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Journal of Function Spaces and Applications
VolumeΒ 2012Β (2012), Article IDΒ 679465, 30 pages
http://dx.doi.org/10.1155/2012/679465
Research Article

A Weak Comparison Principle for Reaction-Diffusion Systems

Centro de InvestigaciΓ³n Operativa, Universidad Miguel HernΓ‘ndez de Elche, Avenida Universidad s/n, 03202 Elche, Spain

Received 4 May 2012; Accepted 17 July 2012

Academic Editor: S.Β Romaguera

Copyright Β© 2012 JosΓ© Valero. 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

We prove a weak comparison principle for a reaction-diffusion system without uniqueness of solutions. We apply the abstract results to the Lotka-Volterra system with diffusion, a generalized logistic equation, and to a model of fractional-order chemical autocatalysis with decay. Moreover, in the case of the Lotka-Volterra system a weak maximum principle is given, and a suitable estimate in the space of essentially bounded functions 𝐿∞ is proved for at least one solution of the problem.

1. Introduction

Comparison results for parabolic equations and ordinary differential equations are well known in the literature (see, e.g., [1–4] among many others). One of the important applications of such kind of results is the theory of monotone dynamical systems, which leads to a more precise characterization of πœ”-limit sets and attractors. In the last years, several authors have been working in this direction (see, e.g., [4–8] for the deterministic case, and [9–12] for the stochastic case). In all these papers, it is considered the classical situation where the initial-value problem possesses a unique solution.

However, the situation is more complicated when we consider a differential equation for which uniqueness of the Cauchy problem fails (or just it is not known to hold). Let us consider an abstract parabolic problem: 𝑑𝑒𝑑𝑑=𝐴(𝑑,𝑒(𝑑)),πœβ‰€π‘‘β‰€π‘‡,𝑒(𝜏)=π‘’πœ,(1.1) for which we can prove that for every initial data in the phase space 𝑋 (with a partial order ≀) there exists at least one solution.

If we try to compare solutions of (1.1) for two ordered initial data 𝑒1πœβ‰€π‘’2𝜏, then we can consider a strong comparison principle and a weak one.

The strong version would imply the existence of a solution 𝑒1 with 𝑒1(𝜏)=𝑒1𝜏 such that 𝑒1(𝑑)≀𝑒2[],(𝑑)forπ‘‘βˆˆπœ,𝑇(1.2) for any solution 𝑒2 with 𝑒2(𝜏)=𝑒2𝜏, and, viceversa, the existence of a solution 𝑒2 with 𝑒2(𝜏)=𝑒2𝜏 such that (1.2) is satisfied for any solution 𝑒1 with 𝑒1(𝜏)=𝑒1𝜏. This kind of result is established in [13] for a delayed ordinary differential equations, defining then a multivalued order-preserving dynamical system.

The weak version of the comparison principle says that if 𝑒1πœβ‰€π‘’2𝜏, then there exist two solutions 𝑒1,𝑒2 of (1.1) such that 𝑒1(𝜏)=𝑒1𝜏, 𝑒2(𝜏)=𝑒2𝜏, and (1.2) hold.

There is in fact an intermediate version of the comparison principle, which says that if we fix a solution 𝑒1 of (1.1) with 𝑒1(𝜏)=𝑒1𝜏, then there exists a solution 𝑒2 with 𝑒2(𝜏)=𝑒2𝜏 such that (1.2) is satisfied (and vice versa). This is proved in [14] for a differential inclusion generated by a subdifferential map.

In this paper, we establish a weak comparison principle for a reaction-diffusion system in which the nonlinear term satisfies suitable dissipative and growth conditions, ensuring existence of solutions but not uniqueness. This principle is applied to several well-known models in physics and biology. Namely, a weak comparison of solutions is proved for the Lotka-Volterra system, the generalized logistic equation and for a model of fractional-order chemical autocatalysis with decay. Moreover, in the case of the Lotka-Volterra system a weak maximum principle is given, and a suitable estimate in the space of essentially bounded functions 𝐿∞ is proved for at least one solution of the problem.

We note that in the papers [15, 16] the existence of a global attractor is proved for such kind of reaction-diffusion systems. In the near future, we will apply these results to obtain theorems concerning the structure of the global attractor.

2. Comparison Results for Reaction-Diffusion Systems

We shall denote by |β‹…| and (β‹…,β‹…) the norm and scalar product in the space β„π‘š, π‘šβ‰₯1. Let 𝑑>0 be an integer and Ξ©βŠ‚β„π‘ be a bounded open subset with smooth boundary. Consider the problem: πœ•π‘’πœ•π‘‘βˆ’π‘ŽΞ”π‘’+𝑓(𝑑,𝑒)=β„Ž(𝑑,π‘₯),(𝑑,π‘₯)∈(𝜏,𝑇)Γ—Ξ©,𝑒|π‘₯βˆˆπœ•Ξ©π‘’|=0,𝑑=𝜏=π‘’πœ(π‘₯),(2.1) where 𝜏,π‘‡βˆˆβ„, 𝑇>𝜏, π‘₯∈Ω, 𝑒=(𝑒1(𝑑,π‘₯),…,𝑒𝑑(𝑑,π‘₯)),𝑓=(𝑓1,…,𝑓𝑑), π‘Ž is a real 𝑑×𝑑 matrix with a positive symmetric part (π‘Ž+π‘Žπ‘‘)/2β‰₯𝛽𝐼,𝛽>0, β„ŽβˆˆπΏ2(𝜏,𝑇;(𝐿2(Ξ©))𝑑). Moreover, 𝑓=(𝑓1(𝑑,𝑒),…,𝑓𝑑(𝑑,𝑒)) is jointly continuous on [𝜏,𝑇]×ℝ𝑑 and satisfies the following conditions: 𝑑𝑖=1||𝑓𝑖||(𝑑,𝑒)𝑝𝑖/(π‘π‘–βˆ’1)≀𝐢11+𝑑𝑖=1||𝑒𝑖||𝑝𝑖ξƒͺ,(𝑓(𝑑,𝑒),𝑒)β‰₯𝛼𝑑𝑖=1||𝑒𝑖||π‘π‘–βˆ’πΆ2,(2.2) where 𝑝𝑖β‰₯2, 𝛼,𝐢1,𝐢2>0.

Let 𝐻=(𝐿2(Ξ©))𝑑, 𝑉=(𝐻10(Ξ©))𝑑, and let π‘‰ξ…ž be the dual space of 𝑉. By β€–β‹…β€–, ‖⋅‖𝑉 we denote the norm by 𝐻 and 𝑉, respectively. For 𝑝=(𝑝1,…,𝑝𝑑), we define the spaces 𝐿𝑝(Ξ©)=𝐿𝑝1(Ξ©)×⋯×𝐿𝑝𝑑(𝐿Ω),𝑝(𝜏,𝑇;𝐿𝑝(Ξ©))=𝐿𝑝1(𝜏,𝑇;𝐿𝑝1(Ξ©))×⋯×𝐿𝑝𝑑(𝜏,𝑇;𝐿𝑝𝑑(Ξ©)).(2.3) We take π‘ž=(π‘ž1,…,π‘žπ‘‘), where (1/𝑝𝑖)+(1/π‘žπ‘–)=1.

We say that the function 𝑒(β‹…) is a weak solution of (2.1) if π‘’βˆˆπΏπ‘(𝜏,𝑇;𝐿𝑝(Ξ©))∩𝐿2(𝜏,𝑇;𝑉)∩𝐢([𝜏,𝑇];𝐻), 𝑑𝑒/π‘‘π‘‘βˆˆπΏ2(𝜏,𝑇;π‘‰ξ…ž)+πΏπ‘ž(𝜏,𝑇;πΏπ‘ž(Ξ©)), 𝑒(𝜏)=π‘’πœ, and ξ€œπ‘‡πœξ‚¬π‘‘π‘’ξ‚­ξ€œπ‘‘π‘‘,πœ‰π‘‘π‘‘+π‘‡πœξ€œΞ©ξ€œ(βˆ‡(π‘Žπ‘’),βˆ‡πœ‰)𝑑π‘₯𝑑𝑑+π‘‡πœξ€œΞ©ξ€œ(𝑓(𝑑,𝑒),πœ‰)𝑑π‘₯𝑑𝑑=π‘‡πœξ€œΞ©(β„Ž,πœ‰)𝑑π‘₯𝑑𝑑,(2.4) for all πœ‰βˆˆπΏπ‘(𝜏,𝑇;𝐿𝑝(Ξ©))∩𝐿2(𝜏,𝑇;𝑉), where βŸ¨β‹…,β‹…βŸ© denotes pairing in the space π‘‰ξ…ž+πΏπ‘ž(Ξ©), and βˆ‘(βˆ‡π‘’,βˆ‡π‘£)=𝑑𝑖=1(βˆ‡π‘’π‘–,βˆ‡π‘£π‘–).

Under conditions (2.2), it is known [17, page 284] that for any π‘’πœβˆˆπ» there exists at least one weak solution 𝑒=𝑒(𝑑,π‘₯) of (2.1), and also that the function 𝑑↦‖𝑒(𝑑)β€–2 is absolutely continuous on [𝜏,𝑇] and (𝑑/𝑑𝑑)‖𝑒(𝑑)β€–2=2βŸ¨π‘‘π‘’/𝑑𝑑,π‘’βŸ© for a.a. π‘‘βˆˆ(𝜏,𝑇).

