NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS OF HIGHER ORDER DIFFERENTIAL EQUATIONS WITH p-laplacian

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1 Electronc Journal of Dfferental Equatons, Vol. 2828, No. 2, pp. 43. ISSN: URL: or ftp ejde.ath.txstate.edu logn: ftp NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS OF HIGHER ORDER DIFFERENTIAL EQUATIONS WITH p-laplacian YUJI LIU Abstract. We establsh suffcent condtons for the exstence of postve solutons to fve ult-pont boundary value probles. These probles have a coon equaton n dfferent functon doans and dfferent boundary condtons. It s nterestng note that the ethods for solvng all these probles and ost of the reference are based on the Mawhn s concdence degree theory. Frst, we present a survey of ult-pont boundary-value probles and the otvaton of ths paper. Then we present the an results whch generalze and prove results n the references. We conclude ths artcle wth exaples of probles that can not solved by ethods known so far.. Introducton Mult-pont boundary-value probles BVPs for dfferental equatons were ntaled by Il n and Moseev 2] and have receved a wde attenton because of ther potental applcatons. There are any exctng results concerned wth the exstence of postve solutons of boundary-value probles of second or hgher order dfferental equatons wth or wthout p-laplacan subjected to the specal hoogeneous ult-pont boundary condtons BCs; we refer the readers to ] ], 9] 24] 27] 47], 49] 52], 55] 79]. The ethods used for fndng postve solutons of these probles at non-resonance cases, or solutons at resonance cases, are crtcal pont theory, fxed pont theores n cones n Banach spaces, fxed pont ndex theory, alternatve of Leray-Schauder, upper and lower soluton ethods wth teratve technques, and so on. There are also several results concerned wth the exstence of postve solutons of ult-pont boundary-value probles for dfferental equatons wth non-hoogeneous BCs; see for exaple 2, 3, 25, 26, 48, 53] and the early paper 79]. For the reader s nforaton and to copare our results wth the known ones, we now gve a sple survey. 2 Matheatcs Subject Classfcaton. 34B, 34B5, 35B. Key words and phrases. One-denson p-laplacan dfferental equaton; postve soluton; ult-pont boundary-value proble; non-hoogeneous boundary condtons; Mawhn s concdence degree theory. c 28 Texas State Unversty - San Marcos. Subtted Septeber 27, 27. Publshed February 2, 28. Supported by grant 6JJ58 fro the Natural Scence Foundaton of Hunan Provnce and by the Natural Scence Foundaton of Guangdong Provnce, P. R. Chna.

2 2 Y. LIU EJDE-28/2 Mult-pont boundary-value probles wth hoogeneous BCs consst of the second order dfferental equaton and the ult-pont hoogeneous boundary condtons. The second order dfferental equaton s ether or one of the followng cases φx t] ft, xt, x t =, t,, x t ft, xt, x t =, t,, φx t ] ft, xt =, t,, x t ft, xt =, t,. The ult-pont hoogeneous boundary condtons are ether n x α xξ = x β xη =, x x x x x x α x ξ = x α xξ = x α x ξ = x α x ξ = x α x ξ = x α x ξ = x n β xη =, n β x η =, n β x η =, n β x η =, n β xη =, n β x η =, or ther specal cases, where < ξ < < ξ < and < η < < η n <, α, β j R are constants. These probles were studed extensvely n papers ] 75] and the references theren.. For the second order dfferental equatons, Gupta 6] studed the followng ult-pont boundary-value proble and x t = ft, xt, x t rt, t,, n x α xξ = x β xη =, x t = ft, xt, x t rt, t,, n x α xξ = x β x η =,..2 where < ξ < < ξ <, < η < < η n <, α, β R wth α ξ n β α n β η for. and wth α n β for.2. Soe exstence results for solutons of.

3 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS 3 and.2 were establshed n 4]. solutons of. for the case α ξ = Lu 36] establshed the exstence results of α = β = β ξ =. Lu and Yu 33, 34, 35, 37] studed the exstence of solutons of. and.2 at soe specal cases. Zhang and Wang 78] studed the ult-pont boundary-value proble x t = ft, xt, t,, n x α xξ = x β xη =,.3 where < ξ < < ξ <, α, β, wth < α < and β <. Under certan condtons on f, they establshed soe exstence results for postve solutons of.3. Lu n 32], and Lu and Ge n 43] studed the four-pont boundary-value proble x t ft, xt =, t,, x αxξ = x βxη =,.4 where < ξ, η <, α, β, f s a nonnegatve contnuous functon. Usng the Green s functon of ts correspondng lnear proble, Lu establshed exstence results for at least one or two postve solutons of.4. Ma n 49], and Zhang and Sun n 77] studed the followng ult-pont boundaryvalue proble x t atfxt =, t,,.5 x = x α xξ =, where < ξ <, α wth α ξ <, a and f are nonnegatve contnuous functons, there s t ξ, ] so that at >. Let fx fx l = l, l x x x x = L. It was proved that f l =, L = or l =, L =, then.5 has at least one postve soluton. Ma and Castaneda 5] studed the proble x t atfxt =, t,, x α x ξ = x β xξ =,.6 where < ξ < < ξ <, α, β wth < α < and < β < and a and f are nonnegatve contnuous functons, there s t ξ, ] so that at >. Ma and Castaneda establshed exstence results for postve solutons of.6 under the assuptons fx fx l =, l x x x x = or l fx fx =, l x x x x =.

4 4 Y. LIU EJDE-28/2 2. For second order dfferental equatons wth p-laplacan, Drabek and Takc 8] studed the exstence of solutons of the proble φx t λφx = ft, t, T, x = xt =,.7 In a recent paper 28], the author establshed ultplcty results for postve solutons of the probles φp x t ] ft, xt =, t,, and x = xsdhs, φ p x = φ p x sdgs, φp x t ] ft, xt =, t,, φ p x = φ p x sdhs, x = xsdgs. Gupta 7] studed the exstence of solutons of the proble φx t ] ft, xt, x t et =, t,, x α xξ = x β xξ =.8 by usng topologcal degree and soe a pror estates. Ba and Fang 6] nvestgated the followng ult-pont boundary-value proble φx t ] atft, xt =, t,, x = x β xξ =,.9 where < ξ < < ξ <, β wth β ξ <, a s contnuous and nonnegatve and there s t ξ, ] so that at >, f s a contnuous nonnegatve functon. The purpose of 6] s to generalze the results n 49]. Wang and Ge 63], J, Feng and Ge 2], Feng, Ge and Jang 9], Rynne 58] studed the exstence of ultple postve solutons of the followng ore general proble φx t ] atft, xt =, t,, x α xξ = x β xξ = by usng fxed pont theores for operators n cones. Sun, Qu and Ge 62] usng the onotone teratve technque establshed exstence results of postve solutons of the proble φx t ] atft, xt, x t =, t,, x α xξ = x β xξ =.

