Lasota–Opial type conditions for periodic problem for systems of higher-order functional differential equations
Abstract
In the paper we study the question of solvability and unique solvability of systems of the higher-order functional differential equations. In the paper in some sense optimal conditions that guarantee the unique solvability of the linear problem are obtained, and on the basis of these results the optimal conditions of the solvability and unique solvability for the nonlinear problem are proved.
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Mukhigulashvili and P˚uža Journal of Inequalities and Applications (2020) 2020:155 https://doi.org/10.1186/s13660-020-02414-9 RESEARCH Open Access Lasota–Opial type conditions for periodic problem for systems of higher-order functional differential equations Sulkhan Mukhigulashvili1,2* and Bedˇ rich P˚uža2 *Correspondence: [email protected] 1Institute of mathematics of the Czech Academy of Sciences, Brno, Czech Republic 2Faculty of Business and Management, Brno University of Technology, Brno, Czech Republic Abstract In the paper we study the question of solvability and unique solvability of systems of the higher-order functional differential equations u(mi) i(t)=i(ui+1)(t)+qi(t)(i=1,n)fort∈I:= [a,b] and u(mi) i(t)=Fi(u)(t)+q0i(t)(i=1,n)fort∈I under the periodic boundary conditions u(j) i(b)–u(j) i(a)=cij (i=1,n,j=0,mi–1), where un+1 =u1,mi≥1, n≥2, cij ∈R,qi,q0i∈L(I;R), i:C0 1(I;R)→L(I;R) are monotone operators and Fiare the local Caratheodory’s class operators. In the paper in some sense optimal conditions that guarantee the unique solvability of the linear problem are obtained, and on the basis of these results the optimal conditions of the solvability and unique solvability for the nonlinear problem are proved. MSC: 34K06; 34K13; 34B05 Keywords: Higher-order systems; Periodic problem; Functional differential equations; Unique solvability 1 Introduction Consider ontheinterval I=[a,b] the system ofhigher-order linear functional differential equations u(mi) i(t)=i(ui+1)(t)+qi(t)(i=1,n), (1) ©The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Mukhigulashvili and P˚uža Journal of Inequalities and Applications (2020) 2020:155 Page 2 of 20 where un+1 := u1, and the system of higher-order nonlinear functional differential equations u(mi) i(t)=Fi(u)(t)+q0i(t)(i=1,n), (2) where mi≥1, qi,q0i∈L(I;R), i:C0 1(I;R)→L(I;R) are linear bounded operators, and Fi∈K(Cm1,...,mn,L) (see Definition 1.1), under the periodic boundary conditions u(j) i(b)–u(j) i(a)=cij (i=1,n,j=0,mi–1). (3) Throughout the paper we use the following notations: Nis the set of the natural numbers; R=]–∞,+∞[, R+=[0,+∞[; Rnis the space of the nth dimensional column vectors x:= (xi)n i=1 with the components xi∈R(i=1,n)andthenormx=n i=1 |xi|;Cn–1 1(I;R) (n∈N)istheBanachspaceofthefunctions u:I→Rwhicharecontinuoustogetherwith their (n–1)th derivatives, with the norm uCn–1 1=maxn j=1 u(j–1)(t):t∈I; Cm1,...,mn(I;Rn) is theBanach spaceofthevector-functions u:= (ui)n i=1 :I→Rn,whereui∈ Cmi–1 1(I;R)(i=1,n), with the norm uCm1,...,mn=maxn i=1 mi j=1 u(j–1) i(t):t∈I, and for the case when mi=1(i=1,n), we will use the notations C0 nI;Rn:=C1,...,1I;Rn,uC0 n:=uC1,...,1; C0(I;R) is the Banach space of the functions u:I→Rwhich are absolutely continuous with the norm u C0=uC+b a|u(s)|ds; Cn–1(I;R)(n∈N)isasetoffunctions u:I→Rwhich are absolutely continuous together with their (n–1)thderivatives; L(I;R) is the Banach space of the Lebesgue integrable functions p:I→Rwith the norm pL= b a|p(s)|ds;Mn(I)isthesetofthevector-functionsτ:=(τi)n i=1 :I→In,withthemeasurable components τi:I→I(i=1,n). For arbitrary x∈R,weassumethat sgnx=⎧ ⎨ ⎩ 1ifx≥0, –1 if x<0. Definition 1.1 We will say that the operator F:Cm1,...,mn(I;Rn)→L(I;R)belongsto Caratheodory’s local class K(Cm1,...,mn,L)ifFis a continuous operator, and for arbitrary r>0, the inclusion supF(x)(·):xCm1,...,mn≤r,x∈Cm1,...,mnI;Rn∈L(I;R+) holds.
