Research Article Existence of Positive Solutions for Fourth-Order Three-Point Boundary Value Problems

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1 Hindawi Publihing Corporation Boundary Value Problem Volume 27, Article ID 68758, 1 page doi:1.1155/27/68758 Reearch Article Exitence of Poitive Solution for Fourth-Order Three-Point Boundary Value Problem Chuanzhi Bai Received 11 July 27; Accepted 7 November 27 Recommended by Jean Mawhin We are concerned with the nonlinear fourth-order three-point boundary value problem u (4) (t) = a(t) f (u(t)), <t<1, u() = u(1) =, αu () βu () =, γu (1) + δu (1) =. By uing Kranoelkii fixed point theorem in a cone, we get ome exitence reult of poitive olution. Copyright 27 Chuanzhi Bai. Thi i an open acce article ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited. 1. Introduction A i pointed out in [1, 2, boundary value problem for econd- and higher-order differential equation play a very important role in both theory and application. Recently, an increaing interet in tudying the exitence of olution and poitive olution to boundary value problem for fourth-order differential equation i oberved; ee, for example, [3 8. In thi paper, we are concerned with the exitence of poitive olution for the following fourth-order three-point boundary value problem (BVP): u (4) (t) = a(t) f ( u(t) ), <t<1, u() = u(1) =, αu () βu () =, γu (1) + δu (1) =, (1.1) where α, β, γ, andδ are nonnegative contant atifying αδ + βγ + αγ > ; <<1, a C[,1, and f C([, ),[, )). We ue Kranoelkii fixed point theorem in cone to etablih ome imple criteria for the exitence of at leat one poitive olution to BVP (1.1). To the bet of our knowledge, no paper in the literature ha invetigated the exitence of poitive olution for BVP (1.1).

2 2 Boundary Value Problem The paper i formulated a follow. In Section 2, ome definition and lemma are given. In Section 3, we prove ome exitence theorem of the poitive olution for BVP (1.1). 2. Preliminarie and lemma In thi ection, we introduce ome neceary definition and preliminary reult that will be ued to prove our main reult. Firt, we lit the following notation and aumption: f (u) f = lim u u, f f (u) = lim u u. (2.1) (H 1 ) f :[, ) [, )icontinuou; (H 2 ) a C[,1, a(t), for all t [,, a(t), for all t [,1, and a(t), for all t (p,) (,q)(<p<<q<1). By routine calculation, we eaily obtain the following lemma. Lemma 2.1. Suppoe that α, β, γ, δ are nonnegative contant atifying αδ + βγ + αγ >. If h C[,1, then the boundary value problem v (t) = h(t), t [,1, αv() βv () =, γv(1) + δv (1) = (2.2) ha a unique olution t v(t) = )h()d+ (t 1 where = αδ + βγ + αγ(1 ) >. Let G(t,) be the Green functionof the differential equation ubject to the boundary condition In particular, ( α( t) β )( γ(1 )+δ ) h()d, (2.3) u (t) =, t (,1), (2.4) u() = u(1) =. (2.5) (1 t), t 1, G(t,) = t(1 ), t< 1. (2.6)

3 Chuanzhi Bai 3 It i rather traightforward that G(t,) G(,), t, 1, (2.7) G(t,) mg(,), t [p,q, [,1, (2.8) where <p<q<1, and <m= min{p,1 q} < 1. Let X be the Banach pace C[,1 endowed with the norm We define the operator T : X X by u =max u(t). (2.9) t 1 [ Tu(t) = G(t,) (τ )a(τ) f + 1 ( )( ) ( ) β α( ) γ(1 τ)+δ a(τ) f u(τ d, (2.1) where G(t,) ain(2.6). From Lemma 2.1, we eaily know that u(t) i a olution of the fourth-order three-point boundary value problem (1.1)ifandonlyifu(t)iafixedpoint of the operator T. Define the cone K in X by K = { u X u, min u(t) m u t [p,q }, (2.11) where <p<<q<1, and <m= max{p,1 q} < 1. (2.12) Lemma 2.2. Aume that (H 1 ) and (H 2 ) hold. If β α, then T : K K i completely continuou. Proof. For any u K,weknowfrom(2.1), (H 1 ), (H 2 ), and β α that [ (Tu)(t) = G(t,) ( τ)a(τ) f + 1 ( )( ) ( ) β α( ) γ(1 τ)+δ a(τ) f u(τ d [ + G(t,) (τ )a(τ) f + 1 ( )( ) ( ) β α( ) γ(1 τ)+δ a(τ) f u(τ + 1 ( )( ) ( ) β α( ) γ(1 τ)+δ a(τ) f u(τ d

