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.
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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
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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
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[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: andersod@cord.edu