Denote π‘Ÿ=(π‘Ÿ1,…,π‘Ÿπ‘‘),π‘Ÿπ‘–=max{1;𝑁(1/π‘žπ‘–βˆ’1/2)}. Any weak solution satisfies (𝑑𝑒/𝑑𝑑)βˆˆπΏπ‘ž(𝜏,𝑇;π»βˆ’π‘Ÿ(Ξ©)) and πΏπ‘ž(0,𝑇;π»βˆ’π‘Ÿ(Ξ©))=πΏπ‘ž1(0,𝑇;π»βˆ’π‘Ÿ1(Ξ©))Γ—β‹―Γ—πΏπ‘žπ‘‘(0,𝑇;π»βˆ’π‘Ÿπ‘‘(Ξ©)).(2.5)

If, additionally, we assume that that 𝑓(𝑑,𝑒) is continuously differentiable with respect to 𝑒 for any π‘‘βˆˆ[𝜏,𝑇],π‘’βˆˆβ„π‘‘, and 𝑓𝑒(𝑑,𝑒)𝑀,𝑀β‰₯βˆ’πΆ3(𝑑)|𝑀|2,βˆ€π‘€,π‘’βˆˆβ„π‘‘,(2.6) where 𝐢3(β‹…)∈𝐿1(𝜏,𝑇), 𝐢3(𝑑)β‰₯0, the weak solution of (2.1) is unique. Here, 𝑓𝑒 denotes the Jacobian matrix of 𝑓.

We consider also the following assumption: there exists 𝑅0>0 such that 𝑓𝑖(𝑑,𝑒)β‰₯𝑓𝑖(𝑑,𝑣),(2.7) for any π‘‘βˆˆ[𝜏,𝑇] and any 𝑒,π‘£βˆˆβ„π‘‘ such that 𝑒𝑖=𝑣𝑖 and 𝑒𝑗≀𝑣𝑗 if 𝑗≠𝑖, and |𝑒|,|𝑣|≀𝑅0, which means that the systems is cooperative in the ball with radius 𝑅0 centered at 0.

Consider the two problems: πœ•π‘’πœ•π‘‘βˆ’π‘ŽΞ”π‘’+𝑓1(𝑑,𝑒)=β„Ž1(𝑑,π‘₯),(𝑑,π‘₯)∈(𝜏,𝑇)Γ—Ξ©,𝑒|π‘₯βˆˆπœ•Ξ©π‘’|=0,𝑑=𝜏=π‘’πœ(π‘₯),(2.8)πœ•π‘’πœ•π‘‘βˆ’π‘ŽΞ”π‘’+𝑓2(𝑑,𝑒)=β„Ž2𝑒|(𝑑,π‘₯),(𝑑,π‘₯)∈(𝜏,𝑇)Γ—Ξ©,π‘₯βˆˆπœ•Ξ©=0,𝑒|𝑑=𝜏=π‘’πœ(π‘₯),(2.9) where 𝑓𝑗 are jointly continuous on [𝜏,𝑇]×ℝ𝑑. Among conditions (2.2) and (2.6)-(2.7), we shall consider the following: β„Ž1(𝑑,π‘₯)β‰€β„Ž2(𝑑,π‘₯),fora.a.(𝑑,π‘₯),𝑓𝑖1(𝑑,𝑒)β‰₯𝑓𝑖2(𝑑,𝑒),βˆ€π‘‘,𝑒.(2.10)

Lemma 2.1. If 𝑓𝑗 satisfy (2.2) and (2.10), then the constants 𝑝𝑖 have to be the same for 𝑓1 and 𝑓2.

Proof. Denote by 𝑝𝑗𝑖, 𝛼𝑗,𝐢𝑗1, and 𝐢𝑗2 the constants corresponding to 𝑓𝑗 in (2.2). By contradiction let, for example, 𝑝21>𝑝11. Take the sequence 𝑒𝑛=(𝑒1𝑛,0,…,0), where 𝑒1𝑛→+∞ as π‘›β†’βˆž. Then by (2.2), (2.10), and Young’s inequality, we have 𝛼2||𝑒1𝑛||𝑝21βˆ’πΆ22≀𝑓2𝑑,𝑒𝑛,𝑒𝑛≀𝑓1𝑑,𝑒𝑛,𝑒𝑛=𝑓11𝑑,𝑒𝑛𝑒1𝑛≀𝐢11ξ‚€||𝑒1+1𝑛||𝑝11(𝑝11βˆ’1)/𝑝11𝑒𝑛1ξ‚€||𝑒≀𝐢1+1𝑛||𝑝11.(2.11) But 𝑝21>𝑝11 implies the existence of 𝑛 such that 𝛼2|𝑒1𝑛|𝑝21βˆ’πΆ22>𝐢(1+|𝑒1𝑛|𝑝11), which is a contradiction. Hence, 𝑝21≀𝑝11.
Conversely, let 𝑝21<𝑝11. Then we take 𝑒𝑛=(𝑒1𝑛,0,…,0) with 𝑒1π‘›β†’βˆ’βˆž as π‘›β†’βˆž so that 𝛼1||𝑒1𝑛||𝑝11βˆ’πΆ12≀𝑓1𝑑,𝑒𝑛,𝑒𝑛≀𝑓2𝑑,𝑒𝑛,𝑒𝑛=𝑓12𝑑,𝑒𝑛𝑒1𝑛≀𝐢21ξ‚€||𝑒1+1𝑛||𝑝21(𝑝21βˆ’1)/𝑝21𝑒𝑛1ξ‚€||𝑒≀𝐢1+1𝑛||𝑝21.(2.12) As before, we obtain a contradiction, so 𝑝21=𝑝11.
Repeating similar arguments for the other 𝑝𝑗𝑖, we obtain that 𝑝1𝑖=𝑝2𝑖 for 𝑖=1,…,𝑑.

We recall [15] that under conditions (2.2) any solution 𝑒(β‹…) of (2.8) satisfies the inequality: ‖𝑒(𝑑)β€–2ξ€œ+2π›½π‘‘π‘ β€–βˆ‡π‘’(𝜏)β€–2π‘‘πœ+𝛼𝑑𝑖=1ξ€œπ‘‘π‘ β€–β€–π‘’π‘–β€–β€–(π‘Ÿ)𝑝𝑖𝐿𝑝𝑖(Ξ©)π‘‘π‘Ÿβ‰€β€–π‘’(𝑠)β€–2ξ€œ+πΆπ‘‘π‘ ξ‚€β€–β€–β„Ž1β€–β€–(π‘Ÿ)2+1π‘‘π‘Ÿ,(2.13) for some constant 𝐢>0. Of course, the same is valid for any solution of (2.9). From (2.13), for any 𝑇>𝜏 we obtain ‖𝑒(𝑑)β€–2β‰€β€–β€–π‘’πœβ€–β€–2ξ€œ+πΆπ‘‡πœξ‚€β€–β€–β„Ž1β€–β€–(π‘Ÿ)2+1π‘‘π‘Ÿ=𝐾2ξ€·β€–β€–π‘’πœβ€–β€–ξ€Έ,𝜏,π‘‡βˆ€πœβ‰€π‘‘β‰€π‘‡.(2.14)

We shall denote by 𝑒1(𝑑) the solution of (2.8) corresponding to the initial data 𝑒1𝜏 and by 𝑒2(𝑑) the solution of (2.9) corresponding to the initial data 𝑒2𝜏. Also, we take 𝑒+=max{𝑒,0} for π‘’βˆˆβ„.

We obtain the following comparison result.

Theorem 2.2. Assume that 𝑓𝑗,β„Žπ‘— satisfy (2.2), (2.6), and (2.10). If 𝑒1πœβ‰€π‘’2𝜏 and we suppose that 𝑓2 satisfies (2.7) with 𝑅20β‰₯2max{𝐾2(‖𝑒1πœβ€–,𝜏,𝑇),𝐾2(‖𝑒2πœβ€–,𝜏,𝑇)}, where 𝐾(β€–π‘’π‘—πœβ€–,𝜏,𝑇) is taken from (2.14), we have 𝑒1(𝑑)≀𝑒2(𝑑), for all π‘‘βˆˆ[𝜏,𝑇].

Remark 2.3. The results remain valid if, instead, 𝑓1 satisfies (2.7) with 𝑅20β‰₯𝐾2(‖𝑒1πœβ€–,𝑇).

Remark 2.4. If 𝑓2 satisfies (2.7) for an arbitrary 𝑅0>0 (i.e., in the whole space ℝ𝑑), then the result is true for any initial data 𝑒1πœβ‰€π‘’2𝜏.