Mukhigulashvili and P˚uža Journal of Inequalities and Applications (2020) 2020:155 Page 3 of 20 Definition 1.2 We will say that a linear operator :C0 1(I;R)→L(I;R) is nonnegative (nonpositive) if, for any nonnegative x∈C0 1(I;R), the inequality (x)(t)≥0((x)(t)≤0) for t∈Iis satisfied. We will say that an operator is monotone if it is nonnegative or nonpositive. By a solution of problem (2), (3) we understand a vector-function u:= (ui)n i=1 where ui∈ Cmi–1(I;R)(i=1,n), which satisfies equation (2)almosteverywhereonIand satisfies conditions (3). Firstofall,wewouldliketogiveahistoricalreviewwhichbeginswithLasotaandOpial’s article [1] from the year 1964, which became the basis for a lot of interesting studies. In this article (see Theorems 5 and 6) authors proved that the problem u(n)(t)=p(t)u(t)+q(t)fort∈I, u(j)(b)–u(j)(a)=cj(j=0,n–1), (4) is uniquely solvable if b ap(s)ds≤0forn=2,p(t)≤0forn≥3, and b ap(s)ds<Ln (b–a)n–1 ,andp≡0, (5) whereL2=16 andthegeneraltermLncanbefoundbyformula(40)fromthepaper[1],in which it is also shown that for n=2 condition (5)isoptimal,whileforn≥3 it is far from being optimal. In the article [2], we announced the Lasota–Opial type optimal results of unique solvability of the periodic problem for second-order linear functional differential equations, which in a more general form are considered in the paper [3] from 2006. In particular, in Theorem 1.1 of the paper [3](seealso[4]) it is proved that the functional differential equation u(n)(t)=n–1 i=0 iu(i)(t)+q(t)(6) under boundary conditions (4)forn=2 is uniquely solvable if the optimal conditions b a0(1)(s)ds=0 and b a0(1)(s)ds≤16 b–a1–b a+ 1(1)(s)+– 1(1)(s)ds hold, where 0:C0 1(I;R)→L(I;R) is a monotone operator, i=+ i–– i(i=1,n–1)and ± i:C0 1(I;R)→L(I;R) are nonnegative operators (for n=3see[5]). For the general case, analogousconditionsoftheuniquesolvabilityofproblem(6),(4)areprovedin[6],fori= + i–– i(i=0,n–1),whichif0is monotone operator and i≡0(i=1,n–1)transforms
Mukhigulashvili and P˚uža Journal of Inequalities and Applications (2020) 2020:155 Page 4 of 20 to the conditions b a0(1)(s)ds=0 and b a0(1)(s)ds≤4Tn–1 (b–a)n–1 ,(7) where T1=4,T2=32,T3= 192, andthe general term Tnof thissequencecan befoundby formulas (9). In [6](seealso[7]) it is proved that the constants Tnare sharp when n≤7; for n>7, the problem of sharpness of the numbers Tnis still opened. InthisbriefhistoricalreviewBravyi’soriginalstudiesofproblem(6),(4)cannotbeomitted. The method developed in these studies turned out to be very fruitful. Particularly in [8] the author proved that the condition Y 1–Y≤X≤2(1+√1–Y), (8) where