’ξ€Έξ€·π‘šβˆ’π‘’π‘šβˆ’1𝑖,(1.3)0<π‘Ÿ<1,0<𝑑<𝑇, with π‘’π‘š satisfying (1.1)2-3. The first term 𝑒0 is chosen as 𝑒0≑0. If𝐡∈𝐢1(ℝ+) and π‘“βˆˆπΆπ‘([0,1]×ℝ+×ℝ), we prove that the sequence {π‘’π‘š} converges at a rate of order 𝑁 to a unique weak solution of the problem (1.1). This result is a relative generalization of [2, 3, 8, 9, 14, 16].

2. Preliminary Results, Notations, Function Spaces

Put Ξ©=(0,1). We omit the definitions of the usual function spaces πΆπ‘š(Ξ©),𝐿𝑝(Ξ©),π»π‘š(Ξ©), and π‘Šπ‘š,𝑝(Ξ©). For any function π‘£βˆˆπΆ0(Ξ©) we define ‖𝑣‖0 as ‖𝑣‖0∫=(10π‘Ÿπ‘£2(π‘Ÿ)π‘‘π‘Ÿ)1/2 and define the space 𝑉0 as completion of the space 𝐢0(Ξ©) with respect to the norm β€–β‹…β€–0. Similarly, for any function π‘£βˆˆπΆ1(Ξ©) we define ‖𝑣‖1 as ‖𝑣‖1=(‖𝑣‖20+β€–π‘£π‘Ÿβ€–20)1/2 and define the space 𝑉1 as completion of the space 𝐢1(Ξ©) with respect to the norm β€–β‹…β€–1. Note that the norms β€–β‹…β€–0 and β€–β‹…β€–1 can be defined, respectively, from the inner products ξ€œβŸ¨π‘’,π‘£βŸ©=10π‘Ÿπ‘’(π‘Ÿ)𝑣(π‘Ÿ)π‘‘π‘Ÿ,βŸ¨π‘’,π‘£βŸ©+βŸ¨π‘’π‘Ÿ,π‘£π‘ŸβŸ©.(2.1) Identifying 𝑉0 with its dual π‘‰ξ…ž0 we obtain the dense and continuous embedding 𝑉1β†ͺ𝑉0β‰‘π‘‰ξ…ž0β†ͺπ‘‰ξ…ž1. The inner product notation will be reutilized to denote the duality pairing between 𝑉1 and π‘‰ξ…ž1.

We then have the following lemmas, the proofs of which can be found in [1].

Lemma 2.1. There exist two constants 𝐾1>0 and 𝐾2>0 such that, for all π‘£βˆˆπΆ1(Ξ©), we have(i)β€–π‘£π‘Ÿβ€–20+𝑣2(1)β‰₯‖𝑣‖20, (ii)|𝑣(1)|≀𝐾1‖𝑣‖1, (iii)βˆšπ‘Ÿ|𝑣(π‘Ÿ)|≀𝐾2‖𝑣‖1,forallπ‘ŸβˆˆΞ©.

Lemma 2.2. The embedding 𝑉1β†ͺ𝑉0 is compact.

Remark 2.3. In Lemma 2.1, the two constants 𝐾1 and 𝐾2 can be given explicitly as 𝐾1=ξ”βˆš1+2 and 𝐾2=ξ”βˆš1+5. We also note that limπ‘Ÿβ†’0+βˆšπ‘Ÿπ‘£(π‘Ÿ)=0 for all π‘£βˆˆπ‘‰1 (see [17, page 128/Lemma 5.40]). On the other hand, by 𝐻1(πœ€,1)β†ͺ𝐢0([πœ€,1]),0<πœ€<1 and βˆšπœ€β€–π‘£β€–π»1(πœ€,1)≀‖𝑣‖1 for all π‘£βˆˆπ‘‰1, it follows that 𝑣|[πœ€,1]∈𝐢0([πœ€,1]). From both relations we deduce that βˆšπ‘Ÿπ‘£βˆˆπΆ0(Ξ©) for all π‘£βˆˆπ‘‰1.
Now, let the bilinear form π‘Ž(β‹…,β‹…) be defined byξ€œπ‘Ž(𝑒,𝑣)=β„Žπ‘’(1)𝑣(1)+10π‘Ÿπ‘’π‘Ÿ(π‘Ÿ)π‘£π‘Ÿ(π‘Ÿ)π‘‘π‘Ÿ,𝑒,π‘£βˆˆπ‘‰1,(2.2) where β„Ž is a positive constant. Then, there exists a unique bounded linear operator π΄βˆΆπ‘‰1β†’π‘‰ξ…ž1 such that π‘Ž(𝑒,𝑣)=βŸ¨π΄π‘’,π‘£βŸ© for all 𝑒,π‘£βˆˆπ‘‰1. We then have the following lemma.

Lemma 2.4. The symmetric bilinear form π‘Ž(β‹…,β‹…) defined by (2.2) is continuous on 𝑉1×𝑉1 and coercive on 𝑉1, that is,(i)|π‘Ž(𝑒,𝑣)|≀𝐢1‖𝑒‖1‖𝑣‖1, (ii)π‘Ž(𝑣,𝑣)β‰₯𝐢0‖𝑣‖21, for all 𝑒,π‘£βˆˆπ‘‰1, where 𝐢0=(1/2)min{1,β„Ž} and 𝐢1√=1+(1+2)β„Ž.

The proof of Lemma 2.4 is straightforward and we omit it.

Lemma 2.5. There exists an orthonormal Hilbert basis {𝑀𝑗} of the space 𝑉0 consisting of eigenfunctions 𝑀𝑗 corresponding to eigenvalues πœ†π‘— such that(i)0<πœ†1β‰€πœ†2β‰€β‹―β‰€πœ†π‘—β†‘+∞asπ‘—β†’βˆž, (ii)π‘Ž(𝑀𝑗,𝑣)=πœ†π‘—βŸ¨π‘€π‘—,π‘£βŸ©forallπ‘£βˆˆπ‘‰1andπ‘—βˆˆβ„•. Note that it follows from (ii) that {𝑀𝑗/βˆšπœ†π‘—} is automatically an orthonormal set in 𝑉1 with respect to π‘Ž(β‹…,β‹…) as inner product. The eigensolutions 𝑀𝑗 are indeed eigensolutions for the boundary value problem π΄π‘€π‘—β‰‘βˆ’1π‘Ÿπ‘‘ξ‚΅π‘Ÿπ‘‘π‘Ÿπ‘‘π‘€π‘—ξ‚Άπ‘‘π‘Ÿ=πœ†π‘—π‘€π‘—||||,inΞ©,limπ‘Ÿβ†’0+βˆšπ‘Ÿπ‘‘π‘€π‘—||||𝑑r(π‘Ÿ)<+∞,π‘‘π‘€π‘—π‘‘π‘Ÿ(1)+β„Žπ‘€π‘—(1)=0.(2.3)

The proof of Lemma 2.5 can be found in ([18, page 87, Theorem 7.7]) with 𝑉=𝑉1,𝐻=𝑉0 and π‘Ž(β‹…,β‹…) as defined by (2.2).

For any function π‘£βˆˆπΆ2(Ξ©) we define ‖𝑣‖2 as ‖𝑣‖2=‖𝑣‖20+β€–β€–π‘£π‘Ÿβ€–β€–20+‖𝐴𝑣‖201/2(2.4) and define the space 𝑉2 as completion of 𝐢2(Ξ©) with respect to the norm β€–β‹…β€–2. Note that 𝑉2 is also a Hilbert space with respect to the scalar product βŸ¨π‘’,π‘£βŸ©+βŸ¨π‘’π‘Ÿ,π‘£π‘ŸβŸ©+βŸ¨π΄π‘’,π΄π‘£βŸ©(2.5) and that 𝑉2 can be defined also as 𝑉2={π‘£βˆˆπ‘‰1βˆΆπ΄π‘£βˆˆπ‘‰0}.

We then have the following two lemmas the proof of which can be found in [1].

Lemma 2.6. The embedding 𝑉2β†ͺ𝑉1 is compact.