5 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS 5 Ba and Fang 5] studed the proble φx t ] ft, xt =, t,, x α x ξ = x β xξ =,. where < ξ < < ξ <, α, β wth < α < and < β <, f s contnuous and nonnegatve. The purpose of 5] s to generalze and prove the results n 5]. In paper Ma, Du and Ge 54] studed.6 by usng the onotone teratve ethods. The exstence of onotone postve solutons of.6 were obtaned. Based upon the fxed pont theore due to Avery and Peterson 4], Wang and Ge 64], Sun, Ge and Zhao 6] establshed exstence results of ultple postve solutons of the followng probles φx t ] atft, xt, x t =, t,, and x α x ξ = x β xξ = φx t ] atft, xt, x t =, t,, x α xξ = x β x ξ =. In 28, 37], the authors studed the exstence of solutons of the followng BVPs at resonance cases x t = ft, xt, x t et, t, T, x = αx ξ, x = β x ξ.. In a recent paper ], the authors nvestgated the exstence of solutons of the followng proble for p-laplacan dfferental equaton φx t = ft, xt, x t, t, T, x =, θx = α θx ξ,.2 where θ and φ are two odd ncreasng hoeoorphss fro R to R wth φ = θ =. In the recent papers 9, 24, 25, 29, 36, 56, 6, 6, 63, 64, 65, 66, 7, 76], the authors studed the exstence of ultple postve solutons of.8,.9,. or other ore general ult-pont boundary-value probles, respectvely, by usng of ultple fxed pont theores n cones n Banach spaces such as the fve functonals fxed pont theore 9], the fxed-pont ndex theory 59], the fxed pont theore due to Avery and Peterson, a two-fxed-pont theore 9, 6, 63, 64], Krasnoselsk s fxed pont theore and the contracton appng prncple 22, 29, 56, 6, 7], the Leggett-Wllas fxed-pont theore 23, 36], the generalzaton of polar coordnates 65], usng the soluton of an plct functonal equaton 22, 23].

6 6 Y. LIU EJDE-28/2 3. For hgher order dfferental equatons, there have been any papers dscussed the exstence of solutons of ult-pont boundary-value probles for thrd order dfferental equatons 5, 47, 55]. Ma 47] studed the solvablty of the proble x t k 2 x t gxt, x t = pt, x =, x = x π =, t, π,.3 where k N, g s contnuous and bounded, p s contnuous. In 5, 55], the authors nvestgated the solvablty of the proble x t k 2 x t gt, xt, x t, x t = pt, x = x =, x =,.4 where g and p are contnuous, k R. It was supposed n 55] that g s bounded and n 5] g satsfes gt, u, v, wv for t, ], u, v, w R 3, gt, u, v, w l < 3π 2 unforly n t, u, w. v v The upper and lower soluton ethods wth onotone teratve technque are used to solve ult-pont boundary-value probles for thrd or fourth order dfferental equatons n papers 76] and 66]. In 4], the authors studed the proble x n t λfxt =, t,, x =, =,..., n 3, x n 2 αx n 2 η = x n 2 βx n 2 η =,.5 the exstence results for postve solutons of.5 were establshed n 4] n the case that the nonlnearty f changes sgn. The exstence of postve solutons of the followng two probles: and x n t φtft, xt,..., x n 2 t =, t,, x =, =,..., n 2, x n =, x n t φtft, xt,..., x n 2 t =, t,, x =, =,..., n 2, x n 2 =,.6.7 were studed n 2, 73]. 4. For Stur-Louvlle type ult-pont boundary condtons, Grossnho 2] studed the proble x t ft, xt, x t, x t =, t,, x =, ax bx = A, cx dx = B..8 By usng theory of Leray-Schauder degree, t was proved that.8 has solutons under the assuptons that there exst super and lower solutons of the correspondng proble.

7 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS 7 Agarwal and Wong 3], Q 57] nvestgated the solvablty of the followng proble wth Stur-Louvlle type boundary condtons x n t = ft, xt,..., x n 2 t, t,, x =, =,..., n 3, αx n 2 βx n = γx n 2 τx n =,.9 The authors n 24] studed the exstence and nonexstence of solutons of a stuaton ore general than.8. Lan and Wong 3] studed the exstence of postve solutons of the followng BVPs consstng of the p-laplacan dfferental equaton and Stur-Louvlle boundary condtons φx n t ] ft, xt,..., x n 2 t =, t,, x =, =,..., n 3, αx n 2 βx n = γx n 2 τx n =,.2 In all above entoned papers, all of the boundary condtons concerned are hoogeneous cases. However, n any applcatons, BVPs are nonhoogeneous BVPc, for exaple, y = λ y2 2, ya = aα, t a, b, yb = β and y = y t 2 2yt α, ya = aα, t a, b, yb = β are very well known BVPs, whch were proposed n 69 and 696, respectvely. In 964, The BVPs studed by Zhdkov and Shrkov n USSR Coputatonal Matheatcs and Matheatcal Physcs, ] and by Lee n Checal Engneerng Scence, ] are nonhoogeneous BVPs too. There are also several papers concernng wth the exstence of postve solutons of BVPs for dfferental equatons wth non-hoogeneous BCs. Ma 48] studed exstence of postve solutons of the followng BVP consstng of second order dfferental equatons and three-pont BC x t atfxt =, t,, x =, x αxη = b,.2 In a recent paper 25, 26], usng lower and upper solutons ethods, Kong and Kong establshed results for solutons and postve solutons of the followng two probles x t ft, xt, x t =, t,, x α x ξ = λ, x β xξ = λ 2,.22

8 8 Y. LIU EJDE-28/2 and x t ft, xt, x t =, t,, x α xξ = λ, x β xξ = λ 2,.23 respectvely. We note that the boundary condtons n.7,.2,.2 and.22 are two-paraeter non-hoogeneous BCs. The purpose of ths paper s to nvestgate the ore generalzed BVPs for hgher order dfferental equaton wth p-laplacan subjected to non-hoogeneous BCs, n whch the nonlnearty f contans t, x,..., x n,.e. the probles φx n t ] ft, xt,..., x n t =, t,, x n 2 α x n 2 ξ = λ, x n 2 β x n 2 ξ = λ 2, x =, =,..., n 3;.24 φx n t ] ft, xt,..., x n t =, t,, x n α x n ξ = λ, x n 2 β x n 2 ξ = λ 2, x =, =,..., n 3;.25 φx n t ] ft, xt,..., x n t =, t,, x n 2 α x n 2 ξ = λ, x n β x n ξ = λ 2, x =, =,..., n 3;.26 φx n t ] ft, xt,..., x n t =, t,, x n 2 α x n ξ = λ, x n 2 β x n ξ = λ 2, x =, =,..., n 3;.27

9 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS 9 and φx n t ] ft, xt,..., x n t =, t,, φx n θx n α φx n ξ = λ, β θx n ξ = λ 2, x =, =,..., n 3;.28 where n 2, < ξ < < ξ <, α, β R, λ, λ 2 R, f s contnuous, φ s called p-laplacan, φx = x p 2 x for x and φ = wth p >, ts nverse functon s denoted by φ x wth φ x = x q 2 x for x and φ =, where /p /q =, θ s an odd ncreasng hoeoorphss fro R to R wth θ =. We establsh suffcent condtons for the exstence of at least one postve soluton of.24,.25,.26,.27, and at least one soluton of.28, respectvely. The frst otvaton of ths paper s that t s of sgnfcance to nvestgate the exstence of postve solutons of.9 and. snce the operators defned n 5, 6, 48, 49] are can not be used; furtherore, t s ore nterestng to establsh exstence results for postve solutons of hgher order BVPs wth non-hoogeneous BCs. The second otvaton to study.24,.25,.26,.27 and.28 coes fro the facts that.24 contans.,.3,.4,.5,.7.8,.9,.3,.4,.5,.7 and.23 as specal cases;.25 contans.6,. and.22 as specal cases;.26 contans.2 and.6 as specal cases; v.27 contans.8 and.9 as specal cases; v.28 contans. and.2 as specal cases. Furtherore, n ost of the known papers, the nonlnearty f only depends on a part of lower dervatves, the proble s that under what condtons probles have solutons when f depends on all lower dervatves, such as n BVPs above, f depends on x, x,..., x n. The thrd otvaton s that there exst several papers dscussng the solvablty of Stur-Louvlle type boundary-value probles for p-laplacan dfferental equatons, whereas there s few paper concerned wth the solvablty of Stur-Louvlle type ult-pont boundary-value probles for p-laplacan dfferental equatons, such as.27. The fourth otvaton coes fro the challenge to fnd sple condtons on the functon f, for the exstence of a soluton of.28, as the nonlnear hoeoorphss φ and θ generatng, respectvely, the dfferental operator and the boundary condtons are dfferent. The technques for studyng the exstence of postve solutons of ult-pont boundary-value probles consstng of the hgher-order dfferental equaton wth p-laplacan and non-hoogeneous BCs are few. Addtonal otvaton s that the concdence degree theory by Mawhn s reported to be an effectve approach to the study the exstence of perodc solutons of dfferental equatons wth or wthout delays, the exstence of solutons of ultpont boundary-value probles at resonance case for dfferental equatons; see

