International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 519684 |

Abdelfatah Bouziani, Rachid Mechri, "The Rothe's Method to a Parabolic Integrodifferential Equation with a Nonclassical Boundary Conditions", International Journal of Stochastic Analysis, vol. 2010, Article ID 519684, 16 pages, 2010.

The Rothe's Method to a Parabolic Integrodifferential Equation with a Nonclassical Boundary Conditions

Academic Editor: Alexander M. Krasnosel'skii
Received26 Feb 2009
Revised17 Aug 2009
Accepted10 Dec 2009
Published25 Mar 2010


This paper is devoted to prove, in a nonclassical function space, the weak solvability of parabolic integrodifferential equations with a nonclassical boundary conditions. The investigation is made by means of approximation by the Rothes method which is based on a semidiscretization of the given problem with respect to the time variable.

1. Introduction

The purpose of this paper is to study the solvability of the following equation: πœ•π‘£(πœ•πœ•π‘‘π‘₯,𝑑)βˆ’2π‘£πœ•π‘₯2(ξ€œπ‘₯,𝑑)=𝑑0π‘Ž(π‘‘βˆ’π‘ )π‘˜ξ…ž([]𝑠,𝑣(π‘₯,𝑠))𝑑𝑠+𝑔(π‘₯,𝑑),(π‘₯,𝑑)∈(0,1)Γ—0,𝑇,(1.1) with the initial condition 𝑣(π‘₯,0)=𝑉0(π‘₯),π‘₯∈(0,1),(1.2) and the integral conditions ξ€œ10[],ξ€œπ‘£(π‘₯,𝑑)𝑑π‘₯=𝐸(𝑑),π‘‘βˆˆ0,𝑇10[],π‘₯𝑣(π‘₯,𝑑)𝑑π‘₯=𝐺(𝑑),π‘‘βˆˆ0,𝑇(1.3) where 𝑣 is an unknown function, 𝐸, 𝐺, and 𝑉0 are given functions supposed to be sufficiently regular, while π‘˜ξ…ž and π‘Ž are suitably defined functions satisfying certain conditions to be specified later and 𝑇 is a positive constant.

Since 1930, various classical types of initial boundary value problems have been investigated by many authors using Rothe time-discretization method; see, for instance, the monographs by Rektorys [1] and Kačur [2] and references cited therein. The linear case of our problem, that is, βˆ«π‘‘0π‘Ž(π‘‘βˆ’π‘ )π‘˜ξ…ž(𝑠,𝑣(π‘₯,𝑠))𝑑𝑠=0, appears, for instance, in the modelling of the quasistatic flexure of a thermoelastic rod (see [3]) and has been studied, firstly, by the second author with a more general second-order parabolic equation or a 2m-parabolic equation in [3–5] by means of the energy-integrals method and, secondly, by the two authors via the Rothe method [6–8]. For other models, we refer the reader, for instance, to [9–12], and references therein.

The paper is organized as follows. In Section 2, we transform problem (1.1)–(1.3) to an equivalent one with homogeneous integral conditions, namely, problem (2.3). Then, we specify notations and assumptions on data before stating the precise sense of the desired solution. In Section 3, by the Rothe discretization in time method, we construct approximate solutions to problem (2.3). Some a priori estimates for the approximations are derived in Section 4, while Section 5 is devoted to establish the existence and uniqueness of the solution.

2. Preliminaries, Notation, and Main Result

It is convenient at the beginning to reduce problem (1.1)–(1.3) with inhomogeneous integral conditions to an equivalent one with homogeneous conditions. For this, we introduce a new unknown function 𝑒 by setting

[]𝑒(π‘₯,𝑑)=𝑣(π‘₯,𝑑)βˆ’π‘…(π‘₯,𝑑),(π‘₯,𝑑)∈(0,1)Γ—0,𝑇,(2.1) where

𝑅(π‘₯,𝑑)=6(2𝐺(𝑑)βˆ’πΈ(𝑑))π‘₯βˆ’2(3𝐺(𝑑)βˆ’2𝐸(𝑑)).(2.2) Then, the function 𝑒 is seen to be the solution of the following problem:

πœ•π‘’(πœ•πœ•π‘‘π‘₯,𝑑)βˆ’2π‘’πœ•π‘₯2(ξ€œπ‘₯,𝑑)=𝑑0[],π‘Ž(π‘‘βˆ’π‘ )π‘˜(𝑠,𝑒(π‘₯,𝑠))𝑑𝑠+𝑓(π‘₯,𝑑),(π‘₯,𝑑)∈(0,1)Γ—0,𝑇𝑒(π‘₯,0)=π‘ˆ0ξ€œ(π‘₯),π‘₯∈(0,1),10[],ξ€œπ‘’(π‘₯,𝑑)𝑑π‘₯=0,π‘‘βˆˆ0,𝑇10[],π‘₯𝑒(π‘₯,𝑑)𝑑π‘₯=0,π‘‘βˆˆ0,𝑇(2.3) where 𝑓(π‘₯,𝑑)=𝑔(π‘₯,𝑑)βˆ’πœ•π‘…(π‘₯,𝑑),π‘ˆπœ•π‘‘0(π‘₯)=𝑉0(π‘₯)βˆ’π‘…(π‘₯,0),π‘˜(𝑠,𝑒(π‘₯,𝑠))=π‘˜ξ…ž(𝑠,𝑒(π‘₯,𝑠))βˆ’π‘…(π‘₯,𝑑).(2.4) Hence, instead of looking for the function 𝑣, we search for the function 𝑒. The solution of problem (1.1)–(1.3) will be simply given by the formula 𝑣(π‘₯,𝑑)=𝑒(π‘₯,𝑑)+𝑅(π‘₯,𝑑).

We introduce the function spaces, which we need in our investigation. Let 𝐿2(0,1) and 𝐿2(0,𝑇;𝐿2(0,1)) be the standard function spaces. We denote by 𝐢0(0,1) the linear space of continuous functions with compact support in (0,1). Since such functions are Lebesgue integrable, we can define on 𝐢0(0,1) the bilinear form given by ξ€œ((𝑒,𝑣))=10β„‘π‘₯𝑒ℑπ‘₯𝑣𝑑π‘₯,(2.5) where

β„‘π‘₯ξ€œπ‘’=π‘₯0𝑒(𝜁,β‹…)π‘‘πœ.(2.6) The bilinear form (2.5) is considered as a scalar product on 𝐢0(0,1) for which 𝐢0(0,1) is not complete.