Y=Nn(b–a)n–1 min(P–,P+), X=Nn(b–a)n–1 max(P–,P+), the numbers Nnare defined by the certain recurrent formula, and + 0(1)L=P+,– 0(1)L=P–,isnecessaryand sufficientforsolvabilityofproblems(6),(4)ifi≡0(i=1,n–1)(Nn=T–1 n–1 forn=2,7,but forn>7 thevalidityofthelastidentityisunknown).Forthecase i≡0(i=1,n–1),these resultsaregeneralizedin [9].Itis interestingthatthenumbersNnarein some connection with Favard’s, Bernoulli’s, and Euler’s numbers (see [10]and[11]) and if the operator 0is monotone, then condition (8) transforms to the condition 0< b a|0(1)(s)|ds≤4 Nn(b–a)n–1 . Other interesting results about problems (6), (4)canbefoundalsointhepapers[10–15]. Thenextstagewasthegeneralization of Lasota–Opial’sresults forthesystemsoflinear functionaldifferentialequations.Inparticularin[16]itisprovedthatproblem(1),(3)with mi=1(i=1,n)isuniquelysolvableifiare linear monotone operators, i=0(i=1,n), and the condition n i=1 i(1)L<4n(which is optimal) holds. In this connection see also the papers [17]and[18]. The aims of this article are to establish Lasota–Opial type sufficient efficient optimal conditionsofthesolvabilityofproblem(1),(3)and onthe basis oftheseresults to findthe optimalefficientsufficientconditionsofsolvabilityanduniquesolvabilityofthenonlinear problem (2), (3). 2 Main results 2.1 Linear problem Let T0=1,T1=4,T2=32,T3= 192, and T2m+2 =1 max{(hm(t)hm(1–t))1/2 :0≤t≤1}, T2m+3 =1 max{(fm(t,s)fm(1–t,1–s))1/2 :0≤t≤1,0≤s≤1} (9) for m≥1, where the functions fm:[0,1]×[0,1]→R+,hm:[0,1]→R+are defined by the equalities fm(t,s)=m–1 j=0 αmjt2(j+1) +αmms2m+3,hm(t)= m j=0 βmjt2(j+1),
Mukhigulashvili and P˚uža Journal of Inequalities and Applications (2020) 2020:155 Page 5 of 20 where αmj =Aj 3·4j+1T2(m–j)+1 ,βmj =Aj 3·4j+1T2(m–j)for j=0,m–1, αmm =Am 3·4m+1 ,βmm =Bm 3·4m+1 , and A0=1, A1=1 15,Aj=A1 2 m1=1 m1+1 m2=1... mj–2+1 mj–1=1 1 η(m1)...η(mj–1), B1=1 8,Bj=A1 2 m1=1 m1+1 m2=1... mj–2+1 mj–1=11 η(m1)...η(mj–1) mj–1+1 i=1 1+ 1 2i for j≥2, with η(t)=(2t+1)(2t+3). Remark 2.1 In Remarks 1.2 and 1.3 of [7]itwasshownthat T4=211 ·3 5,T5=29·3·5, T6=216 ·32·5 61 ,T7=214 ·32·5·7 17 , and Tn<(2π)n(n∈N). Now we can formulate the first of our main theorems. Theorem2.1 Let the operators i:C0 1(I;R)→L(I;R)(i=1,n)be monotone, b ai(1)(s)ds=0 (i=1,n), (10) and the condition n i=1 1 Tmi–1 b ai(1)(s)ds≤4n(b–a)n–n i=1 mi(11) hold.Then problem (1), (3)is uniquely solvable. For the system u(mi) i(t)=pi(t)ui+1τi(t)+qi(t)(i=1,n), (12) where un+1 := u1,τ:= (τi)n i=1 ∈Mn(I), and pi,qi∈L(I;R), from Theorem 2.1 we have the following. Corollary2.1 Let the function p∈L(I;Rn)be such that the conditions b api(s)ds=0, 0≤σipi(t)(i=1,n)for t ∈I,