10 Y. LIU EJDE-28/2 for exaple 33, 35, 37, 39, 45] and the references theren, but there s few paper concernng the exstence of postve solutons of non-hoogeneous ult-pont boundary-value probles for hgher order dfferental equatons wth p-laplacan by usng the concdence degree theory. The followng of ths paper s organzed as follows: the an results and rearks are presented n Secton 2, and soe exaples are gven n Secton 3. The ethods used and the results obtaned n ths paper are dfferent fro those n known papers. Our theores generalze and prove the known ones. 2. Man Results In ths secton, we present the an results n ths paper, whose proofs wll be done by usng the followng fxed pont theore due to Mawhn. Let X and Y be real Banach spaces, L : DL X Y be a Fredhol operator of ndex zero, P : X X, Q : Y Y be projectors such that I P = Ker L, Ker Q = I L, X = Ker L Ker P, Y = I L I Q. It follows that L DL Ker P : DL Ker P I L s nvertble, we denote the nverse of that ap by K p. If Ω s an open bounded subset of X, DL Ω, the ap N : X Y wll be called L-copact on Ω f QNΩ s bounded and K p I QN : Ω X s copact. Lea 2. ]. Let L be a Fredhol operator of ndex zero and let N be L- copact on Ω. Assue that the followng condtons are satsfed: Lx λnx for every x, λ DL \ Ker L Ω], ; Nx / I L for every x Ker L Ω; deg QN Ker L, Ω Ker L,, where : Y/ I L Ker L s an soorphs. Then the equaton Lx = Nx has at least one soluton n DL Ω. In ths paper, we choose X = C n 2, ] C, ] wth the nor x, y = ax{ x,..., x n 2, y }, and Y = C, ] C, ] R 2 wth the nor x, y, a, b = ax{ x, y, a, b }, then X and Y are real Banach spaces. Let DL = { x, x 2 C n, ] C, ] : x =, =,..., n 3}. Now we prove an portant lea. Then we wll establsh exstence results for postve solutons of.24,.25,.26,.27 and.28 n sub-secton 2., 2.2, 2.3, 2.4 and 2.5, respectvely. Lea 2.2. aσ Kσ a σ for all a and σ >, where K σ s defned by K σ = for σ and K σ = 2 for σ,. Proof. Case. = 2. Wthout loss of generalty, suppose a a 2. Let gx = K σ x σ x σ, x,, then { g = K σ 2 σ 2 σ 2, σ, 2 = 2 σ 2, σ,

11 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS and for x,, we get g x = σx σ K σ /x σ ] {, σ, σx σ 2 / ] =, σ,. We get that gx g for all x and so x σ K σ x σ for all x,. Hence a σ a σ 2 = a σ 2 a /a 2 σ ] K σ a σ 2 a /a 2 ] σ = K σ a a 2 σ. Case 2. > 2. It s easy to see that a σ = a σ a σ 2 The proof s coplete. =3 a σ K σ a a 2 σ K σ a a 2 σ =3 a σ =3 K σ a a 2 σ a σ 3 a σ =4 K σ K σ a a 2 a 3 σ K 2 σ... K σ Reark 2.3. It s easy to see that φa Kp φ a, a a 2 a 3 σ σ. a 2.. Postve solutons of Proble.24. Let =4 a σ =4 a σ a σ φ a K q φ a. f t, x,..., x n = ft, x,..., x n 2, x n, t, x,..., x n, ] R n, where x = ax{, x}. The followng assuptons, whch wll be used n the proofs of all leas n ths sub-secton, are supposed. H f :, ], n R, s contnuous wth ft,,..., on each sub-nterval of,]; H2 λ, λ 2, α, β satsfy < α <, < β < and λ / α = λ 2 / β ; H3 there exst contnuous nonnegatve functons a, b and c so that n 2 ft, x,..., x n 2, x n at b tφ x ctφ x n, for t, x,..., x n, ] R n ; =

12 2 Y. LIU EJDE-28/2 H4 The followng nequalty holds Kq φ α ξ ] b n 2 sds α = csds <. φ n 2! b sds We consder the proble φx n t ] f t, xt,..., x n t =, t,, x =, =,..., n 3, x n 2 α x n 2 ξ = λ, x n 2 β x n 2 ξ = λ Lea 2.4. If H H2 hold and x s a soluton of 2., then xt > for all t,, and x s a postve soluton of.24. Proof. H ples that φx n t] = f t, xt,..., x n t, and then x n t s decreasng and so x n 2 s concave on,], thus n t,] xn 2 t = n{x n 2, x n 2 }. Together wth the boundary condtons n 29 and H2, we get that x n 2 = α x n 2 ξ λ α n{x n 2, x n 2 }, 2.2 and x n 2 = β x n 2 ξ λ 2 β n{x n 2, x n 2 }. Wthout loss of generalty, assue that α β. If n{x n 2, x n 2 } <, then x n 2 β n{x n 2, x n 2 } α n{x n 2, x n 2 }. Together wth 3, we have n{x n 2, x n 2 } α n{x n 2, x n 2 }. Hence n{x n 2, x n 2 }. It follows that n{x n 2, x n 2 }. So H ples that x n 2 t > for all t,. Then fro the boundary condtons, we get x t > for all t, and =,..., n 3. Then f t, xt,..., x n t = ft, xt,..., x n t. Thus x s a postve soluton of.24. The proof s coplete. Lea 2.5. If H H2 hold and x s a solutons of 2., then there exsts ξ, ] such that x n ξ =.

13 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS 3 Proof. In fact, f x n t > for all t, ], then x n 2 = α x n 2 ξ λ > α x n 2 λ, then x n 2 > λ / α, t follows that x n 2 > λ / α. On the other hand, x n 2 = β x n 2 ξ λ 2 < β x n 2 λ 2, thus x n 2 < λ 2 / β = λ / α < x n 2, a contradcton. f x n t < for all t, ], then x n 2 = β x n 2 ξ λ 2 > β x n 2 λ 2, then x n 2 > λ 2 / β, t follows that x n 2 > λ 2 / β. On the other hand, x n 2 = α x n 2 ξ λ < α x n 2 λ, thus x n 2 < λ / α = λ 2 / β < x n 2, contradcton too. Hence there s ξ, ] so that x n ξ =. The proof s coplete. Lea 2.6. If x, x 2 s a soluton of the proble x n t = φ x 2 t, t, ], x 2t = f t, x t,..., x n 2 t, φ x 2 t, t, ], x n 2 α x n 2 ξ = λ, then x s a soluton of 2.. x n 2 n β x n 2 ξ = λ 2, x =, =,..., n 3, 2.3 The proof of the above lea s sple; os t s otted. Defne the operators n Lx, x 2 = x n, x 2, x n 2 α x n 2 ξ, x n 2 β x n 2 ξ, x, x 2 X DL; Nx, x 2 = φ x 2, f t, x,..., x n 2, φ x 2, λ, λ 2, x, x 2 X.

