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SECOND-ORDER n-POINT EIGENVALUE PROBLEMS ON TIME SCALES DOUGLAS R. ANDERSON AND RUYUN MA Received 10 December 2004; Revised 3 November 2005; Accepted 6 November 2005 We discuss conditions for the existence of at least one positive solution to a nonlinear second-order Sturm-Liouville-type multipoint eigenvalue problem on time scales. The results extend previous work on both the continuous case and more general time scales, and are based on the Guo-Krasnosel’ski ˘ i fixed point theorem. Copyright © 2006 D. R. Anderson and R. Ma. This is an open access article distributed under the Creative Commons Att ribution License, which permits unrestricted use, dis- tribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction We are interested in the second-order multipoint time-scale eigenvalue problem  py ∇  Δ (t) − q(t)y(t)+λh(t) f (y) = 0, t 1 <t<t n , (1.1) αy  t 1  − βp  t 1  y ∇  t 1  = n−1  i=2 a i y  t i  , γy  t n  + δp  t n  y ∇  t n  = n−1  i=2 b i y  t i  , (1.2) where p, q :  t 1 ,t n  −→ (0,∞), p ∈ C Δ  t 1 ,t n  , q ∈ C  t 1 ,t n  ; (1.3) the points t i ∈ T κ κ for i ∈{1,2, ,n} with t 1 <t 2 < ··· <t n ; α,β,γ,δ ∈ [0,∞), αγ + αδ + βγ > 0, a i ,b i ∈ [0,∞), i ∈{2, ,n − 1}. (1.4) The continuous function f :[0, ∞) → [0,∞) is such that the following exist: f 0 := lim y→0 + f (y) y , f ∞ := lim y→∞ f (y) y ; (1.5) Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 59572, Pages 1–17 DOI 10.1155/ADE/2006/59572 2 Second-order n-point eigenvalue problems on time scales and the right-dense continuous function h :[t 1 ,t n ] → [0,∞) satisfies some suitable con- ditions to be developed. Problem (1.1), (1.2) is a generalization to time scales of the prob- lem when T is restricted to R on the unit interval in Ma and Thompson [19], and extends the type of time-scale boundar y value problem found in Anderson [2], Atici and Gu- seinov [6], Kaufmann [15], Kaufmann and Raffoul [16], and Sun and Li [21, 22]. Other related t hree-point problems on time scales include Anderson and Avery [4], Anderson et al. [5], Peterson et al. [20], and a singular problem in DaCunha et al. [12]. Some of the work on multipoint time-scale problems includes Anderson [1, 3] and Kong and Kong [17], and a recent singular multipoint problem in Bohner and Luo [8]. For more general information concerning dynamic equations on time scales, introduced by Aulbach and Hilger [7]andHilger[14], see the excellent text by Bohner and Peterson [9] and their edited text [10]. 