Multiple positive solutions of nonlinear third-order boundary value problems with integral boundary conditions on time scales

Transcription

1 Li n Wng Avnce in Difference qution 25 25:9 DOI.86/ R S A R C H Open Acce Multiple poitive olution of nonliner thir-orer bounry vlue problem with integrl bounry conition on time cle Yongkun Li * n Lingyn Wng * Correponence: yklie@ynu.eu.cn Deprtment of Mthemtic, Yunnn Univerity, Kunming, Yunnn 659, People Republic of Chin Abtrct In thi pper, we conier cl of nonliner thir-orer bounry vlue problem with integrl bounry conition on time cle. By pplying generliztion of the Leggett-Willim fixe point theorem, we etblih the exitence of t let three poitive olution. The icue problem involve both n increing n poitive homomorphim, which generlize the p-lplcin opertor. A n ppliction, we give n exmple to illutrte our reult. Keywor: time cle; fixe point theorem; integrl bounry conition; multiple poitive olution; cone Introuction In recent yer, much ttention h been pi to bounry vlue problem with integrl bounry conition ue to their vriou ppliction in chemicl engineering, thermoelticity, popultion ynmic, het conuction, chemicl engineering unergroun wter flow, thermo-elticity, n plm phyic. Mny reult for the bounry vlue problem with integrl bounry conition hve been reporte in the literture ee [ 5] n reference therein. For exmple, in [5], Guo et l. coniere the following bounry vlue problem: u t+ft, u t, u t =, t,, u =, u =, u = gtut t, where f :[,] R + R R + i nonnegtive function. By pplying the fixe point inex theory in cone n pectrl riu of liner opertor, they foun tht ht let one poitive olution. On the other hn, the tuy of ynmic eqution on time cle, which h been crete in orer to unify the tuy of ifferentil n ifference eqution, i n re of mthemtic tht h recently receive lot of ttention; moreover, mny reult on thi iue hve been well ocumente in the monogrph [6 8]. In ition, bounry vlue problem with integrl bounry conition on time cle repreent very intereting n importnt cl of problem. The tuy of bounry vlue problem on time cle h receive lot ttention in the literture ee [9 8]. For exmple, in [7], Li n Zhng tuie the econ-orer p-lplcin ynmic eqution on 25 Li n Wng; licenee Springer. Thi i n Open Acce rticle itribute uner the term of the Cretive Common Attribution Licene which permit unretricte ue, itribution, n reprouction in ny meium, provie the originl work i properly creite.

2 Li n Wng Avnce in Difference qution 25 25:9 Pge 2 of 5 time cle, ϕp x t + λf t, xt, x t =, t, T T, 2 with integrl bounry conition x =, αxt βx = T gx. 3 By uing the Legget-Willim fixe point theorem, they obtine ome criteri for the exitence of t let three poitive olution to problem 2-3. In [8], Hn n Kng coniere the exitence of multiple poitive olution for the following thir-orer p-lplcin ynmic eqution on time cle: p u t + f t, ut =, t [, b], αu βu =, 4 γ ub+δu b=, u =. By uing fixe point theorem in cone, they obtine the exitence criteri of t let two poitive olution to problem 4. However, to the bet of our knowlege, there i no pper publihe on the exitence of multiple poitive olution to nonliner thir-orer bounry vlue problem with integrl bounry conition on time cle. Thi pper ttempt to fill thi gp in the literture. Motivte by the bove, in thi pper, we re concerne with the following thir-orer bounry vlue problem with integrl bounry conition on time cle T: u t + qtf t, ut, u t =, t [, ] T, u bu = g, u, cu + u = g 5 2u, u =, where T i time cle, n re point in T,[,] T := [, ] T, : R R i n increing n poitive homeomorphim ee Definition 2.3with =. Our min purpoe of thi pper i to tuy the exitence of multiple poitive olution to 5. Our metho of thi pper i be on ome recent fixe point theorem erive by Bi n Ge ee [9]. A n ppliction, we give n exmple to illutrte our reult. Throughout thi pper, we ume tht the following conition re tifie: C, b, c, [, + with := c + + bc >, C 2 f C[, ] T R + R +, R + with f t,, for ll t [, ] T, C 3 g C[, ] T R +, R +, g 2 C[, ] T, R + n q C[, ] T, R +. C 4 g 2b + >. 2 Preliminrie n ttement In thi ection, we provie ome bckgroun mteril from theory of cone in Bnch pce. The following efinition cn be foun in the book by Deimling [2]wellin the book by Guo n Lkhmiknthm [2].