14 4 Y. LIU EJDE-28/2 Under the assuptons H H2, t s easy to show the followng results: Ker L = {, c : c R} and I L = { ξ y, y 2, a, b : α α y sds a ξ β y sds β y sds b = } L s a Fredhol operator of ndex zero; There exst projectors P : X X and Q : Y Y such that Ker L = I P and Ker Q = I L. Furtherore, let Ω X be an open bounded subset wth Ω DL, then N s L-copact on Ω; v x = x, x 2 s a soluton of 2.3 f and only f x s a soluton of the operator equaton Lx = Nx n DL. We present the projectors P and Q as follows: P x, x 2 =, x 2 for all x = x, x 2 X and Qy, y 2, a, b = ξ α α ξ β y sds β where = α y sds a α ξ β ] y sds b,,,, β ξ. The generalzed nverse of L : DL Ker P I L s defned by t t s n 2 K P y, y 2, a, b = y sds n 2! n 2! ξ α α tn 2 y sds a, the soorphs : Y/ I L Ker L s defned by c,,, =, c. Lea 2.7. Suppose that H-H4 hold, and let t y 2 sds, Ω = {x, x 2 DL \ Ker L : Lx, x 2 = λnx, x 2 for soe λ, }. Then Ω s bounded. Proof. For x, x 2 Ω, we get Lx, x 2 = λnx, x 2. Then x n t = λφ x 2 t, t, ], x 2t = λf t, x t,..., x n 2 t, φ x 2 t, t, ], x n 2 α x n 2 ξ = λλ, x n 2 β x n 2 ξ = λλ 2, x =, =,..., n 3,

15 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS 5 where λ,. If x, x 2 s a soluton of Lx, x 2 = λnx, x 2 and x, x 2, c, t follows fro Lea 2.5 that there s ξ, ] so that x 2 ξ =. Then H3 ples x 2 t = λ t ξ f s, x s,..., x n 2 s, φ x 2 sds f s, x s,..., x n 2 s, φ x 2 s ds n 2 as = x n 2 = α x n 2 α α b sφ x s cs x 2s ds, α x n 2 α x n 2 x n 2 ξ λ α ξ xn θ λ, θ, ξ ], α α ξ φ x 2 λ. Then Lea 2.2;.e., Reark 2.3, ples x n 2 t x n 2 t x n sds α ξ φ α x 2 K q φ α ξ α Slarly, for =,..., n 3, we get x t x t t t s n 3! xn 2 sds t s n 3 n 3! ds x n 2 n 2! xn 2 n 2! K q φ α ξ φ α x 2 n 2! ]. λ α φ α ξ α λ α x2 φ λ ] α. λ n 2! x2 φ n 2! α

16 6 Y. LIU EJDE-28/2 It follows that x 2 t asds = b sdsφ λ n 2! b n 2 sdsφ α ξ α α ξ n 2! α α asds φk q φk q φk q φk q It follows that x 2 Then = = b n 2 sdsφ asds φk q φk q φk q φk q = csds x 2 b sdsφ n 2! α ξ α b sdsφ n 2! b n 2 sdsφ λ. α = b n 2 sdsφ φk q φk q φ x 2 φ x 2 λ α φ α ξ α x2 x2 csds x 2 λ α b sdsφ φ α ξ n 2! α x2 α ξ α x2 csds x 2 b sdsφ n 2! b n 2 sdsφ λ. α = φ n 2! b n 2 sdsφ asds φk q φk q = λ α b sdsφ α ξ α b sdsφ n 2! b n 2 sdsφ λ. α α ξ α ] csds x 2 λ α

17 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS 7 It follow fro H4 that there s a constant M > so that x 2 M. Snce x t! xn 2 and x n 2 P t αξ P φ x 2 α λ P, there exst constants M α > so that x M for all =,..., n 2. Then Ω s bounded. The proof s coplete. Lea 2.8. Suppose that H2 holds. Then there exsts a constant M > such that for each x =, c Ker L, f N, c I L, we get that c M. Proof. For each x =, c Ker L, f N, c I L, we get φ c, f t,,...,, φ c, λ, λ 2 I L. Then ξ α α φ cds λ ξ β φ cds β φ cds λ 2 =. It follows that φ c = α ξ α β ξ λ 2 β λ β α. So there exsts a constant M > such that c M. The proof s coplete. Lea 2.9. Suppose that H2 holds. Then there exsts a constant M 2 > such that for each x =, c Ker L, f λ, c λ sgn QN, c =, then c M 2. Proof. For each x =, c Ker L, f λ, c λ sgn QN, c =, we get λc = λ sgn α ξ α β ξ β φ c λ λ ] 2 α β. Thus λc 2 = λ sgn λ α If λ =, then c =. If λ,, snce q >, α ξ α λ 2 β α ξ α one sees, for suffcently large c, that λc 2 = λ sgn λ α c β ξ β ]. β ξ β >, α ξ α λ 2 β c β ξ β ] < φ cc c q

18 8 Y. LIU EJDE-28/2 a contradcton. So there exsts a constant M 2 > such that c M 2. The proof s coplete. Theore 2.. Suppose H H4 hold. Then.24 has at least one postve soluton. Proof. Let Ω Ω be a bounded open subset of X centered at zero wth ts daeter greater than ax{m, M 2}. It follows fro Leas 2.7, 2.8, 2.9 that Lx λnx for all x, λ DL \ Ker L Ω], ; Nx / I L for every x Ker L Ω; deg QN Ker L, Ω Ker L,. Snce H holds, L be a Fredhol operator of ndex zero and N be L-copact on Ω. It follows fro Lea 2. that Lx = Nx has at least one soluton x = x, x 2. Then x s a soluton of 2.. We note that x t for t, ] and =,..., n 2, so f t, x t,..., x n 2 t, φ x 2 t = f t, x t,..., x n 2 t, φ x 2 t. Hence x s a postve soluton of.24. The proof s coplete. Reark 2.. The operator defned n 6] can not be used, so we follow a dfferent ethod. Theore 2. also generalzes and proves the results n 8, 4, 32, 4, 49] Postve solutons of Proble.25. Let f t, x,..., x n = ft, x,..., x n 2, x n, t, x,..., x n, ] R n, and x = ax{, x} and y = n{, y}. We consder the proble φx n t ] f t, xt,..., x n t =, t,, x n x n 2 α x n ξ = λ, β x n 2 ξ = λ 2, x =, =,..., n 3, 2.4 Suppose H3 and the followng assuptons, whch wll be used n the proof of all leas n ths sub-secton. H5 f :, ], n, ], s contnuous and ft,,..., on each sub-nterval of,]; H6 λ, λ 2, α, β wth φα < /φkq, α < and β <. H7 The followng nequalty holds φkq φkq φα φα ξ b n 2 φk q φ β β ξ b φk q φ φ β ξ n 2! ] β c <. =