Proof. Let 𝑔2(𝑑,𝑒)=𝑓2(𝑑,𝑒)+𝐢3(𝑑)𝑒. The function 𝑔2(𝑑,β‹…) satisfies (2.6) with 𝐢3≑0. For any 𝑒1,𝑒2βˆˆβ„π‘‘ define 𝑣2(𝑒1,𝑒2) by 𝑣𝑖2=𝑒𝑖2,if𝑒𝑖1β‰₯𝑒𝑖2,𝑒𝑖1,if𝑒𝑖1<𝑒𝑖2.(2.15) Note that 𝑒1βˆ’π‘£2=(𝑒1βˆ’π‘’2)+ and 𝑣2βˆ’π‘’2=βˆ’(𝑒2βˆ’π‘’1)+, so 𝑣𝑖2≀𝑒𝑖2 for all 𝑖. For the function (𝑒1βˆ’π‘’2)+, we can obtain by (2.10) and the mean value theorem that 12𝑑‖‖𝑒𝑑𝑑1βˆ’π‘’2ξ€Έ+β€–β€–2‖‖𝑒+π‘Ž1βˆ’π‘’2ξ€Έ+β€–β€–2π‘‰ξ€œβ‰€βˆ’Ξ©ξ‚€π‘“1𝑑,𝑒1ξ€Έβˆ’π‘“2𝑑,𝑒2ξ€Έ,𝑒1βˆ’π‘’2ξ€Έ+ξ‚ξ€œπ‘‘π‘₯β‰€βˆ’Ξ©ξ‚€π‘“2𝑑,𝑒1ξ€Έβˆ’π‘“2𝑑,𝑣2ξ€Έ,𝑒1βˆ’π‘’2ξ€Έ+ξ‚ξ€œπ‘‘π‘₯βˆ’Ξ©ξ‚€π‘“2𝑑,𝑣2ξ€Έβˆ’π‘“2𝑑,𝑒2ξ€Έ,𝑒1βˆ’π‘’2ξ€Έ+ξ‚ξ€œπ‘‘π‘₯=βˆ’Ξ©ξ‚€π‘”2𝑒𝑑,𝑣𝑑,π‘₯,𝑒1,𝑒2𝑒1βˆ’π‘£2ξ€Έ,𝑒1βˆ’π‘’2ξ€Έ+𝑑π‘₯+𝐢3(‖‖𝑒𝑑)1βˆ’π‘’2ξ€Έ+β€–β€–2βˆ’ξ€œΞ©ξ‚€π‘“2𝑑,𝑣2ξ€Έβˆ’π‘“2𝑑,𝑒2ξ€Έ,𝑒1βˆ’π‘’2ξ€Έ+ξ‚ξ€œπ‘‘π‘₯=βˆ’Ξ©ξ‚€π‘”2𝑒𝑑,𝑣𝑑,π‘₯,𝑒1,𝑒2𝑒1βˆ’π‘’2ξ€Έ+,𝑒1βˆ’π‘’2ξ€Έ+𝑑π‘₯+𝐢3‖‖𝑒(𝑑)1βˆ’π‘’2ξ€Έ+β€–β€–2βˆ’ξ€œΞ©ξ‚€π‘“2𝑑,𝑣2ξ€Έβˆ’π‘“2𝑑,𝑒2ξ€Έ,𝑒1βˆ’π‘’2ξ€Έ+𝑑π‘₯≀𝐢3‖‖𝑒(𝑑)1βˆ’π‘’2ξ€Έ+β€–β€–2βˆ’ξ€œΞ©ξ‚€π‘“2𝑑,𝑣2ξ€Έβˆ’π‘“2𝑑,𝑒2ξ€Έ,𝑒1βˆ’π‘’2ξ€Έ+𝑑π‘₯.(2.16)
For all 𝑑, we have 𝑓2𝑑,𝑣2ξ€Έβˆ’π‘“2𝑑,𝑒2ξ€Έ,𝑒1βˆ’π‘’2ξ€Έ+=ξ“π‘–βˆˆπ½ξ€·π‘“π‘–2𝑑,𝑣2ξ€Έβˆ’π‘“π‘–2𝑑,𝑒2𝑒𝑖1βˆ’π‘’π‘–2ξ€Έ+,(2.17) where 𝑒𝑖1βˆ’π‘’π‘–2>0, for π‘–βˆˆπ½, and 𝑒𝑖1βˆ’π‘’π‘–2≀0 if π‘–βˆ‰π½. For any π‘–βˆˆπ½, we have that 𝑣𝑖2=𝑒𝑖2, and then by 𝑣2≀𝑒2, |𝑣2|2≀|𝑒1|2+|𝑒2|2, (2.14), and (2.7), we get 𝑓𝑖2𝑑,𝑣2ξ€Έβˆ’π‘“π‘–2𝑑,𝑒2ξ€Έβ‰₯0.(2.18)
By Gronwall’s lemma, we get ‖‖𝑒1(𝑑)βˆ’π‘’2ξ€Έ(𝑑)+β€–β€–2≀‖‖𝑒1πœβˆ’π‘’2πœξ€Έ+β€–β€–2π‘’βˆ«π‘‘πœ2𝐢3(𝑠)𝑑𝑠=0.(2.19) Thus β€–(𝑒1βˆ’π‘’2)+β€–=0, which means that 𝑒𝑖1(π‘₯,𝑑)βˆ’π‘’π‘–2(π‘₯,𝑑)≀0, for a.a. π‘₯∈Ω, and all π‘–βˆˆ{1,…,𝑑}, π‘‘βˆˆ[𝜏,𝑇].

Remark 2.5. In the scalar case, that is, 𝑑=1, condition (2.7) is trivially satisfied.

When condition (2.6) fails to be true, we will obtain a weak comparison principle.

Define a sequence of smooth functions πœ“π‘˜βˆΆβ„+β†’[0,1] satisfying πœ“π‘˜(⎧βŽͺ⎨βŽͺβŽ©π‘ )=1,if0β‰€π‘ β‰€π‘˜,0β‰€πœ“π‘˜(𝑠)≀1,ifπ‘˜β‰€π‘ β‰€π‘˜+1,0,if𝑠β‰₯π‘˜+1.(2.20)

For every π‘˜β‰₯1 we put π‘“π‘–π‘˜(𝑑,𝑒)=πœ“π‘˜(|𝑒|)𝑓𝑖(𝑑,𝑒)+(1βˆ’πœ“π‘˜(|𝑒|))𝑔𝑖(𝑒), where 𝑔𝑖(𝑒)=|𝑒𝑖|π‘π‘–βˆ’2𝑒𝑖. Then π‘“π‘˜βˆˆβ„‚([𝜏,𝑇]×ℝ𝑑;ℝ𝑑) and for any 𝐴>0, supπ‘‘βˆˆ[𝜏,𝑇]sup|𝑒|≀𝐴||π‘“π‘˜||(𝑑,𝑒)βˆ’π‘“(𝑑,𝑒)β†’0,asπ‘˜β†’βˆž.(2.21)

Let πœŒπœ€βˆΆβ„π‘‘β†’β„+ be a mollifier, that is, πœŒπœ–βˆˆβ„‚βˆž0(ℝ𝑑;ℝ), suppπœŒπœ–βŠ‚π΅πœ–={π‘₯βˆˆβ„π‘‘βˆΆ|π‘₯|<πœ–}, βˆ«β„π‘‘πœŒπœ–(𝑠)𝑑𝑠=1 and πœŒπœ–(𝑠)β‰₯0 for all π‘ βˆˆβ„π‘‘. We define the functions π‘“πœ–π‘˜(ξ€œπ‘‘,𝑒)=β„π‘‘πœŒπœ–(𝑠)π‘“π‘˜(𝑑,π‘’βˆ’π‘ )𝑑𝑠.(2.22) Since for any π‘˜β‰₯1π‘“π‘˜ is uniformly continuous on [0,𝑇]Γ—[βˆ’π‘˜βˆ’1,π‘˜+1], there exist πœ–π‘˜βˆˆ(0,1) such that for all 𝑒 satisfying |𝑒|β‰€π‘˜, and for all 𝑠 for which |π‘’βˆ’π‘ |<πœ–π‘˜ we have supπ‘‘βˆˆ[𝜏,𝑇]||π‘“π‘˜(𝑑,𝑒)βˆ’π‘“π‘˜||≀1(𝑑,𝑠)π‘˜.(2.23) We put π‘“π‘˜(𝑑,𝑒)=π‘“πœ–π‘˜π‘˜(𝑑,𝑒). Then π‘“π‘˜(𝑑,β‹…)βˆˆβ„‚βˆž(ℝ𝑑;ℝ𝑑), for all π‘‘βˆˆ[𝜏,𝑇],π‘˜β‰₯1.

For further arguments we need the following technical result [16, Lemma 2].

Lemma 2.6. Let 𝑓 satisfy (2.2). For all π‘˜β‰₯1, the following statements hold: supπ‘‘βˆˆ[𝜏,𝑇]sup|𝑒|≀𝐴||π‘“π‘˜||(𝑑,𝑒)βˆ’π‘“(𝑑,𝑒)β†’0,asπ‘˜β†’βˆž,βˆ€π΄>0,(2.24)𝑑𝑖=1||π‘“π‘˜π‘–||(𝑑,𝑒)𝑝𝑖/(π‘π‘–βˆ’1)≀𝐷11+𝑑𝑖=1||𝑒𝑖||𝑝𝑖ξƒͺ,ξ€·π‘“π‘˜ξ€Έ(𝑑,𝑒),𝑒β‰₯𝛾𝑑𝑖=1||𝑒𝑖||π‘π‘–βˆ’π·2,𝑓(2.25)π‘˜π‘’ξ€Έ(𝑑,𝑒)𝑀,𝑀β‰₯βˆ’π·3(π‘˜)|𝑀|2,βˆ€π‘’,𝑀,(2.26) where 𝐷3(π‘˜) is a nonnegative number, and the positive constants 𝐷1, 𝐷2β‰₯𝐢2, 𝛾 do not depend on π‘˜.

Consider first the scalar case.

Theorem 2.7. Let 𝑑=1. Assume that 𝑓𝑗,β„Žπ‘— satisfy (2.2) and (2.10). If 𝑒1πœβ‰€π‘’2𝜏, there exist two solutions 𝑒1,𝑒2 (of (2.8) and (2.9), resp.) such that 𝑒1(𝑑)≀𝑒2(𝑑) for all π‘‘βˆˆ[𝜏,𝑇].