3 Li n Wng Avnce in Difference qution 25 25:9 Pge 3 of 5 Definition 2. Let be rel Bnch pce. A nonempty cloe convex et P i clle cone if it tifie the following two conition: i x P, λ implie λx P; ii x P, x P implie x =. very cone P inuce n orering in given by x y if n only if y x P. Definition 2.2 Ampψ i i to be nonnegtive continuou concve functionl on cone P of rel Bnch pce if ψ : P [, i continuou n ψ tx + ty tψx+ tψy for ll x, y P n t [, ]. Similrly, we y tht mp α i nonnegtive continuou convex functionl on cone P of rel Bnch pce if α : P [, i continuou n α tx + ty tαx+ tαy for ll x, y P n t [, ]. Definition 2.3 Aprojection : R R i clle n increing n poitive homomorphim if the following conition re tifie: i If x y,thenx y for ll x, y R; ii i continuou bijection, n it invere mpping i lo continuou; iii xy=xy for ll x, y R +,wherer + =[,+. Let ψ be nonnegtive continuou concve functionl on P,n α n β be nonnegtive continuou convex functionl on P. For nonnegtive rel number r,, nl, weefine the following convex et: Pα, r; β, l= u P : αu<r, βu<l }, Pα, r; β, l= u P : αu r, βu l }, Pα, r; β, l; ψ, = u P : αu<r, βu<l, ψu> }, Pα, r; β, l; ψ, = u P : αu r, βu l, ψu }. To prove our min reult, we nee the following fixe point theorem, which come from Bi n Ge in [9]. Lemm 2. [9] Let P be cone in rel Bnch pce. Aume tht contnt r, b,, r 2, l, n l 2 tify <r < b < r 2 n <l l 2. Ifthereexittwononnegtivecontinuou convex functionl α n β on P n nonnegtive continuou concve functionl ψ on P uch tht: A there exit M >uch tht u M mxαu, βu} for ll u P; A 2 Pα, r; β, l for ny r >n l >; A 3 ψu αu for ll u Pα, r 2 ; β, l 2 ; n if F : Pα, r 2 ; β, l 2 Pα, r 2 ; β, l 2 i completely continuou opertor which tifie B u Pα, ; β, l 2 ; ψ, b : ψu>b}, ψfu>b for u Pα, ; β, l 2 ; ψ, b;

4 Li n Wng Avnce in Difference qution 25 25:9 Pge 4 of 5 B 2 αfu<r, βfu<l for u Pα, r ; β, l ; B 3 ψfu>b for u Pα, r 2 ; β, l 2 ; ψ, b with αfu>, then F h t let three ifferent fixe point u, u 2, n u 3 in Pα, r 2 ; β, l 2 with u Pα, r ; β, l, u 2 P α, r 2 ; β, l 2 ; ψ, b : ψu>b }, u 3 Pα, r 2 ; β, l 2 \ Pα, r 2 ; β, l 2 ; ψ, b Pα, r ; β, l. Let = C [, ] = u n u re continuou on [, ] T }.Then i Bnch pce with repect to the norm u = mx mx ut, mx u t }. t [,] T t [,] T Define P = u : ut, u t, ut i concve on [, ] T }. Clerly, P i cone. Lemm 2.2 Aume tht C -C 3, n g 2b + hol. Then u i olution of the bounry vlue problem 5 if n only if ut i olution of the following integrl eqution: ut= Gt, qτf τ, uτ, u τ τ + t f +b + t f, where Gt, = b + σ t, σ t, b + t σ, t, 6 f := g, u, 7 [ ] [ f := g 2 b + g 2 Ɣ f + g 2 g, u ], 8 Ɣ f t:= Gt, qτf τ, uτ, u τ τ. Proof Let t ut= qτf τ, uτ, u τ τ b + σ t + t b + t σ qτf τ, uτ, u τ τ + t f +b + t f. 9

5 Li n Wng Avnce in Difference qution 25 25:9 Pge 5 of 5 Tking the -erivtive of 9, we get t u c t= + t c f + f. b + σ σ qτf τ, uτ, u τ τ qτf τ, uτ, u τ τ It follow tht u t= c b + σ t σ t qτf τ, uτ, u τ τ t = c + + bc qτf τ, uτ, u τ τ t t = qτf τ, uτ, u τ τ, hence, u = = n o u t = t qτf τ, uτ, u τ τ, u t = qtf t, ut, u t, tht i, u t + qtf t, ut, u t =. Furthermore, we hve hence b u = u = σ σ qτf τ, uτ, u τ τ + f + b f, qτf τ, uτ, u τ τ c f + f, Since u bu = f u = = g, u. b + σ qτf τ, uτ, u τ τ + f +b + f,