Lemma 2.7. For all π‘£βˆˆπ‘‰2 we have ‖‖𝑣(i)π‘Ÿβ€–β€–πΏβˆž(Ξ©)≀1√2‖𝐴𝑣‖0,‖‖𝑣(ii)π‘Ÿπ‘Ÿβ€–β€–0≀32‖𝐴𝑣‖0,(iii)‖𝑣‖2𝐿∞(Ξ©)≀2‖𝑣‖0+1√2‖𝐴𝑣‖0ξƒͺ‖𝑣‖0.(2.6)

For a Banach space 𝑋, we denote by ‖⋅‖𝑋 its norm, by π‘‹ξ…ž its dual space, and by 𝐿𝑝(0,𝑇;𝑋),1β‰€π‘β‰€βˆž the Banach space of all real measurable functions π‘’βˆΆ(0,𝑇)→𝑋 such that ‖𝑒‖𝐿𝑝(0,𝑇;𝑋)=ξ‚΅ξ€œπ‘‡0‖𝑒(𝑑)‖𝑝𝑋𝑑𝑑1/𝑝‖<∞,for1≀𝑝<∞,π‘’β€–πΏβˆž(0,𝑇;𝑋)=esssup0<𝑑<𝑇‖𝑒(𝑑)‖𝑋for𝑝=∞.(2.7)

Let 𝑒(𝑑),π‘’ξ…ž(𝑑)=𝑒𝑑(𝑑)=̇𝑒(𝑑),π‘’ξ…žξ…ž(𝑑)=𝑒𝑑𝑑(𝑑)=Μˆπ‘’(𝑑),π‘’π‘Ÿ(𝑑)=βˆ‡π‘’(𝑑),π‘’π‘Ÿπ‘Ÿ(𝑑)(2.8) denote 𝑒(π‘Ÿ,𝑑),πœ•π‘’πœ•πœ•π‘‘(π‘Ÿ,𝑑),2π‘’πœ•π‘‘2(π‘Ÿ,𝑑),πœ•π‘’πœ•πœ•π‘Ÿ(π‘Ÿ,𝑑),2π‘’πœ•π‘Ÿ2(π‘Ÿ,𝑑),(2.9) respectively.

With π‘“βˆˆπΆπ‘˜(Ω×ℝ+×ℝ),𝑓=𝑓(π‘Ÿ,𝑑,𝑒), we put 𝐷1𝑓=πœ•π‘“/πœ•π‘Ÿ,𝐷2𝑓=πœ•π‘“/πœ•π‘‘,𝐷3𝑓=πœ•π‘“/πœ•π‘’, and 𝐷𝛾𝑓=𝐷𝛾11𝐷𝛾22𝐷𝛾33𝑓,𝛾=(𝛾1,𝛾2,𝛾3)βˆˆβ„€3+,|𝛾|=𝛾1+𝛾2+𝛾3=π‘˜.

3. The Hight Order Iterative Schemes

Fix π‘‡βˆ—>0, we make the following assumptions: (𝐻1)̃𝑒0βˆˆπ‘‰2and̃𝑒1βˆˆπ‘‰1; (𝐻2)𝐡∈𝐢1(ℝ+)andthereexistconstantsπ‘βˆ—>0,𝛼>1,𝑑0,𝑑1>0suchthat(i)π‘βˆ—β‰€π΅(πœ‚)≀𝑑0(1+πœ‚π›Ό),forallπœ‚β‰₯0, (ii)|π΅ξ…ž(πœ‚)|≀𝑑1(1+πœ‚π›Όβˆ’1),forallπœ‚β‰₯0; (𝐻3)π‘“βˆˆπΆπ‘(Ω×[0,π‘‡βˆ—]×ℝ)andsatisfiesthefollowingcondition∢forall𝑀>0, 𝐾(1)βˆ—π‘–(𝑀,𝑓)=sup(π‘Ÿ,𝑑,𝑒)βˆˆπ΄π‘€||||ξ‚€βˆšπ‘Ÿξ‚βˆ’π‘–πœ•π‘–π‘“πœ•π‘’π‘–||||𝐾(π‘Ÿ,𝑑,𝑒)<+∞,𝑖=0,1,…,π‘βˆ’1,(2)βˆ—π‘–(𝑀,𝑓)=sup(π‘Ÿ,𝑑,𝑒)βˆˆπ΄π‘€||||ξ‚€βˆšπ‘Ÿξ‚βˆ’π‘–πœ•π‘–π‘“πœ•π‘Ÿπœ•π‘’π‘–βˆ’1||||(π‘Ÿ,𝑑,𝑒)<+∞,𝑖=1,…,π‘βˆ’1,(3.1)

where 𝐴𝑀={(π‘Ÿ,𝑑,𝑒)∈[0,1]Γ—[0,π‘‡βˆ—ξ”]Γ—β„βˆΆ|𝑒|β‰€π‘€βˆš2+1/2}. We put ξπΎβˆ—π‘–βŽ§βŽͺ⎨βŽͺ⎩𝐾(𝑀,𝑓)=(1)βˆ—0𝐾(𝑀,𝑓),𝑖=0,max(1)βˆ—π‘–ξπΎ(𝑀,𝑓),(2)βˆ—π‘–ξ‚‡(𝑀,𝑓),𝑖=1,…,π‘βˆ’1.(3.2)

With 𝐡 and 𝑓 satisfying assumptions (𝐻2) and (𝐻3), respectively, for each 𝑀>0 given, we introduce the following constants:𝐾𝑀(𝐡)=sup0β‰€πœ‚β‰€π‘€2𝐡||𝐡(πœ‚)+ξ…ž||ξ€Έ,(πœ‚)𝐾0(𝑀,𝑓)=sup(π‘Ÿ,𝑑,𝑒)βˆˆπ΄π‘€||||,𝑓(π‘Ÿ,𝑑,𝑒)𝐾𝑁(𝑀,𝑓)=||𝛾||≀𝑁𝐾0(𝑀,𝐷𝛾𝑓).(3.3)

For each π‘‡βˆˆ(0,π‘‡βˆ—] and 𝑀>0 we get ξ€½π‘Š(𝑀,𝑇)=π‘£βˆˆπΏβˆžξ€·0,𝑇;𝑉2ξ€ΈβˆΆπ‘£ξ…žβˆˆπΏβˆžξ€·0,𝑇;𝑉1ξ€Έ,π‘£ξ…žξ…žβˆˆπΏ2ξ€·0,𝑇;𝑉0ξ€Έ,withβ€–π‘£β€–πΏβˆž(0,𝑇;𝑉2),β€–β€–π‘£ξ…žβ€–β€–πΏβˆž(0,𝑇;𝑉1),β€–β€–π‘£ξ…žξ…žβ€–β€–πΏ2(0,𝑇;𝑉0),π‘Šβ‰€π‘€1ξ€½(𝑀,𝑇)=π‘£βˆˆπ‘Š(𝑀,𝑇)βˆΆπ‘£ξ…žξ…žβˆˆπΏβˆžξ€·0,𝑇;𝑉0.ξ€Έξ€Ύ(3.4)

We will choose as first initial term 𝑒0≑0, suppose thatπ‘’π‘šβˆ’1βˆˆπ‘Š1(𝑀,𝑇),(3.5) and associate with the problem (1.1) the following variational problem.

Find π‘’π‘šβˆˆπ‘Š1(𝑀,𝑇)(π‘šβ‰₯1) so thatξ«π‘’π‘šξ…žξ…žξ¬(𝑑),𝑣+π‘π‘šξ€·π‘’(𝑑)π‘Žπ‘šξ€Έ(𝑑),𝑣=βŸ¨πΉπ‘š(𝑑),π‘£βŸ©,βˆ€π‘£βˆˆπ‘‰1,π‘’π‘š(0)=̃𝑒0,π‘’ξ…žπ‘š(0)=̃𝑒1,(3.6) whereπ‘π‘šξ‚€β€–β€–(𝑑)=π΅βˆ‡π‘’π‘šβ€–β€–(𝑑)20,πΉπ‘š(π‘Ÿ,𝑑)=π‘βˆ’1𝑖=01𝐷𝑖!𝑖3π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1π‘’ξ€Έξ€·π‘šβˆ’π‘’π‘šβˆ’1𝑖.(3.7)

Then, we have the following theorem.

Theorem 3.1. Let assumptions (𝐻1)-(𝐻3) hold. Then there exist a constant 𝑀>0 depending on π‘‡βˆ—,̃𝑒0,̃𝑒1,𝐡,β„Ž and a constant 𝑇>0 depending on π‘‡βˆ—,̃𝑒0,̃𝑒1,𝐡,β„Ž,𝑓 such that, for 𝑒0≑0, there exists a recurrent sequence {π‘’π‘š}βŠ‚π‘Š1(𝑀,𝑇) defined by (3.6), (3.7).