Proof. For the functions 𝑓𝑗 we take the approximations π‘“π‘˜π‘— (defined in Lemma 2.6), which satisfy (2.24)–(2.26), and consider the problems πœ•π‘’πœ•π‘‘βˆ’π‘ŽΞ”π‘’+π‘“π‘˜π‘—(𝑑,𝑒)=β„Žπ‘—(𝑑,π‘₯),(𝑑,π‘₯)∈(𝜏,𝑇)Γ—Ξ©,𝑒|π‘₯βˆˆπœ•Ξ©π‘’|=0,𝑑=𝜏=π‘’πœ,(2.27) for 𝑗=1,2. Problem (2.27) has a unique solution for any initial data π‘’πœβˆˆπ». In view of Lemma 2.1, the constant 𝑝 is the same for 𝑓1 and 𝑓2. We note that 𝑓1π‘˜(𝑑,𝑒)=πœ“π‘˜(|𝑒|)𝑓1ξ€·(𝑑,𝑒)+1βˆ’πœ“π‘˜ξ€Έ(|𝑒|)|𝑒|π‘βˆ’2𝑒β‰₯πœ“π‘˜(|𝑒|)𝑓2ξ€·(𝑑,𝑒)+1βˆ’πœ“π‘˜ξ€Έ(|𝑒|)|𝑒|π‘βˆ’2𝑒=𝑓2π‘˜(𝑑,𝑒).(2.28) Then, it is clear that π‘“π‘˜1(𝑑,𝑒)β‰₯π‘“π‘˜2(𝑑,𝑒) for every (𝑑,𝑒).
By Theorem 2.2 we know that as 𝑒1πœβ‰€π‘’2𝜏, we have π‘’π‘˜1(𝑑)β‰€π‘’π‘˜2(𝑑), for all π‘‘βˆˆ[𝜏,𝑇], for the corresponding solutions of (2.27).
In view of Lemma 2.6, one can obtain in a standard way that (2.13) is satisfied for the solutions of (2.27) with a constant 𝐢 not depending on π‘˜ and replacing 𝛼 by 𝛾. Hence, the sequences π‘’π‘˜π‘—(β‹…) are bounded in 𝐿∞(𝜏,𝑇;𝐻)∩𝐿2(𝜏,𝑇;𝑉)βˆ©πΏπ‘(𝜏,𝑇;𝐿𝑝(Ξ©)). It follows from (2.25) that π‘“π‘—π‘˜(β‹…,π‘’π‘˜π‘—(β‹…)) are bounded in πΏπ‘ž(𝜏,𝑇;πΏπ‘ž(Ξ©)) and also that {(π‘‘π‘’π‘˜/𝑑𝑑)(β‹…)} is bounded in πΏπ‘ž(𝜏,𝑇;π»βˆ’π‘Ÿ(Ξ©)), where π‘Ÿπ‘–=max{1;(1/2βˆ’1/𝑝𝑖)𝑁}. By the Compactness Lemma [18], we have that for some functions 𝑒𝑗=𝑒𝑗(𝑑,π‘₯), 𝑗=1,2: π‘’π‘˜π‘—β†’π‘’π‘—weaklystarinπΏβˆžπ‘’(𝜏,𝑇;𝐻),(2.29)π‘˜π‘—β†’π‘’π‘—in𝐿2(𝜏,𝑇;𝐻),π‘’π‘˜π‘—(𝑑)→𝑒𝑗𝑒(𝑑)in𝐻fora.a.π‘‘βˆˆ(𝜏,𝑇),(2.30)π‘˜π‘—(𝑑,π‘₯)→𝑒𝑗𝑒(𝑑,π‘₯)fora.a.(𝑑,π‘₯)∈(𝜏,𝑇)Γ—Ξ©,(2.31)π‘˜π‘—β†’π‘’π‘—weaklyin𝐿2(𝜏,𝑇;𝑉),(2.32)π‘‘π‘’π‘˜π‘—β†’π‘‘π‘‘π‘‘π‘’π‘—π‘‘π‘‘weaklyinπΏπ‘ž(𝜏,𝑇;π»βˆ’π‘Ÿπ‘’(Ξ©)),(2.33)π‘˜π‘—β†’π‘’π‘—weaklyin𝐿𝑝(𝜏,𝑇;𝐿𝑝(Ξ©)).(2.34) Also, arguing as in [19, page 3037] we obtain π‘’π‘˜π‘—(𝑑)→𝑒𝑗[].(𝑑)weaklyinπ»βˆ€π‘‘βˆˆπœ,𝑇(2.35) Moreover, by (2.24) and (2.31) we have π‘“π‘—π‘˜(𝑑,π‘’π‘˜(𝑑,π‘₯))→𝑓𝑗(𝑑,𝑒(𝑑,π‘₯)) for a.a. (𝑑,π‘₯)∈(𝜏,𝑇)Γ—Ξ© and then the boundedness of π‘“π‘—π‘˜(β‹…,π‘’π‘˜π‘—(β‹…)) in πΏπ‘ž(𝜏,𝑇;πΏπ‘ž(Ξ©)) implies that π‘“π‘—π‘˜(β‹…,π‘’π‘˜π‘—(β‹…)) converges to 𝑓(β‹…,𝑒(β‹…)) weakly in πΏπ‘ž(𝜏,𝑇;πΏπ‘ž(Ξ©)) [18]. It follows that 𝑒1(β‹…),𝑒2(β‹…) are weak solutions of (2.8) and (2.9), respectively, with 𝑒1(𝜏)=𝑒1𝜏, 𝑒2(𝜏)=𝑒2𝜏.
Moreover, one can prove that π‘’π‘˜π‘—(𝑑)→𝑒𝑗[].(𝑑)stronglyinπ»βˆ€π‘‘βˆˆπœ,𝑇(2.36) Indeed, we define the functions π½π‘—π‘˜(𝑑)=β€–π‘’π‘˜π‘—(𝑑)β€–2βˆ«βˆ’πΆπ‘‘πœ(β€–β„Žπ‘—(𝑠)β€–2+1)𝑑𝑠, 𝐽𝑗(𝑑)=‖𝑒𝑗(𝑑)β€–2βˆ«βˆ’πΆπ‘‘πœ(β€–β„Žπ‘—(𝑠)β€–2+1)𝑑𝑠, which are nonincreasing in view of (2.13). Also, from (2.30) we have π½π‘—π‘˜(𝑑)→𝐽𝑗(𝑑) for a.a. π‘‘βˆˆ(𝜏,𝑇). Then one can prove that limsupπ‘˜β†’βˆžπ½π‘—π‘˜(𝑑)≀𝐽𝑗(𝑑) for all π‘‘βˆˆ[𝜏,𝑇] (see [15, page 623] for the details). Hence, limsupπ‘˜β†’βˆžβ€–π‘’π‘˜π‘—(𝑑)‖≀‖𝑒𝑗(𝑑)β€–. Together with (2.35) this implies (2.36) (see again [15, page 623] for more details).
Hence, passing to the limit we obtain 𝑒1(𝑑)≀𝑒2[].(𝑑),βˆ€π‘‘βˆˆπœ,𝑇(2.37)

Further, let us prove the general case for an arbitrary π‘‘βˆˆβ„•.

Theorem 2.8. Assume that 𝑓𝑗,β„Žπ‘— satisfy (2.2) and (2.10). Also, suppose that either 𝑓1 or 𝑓2 satisfies (2.7) for an arbitrary 𝑅0>0. If 𝑒1πœβ‰€π‘’2𝜏, there exist two solutions 𝑒1,𝑒2 (of (2.8) and (2.9), resp.) such that 𝑒1(𝑑)≀𝑒2(𝑑), for all π‘‘βˆˆ[𝜏,𝑇].

Proof. Let 𝑓1 be the function which satisfies (2.7). We take the approximations π‘“π‘˜1,π‘“π‘˜2 (defined in Lemma 2.6), which satisfy (2.24)–(2.26). Then, we consider problems (2.27).
In view of Lemma 2.1, the constants 𝑝𝑖 are the same for 𝑓1 and 𝑓2. We note that 𝑓𝑖1π‘˜(𝑑,𝑒)=πœ“π‘˜(|𝑒|)𝑓𝑖1ξ€·(𝑑,𝑒)+1βˆ’πœ“π‘˜ξ€Έ||𝑒(|𝑒|)𝑖||π‘π‘–βˆ’2𝑒𝑖β‰₯πœ“π‘˜(|𝑒|)𝑓𝑖2ξ€·(𝑑,𝑒)+1βˆ’πœ“π‘˜ξ€Έ||𝑒(|𝑒|)𝑖||π‘π‘–βˆ’2𝑒𝑖=𝑓𝑖2π‘˜(𝑑,𝑒).(2.38) Then, it is clear that π‘“π‘˜1(𝑑,𝑒)β‰₯π‘“π‘˜2(𝑑,𝑒) for every (𝑑,𝑒).
Using Lemma 2.6 it is standard to obtain estimate (2.14) with a constant 𝐢 not depending on π‘˜. Hence, the solutions π‘’π‘˜π‘—(β‹…) of (2.27) satisfy β€–β€–π‘’π‘˜π‘—β€–β€–(𝑑)2β‰€β€–β€–π‘’π‘—πœβ€–β€–2ξ€œ+πΆπ‘‡πœξ‚€β€–β€–β„Žπ‘—β€–β€–(π‘Ÿ)2+1π‘‘π‘Ÿ=𝐾2ξ€·β€–β€–π‘’π‘—πœβ€–β€–ξ€Έ.,𝜏,𝑇(2.39) We note that 𝑓1π‘˜(𝑑,𝑒)=𝑓1(𝑑,𝑒),(2.40) if |𝑒|β‰€π‘˜, since in such a case πœ“π‘˜(|𝑒|)=1. Hence, if π‘˜2β‰₯2max{𝐾2(‖𝑒1πœβ€–,𝜏,𝑇),𝐾2(‖𝑒2πœβ€–,𝜏,𝑇)}, the functions 𝑓1π‘˜ satisfy condition (2.7) with 𝑅0=π‘˜. Therefore, for any π‘‘βˆˆ[𝜏,𝑇] and any 𝑒,π‘£βˆˆβ„π‘‘ such that 𝑒𝑖=𝑣𝑖 and 𝑒𝑗≀𝑣𝑗 if 𝑗≠𝑖, and |𝑒|,|𝑣|β‰€π‘˜βˆ’1, we have π‘“π‘˜1(ξ€œπ‘‘,𝑒)=β„π‘‘πœŒπœ–π‘˜(𝑠)𝑓1π‘˜(ξ€œπ‘‘,π‘’βˆ’π‘ )𝑑𝑠β‰₯β„π‘‘πœŒπœ–π‘˜(𝑠)𝑓1π‘˜(𝑑,π‘£βˆ’π‘ )𝑑𝑠=π‘“π‘˜2(𝑑,𝑒).(2.41) Thus, if (π‘˜βˆ’1)2β‰₯2max{𝐾2(‖𝑒1πœβ€–,𝑇),𝐾2(‖𝑒2πœβ€–,𝑇)}, the functions π‘“π‘˜1 satisfy condition (2.7) with 𝑅0=π‘˜βˆ’1.
By Theorem 2.2 we know that as 𝑒1πœβ‰€π‘’2𝜏, we have π‘’π‘˜1(𝑑)β‰€π‘’π‘˜2(𝑑), for all π‘‘βˆˆ[𝜏,𝑇], π‘˜β‰₯1+(2max{𝐾2(‖𝑒1πœβ€–,𝑇),𝐾2(‖𝑒2πœβ€–,𝑇)})1/2, for the corresponding solutions of (2.27).
Repeating the same proof of Theorem 2.7, we obtain that the sequences π‘’π‘˜1,π‘’π‘˜2 converge (up to a subsequence) in the sense of (2.29)–(2.36) to the solutions 𝑒1,𝑒2 of problems (2.8) and (2.9), respectively. Also, it holds 𝑒1(𝑑)≀𝑒2[].(𝑑),βˆ€π‘‘βˆˆπœ,𝑇(2.42)