19 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS 9 Lea 2.2. If x, x 2 s a soluton of the proble x n t = φ x 2 t, t, ], x 2t = f t, x t,..., x n 2 t, φ x 2 t, t, ], φ x 2 α φ x 2 ξ = λ, then x s a soluton of 2.4. x n 2 n β x n 2 ξ = λ 2, x =, =,..., n 3, The proof of the above lea s sple and s otted. 2.5 Lea 2.3. If H5 H6 hold and x s a soluton of 2.4, then xt > for all t,, and x s a postve soluton of.25. Proof. Frstly, snce φx n t] = f t, xt,..., x n t and α, α < and λ, we have, usng.6, that x n = α x n ξ λ α x n. Hence x n and H6. We get x n t for all t, ]. Snce x n t for all t, ], we get x n 2 = β x n 2 ξ λ 2 β x n 2. So one gets x n 2. Thus we get x n 2 t > for all t, snce x n t for all t, ]. It follows fro the boundary condtons that x t > for all t,, =,..., n 3. Then f t, xt,..., x n t = ft, xt,..., x n t. Thus x s a postve soluton of.25. The proof s coplete. Let λ,, consder the proble x n t = λφ x 2 t, t, ], x 2t = λf t, x t,..., x n 2 t, φ x 2 t, t, ], = λφ x 2 α φ x 2 ξ λ, x n 2 n β x n 2 ξ = λλ 2, x =, =,..., n 3. Lea 2.4. Suppose H5 H6 hold. If x, x 2 s a soluton of 2.6, then x 2 x 2 φk q φk q φα 2.6 φkq λ φα ξ φkq φα.

20 2 Y. LIU EJDE-28/2 Proof. Snce H5 H6 and 2.6 ply that x 2t for all t, ]. Slar to the dscusson of Lea 2.3, λ and usng 2.6, we have φ x 2 = α φ x 2 ξ λ α φ x 2 ξ α φ x 2. Ths together wth H6, one sees that x 2. Then x 2 t for all t, ]. It follows fro φ x 2 α φ x 2 ξ λ = and Lea 2.2 that φ x 2 K q φ φα x 2 ξ φλ. Hence we get x 2 φkq φα x 2 ξ φλ. Thus, fro H6, we see, there s η, ξ ], that x 2 = = φkq φα φk q φk q x 2 φk q φα φkq φα x 2 ξ φα x 2 φkq λ φk q φk q φα x 2 Hence we get φk q φk q φα φα x 2 φα ξ x φkq λ 2η ] φkq φα φα ξ φk q λ φk q φα. Thus x 2 t x 2 t x 2 x 2 x 2 x 2 φk q λ φk q φα. φk q φk q φα φα ξ x 2 x 2 φk q φk q φα φkq λ φα ξ φkq φα.

21 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS 2 Lea 2.5. Suppose H5 H6 hold. If x, x 2 s a soluton of 2.6, then x n 2 φ x 2 β ξ β λ 2 β. Proof. In fact, x n 2 = β xn 2 β x n 2 β β ξ x n λ 2 β Then we get β β ξ φ x 2 λ 2 β. x n 2 t x n 2 t x n 2 x n 2 β β ξ φ λ 2 x 2 β x n β β ξ φ x 2 Then x n 2 φ x 2 and for =,..., n 3, x n 2! xn 2 n 2! Defne the operators λ ] 2 β. β φ x 2 Lx, x 2 = x n, x 2,, x n 2 n λ 2 β φ x 2. λ 2 β ξ β. β β ξ β x n 2 ξ, x, x 2 X DL, Nx, x 2 = φ x 2, f t, x, φ x 2, φ x 2 α φ x 2 ξ λ, λ 2, x, x 2 X. Suppose H5 H6 hold. It s easy to show the followng results: Ker L = {, c : c R} and I L = {y, y 2, a, b : a = }; L s a Fredhol operator of ndex zero;

22 22 Y. LIU EJDE-28/2 There are projectors P : X X and Q : Y Y such that Ker L = I P and Ker Q = I L. Furtherore, let Ω X be an open bounded subset wth Ω DL, then N s L-copact on Ω; v x = x, x 2 s a soluton of 2.5 f and only f x s a soluton of the operator equaton Lx = Nx n DL. We present the projectors P and Q as follows: P x, x 2 =, x 2 for all x = x, x 2 X and Qy, y 2, a, b =,, a,. The generalzed nverse of L : DL Ker P I L s defned by t K P y, y 2, a, b = t s n 2 y sds n 2! ξ β y sds b tn 2 n 2! β y sds t, y 2 sds, the soorphs : Y/ I L Ker L s defned by,, c, =, c. Lea 2.6. Suppose H3, H5 H7 hold. Then the set Ω = { x, x 2 DL \ Ker L : Lx, x 2 = λnx, x 2 for soe λ, } s bounded. Proof. It follows fro 2.6, H3, Leas 2.2, 2.4 and 2.5 that x 2 x 2 φk q φk q φα ax f t, x t,..., x n 2 t, φ x 2 t t,] φkq φkq φα φkq φkq φα φα ξ φkq λ φα ξ φkq φα φα ξ n 2 a b φ x c φφ x 2 = φk q λ φk q φα φk q φk q { a b n 2 φ λ 2 β b φ = φα φα ξ φ x 2 n 2! φ x 2 β φk q λ φk q φα β ξ β β ξ

23 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS 23 λ ] } 2 β c x 2 φk q φk q φα a b n 2 φk q φ b n 2 φk q φ λ 2 β φ = β β ξ φkq λ φkq φα φα ξ β b φ φkq φ n 2! φkq λ φkq φα. We get { φk q φk q φα = β ξ c x 2 b φk q φ x2 n 2! λ 2 β φα ξ b n 2 φk q φ β = b φk q φ φ n 2! c ]} x 2 φkq φkq φα ] β ξ β β ξ φα ξ a φk q φ λ 2 β b n 2 b φ φkq φ λ 2 n 2! β = φk q λ φk q φα. It follows fro H7 that there s a constant M > so that x 2 M. Thus, fro Lea 2.5, x n 2 φ M λ 2 β β ξ β =: M n 2, and there exst constants M > such that x M for =,..., n 3. So Ω s bounded. The proof s coplete. Lea 2.7. Suppose H5 H7 hold. If, c Ker L and N, c I L, then there exsts a constant M > such that c M.

24 24 Y. LIU EJDE-28/2 Proof. If, c Ker L and N, c I L, we get φ c α φ c = λ. Then H6 ples that there s M > such that c M. Lea 2.8. Suppose H5 H7 hold. If, c Ker L wth λ, c λqn, c =, then there exsts a constant M 2 > such that c M 2. Proof. If, c Ker L wth λ, c λqn, c =, then It follows that λc = λ φ c α φ c λ. λc 2 = cφ c λ It s easy to see that there s M 2 > so that c M 2. α λ φ. c Theore 2.9. Suppose H3, H5 H7 hold. postve soluton. Then.25 has at least one Proof. Let Ω Ω be a bounded open subset of X centered at zero wth ts daeter greater than ax{m, M 2}. Then Leas 2.6, 2.7 and 2.8 ply that Lx λnx for all x, λ DL \ Ker L Ω], ; Nx / I L for every x Ker L Ω; deg QN Ker L, Ω Ker L,. Snce H5 holds, we let L be a Fredhol operator of ndex zero and N be L- copact on Ω. It follows fro Lea 2. that Lx = Nx has at least one soluton x = x, x 2. Then x s a soluton of 2.4. We note that x t for t, ] and =,..., n 2, and x 2 t for all t, ], so f t, x t,..., x n 2 t, φ x 2 t = f t, x t,..., x n 2 t, φ x 2 t. Hence x s a postve soluton of.25. Reark 2.2. The operator defned n 5] can not be used, so follow a dfferent ethod. Theore 2.9 generalzes and proves the theores n 5, 4, 26, 5] Postve solutons of Proble.26. Let f t, x,..., x n = ft, x,..., x n 2, x n, t, x,..., x n, ] R n, and x = ax{, x}. We consder the proble φx n t ] f t, xt,..., x n t =, t,, x n 2 α x n 2 ξ = λ, x n β x n ξ = λ 2, x =, =,..., n 3, 2.7

