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Electronic Journal of Differential Equations, Vol. 2003(2003), No. 40, pp. 1–7. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) EXISTENCE OF SOLUTIONS TO HIGHER-ORDER DISCRETE THREE-POINT PROBLEMS DOUGLAS R. ANDERSON Abstract. We are concerned with the higher-order discrete three-point boundaryvalue problem (∆n x)(t) = f (t, x(t + θ)), i 0 ≤ i ≤ n − 4, (∆ x)(t1 ) = 0, α(∆ n−3 x)(t) − β(∆ n−2 t1 ≤ t ≤ t3 − 1, x)(t) = η(t), −τ ≤ θ ≤ 1 n≥4 t1 − τ − 1 ≤ t ≤ t1 (∆n−2 x)(t2 ) = (∆n−1 x)(t3 ) = 0. By placing certain restrictions on the nonlinearity and the distance between boundary points, we prove the existence of at least one solution of the boundary value problem by applying the Krasnoselskii fixed point theorem. 1. Introduction We are concerned with the existence of solutions to the higher-order discrete three-point problem (∆n x)(t) = f (t, x(t + θ)), i (∆ x)(t1 ) = 0, α(∆ n−3 x)(t) − β(∆ (∆ n−2 n−2 t1 ≤ t ≤ t3 − 1, 0 ≤ i ≤ n − 4, x)(t) = η(t), x)(t2 ) = (∆ n−1 −τ ≤ θ ≤ 1 (1.1) n≥4 t 1 − τ − 1 ≤ t ≤ t1 x)(t3 ) = 0. (1.2) Here we assume (i) (ii) (iii) (iv) (v) (vi) any interval [a, b] is the set of integers {a, a + 1, · · · , b − 1, b}; ti+1 > ti + n − 1 to avoid overlap in boundary conditions, i ∈ {1, 2}; f : [t1 , t3 − 1] × [0, ∞) → [0, ∞); α, β > 0, t3 − t1 ≥ τ ≥ −1, and θ ∈ [−τ, 1] is constant; η : [t1 − τ − 1, t1 ] → R with η(t1 ) = 0; x is defined on [t1 − τ − 1, t3 + n − 1]. For the rest of this paper we also have the hypotheses 2000 Mathematics Subject Classification. 39A10. Key words and phrases. Difference equations, boundary-value problem, Green’s function, fixed points, cone. c 2003 Southwest Texas State University. Submitted August 19, 2002. Published April 15, 2003. 1 2 DOUGLAS R. ANDERSON EJDE–2003/40 (H1) G(t, s) on [t1 , t3 +n−1]×[t1 , t3 −1] is the Green’s function for the difference equation (∆n u)(t) = 0, t ∈ [t1 , t3 − 1] subject to the boundary conditions (1.2) with τ = −1. (H2) g(t, s) on [t1 , t3 + 2] × [t1 , t3 − 1] is the Green’s function for the difference equation (∆3 u)(t) = 0, t ∈ [t1 , t3 − 1] subject to the boundary conditions αu(t1 ) − β(∆u)(t1 ) = 0 (∆u)(t2 ) = (∆2 u)(t3 ) = 0 for α, β as in (iv). (H3) kxk[t1 −τ −1,t3 +2] := sup (1.3) |(∆n−3 x)(t)|. t1 −τ −1≤t≤t3 +2 (H4) For Ξ := {t ∈ [t1 , t3 + n − 1] : t1 ≤ t + θ ≤ t3 − 1}, Ξh := {t ∈ Ξ : t2 − h ≤ t + θ ≤ t2 + h} is nonempty for some h ∈ (0, t3 − t2 − 2), which is nonempty by (ii). The corresponding Green’s function for the discrete homogeneous problem (∆3 u)(t) = 0 satisfying the boundary conditions (1.3), a slight generalization of that in [1, 2, 3, 4], is given via ( u1 (t, s) : t ≤ s + 1 s ∈ [t1 , t2 − 1] : v1 (t, s) : t ≥ s + 1 (1.4) g(t, s) = ( u2 (t, s) : t ≤ s + 1 s ∈ [t2 − 1, t3 − 1] : v2 (t, s) : t ≥ s + 1 for t ∈ [t1 , t3 + 2] and s ∈ [t1 , t3 − 1], where 1 β u1 (t, s) := (t − t1 )(2s − t − t1 + 3) + (s − t1 + 1), 2 α 1 β v1 (t, s) := (s − t1 + 2)(s − t1 + 1) + (s − t1 + 1), 2 α 1 β u2 (t, s) := (t − t1 )(2t2 − t − t1 + 1) + (t2 − t1 ), 2 α 1 β 1 v2 (t, s) := (t−t1 )(2t2 − t − t1 + 1) + (t2 −t1 ) + (t−s−1)(t−s−2). 