Definition 2.1. We denote by 𝐡12(0,1) a completion of 𝐢0(0,1) for the scalar product (2.5), which is denoted by (β‹…,β‹…)𝐡12(0,1), called the Bouziani space or the space of square integrable primitive functions on (0,1). By the norm of function 𝑒 from 𝐡12(0,1), we understand the nonnegative number ‖𝑒‖𝐡12(0,1)=(𝑒,𝑒)𝐡12(0,1)=β€–β€–β„‘π‘₯𝑒‖‖,(2.7)where ‖𝑣‖ denotes the norm of 𝑣 in 𝐿2(0,1).

For π‘’βˆˆπΏ2(0,1), we have the elementary inequality ‖𝑒‖𝐡12(0,1)≀1√2‖𝑒‖.(2.8)

We denote by 𝐿2(0,𝑇;𝐡12(0,1)) the space of functions which are square integrable in the Bochner sense, with the scalar product


Since the space 𝐡12(0,1) is a Hilbert space, it can be shown that 𝐿2(0,𝑇;𝐡12(0,1)) is a Hilbert space as well. The set of all continuous abstract functions in [0,𝑇] equipped with the norm

sup0β‰€πœβ‰€π‘‡β€–π‘’(β‹…,𝜏)‖𝐡12(0,1)(2.10) is denoted 𝐢(0,𝑇;𝐡12(0,1)). Let 𝑉 be the set which we define as follows:


Since 𝑉 is the null space of the continuous linear mapping 𝑙: 𝐿2(0,1)→ℝ2, βˆ«πœ‘β†’π‘™(πœ‘)=(10βˆ«πœ‘(π‘₯)𝑑π‘₯,10π‘₯πœ‘(π‘₯)𝑑π‘₯), it is a closed linear subspace of 𝐿2(0,1), consequently 𝑉 is a Hilbert space endowed with the inner product (β‹…,β‹…). Strong or weak convergence is denoted by β†’ or ⇀, respectively. The letter 𝐢 will stand for a generic positive constant which may be different in the same discussion.

Lemma 2.2 (Gronwall's lemma). (a1) Let π‘₯(𝑑)β‰₯0, β„Ž(𝑑), 𝑦(𝑑) be real integrable functions on the interval [π‘Ž,𝑏]. If ξ€œπ‘¦(𝑑)β‰€β„Ž(𝑑)+π‘‘π‘Žπ‘₯(𝑠)𝑦(𝑠)𝑑𝑠,βˆ€π‘‘βˆˆ(π‘Ž,𝑏),(2.12) then ξ€œπ‘¦(𝑑)β‰€β„Ž(𝑑)+π‘‘π‘Žξ‚΅ξ€œβ„Ž(𝑠)π‘₯(𝑠)expπ‘‘π‘Žξ‚Άπ‘₯(𝜏)π‘‘πœπ‘‘π‘ ,βˆ€π‘‘βˆˆ(0,𝑇).(2.13)
In particular, if π‘₯(𝑑)≑𝐢 is a constant and β„Ž(𝑑) is nondecreasing, then 𝑦(𝑑)β‰€β„Ž(𝑑)𝑒𝑐(π‘‘βˆ’π‘Ž),βˆ€π‘‘βˆˆ(0,𝑇).(2.14)
(a2) Let {π‘Žπ‘–}𝑖be a sequence of real nonnegative numbers satisfying π‘Žπ‘–β‰€π΄+π΅β„Žπ‘–βˆ’1ξ“π‘˜=1π‘Žπ‘˜,βˆ€π‘–=1,2,…,(2.15)where𝐴, 𝐡, and β„Ž are positive constants, such that π΅β„Ž<1. Then π‘Žπ‘–[]≀𝐴exp𝐡(π‘–βˆ’1)β„Ž,(2.16)takes place for all 𝑖=1,2,….

In the sequel, we make the following assumptions.

(𝐻1) Functions π‘“βˆΆ[0,𝑇]→𝐿2(0,1) and π‘ŽβˆΆ[0,𝑇]→ℝ are Lipschitz continuous, that is,


(𝐻2) The mapping π‘˜βˆΆ[0,𝑇]×𝑉→𝐿2(0,1) is Lipschitz continuous in both variables, that is,

βˆƒπ‘™3βˆˆβ„+;β€–β€–π‘˜ξ€·π‘‘(𝑑,𝑒)βˆ’π‘˜ξ…ž,π‘’ξ…žξ€Έβ€–β€–β‰€π‘™3ξ€Ί||π‘‘βˆ’π‘‘ξ…ž||+β€–β€–π‘’βˆ’π‘’ξ…žβ€–β€–ξ€»,(2.18) for all 𝑑,π‘‘ξ…žβˆˆπΌ, 𝑒,π‘’ξ…žβˆˆπ‘‰, and satisfies

βˆƒπ‘™4,𝑙5βˆˆβ„+;β€–π‘˜(𝑑,𝑒)‖𝐡12(0,1)≀𝑙4‖𝑒‖𝐡12(0,1)+𝑙5,(2.19) for all π‘‘βˆˆπΌ and all π‘’βˆˆπ‘‰, where 𝑙4 and 𝑙5 are positive constants.

(𝐻3)π‘ˆ0∈𝐻2(0,1) and


We will be concerned with a weak solution in the following sense.

Definition 2.3. A function π‘’βˆΆπΌβ†’πΏ2(0,1) is called a weak solution to problem (2.3) if the following conditions are satisfied:(i)π‘’βˆˆπΏβˆž(𝐼,𝑉)∩𝐢(𝐼,𝐡12(0,1)),(ii)𝑒 is strongly differentiable a.e. in 𝐼 and 𝑑𝑒/π‘‘π‘‘βˆˆπΏβˆž(𝐼,𝐡12(0,1)),(iii)𝑒(0)=π‘ˆ0 in 𝑉,(iv)the identity 𝑑𝑒𝑑𝑑(𝑑),𝑣𝐡12(0,1)=ξ‚΅ξ€œ+(𝑒(𝑑),𝑣)𝑑0ξ‚Άπ‘Ž(π‘‘βˆ’π‘ )π‘˜(𝑠,𝑒(𝑠))𝑑𝑠,𝑣𝐡12(0,1)+(𝑓(𝑑),𝑣)𝐡12(0,1)(2.21)holds for all π‘£βˆˆπ‘‰ and a.e. π‘‘βˆˆ[0,𝑇].

To close this section, we announce the main result of the paper.

Theorem 2.4. Under assumptions (𝐻1)-(𝐻3), problem (2.3) admits a unique weak solution 𝑒, in the sense of Definition (2.3).