Mukhigulashvili and P˚uža Journal of Inequalities and Applications (2020) 2020:155 Page 6 of 20 where σi∈{–1,1}(i=1,n), and n i=1 1 Tmi–1 b api(s)ds≤4n(b–a)n–n i=1 mi hold.Then,for arbitrary τ∈Mn(I), problem (12), (3)is uniquely solvable. Remark 2.2 For some values of the numbers n,m1,...,mn, condition (11) is optimal. For example, if n=2andn=2,m1=m2=1,thenproblem(1), (3)with1(u)≡u,2=0, q1≡0transformstoproblem(6), (4)with1≡0, q≡q2and condition (11)transforms to condition (7) which, as we have already said, is optimal for the unique solvability of problem (6),(4)withn=2. 2.2 Nonlinear problem Definition 2.1 Let hi:C0 1(I;R)→L(I;R)(i=1,n) be nonnegative linear operators, then we will say that h:=(hi)n i=1 ∈P(I) (13) if b ahi(1)(s)ds=0 (i=1,n), (14) and for arbitrary monotone operators i:C0 1(I;R)→L(I;R)(i=1,n) such that the conditions b ai(1)(s)ds=0, 0≤σii(1)(t)≤hi(1)(t)(i=1,n)fort∈I(15) hold, where σi∈{–1,1}(i=1,n), the homogeneous problem u(mi) i(t)=i(ui+1)(t)(i=1,n), (16) u(j) i(b)–u(j) i(a)=0 (i=1,n,j=0,mi–1), (17) where un+1 :=u1, has no nontrivial solution. Also note that in all our propositions below the functions ηi:I×R+→R+(i=1,n) are summable in the first argument, nondecreasing in the second one, and admit to the conditions lim ρ→+∞ 1 ρb aηi(s,ρ)ds=0 (i=1,n). (18) Theorem 2.2 Let the linear nonnegative operators hi:C0 1(I;R)→L(I;R)(i=1,n), the function g0∈L(I;Rn), and the numbers σi∈{–1,1}(i=1,n), r0>0,be such that,for all
Mukhigulashvili and P˚uža Journal of Inequalities and Applications (2020) 2020:155 Page 7 of 20 i∈{1,...,n}on I,the conditions g0i(t)≤σiFi(x)(t)sgnhi(xi+1)(t)≤hi(xi+1)(t)+ηit,xCm1,...,mn if xCm1,...,mn≥r0, (19) with x:=(xi)n i=1 ∈Cm1,...,mn(I;Rn), xn+1 =x1,and inclusion (13)hold.Moreover,let the function g ∈L(I;Rn +)be such that,for all i∈{1,...,n}on I,the conditions gi(t)≤σiFi(x)(t)sgnhi(xi+1)(t)if minxi+1(t):t∈I≥r0(20) are fulfilled,and b agi(s)ds–b aq0i(s)ds≥|cimi–1|.(21) Then problem (2), (3)has at least one solution. Remark 2.3 From the inequality xCm1,...,mn≥min{|xi+1(t)|:t∈I}it follows that g0i≤gi. Therefore, if we assume that instead of (21) inequalities b ag0i(s)ds–b aq0i(s)ds≥|cimi–1|(i=1,n) (22) hold, then condition (20) can be omitted. On the other hand, in the example below we construct the operator Ffor which conditions (21)holdand(22) do not hold, and therefore condition (20) cannot be omitted because, as it follows from Remark 2.4, condition (21)isimprovable. Example2.1 Consider system (2)with Fi(u)(t)=⎧ ⎨ ⎩ σi hi(t)cos(πy (xi+1(t)+y)xi+1(t))xi+1(t)ifxi+1(t)=0, 0ifxi+1(t)=0, where y:= uCm1,...,mn,xi+1 := |ui+1(τi(·))|,un+1 := u1,σi∈{–1,1},andτ∈Mn(I). Let also the numbers δ1∈(0,1), cimi–1 and the functions h∈L(I;Rn +), q0∈L(I;Rn)besuchthat inclusion (13)withhi(z)(·)= hi(·)z(τi(·)) holds, and b a hi(s)ds–1b aq0i(s)ds+|cimi–1|≤δ1(i=1,n). Then, if α(x)=xcos(π/x), in view the facts that α(2)=0, α(x)>0 ifx>2, and lim x→+∞α(x)=1, there exist r0>2andδ0∈(0,r0)suchthatr0cos(π/r0)=δ1, and the inequalities δ1≤cosπ xi+1(t)xi+1(t)<cosπy (xi+1(t)+y)xi+1(t)xi+1(t) if xi+1(t)≥r0,y≥r0,
Mukhigulashvili and P˚uža Journal of Inequalities and Applications (2020) 2020:155 Page 8 of 20 and –δ0≤cosπy (xi+1(t)+y)xi+1(t)xi+1(t)≤xi+1(t) if 0< xi+1(t)<r0,y≥r0, are valid on I. Therefore, the validity of conditions (19)–(21)withgi≡δ1 hi,g0i≡–δ0 hi, ηi≡0isobvious,andfromTheorem2.2 the solvability of problem (2), (3) follows. Corollary2.2 Let the linear nonnegative operators hi:C0 1(I;R)→L(I;R)(i=1,n)be such that conditions (14)and n i=1 1 Tmi–1 b ahi(1)(s)ds≤4n(b–a)n–n i=1 mi(23) hold.Moreover,let the functions g0∈L(I;Rn), g∈L(I;Rn +)and the numbers σi∈{–1,1} (i=1,n), r0>0be such that conditions (19)–(21)are fulfilled.Then problem (2), (3)has at least one solution. For the case when (2)is the systemof higher-orderdifferentialequationswith the argument deviation of the form u(mi) i(t)=fit,ui+1τi(t)+q0i(t)(i=1,n), (24) where un+1 := u1and fi:I×R→R(i=1,n) are the functions from Caratheodory’s class, the following corollary is true. Corollary2.3 Let the function h∈L(I;Rn +)be such that the conditions hi(t)>0 (i=1,n)a.e.onI, n i=1 1 Tmi–1 b a hi(s)ds≤4n(b–a)n–n i=1 mi(25) hold.Moreover,letthe functiong ∈L(I;Rn +)and the numbers σi∈{–1,1}(i=1,n), r0>0be such that conditions (21)and gi(t)≤σifi(t,x)sgnx≤ hi(t)|x|+ηit,|x| for |x|≥r0,t∈I(i=1,n), (26) hold.Then,for arbitrary τ∈Mn(I), problem (24), (3)has at least one solution. Remark 2.4 Theorem 2.2 is optimal in the sense that there does not exist such i0∈ {1,...,n},forwhichi0th inequality of condition (21) can be replaced by the inequality ε+b agi0(s)ds–b aq0i0(s)ds≥|ci0mi0–1|,(27)
Mukhigulashvili and P˚uža Journal of Inequalities and Applications (2020) 2020:155 Page 9 of 20 no matter how small ε> 0 would be. Indeed, let I=[0,1], σi∈{–1,1},mi=1, ci0=0, τi≡t,ηi≡0(i=1,n), hi≡1, gi≡1, q0i≡0, fi(t,x)≡σix(i=1,n;i=i0), and gi0≡0, q0i0=ε 1+ε,fi0(t,x)≡0, hi0≡4n–ε. Then, for the arbitrary functions pi∈L(I;R)suchthatpi≡ 0, 0 ≤σipi(t)≤ hi(t), it is clear that n i=1 b a|pi(s)|ds ≤4n, and due to Corollary 2.1 inclusion (13)withhi(x)(t)= hi(t)x(τi(t)) holds. Also it is not difficult to verify that instead of the i0th inequality of condition (21)inequality(27) is