25 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS 25 Proble 2.3 can be transfored nto x n t = φ x 2 t, t, ], x 2t = f t, x t,..., x n 2 t, φ x 2 t, t, ], x n 2 α x n 2 ξ = λ, = φ x 2 n β φ x 2 ξ λ 2, x =, =,..., n Suppose that H3 and the followng assuptons hold. H8 f :, ], n, s contnuous wth ft,,..., on each sub-nterval of,]; H9 λ, λ 2, α, β wth φβ < /φkq, α < and β <. H The followng nequalty holds φkq φkq φβ b n 2 φk q φ = Defne the operators Lx, x 2 = α φβ ξ α ξ b φk q φ φ α ξ n 2! ] α c <. x n, x 2, x n 2 Nx, x 2 = n α x n 2 ξ,, x, x 2 X DL, φ x 2, f t, x t,..., x n 2 t, λ, φ x 2 β φ x 2 ξ λ 2 for x, x 2 X. It s easy to show the followng results: Ker L = {, c : c R} and I L = {y, y 2, a, b : b = }; L s a Fredhol operator of ndex zero; There are projectors P : X X and Q : Y Y such that Ker L = I P and Ker Q = I L. Furtherore, let Ω X be an open bounded subset wth Ω DL, then N s L-copact on Ω; v x = x, x 2 s a soluton of 2.8 f and only f x s a soluton of the operator equaton Lx = Nx n DL. We defne the projectors P and Q as follows: P x, x 2 =, x 2 for all x = x, x 2 X and Qy, y 2, a, b =,,, b. The generalzed nverse of L s defned

26 26 Y. LIU EJDE-28/2 by t K P y, y 2, a, b = tn 2 t s n 2 y sds n 2! n 2! ξ α α y sds a, t y 2 sds, the soorphs : Y/ I L Ker L s defned by,,, b =, b. Slar to the leas n sub-secton 2.2, t s easy to prove the followng Leas. Lea 2.2. If x, x 2 s a soluton of proble 2.8, then x s a soluton of 2.7. The proof s easy; t s otted. Lea Suppose that H8 H9 hold. If x s a soluton of the proble 2.7, then xt > for all t,. The proof s slar to that of Lea 2.3; t s otted. Let λ,, consder the proble x n t = λφ x 2 t, t, ], x 2t = λf t, x t,..., x n 2 t, φ x 2 t, t, ], = λφ x 2 β φ x 2 ξ λ 2, x n 2 n α x n 2 ξ = λλ, x =, =,..., n Lea Suppose that H8 H9 hold. If x, x 2 s a soluton of 2.9, then x 2 x φkq 2 φkq φβ φβ ξ φk q λ 2 φk q φβ. Lea Suppose that H8 H9 hold. If x, x 2 s a soluton of 2.5, then x n 2 φ x 2 λ α α ξ α, and for =,..., n 3, x n 3! xn 2 n 3! φ x 2 α λ α. Slar to Theore 2.9, we obtan the followng theore. α ξ

27 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS 27 Theore Suppose H3, H8 H hold. postve soluton. Then.26 has at least one We reark that Theore 2.25 generalzes the theores n 2, 4, 73] Solutons of Proble.27. We consder.27, H, H3 and the followng assuptons are supposed n ths sub-secton. H α, β for all =,..., and λ, λ 2 R; H2 The followng nequalty holds φk q φ α b sds n 2! = φ ] α b n 2 sds csds <. Let x = x and x 2 = φx, then.24 s transfored nto x n t = φ x 2 t, t, ], x 2t = ft, x t,..., x n 2 t, φ x 2 t, t, ], x n 2 = α φ x 2 ξ λ, x n 2 = n β φ x 2 ξ λ 2, x =, =..., n 3, Suppose λ,, we consder the proble x n t = λφ x 2 t, t, ], x 2t = λft, x t,..., x n 2 t, φ x 2 t, t, ], x n 2 = λ α φ x 2 ξ λ, Defne the operators x n 2 = λ n β φ x 2 ξ λ 2, x =,..., n 3, Lx, x 2 = x n, x 2, x n 2, x n 2, x, x 2 X DL, Nx, x 2 = φ x 2, f t, x t,..., x n 2 t, φ x 2 t, n α φ x 2 ξ λ, β φ x 2 ξ λ 2, for x, x 2 X. Suppose H H2 hold. It s easy to show the followng results: Ker L = {, c : c R} and I L = {y, y 2, a, b : y sds = b a};

28 28 Y. LIU EJDE-28/2 L s a Fredhol operator of ndex zero; There are projectors P : X X and Q : Y Y such that Ker L = I P and Ker Q = I L. Furtherore, let Ω X be an open bounded subset wth Ω DL, then N s L-copact on Ω; v x = x, x 2 s a soluton of 2. f and only f x s a soluton of the operator equaton Lx = λnx n DL. We defne the projectors P and Q as follows: P x, x 2 =, x 2 for all x = x, x 2 X and Qy, y 2, a, b = y sds b a,,,. The generalzed nverse of L s K P y, y 2, a, b = a n 2! tn 2 t t s n 2 y sds, n 2! t y 2 sds, the soorphs : Y/ I L Ker L s defned by c,,, =, c. Slar to Leas n sub-secton 2.2, t s easy to prove the followng Leas. Lea Suppose H, H2 hold. If x = x, x 2 s a soluton of 2., then there exsts ξ, ] such that φ x n ξ M =: { λ λ 2 P αp, β α β, λ λ 2, α β =. Proof. Case. α β =. Then α = β =. In ths case, x n 2 = λλ and x n 2 = λλ 2, t s easy to see that there s ξ, ] so that x n ξ = λ λ λ 2. Then λ φ x 2 ξ = λ λ λ 2. So φ x 2 ξ = λ λ 2. Case 2. α β. In ths case, f φ x 2 t > M for all t,, then x n 2 < x n 2. If λ λ 2, fro the boundary condtons, usng H, we obtan x n 2 > λ λ α M λλ λ λ 2 α α β λλ = λ λ 2 α λ β α β. On the other hand, x n 2 λ λ 2 < λ β M λλ 2 λ β α β λλ 2 < x n 2, a contradcton. Slar to above dscusson, f x n t < M for all t,, we can get a contradcton. Then there s ξ, so that φ x 2 ξ M. If λ < λ 2, we can get that there s ξ, so that φ x 2 ξ M. Lea Suppose H H2 hold. If x, x 2 s a soluton of 2., then x n 2 α φ x 2 λ.

29 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS 29 Proof. Fro the boundary condtons, we get x n 2 α φ x 2 λ. So we get x n 2 t x n 2 t x n 2 x n 2 Ths copletes the proof. We also get, for =,..., n 3, that α φ x 2 λ. x t α φ x 2 λ ]. n 2! Lea Let Ω = {x DL \ Ker L : Lx = λnx for λ, }. Then Ω s bounded. Proof. In fact, f x Ω, we get 2.. It follows fro Lea 2.26 that there s ξ, so that φ x 2 ξ M. Thus usng H3 and Lea 2.2 we get x 2 t φm λ Then φm φm φm t ξ ft, x t,..., x n 2 t, φ x 2 tdt ft, x t,..., x n 2 t, φ x 2 t dt n 2 asds = asds = φ n 2! φm b sφ x s ds cs x 2 s ds b sds α φ x 2 n 2! λ b n 2 sdsφ α φ x 2 λ asds = b sds α x2 φ φk q φ φ n 2! b n 2 sdsφ α x2 φ λ x 2 φm φ asds = α x2 φ n 2! λ csds x 2 n 2! λ csds x 2. b sdsφk q φ n 2!