3. Construction of an Approximate Solution

In order to solve problem (2.3) by the Rothe method, we proceed as follows. Let 𝑛 be a positive integer, we divide the time interval 𝐼=[0,𝑇] into 𝑛 subintervals πΌπ‘›π‘—βˆΆ=[π‘‘π‘›π‘—βˆ’1,𝑑𝑛𝑗], 𝑗=1,…,𝑛, where π‘‘π‘›π‘—βˆΆ=π‘—β„Žπ‘› and β„Žπ‘›βˆΆ=𝑇/𝑛. Then, for each 𝑛β‰₯1, problem (2.3) may be approximated by the following recurrent sequence of time-discretized problems. Successively, for 𝑗=1,…,𝑛, we look for functions π‘’π‘›π‘—βˆˆπ‘‰ such that π‘’π‘›π‘—βˆ’π‘’π‘›π‘—βˆ’1β„Žπ‘›βˆ’π‘‘2𝑒𝑛𝑗𝑑π‘₯2=β„Žπ‘›π‘—βˆ’1𝑖=0π‘Žξ€·π‘‘π‘›π‘—βˆ’π‘‘π‘›π‘–ξ€Έπ‘˜ξ€·π‘‘π‘›π‘–,𝑒𝑛𝑖+𝑓𝑛𝑗,ξ€œ(3.1)10π‘’π‘›π‘—ξ€œ(π‘₯)𝑑π‘₯=0,(3.2)10π‘₯𝑒𝑛𝑗(π‘₯)𝑑π‘₯=0,(3.3) starting from

𝑒𝑛0=π‘ˆ0,𝛿𝑒𝑛0=𝑑2𝑑π‘₯2π‘ˆ0+𝑓(0),(3.4) where 𝑒𝑛𝑗(π‘₯)∢=𝑒(π‘₯,𝑑𝑛𝑗), π›Ώπ‘’π‘›π‘—βˆΆ=(π‘’π‘›π‘—βˆ’π‘’π‘›π‘—βˆ’1)/β„Žπ‘›, 𝑓𝑛𝑗(π‘₯)∢=𝑓(π‘₯,𝑑𝑛𝑗). For this, multiplying for all 𝑗=1,…,𝑛, (3.1) by β„‘2π‘₯βˆ«π‘£βˆΆ=π‘₯0(βˆ«πœ‰0𝑣(𝜏)π‘‘πœ)π‘‘πœ‰ and integrating over (0,1), we get

ξ€œ10𝛿𝑒𝑛𝑗(π‘₯)β„‘2π‘₯ξ€œπ‘£π‘‘π‘₯βˆ’10𝑑2𝑒𝑛𝑗𝑑π‘₯2(π‘₯)β„‘2π‘₯𝑣𝑑π‘₯=β„Žπ‘›ξ€œ10π‘—βˆ’1𝑖=0π‘Žξ€·π‘‘π‘›π‘—βˆ’π‘‘π‘›π‘–ξ€Έπ‘˜ξ€·π‘‘π‘›π‘–,𝑒𝑛𝑖ℑ2π‘₯ξ€œπ‘£π‘‘π‘₯+10𝑓𝑛𝑗ℑ2π‘₯𝑣𝑑π‘₯.(3.5) Noting that, using a standard integration by parts, we have

β„‘21ξ€œπ‘£=10ξ€œ(1βˆ’πœ‰)𝑣(πœ‰)π‘‘πœ‰=10ξ€œπ‘£(πœ‰)π‘‘πœ‰βˆ’10πœ‰π‘£(πœ‰)π‘‘πœ‰=0,βˆ€π‘£βˆˆπ‘‰.(3.6) Carrying out some integrations by parts and invoking (3.6), we obtain for each term in (3.5) ξ€œ10𝛿𝑒𝑛𝑗ℑ2π‘₯𝑣𝑑π‘₯=βˆ’π›Ώπ‘’π‘›π‘—ξ€Έ,𝑣𝐡12(0,1),ξ€œ10𝑑2𝑒𝑛𝑗𝑑π‘₯2(π‘₯)β„‘2π‘₯𝑒𝑣𝑑π‘₯=𝑛𝑗,β„Ž,π‘£π‘›ξ€œ10π‘—βˆ’1𝑖=0π‘Žξ€·π‘‘π‘›π‘—βˆ’π‘‘π‘›π‘–ξ€Έπ‘˜ξ€·π‘‘π‘›π‘–,𝑒𝑛𝑖ℑ(π‘₯)2π‘₯𝑣𝑑π‘₯=βˆ’β„Žπ‘›π‘—βˆ’1𝑖=0π‘Žξ€·π‘‘π‘›π‘—βˆ’π‘‘π‘›π‘–π‘˜ξ€·π‘‘ξ€Έξ€·π‘›π‘–,𝑒𝑛𝑖,𝑣𝐡12(0,1),(3.7) and for the last one

ξ€œ10𝑓𝑛𝑗(π‘₯)β„‘2π‘₯𝑓𝑣(π‘₯)𝑑π‘₯=βˆ’π‘›π‘—ξ€Έ,𝑣𝐡12(0,1).(3.8) By virtue of (3.7) and (3.8), (3.5) becomes 𝛿𝑒𝑛𝑗,𝑣𝐡12(0,1)+𝑒𝑛𝑗,𝑣=β„Žπ‘›π‘—βˆ’1𝑖=0π‘Žξ€·π‘‘π‘›π‘—βˆ’π‘‘π‘›π‘–π‘˜ξ€·π‘‘ξ€Έξ€·π‘›π‘–,𝑒𝑛𝑖,𝑣𝐡12(0,1)+𝑓𝑛𝑗,𝑣𝐡12(0,1),(3.9) or

𝑒𝑛𝑗,𝑣𝐡12(0,1)+β„Žπ‘›ξ€·π‘’π‘›π‘—ξ€Έ,𝑣=β„Ž2π‘›π‘—βˆ’1𝑖=0π‘Žξ€·π‘‘π‘›π‘—βˆ’π‘‘π‘›π‘–π‘˜ξ€·π‘‘ξ€Έξ€·π‘›π‘–,𝑒𝑛𝑖,𝑣𝐡12(0,1)+β„Žπ‘›ξ€·π‘“π‘›π‘—ξ€Έ,𝑣𝐡12(0,1)+ξ‚€π‘’π‘›π‘—βˆ’1,𝑣𝐡12(0,1).(3.10) Let πœ‚(β‹…,β‹…)βˆΆπ‘‰Γ—π‘‰β†’β„and 𝐿𝑗(β‹…)βˆΆπ‘‰β†’β„ be two functions defined by


It is easy to see that the bilinear form πœ‚(β‹…,β‹…) is continuous on 𝑉 and V-elliptic, and the form 𝐿𝑗(β‹…) is continuous for each 𝑗=1,…,𝑛. Then, Lax-Milgram lemma guarantees the existence and uniqueness of 𝑒𝑛𝑗, for all 𝑗=1,…,𝑛.