satisfied, but all the other assumptions of Corollary 2.3 hold with r0= 1. Nevertheless, in that case problem (24), (3) is not solvable because ui0(1)–ui0(0)=ε/(1+ε)>0=ci00. Example2.2 Consider the system of the differential equations u(mi) i(t)=σi hi(t)sinuCm1,...,mnui+1τi(t)+q0i(t)(i=1,n), (28) where un+1 =u1,σi∈{–1,1}, the functions τ∈Mn(I), h∈L(I;Rn +) are such that inclusion (13)holdswithhi(x)(t)= hi(t)x(τi(t)), and b aq0i(s)ds=0(i=1,n). Then, from Theorem 2.2 with g0i≡gi≡0, the solvability of problems (28), (3)withcimi–1 =0(i=1,n) follows. Also,onthebasisofCorollary 2.3,wecanprovethefollowingexistenceanduniqueness theorem. Theorem 2.3 Let the function h∈L(I;Rn +)be such that conditions (25)hold and fi(t,0)≡ 0(i=1,n). Moreover,let the functions h0∈L(I;Rn +), βi:R2→R+and the numbers σi∈ {–1,1},r>0be such that,for all i ∈{1,...,n},conditions (21), h0i(t)βi(x,y)≤σifi(t,x)–fi(t,y)sgn(x–y)≤ hi(t)|x–y| for t ∈I,x,y∈R, (29) and h0i(t)≥0for t ∈I, h0i(t)≡0, βi(x,y)>0 for x =y(30) hold,where gi(t)=minfi(t,r),fi(t,–r).(31) Then,for arbitrary τ∈Mn(I),problem (24), (3)is uniquely solvable.
Mukhigulashvili and P˚uža Journal of Inequalities and Applications (2020) 2020:155 Page 16 of 20 Now if we multiply the least two expressions by the well-known inequality A·B≤(A+ B)2/4 if A≥0, B≥0, and (34), we obtain 0<v(mi–1) i≤1 4b ai(1)(s)ds(vi+1). (57) It is not difficult to verify that inequality (57) holds even when the operator σiiis nonpositive. On the other hand, due to (17)and(54), all the assumptions of Lemma 3.1 hold for the functions viwith m=mi+1 –1; and consequently, (57)by(33)implies 0<v(mi–1) i<(b–a)mi+1–1 4Tmi+1–1 b ai(1)(s)dsv(mi+1–1) i+1 .(58) Finally, if we multiply inequalities (58) for all i∈1,nand take into account notations (53), weget the contradictionto condition(11). Therefore ourassumption is invalid and vi≡0 (i=1,n), which definitely proves our theorem. ProofofTheorem2.2 Firstofall,noticethatinviewofinclusion(13)inequalities(14)hold, and then there exists r1>r0such that r1hi(1)L>|cimi–1|(i=1,n). (59) Let now λ∈(0,1) be an arbitrary fixed number and u:= (ui)n i=1 be a solution of problem (40), (41)andshowthat μ(ui)=minui(t):t∈I≤r1(i=1,n). (60) Assume to the contrary that there exists i∈{1,...,n}such that |ui+1(t)|>r1on I.Then there existσi∈{–1,1}such that σisgnhi(xi+1)(t)≥0onI, and due to (20), (21), (40), (41), and (59), we obtain the contradiction |cimi–1|≥λcimi–1σiσi=σiσiu(mi–1) i(b)–u(mi–1) i(a) =σib au(mi) i(s)σids=λb ahi(ui+1)(s)ds +(1–λ)σib aFi(u)(s)+q0i(s)σids≥λr1hi(1)L +(1–λ)b agi(s)ds–b aq0i(s)ds>λ|cimi–1|+(1–λ)|cimi+1|=|cimi–1|, which proves (60).