In the applications we need to generalize this theorem to the case where the constant 𝛼 can be negative. We shall do this when 𝑓1,𝑓2 have sublinear growth (i.e., 𝑝𝑖=2 for all 𝑖). Consider for (2.1) the following conditions: ||||𝑓(𝑑,𝑒)≀𝐢1(1+|𝑒|),(𝑓(𝑑,𝑒),𝑒)β‰₯𝛼|𝑒|2βˆ’πΆ2,(2.43) where π›Όβˆˆβ„, and 𝐢1,𝐢2>0.

Let 𝑓1,𝑓2 satisfy (2.43) with constants 𝛼𝑗,𝐢𝑗1,𝐢𝑗2,𝑗=1,2. Then if min{𝛼1,𝛼2}≀0, we make in (2.1) the change of variable 𝑣=π‘’βˆ’π›½π‘‘π‘’, where 𝛽>βˆ’min{𝛼1,𝛼2}. Hence, multiplying (2.8) and (2.9) by π‘’βˆ’π›½π‘‘ we have πœ•π‘£πœ•π‘‘βˆ’π‘ŽΞ”π‘£+π‘’βˆ’π›½π‘‘π‘“1𝑑,𝑒𝛽𝑑𝑣+𝛽𝑣=π‘’βˆ’π›½π‘‘β„Ž1(𝑑,π‘₯),(𝑑,π‘₯)∈(𝜏,𝑇)Γ—Ξ©,𝑣|π‘₯βˆˆπœ•Ξ©=0,𝑣|𝑑=𝜏=π‘’βˆ’π›½πœπ‘’πœ(π‘₯),(2.44)πœ•π‘£πœ•π‘‘βˆ’π‘ŽΞ”π‘£+π‘’βˆ’π›½π‘‘π‘“2𝑑,𝑒𝛽𝑑𝑣+𝛽𝑣=π‘’βˆ’π›½π‘‘β„Ž2(𝑑,π‘₯),(𝑑,π‘₯)∈(𝜏,𝑇)Γ—Ξ©,𝑣|π‘₯βˆˆπœ•Ξ©=0,𝑣|𝑑=𝜏=π‘’βˆ’π›½πœπ‘’πœ(π‘₯).(2.45)

It is easy to check that if 𝑣(𝑑) is a weak solution of (2.44), then 𝑒(𝑑)=𝑒𝛽𝑑𝑣(𝑑) is a weak solution of (2.8) (and the same is true, of course, for (2.45) and (2.9)). Conversely, if 𝑒(𝑑) is a weak solution of (2.8), then 𝑣(𝑑)=π‘’βˆ’π›½π‘‘π‘’(𝑑) is a weak solution of (2.44).

The functions 𝑓𝑗(𝑑,𝑣)=π‘’βˆ’π›½π‘‘π‘“π‘—(𝑑,𝑒𝛽𝑑𝑣)+𝛽𝑣 satisfy (2.2) with 𝑝𝑖=2 for all 𝑖. Indeed, ||𝑓𝑗||(𝑑,𝑣)β‰€π‘’βˆ’π›½π‘‘πΆπ‘—1ξ€·1+𝑒𝛽𝑑𝐢|𝑣|+𝛽|𝑣|≀𝑗1𝑓(1+|𝑣|),𝑗(𝑑,𝑣),𝑣β‰₯π‘’βˆ’2𝛽𝑑𝑓𝑗𝑑,𝑒𝛽𝑑𝑣,𝑒𝛽𝑑𝑣+𝛽|𝑣|2β‰₯𝛼𝑗+𝛽|𝑣|2βˆ’πΆπ‘—2,(2.46) where 𝛼𝑗+𝛽>0.

Then, we obtain the following.

Theorem 2.9. Assume that 𝑓𝑗,β„Žπ‘— satisfy (2.43) and (2.10). Also, suppose that either 𝑓1 or 𝑓2 satisfies (2.7) for an arbitrary 𝑅0>0. If 𝑒1πœβ‰€π‘’2𝜏, there exist two solutions 𝑒1,𝑒2 (of (2.8) and (2.9), resp.) such that 𝑒1(𝑑)≀𝑒2(𝑑), for all π‘‘βˆˆ[𝜏,𝑇].

Proof. We consider problems (2.44) and (2.45). In view of (2.46) 𝑓𝑗(𝑑,𝑣)=π‘’βˆ’π›½π‘‘π‘“π‘—(𝑑,𝑒𝛽𝑑𝑣)+𝛽𝑣 satisfy (2.2). Also, defining ξ‚β„Žπ‘—(𝑑,π‘₯)=π‘’βˆ’π›½π‘‘β„Žπ‘—(𝑑,π‘₯) it is clear that (2.10) holds. Finally, if, for example, 𝑓1 satisfies (2.7) for any 𝑅0>0, then it is obvious that for 𝑓1 is true as well.
Hence, by Theorem 2.8 there exist two solutions 𝑣1,𝑣2 (of (2.44) and (2.45), resp.), with 𝑣𝑗(𝜏)=π‘’βˆ’π›½πœπ‘’π‘—πœ such that 𝑣1(𝑑)≀𝑣2(𝑑) for all π‘‘βˆˆ[𝜏,𝑇]. Thus 𝑒1(𝑑)=𝑒𝛽𝑑𝑣1(𝑑)≀𝑒𝛽𝑑𝑣2(𝑑)=𝑒2[],(𝑑),forπ‘‘βˆˆπœ,𝑇(2.47) and 𝑒1,𝑒2 are solutions of (2.8) and (2.9), respectively such that 𝑒𝑗(𝜏)=π‘’π‘—πœ.

Remark 2.10. If 𝑓𝑗 satisfy (2.6), then the solutions 𝑒1,𝑒2 given in Theorem 2.9 are unique for the corresponding initial data.

3. Comparison for Positive Solutions

Denote ℝ𝑑+={π‘’βˆˆβ„π‘‘βˆΆπ‘’π‘–β‰₯0}. Let us consider the previous results in the case where the solutions have to be positive. Consider now the following conditions. β„ŽThematrixπ‘Žisdiagonal,(3.1)𝑖(𝑑,π‘₯)βˆ’π‘“π‘–ξ€·π‘‘,𝑒1,…,π‘’π‘–βˆ’1,0,𝑒𝑖+1,…,𝑒𝑑β‰₯0,(3.2) for all 𝑖, a.e. (𝑑,π‘₯)∈(𝜏,𝑇)Γ—Ξ© and 𝑒𝑗β‰₯0 if 𝑗≠𝑖. Obviously, in the scalar case these conditions just mean that β„Ž(𝑑,π‘₯)βˆ’π‘“(𝑑,0)β‰₯0,(3.3) for a.e. (𝑑,π‘₯)∈(𝜏,𝑇)Γ—Ξ©.

It is well known (see [16, Lemma 5] for a detailed proof) that if we assume conditions (2.2) only for π‘’βˆˆβ„π‘‘+, and also (2.6) and (3.1)-(3.2), then for any π‘’πœβ‰₯0 there exists a unique weak solution 𝑒(β‹…) of (2.1). Moreover, 𝑒(β‹…) is such that 𝑒(𝑑)β‰₯0 for all π‘‘βˆˆ[𝜏,𝑇].

On the other hand, if we assume these conditions except (2.6), then there exists at least one weak solution 𝑒(β‹…) of (2.1) such that 𝑒(𝑑)β‰₯0 for all π‘‘βˆˆ[𝜏,𝑇] [16, Theorem 4]. Moreover, we can prove the following.

Lemma 3.1. Assume conditions (2.2), (2.6) only for π‘’βˆˆβ„π‘‘+, and also (3.1)-(3.2). Then there exists a weak solution 𝑒(β‹…) of (2.1), which is unique in the class of solutions satisfying 𝑒(𝑑)β‰₯0 for all π‘‘βˆˆ[𝜏,𝑇].