4. A Priori Estimates

Lemma 4.1. There exists 𝐢>0 such that, for all 𝑛β‰₯1 and all 𝑗=1,…,𝑛, the solution 𝑒𝑗 of the discretized problem (3.1)–(3.4) satisfies the estimates ‖‖𝑒𝑛𝑗‖‖‖‖≀𝐢,(4.1)𝛿𝑒𝑛𝑗‖‖𝐡12(0,1)≀𝐢.(4.2)

Proof. Testing the difference (3.9)π‘—βˆ’1-(3.9)𝑗 with 𝑣=𝛿𝑒𝑛𝑗(βˆˆπ‘‰), taking into account assumptions (𝐻1)-(𝐻3) and the Cauchy-Schwarz inequality, we obtain ‖‖𝛿𝑒𝑛𝑗‖‖𝐡12(0,1)+β€–β€–π‘’π‘›π‘—βˆ’π‘’π‘›π‘—βˆ’1‖‖𝐡12(0,1)β‰€β€–β€–π›Ώπ‘’π‘›π‘—βˆ’1‖‖𝐡12(0,1)+𝐢13β„Ž2π‘›π‘—βˆ’2𝑖=0‖‖𝑒𝑛𝑖‖‖𝐡12(0,1)+𝐢13β„Žπ‘›+𝐢13β„Žπ‘›β€–β€–π‘’π‘›π‘—βˆ’1‖‖𝐡12(0,1),(4.3)where 𝐢1ξ€½π‘™βˆΆ=3max2𝜁,𝑇𝑙2𝜁+𝑀1𝜁+𝑙1ξ€Ύ,𝑀1∢=maxπ‘‘βˆˆπΌ||π‘Ž||𝑙(𝑑),𝜁∢=max4,𝑙5ξ€Ύ.(4.4)Multiplying the left-hand side of the last inequality with (1βˆ’(𝐢1/3)β„Žπ‘›)(<1) and adding the terme 23𝐢1β„Žπ‘›ξ‚ƒβ€–β€–π‘’π‘›π‘—βˆ’π‘’π‘›π‘—βˆ’1‖‖𝐡12(0,1)βˆ’β€–β€–π›Ώπ‘’π‘›π‘—β€–β€–π΅12(0,1)ξ‚„(<0),(4.5)we get ξ€·1βˆ’πΆ1β„Žπ‘›ξ€Έξ‚ƒβ€–β€–π›Ώπ‘’π‘›π‘—β€–β€–π΅12(0,1)+‖‖𝑒𝑛𝑗‖‖𝐡12(0,1)ξ‚„β‰€ξ‚ƒβ€–β€–π‘’π‘›π‘—βˆ’1‖‖𝐡12(0,1)+β€–β€–π›Ώπ‘’π‘›π‘—βˆ’1‖‖𝐡12(0,1)ξ‚„+𝐢1β„Ž2π‘›π‘—βˆ’2𝑖=0‖‖𝑒𝑛𝑖‖‖𝐡12(0,1)+𝐢1β„Žπ‘›.(4.6) Applying the last inequality recursively, it follows that ξ€·1βˆ’πΆ1β„Žπ‘›ξ€Έπ‘—ξ‚ƒβ€–β€–π›Ώπ‘’π‘›π‘—β€–β€–π΅12(0,1)+‖‖𝑒𝑛𝑗‖‖𝐡12(0,1)≀‖‖𝑒𝑛0‖‖𝐡12(0,1)+‖‖𝛿𝑒𝑛0‖‖𝐡12(0,1)+𝐢1𝑇+𝑇𝐢1β„Žπ‘›π‘—βˆ’2𝑖=0‖‖𝑒𝑛𝑖‖‖𝐡12(0,1),(4.7) or, by virtue of Lemma 2.2, there exists 𝑛0βˆˆβ„•βˆ— such that ‖‖𝛿𝑒𝑛𝑗‖‖𝐡12(0,1)+‖‖𝑒𝑛𝑗‖‖𝐡12(0,1)≀𝐢2,βˆ€π‘›β‰₯𝑛0,(4.8)where 𝐢2ξ€·ξ€·βˆΆ=exp𝑇𝐢1‖‖+1𝛿𝑒𝑛0‖‖𝐡12(0,1)+‖‖𝑒𝑛0‖‖𝐡12(0,1)+𝑇𝐢1ξ‚„ξ€·Γ—expξ€Ίξ€·exp𝑇𝐢1ξ€Έξ€Έ+1𝑇𝐢1ξ€»,(4.9)and so our proof is complete.

We address now the question of convergence and existence.

5. Convergence and Existence

Now let us introduce the Rothe function 𝑒𝑛(𝑑)βˆΆπΌβ†’π‘‰ obtained from the functions 𝑒𝑗 by piecewise linear interpolation with respect to time

𝑒𝑛(𝑑)=π‘’π‘›π‘—βˆ’1+π›Ώπ‘’π‘›π‘—ξ‚€π‘‘βˆ’π‘‘π‘›π‘—βˆ’1,in𝐼𝑛𝑗,(5.1) as well the step functions ̃𝑒𝑛(𝑑), ̂𝑒𝑛(𝑑), 𝑓𝑛(𝑑), and Μƒπ‘˜(𝑑,̃𝑒𝑛(𝑑)) defined as follows:


Corollary 5.1. There exist 𝐢>0 such that the estimates ‖𝑒𝑛‖‖(𝑑)‖≀𝐢,̃𝑒𝑛‖‖‖‖‖(𝑑)≀𝐢,βˆ€π‘‘βˆˆπΌ,(5.5)𝑑𝑒𝑛‖‖‖𝑑𝑑(𝑑)𝐡12(0,1)‖‖≀𝐢,fora.e.π‘‘βˆˆπΌ,(5.6)̃𝑒𝑛(𝑑)βˆ’π‘’π‘›β€–β€–(𝑑)𝐡12(0,1)β‰€πΆβ„Žπ‘›,‖‖̂𝑒𝑛(𝑑)βˆ’π‘’π‘›β€–β€–(𝑑)𝐡12(0,1)β‰€πΆβ„Žπ‘›β€–β€–Μƒπ‘˜,βˆ€π‘‘βˆˆπΌ,(5.7)𝑛‖‖(𝑑)≀𝐢,βˆ€π‘‘βˆˆπΌ,(5.8) hold for all π‘›βˆˆβ„•βˆ—.