6 Li n Wng Avnce in Difference qution 25 25:9 Pge 6 of 5 u c = we obtin b + σ qτf τ, uτ, u τ τ c f + f, cu + u = f = g 2 ζ Gζ, qτf τ, uτ, u τ τ + ζ f +b + ζ f ζ. 2 From n2, we fin tht f n f tify f = g, u, [ g 2 ] f +[ g 2b + ] f = g 2Ɣ f, which implie tht f n f re efine by 7 n8, repectively. The proof i complete. Define n opertor F : P by Fut= Gt, qτf τ, uτ, u τ τ + t f +b + t f, 3 where G, f,n f re efine by 6, 7, n 8, repectively. Lemm 2.3 Aume tht C -C 4 hol. Then for u P, we hve: i Fut i concve on [, ] T. ii Fut for t [, ] T. Proof For u P: i By the efinition of F n imilr to the proof of,wehve Fu t= t qτf τ, uτ, u τ τ, 4 o Fu t i nonincreing. Thi implie tht Fut i concve. ii From 6 n 3,wecnverifythtFut for t [, ] T. The proof i complete. Lemm 2.4 Let C -C 4 hol. Aume tht C 5 c g 2 <. Then Fu t for u Pnt [, ] T.

7 Li n Wng Avnce in Difference qution 25 25:9 Pge 7 of 5 Proof On the contrry, we ume tht the inequlity Fu t <holforu P n t [, ] T. By Lemm 2.3,weeethtFu t i nonincreing on [, ] T.Hence Fu Fu t, t [, ] T. Similrtothe proofof2, we eily obtin Hence cfu + Fu = c Fu + Clerly, we hve Fu tht i, c g 2 = g 2 Fu. g 2 Fu =Fu Fu t<. g 2 Fu >. g 2 Fu g 2 Fu < cfu, Accoring to Lemm 2.3, wehvefu. So, c g 2 >. However, thi contrict conition C 5. Conequently, Fu t fort [, ] T. The proof i complete. Lemm 2.5 Suppoe tht C -C 5 hol. Then F : P P i completely continuou opertor. Proof From Lemm 2.3 n 2.4 it follow tht F : P P i well efine. Next, we how tht F i completelycontinuou. To thien, we umetht m i poitive contnt n u P m = u P : u m}. Note tht the continuity of f t, ut, u t n qtgurntee tht there i n M >uchthtqtf t, ut, u t M for ll t [, ] T. Therefore, ccoring to Lemm 2.4 n Lemm 2.3i, we hve mx Fut =Fu t [,] T = b + σ + f +b + f b + b + M b + + f +b + f qτf τ, uτ, u τ τ qτf τ, uτ, u τ τ + f +b + f Mτ + f +b + f

8 Li n Wng Avnce in Difference qution 25 25:9 Pge 8 of 5 n mx t [,] T Fu t =Fu = σ c f + f M + f, qτf τ, uτ, u τ τ qτf τ, uτ, u τ τ + f Mτ + f which imply tht F P m i uniformly boune. In ition, ince F P m i uniformly boune, ccoring to the men vlue theorem Theorem.4 in [7], one cn eily ee tht, for u P m, t, t 2 [, ] T, Fut2 Fut, t2 t. 5 From 4, for ll u P m, t [, ] T, it follow tht t Fu t=fu qτf τ, uτ, u τ τ. Hence, for ll u P m, t, t 2 [, ] T,wehve Fu t 2 Fu t t = qτf τ, uτ, u τ τ t2 qτf τ, uτ, u τ τ t = qτf τ, uτ, u τ τ t 2 t M t 2 = M t 2 t, which implie tht Fu t 2 Fu t, t 2 t. 6 Therefore, 5 n6 implythtfu i equicontinuou for ll u P m. By pplying the Arzel-Acoli theorem on time cle, we cn ee tht F P m i reltively compct. In view of Lebegue ominte convergence theorem on time cle, it i cler tht F i continuou opertor. Hence, F : P P i completelycontinuou opertor. The proofi complete.