Proof. The proof consists of several steps.
Step 1. The Faedo-Galerkin approximation (introduced by Lions [19]). Consider as in Lemma 2.5 the basis {𝑀𝑗} for 𝑉1 and put π‘’π‘š(π‘˜)(𝑑)=π‘˜ξ“π‘—=1𝑐(π‘˜)π‘šπ‘—(𝑑)𝑀𝑗,(3.8) where the coefficients 𝑐(π‘˜)π‘šπ‘— satisfy the system of the following nonlinear differential equations: ξ‚¬Μˆπ‘’π‘š(π‘˜)(𝑑),𝑀𝑗+π‘π‘š(π‘˜)𝑒(𝑑)π‘Žπ‘š(π‘˜)(𝑑),𝑀𝑗=ξ‚¬πΉπ‘š(π‘˜)(𝑑),𝑀𝑗𝑒,1β‰€π‘—β‰€π‘˜,π‘š(π‘˜)(0)=̃𝑒0π‘˜,Μ‡π‘’π‘š(π‘˜)(0)=̃𝑒1π‘˜,(3.9) where ̃𝑒0π‘˜=π‘˜ξ“π‘—=1𝛼𝑗(π‘˜)π‘€π‘—βŸΆΜƒπ‘’0stronglyin𝑉2,̃𝑒1π‘˜=π‘˜ξ“π‘—=1𝛽𝑗(π‘˜)π‘€π‘—βŸΆΜƒπ‘’1stronglyin𝑉1.𝑏(3.10)π‘š(π‘˜)ξ‚€β€–β€–(𝑑)=π΅βˆ‡π‘’π‘š(π‘˜)β€–β€–(𝑑)20,πΉπ‘š(π‘˜)(π‘Ÿ,𝑑)=π‘βˆ’1𝑖=01𝐷𝑖!𝑖3π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έξ‚€π‘’π‘š(π‘˜)βˆ’π‘’π‘šβˆ’1𝑖=𝑁𝑗=0Ξ¨π‘—ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έξ‚€π‘’π‘š(π‘˜)𝑗,Ξ¨π‘—ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έ=π‘βˆ’1𝑖=𝑗1𝑗!(π‘–βˆ’π‘—)!(βˆ’1)π‘–βˆ’π‘—π·π‘–3π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έπ‘’π‘–βˆ’π‘—π‘šβˆ’1,0β‰€π‘—β‰€π‘βˆ’1.(3.11)
Let us suppose that π‘’π‘šβˆ’1 satisfies (3.5). Then we have the following lemma.