Proof. For the inequalities (5.5), (5.6), and (5.7) see [6, Corollary  4.2.], whereas for the last inequality, assumption (𝐻2) and estimate (4.1) guarantee the desired result.

Proposition 5.2. The sequence (𝑒𝑛)𝑛 converges in the norm of the space 𝐢(𝐼,𝐡12(0,1)) to some function π‘’βˆˆπΆ(𝐼,𝐡12(0,1)) and the error estimate β€–π‘’π‘›β€–βˆ’π‘’πΆ(𝐼,𝐡12(0,1))βˆšβ‰€πΆβ„Žπ‘›(5.9) takes place for all 𝑛β‰₯𝑛0.

Proof. By virtue of (5.2), (5.3), and (5.4) the variational equation (3.9) may be rewritten in the form 𝑑𝑒𝑛𝑑𝑑(𝑑),𝑣𝐡12(0,1)+̃𝑒𝑛=ξ€·Μƒπ‘˜(𝑑),𝑣𝑛(𝑑),𝑣𝐡12(0,1)+𝑓𝑛(𝑑),𝑣𝐡12(0,1),(5.10) for a.e. π‘‘βˆˆ[0,𝑇].In view of (5.10), using (5.6) and (5.8) with the fact that ‖‖𝑓𝑛(‖‖𝑑)𝐡12(0,1)≀𝑀2∢=maxπ‘‘βˆˆπΌ(‖𝑓𝑑)‖𝐡12(0,1)<∞,(5.11)we obtain ||̃𝑒𝑛||β‰€ξ‚΅β€–β€–Μƒπ‘˜(𝑑),𝑣𝑛‖‖(𝑑)𝐡12(0,1)+‖‖𝑓𝑛‖‖(𝑑)𝐡12(0,1)+‖‖‖𝑑𝑒𝑛‖‖‖𝑑𝑑(𝑑)𝐡12(0,1)‖𝑣‖𝐡12(0,1)≀𝐢‖𝑣‖𝐡12(0,1)[].,a.e.π‘‘βˆˆ0,𝑇(5.12) Now, for 𝑛, π‘š being two positive integers, testing the difference (5.10)𝑛-(5.10)π‘š with 𝑣=𝑒𝑛(𝑑)βˆ’π‘’π‘š(𝑑) which is in 𝑉, with the help of the Cauchy-Schwarz inequality and taking into account that 2𝑑𝑑𝑑𝑒(𝑑),𝑒(𝑑)𝐡12(0,1)=𝑑‖𝑑𝑑‖𝑒(𝑑)2𝐡12(0,1)[],a.e.π‘‘βˆˆ0,𝑇,(5.13)and, by virtue of (5.12) we obtain after some rearrangements 12𝑑𝑑𝑑‖𝑒𝑛(𝑑)βˆ’π‘’π‘šβ€–(𝑑)2𝐡12(0,1)+‖̃𝑒𝑛(𝑑)βˆ’Μƒπ‘’π‘šβ€–(𝑑)2β€–β€–π‘’β‰€πΆπ‘š(𝑑)βˆ’Μƒπ‘’π‘šβ€–β€–(𝑑)𝐡12(0,1)β€–β€–+𝐢̃𝑒𝑛(𝑑)βˆ’π‘’π‘›β€–β€–(𝑑)𝐡12(0,1)+β€–β€–Μƒπ‘˜π‘›Μƒπ‘˜(𝑑)βˆ’π‘šβ€–β€–(𝑑)𝐡12(0,1)‖𝑒𝑛(𝑑)βˆ’π‘’π‘šβ€–(𝑑)𝐡12(0,1)+‖‖𝑓𝑛𝑓(𝑑)βˆ’π‘šβ€–β€–(𝑑)𝐡12(0,1)‖𝑒𝑛(𝑑)βˆ’π‘’π‘š(𝑑)‖𝐡12(0,1)[].,a.e.π‘‘βˆˆ0,𝑇(5.14) To derive the required result, we need to estimate the third and the last terms in the right-hand side, for this, let 𝑑 be arbitrary but fixed in (0,𝑇], without loss of generality we can suppose that there exist three positive integers 𝑝,π‘ž and 𝛽, such that ξ‚€π‘‘π‘‘βˆˆπ‘›π‘βˆ’1,π‘‘π‘›π‘ξ‚„βˆ©ξ‚€π‘‘π‘šπ‘žβˆ’1,π‘‘π‘šπ‘žξ‚„,𝑛=π›½π‘š,𝑑𝑛𝑝=π‘‘π‘šπ‘ž.(5.15)Hence, using (5.4) we can write β€–β€–Μƒπ‘˜π‘›Μƒπ‘˜(𝑑)βˆ’π‘šβ€–β€–(𝑑)𝐡12(0,1)=β„Žπ‘šβ€–β€–β€–β€–π‘βˆ’1𝑗=0𝛽(𝑗+1)βˆ’1𝑖=π‘—π›½ξ€·π‘Žξ€·π‘‘π‘›π‘βˆ’π‘‘π‘›π‘—ξ€Έπ‘˜ξ€·π‘‘π‘›π‘—,π‘’π‘›π‘—ξ€Έξ€·π‘‘βˆ’π‘Žπ‘šπ‘žβˆ’π‘‘π‘šπ‘–ξ€Έπ‘˜ξ€·π‘‘π‘šπ‘–,π‘’π‘šπ‘–ξƒ­β€–β€–β€–β€–ξ€Έξ€Έπ΅12(0,1).(5.16)By virtue of assumption (𝐻1) and the fact that |π‘Ž(π‘‘π‘›π‘βˆ’π‘‘π‘›π‘—)βˆ’π‘Ž(π‘‘π‘šπ‘žβˆ’π‘‘π‘šπ‘–)|β‰€πΆβ„Žπ‘›, there exist πœ€π‘›βˆˆ[0,πΆβ„Žπ‘›] such that β€–β€–Μƒπ‘˜π‘›Μƒπ‘˜(𝑑)βˆ’π‘šβ€–β€–(𝑑)𝐡12(0,1)β‰€β„Žπ‘šπ‘βˆ’1𝑗=0𝛽(𝑗+1)βˆ’1𝑖=π‘—π›½β€–β€–ξ€·πΆβ„Žπ‘›βˆ’πœ€π‘›ξ€Έπ‘˜ξ€·π‘‘π‘›π‘—,𝑒𝑛𝑗‖‖𝐡12(0,1)+||π‘Žξ€·π‘‘π‘šπ‘žβˆ’π‘‘π‘šπ‘–ξ€Έ||β€–β€–π‘˜ξ€·π‘‘π‘›π‘—,π‘’π‘›π‘—ξ€Έξ€·π‘‘βˆ’π‘˜π‘šπ‘–,π‘’π‘šπ‘–ξ€Έβ€–β€–π΅12(0,1)ξƒ­.