Proof. Let 𝑒1,𝑒2 be two solutions with 𝑒𝑖(𝜏)=π‘’πœ, 𝑖=1,2 such that 𝑒𝑖(𝑑)β‰₯0 for all 𝑑. Denote 𝑀(𝑑)=𝑒1(𝑑)βˆ’π‘’2(𝑑). Then in a standard way by the mean value theorem, we obtain 12𝑑(𝑑𝑑‖𝑀𝑑)β€–2ξ€œβ‰€βˆ’Ξ©ξ€·π‘“ξ€·π‘‘,𝑒1(𝑑,π‘₯)βˆ’π‘“π‘‘,𝑒2(ξ€Έξ€Έξ€œπ‘‘,π‘₯),𝑀(𝑑,π‘₯)𝑑π‘₯=βˆ’Ξ©ξ€·π‘“π‘’ξ€·ξ€·π‘‘,𝑣𝑑,π‘₯,𝑒1,𝑒2𝑀(𝑑,π‘₯),𝑀(𝑑,π‘₯)𝑑π‘₯≀𝐢3(𝑑)‖𝑀(𝑑)β€–2,(3.4) where 𝑣(𝑑,π‘₯,𝑒1,𝑒2)∈𝐿(𝑒1(𝑑,π‘₯),𝑒2(𝑑,π‘₯))={𝛼𝑒1(𝑑,π‘₯)+(1βˆ’π›Ό)𝑒2(𝑑,π‘₯)βˆΆπ›Όβˆˆ0,1]} so that 𝑣(𝑑,π‘₯,𝑒1,𝑒2)β‰₯0. The uniqueness follows from Gronwall’s lemma

We prove also a result, which is similar to Lemma 2.1. Denote by 𝑝𝑗𝑖, 𝛼𝑗,𝐢𝑗1, and 𝐢𝑗2 the constants corresponding to 𝑓𝑗 in (2.2) for problems (2.8) and (2.9), respectively. Arguing as in the proof of Lemma 2.1 we obtain the following lemma.

Lemma 3.2. If 𝑓𝑗 satisfy (2.2) and (2.10) for π‘’βˆˆβ„+𝑑, then 𝑝1𝑖β‰₯𝑝2𝑖 for all 𝑖.

Theorem 3.3. Let 𝑓𝑗,β„Žπ‘— satisfy (2.6) and (3.1)-(3.2). Assume that 𝑓𝑗,β„Žπ‘— satisfy (2.2) and (2.10) for π‘’βˆˆβ„+𝑑. If 0≀𝑒1πœβ‰€π‘’2𝜏 and one supposes that 𝑓2 satisfies (2.7) for π‘’βˆˆβ„π‘‘+ with 𝑅20β‰₯2max{𝐾2(‖𝑒1πœβ€–,𝜏,𝑇),𝐾2(‖𝑒2πœβ€–,𝜏,𝑇)}, where 𝐾(β€–π‘’π‘—πœβ€–,𝜏,𝑇) is taken from (2.14), one has 0≀𝑒1(𝑑)≀𝑒2(𝑑) for all π‘‘βˆˆ[𝜏,𝑇], where 𝑒1(β‹…),𝑒2(β‹…) are the solutions corresponding to 𝑒1𝜏 and 𝑒2𝜏, respectively.

Proof. As the solutions 𝑒1(β‹…), 𝑒2(β‹…) corresponding to 𝑒1𝜏 and 𝑒2𝜏 are nonnegative, repeating exactly the same steps of the proof of Theorem 2.2 we obtain the desired result.

Remark 3.4. The results remain valid if, instead, 𝑓1 satisfies (2.7) with the same 𝑅0.

Remark 3.5. If 𝑓2 satisfies (2.7) for an arbitrary 𝑅0>0 (i.e., in the whole space ℝ𝑑), then the result is true for any initial data 0≀𝑒1πœβ‰€π‘’2𝜏.

We shall need also the following modification of Theorem 3.3.

Theorem 3.6. Let 𝑓𝑗,β„Žπ‘— satisfy (2.6) and (3.1)-(3.2). Assume that 𝑓𝑗,β„Žπ‘— satisfy (2.2) and (2.10) for π‘’βˆˆβ„π‘‘+. Let 0≀𝑒1πœβ‰€π‘’2𝜏. One supposes that 𝑓2 satisfies 𝑓𝑖2(𝑑,𝑒)β‰₯𝑓𝑖2(𝑑,𝑣)βˆ’πœ€,(3.5) for any π‘‘βˆˆ[𝜏,𝑇] and any 𝑒,π‘£βˆˆβ„π‘‘+ such that 𝑒𝑖=𝑣𝑖 and 𝑒𝑗≀𝑣𝑗 if 𝑗≠𝑖, and |𝑒|,|𝑣|≀𝑅0 with 𝑅20β‰₯2max{𝐾2(‖𝑒1πœβ€–,𝜏,𝑇),𝐾2(‖𝑒2πœβ€–,𝜏,𝑇)}, where 𝐾(β€–π‘’π‘—πœβ€–,𝜏,𝑇) is taken from (2.14).
Then there exists a constant 𝐢(𝜏,𝑇) such that ‖‖𝑒1(𝑑)βˆ’π‘’2(𝑑)+β€–β€–[],≀𝐢(𝜏,𝑇)πœ€,βˆ€π‘‘βˆˆπœ,𝑇(3.6) where 𝑒1(β‹…),𝑒2(β‹…) are the solutions corresponding to 𝑒1𝜏 and 𝑒2𝜏, respectively.

Proof. Arguing as in the proof of Theorem 2.2 we obtain the inequality 12𝑑‖‖𝑒𝑑𝑑1βˆ’π‘’2ξ€Έ+β€–β€–2‖‖𝑒+π‘Ž1βˆ’π‘’2ξ€Έ+β€–β€–2𝑉≀𝐢3‖‖𝑒(𝑑)1βˆ’π‘’2ξ€Έ+β€–β€–2βˆ’ξ€œΞ©ξ‚€π‘“2𝑑,𝑣2ξ€Έβˆ’π‘“2𝑑,𝑒2ξ€Έ,𝑒1βˆ’π‘’2ξ€Έ+𝑑π‘₯,(3.7) where 𝑣2 is defined in (2.15).
Using (2.17), 𝑣2≀𝑒2,|𝑣2|2≀|𝑒1|2+|𝑒2|2, (2.14), and (3.5), we get 𝑓𝑖2𝑑,𝑣2ξ€Έβˆ’π‘“π‘–2𝑑,𝑒2ξ€Έβ‰₯βˆ’πœ€.(3.8) Thus 𝑑‖‖𝑒𝑑𝑑1βˆ’π‘’2ξ€Έ+β€–β€–2≀2𝐢3(‖‖𝑒𝑑)1βˆ’π‘’2ξ€Έ+β€–β€–2ξ€œ+2πœ€Ξ©ξ“π‘–βˆˆπ½ξ€·π‘’π‘–1βˆ’π‘’π‘–2ξ€Έ+≀𝑑π‘₯2𝐢3‖‖𝑒(𝑑)+11βˆ’π‘’2ξ€Έ+β€–β€–2+πΎπœ€2,(3.9) for some constant 𝐾>0. By Gronwall’s lemma, we get ‖‖𝑒1(𝑑)βˆ’π‘’2(𝑑)+β€–β€–2≀‖‖𝑒1πœβˆ’π‘’2πœξ€Έ+β€–β€–2π‘’βˆ«π‘‘πœ(2𝐢3(𝑠)+1)𝑑𝑠+πΎπœ€2ξ€œπ‘‘πœπ‘’βˆ«π‘‘π‘Ÿ(2𝐢3(𝑠)+1)π‘‘π‘ π‘‘π‘Ÿβ‰€πΆ2(𝜏,𝑇)πœ€2.(3.10)

Let us consider now the multivalued case. We will obtain first some auxiliary statements.

We shall define suitable approximations. For any 𝑛β‰₯1 we put 𝑓𝑖𝑛(𝑑,𝑒)=πœ“π‘›(|𝑒|)𝑓𝑖(𝑑,𝑒)+(1βˆ’πœ“π‘›(|𝑒|))𝑔𝑖(𝑑,𝑒), where 𝑔𝑖(𝑑,𝑒)=|𝑒𝑖|π‘π‘–βˆ’2𝑒𝑖+𝑓𝑖(𝑑,0,…,0), and πœ“π‘› was defined in (2.20). Then π‘“π‘›βˆˆβ„‚([𝜏,𝑇]×ℝ𝑑;ℝ𝑑) and for any 𝐴>0, supπ‘‘βˆˆ[𝜏,𝑇]sup|𝑒|≀𝐴||𝑓𝑛||(𝑑,𝑒)βˆ’π‘“(𝑑,𝑒)β†’0,asπ‘›β†’βˆž.(3.11) We will check first that 𝑓𝑛 satisfy conditions (2.2) for π‘’βˆˆβ„π‘‘+, where the constants do not depend on 𝑛.

Lemma 3.7. Let 𝑓 satisfy (2.2) for π‘’βˆˆβ„π‘‘+. For all 𝑛β‰₯1 one has 𝑑𝑖=1||𝑓𝑖𝑛||(𝑑,𝑒)𝑝𝑖/(π‘π‘–βˆ’1)≀𝐷11+𝑑𝑖=1||𝑒𝑖||𝑝𝑖ξƒͺ,𝑓𝑛(𝑑,𝑒),𝑒β‰₯𝛾𝑑𝑖=1||𝑒𝑖||π‘π‘–βˆ’π·2,(3.12) for π‘’βˆˆβ„+𝑑, where the positive constants 𝐷1, 𝐷2, and 𝛾 do not depend on 𝑛.
If |𝑒|>𝑛+1, then for any π‘€βˆˆβ„π‘‘ one has 𝑓𝑛𝑒(𝑑,𝑒)𝑀,𝑀β‰₯0.(3.13) Moreover, if 𝑓,β„Ž satisfy (3.2), then 𝑓𝑛,β„Ž also satisfies this condition.