9 Li n Wng Avnce in Difference qution 25 25:9 Pge 9 of 5 3 Min reult For u P,weefine αu= mx t [,] T ut = u, βu= mx t [,] T u t = u, ψu= min t [ω,] T ut=uω. It i ey to ee tht α, β : P [, re nonnegtive continuou convex functionl with u = mxαu, βu}; ψ : P [, i nonnegtive concve functionl. We hve ψu αuforu P, thi men tht umption A -A 3 in Lemm 2. hol. Suppoe tht ω T with < ω <. For convenience, we introuce the following nottion: ω := Gω, qττ, ω Ɣ:= G, γ qττ γ, := qττ [ ] + g 2 b + g 2 Ɣ+, := + b [ +b + qττ + ] g 2 b + g 2 Ɣ+. Theorem 3. Aume tht C -C 4 hol. If there exit contnt r, r, r 2, l, n l 2 with <r < r < r ω r 2,<l l 2 uch tht r min r 2, l 2 }. Suppoe further tht f, g tify the following three conition: i f t, u, v min r 2, l 2 }, g, u min r 2, l 2 }, for t, u, v [, ] T [, r 2 ] [ l 2, l 2 ]; ii f t, u, v> r for t, u, v [ω,] T [r, r ω ] [ l 2, l 2 ]; iii f t, u, v<min r, l }, g, u min r, l }, for t, u, v [, ] T [, r ] [ l, l ]. Then problem 5 h t let three nonnegtive olution u, u 2, u 3, which tify mx t [,] T u t } < r, mx u t < l ; t [,] T r < min t [ω,] T u2 t } mx t [,] T u2 t } r 2, mx u 2 t l 2 ; t [,] T min u3 t } < r, t [ω,] T r < mx t [,] T u3 t } < r ω, l < mx t [,] T u 3 t l 2. Proof The bounry vlue problem 5 holutionu = ut ifnonlyifu olve the opertor eqution Fu = u. Thu, we et out to verify tht the opertor F tifie the generliztion of the Leggett-Willim fixe point theorem, which will prove the exitence of fixe point of F.

10 Li n Wng Avnce in Difference qution 25 25:9 Pge of 5 We firt prove tht if umption i i tifie, then F : Pα, r 2 ; β, l 2 Pα, r 2 ; β, l 2. Let u Pα, r 2 ; β, l 2, then αu= mx t [,] T ut r 2, βu= mx t [,] T u t l 2, n umption i implie f t, ut, u t min r2, For ll u P,wehveFu P, therefore, n αfu = mx Fut =Fu t [,] T = b + σ b + σ b + l2 qτ }, t [, ] T. qτf τ, uτ, u τ τ + f +b + f qτf τ, uτ, u τ τ + f +b + f r2 τ + r 2 ] g b r [ 2 g 2 b + = r 2 b + [ ] + + b g 2 b + = r 2 βfu = mx Fu t =Fu t [,] T = + l 2 = l 2 [ + = l 2. σ σ qττ + σ qτ [ Ɣ+ g 2 Ɣ+ qτf τ, uτ, u τ τ c f + f qτf τ, uτ, u τ τ + f τ Ɣ+ l2 ] g 2 b + g 2 σ qττ Ɣ+ ] g 2 b + g 2 Thu, F Pα, r 2 ; β, l 2 nf Pα, r 2 ; β, l 2 Pα, r 2 ; β, l 2.

11 Li n Wng Avnce in Difference qution 25 25:9 Pge of 5 Seconly, we how tht conition B of Lemm 2. hol. We let ut= r ω for t [, ] T. It i obviou tht ut= r ω Pα, r ω ; β, l nψu= r > r, n conequently ω u P α, r } ω ; β, l 2; ψ, r : ψu>r. For ll u Pα, r ω ; β, l 2; ψ, r, we hve r ut r ω, u t l 2 for t [ω,] T.Thu,by umption ii we get f t, ut, u t r >, fort [ω,] T. From the efinition of the functionl ψ we ee tht ψfu = min Fut =Fuω t [ω,] T = Gω, qτf τ, uτ, u τ τ ω + ω f +b + ω f Gω, qτf τ, uτ, u τ τ > ω = r ω Gω, ω Gω, ω Gω, qτf τ, uτ, u τ τ r qτ τ qττ = r. ω So, we obtin ψfu>r for u Pα, r ω ; β, l 2; ψ, r. Therefore, conition B of Lemm 2. i tifie. Thirly, we how tht the conition B 2 of Lemm 2. i tifie. For ll u Pα, r ; β, l, we hve ut r, l u t l for t [, ] T. From umption iii we obtin Thu f t, ut, u t < min r, l }, t [, ] T. αfu = mx Fut =Fu t [,] T = < b + σ b + σ b + σ qτ qτf τ, uτ, u τ τ + f +b + f qτf τ, uτ, u τ τ + f +b + f r τ + r