(5.17)Therefore, recalling assumptions (𝐻1), (𝐻2) and having in mind that πœ€π‘›βˆˆ[0,πΆβ„Žπ‘›], we estimate β€–β€–Μƒπ‘˜π‘›Μƒπ‘˜(𝑑)βˆ’π‘šβ€–β€–(𝑑)𝐡12(0,1)β‰€β„Žπ‘šπ‘βˆ’1𝑗=0𝛽(𝑗+1)βˆ’1𝑖=π‘—π›½πΆβ„Žπ‘›ξ‚€β„Ž+𝐢𝑛+β€–β€–π‘’π‘›π‘—βˆ’π‘’π‘šπ‘–β€–β€–π΅12(0,1),(5.18)from where, we derive for all π‘ βˆˆ(π‘‘π‘šπ‘–,π‘‘π‘šπ‘–+1]β€–β€–Μƒπ‘˜π‘›Μƒπ‘˜(𝑑)βˆ’π‘šβ€–β€–(𝑑)𝐡12(0,1)β‰€β„Žπ‘šπ‘βˆ’1𝑗=0𝛽(𝑗+1)βˆ’1𝑖=π‘—π›½πΆβ„Žπ‘›ξ‚€β„Ž+𝐢𝑛+‖‖̃𝑒𝑛(𝑠)βˆ’π‘’π‘›β€–β€–(𝑠)𝐡12(0,1)+‖𝑒𝑛(𝑠)βˆ’π‘’π‘šβ€–(𝑠)𝐡12(0,1)+β€–β€–π‘’π‘š(𝑠)βˆ’Μƒπ‘’π‘šβ€–β€–(𝑠)𝐡12(0,1).(5.19)Taking the supremum with respect to 𝑠 from 0 to 𝑑 in the right-hand side, invoking the fact that π‘ βˆˆ(π‘‘π‘šπ‘–,π‘‘π‘šπ‘–+1]βŠ‚(π‘‘π‘›π‘—βˆ’1,𝑑𝑛𝑗] and estimate (5.7), we obtain β€–β€–Μƒπ‘˜π‘›Μƒπ‘˜(𝑑)βˆ’π‘šβ€–β€–(𝑑)𝐡12(0,1)β‰€β„Žπ‘šπ‘žβˆ’1𝑖=0ξ‚΅πΆβ„Žπ‘›+𝐢sup0≀𝑠≀𝑑‖𝑒𝑛(𝑠)βˆ’π‘’π‘š(𝑠)‖𝐡12(0,1)ξ‚Ά,(5.20)so that β€–β€–Μƒπ‘˜π‘›Μƒπ‘˜(𝑑)βˆ’π‘šβ€–β€–(𝑑)𝐡12(0,1)β‰€πΆβ„Žπ‘›+𝐢sup0≀𝑠≀𝑑‖𝑒𝑛(𝑠)βˆ’π‘’π‘š(𝑠)‖𝐡12(0,1).(5.21) Let π‘‘βˆˆ(π‘‘π‘›π‘βˆ’1,𝑑𝑛𝑝]∩(π‘‘π‘šπ‘žβˆ’1,π‘‘π‘šπ‘ž], from assumption (𝐻1) it follows that ‖‖𝑓𝑛(𝑓𝑑)βˆ’π‘š(‖‖𝑑)𝐡12(0,1)=β€–β€–π‘“ξ€·π‘‘π‘›π‘ξ€Έξ€·π‘‘βˆ’π‘“π‘šπ‘žξ€Έβ€–β€–π΅12(0,1)≀𝑙1||π‘‘π‘›π‘βˆ’π‘‘π‘šπ‘ž||≀𝑙1β„Žπ‘›.(5.22) Ignoring the second term in the left-hand side of (5.14) which is clearly positive and using estimates (5.5), (5.7), (5.21), and (5.22) yield 𝑑𝑑𝑑‖𝑒𝑛(𝑑)βˆ’π‘’π‘šβ€–(𝑑)2𝐡12(0,1)ξ€·β„Žβ‰€πΆπ‘›+β„Žπ‘šξ€Έ+𝐢sup0≀𝑠≀𝑑‖𝑒𝑛(𝑠)βˆ’π‘’π‘šβ€–(𝑠)2𝐡12(0,1)[],a.e.π‘‘βˆˆ0,𝑇.(5.23)Integrating this inequality with respect to time from 0 to 𝑑 and invoking the fact that 𝑒𝑛(0)=π‘’π‘š(0)=π‘ˆ0, we get ‖𝑒𝑛(𝑑)βˆ’π‘’π‘š(𝑑)β€–2𝐡12(0,1)ξ€·β„Žβ‰€πΆπ‘›+β„Žπ‘šξ€Έξ€œ+𝐢𝑑0sup0β‰€πœ‰β‰€π‘‘β€–π‘’π‘›(πœ‰)βˆ’π‘’π‘š(πœ‰)β€–2𝐡12(0,1)π‘‘πœ‰,(5.24)whence sup0≀𝑠≀𝑑‖𝑒𝑛(𝑠)βˆ’π‘’π‘š(𝑠)β€–2𝐡12(0,1)ξ€·β„Žβ‰€πΆπ‘›+β„Žπ‘šξ€Έξ€œ+𝐢𝑑0sup0β‰€πœ‰β‰€π‘‘β€–π‘’π‘›(πœ‰)βˆ’π‘’π‘š(πœ‰)β€–2𝐡12(0,1)π‘‘πœ‰.(5.25)Accordingly, by Gronwall's lemma we obtain sup0≀𝑠≀𝑑‖𝑒𝑛(𝑠)βˆ’π‘’π‘šβ€–(𝑠)2𝐡12(0,1)ξ€·β„Žβ‰€πΆπ‘›+β„Žπ‘šξ€Έ[]exp(𝐢𝑑),βˆ€π‘‘βˆˆ0,𝑇,(5.26)consequently sup0≀𝑠≀𝑑‖𝑒𝑛(𝑠)βˆ’π‘’π‘šβ€–(𝑠)𝐡12(0,1)βˆšβ‰€πΆβ„Žπ‘›+β„Žπ‘š(5.27) takes place for all 𝑛,π‘šβˆˆβ„•βˆ—. This implies that (𝑒𝑛(𝑑))𝑛 is a Cauchy sequence in the Banach space 𝐢(𝐼,𝐡12(0,1)), and hence it converges in the norm of this latter to some function π‘’βˆˆπΆ(𝐼,𝐡12(0,1)). Besides, passing to the limit π‘šβ†’βˆž in (5.27), we obtain the desired error estimate, which finishes the proof.