Proof. In view of (2.2) we get 𝑓𝑛(𝑑,𝑒),𝑒=πœ“π‘›ξ€·(|𝑒|)(𝑓(𝑑,𝑒),𝑒)+1βˆ’πœ“π‘›ξ€Έ(|𝑒|)(𝑔(𝑒),𝑒)β‰₯πœ“π‘›ξƒ©π›Ό(|𝑒|)𝑑𝑖=1||𝑒𝑖||π‘π‘–βˆ’πΆ2ξƒͺ+ξ€·1βˆ’πœ“π‘›ξ€Έ(|𝑒|)𝑑𝑖=1||𝑒𝑖||𝑝𝑖+ξ€·1βˆ’πœ“π‘›ξ€Έ(|𝑒|)𝑑𝑖=1𝑓𝑖(𝑑,0,…,0)𝑒𝑖β‰₯πœ“π‘›(|𝑒|)𝛼𝑑𝑖=1||𝑒𝑖||π‘π‘–βˆ’πΆ2+ξ€·1βˆ’πœ“π‘›ξ€Έ1(|𝑒|)2𝑑𝑖=1||𝑒𝑖||π‘π‘–βˆ’πΎ1ξ€·1βˆ’πœ“π‘›ξ€Έ(|𝑒|)𝑑𝑖=1||𝑓𝑖||(𝑑,0,…,0)𝑝𝑖/(π‘π‘–βˆ’1)β‰₯𝛼𝑑𝑖=1||𝑒𝑖||π‘π‘–βˆ’πΆ2βˆ’πΎ1𝐢1,(3.14) where 𝛼=min{1/2,𝛼}, for some constant 𝐾1>0. Also, 𝑑𝑖=1||𝑓𝑖𝑛||(𝑑,𝑒)𝑝𝑖/(π‘π‘–βˆ’1)≀𝐾2𝑑𝑖=1||𝑓𝑖||(𝑑,𝑒)𝑝𝑖/(π‘π‘–βˆ’1)+𝑑𝑖=1||𝑔𝑖||(𝑒)𝑝𝑖/(π‘π‘–βˆ’1)ξƒͺ≀𝐾3𝐢11+𝑑𝑖=1||𝑒𝑖||𝑝𝑖ξƒͺ+𝑑𝑖=1||𝑒𝑖||𝑝𝑖+𝑑𝑖=1||𝑓𝑖||(𝑑,0,…,0)𝑝𝑖/(π‘π‘–βˆ’1)ξƒͺ≀𝐾4𝑑𝑖=1||𝑒𝑖||𝑝𝑖ξƒͺ,+1(3.15) for some constant 𝐾4>0. Thus, for 𝐷1=𝐾4, 𝐷2=𝐢2+𝐾1𝐢1, 𝛼=min{1/2,𝛼} we have (3.12) for the functions 𝑓𝑛.
Moreover, if |𝑒|>𝑛+1, then for any π‘€βˆˆβ„π‘‘, 𝑓𝑛𝑒=𝑔(𝑑,𝑒)𝑀,𝑀𝑒=(𝑑,𝑒)𝑀,𝑀𝑑𝑖=1𝑝𝑖||π‘’βˆ’1𝑖||π‘π‘–βˆ’2𝑀2𝑖β‰₯0.(3.16)
Finally, if (3.2) is satisfied, then β„Žπ‘–(𝑑,π‘₯)βˆ’π‘“π‘–π‘›(𝑑,𝑒)=πœ“π‘›ξ€·β„Ž(|𝑒|)𝑖(𝑑,π‘₯)βˆ’π‘“π‘–ξ€Έ+ξ€·(𝑑,𝑒)1βˆ’πœ“π‘›β„Ž(|𝑒|)𝑖(𝑑,π‘₯)βˆ’π‘“π‘–ξ€Έ(𝑑,0,…,0)β‰₯0,(3.17) for all 𝑖, a.e. (𝑑,π‘₯)∈(𝜏,𝑇)Γ—Ξ© and 𝑒 such that 𝑒𝑖=0 and 𝑒𝑗β‰₯0 if 𝑗≠𝑖.

Let 2β‰€π‘žπ‘–β‰€π‘π‘–, 𝑖=1,…,𝑑. We define also the following approximations 𝑓𝑖𝑛(𝑑,𝑒)=πœ“π‘›(|𝑒|)𝑓𝑖(𝑑,𝑒)+(1βˆ’πœ“π‘›(|𝑒|))̃𝑔𝑖(𝑑,𝑒), where ̃𝑔𝑖(𝑑,𝑒)=|𝑒𝑖|π‘π‘–βˆ’2𝑒𝑖+|𝑒𝑖|π‘žπ‘–βˆ’2𝑒𝑖+𝑓𝑖(𝑑,0,…,0). Then (3.11) holds. We check that 𝑓𝑛 satisfy conditions (2.2) for π‘’βˆˆβ„π‘‘+, where the constants do not depend on 𝑛.

Lemma 3.8. Let 𝑓 satisfy (2.2) for π‘’βˆˆβ„π‘‘+. For all 𝑛β‰₯1 one has 𝑑𝑖=1|||𝑓𝑖𝑛|||(𝑑,𝑒)𝑝𝑖/(π‘π‘–βˆ’1)≀𝐷11+𝑑𝑖=1||𝑒𝑖||𝑝𝑖ξƒͺ,𝑓𝑛(𝑑,𝑒),𝑒β‰₯𝛾𝑑𝑖=1||𝑒𝑖||π‘π‘–βˆ’π·2,(3.18) for π‘’βˆˆβ„+𝑑, where the positive constants 𝐷1, 𝐷2, and 𝛾 do not depend on 𝑛.
If |𝑒|>𝑛+1, then for any π‘€βˆˆβ„π‘‘ one has 𝑓𝑛𝑒(𝑑,𝑒)𝑀,𝑀β‰₯0.(3.19)
Moreover, if 𝑓,β„Ž satisfy (3.2), then 𝑓𝑛,β„Ž also satisfy this condition.

Proof . In view of (3.12), we have 𝑓𝑛=𝑓(𝑑,𝑒),𝑒𝑛+ξ€·(𝑑,𝑒),𝑒1βˆ’πœ“π‘›ξ€Έ(|𝑒|)𝑑𝑖=1||𝑒𝑖||π‘žπ‘–β‰₯𝛾𝑑𝑖=1||𝑒𝑖||π‘π‘–βˆ’π·2,𝑑𝑖=1|||𝑓𝑖𝑛|||(𝑑,𝑒)𝑝𝑖/(π‘π‘–βˆ’1)≀𝐾1𝑑𝑖=1||𝑓𝑖𝑛||(𝑑,𝑒)𝑝𝑖/(π‘π‘–βˆ’1)+𝑑𝑖=1||𝑒𝑖||𝑝𝑖(π‘žπ‘–βˆ’1)/(π‘π‘–βˆ’1)ξƒͺ≀𝐾1𝐷11+𝑑𝑖=1||𝑒𝑖||𝑝𝑖ξƒͺ+𝑑𝑖=1||𝑒𝑖||π‘žπ‘–ξƒͺ≀𝐾21+𝑑𝑖=1||𝑒𝑖||𝑝𝑖ξƒͺ,(3.20) where we have used that 𝑝𝑖β‰₯π‘žπ‘– implies 𝑝𝑖/(π‘π‘–βˆ’1)β‰€π‘žπ‘–/(π‘žπ‘–βˆ’1). Finally, (3.19) and condition (3.2) are proved in the same way as in Lemma 3.7.

For every 𝑛β‰₯1 consider the sequence π‘“πœ€π‘›(𝑑,𝑒) defined by π‘“πœ–π‘›βˆ«(𝑑,𝑒)=β„π‘‘πœŒπœ–(𝑠)𝑏𝑛(𝑑,π‘’βˆ’π‘ )𝑑𝑠, where either 𝑏𝑛=𝑓𝑛 or 𝑏𝑛=𝑓𝑛, defined before. Since any 𝑏𝑛 are uniformly continuous on [𝜏,𝑇]Γ—[βˆ’π‘˜βˆ’1,π‘˜+1], for any π‘˜β‰₯1, there exist πœ–π‘˜,π‘›βˆˆ(0,1) such that for all 𝑒 satisfying |𝑒|β‰€π‘˜, and for all 𝑠 for which |π‘’βˆ’π‘ |<πœ–π‘˜,𝑛 we have supπ‘‘βˆˆ[𝜏,𝑇]||𝑏𝑛(𝑑,𝑒)βˆ’π‘π‘›||≀1(𝑑,𝑠)π‘˜.(3.21) We put π‘“π‘˜π‘›(𝑑,𝑒)=π‘“πœ–π‘˜,𝑛𝑛(𝑑,𝑒). Then, π‘“π‘˜π‘›(𝑑,β‹…)βˆˆβ„‚βˆž(ℝ𝑑;ℝ𝑑) for all π‘‘βˆˆ[𝜏,𝑇],π‘˜,𝑛β‰₯1. Since for any compact subset π΄βŠ‚β„π‘‘ and any 𝑛 we have π‘“π‘˜π‘›β†’π‘π‘› uniformly on [𝜏,𝑇]×𝐴, we obtain the existence of a sequence π›Ώπ‘›π‘˜βˆˆ(0,1) such that π›Ώπ‘›π‘˜β†’0, as π‘˜β†’βˆž, and |π‘“π‘›π‘˜π‘–(𝑑,𝑒)βˆ’π‘π‘–π‘›(𝑑,𝑒)|β‰€π›Ώπ‘›π‘˜, for any 𝑖,𝑛 and any 𝑒 satisfying |𝑒|≀𝑛+2. We define the function πΉπ‘˜π‘›=(πΉπ‘›π‘˜1,…,πΉπ‘›π‘˜π‘‘) given by πΉπ‘›π‘˜π‘–(𝑑,𝑒)=π‘“π‘›π‘˜π‘–(𝑑,𝑒)βˆ’π‘π›Ώπ‘›π‘˜,(3.22) where π‘βˆˆβ„•.