Lemma 3.2. Let assumptions (𝐻1)-(𝐻3) hold. For fixed 𝑀>0 and 𝑇>0, then, the system (3.8)–(3.11) has a unique solution π‘’π‘š(π‘˜)(𝑑) on an interval [0,π‘‡π‘š(π‘˜)]βŠ‚[0,𝑇].
Proof of Lemma 3.2. The system of (3.8)–(3.11) is rewritten in the form Μˆπ‘(π‘˜)π‘šπ‘—(𝑑)=βˆ’πœ†π‘—π‘π‘š(π‘˜)(𝑑)𝑐(π‘˜)π‘šπ‘—ξ‚¬πΉ(𝑑)+π‘š(π‘˜)(𝑑),𝑀𝑗𝑐,1β‰€π‘—β‰€π‘˜,(π‘˜)π‘šπ‘—(0)=𝛼𝑗(π‘˜),̇𝑐(π‘˜)π‘šπ‘—(0)=𝛽𝑗(π‘˜),(3.12) and it is equivalent to the system of integral equations 𝑐(π‘˜)π‘šπ‘—(𝑑)=𝛼𝑗(π‘˜)+𝛽𝑗(π‘˜)π‘‘βˆ’πœ†π‘—ξ€œπ‘‘0ξ€œπ‘‘πœπœ0π‘π‘š(π‘˜)(𝑠)𝑐(π‘˜)π‘šπ‘—(ξ€œπ‘ )𝑑𝑠+𝑑0ξ€œπ‘‘πœπœ0ξ‚¬πΉπ‘š(π‘˜)(𝑠),𝑀𝑗𝑑𝑠,(3.13) for 1β‰€π‘—β‰€π‘˜. Omitting the index π‘š, it is written as follows: [𝑐],𝑐=β„±(3.14) where β„±[𝑐]=(β„±1[𝑐],…,β„±π‘˜[𝑐]),𝑐=(𝑐1,…,π‘π‘˜), ℱ𝑗[𝑐](𝑑)=π‘žπ‘—(𝑑)βˆ’πœ†π‘—ξ€œπ‘‘0ξ€œπ‘‘πœπœ0̃𝑏(𝑠)𝑐𝑗(+𝑠)π‘‘π‘ π‘βˆ’1𝑖=1ξ€œπ‘‘0ξ€œπ‘‘πœπœ0Ψ𝑖⋅,𝑠,π‘’π‘šβˆ’1𝑒𝑖(𝑠),π‘€π‘—ξ¬π‘žπ‘‘π‘ ,1β‰€π‘—β‰€π‘˜,𝑗(𝑑)=𝛼𝑗(π‘˜)+𝛽𝑗(π‘˜)ξ€œπ‘‘+𝑑0ξ€œπ‘‘πœπœ0Ψ0ξ€·β‹…,𝑠,π‘’π‘šβˆ’1ξ€Έ,𝑀𝑗̃̃𝑏[𝑐]𝑑𝑠,1β‰€π‘—β‰€π‘˜,𝑏(𝑑)=(𝑑)=π΅β€–βˆ‡π‘’(𝑑)β€–20ξ€Έ,𝑒(𝑑)=π‘˜ξ“π‘—=1𝑐𝑗(𝑑)𝑀𝑗.(3.15)
For every π‘‡π‘š(π‘˜)∈(0,𝑇] and 𝜌>0 that will be chosen later, we put 𝑋=𝐢0([0,π‘‡π‘š(π‘˜)];β„π‘˜),𝑆={π‘βˆˆπ‘ŒβˆΆβ€–π‘β€–π‘‹β‰€πœŒ}, where ‖𝑐‖𝑋=sup0β‰€π‘‘β‰€π‘‡π‘š(π‘˜)|𝑐(𝑑)|1,|𝑐(𝑑)|1=βˆ‘π‘˜π‘—=1|𝑐𝑗(𝑑)|, for each 𝑐=(𝑐1,…,π‘π‘˜)βˆˆπ‘‹. Clearly 𝑆 is a closed nonempty subset in 𝑋, and we have the operator β„±βˆΆπ‘‹β†’π‘‹. In what follows, we will choose 𝜌>0 and π‘‡π‘š(π‘˜)>0 such that(i)𝑆ismappedintoitselfbyβ„±, (ii)β„±βˆΆπ‘†β†’π‘†iscontractive.
Proof (i). First we note that, for all 𝑐=(𝑐1,…,π‘π‘˜)βˆˆπ‘†, 𝑒(𝑑)=π‘˜ξ“π‘—=1𝑐𝑗(𝑑)𝑀𝑗,‖𝑒(𝑑)β€–0=ξ„Άξ„΅ξ„΅ξ„΅βŽ·π‘˜ξ“π‘—=1𝑐2𝑗||||(𝑑)≀𝑐(𝑑)1,β€–β€–βˆ‡π‘’(𝑑)20β‰€π‘Ž(𝑒(𝑑),𝑒(𝑑))=π‘˜ξ“π‘–,𝑗=1𝑐𝑖(𝑑)𝑐𝑗𝑀(𝑑)π‘Žπ‘–,𝑀𝑗=π‘˜ξ“π‘—=1πœ†π‘—π‘2𝑗(𝑑)β‰€πœ†π‘˜β€–β€–π‘’(𝑑)20,‖𝑒(𝑑)β€–0≀||||𝑐(𝑑)1β‰€β€–π‘β€–π‘‹β‰€πœŒ,β€–βˆ‡π‘’(𝑑)β€–0β‰€βˆšπœ†π‘˜||||𝑐(𝑑)1β‰€βˆšπœ†π‘˜πœŒ,‖𝑒(𝑑)β€–1β‰€ξƒŽ1𝐢0βˆšξƒŽπ‘Ž(𝑒(𝑑),𝑒(𝑑))≀1𝐢0βˆšπœ†π‘˜β€–π‘’(𝑑)β€–0β‰€ξƒŽπœ†π‘˜πΆ0||||𝑐(𝑑)1β‰€ξƒŽπœ†π‘˜πΆ0𝜌,(3.16) so ̃𝑏‖(𝑑)=π΅β€–βˆ‡π‘’(𝑑)20≀𝑑0ξ€·β€–1+β€–βˆ‡π‘’(𝑑)02𝛼≀𝑑0ξ€·1+πœ†π‘˜2π›ΌπœŒ2𝛼.(3.17) On the other hand, by ||π‘’π‘šβˆ’1||ξƒŽβ‰€π‘€12+√2||Ξ¨β‰‘πœƒ,0ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έ||=|||||π‘βˆ’1𝑖=01𝑖!(βˆ’1)𝑖𝐷𝑖3π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έπ‘’π‘–π‘šβˆ’1|||||β‰€π‘βˆ’1𝑖=0|||1𝐷𝑖!𝑖3π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έπ‘’π‘–π‘šβˆ’1|||≀𝐾𝑁(𝑀,𝑓)π‘βˆ’1𝑖=01ξƒ©π‘€ξƒŽπ‘–!12+√2ξƒͺ𝑖=𝐾𝑁(𝑀,𝑓)π‘βˆ’1𝑖=0πœƒπ‘–,𝑖!(3.18) we have ||Ψ0𝑑,π‘’π‘šβˆ’1ξ€Έ,𝑀𝑗||≀‖‖Ψ0𝑑,π‘’π‘šβˆ’1ξ€Έβ€–β€–0‖‖𝑀𝑗‖‖0=β€–β€–Ξ¨0𝑑,π‘’π‘šβˆ’1ξ€Έβ€–β€–0≀1√2𝐾𝑁(𝑀,𝑓)π‘βˆ’1𝑖=0πœƒπ‘–.𝑖!(3.19) By Lemma 2.1, (iii), and the assumption (𝐻3), we deduce from (3.16) that ||Ψ𝑖𝑠,π‘’π‘šβˆ’1𝑒𝑖(𝑠),𝑀𝑗||=|||||π‘βˆ’1𝑙=𝑖1𝑖!(π‘™βˆ’π‘–)!(βˆ’1)π‘™βˆ’π‘–ξ«π·π‘™3π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έπ‘’π‘™βˆ’π‘–π‘šβˆ’1𝑒𝑖(𝑠),𝑀𝑗|||||=|||||π‘βˆ’1𝑙=𝑖1𝑖!(π‘™βˆ’π‘–)!(βˆ’1)π‘™βˆ’π‘–ξƒ‘ξ‚€βˆšπ‘Ÿξ‚βˆ’π‘™π·π‘™3π‘“ξ€·π‘Ÿ,𝑠,π‘’π‘šβˆ’1ξ€Έξ‚€βˆšπ‘Ÿξ‚π‘™βˆ’π‘–π‘’π‘™βˆ’π‘–π‘šβˆ’1ξ‚€βˆšπ‘Ÿξ‚π‘–π‘’π‘–(𝑠),𝑀𝑗|||||β‰€π‘βˆ’1𝑙=𝑖1𝐾𝑖!(π‘™βˆ’π‘–)!βˆ—π‘™(𝑀,𝑓)πœƒπ‘™βˆ’π‘–πΎπ‘–2‖𝑒(𝑠)‖𝑖1ξƒ‘ξ‚€βˆšπ‘Ÿξ‚π‘™βˆ’π‘–,||𝑀𝑗||ξƒ’β‰€π‘βˆ’1𝑙=𝑖1𝐾𝑖!(π‘™βˆ’π‘–)!βˆ—π‘™(𝑀,𝑓)πœƒπ‘™βˆ’π‘–πΎπ‘–2ξƒ©ξƒŽπœ†π‘˜πΆ0𝜌ξƒͺ𝑖1βˆšβ‰€2+π‘™βˆ’π‘–π‘βˆ’1𝑙=𝑖11𝑖!(π‘™βˆ’π‘–)!√𝐾2+π‘™βˆ’π‘–βˆ—π‘™(𝑀,𝑓)πœƒπ‘™βˆ’π‘–πΎπ‘–2ξƒ©ξƒŽπœ†π‘˜πΆ0𝜌ξƒͺ𝑖,1β‰€π‘–β‰€π‘βˆ’1.(3.20)
It follows that||ℱ𝑗[𝑐](||≀||π‘žπ‘‘)𝑗(||𝑑)+πœ†π‘˜π‘‘0ξ€·1+πœ†π‘˜2π›ΌπœŒ2π›Όξ€Έξ€œπ‘‘0ξ€œπ‘‘πœπœ0||𝑐𝑗(||+1𝑠)𝑑𝑠2𝑑2π‘βˆ’1𝑖=1π‘βˆ’1𝑙=𝑖11𝑖!(π‘™βˆ’π‘–)!√𝐾2+π‘™βˆ’π‘–βˆ—π‘™(𝑀,𝑓)πœƒπ‘™βˆ’π‘–πΎπ‘–2ξƒ©ξƒŽπœ†π‘˜πΆ0𝜌ξƒͺ𝑖.(3.21) Thus ||β„±[𝑐](||𝑑)1≀||||π‘ž(𝑑)1+πœ†π‘˜π‘‘0ξ€·1+πœ†π‘˜2π›ΌπœŒ2π›Όξ€Έξ€œπ‘‘0ξ€œπ‘‘πœπœ0||||𝑐(𝑠)1+1𝑑𝑠2π‘˜π‘‘2π‘βˆ’1𝑖=1π‘βˆ’1𝑙=𝑖11𝑖!(π‘™βˆ’π‘–)!√𝐾2+π‘™βˆ’π‘–βˆ—π‘™(𝑀,𝑓)πœƒπ‘™βˆ’π‘–πΎπ‘–2ξƒ©ξƒŽπœ†π‘˜πΆ0𝜌ξƒͺπ‘–β‰€β€–π‘žβ€–π‘‡+12π·πœŒξ‚€π‘‡π‘š(π‘˜)2,(3.22) where β€–π‘žβ€–π‘‡=sup0≀𝑑≀𝑇|π‘ž(𝑑)|1 and 𝐷𝜌=𝐷𝜌(𝑓,𝜌,π‘˜,𝑀,π‘š,𝑁)=πœ†π‘˜π‘‘0ξ€·1+πœ†π‘˜2π›ΌπœŒ2π›Όξ€ΈπœŒ+π‘˜π‘βˆ’1𝑖=1π‘βˆ’1𝑙=𝑖11𝑖!(π‘™βˆ’π‘–)!√𝐾2+π‘™βˆ’π‘–βˆ—π‘™(𝑀,𝑓)πœƒπ‘™βˆ’π‘–πΎπ‘–2ξƒ©ξƒŽπœ†π‘˜πΆ0𝜌ξƒͺ𝑖.(3.23)
Hence, we obtain[𝑐]β€–β€–β„±π‘‹β‰€β€–π‘žβ€–π‘‡+12π·πœŒξ‚€π‘‡π‘š(π‘˜)2,(3.24)
choosing 𝜌>β€–π‘žβ€–π‘‡ andπ‘‡π‘š(π‘˜)∈(0,𝑇], such that12π·πœŒξ‚€π‘‡π‘š(π‘˜)2β‰€πœŒβˆ’β€–π‘žβ€–π‘‡,12ξ‚π·πœŒξ‚€π‘‡π‘š(π‘˜)2<1,(3.25) where ξ‚π·πœŒ=ξ‚π·πœŒ(𝜌,π‘˜,𝑀,𝑇,π‘š,𝑁,𝑓)≑𝑑0πœ†π‘˜ξ€·1+πœ†π‘˜2π›ΌπœŒ2𝛼+2𝑑1πœ†2π‘˜πœŒ2ξ€·1+πœ†π‘˜π›Όβˆ’1𝜌2π›Όβˆ’2ξ€Έ+π‘˜π‘βˆ’1𝑖=1π‘–ξƒ©ξƒŽπœ†π‘˜πΆ0𝐾2ξƒͺπ‘–πœŒπ‘–βˆ’1π‘βˆ’1𝑙=𝑖11𝑖!(π‘™βˆ’π‘–)!√𝐾2+π‘™βˆ’π‘–βˆ—π‘™(𝑀,𝑓)πœƒπ‘™βˆ’π‘–.(3.26) Then [𝑐]β€–β€–β„±π‘‹β‰€β€–π‘žβ€–π‘‡+12π·πœŒξ‚€π‘‡π‘š(π‘˜)2β‰€πœŒ,βˆ€π‘βˆˆπ‘†,(3.27) which means that β„± maps 𝑆 into itself.