2 α 2 Remark 1.1. As in [2], it can be shown that if β 1 (t2 − t1 ) + 1 > (t3 − t1 + 2)(t3 + t1 − 2t2 + 1), α 2 then g(t, s) > 0 for all t ∈ [t1 , t3 + 2], s ∈ [t1 , t3 − 1]. Note that if the boundary points satisfy t3 − t2 ≤ t2 − t1 − 1, (1.5) then the above inequality holds for any choice of α, β > 0. Thus throughout this paper we assume that (1.5) holds. Moreover, as in [3], we have the following boundedness result. EJDE–2003/40 DISCRETE THREE-POINT PROBLEM 3 Lemma 1.2. For all t ∈ [t1 , t3 + 2] and s ∈ [t1 , t3 − 1], `(t)g(t2 , s) ≤ g(t, s) ≤ g(t2 , s) where `(t) := min t − t1 t 3 − t + 2 , t 2 − t 1 t 3 − t2 + 2 (1.6) . (1.7) Remark 1.3. The following discussion is similar to that found in [6] for a continuous two-point problem on the unit interval. If x is a solution of (1.1), (1.2), it can be written as : t1 − τ − 1 ≤ t ≤ t 1 x(−τ ; t) tX 3 −1 x(t) = G(t, s)f (s, x(s + θ)) : t1 ≤ t ≤ t3 + n − 1 s=t1 where, using standard first-order linear difference equation methods [7], x(−τ ; t) satisfies t−t1 t−s−1 t1 −1 α 1 X α n−3 n−3 (∆ x)(t1 ) + 1+ (∆ x)(−τ ; t) = 1 + η(s) β β s=t β for t ∈ [t1 − τ − 1, t1 ]. If u0 is the solution of (1.1), (1.2) with f ≡ 0, then u0 satisfies t −1 t−s−1 1 1 X 1+ α η(s) : t1 − τ − 1 ≤ t ≤ t1 β (∆n−3 u0 )(t) = β s=t 0 : t1 ≤ t ≤ t3 + 2; (1.8) note that actually, using the Green’s function, u0 ≡ 0 on [t1 , t3 + n − 1]. If x is any solution of (1.1), (1.2) set u(t) := x(t) − u0 (t). Then u(t) ≡ x(t) on [t1 , t3 + n − 1], and u satisfies t−t1 α (∆n−3 u)(t1 ) : t1 − τ − 1 ≤ t ≤ t1 1+ β n−3 3 −1 (∆ u)(t) = tX g(t, s)f (s, u(s + θ) + u0 (s + θ)) : t1 ≤ t ≤ t3 + 2. s=t1 But this implies n−3 t−t1 β 1+ α (∆n−3 u)(t1 ) : t 1 − τ − 1 ≤ t ≤ t1 α β 3 −1 u(t) = tX G(t, s)f (s, u(s + θ) + u0 (s + θ)) : t1 ≤ t ≤ t3 + n − 1. s=t1 2. Existence of at Least One Solution We are concerned with proving the existence of solutions of the higher-order discrete nonlinear boundary value problem (1.1), (1.2). In light of the above discussion in Remark 1.3, consider the fixed points of the operator A defined by n−3 t−t1 β 1+ α (∆n−3 u)(t1 ) : t1 − τ − 1 ≤ t ≤ t 1 α β t −1 3 Au(t) = X G(t, s)f (s, u(s + θ) + u0 (s + θ)) : t1 ≤ t ≤ t3 + n − 1, s=t1 4 DOUGLAS R. ANDERSON EJDE–2003/40 with domain {u : [t1 − τ − 1, t3 + n − 1] → R}. If Au = u, then a solution x of (1.1), (1.2) would be given by t−t1 β n−3 1+ α (∆n−3 u)(t1 ) + u0 (t) : t1 − τ − 1 ≤ t ≤ t1 α β