30 3 Y. LIU EJDE-28/2 b n 2 sdsφ α x2 φ λ csds x 2. Hence φk q φ α b sds φ α b n 2 sds n 2! = ] csds x 2 φm asds φ n 2! λ φ λ. Fro H2, we get that there s A > so that x 2 A. Hence x n 2 And for =,..., n 3, we get α φ A λ. x α φ A λ ]. n 2! The above nequaltes ply that Ω s bounded. Lea Let Ω = {x Ker L : Nx I L}. Then Ω s bounded. Proof. In fact,, c Ω, then, c Ker L and N, c I L, then we get φ c = α φ c β φ c λ 2 λ. Hence there s M > so that c M. Lea 2.3. Let Ω 2 = {x Ker L : λ x λqnx = }. Then Ω 2 s bounded. Proof. In fact, f, c Ω 2, then Thus λc = λ α φ c β φ c λ 2 λ. λc 2 = λcφ c α β λ 2 λ φ. c Hence there s M > so that c M. The followng theore has proof slar to that of Theore 2.; ts proof s otted. Theore 2.3. Suppose H, H3, H, H2 hold. Then.27 has at least one soluton.

31 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS 3 Reark Consder the probles φx n t ] ft, xt,..., x n t =, t,, x n 2 αx n = λ, x n 2 βx n = λ 2, x =, =,..., n 3, 2.2 where α, β, λ, λ 2, and f s nonnegatve and contnuous. If xt s a soluton of 2.2, then x n s decreasng on, ]. Case. x n and x n ; At ths case, we see that x n 2 t s ncreasng on, ]. It follows fro x n 2 = αx n λ that x n 2 t > for all t,. Then xt s a postve soluton of 2.2. Case 2. x n and x n ; At ths case, one sees that and x n 2 = βx n λ 2 x n 2 = αx n λ 2. It follows fro x n t that x n 2 t for all t, ]. Then x s a postve soluton of 2.2. Case 2. x n and x n ; At ths case, one sees that x n 2 t s decreasng on, ]. It follows fro x n 2 = βx n λ 2 that x n 2 t > for all t,. Then xt s a postve soluton of 2.2. We can establsh slar results for the exstence of postve solutons of 2.2 and the detals are otted. Reark Consder the probles φx n t ] ft, xt,..., x n t =, t,, x n 2 αx n = λ, x n 2 β x n ξ = λ 2, x =, =,..., n 3, 2.3 where α, β, λ, λ 2, and f s nonnegatve and contnuous. 2.3 need not has postve soluton. It s easy to show that the proble x 8 6t =, t,, x x =, x θx 8 = has no postve soluton snce the soluton of the above proble s xt = 4t 2 t θ 2 θ t θ 2 θ.

32 32 Y. LIU EJDE-28/2 Then x = θ 2 θ = θ < f θ >. 2 θ 2.5. Solutons of.28. We consder.28, assung H3 and the followng condtons: H3 there are nonnegatve nubers α, θ and L so that ft, x,..., x n αφ x n 2 = θ φ x θ n φ x n L; H4 t ft, n 2 n 2! l a,..., a, φ λ P a φ a and the followng three nequaltes hold: and α ξ α φ K n p = α = µ φ θ α φ <, n 2! µ θ φ < α, n 2! = = b φ bn 2 n 2! φ φ Kp n φ β α φ φ K p φ β n α λ 2 λ P α ; φkp n 2! α ξ α c <. H5 φ θ P = β H6 λ, λ 2 R, α, β for =,..., wth α < and β <. Let x = x and x 2 = φx, then.28 s transfored nto x n t = φ x 2 t, t, ], x 2t = ft, x t,..., x n 2 t, φ x 2 t, t, ], = x 2 α x 2 ξ λ, = θφ x 2 β θφ x 2 ξ λ 2, x =, =..., n 3, 2.4

33 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS 33 Suppose λ,, we consder the followng proble x n t = λφ x 2 t, t, ], x 2t = λft, x t,..., x n 2 t, φ x 2 t, t, ], = λx 2 α x 2 ξ λ, Defne the operators = λθφ x 2 β θφ x 2 ξ λ 2, x =, =..., n 3, 2.5 Lx, x 2 = x n, x 2,,, x, x 2 X DL, φ T x 2 Nx, x 2 = f t, x,..., x n 2 t, φ x 2 x 2 α x 2 ξ λ θφ x 2, x, x 2 X. n β θφ x 2 ξ λ 2 Suppose that H4, H5, H6 hold. It s easy to show the followng results: Ker L = { tn 2 n 2! a, b : a, b R} and I L = {y, y 2, a, b : a = b = }; L s a Fredhol operator of ndex zero; There are projectors P : X X and Q : Y Y such that Ker L = I P and Ker Q = I L. Furtherore, let Ω X be an open bounded subset wth Ω DL, then N s L-copact on Ω; v x = x, x 2 s a soluton of 2.4 f and only f x s a soluton of the operator equaton Lx = Nx n DL. We present the projectors P and Q as follows: P x, x 2 = tn 2 n 2! xn 2, x 2 for all x = x, x 2 X and Qy, y 2, a, b =,, a, b. The generalzed nverse of L s t K P y, y 2, a, b = t s n 2 y sds, n 2! t y 2 sds, the soorphs : Y/ I L Ker L s defned by,, a, b = tn 2 n 2! a, b. Lea If x, x 2 s a soluton of proble 42, then x s a soluton of.28. Lea Suppose that H 4, H 5, H 6 hold. If x, x 2 s a soluton of proble 2.5, then there s a ξ, so that x 2ξ =. Proof. In fact, f x 2t > for all t,, we get x 2 = α x 2 ξ λ > α x 2 λ. So x 2 > λ / α. It follows fro x 2t > that x 2 > λ / α. On the other hand, θφ x 2 = β θφ x 2 ξ λ 2 < β θφ x 2 λ 2.