Proof (ii). We now prove that, for all 𝑐,π‘‘βˆˆπ‘†, for all π‘‘βˆˆ[0,π‘‡π‘š(π‘˜)], ||β„±[𝑐][𝑑]||(𝑑)βˆ’β„±(𝑑)1≀12ξ‚π·πœŒπ‘‘2β€–π‘βˆ’π‘‘β€–π‘‹,βˆ€π‘›βˆˆβ„•,(3.28) where ξ‚π·πœŒ is defined as (3.26).
Proof of (3.28) is as follows.
For all 𝑗=1,2,…,π‘˜, for all π‘‘βˆˆ[0,π‘‡π‘š(π‘˜)], we have||ℱ𝑗[𝑐](𝑑)βˆ’β„±π‘—[𝑑](||𝑑)β‰€πœ†π‘—ξ€œπ‘‘0ξ€œπ‘‘πœπœ0||̃𝑏[𝑐](𝑐𝑠)𝑗(𝑠)βˆ’π‘‘π‘—(ξ€Έ||𝑠)𝑑𝑠+πœ†π‘—ξ€œπ‘‘0ξ€œπ‘‘πœπœ0||̃𝑏[𝑐]̃𝑏[𝑑]𝑑(𝑠)βˆ’(𝑠)𝑗||+(𝑠)π‘‘π‘ π‘βˆ’1𝑖=1ξ€œπ‘‘0ξ€œπ‘‘πœπœ0||Ψ𝑖𝑠,π‘’π‘šβˆ’1𝑒𝑖(𝑠)βˆ’π‘£π‘–ξ€Έ(𝑠),𝑀𝑗||𝑑𝑠,(3.29) where ̃𝑏[𝑐]ξ€·β€–(𝑑)=π΅β€–βˆ‡π‘’(𝑑)20ξ€Έ,̃𝑏[𝑑]ξ€·β€–(𝑑)=π΅β€–βˆ‡π‘£(𝑑)20ξ€Έ,𝑒(𝑑)=π‘˜ξ“π‘—=1𝑐𝑗(𝑑)𝑀𝑗,𝑣(𝑑)=π‘˜ξ“π‘—=1𝑑𝑗(𝑑)𝑀𝑗,(3.30) so ||β„±[𝑐]([𝑑](||𝑑)βˆ’β„±π‘‘)1β‰€πœ†π‘˜ξ€œπ‘‘0ξ€œπ‘‘πœπœ0̃𝑏[𝑐](||||𝑠)𝑐(𝑠)βˆ’π‘‘(𝑠)1𝑑𝑠+πœ†π‘˜ξ€œπ‘‘0ξ€œπ‘‘πœπœ0||̃𝑏[𝑐]̃𝑏[𝑑]||||||(𝑠)βˆ’(𝑠)𝑑(𝑠)1+π‘‘π‘ π‘˜ξ“π‘—=1π‘βˆ’1𝑖=1ξ€œπ‘‘0ξ€œπ‘‘πœπœ0||Ψ𝑖𝑠,π‘’π‘šβˆ’1𝑒𝑖(𝑠)βˆ’π‘£π‘–ξ€Έ(𝑠),𝑀𝑗||π‘‘π‘ β‰€πœ†π‘˜ξ€œπ‘‘0ξ€œπ‘‘πœπœ0̃𝑏[𝑐]||||(𝑠)𝑐(𝑠)βˆ’π‘‘(𝑠)1𝑑𝑠+πœ†π‘˜πœŒξ€œπ‘‘0ξ€œπ‘‘πœπœ0||̃𝑏[𝑐]̃𝑏[𝑑]||+(𝑠)βˆ’(𝑠)π‘‘π‘ π‘˜ξ“π‘—=1π‘βˆ’1𝑖=1ξ€œπ‘‘0ξ€œπ‘‘πœπœ0||Ψ𝑖𝑠,π‘’π‘šβˆ’1𝑒𝑖(𝑠)βˆ’π‘£π‘–ξ€Έ(𝑠),𝑀𝑗||𝑑𝑠≑𝐽1+𝐽2+𝐽3,(3.31) in which 𝐽1=πœ†π‘˜ξ€œπ‘‘0ξ€œπ‘‘πœπœ0̃𝑏[𝑐](||||𝑠)𝑐(𝑠)βˆ’π‘‘(𝑠)1π‘‘π‘ β‰€πœ†π‘˜π‘‘0ξ€·1+πœ†π‘˜2π›ΌπœŒ2π›Όξ€Έξ€œπ‘‘0ξ€œπ‘‘πœπœ0||||𝑐(𝑠)βˆ’π‘‘(𝑠)1π‘‘π‘ β‰‘πœ1ξ€œπ‘‘0ξ€œπ‘‘πœπœ0||||𝑐(𝑠)βˆ’π‘‘(𝑠)1𝑑𝑠.(3.32) In order to consider 𝐽2, we also note that ̃𝑏[𝑐]̃𝑏[𝑑](𝑠)βˆ’(𝑠)=π΅ξ…žξ€·β€–(πœ‰)β€–βˆ‡π‘’(𝑠)20β€–βˆ’β€–βˆ‡π‘£(𝑠)20ξ€Έ,(3.33) where β€–πœ‰=πœƒβ€–βˆ‡π‘’(𝑠)20+β€–(1βˆ’πœƒ)β€–βˆ‡π‘£(𝑠)20,0β‰€πœ‰β‰€πœ†π‘˜πœŒ2,0<πœƒ<1,(3.34) and π΅ξ…ž(πœ‰) satisfy the following inequality: ||π΅ξ…ž||(πœ‰)≀𝑑1ξ€·1+πœ‰π›Όβˆ’1≀𝑑1ξ€·1+πœ†π‘˜π›Όβˆ’1𝜌2π›Όβˆ’2ξ€Έ.(3.35)
It implies that||̃𝑏[𝑐]̃𝑏[𝑑]||=||𝐡(𝑠)βˆ’(𝑠)ξ…žξ€·β€–(πœ‰)β€–βˆ‡π‘’(𝑠)20β€–βˆ’β€–βˆ‡π‘£(𝑠)20ξ€Έ||≀𝑑1ξ€·1+πœ†π‘˜π›Όβˆ’1𝜌2π›Όβˆ’2ξ€Έ2πœ†π‘˜πœŒ||||𝑐(𝑠)βˆ’π‘‘(𝑠)1=2πœ†π‘˜πœŒπ‘‘1ξ€·1+πœ†π‘˜π›Όβˆ’1𝜌2π›Όβˆ’2ξ€Έ||||𝑐(𝑠)βˆ’π‘‘(𝑠)1,(3.36) and then 𝐽2=πœ†π‘˜πœŒξ€œπ‘‘0ξ€œπ‘‘πœπœ0||̃𝑏[𝑐](̃𝑏[𝑑](||𝑠)βˆ’π‘ )𝑑𝑠≀2πœ†2π‘˜πœŒ2𝑑1ξ€·1+πœ†π‘˜π›Όβˆ’1𝜌2π›Όβˆ’2ξ€Έξ€œπ‘‘0ξ€œπ‘‘πœπœ0||||𝑐(𝑠)βˆ’π‘‘(𝑠)1π‘‘π‘ β‰‘πœ2ξ€œπ‘‘0ξ€œπ‘‘πœπœ0||||𝑐(𝑠)βˆ’π‘‘(𝑠)1𝑑𝑠.(3.37)
It remains to estimate 𝐽3. Byξ‚€βˆšξ‚π‘Ÿπ‘’(𝑠)π‘–βˆ’ξ‚€βˆšξ‚π‘Ÿπ‘£(𝑠)𝑖=π‘–βˆ’1𝑗=0ξ‚€βˆšξ‚π‘Ÿπ‘’(𝑠)π‘—ξ‚€βˆšξ‚π‘Ÿπ‘£(𝑠)π‘–βˆ’π‘—βˆ’1βˆšπ‘Ÿ(𝑒(𝑠)βˆ’π‘£(𝑠)),(3.38) we obtain |||ξ‚€βˆšξ‚π‘Ÿπ‘’(𝑠)π‘–βˆ’ξ‚€βˆšξ‚π‘Ÿπ‘£(𝑠)𝑖|||≀𝐾𝑖2π‘–βˆ’1𝑗=0‖𝑒(𝑠)‖𝑗1‖𝑣(𝑠)β€–1π‘–βˆ’π‘—βˆ’1‖𝑒(𝑠)βˆ’π‘£(𝑠)β€–1≀𝐾𝑖2π‘–βˆ’1𝑗=0‖‖𝑒(𝑠)𝑗1‖‖𝑣(𝑠)1π‘–βˆ’π‘—βˆ’1‖‖𝑒(𝑠)βˆ’π‘£(𝑠)1≀𝐾𝑖2π‘–βˆ’1𝑗=0ξƒ©ξƒŽπœ†π‘˜πΆ0𝜌ξƒͺπ‘—ξƒ©ξƒŽπœ†π‘˜πΆ0𝜌ξƒͺπ‘–βˆ’π‘—βˆ’1ξƒŽπœ†π‘˜πΆ0||||𝑐(𝑠)βˆ’π‘‘(𝑠)1=𝐾𝑖2π‘–βˆ’1𝑗=0ξƒ©ξƒŽπœ†π‘˜πΆ0ξƒͺπ‘–πœŒπ‘–βˆ’1||||𝑐(𝑠)βˆ’π‘‘(𝑠)1ξƒ©ξƒŽ=π‘–πœ†π‘˜πΆ0𝐾2ξƒͺπ‘–πœŒπ‘–βˆ’1||||𝑐(𝑠)βˆ’π‘‘(𝑠)1.(3.39) On the other hand, Ξ¨π‘–ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έ=π‘βˆ’1𝑙=𝑖1𝑖!(π‘™βˆ’π‘–)!(βˆ’1)π‘™βˆ’π‘–π·π‘™3π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έπ‘’π‘™βˆ’π‘–π‘šβˆ’1,ξ‚€βˆšπ‘Ÿξ‚βˆ’π‘–||Ξ¨π‘–ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έ||=|||||π‘βˆ’1𝑙=𝑖1(𝑖!(π‘™βˆ’π‘–)!βˆ’1)π‘™βˆ’π‘–ξ‚€βˆšπ‘Ÿξ‚βˆ’π‘™π·π‘™3π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έξ‚€βˆšπ‘Ÿξ‚π‘™βˆ’π‘–π‘’π‘™βˆ’π‘–π‘šβˆ’1|||||≀|||||π‘βˆ’1𝑙=𝑖1ξ‚€βˆšπ‘–!(π‘™βˆ’π‘–)!π‘Ÿξ‚βˆ’π‘™||𝐷𝑙3π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έ||ξ‚€βˆšπ‘Ÿξ‚π‘™βˆ’π‘–||π‘’π‘™βˆ’π‘–π‘šβˆ’1|||||||β‰€π‘βˆ’1𝑙=𝑖1𝐾𝑖!(π‘™βˆ’π‘–)!βˆ—π‘™(𝑀,𝑓)πœƒπ‘™βˆ’π‘–ξ‚€βˆšπ‘Ÿξ‚π‘™βˆ’π‘–.(3.40)
Hence, we deduce from (3.39), (3.40) that𝐽3=π‘˜ξ“π‘—=1π‘βˆ’1𝑖=1ξ€œπ‘‘0ξ€œπ‘‘πœπœ0||||ξƒ‘ξ‚€βˆšπ‘Ÿξ‚βˆ’π‘–Ξ¨π‘–ξ€·π‘ ,π‘’π‘šβˆ’1ξ€Έξ‚΅ξ‚€βˆšξ‚π‘Ÿπ‘’(𝑠)π‘–βˆ’ξ‚€βˆšξ‚π‘Ÿπ‘£(𝑠)𝑖,𝑀𝑗||||β‰€π‘‘π‘ π‘˜ξ“π‘—=1π‘βˆ’1𝑖=1π‘βˆ’1𝑙=𝑖1𝐾𝑖!(π‘™βˆ’π‘–)!βˆ—π‘™(𝑀,𝑓)πœƒπ‘™βˆ’π‘–π‘–ξƒ©ξƒŽπœ†π‘˜πΆ0𝐾2ξƒͺπ‘–πœŒπ‘–βˆ’1Γ—ξ€œπ‘‘0ξ€œπ‘‘πœπœ0||||ξƒ‘ξ‚€βˆšπ‘Ÿξ‚π‘™βˆ’π‘–,||𝑀𝑗||ξƒ’||||||||𝑐(𝑠)βˆ’π‘‘(𝑠)1=π‘‘π‘ π‘˜ξ“π‘—=1π‘βˆ’1𝑖=1π‘βˆ’1𝑙=𝑖1𝐾𝑖!(π‘™βˆ’π‘–)!βˆ—π‘™(𝑀,𝑓)πœƒπ‘™βˆ’π‘–π‘–ξƒ©ξƒŽπœ†π‘˜πΆ0𝐾2ξƒͺπ‘–πœŒπ‘–βˆ’11βˆšΓ—ξ€œ2+π‘™βˆ’π‘–π‘‘0ξ€œπ‘‘πœπœ0||||𝑐(𝑠)βˆ’π‘‘(𝑠)1𝑑𝑠=π‘˜π‘βˆ’1𝑖=1π‘–ξƒ©ξƒŽπœ†π‘˜πΆ0𝐾2ξƒͺπ‘–πœŒπ‘–βˆ’1π‘βˆ’1𝑙=𝑖1𝐾𝑖!(π‘™βˆ’π‘–)!βˆ—π‘™(𝑀,𝑓)πœƒπ‘™βˆ’π‘–1βˆšΓ—ξ€œ2+π‘™βˆ’π‘–π‘‘0ξ€œπ‘‘πœπœ0||||𝑐(𝑠)βˆ’π‘‘(𝑠)1π‘‘π‘ β‰‘πœ3ξ€œπ‘‘0ξ€œπ‘‘πœπœ0||||𝑐(𝑠)βˆ’π‘‘(𝑠)1𝑑𝑠.(3.41)
We deduce that||β„±[𝑐]([𝑑](||𝑑)βˆ’β„±π‘‘)1β‰€ξ€·πœ1+𝜁2+𝜁3ξ€Έξ€œπ‘‘0ξ€œπ‘‘πœπœ0||||𝑐(𝑠)βˆ’π‘‘(𝑠)11𝑑𝑠≀2ξ‚π·πœŒπ‘‘2β€–π‘βˆ’π‘‘β€–π‘‹.(3.42) We note that 𝜁1+𝜁2+𝜁3=ξ‚π·πœŒξ‚π·(𝜌,π‘˜,𝑀,𝑇,π‘š,𝑁,𝑓)=𝜌.(3.43)
It follows from (3.28) that[𝑐][𝑑]β€–β€–β„±βˆ’β„±π‘‹β‰€12ξ‚π·πœŒξ‚€π‘‡π‘š(π‘˜)2β€–π‘βˆ’π‘‘β€–π‘‹,βˆ€π‘,π‘‘βˆˆπ‘†.(3.44)
By (3.25), it follows that β„±βˆΆπ‘†β†’π‘† is contractive. We deduce that β„± has a unique fixed point in 𝑆; that is, the system (3.8)–(3.11) has a unique solution π‘’π‘š(π‘˜)(𝑑) on an interval [0,π‘‡π‘š(π‘˜)]. The proof of Lemma 3.2 is complete.