Lemma 3.9. Let 𝑓 satisfy (2.2) for π‘’βˆˆβ„π‘‘+. For all 𝑛,π‘˜β‰₯1 we have 𝑑𝑖=1||πΉπ‘›π‘˜π‘–||(𝑑,𝑒)𝑝𝑖/(π‘π‘–βˆ’1)≀𝐷31+𝑑𝑖=1||𝑒𝑖||𝑝𝑖ξƒͺ,ξ€·πΉπ‘˜π‘›ξ€Έ(𝑑,𝑒),𝑒β‰₯πœˆπ‘‘ξ“π‘–=1||𝑒𝑖||π‘π‘–βˆ’π·4,(3.23) for π‘’βˆˆβ„π‘‘+, where the positive constants 𝐷3,𝐷4, and β€‰β€‰πœˆ do not depend neither on 𝑛 nor π‘˜.
Moreover, if 𝑓,β„Ž satisfy (3.2), then πΉπ‘˜π‘›,β„Ž also satisfy this condition if |𝑒|≀𝑛+2.

Proof. Since 𝑓𝑛 satisfy (3.12) and 𝑓𝑛 satisfies (3.18), we have 𝑑𝑖=1||πΉπ‘›π‘˜π‘–||(𝑑,𝑒)𝑝𝑖/(π‘π‘–βˆ’1)≀𝑅1𝑑𝑖=1||π‘“π‘›π‘˜π‘–||(𝑑,𝑒)𝑝𝑖/(π‘π‘–βˆ’1)ξƒͺ+1≀𝑅1𝑑𝑖=1ξƒ©ξ‚΅ξ€œβ„π‘‘πœŒπœ–π‘˜ξ‚Ά(𝑠)𝑑𝑠1/(π‘π‘–βˆ’1)ξ€œβ„π‘‘πœŒπœ–π‘˜||𝑏(𝑠)𝑖𝑛||(𝑑,π‘’βˆ’π‘ )𝑝𝑖/(π‘π‘–βˆ’1)ξƒͺξƒͺ𝑑𝑠+1≀𝑅2𝑑𝑖=1ξ€œβ„π‘‘πœŒπœ–π‘˜ξ€·||𝑒(𝑠)1+π‘–βˆ’π‘ π‘–||𝑝𝑖ξƒͺ𝑑𝑠+1≀𝑅3𝑑𝑖=1ξ€œβ„π‘‘πœŒπœ–π‘˜(ξ€·||𝑒𝑠)𝑖||𝑝𝑖+πœ–π‘π‘–π‘˜ξ€Έξƒͺ𝑑𝑠+1≀𝑅41+𝑑𝑖=1||𝑒𝑖||𝑝𝑖ξƒͺ,(3.24) for some constant 𝑅4>0.
On the other hand, ξ€·πΉπ‘˜π‘›ξ€Έ=ξ€œ(𝑑,𝑒),π‘’β„π‘‘πœŒπœ–π‘˜ξ€·π‘(𝑠)π‘›ξ€Έξ€œ(𝑑,π‘’βˆ’π‘ ),π‘’βˆ’π‘ π‘‘π‘ +β„π‘‘πœŒπœ–π‘˜ξ€·π‘(𝑠)𝑛(𝑑,π‘’βˆ’π‘ ),π‘ π‘‘π‘ βˆ’π‘π›Ώπ‘›π‘›π‘˜ξ“π‘–=1𝑒𝑖β‰₯ξ€œβ„π‘‘πœŒπœ–π‘˜ξƒ©π›Ύ(𝑠)𝑑𝑖=1||π‘’π‘–βˆ’π‘ π‘–||π‘π‘–βˆ’π·2ξƒͺβˆ’ξ€œπ‘‘π‘ β„π‘‘πœŒπœ–π‘˜(𝑠)𝑑𝑖=1𝛾2𝐷1||𝑏𝑖𝑛||(𝑑,π‘’βˆ’π‘ )𝑝𝑖/(π‘π‘–βˆ’1)+𝑅5||𝑠𝑖||π‘π‘–ξ‚Άπ‘‘π‘ βˆ’π‘π›Ώπ‘›π‘›π‘˜ξ“π‘–=1𝑒𝑖β‰₯ξ€œβ„π‘‘πœŒπœ–π‘˜ξƒ©π›Ύ(𝑠)𝑑𝑖=1||π‘’π‘–βˆ’π‘ π‘–||π‘π‘–βˆ’π·2ξƒͺβˆ’ξ€œπ‘‘π‘ β„π‘‘πœŒπœ–π‘˜ξƒ©π›Ύ(𝑠)2𝑑𝑖=1||π‘’π‘–βˆ’π‘ π‘–||𝑝𝑖+𝑅6ξƒͺπ‘‘π‘ βˆ’π‘π›Ώπ‘›π‘›π‘˜ξ“π‘–=1𝑒𝑖β‰₯𝛾2ξ€œβ„π‘‘πœŒπœ–π‘˜(𝑠)𝑑𝑖=1||π‘’π‘–βˆ’π‘ π‘–||π‘π‘–π‘‘π‘ βˆ’π‘…7βˆ’π‘π›Ώπ‘›π‘›π‘˜ξ“π‘–=1𝑒𝑖β‰₯πœˆπ‘‘ξ“π‘–=1||𝑒𝑖||π‘π‘–βˆ’π‘…8,(3.25) for some constants 𝜈,𝑅8>0, where in the last inequality we have used that for some 𝐷>0, ||𝑒𝑖||𝑝𝑖=||π‘’π‘–βˆ’π‘ π‘–+𝑠𝑖||𝑝𝑖||π‘’β‰€π·π‘–βˆ’π‘ π‘–||𝑝𝑖+||𝑠𝑖||𝑝𝑖||π‘’β‰€π·π‘–βˆ’π‘ π‘–||𝑝𝑖+πœ–π‘π‘–π‘˜ξ€Έ.(3.26) Hence, (3.23) holds.
In view of Lemmas 3.7 and 3.8 the functions 𝑏𝑛,β„Ž satisfy (3.2). Hence, |π‘“π‘›π‘˜π‘–(𝑑,𝑒)βˆ’π‘π‘–π‘›(𝑑,𝑒)|β‰€π›Ώπ‘›π‘˜, for any 𝑖,𝑛 and any 𝑒 satisfying |𝑒|≀𝑛+2, implies that β„Žπ‘–(𝑑,π‘₯)βˆ’πΉπ‘›π‘˜π‘–(𝑑,𝑒)=β„Žπ‘–(𝑑,π‘₯)βˆ’π‘“π‘›π‘˜π‘–(𝑑,𝑒)+π‘π›Ώπ‘›π‘˜β‰₯β„Žπ‘–(𝑑,π‘₯)βˆ’π‘π‘–π‘›(𝑑,𝑒)β‰₯0,(3.27) for 𝑒 such that 𝑒𝑖=0, 𝑒𝑗β‰₯0, 𝑗≠𝑖, and |𝑒|≀𝑛+2.

Define a smooth function πœ™π‘›βˆΆβ„+β†’[0,1] satisfying πœ™π‘›(⎧βŽͺ⎨βŽͺβŽ©π‘ )=1,if0≀𝑠≀𝑛+1+𝛾,0β‰€πœ™π‘›(𝑠)≀1,if𝑛+1+𝛾≀𝑠≀𝑛+2,0,if𝑠β‰₯𝑛+2,(3.28) where 0<𝛾<1 is fixed. Let π‘™π‘˜π‘›(𝑑,𝑒) be given by π‘™π‘˜π‘›(𝑑,𝑒)=πœ™π‘›(|𝑒|)πΉπ‘˜π‘›ξ€·(𝑑,𝑒)+1βˆ’πœ™π‘›ξ€Έπ‘(|𝑒|)𝑛(𝑑,𝑒).(3.29) Since for any compact subset π΄βŠ‚β„π‘‘ and any 𝑛 we have π‘“π‘˜π‘›β†’π‘π‘› uniformly on [𝜏,𝑇]×𝐴 as π‘˜β†’βˆž, it is clear that supπ‘‘βˆˆ[𝜏,𝑇]sup|𝑒|≀𝐴||π‘™π‘˜π‘›(𝑑,𝑒)βˆ’π‘π‘›||(𝑑,𝑒)β†’0,asπ‘˜β†’βˆž.(3.30)

Lemma 3.10. Let 𝑓 satisfy (2.2) for π‘’βˆˆβ„π‘‘+. For all 𝑛,π‘˜β‰₯1 we have 𝑑𝑖=1||π‘™π‘›π‘˜π‘–||(𝑑,𝑒)𝑝𝑖/(π‘π‘–βˆ’1)≀𝐷51+𝑑𝑖=1||𝑒𝑖||𝑝𝑖ξƒͺ,ξ€·π‘™π‘˜π‘›ξ€Έ(𝑑,𝑒),𝑒β‰₯πœ†π‘‘ξ“π‘–=1||𝑒𝑖||π‘π‘–βˆ’π·6,(3.31) for π‘’βˆˆβ„π‘‘+, where the positive constants 𝐷5,𝐷6, and πœ† do not depend neither on 𝑛 nor π‘˜. Also, ξ€·π‘™π‘˜π‘›π‘’ξ€Έ(𝑑,𝑒)𝑀,𝑀β‰₯βˆ’π·7(π‘˜,𝑛)|𝑀|2,βˆ€π‘’,𝑀,(3.32) where 𝐷7(π‘˜,𝑛) is a nonnegative number.
Moreover, if 𝑓,β„Ž satisfy (3.2), then π‘™π‘˜π‘›,β„Ž also satisfies this condition.

Proof. The inequalities given in (3.31) are an easy consequence of (3.12), (3.18), and (3.23).
On the other hand, if 𝑒 is such that 𝑒𝑖=0, 𝑒𝑗β‰₯0, 𝑗≠𝑖, then in view of Lemmas 3.7, 3.8, and 3.9 we have πœ™π‘›ξ€·β„Ž