x(t) = u(t) : t1 ≤ t ≤ t3 + n − 1, where u0 satisfies (1.8). Remark 2.1. In the following discussion we will need an h ∈ (0, t3 − t2 − 2); note that for all t ∈ [t2 − h, t2 + h], we then have h+1 `(t) ≥ `(t2 + h + 1) = 1 − (2.1) t3 − t 2 + 2 for all h ∈ (0, t3 − t2 − 2), where ` is given in (1.7). Moreover, let k, m > 0 such that tX 3 −1 k −1 : = g(t2 , s) (2.2) s=t1 = 1 (t2 − t1 + 1)(t2 − t1 )(3t3 − 2t2 − t1 + 2) 6 β (t2 − t1 )(2t3 − t2 − t1 + 1) + 2α and m −1 : = `(t2 + h + 1) tX 2 +h g(t2 , s) (2.3) s=t2 −h 1 = 6 h+1 1− t3 − t 2 + 2 (t2 − t1 + 1)2 (t2 − t1 + 3h + 5) 3β 2 −(t2 − t1 − h + 2) + (4ht2 + 2t2 − 4ht1 − 2t1 − h + h) , α 3 where we have used the so-called falling factorial power [7] br := b(b − 1)(b − 2) · · · (b − r + 1). Finally, set M0 := ku0 k[t1 −τ −1,t3 +2] (2.4) for u0 as in (1.8). We will employ the following fixed point theorem due to Krasnoselskii [8]. Theorem 2.2. Let E be a Banach space, P ⊆ E be a cone, and suppose that Ω1 , Ω2 are bounded open balls of E centered at the origin with Ω1 ⊂ Ω2 . Suppose further that A : P ∩ (Ω2 \ Ω1 ) → P is a completely continuous operator such that either (i) kAuk ≤ kuk, u ∈ P ∩ ∂Ω1 and kAuk ≥ kuk, u ∈ P ∩ ∂Ω2 , or (ii) kAuk ≥ kuk, u ∈ P ∩ ∂Ω1 and kAuk ≤ kuk, u ∈ P ∩ ∂Ω2 holds. Then A has a fixed point in P ∩ (Ω2 \ Ω1 ). Theorem 2.3. Let k, m, M0 be as in (2.2), (2.3), (2.4), respectively, and suppose the following conditions are satisfied. (C1 ) There exists p > 0 such that f (t, w) ≤ kp for t ∈ [t1 , t3 − 1] and 0 ≤ kwk ≤ p + M0 . EJDE–2003/40 DISCRETE THREE-POINT PROBLEM 5 (C2 ) There exists q > 0 such that f (t, w) ≥ mq for t ∈ Ξh and q`(t2 + h + 1) ≤ kwk ≤ q, for h ∈ (0, t3 − t2 − 2) and Ξh as in (H4). Then (1.1), (1.2) has a solution x = u + u0 such that kxk[t1 −τ −1,t3 +2] lies between max{0, p − M0 } and q + M0 . Proof. Many of the techniques employed here are as in [5, 6]. Let B denote the Banach space {u : [t1 − τ − 1, t3 + n − 1] → R} with the norm kuk[t1 −τ −1,t3 +2] = sup |(∆n−3 u)(t)|. t∈[t1 −τ −1,t3 +2] Define the cone P ⊂ B by P = {u ∈ B : min t∈[t2 −h,t2 +h] (∆n−3 u)(t) ≥ `(t2 + h + 1)kuk[t1 −τ −1,t3 +2] }. Consider the mapping A : P → B via n−3 t−t1 β α 1 + (∆n−3 u)(t1 ) : t1 − τ − 1 ≤ t ≤ t 1 α β 3 −1 Au(t) = tX G(t, s)f (s, u(s + θ) + u0 (s + θ)) : t1 ≤ t ≤ t3 + n − 1. s=t1 Then ∆n−3 (Au)(t) = 3 −1 t−t1 tX α 1 + g(t1 , s)f (s, u(s + θ) + u0 (s + θ)) β s=t1 tX 3 −1 g(t, s)f (s, u(s + θ) + u0 (s + θ)) s=t1 n−3 so that ∆ (Au)(t) ≤ ∆n−3 (Au)(t1 ) for t1 − τ − 1 ≤ t ≤ t1 . In other words, kAuk[t1 −τ −1,t3 +2] = kAuk[t1 ,t3 +2] . It follows for h ∈ (0, t3 − t2 − 2) and t ∈ [t2 − h, t2 + h] that ∆n−3 (Au)(t) = tX 3 −1 g(t, s)f (s, u(s + θ) + u0 (s + θ)) s=t1 ≥ `(t) tX 3 −1 g(t2 , s)f (s, u(s + θ) + u0 (s + θ)) s=t1 ≥ `(t2 + h + 1)kAuk[t1 −τ −1,t3 +2] by properties of the Green’s function (1.6), so that A : P → P. Without loss of generality, we may assume 0 < p < q. Define the bounded open balls Ωp = {u ∈ B : kuk[t1 −τ −1,t3 +2] < p}, and Ωq = {u ∈ B : kuk[t1 −τ −1,t3 +2] < q}; then 0 ∈ Ωp ⊂ Ωq . If u ∈ P ∩ ∂Ωp , then kuk = p and |(∆n−3 u)(t) + (∆n−3 u0 )(t)| ≤ p + M0 6 DOUGLAS R. ANDERSON EJDE–2003/40 for all t ∈ [t1 , t3 + 2]. As a result, kAuk = tX 3 −1 g(t2 , s)f (s, u(s + θ) + u0 (s + θ)) s=t1 ≤ kp tX 3 −1 g(t2 , s) s=t1 =p = kuk using (C1 ) and (2.2). Thus, kAuk ≤ kuk for u ∈ P ∩ ∂Ωp . Similarly, let u ∈ P ∩ ∂Ωq , so that kuk = q. Then for s ∈ Ξh , (∆n−3 u)(s + θ) ≥ min t∈[t2 −h,t2 +h] (∆n−3 u)(t) ≥ kuk`(t2 + h + 1) for all h ∈ (0, t3 − t2 − 2) and `(·) as in (2.1). As a result, q`(t2 + h + 1) ≤ (∆n−3 u)(s + θ) + (∆n−3 u0 )(s + θ) ≤ q for s ∈ Ξh , since ∆n−3 u0 ≡ 0 on [t1 , t3 + 2] by Remark 1.3. It follows that kAuk = tX 3 −1 g(t2 , s)f (s, u(s + θ) + u0 (s + θ)) s=t1 ≥ X g(t2 , s)f (s, u(s + θ) + u0 (s + θ)) s∈Ξh ≥ mq`(t2 + h + 1) tX 2 +h g(t2 , s) s=t2 −h =q = kuk by (C2 ) and (2.3). Consequently, kAuk ≥ kuk for u ∈ P ∩ ∂Ωq . By Theorem 2.2, A has a fixed point u ∈ P ∩ (Ωq \ Ωp ); i.e., p ≤ kuk ≤ q. Therefore the discrete problem (1.1), (1.2) has a solution x = u + u0 such that p − M0 ≤ kxk ≤ q + M0 , if M0 < p. References [1] D. R. Anderson and R. I. Avery, Multiple positive solutions to a third order discrete focal boundary value problem, Computers and Mathematics with Applications, 42 (2001), 333-340. [2] D. R. Anderson, Positivity of Green’s function for an n-point right focal boundary value problem on measure chains, Math. Comput. Modelling 31 (2000), 29-50. [3] D. R. Anderson, Discrete third-order three-point right focal boundary value problems, Computers and Mathematics with Applications, to appear. [4] D. R. Anderson, R. I. Avery and A. C. Peterson, Three positive solutions to a discrete focal boundary value problem, Journal of Computational and Applied Mathematics, 88 (1998), 103-118. [5] J. Henderson, Multiple solutions for 2mth order Sturm-Liouville boundary value problems on a measure chain, J. Diff. Equations Appl. 6 (2000), 417-429. [6] Chen-Huang Hong, Fu-Hsiang Wong, and Cheh-Chih Yeh. Existence of Positive Solutions for Higher-Order Functional Differential Equations. Journal of Mathematical Analysis and Applications (2002), in press. [7] Walter G. Kelley and Allan C. Peterson. Difference Equations: An Introduction with Applications, 2e, Harcourt/Academic Press, San Diego, 2001. EJDE–2003/40 DISCRETE THREE-POINT PROBLEM 7 [8] M.A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. Douglas R. Anderson Department of Mathematics and Computer Science, Concordia College, Moorhead, MN 56562 USA E-mail address: [email protected]