The following estimates allow one to take constant π‘‡π‘š(π‘˜)=𝑇 for all π‘š and π‘˜.
Step 2. A priori estimates. Put π‘†π‘š(π‘˜)(𝑑)=π‘‹π‘š(π‘˜)(𝑑)+π‘Œπ‘š(π‘˜)(ξ€œπ‘‘)+𝑑0β€–β€–Μˆπ‘’π‘š(π‘˜)(‖‖𝑠)20𝑑𝑠,(3.45) where π‘‹π‘š(π‘˜)β€–β€–(𝑑)=Μ‡π‘’π‘š(π‘˜)β€–β€–(𝑑)20+π‘π‘š(π‘˜)𝑒(𝑑)π‘Žπ‘š(π‘˜)(𝑑),π‘’π‘š(π‘˜),π‘Œ(𝑑)π‘š(π‘˜)ξ‚€(𝑑)=π‘ŽΜ‡π‘’π‘š(π‘˜)(𝑑),Μ‡π‘’π‘š(π‘˜)(𝑑)+π‘π‘š(π‘˜)β€–β€–(𝑑)π΄π‘’π‘š(π‘˜)β€–β€–(𝑑)20,(3.46) with 𝐴 is defined by (2.2). Then it follows that π‘†π‘š(π‘˜)(𝑑)=π‘†π‘š(π‘˜)(ξ€œ0)+𝑑0Μ‡π‘π‘š(π‘˜)(ξ‚ƒπ‘Žξ‚€π‘’π‘ )π‘š(π‘˜)(𝑠),π‘’π‘š(π‘˜)(+‖‖𝑠)π΄π‘’π‘š(π‘˜)(‖‖𝑠)20ξ‚„ξ€œπ‘‘π‘ +2𝑑0ξ‚¬πΉπ‘š(π‘˜)(𝑠),Μ‡π‘’π‘š(π‘˜)ξ‚­ξ€œ(𝑠)𝑑𝑠+2𝑑0π‘Žξ‚€πΉπ‘š(π‘˜)(𝑠),Μ‡π‘’π‘š(π‘˜)+ξ€œ(𝑠)𝑑𝑠𝑑0ξ‚¬πΉπ‘š(π‘˜)(𝑠),Μˆπ‘’π‘š(π‘˜)ξ‚­ξ€œ(𝑠)π‘‘π‘ βˆ’π‘‘0π‘π‘š(π‘˜)(𝑠)π΄π‘’π‘š(π‘˜)(𝑠),Μˆπ‘’π‘š(π‘˜)ξ‚­(𝑠)π‘‘π‘ β‰‘π‘†π‘š(π‘˜)(0)+5𝑗=1𝐼𝑗.(3.47)
We will now require the following lemma.

Lemma 3.3. We have (i)0<π‘βˆ—β‰€π‘π‘š(π‘˜)(𝑑)≀𝑑0ξ‚€β€–β€–1+βˆ‡π‘’π‘š(π‘˜)β€–β€–(𝑑)02𝛼,||̇𝑏(ii)π‘š(π‘˜)||≀(𝑑)2𝑑1βˆšπ‘βˆ—ξ‚ƒπ‘†π‘š(π‘˜)(𝑑)+π‘βˆ—1βˆ’π›Όξ‚€π‘†π‘š(π‘˜)(𝑑)𝛼,‖‖𝐹(iii)π‘š(π‘˜)β€–β€–(𝑑)0β‰€π‘βˆ’1𝑗=0Μƒπ‘Žπ‘—(0)ξ‚΅ξ”π‘†π‘š(π‘˜)ξ‚Ά(𝑑)𝑗,(β€–β€–β€–πœ•iv)πΉπœ•π‘Ÿπ‘š(π‘˜)(‖‖‖𝑑)0β‰€π‘βˆ’1𝑗=0Μƒπ‘Žπ‘—(1)ξ‚΅ξ”π‘†π‘š(π‘˜)(𝑑)𝑗,(3.48) where Μƒπ‘Žπ‘—(0),Μƒπ‘Žπ‘—(1),𝑗=0,1,…,π‘βˆ’1 are defined as follows: Μƒπ‘Žπ‘—(0)=⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽ©Μƒπ‘Ž0(0)=1√2𝐾𝑁(𝑀,𝑓)π‘βˆ’1𝑖=0πœƒπ‘–π‘–!,𝑗=0,Μƒπ‘Žπ‘—(0)=𝐾𝑗2ξ‚€βˆšπ‘βˆ—πΆ0ξ‚π‘—π‘βˆ’1𝑖=π‘—πœƒπ‘–βˆ’π‘—1𝑗!(π‘–βˆ’π‘—)!√𝐾2+π‘–βˆ’π‘—βˆ—π‘–(𝑀,𝑓),1β‰€π‘—β‰€π‘βˆ’1,Μƒπ‘Žπ‘—(1)=⎧βŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽ©ξƒ©1√2ξƒͺ+2𝑀𝐾𝑁(𝑀,𝑓)π‘βˆ’1𝑖=0πœƒπ‘–πΎπ‘–!,𝑗=0,𝑗2ξ‚€βˆšπ‘βˆ—πΆ0ξ‚π‘—π‘βˆ’1𝑖=π‘—πœƒπ‘–βˆ’π‘—ξƒ¬π‘—!(π‘–βˆ’π‘—)!𝑗𝐾2βˆ’1ξπΎβˆ—π‘–+1(𝑀,𝑓)√ξƒͺξπΎπ‘–βˆ’π‘—+3+2π‘€βˆ—π‘–+1ξƒ­ξƒŽ(𝑀,𝑓),1β‰€π‘—β‰€π‘βˆ’1,πœƒ=𝑀12+√2.(3.49)
Proof of Lemma 3.3. Proof (i), (ii). Note thatπ‘†π‘š(π‘˜)(𝑑)β‰₯π‘‹π‘š(π‘˜)(𝑑)β‰₯π‘βˆ—π‘Žξ‚€π‘’π‘š(π‘˜)(𝑑),π‘’π‘š(π‘˜)(𝑑)β‰₯π‘βˆ—β€–β€–βˆ‡π‘’π‘š(π‘˜)β€–β€–(𝑑)20,π‘†π‘š(π‘˜)(𝑑)β‰₯π‘Œπ‘š(π‘˜)ξ‚€(𝑑)β‰₯π‘ŽΜ‡π‘’π‘š(π‘˜)(𝑑),Μ‡π‘’π‘š(π‘˜)β‰₯β€–β€–(𝑑)βˆ‡Μ‡π‘’π‘š(π‘˜)β€–β€–(𝑑)20.(3.50) We deduce that π‘π‘š(π‘˜)ξ‚€β€–β€–(𝑑)=π΅βˆ‡π‘’π‘š(π‘˜)β€–β€–(𝑑)20≀𝑑0ξ‚€β€–β€–1+βˆ‡π‘’π‘š(π‘˜)β€–β€–(𝑑)02𝛼,||Μ‡π‘π‘š(π‘˜)|||||𝐡(𝑑)=2ξ…žξ‚€β€–β€–βˆ‡π‘’π‘š(π‘˜)β€–β€–(𝑑)20||||||ξ‚¬βˆ‡π‘’π‘š(π‘˜)(𝑑),βˆ‡Μ‡π‘’π‘š(π‘˜)ξ‚­|||(𝑑)≀2𝑑1ξ‚€β€–β€–1+βˆ‡π‘’π‘š(π‘˜)β€–β€–(𝑑)02π›Όβˆ’2ξ‚β€–β€–βˆ‡π‘’π‘š(π‘˜)β€–β€–(𝑑)0β€–β€–βˆ‡Μ‡π‘’π‘š(π‘˜)β€–β€–(𝑑)0≀2𝑑1ξ‚΅1+π‘βˆ—1βˆ’π›Όξ‚€π‘†π‘š(π‘˜)(𝑑)π›Όβˆ’1ξ‚Ά1βˆšπ‘βˆ—π‘†π‘š(π‘˜)=(𝑑)2𝑑1βˆšπ‘βˆ—ξ‚ƒπ‘†π‘š(π‘˜)(𝑑)+π‘βˆ—1βˆ’π›Όξ‚€π‘†π‘š(π‘˜)(𝑑)𝛼.(3.51)
Proof (iii). We have β€–β€–πΉπ‘š(π‘˜)β€–β€–(𝑑)0≀‖‖Ψ0ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έβ€–β€–0+π‘βˆ’1𝑗=1β€–β€–β€–Ξ¨π‘—ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έξ‚€π‘’π‘š(π‘˜)𝑗‖‖‖0.(3.52) By (3.18)3, we have β€–β€–Ξ¨0ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έβ€–β€–0≀1√2𝐾𝑁(𝑀,𝑓)π‘βˆ’1𝑖=0πœƒπ‘–π‘–!β‰‘Μƒπ‘Ž0(0).(3.53) On the other hand, it follows from (3.49) and π‘’π‘šβˆ’1βˆˆπ‘Š1(𝑀,𝑇) that β€–β€–β€–Ξ¨π‘—ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έξ‚€π‘’π‘š(π‘˜)𝑗‖‖‖0β‰€π‘βˆ’1𝑖=𝑗1‖‖‖𝐷𝑗!(π‘–βˆ’π‘—)!𝑖3π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έπ‘’π‘–βˆ’π‘—π‘šβˆ’1ξ‚€π‘’π‘š(π‘˜)𝑗‖‖‖0=π‘βˆ’1𝑖=𝑗1β€–β€–β€–ξ‚€βˆšπ‘—!(π‘–βˆ’π‘—)!π‘Ÿξ‚π‘–βˆ’π‘—ξ‚€βˆšπ‘Ÿξ‚βˆ’π‘–π·π‘–3π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έπ‘’π‘–βˆ’π‘—π‘šβˆ’1ξ‚€βˆšπ‘Ÿξ‚π‘—ξ‚€π‘’π‘š(π‘˜)𝑗‖‖‖0β‰€π‘βˆ’1𝑖=𝑗1β€–β€–β€–ξ‚€βˆšπ‘—!(π‘–βˆ’π‘—)!π‘Ÿξ‚π‘–βˆ’π‘—β€–β€–β€–0ξπΎβˆ—π‘–(𝑀,𝑓)πœƒπ‘–βˆ’π‘—πΎπ‘—2β€–β€–π‘’π‘š(π‘˜)β€–β€–(𝑑)𝑗1=π‘βˆ’1𝑖=𝑗11𝑗!(π‘–βˆ’π‘—)!√𝐾2+π‘–βˆ’π‘—βˆ—π‘–(𝑀,𝑓)πœƒπ‘–βˆ’π‘—πΎπ‘—2β€–β€–π‘’π‘š(π‘˜)β€–β€–(𝑑)𝑗1≀𝐾𝑗2ξ‚€βˆšπ‘βˆ—πΆ0ξ‚π‘—π‘βˆ’1𝑖=π‘—πœƒπ‘–βˆ’π‘—1𝑗!(π‘–βˆ’π‘—)!√𝐾2+π‘–βˆ’π‘—βˆ—π‘–ξ‚΅ξ”(𝑀,𝑓)π‘†π‘š(π‘˜)ξ‚Ά(𝑑)π‘—β‰‘Μƒπ‘Žπ‘—(0)ξ‚΅ξ”π‘†π‘š(π‘˜)ξ‚Ά(𝑑)𝑗.(3.54)
It follows from (3.52)–(3.54) that β€–β€–πΉπ‘š(π‘˜)β€–β€–(𝑑)0β‰€π‘βˆ’1𝑗=0Μƒπ‘Žπ‘—(0)ξ‚΅ξ”π‘†π‘š(π‘˜)ξ‚Ά(𝑑)𝑗,(3.55) where Μƒπ‘Žπ‘—(0),0β‰€π‘—β‰€π‘βˆ’1 are defined by (3.49)1.