4 4 Boundary Value Problem [ = G(t,) ( τ)a(τ) f + 1 ( )( ) ( ) β α( ) γ(1 τ)+δ a(τ) f u(τ d [ 1 + G(t,), t [, [ αδ(τ )+βγ(1 )+αγ(1 )(τ )+βδ a(τ) f ( β + α( ) ) d (2.13) Hence, in view of (2.13)and(2.7), we have [ Tu = max (Tu)(t) G(,) ( τ)a(τ) f t [,1 + 1 ( )( ) ( ) β α( ) γ(1 τ)+δ a(τ) f u(τ d [ 1 + G(,) + 1 Thu from (2.8), (2.13), and (2.14), we get min t [p,q (Tu)(t) m [ αδ(τ )+βγ(1 )+αγ(1 )(τ )+βδ a(τ) f ( β + α( ) ) d. [ G(,) ( τ)a(τ) f ( )( ) ( ) β α( ) γ(1 τ)+δ a(τ) f u(τ d + 1 [ 1 + m G(,) + 1 (2.14) [ αδ(τ )+βγ(1 )+αγ(1 )(τ )+βδ a(τ) f ( β + α( ) ) d = m Tu, (2.15) where m a in (2.12). So T : K K. Moreover, it i eay to check by the Arzela-Acoli theorem that the operator T i completely continuou. Remark 2.3. By = αδ + βγ + αγ(1 ) > andβ α,wehaveβ>. Recently, Kranoelkii theorem of cone expanion/compreion type ha been ued to tudy the exitence of poitive olution of boundary value problem in many paper;

5 Chuanzhi Bai 5 ee, for example, Liu [7, Ma [9, Torre [1, and the reference contained therein. The following lemma (Kranoelkii fixed point theorem) will play an important role in the proofofourtheorem. Lemma 2.4 [11. Let X be a Banach pace, and let K X beaconeinx. Aume that Ω 1, Ω 2 are open ubet of X with Ω 1, Ω 1 Ω 2 and let A : K (Ω 2 \ Ω 1 ) K be a completely operator uch that either (i) Au u, u K Ω 1 and Au u, u K Ω 2 ;or (ii) Au u, u K Ω 1 and Au u, u K Ω 2. Then A ha a fixed point in K (Ω 2 \ Ω 1 ). 3. Main reult We are now in a poition to preent and prove our main reult. Theorem 3.1. Let β α. Aume that (H 1 )-(H 2 ) hold. If f = and f =, then(1.1) ha at leat a poitive olution. Proof. Since f =,wecanchooer>ufficiently mall o that where ε atifie ε f (u) εu for u r, (3.1) 6 m(1 ) a(τ)τ3 dτ, if a( ) t <, for ome t (p,), βm (τ )(1 τ) ) a(τ)dτ, if a( t 1 ) >, for ome t1 (,q). (3.2) Set Ω r ={u K u <r}. From condition (H 2 ), we conider two cae a follow. Cae 1. If a(t ) < foromet (p,), then, for u Ω r,wehavefrom(2.13), (3.1), and (3.2)that [ (Tu)() G(,) ( τ)a(τ) f + 1 ( )( ) ( ) β α( ) γ(1 τ)+δ a(τ) f u(τ d G(,) ( τ)a(τ) f ) dτ d ε G(,) ( τ)a(τ)u(τ)dτ d mε u G(,) = mε u a(τ)dτ τ ( τ)a(τ)dτ d = mε u (1 )( τ)d = mε u 1 6 a(τ)dτ τ G(,)( τ)d a(τ)τ 3 dτ u, (3.3)