Mukhigulashvili and P˚uža Journal of Inequalities and Applications (2020) 2020:155 Page 17 of 20 Let ρ0be a number defined in Lemma 3.4, then due to condition (18)thereexistssuch a constant ρ1>r0that ρ0 n i=1 μ(ui)+mi–1 j=0 |cij|+3αL+q0iL+b aηi(s,ρ)ds <ρfor ρ≥ρ1, (61) whereα(t)=1+n i=1 |g0i(s)|.NowassumethatuCm1,...,mn≥ρ1andintroducethenotation νi(t):= σiFi(u)(t)sgnhi(ui+1)(t)+α(t) |hi(ui+1)(t)|+ηi(t,uCm1,...,mn)+2α(t)for t∈I. Then, in view of conditions (19), we have 0<νi(t)<1 (i=1,n)onI. (62) Ontheotherhanditisnotdifficulttoverifythat uisasolutionalsoofthesystem(1),with i(x)(t)=σiλ+(1–λ)νi(t)hi(x)(t), qi(t)=(1–λ)νi(t)2α(t)+ηit,uCm1,...,mn–α(t)σisgnhi(ui+1)(t)+q0i(t), where due to inequalities (62) the following estimations are valid: 0≤σii(1)(t)≤hi(1)(t), qi(t)≤3α(t)+q0i(t)+ηit,uCm1,...,mn on I. Then, for the function u, as for a solution of problem (1), (41), by Lemma 3.4 we get the estimation uCm1,...,mn ≤ρ0 n i=1 μ(ui)+mi–1 j=0 |cij|+3αL+q0iL+b aηis,uCm1,...,mnds, which due to the assumption uCm1,...,mn≥ρ1contradicts (61). Therefore our assumption is invalid and estimation (42) holds, and then from Proposition 3.2 the solvability of problem (2), (3) follows. Proof of Corollary 2.2 In view of Corollary 2.1, from conditions (14)and(23), it follows that, for arbitrary monotone operators i:C(I;R)→L(I;R)(i=1,n)whichadmittoinequalities(15),problem(16),(17)hasonlythezerosolution,andtheninclusion(13)holds. Therefore all the assumptions of Theorem 2.2 are fulfilled, and then problem (2), (3)is solvable. Proof of Corollary 2.3 Let hi(x)(t)= hi(t)x(τi(t)), then from conditions (25)andTheorem 2.1 it follows that inclusion (13)holds,and sgnhi(x)(t)=sgnxτi(t)(i=1,n)a.e.onI.