Proof (iv). We have πœ•πΉπœ•π‘Ÿπ‘š(π‘˜)πœ•(π‘Ÿ,𝑑)=Ξ¨πœ•π‘Ÿ0ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έ+π‘βˆ’1𝑗=1π‘—Ξ¨π‘—ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έξ‚€π‘’π‘š(π‘˜)ξ‚π‘—βˆ’1βˆ‡π‘’π‘š(π‘˜)+π‘βˆ’1𝑗=1ξ‚€πœ•Ξ¨πœ•π‘Ÿπ‘—ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έπ‘’ξ‚ξ‚€π‘š(π‘˜)𝑗.(3.56)
Henceβ€–β€–β€–πœ•πΉπœ•π‘Ÿπ‘š(π‘˜)β€–β€–β€–(𝑑)0β‰€β€–β€–β€–πœ•Ξ¨πœ•π‘Ÿ0ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έβ€–β€–β€–0+π‘βˆ’1𝑗=1π‘—β€–β€–β€–Ξ¨π‘—ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έξ‚€π‘’π‘š(π‘˜)ξ‚π‘—βˆ’1βˆ‡π‘’π‘š(π‘˜)β€–β€–β€–0+π‘βˆ’1𝑗=1β€–β€–β€–ξ‚€πœ•Ξ¨πœ•π‘Ÿπ‘—ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έπ‘’ξ‚ξ‚€π‘š(π‘˜)𝑗‖‖‖0≑𝐿1+𝐿2+𝐿3.(3.57) We shall estimate step by step the terms on the right-hand side of (3.57) as follows.(iv.1) Estimating 𝐿1=β€–(πœ•/πœ•π‘Ÿ)Ξ¨0(π‘Ÿ,𝑑,π‘’π‘šβˆ’1)β€–0. We haveπœ•Ξ¨πœ•π‘Ÿ0ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έ=π‘βˆ’1𝑖=11𝑖!(βˆ’1)𝑖𝐷𝑖3π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έπ‘–π‘’π‘–βˆ’1π‘šβˆ’1βˆ‡π‘’π‘šβˆ’1+π‘βˆ’1𝑖=01𝑖!(βˆ’1)𝑖𝐷1𝐷𝑖3π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έπ‘’π‘–π‘šβˆ’1+π‘βˆ’1𝑖=01𝑖!(βˆ’1)𝑖𝐷3𝑖+1π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έπ‘’π‘–π‘šβˆ’1βˆ‡π‘’π‘šβˆ’1=π‘Ž(1)+π‘Ž(2)+π‘Ž(3).(3.58) We will estimate step by step the terms π‘Ž(1),π‘Ž(2),π‘Ž(3) as follows.(iv.1.1) Estimating β€–π‘Ž(1)β€–0. We have||||=|||||π‘Ž(1)π‘βˆ’1𝑖=11𝑖!(βˆ’1)𝑖𝐷𝑖3π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έπ‘–π‘’π‘–βˆ’1π‘šβˆ’1βˆ‡π‘’π‘šβˆ’1|||||≀𝐾𝑁(𝑀,𝑓)π‘βˆ’1𝑖=11𝑖!π‘–πœƒπ‘–βˆ’1||βˆ‡π‘’π‘šβˆ’1||=𝐾𝑁(𝑀,𝑓)π‘βˆ’2𝑖=0πœƒπ‘–||𝑖!βˆ‡π‘’π‘šβˆ’1||≀𝐾𝑁(𝑀,𝑓)π‘βˆ’1𝑖=0πœƒπ‘–||𝑖!βˆ‡π‘’π‘šβˆ’1||.(3.59) Hence β€–π‘Ž(1)β€–0≀𝐾𝑁(𝑀,𝑓)π‘βˆ’1𝑖=0πœƒπ‘–β€–β€–π‘–!βˆ‡π‘’π‘šβˆ’1β€–β€–0≀𝑀𝐾𝑁(𝑀,𝑓)π‘βˆ’1𝑖=0πœƒπ‘–.𝑖!(3.60)(iv.1.2) Estimating β€–π‘Ž(2)β€–0. It follows from||||=|||||π‘Ž(2)π‘βˆ’1𝑖=01𝑖!(βˆ’1)𝑖𝐷1𝐷𝑖3π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έπ‘’π‘–π‘šβˆ’1|||||≀𝐾𝑁(𝑀,𝑓)π‘βˆ’1𝑖=0πœƒπ‘–π‘–!(3.61) that β€–π‘Ž(2)β€–0≀1√2𝐾𝑁(𝑀,𝑓)π‘βˆ’1𝑖=0πœƒπ‘–.𝑖!(3.62)(iv.1.3) Estimating β€–π‘Ž(3)β€–0. Similarly, with ||||=|||||π‘Ž(3)π‘βˆ’1𝑖=01𝑖!(βˆ’1)𝑖𝐷3𝑖+1π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έπ‘’π‘–π‘šβˆ’1βˆ‡π‘’π‘šβˆ’1|||||≀𝐾𝑁(𝑀,𝑓)π‘βˆ’1𝑖=0πœƒπ‘–||𝑖!βˆ‡π‘’π‘šβˆ’1||,(3.63) we obtainβ€–π‘Ž(3)β€–0≀𝐾𝑁(𝑀,𝑓)π‘βˆ’1𝑖=0πœƒπ‘–β€–β€–π‘–!βˆ‡π‘’π‘šβˆ’1β€–β€–0≀𝑀𝐾𝑁(𝑀,𝑓)π‘βˆ’1𝑖=0πœƒπ‘–.𝑖!(3.64)
It follows from (3.58), (3.60), (3.62), (3.64) that 𝐿1=β€–β€–β€–πœ•Ξ¨πœ•π‘Ÿ0ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έβ€–β€–β€–0β‰€β€–β€–π‘Ž(1)0+β€–β€–π‘Ž(2)0+β€–β€–π‘Ž(3)0≀1√2𝐾𝑁(𝑀,𝑓)π‘βˆ’1𝑖=0πœƒπ‘–π‘–!+2𝑀𝐾𝑁(𝑀,𝑓)π‘βˆ’1𝑖=0πœƒπ‘–β‰€ξƒ©1𝑖!√2ξƒͺ+2𝑀𝐾𝑁(𝑀,𝑓)π‘βˆ’1𝑖=0πœƒπ‘–.𝑖!(3.65)
(iv.2) Estimating 𝐿2=βˆ‘π‘βˆ’1𝑗=1𝑗‖Ψ𝑗(π‘Ÿ,𝑑,π‘’π‘šβˆ’1)(π‘’π‘š(π‘˜))π‘—βˆ’1βˆ‡π‘’π‘š(π‘˜)β€–0. By the assumption (𝐻3), we deduce that𝐿2=π‘βˆ’1𝑗=1π‘—β€–β€–β€–Ξ¨π‘—ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έξ‚€π‘’π‘š(π‘˜)ξ‚π‘—βˆ’1βˆ‡π‘’π‘š(π‘˜)β€–β€–β€–0β‰€π‘βˆ’1𝑗=1π‘—π‘βˆ’1𝑖=𝑗1‖‖‖𝐷𝑗!(π‘–βˆ’π‘—)!𝑖3π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1ξ€Έπ‘’π‘–βˆ’π‘—π‘šβˆ’1ξ‚€π‘’π‘š(π‘˜)ξ‚π‘—βˆ’1βˆ‡π‘’π‘š(π‘˜)β€–β€–β€–0=π‘βˆ’1𝑗=1π‘—π‘βˆ’1𝑖=𝑗1β€–β€–β€–ξ‚€βˆšπ‘—!(π‘–βˆ’π‘—)!π‘Ÿξ‚βˆ’π‘–π·π‘–3π‘“ξ€·π‘Ÿ,𝑑,π‘’π‘šβˆ’1