Now, we present some properties of the obtained solution.

The limit-function 𝑒 from Proposition  5.2, possesses the following properties:

(i)π‘’βˆˆπΆ(𝐼,𝐡12(0,1))∩𝐿∞(𝐼,𝑉)),(ii)𝑒 is strongly differentiable a.e. in 𝐼 and 𝑑𝑒/π‘‘π‘‘βˆˆπΏβˆž(𝐼,𝐡12(0,1)),(iii)̃𝑒𝑛(𝑑)→𝑒(𝑑) in 𝐡12(0,1) for all π‘‘βˆˆπΌ,(iv)𝑒𝑛(𝑑), ̃𝑒𝑛(𝑑)⇀𝑒(𝑑) in 𝑉 for all π‘‘βˆˆπΌ,(v)(𝑑𝑒𝑛/𝑑𝑑)(𝑑)⇀(𝑑𝑒/𝑑𝑑)(𝑑) in 𝐿2(𝐼,𝐡12(0,1)).

Proof. On the basis of estimates (5.5) and (5.6), uniform convergence statement from Proposition  5.2, and the continuous embedding 𝑉β†ͺ𝐡12(0,1), the assertions of the present theorem are a direct consequence of [2, Lemma  1.3.15].

Theorem 5.3. Under Assumptions (𝐻1)-(𝐻3), (2.3) admits a unique weak solution, namely, the limit function 𝑒 from Proposition  5.2, in the sense of Definition  2.3.