Mukhigulashvili and P˚uža Journal of Inequalities and Applications (2020) 2020:155 Page 18 of 20 Also, it is clear that due to (26), for arbitrary x∈Cm1,...,mn(I;Rn), the conditions gi(t)≤σifit,xi+1τi(t)sgnxi+1τi(t) for t∈s∈I:xi+1τi(s)≥r0(i=1,n) are fulfilled. On the other hand, if gi(t):=maxfi(t,y):|y|≤r0,g0i(t):=min– gi(t),gi(t)(i=1,n), then, in view of (26) and the fact that the functions ηiare nondecreasing in the second argument, we obtain g0i(t)≤σifit,xi+1τi(t)sgnxi+1τi(t)≤ hi(t)xi+1τi(t)+ gi(t)+ηit,xCm1,...,mn for xCm1,...,mn≥r0,t∈I(i=1,n), and due to (18) the following equalities hold: lim ρ→+∞ 1 ρb a gi(s)+ηis,xCm1,...,mnds=0 (i=1,n). Consequently, if we introduce the notations Fi(x)(t)=fi(t,xi+1(τi(t))), we get that all the assumptionsofTheorem2.2arevalid,fromwhichthevalidityofourcorollaryimmediately follows. Proof of Theorem 2.3 First note that from conditions (25), analogously as in the proof of previous Corollary 2.3, validity of inclusion (13)withhi(x)= hi(t)x(τi(·)) follows. Also, for arbitrary r≥0, from condition (29), (30)andthefactthat sgnx–(–1)jr=sgnxif (–1)jx>r(j=0,1), we have σifit,(–1)jrsgnx≤ h0i(t)βi(x,r)+σifit,(–1)jrsgnx≤σifi(t,x)sgnx ≤ hi(t)|x|+r+σifit,(–1)jrsgnxfor t∈I,(–1)jx>r. On theother hand, due to theconditions fi(t,0)≡0(i=1,n)and(30), from(29) it follows that σifit,(–1)jrsgnx≥0fort∈I,(–1)jx>r. From the last two inequalities and (21) it is clear that all the assumptions of Corollary 2.3 hold when the functions giare defined by (31)andηi(t,x)= hi(t)r+max{|f(t,r)|,|f(t,–r)|} for t∈I,|x|>r.Thereforeit remainstoprovethatproblem(24),(3)hasnomorethanone solution.
Mukhigulashvili and P˚uža Journal of Inequalities and Applications (2020) 2020:155 Page 19 of 20 Let u1:= (u1i)n i=1 and u2:= (u2i)n i=1 be the arbitrary solutions of problem (24), (3), and u:= u1– u2. Then the function uadmits to conditions (17), and from (29)weobtainthat, for all i=1,n,theinequalities h0i(t)βi u1i+1(t), u2i+1(t) ≤σiu(mi) i(t)sgnui+1τi(t)≤ hi(t)ui+1τi(t)(63) hold on I. Now, if we assume that there exists i0∈{1,...,n}such that |ui0+1(t)|>0onI, then sgnui0+1(τi0(t))=sgnui0+1(a), and also due to (30)weobtainβ0i0:=min{βi0( u1i0+1(t), u2i0+1(t)): t∈I}>0, h0i0L>0. Therefore from (17)and(63) we get the contradiction 0=σi0u(mi0–1) i0(b)–u(mi0–1) i0(a)sgnui0+1(a) =σi0b au(mi0) i0(s)sgnui0+1τi0(s)ds≥β0i0 h0i0L>0 a.e.onI, which shows that minui(t):t∈I=0 (i=1,n). (64) Suppose that pi(t)=⎧ ⎨ ⎩ u(mi) i(t)/ui+1(τi(t)) if ui+1(τi(t))=0, 0ifui+1(τi(t))=0, then uis a solution of the linear homogeneous problem (16), (17)withi(x)(t)= pi(t)x(τi(t)), which due to condition (63) admits to the inequalities 0≤σii(1)(t)≤hi(1)(t)(i=1,n)fort∈I. Let pi≡0(i=1,n), then in view of the inclusion h∈P(I)weobtainu≡0. Now assume that there exists i0∈{1,...,n}such that pi0≡0. Then from (16) the identity u(mi0) i0≡0 follows, and in view of conditions (17)and(64)wegetui0≡0, fromwhichonthebasisof (16)itfollowsthatu(mi0–1) i0–1 ≡0.Afteranalogousn–1stepswegetthatu≡0,andtherefore u1≡ u2. Acknowledgements The research was supported by RVO: 67985840. Funding Not applicable. Availability of data and materials Datacitations(suchasaDOI,URL,MRnumbr,...)whentheyexistareincludedinthereferencelist. Competing interests The authors declare that they have no competing interests. Authors’ contributions SM contributed with the results on linear problem and partly on nonlinear problem. BP contributed partly with the results on nonlinear problem and principally in preparation of the manuscript. All authors read and approved the final manuscript.
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