2. Time-scale primer Any arbitrary nonempty closed subset of the reals R can serve as a time-scale T;see[9, 10]. For t ∈ T define the forward jump operator σ : T → T by σ(t) = inf{s ∈ T : s>t}, and the backward jump operator ρ : T → T by ρ(t) = sup{s ∈ T : s<t}. The graininess operators μ σ ,μ ρ : T → [0,∞)aredefinedbyμ σ (t) = σ(t) − t and μ ρ (t) = ρ(t) − t. A function f : T → R is right-dense continuous (rd-continuous) provided it is con- tinuous at all right-dense points of T and its left-sided limit exists (is finite) at left- dense points of T. The set of all right-dense continuous functions on T is denoted by C rd = C rd (T) = C rd (T,R). Define the set T κ by T κ = T −{ m} if T has a right scattered minimum m and T κ = T otherwise. In a similar vein, T κ = T −{ M} if T has a left scattered maximum M and T κ = T otherwise. We take T κ κ = T κ ∩ T κ . Definit ion 2.1 (delta derivative). Assume f : T → R is a function and let t ∈ T κ .Define f Δ (t) to be the number (provided it exists) with the property that given any  > 0, there is a neighborhood U ⊂ T of t such that    f  σ(t)  − f (s)  − f Δ (t)  σ(t) − s    ≤    σ(t) − s   ∀ s ∈ U. (2.1) The function f Δ (t) is the delta derivative of f at t. Definit ion 2.2 (nabla derivative). For f : T → R and t ∈ T κ ,define f ∇ (t)tobethenumber (provided it exists) with the property that given any  > 0, there is a neighborhood U of t such that   f  ρ(t)  − f (s) − f ∇ (t)  ρ(t) − s    ≤    ρ(t) − s   ∀ s ∈ U. (2.2) The function f ∇ (t) is the nabla derivative of f at t. In the case T = R, f Δ (t) = f  (t) = f ∇ (t). When T = Z, f Δ (t) = f (t +1)− f (t)and f ∇ (t) = f (t) − f (t − 1). D. R. Anderson and R. Ma 3 Definit ion 2.3 (delta integ ral). Let f : T → R be a function, and let a,b ∈ T. If there exists a function F : T → R such that F Δ (t) = f (t)forallt ∈ T κ ,thenF is a delta antiderivative of f . In this case the integral is given by the formula  b a f (t)Δt = F(b) − F(a)fora,b ∈ T. (2.3) All right-dense continuous functions are delta integrable; see [9, Theorem 1.74]. 3. Linear preliminaries We first construct Green’s function for the second-order boundary value problem  py ∇  Δ (t) − q(t)y(t)+u(t) = 0, t 1 <t<t n , (3.1) αy  t 1  − βp  t 1  y ∇  t 1  = 0, γy  t n  + δp  t n  y ∇  t n  = 0, (3.2) where α, β, γ, δ are real numbers such that |α| + |β| = 0, |γ| + |δ| = 0. The techniques here are similar to those found in [6, 19]. Denote by φ and ψ the solutions of the corresponding homogeneous equation  py ∇  Δ (t) − q(t)y(t) = 0, t ∈  t 1 ,t n  , (3.3) under the initial conditions ψ  t 1  = β, p  t 1  ψ ∇  t 1  = α, (3.4) φ  t n  = δ, p  t n )φ ∇  t n  =− γ, (3.5) so that ψ and φ satisfy the first and second boundary conditions in (3.2), respectively. Set d =−W t (ψ,φ) = p(t)ψ ∇ (t)φ(t) − ψ(t)p(t)φ ∇ (t). (3.6) Since the Wronskian of any two solutions is independent of t, evaluating at t = t 1 , t = t n , and using the boundary conditions (3.4), (3.5)yields d = αφ  t 1  − βp  t 1  φ ∇  t 1  = γψ  t n  + δp  t n  ψ ∇  t n  . (3.7) In addition d = 0 if and only if the homogeneous equation (3.3) has only the trivial so- lution satisfying the boundary conditions (3.2). For the proof of the following theorem, see [6, Theorem 4.2]. Lemma 3.1. Assume (1.3)and(1.4). If d = 0, then the nonhomogeneous