6 6 Boundary Value Problem which implie Tu u, u Ω r. (3.4) Cae 2. If a(t 1 ) > foromet 1 (,q), then, for u Ω r,wehavefrom(2.13), (3.1), and (3.2)that [ [ ( ) (Tu)() G(,) αδ(τ )+βγ(1 )+αγ(1 )(τ )+βδ a(τ) f u(τ + 1 ( )( ) ( ) β + α( ) γ(1 τ)+δ a(τ) f u(τ d that i, 1 β G(,) G(,) εβm u = εβm u = εβm u εβm u ( β + α( ) ) d d G(,)d ) a(τ)dτ ( ) τ γ(1 τ)+δ a(τ)dτ (1 )d ( (τ ) 1 1 ) (γ(1 2 (τ + ) ) τ)+δ a(τ)dτ (τ )(1 τ) ) a(τ)dτ u, Next, define a function f (v):[, ) [, )by (3.5) Tu u, u Ω r. (3.6) f (v) = max f (u). (3.7) u v It i eay to ee that f (v) inondecreaing.since f =, we have lim v f (v)/v =. Thu, thereexitr>ruch that f (R) θr, (3.8) where θ atifie [ θ a(τ)τ 3 dτ + 1 [ ( (1 ) + βδ 1 2 ) a(τ)dτ ( ) ( ) β + α(1 ) γ(1 τ)+δ a(τ)dτ 1. 6 (3.9)

7 Chuanzhi Bai 7 Hence, we obtain Thu from (2.14)and(3.1), for all u Ω R,wehave f (u) f (R) θr, u R. (3.1) [ [ Tu θr G(,) ( τ)a(τ)dτ + 1 (β α + α) ) a(τ)dτ d [ [ + G(,) αδ(τ )+βγ(1 )+αγ(1 )(τ )+βδ a(τ)dτ + 1 ( )( ) β + α( ) γ(1 τ)+δ a(τ)dτ d [ τ θr a(τ)dτ (1 )( τ)d+ β ( ) γ(1 τ)+δ a(τ)dτ (1 )d + 1 [ (1 )d αδ(1 )+βγ(1 )+αγ(1 ) 2 + βδ a(τ)dτ + 1 ( )( ) (1 )d β + α(1 ) γ(1 τ)+δ a(τ)dτ that i, [ = θr a(τ)τ 3 dτ + β ( ) ( ) γ(1 τ)+δ a(τ)dτ [ ( (1 ) + βδ ) a(τ)dτ + 1 ( )( β + α(1 ) ) ( ) γ(1 τ)+δ a(τ)dτ 6 [ θr a(τ)τ 3 dτ + 1 [ ( (1 ) + βδ 1 2 ) a(τ)dτ ( ) ( ) β + α(1 ) γ(1 τ)+δ a(τ)dτ R = u, 6 (3.11) Tu u, foru Ω R. (3.12) Hence, from (3.6), (3.12), and Lemma 2.4, T ha a fixed point u Ω R \ Ω r, which mean that u i a poitive olution of BVP (1.1). Theorem 3.2. Let β α. Aume that (H 1 )-(H 2 )hold.if f = and f =,then(1.1) ha at leat a poitive olution.

8 8 Boundary Value Problem Proof. Since f =,wecanchooer 1 > ufficiently large o that f (u) Au, u R 1, (3.13) where A atifie 6 m(1 ) p a(τ)(τ p)( τ 2 + τp τp 2, A βm q (τ )(1 τ) ) a(τ)dτ, if a( t ) <, for ome t (p,), if a( t 1 ) >, for ome t1 (,q). (3.14) Chooe R R 1 m, (3.15) where m>ain(2.12). Let u Ω R.Sinceu(t) m u =mr R 1 for t [p,q, from (3.13), we ee that f ( u(t) ) Au(t) AmR, t [p,q, u Ω R. (3.16) For u Ω R, we conider two cae a follow. Cae 1. If a(t ) < foromet (p,), then we have from (3.3), (3.14), and (3.16)that (Tu)() p G(,) ( τ)a(τ) f d G(,) ( τ)a(τ) f d AmR G(,) ( τ)a(τ)dτ d p p τ = AmR(1 ) a(τ)dτ ( τ)d p = 1 p 6 AmR(1 ) a(τ)(τ p) ( τ 2 + τp 2p 2 R = u, (3.17) which implie Tu u, u Ω R. (3.18)