12 Li n Wng Avnce in Difference qution 25 25:9 Pge 2 of 5 +b + r [ ] g 2 b + = r + b qττ + [ ] +b + g 2 b + = r g 2 Ɣ+ g 2 Ɣ+ n βfu = mx Fu t =Fu t [,] T = < = l σ σ l qτ + l [ ] g 2 b + qττ [ ] + g 2 b + = l. qτf τ, uτ, u τ τ c f + f qτf τ, uτ, u τ τ + f τ g 2 Ɣ+ g 2 Ɣ+ We get F : Pα, r ; β, l Pα, r ; β, l, which men tht B 2 in Lemm 2. i tifie. Finlly, we how tht ψfu>b for u Pα, r 2 ; β, l 2 ; ψ, bwithαfu>. It i ey to ee tht Fu t for ny t [, ] T.HenceFu i ecreing function on [, ] T. Thi men tht grph of Fu i concve own on ω, T.So,wehve nmely, Fuω Fu ω Fu Fu, Fuω ωfu + ωfu ωfu. 7 For ll u Pα, r 2 ; β, l 2 ; ψ, rwithαfu> r n 7, we hve ω ψfu= min t [ω,] T Fut=Fuω ωfu = ω mx t [,] T Fut=ωαFu>r.

13 Li n Wng Avnce in Difference qution 25 25:9 Pge 3 of 5 Therefore, conition B 3 of Lemm 2. i tifie. So, ll the conition of Lemm 2. re tifie. It follow from Lemm 2. n the umption tht f t,, on[,] T tht F h t let three fixe point u, u 2, u 3 tifying mx t [,] T u t } < r, mx u t < l ; t [,] T r < min t [ω,] T u2 t } mx t [,] T u2 t } r 2, mx u 2 t l 2 ; t [,] T min u3 t } < r, t [ω,] T r < mx t [,] T u3 t } < r ω, l < mx t [,] T u 3 t l 2. The proof i complete. If x= p x= x p 2 x for ome p >,where p with p-lplce opertor: = q,then5cnbewrittenbvp p u t + qtf t, ut, u t =, t [, ] T, u bu = g, u, cu + u = g 2u, u =. 8 Then, by Theorem 3., we hve the following. Corollry 3. Aume tht C -C 5 hol. If there exit contnt r, r, r 2, l, n l 2 with <r < r < r ω r 2,<l l 2 uch tht r min r 2, l 2 }. Suppoe further tht f, g tify the following three conition: i f t, u, v min p r 2, p l 2 }, g, u min r 2, l 2 }, for t, u, v [, ] T [, r 2 ] [ l 2, l 2 ]; ii f t, u, v> p r for t, u, v [ω,] T [r, r ω ] [ l 2, l 2 ]; iii f t, u, v<min p r, p l }, g, u min r, l }, for t, u, v [, ] T [, r ] [ l, l ]. Then problem 8 h t let three nonnegtive olution u, u 2, u 3, which tify mx t [,] T u t } < r, mx u t < l ; t [,] T r < min t [ω,] T u2 t } mx t [,] T u2 t } r 2, mx u 2 t l2 ; t [,] T min u3 t } < r, t [ω,] T r < mx t [,] T u3 t } < r ω, l < mx t [,] T u 3 t l2. 4 Anexmple Let T = [, ]. Tke qt=, =3,b =,c =,n =4fort [, ]. Conier the following 6 BVP: u t + f t, ut, u t =, t [, ], 3u u = g, u, 6 u + 4u = g 2u, u =, 9