boundary value problem (3.1)-(3.2)hasauniquesolutiony for which the formula y(t) =  t n t 1 G(t,s)u(s)Δs, t ∈  ρ  t 1  ,t n  (3.8) 4 Second-order n-point eigenvalue problems on time scales holds, where the function G(t,s) is given by G(t,s) = 1 d ⎧ ⎪ ⎨ ⎪ ⎩ ψ(t)φ(s), ρ  t 1  ≤ t ≤ s ≤ t n , ψ(s)φ(t), ρ  t 1  ≤ s ≤ t ≤ t n , (3.9) and G(t, s) is Green’s function of the boundary value problem (3.1)-(3.2). Furthermore Green’s function is symmetric, that is, G(t,s) = G(s,t) for t, s ∈ [ρ(t 1 ),t n ]. Lemma 3.2. Assume (1.3)and(1.4). Then the functions ψ and φ satisfy ψ(t) ≥ 0, t ∈  ρ  t 1 ),t n  , ψ(t) > 0, t ∈  ρ  t 1  ,t n  , p(t)ψ ∇ (t) ≥ 0, t ∈  ρ  t 1  ,t n  , φ(t) ≥ 0, t ∈  ρ  t 1  ,t n  , φ(t) > 0, t ∈  ρ  t 1  ,t n  , p(t)φ ∇ (t) ≤ 0, t ∈  ρ  t 1  ,t n  . (3.10) Proof. The proof is very similar to the proof of [6, Lemma 5.1] and is omitted.  Set D : =            − n−1  i=2 a i ψ  t i  d − n−1  i=2 a i φ  t i  d − n−1  i=2 b i ψ  t i  − n−1  i=2 b i φ  t i             . (3.11) Lemma 3.3. Assume (1.3)and(1.4). If D = 0 and u ∈ C rd [t 1 ,t n ], then the nonhomogeneous dynamic equation (3.1) with boundary conditions (1.2)hasauniquesolutiony for which the formula y(t) =  t n t 1 G(t,s)u(s)Δs + A(u)ψ(t)+B(u)φ(t), t ∈  ρ  t 1  ,t n  , (3.12) holds, where the function G(t, s) is Green’s function (3.9)oftheboundaryvalueproblem (3.1)-(3.2)andthefunctionalsA and B are defined by A(u): = 1 D            n−1  i=2 a i  t n t 1 G  t i ,s  u(s)Δsd− n−1  i=2 a i φ  t i  n−1  i=2 b i  t n t 1 G  t i ,s  u(s)Δs − n−1  i=2 b i φ  t i             , (3.13) B(u): = 1 D            − n−1  i=2 a i ψ  t i  n−1  i=2 a i  t n t 1 G  t i ,s  u(s)Δs d − n−1  i=2 b i ψ  t i  n−1  i=2 b i  t n t 1 G  t i ,s  u(s)Δs            . (3.14) D. R. Anderson and R. Ma 5 Proof. It can be verified that for a solution y of the nonhomogeneous equation (3.1) under the nonhomogeneous boundary conditions (1.2), the formula (3.12)holds,where G(t,s)isgivenby(3.9). We thus show that the function y given in (3.12)isasolutionof (3.1) with conditions (1.2)onlyifA and B are given by (3.13)and(3.14), respectively. If y as in (3.12)isasolutionof(3.1), (1.2), then y(t) = 1 d  t t 1 φ(t)ψ(s)u(s)Δs + 1 d  t n t ψ(t)φ(s)u(s)Δs + Aψ(t)+Bφ(t) (3.15) for some constants A and B. Taking the nabla derivative and multiplying by p yields py ∇ = pφ ∇ d  t t 1 ψ(s)u(s)Δs + pψ ∇ d  t n t φ(s)u(s)Δs + Apψ ∇ + Bpφ ∇ ; (3.16) the delta derivative of this expression i s  py ∇  Δ =  pφ ∇ d  Δ  σ(t) t 1 ψ(s)u(s)Δs + pφ ∇ d ψ(t)u(t)+A  pψ ∇  Δ + B  pφ ∇  Δ +  pψ ∇ d  Δ  t n σ(t) φ(s)u(s)Δs − pψ ∇ d φ(t)u(t). (3.17) Using [9, Theorem 1.75], and the fact that ψ and φ are solutions to (3.3), we obtain  py ∇  Δ (t) = q(t) d  t t 1 φ(t)ψ(s)u(s)Δs + q(t) d φ(t)μ σ (t)ψ(t)u(t)+ u(t) d p(t)φ ∇ (t)ψ(t) + q(t) d  t n t ψ(t)φ(s)u(s)Δs − q(t) d ψ(t)μ σ (t)φ(t)u(t) − u(t) d p(t)ψ ∇ (t)φ(t)+q(t)  Aψ(t)+bφ(t)  . (3.18) Recall that d is in terms