9 Chuanzhi Bai 9 Cae 2. If a(t 1 ) > foromet 1 (,q), then we have from (3.5), (3.14), and (3.16)that (Tu)() β which implie β q AmR β G(,) G(,) q q q G(,) d d q ) a(τ)dτ d = AmR β ( ) τ γ(1 τ)+δ a(τ)dτ (1 )d AmR β q (τ )(1 τ) ) a(τ)dτ R = u, (3.19) Tu u, u Ω R. (3.2) Since f =,wecanchooe<r<ruch that f (u) θu, u r, (3.21) where θ a in (3.9). For u Ω r,wehavefrom(3.11)and(3.21)that So, [ Tu θ u a(τ)τ 3 dτ + 1 [ ( (1 ) + βδ 1 2 ) a(τ)dτ ( β + α(1 ) ) ) a(τ)dτ u. (3.22) Tu u, u Ω r. (3.23) Therefore, from (3.2), (3.23), and Lemma 2.4, T haafixedpointu Ω R \ Ω r, which mean u i a poitive olution of BVP (1.1). Finally, we conclude thi paper with the following example. Example 3.3. Conider the following fourth-order three-point boundary value problem: u (4) (t) = inπ(1 + 2t)u r (t), <t<1, αu ( 1 2 u() = u(1) =, ) ( ) 1 βu =, γu 2 (1) + δu (1) =, (3.24) where <r<1, α, β, γ, andδ are nonnegative contant atifying αδ + βγ + αγ > and β (1/2)α.ThenBVP(3.24) ha at leat one poitive olution.

10 1 Boundary Value Problem To ee thi, we will apply Theorem 3.1.Set f (u) = u r, a(t) = inπ(1 + 2t), = 1 2. (3.25) With the above function f and a,weeethat(h 1 )and(h 2 )hold.moreover,itieayto ee that The reult now follow from Theorem 3.1. Acknowledgment f =, f =, β α. (3.26) Thi work i upported by the National Natural Science Foundation of China ( ) and the Natural Science Foundation of Jiangu Education Office (6KJB111). The author i grateful to the referee for their valuable uggetion and comment. Reference [1 R. P. Agarwal, Focal Boundary Value problem for Differential and Difference Equation, vol. 436 of MathematicandItApplication, Kluwer Academic Publiher, Dordrecht, The Netherland, [2 D. G. Zill and M. R. Cullen, Differential Equation with Boundary-Value Problem, Brook/Cole, Belmont, Calif, USA, 5th edition, 21. [3 Z. Bai and H. Wang, On poitive olution of ome nonlinear fourth-order beam equation, Journal of Mathematical Analyi and Application, vol. 27, no. 2, pp , 22. [4 J. R. Graef and B. Yang, On a nonlinear boundary value problem for fourth order equation, Applicable Analyi, vol. 72, no. 3 4, pp , [5 Z. Hao, L. Liu, and L. Debnath, A neceary and ufficient condition for the exitence of poitive olution of fourth-order ingular boundary value problem, Applied Mathematic Letter, vol. 16, no. 3, pp , 23. [6 F. Li, Q. Zhang, and Z. Liang, Exitence and multiplicity of olution of a kind of fourth-order boundary value problem, Nonlinear Analyi: Theory, Method & Application,vol.62,no.5,pp , 25. [7 B. Liu, Poitive olution of fourth-order two point boundary value problem, Applied Mathematic and Computation, vol. 148, no. 2, pp , 24. [8 D. Yang, H. Zhu, and C. Bai, Poitive olution for emipoitone fourth-order two-point boundary value problem, Electronic Journal of Differential Equation, no. 16, pp. 1 8, 27. [9 R. Ma, Poitive olution of fourth-order two-point boundary value problem, Annal of Differential Equation, vol. 15, no. 3, pp , [1 P. J. Torre, Exitence of one-igned periodic olution of ome econd-order differential equation via a Kranoel kii fixed point theorem, JournalofDifferential Equation, vol. 19, no. 2, pp , 23. [11 M. A. Kranoel kii, Poitive Solution of Operator Equation, P. Noordhoff, Groningen, The Netherland, Chuanzhi Bai: Department of Mathematic, Huaiyin Teacher College, Huaian 2233, China addre: czbai8@ohu.com