14 Li n Wng Avnce in Difference qution 25 25:9 Pge 4 of 5 where n f t, u, v= t u3 + v 3, u 3, t v 3, u >3, u 3 u=u, g t, u=t 2 u + t, g 2 t=. By clculting, we hve = 38 3,n Gt, = t, t, t 25 6, t. 6 Set ω = 2,thenweobtin = 27,78, = 68 9,272, = 4,537 2,38. Clerly, umption C -C 5 holnft,, on[,].wechooer = 2, r =3, r 2 =2,nl = 4, l 2 =9.So<r < r < r ω n < l < l 2,nitieytochecktht r min r 2, l 2 }. Now, we how tht conition i-iii re tifie: i f t, u, v 8.2 < 6.39 min r 2, l 2 }, g, u 2t t =4.864<6.39 min r 2, l 2 },for t, u, v [, ] [, 2] [ 4, 4 ]; ii f t, u, v 7.28 > 8.82 r,fort, u, v [,] [3, 6] [ 9, 9]; 2 iii f t, u, v.33 <.97 min r, l }, g, u 2 t2.667 <.97 min r, l },for t, u, v [, ] [, 2 ] [ 4, 4 ]. From the bove, we ee tht ll the conition of Theorem 3. re tifie. Hence, by Theorem 3.,BVP9 h t let three nonnegtive olution u, u 2, u 3 uch tht mx u t } < t [,] T 2, mx u t < t [,] T 4 ; 3< min t [,] T u2 t } mx t [,] T u2 t } 2, min t [ω,] T u3 t } <3, 2 < mx t [,] T u3 t } <6, mx u 2 t 9; t [,] T 4 < mx t [,] T u 3 t 9. Competing interet The uthor eclre tht they hve no competing interet. Author contribution All uthor wrote, re, n pprove the finl mnucript. Acknowlegement The uthor woul like to thnk the referee for their vluble comment n uggetion. Thi tuy w upporte by the Ntionl Nturl Science Fountion of People Republic of Chin uner Grnt Receive: 6 Februry 25 Accepte: 3 Mrch 25

15 Li n Wng Avnce in Difference qution 25 25:9 Pge 5 of 5 Reference. Timohenko, S: Theory of ltic Stbility. McGrw-Hill, New York Yng, ZL: Poitive olution of econ orer integrl bounry vlue problem. J. Mth. Anl. Appl. 32, Zhng, XM, Feng, MQ, Ge, WG: xitence of olution of bounry vlue problem with integrl bounry conition for econ-orer impulive integro-ifferentil eqution in Bnch pce. Comput. Appl. Mth. 233, Feng, MQ: xitence of ymmetric poitive olution for bounry vlue problem with integrl bounry conition. Appl. Mth. Lett. 24, Guo, YP, Liu, YJ, Ling, YC: Poitive olution for the thir-orer bounry vlue problem with the econ erivtive. Boun. Vlue Probl. 22,ArticleID Bohner, M, Peteron, A: Dynmic qution on Time Scle: An Introuction with Appliction. Birkhäuer, Boton 2 7. Bohner, M, Peteron, A: Avnce in Dynmic qution on Time Scle. Birkhäuer, Boton Lkhmiknthm, V, Sivunrm, S, Kymkcln, B: Dynmic Sytem on Meure Chin. Kluwer Acemic, Boton Li, YK, Shu, JY: Multiple poitive olution for firt-orer impulive integrl bounry vlue problem on time cle. Boun. Vlue Probl. 2,ArticleID2 2. Li, YK, Sun, LJ: Infinite mny poitive olution for nonliner firt-orer BVP with integrl bounry conition on time cle. Topol. Metho Nonliner Anl. 4, Agrwl, RP, Otero-pinr, V, Perer, K, Vivero, DR: Multiple poitive olution of ingulr Dirichlet problem on time cle vi vritionl metho. Nonliner Anl. 67, Jing, LQ, Zhou, Z: xitence of wek olution of two-point bounry vlue problem for econ-orer ynmic eqution on time cle. Nonliner Anl. 69, Rynne, BP: L 2 Spce n bounry vlue problem on time-cle. J. Mth. Anl. Appl. 328, Zhou, JW, Li, YK: Sobolev pce on time cle n it ppliction to cl of econ orer Hmiltonin ytem on time cle. Nonliner Anl. 73, Zhou, JW, Li, YK: Vritionl pproch to cl of econ orer Hmiltonin ytem on time cle. Act Appl. Mth. 7, Tokmk, F, Krc, IY: xitence of poitive olution for thir-orer bounry vlue problem with integrl bounry conition on time cle. J. Inequl. Appl. 23, Article ID Li, YK, Zhng, TW: Multiple poitive olution for econ-orer p-lplcin ynmic eqution with integrl bounry conition. Boun. Vlue Probl. 2, ArticleID Hn, W, Kng, SG: Multiple poitive olution of nonliner thir-orer BVP for cl of p-lplcin ynmic eqution on time cle. Mth. Comput. Moel. 49, Bi, ZB, Ge, WG: xitence of three poitive olution for ome econ-orer bounry vlue problem. Comput. Mth. Appl. 48, Deimling, K: Nonliner Functionl Anlyi. Springer, New York Guo, D, Lkhmiknthm, V: Nonliner Problem in Abtrct Cone. Acemic Pre, New York 988