of the Wronskian of ψ and φ in (3.6); it follows that  py ∇  Δ (t) = q(t)y(t) − u(t). (3.19) Now y  t 1  = ψ  t 1  d  t n t 1 φ(s)u(s)Δs + Aψ  t 1  + Bφ  t 1  , p  t 1  y ∇  t 1  = p  t 1  ψ ∇  t 1  d  t n t 1 φ(s)u(s)Δs + Ap  t 1  ψ ∇  t 1  + Bp  t 1  φ ∇  t 1  ; (3.20) 6 Second-order n-point eigenvalue problems on time scales multiply the first line by α and the second by −β,anduse(1.2)and(3.4)toseethat B  αφ  t 1  − βp  t 1  φ ∇  t 1  = n−1  i=2 a i   t n t 1 G  t i ,s  u(s)Δs + Aψ  t i  + Bφ  t i   . (3.21) At the other end, y  t n  = φ  t n  d  t n t 1 ψ(s)u(s)Δs + Aψ  t n  + Bφ  t n  , p  t n  y ∇  t n  = p  t n  φ ∇  t n  d  t n t 1 ψ(s)u(s)Δs + Ap  t n  ψ ∇  t n  + Bp  t n  φ ∇  t n  ; (3.22) consequently A  γψ  t n  + δp  t n  ψ ∇  t n  = n−1  i=2 b i   t n t 1 G  t i ,s  u(s)Δs + Aψ  t i  + Bφ  t i   . (3.23) Combining (3.21)and(3.23) and using (3.6), we arrive at the system of equations −A n−1  i=2 a i ψ  t i  + B  αφ  t 1  − βp  t 1  φ ∇  t 1  − n−1  i=2 a i φ  t i   = n−1  i=2 a i  t n t 1 G  t i ,s  u(s)Δs, A  γψ  t n  + δp  t n  ψ ∇ (t n ) − n−1  i=2 b i ψ  t i   − B n−1  i=2 b i φ  t i  = n−1  i=2 b i  t n t 1 G  t i ,s  u(s)Δs. (3.24) Again using (3.6) at both t 1 and t n ,weverify(3.13)and(3.14).  Lemma 3.4. Let (1.3)and(1.4) hold, and assume D<0, d − n−1  i=2 a i φ  t i  > 0, d − n−1  i=2 b i ψ  t i  > 0 (3.25) for D and d givenin(3.11 )and(3.6), respect ively. If u ∈ C rd [t 1 ,t n ] with u ≥ 0, the unique solution y as in (3.12)oftheproblem(3.1), (1.2)satisfiesy(t) ≥ 0 for t ∈ [t 1 ,t n ]. Proof. From the previous lemmas and assumptions we know that Green’s function (3.9) satisfies G(t,s) ≥ 0on[ρ(t 1 ),t n ] × [ρ( t 1 ),t n ]. Hypotheses (1.3), (1.4), and (3.25)applied to (3.13)and(3.14)implythatA(u), B(u) ≥ 0.  Suppose (3.25) does not hold. For example, let n = 3, p(t) ≡ 1 = α = γ, q(t) ≡ 0 = β = δ = a 2 ,andt 1 = 0. Then (3.1), (1.2)becomes y ∇Δ (t)+u(t) = 0, t 1 <t<t 3 , y  t 1  = 0, y  t 3  = b 2 y  t 2  . (3.26) D. R. Anderson and R. Ma 7 Note that ψ(t) = t, d = t 3 ,andD = t 3 (b 2 t 2 − t 3 ). If D>0, then b 2 t 2 >t 3 , and there is no positive solution; see [15, Lemma 4]. Lemma 3.5. Let (1.3), (1.4), and (3.25)hold,andfix ξ 1 ,ξ 2 ∈ T κ κ , ρ  t 1  <ξ 1 <ξ 2 <t n . (3.27) If u ∈ C rd [t 1 ,t n ] with u ≥ 0, the unique s olution y as in (3.12) of the time-scale boundary value problem (3.1), (1.2)satisfies min t∈[ξ 1 ,ξ 2 ] y(t) ≥ Γy, y := max t∈[ρ(t 1 ),t n ] y(t), (3.28) where Γ : = min  φ  ξ 2  φ  ρ  t 1  , ψ  ξ 1  ψ  t n   ∈ (0,1). (3.29) Proof. From (1.3), (3.9), and Lemma 3.2, 0 ≤ G(t,s) ≤ G(s,s), t ∈  ρ  t 1  ,t n  , (3.30) so that y(t) ≤  t n t 1 G(s,s)u(s)Δs + A(u)ψ(t n )+B(u)φ  ρ  t 1  ∀ t ∈  ρ  t 1  ,t n  . (3.31) For t ∈ [ξ 1 ,ξ 2 ], Green’s function (3.9) satisfies G(t,s) G(s,s) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ φ(t) φ(s) : ρ  t 1  ≤ s ≤ t ≤ t n ψ(t) ψ(s) : ρ(t 1 ) ≤ t ≤ s ≤ t n ≥ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ φ  ξ 2  φ  ρ  t 1  : ρ  t 1  ≤ s ≤ t ≤ t n ψ  ξ 1  ψ  t n  : ρ  t 1  ≤ t ≤ s ≤ t n ≥ Γ (3.32) for Γ as in (3.29), and y(t) =  t n t 1 G(t,s) G(s,s) G(s,s)u(s)Δs + A(u)ψ(t)+B(u)φ(t) ≥  t n t 1 ΓG(s,s)u(s)Δs + A(u)ψ  ξ 1  + B(u)φ  ξ 2  ≥ Γ   t n t 1 G(s,s)u(s)Δs + A(u)ψ  t n  + B(u)φ  ρ  t 1   ≥ Γy. (3.33)  4. Eigenvalue intervals To establish eigenvalue intervals we will employ the following fixed point theorem due to Krasnosel’ski ˘ i[18]; for more on the establishment of eigenvalue intervals for time-scale boundary value problems, see, for example, Chyan and Henderson [11]andDavisetal. [13]. 8 Second-order n-point eigenvalue problems on time scales Theorem 4.1. Let E be a Banach space, P ⊆ E a cone, and suppose that Ω 1 , Ω 2 are bounded open balls of E centered at the or igin with Ω 1 ⊂ Ω 2 . Suppose further that L : P ∩ (Ω 2 \ Ω 1 ) → P is a completely continuous operator such that either (i) Ly≤y, y ∈ P ∩ ∂Ω 1 and Ly≥y, y ∈ P ∩ ∂Ω 2 ,or (ii) Ly≥y, y ∈ P ∩ ∂Ω 1 and Ly≤y, y ∈ P ∩ ∂Ω 2 holds. Then L has a fixed point in P ∩ (Ω 2 \ Ω 1 ). Assume that the right-dense continuous function h satisfies h :  t 1 ,t n  −→ [0,∞), ∃t ∗ ∈  σ  t 1  ,ρ  t n   h(t ∗ ) > 0. (4.1) Then there exist ξ 1 , ξ 2 as in Lemma 3.5 such that ξ 1 <t ∗ <ξ 2 ,  ξ 2 ξ 1 G(t,s)h(s)Δs>0, t ∈  ρ  t 1  ,t n  . (4.2) In the following , let Γ be the constant defined in (3.29) with respect to such constants ξ 1 , ξ 2 .Letτ ∈ [ρ(t 1 ),t n ]bedeterminedby  ξ 2 ξ 1 G(τ,s)h(s)Δs = max ρ(t 1 )≤t≤t n  ξ 2 ξ 1 G(t,s)h(s)Δs>0. (4.3) For G(t,s)in(3.9)andA,B as in (3.13), (3.14), respectively, define the constant K : =  t n t 1 G(s,s)h(s)Δs + A(h)ψ  t n  + B(h)φ  ρ  t 1  . (4.4) Let Ꮾ denote the Banach space C[ρ(t 1 ),t n ] with the norm y=sup t∈[ρ(t 1 ),t n ] |y(t)|.De- fine the cone ᏼ ⊂ Ꮾ by ᏼ =  y ∈ Ꮾ : y(t) ≥ 0on  ρ  t 1  ,t n  , y(t) ≥ Γy on  ξ 1 ,ξ 2  , (4.5) where Γ is given in (3.29). Since y is a solution of (1.1), (1.2)ifandonlyif y(t) = λ   t n t 1 G(t,s)h(s) f  y(s)  Δs + A  hf(y)  ψ(t)+B  hf(y)  φ(t)  , t ∈  ρ  t 1  ,t n  , (4.6) D. R. Anderson and R. Ma 9 define for y ∈ ᏼ the operator T : ᏼ → Ꮾ by (Ty)(t): = λ   t n t 1 G(t,s)h(s) f  y(s)  Δs + A  hf(y)  ψ(t)+B  hf(y)  φ(t)  . (4.7) We seek a fixed point of T in ᏼ by establishing the hypotheses of Theorem 4.1. Theorem 4.2. Suppose (1.3), (1.4), (3.25), (4.1), and (4.3) hold. Then for each λ satisfying 1 f ∞ Γ  ξ 2 ξ 1 G(τ,s)h(s)Δs <λ< 1 f 0 K , (4.8) thereexistsatleastonepositivesolutionof(1.1), (1.2)inᏼ. Proof. Let ξ 1 , ξ 2 be as in Lemma 3.5,letτ be as in (4.3), let K be as in (4.4), let λ be as in (4.8), and let  > 0besuchthat 1  f ∞ −   Γ  ξ 2 ξ 1 G(τ,s)h(s)Δs ≤ λ ≤ 1  f 0 +   K . (4.9) Consider the integral operator T in (4.7). If y ∈ ᏼ,thenby(3.30)wehave (Ty)(t) = λ   t n t 1 G(t,s)h(s) f  y(s)  Δs + A  hf(y)  ψ(t)+B  hf(y)  φ(t)  ≤ λ   t n t 1 G(s,s)h(s) f  y(s)  Δs + A  hf(y)  ψ  t n  + B  hf(y)  φ  ρ  t 1   , (4.10) so that for t ∈ [ξ 1 , ξ 2 ], (Ty)(t) = λ   t n t 1 G(t,s)h(s) f  y(s)  Δs + A  hf(y)  ψ(t)+B  hf(y)  φ(t)  ≥ λ   t n t 1 G(t,s) G(s,s) G(s,s)h(s) f  y(s)  Δs + A  hf(y)  ψ  ξ 1  + B  hf(y)  φ  ξ 2   ≥ λΓ   t n t 1 G(s,s)h(s)f  y(s)  Δs+A  hf(y)  ψ  t n  +B  hf(y)  φ  ρ  t 1   ≥ ΓTy. (4.11) Therefore T : ᏼ → ᏼ.Moreover,T is completely continuous by a ty pical application of the Ascoli-Arzela theorem. 