Proof. We have to show that the limit function 𝑒 satisfies all the conditions (i), (ii), (iii), and (iv) of Definition  2.3. Obviously, in light of the properties of the function 𝑒 listed in Theorem 5.3, the first two conditions of Definition  2.3 are already seen. On the other hand, since 𝑒𝑛→𝑒 in 𝐢(𝐼,𝑉) as π‘›β†’βˆž and, by construction, 𝑒𝑛(0)=π‘ˆ0, it follows that 𝑒(0)=π‘ˆ0, so the initial condition is also fulfilled, that is, Definition  2.3(iii) takes place. It remains to see that the integral identity (2.21) is obeyed by 𝑒. For this, integrating (5.10) over (0,𝑑) and using the fact that 𝑒𝑛(0)=π‘ˆ0, we get 𝑒𝑛(𝑑)βˆ’π‘ˆ0ξ€Έ,𝑣𝐡12(0,1)+ξ€œπ‘‘0̃𝑒𝑛(ξ€Έξ€œπœ),π‘£π‘‘πœ=𝑑0ξ€·Μƒπ‘˜ξ€·πœ,̃𝑒𝑛(𝜏),𝑣𝐡12(0,1)ξ€œπ‘‘πœ+𝑑0𝑓𝑛(ξ‚πœ),𝑣𝐡12(0,1)π‘‘πœ,(5.28)consequently, after some rearrangements 𝑒𝑛(𝑑)βˆ’π‘ˆ0ξ€Έ,𝑣𝐡12(0,1)+ξ€œπ‘‘0̃𝑒𝑛(ξ€Έ=ξ€œπœ),π‘£π‘‘πœπ‘‘0ξ‚΅ξ€œπœ0ξ‚Άπ‘Ž(πœβˆ’π‘ )π‘˜(𝑠,𝑒(𝑠))𝑑𝑠,𝑣𝐡12(0,1)ξ€œπ‘‘πœ+𝑑0(𝑓(𝜏),𝑣)𝐡12(0,1)+ξ€œπ‘‘πœπ‘‘0ξ‚΅Μƒπ‘˜ξ€·πœ,Μƒπ‘’π‘›ξ€Έβˆ’ξ€œ(𝜏)𝜏0ξ‚Άπ‘Ž(πœβˆ’π‘ )π‘˜(𝑠,𝑒(𝑠))𝑑𝑠,𝑣𝐡12(0,1)+ξ€œπ‘‘πœπ‘‘0𝑓𝑛(𝜏)βˆ’π‘“(𝜏),𝑣𝐡12(0,1)π‘‘πœ.(5.29) Let Μ‚π‘ π‘›βˆΆπΌβ†’πΌ and Μ‚π‘ π‘›βˆΆπΌβ†’πΌ denote the functions ̂𝑠𝑛𝑑(𝑑)=0,for𝑑=0,π‘›π‘—βˆ’1𝐼,in𝑛𝑗,̃𝑠𝑛𝑑(𝑑)=0,for𝑑=0,𝑛𝑗𝐼,in𝑛𝑗.(5.30) To investigate the desired result, we prove some convergence statements. Using (5.2), (5.4), and (5.30) we have for allπ‘‘βˆˆ(π‘‘π‘›π‘—βˆ’1,𝑑𝑛𝑗]Μƒπ‘˜ξ€·π‘‘,̃𝑒𝑛(ξ€Έβˆ’ξ€œπ‘‘)𝑑0=ξ€œπ‘Ž(π‘‘βˆ’π‘ )π‘˜(𝑠,𝑒(𝑠))𝑑𝑠𝑑𝑛𝑗0ξ€Ίπ‘Žξ€·π‘‘π‘›π‘—βˆ’Μ‚π‘ π‘›ξ€Έπ‘˜ξ€·(𝑠)̂𝑠𝑛(𝑠),Μ‚π‘’π‘›ξ€Έξ€»ξ€œ(𝑠)βˆ’π‘Ž(π‘‘βˆ’π‘ )π‘˜(𝑠,𝑒(𝑠))𝑑𝑠+π‘‘π‘›π‘—π‘‘π‘Ž(π‘‘βˆ’π‘ )π‘˜(𝑠,𝑒(𝑠))𝑑𝑠.(5.31) Taking into account (5.5), (5.9), and assumptions (𝐻1), (𝐻2) it follows that β€–β€–π‘Žξ€·π‘‘π‘›π‘—βˆ’Μ‚π‘ π‘›ξ€Έπ‘˜ξ€·(𝑠)̂𝑠𝑛(𝑠),̂𝑒𝑛‖‖(𝑠)βˆ’π‘Ž(π‘‘βˆ’π‘ )π‘˜(𝑠,𝑒(𝑠))𝐡12(0,1)βˆšβ‰€πΆβ„Žπ‘›.(5.32) Thanks to (5.31) and (5.32) we obtain β€–β€–β€–Μƒπ‘˜ξ€·π‘‘,Μƒπ‘’π‘›ξ€Έβˆ’ξ€œ(𝑑)𝑑0β€–β€–β€–π‘Ž(π‘‘βˆ’π‘ )π‘˜(𝑠,𝑒(𝑠))𝑑𝑠𝐡12(0,1)βˆšβ‰€πΆβ„Žπ‘›.(5.33) On the other hand, in view of the assumed Lipschitz continuity of 𝑓, we have ‖‖𝑓𝑛(β€–β€–πœ)βˆ’π‘“(𝜏)𝐡12(0,1)≀‖‖𝑓̃𝑠𝑛(ξ€Έβ€–β€–πœ)βˆ’π‘“(𝜏)𝐡12(0,1)≀𝑙1β„Žπ‘›.(5.34) Now, the sequences {(̃𝑒𝑛(𝜏),𝑣)}, 𝑓{(𝑛(𝜏),𝑣)𝐡12(0,1)}, and Μƒ{(π‘˜(𝜏,̃𝑒𝑛(𝜏)),𝑣)𝐡12(0,1)} are uniformly bounded with respect to both 𝜏 and 𝑛, so the Lebesgue theorem of majorized convergence is applicable to (5.29). Thus, having in mind (5.7), (5.9), (5.33), and (5.34), we derive that 𝑒(𝑑)βˆ’π‘ˆ0ξ€Έ,𝑣𝐡12(0,1)+ξ€œπ‘‘0(=ξ€œπ‘’(𝜏),𝑣)π‘‘πœπ‘‘0ξ‚΅ξ€œπœ0ξ‚Άπ‘Ž(πœβˆ’π‘ )π‘˜(𝑠,𝑒(𝑠))𝑑𝑠,𝑣𝐡12(0,1)ξ€œπ‘‘πœ+𝑑0(𝑓(𝜏),𝑣)𝐡12(0,1)π‘‘πœ(5.35) takes place for all π‘£βˆˆπ‘‰ and π‘‘βˆˆ[0,𝑇]. Finally, differentiating (5.35) with respect to 𝑑, we get 𝑑𝑑𝑑𝑒(𝑑),𝑣𝐡12(0,1)=ξ‚΅ξ€œ+(𝑒(𝑑),𝑣)𝑑0ξ‚Άπ‘Ž(π‘‘βˆ’π‘ )π‘˜(𝑠,𝑒(𝑠))𝑑𝑠,𝑣𝐡12(0,1)+(𝑓(𝑑),𝑣)𝐡12(0,1)[].,a.e.π‘‘βˆˆ0,𝑇(5.36)The uniqueness may be argued in the usual manner. Indeed, exploiting an idea in [11], consider 𝑒1 and 𝑒2 two different solutions of (2.3), and define 𝑀=𝑒1βˆ’π‘’2 then, we have 𝑑𝑑𝑑𝑀(𝑑),𝑣𝐡12(0,1)ξ‚΅ξ€œ+(𝑀(𝑑),𝑣)=𝑑0ξ€Ίπ‘˜ξ€·π‘Ž(π‘‘βˆ’π‘ )𝑠,𝑒1(𝑠)βˆ’π‘˜π‘ ,𝑒2(𝑠)𝑑𝑠,𝑣𝐡12(0,1).(5.37)Choosing 𝑣=𝑀(𝑑) as a test function, with the aid of Cauchy-Schwarz inequality and assumption (𝐻1), we obtain 12𝑑(𝑑𝑑‖𝑀𝑑)β€–2𝐡12(0,1)+‖𝑀(𝑑)β€–2ξ€œβ‰€πΆπ‘‘0ξ‚ƒβ€–β€–π‘˜ξ€·π‘ ,𝑒1(𝑠)βˆ’π‘˜π‘ ,𝑒2(‖‖𝑠)𝐡12(0,1)𝑑𝑠‖𝑀(𝑑)‖𝐡12(0,1).(5.38) Let πœ‰βˆˆ[0,𝑝] such that ‖𝑀(πœ‰)‖𝐡12(0,1)=max[]π‘ βˆˆ0,𝑝‖𝑀(𝑠)‖𝐡12(0,1),(5.39) integrating (5.38) over (0,𝑝), 0≀𝑝≀𝑇, using (5.39), and invokingassumption (𝐻2), we get ξ€œπ‘012𝑑𝑑𝑑‖𝑀(𝑑)β€–2𝐡12(0,1)+‖𝑀(𝑑)β€–2𝑑𝑑≀𝐢𝑝2‖𝑀(πœ‰)β€–2𝐡12(0,1),(5.40)consequently, with the fact that 𝑀(0)=0ξ€œπ‘012𝑑‖𝑑𝑑‖𝑀(𝑑)2𝐡12(0,1)+‖‖𝑀(𝑑)2𝑑𝑑≀𝐢𝑝2ξ€œπœ‰0𝑑‖𝑑𝑑‖𝑀(𝑑)2𝐡12(0,1)𝑑𝑑.(5.41) Choosing 𝑝 as a constant verifying the condition βˆƒπ›Όβˆˆβ„•,𝑇=𝛼𝑝,𝐢𝑝2≀12,(5.42)we have, by virtue of (5.41) ξ€œπ‘012𝑑‖𝑑𝑑‖𝑀(𝑑)2𝐡12(0,1)ξ€œπ‘‘π‘‘+𝑝0‖‖𝑀(𝑑)2ξ€œπ‘‘π‘‘β‰€πœ‰012𝑑‖𝑑𝑑‖𝑀(𝑑)2𝐡12(0,1)𝑑𝑑,(5.43)taking into account that πœ‰β‰€π‘, we obtain []‖𝑀(𝑑)β€–=0,on0,𝑝.(5.44)Following the same lines as for [0,𝑝], we deduce that []‖𝑀(𝑑)β€–=0,on𝑖𝑝,(𝑖+1)𝑝,𝑖=1,2,3,…,(5.45)therefore, we derive 𝑀(𝑑)≑0, on [0,𝑇], then 𝑒1≑𝑒2. This achives the proof.


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