34 34 Y. LIU EJDE-28/2 Then θφ x 2 < λ 2 / β. So we get x 2 < φθ λ 2 / β = λ / α < x 2, a contradcton. If x 2 t < for all t,, the sae contradcton can be derved. So there s ξ, ] such that x 2 ξ =. Lea Suppose that H4, H5, H6 hold. If x, x 2 s a soluton of proble 2.5, then x 2 t α ξ α x 2 λ α. Proof. In fact, x 2 = α x 2 α x 2 α α x 2 ξ x 2 λ α α ξ x 2 λ. Hence x 2 t x 2 t x 2 x 2 α ξ x λ α 2 α. Lea Suppose that H3, H3-H6 hold. Let Ω = {x, x 2 DL \ Ker L : Lx, x 2 = λnx, x 2 for soe λ, }. Then Ω s bounded. Proof. In fact, f x, x 2 Ω, we get 2.5. It follows fro Leas 2.34, 2.35 and 2.36 that x 2 t α ξ α x λ 2 α and there s ξ, ] so that x 2 ξ =. Then x 2 t α ξ ax α ft, x t,..., x n 2 t, φ x 2 t t,] λ α and fξ, x ξ,..., x n 2 ξ, φ x 2 ξ =. Then H3 ples φ x n 2 ξ θ φ x α ξ θ n α x 2ξ L α α = = = θ φ n 2! xn 2 θ n α x 2 L α. So fro Lea 2.2 we have x n 2 ξ φ θ φ θ n α n 2! xn 2 α x 2 L α

35 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS 35 So φ Kp n φ φ θ /α n 2! xn 2 = ] φ θ n /αφ x 2 φ L/α φ Kp n φ θ /α n 2! xn 2 = φ θ n /αφ x 2 φ L/α. x n 2 t x n 2 t x n 2 ξ x n 2 ξ φ x 2 φ Kp n φ θ /α n 2! xn 2 = φ θ n φ x 2 φ L/α α φ Kp n φ θ /αφ n 2 x n 2! = φ Kp n φ θ n φ x 2 φ Kp n α φ L/α. We get, fro H3 and that φ K n p = x n 2 φ K n φ θ α φ <, n 2! q φ K n = φ K n q φ L/α φ θ /αφ n 2! q φ θ n /αφ x 2 ]. It follows fro Lea 2.2 that φ x n 2 φ φ Kp n = φ θ /αφ n 2! φ φ Kp n φ θ n /αφ x 2 φ K n φ φ Kp n = φ θ /αφ n 2! ] φk p. φ φ K n p φ θ n α x2 K p L α p φ L α

36 36 Y. LIU EJDE-28/2 On the other hand, x 2 t α ξ α ax ft, x t,..., x n 2 t, φ x 2 t λ t,] α α n 2 ξ α a b φ x c x 2 Then x 2 We get λ α α ξ α = a = b φ/n 2!φ x n 2 b n 2 φ x n 2 λ c x 2 α. α { ξ α a b φ/n 2! b n 2 φ φ K p = φ θ /α φ n 2! φk p φ φ Kp n φ θ n x2 Kn p L ] α α } λ c x 2 α. φ α ξ α = = b φ bn 2 n 2! φ K p = φ θ /αφ n 2! φ φ Kp n φ θ n α α ξ α a α ξ α λ α. φk p c ] x 2 It follows that there s a constant M > so that x 2 M. Hence x n 2 φ K p φ θ /α n 2! = φ K p φ θ n /αφ M φ K p φ L/α ] =: M n 2,

37 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS 37 and for =,..., n 3, we get x Hence Ω s bounded. n 2! xn 2 n 2! M n 2 =: M. Lea Suppose that H3, H4, H5, H6 hold. Let Ω = {x Ker L, Nx I L}. Then Ω s bounded. Proof. In fact, f x = tn 2 n 2! a, b Ker L and Nx I L, then we get b α b λ =, θφ b β θφ b λ 2 =. So we have b = λ / α. Fro H4, choose ɛ > so that = θ φ µ ɛ < α, n 2! and then there s a δ > so that t n 2 f t, n 2! x,..., x, φ λ / Let A = Then one sees that ax t,], x δ µ ɛφ a A f t, So θ n λ / α < µ ɛφ x, x > δ. t n 2 f t, n 2! x,..., x, φ λ / t n 2 n 2! a,..., a, φ λ / = α. α αφ a θ φ t n 2 n 2! a θ n λ / α L. α L φ a α θ φ = t n 2 n 2! µ ɛ φ a α θ φ µ ɛ n 2! Then there s M > so that a M. Hence a, b are bounded. Then Ω s bounded. Lea Suppose that H3, H4, H5, H6 hold. Then the set Ω 2 = {x Ker L, λ x λqnx =, λ, ]} s bounded. Proof. In fact, f Ω 2 s unbounded, then there are sequences {λ n, ]} and {x n = tn 2 n 2! a n, b n } such that λ n,, a n, b n λ n,, b n α b n λ, θφ b n β θφ b n λ 2 =

38 38 Y. LIU EJDE-28/2 and ether b n as n tends to nfnty or {b n } s bounded and a n as n tends to nfnty. It follows that λ n a n = λ n b n α b n λ, 2.6 Then λ n b n = λ n θφ b n β θφ b n λ λ n b 2 n = λ n θφ b n b n λ 2 ] β θφ b n ples that there s a constant B > so that b n B snce φ b n b n >. Thus we get that a n as n tends to nfnty. It follows fro 2.6 that λ n as n tends to nfnty. Thus 2.7 ples that b n b = φθ λ 2 / β = λ / α. Then So µ ɛφ a n A f t, θ n λ / It follows fro t n 2 n 2! a n,..., a n, φ λ / αφ a n θ φ = θ n λ / α t n 2 n 2! φ a n α L. t n 2 α L φ a n α θ φ µ ɛ n 2! = = φ a n α θ φ µ ɛ. n 2! = θ φ µ ɛ < α n 2! that there s a constant C > so that a n C, a contradcton. bounded. Hence Ω 2 s Theore 2.4. Suppose that H3, H3 H6 hold. Then.28 has at least one soluton. Proof. Let Ω Ω Ω Ω 2 be a bounded open subset of X centered at zero Then Lx λnx for all x, λ DL \ Ker L Ω], ; Nx / I L for every x Ker L Ω; deg QN Ker L, Ω Ker L,. It follows fro Lea 2. that Lx = Nx has at least one soluton x = x, x 2. Then x s a soluton of 2.4. Hence x s a soluton of.28. We reark that Theore 2.4 generalzes the results n, 36].

39 EJDE-28/2 NON-HOMOGENEOUS BOUNDARY-VALUE PROBLEMS Exaples Now, we present soe exaples to llustrate the an results. These BVPs can not be solved by known results. Exaple 3.. Consder the proble x t bt x t ctxt rt =, t,, x = 2 x/4 2, x = 2 x 2 2, 3. where b, c and r are nonnegatve contnuous functons. Correspondng to.24, t s easy to fnd that H, H2, H3 hold. We fnd fro Theore 2. that f 5 4 csds bsds <, then 3. has at least one postve soluton for each r C, ] wth rt and on each subnterval of,]. Exaple 3.2. Consder the proble φ 3 x atφ 3 x btφ 3 x rt =, t,, x = 2 x/2 6, x = 4 x/4 3 x/2 7, 3.2 where a, b and r are nonnegatve contnuous functons. We fnd p = 3 and q = 3/2. Then by applcaton of Theore 2., 3.2 has at least one postve soluton f φ 3 2φ asds bsds < for each r C, ] wth rt and on each subnterval of,]. Exaple 3.3. Consder the proble φ 3 x atφ 3 x btφ 3 x rt =, t,, x = 2 x /2 3, x = 4 x/4 3 x/2 4, 3.3 where a, b and r are nonnegatve contnuous functons. We fnd p = 3, q = 3/2, = 2. Then by applcaton of Theore 2.9, 3.3 has at least one postve soluton f φ 3 4 φ 3 /2 b φ 3 2φ ] φ 3 4φ 3 / a < for each r C, ] wth rt and on each subnterval of,]. Exaple 3.4. Consder the proble φ 3 x atφ 3 x btφ 3 x rt =, t,, x = 2 x/2 3 x3/4 6, x = 4 x/2 3 x 3/4 7, 3.4 where a, b and r are nonnegatve contnuous functons. Then by applcaton of Theore 2.25, 3.4 has at least one postve soluton f φ4 φ4φ/2 φ/3 φ/4 2 φ/3 4 ] b φ2φ 3 4/3 a < for each r C, ] wth rt and on each subnterval of,].

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