10 Second-order n-point eigenvalue problems on time scales Now consider f 0 . There exists an R 1 > 0suchthat f (y) ≤ ( f 0 + )y for 0 <y≤ R 1 by the definition of f 0 .Picky ∈ ᏼ with y=R 1 .From(3.13)and(3.14),   A  hf(y)    ≤ A(h)   f (y)   ,   B  hf(y)    ≤ B(h)   f (y)   . (4.12) Using (3.30), we have (Ty)(t) = λ   t n t 1 G(t,s)h(s) f  y(s)  Δs + A  hf(y)  ψ(t)+B  hf(y)  φ(t)  ≤ λ   f (y)     t n t 1 G(s,s)h(s)Δs + A(h)ψ  t n  + B(h)φ  ρ(t 1 )   ≤ λ  f 0 +    yK ≤y (4.13) from the right-hand side of (4.9). As a result, Ty≤y.Thus,take Ω 1 :=  y ∈ Ꮾ : y <R 1  (4.14) so that Ty≤y for y ∈ ᏼ ∩ ∂Ω 1 . Next consider f ∞ . Again by definition, there exists an R  2 >R 1 such that f (y) ≥ ( f ∞ −  )y for y ≥ R  2 ;takeR 2 = max{2R 1 ,R  2 /Γ}.Ify ∈ ᏼ with y=R 2 ,thenfors ∈ [ξ 1 , ξ 2 ] we have y(s) ≥ Γy=ΓR 2 . (4.15) Define Ω 2 :={y ∈ Ꮾ : y <R 2 }; using (4.3)and(4.15)fors ∈ [ξ 1 , ξ 2 ], we get (Ty)(τ) = λ   t n t 1 G(τ,s)h(s) f  y(s)  Δs + A  hf(y)  ψ(τ)+B  hf(y)  φ(τ)  ≥ λ  ξ 2 ξ 1 G(τ,s)h(s) f  y(s)  Δs ≥ λ  f ∞ −    ξ 2 ξ 1 G(τ,s)h(s)y(s)Δs ≥ λ  f ∞ −   ΓR 2  ξ 2 ξ 1 G(τ,s)h(s)Δs ≥ R 2 =y, (4.16) where we have used the left-hand side of (4.9). Hence we have shown that Ty≥y, y ∈ ᏼ ∩ ∂Ω 2 . (4.17) An application of Theorem 4.1 yields the conclusion of the theorem; in other words, T has a fixed point y in ᏼ ∩ (Ω 2 \ Ω 1 )withR 1 ≤y≤R 2 .  [...]... 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R1 , (4.22) whereby our work above confirms Ty ≥ y , y ∈ ᏼ ∩ ∂Ω1 (4.23) Next consider f∞ Again by definition there exists an R2 > R1 such that f (y) ≤ ( f∞ + η)y for y ≥ R2 If f is bounded, there exists M > 0 with f (y) ≤ M for all y ∈ (0, ∞) Let R2 := max 2R2 ,λM tn t1 G(s,s)h(s)Δs + A(h)ψ tn + B(h)φ ρ t1 (4.24) 12 Second-order n-point eigenvalue problems on time scales If y ∈ ᏼ with y = R2 , then . 10.1155/ADE/2006/59572 2 Second-order n-point eigenvalue problems on time scales and the right-dense continuous function h :[t 1 ,t n ] → [0,∞) satisfies some suitable con- ditions to be developed. Problem. (5.19) Acknowledgment The second author is partly supported by NSFC (no. 10271095). 16 Second-order n-point eigenvalue problems on time scales References [1] D.R.Anderson,Extension of a second-order multi-point. SECOND-ORDER n-POINT EIGENVALUE PROBLEMS ON TIME SCALES DOUGLAS R. ANDERSON AND RUYUN MA Received 10 December 2004; Revised 3 November 2005; Accepted 6 November 2005 We discuss conditions for

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