MATHEMATICS IN ENGINEERING, SCIENCE AND AEROSPACE
MESA - www.journalmesa.com
Vol. 1, No. 1, pp. 1-13, 2010
c CSP - Cambridge, UK; I&S - Florida, USA, 2010
°
Multiple symmetric positive solutions for two-point even order
boundary value problems on time scales
J. Henderson1,? , P. Murali2 , K.R. Prasad2
1
2
Department of Mathematics, Baylor University, Waco, Texas 76798-7328, USA.
Department of Applied Mathematics, Andhra University, Visakhapatnam, 530003, India.
?
Corresponding Author.
E-mail addresses: Johnny− [email protected]; murali− [email protected]; [email protected]
Abstract. This paper is concerned with symmetric positive solutions of the even order dynamic equation
on a time scale,
n
i
n−1
(−1)n y(∆∇) (t) = f (y(t), y∆∇ (t), ..., y(∆∇) (t), ..., y(∆∇)
(t)), t ∈ [a, b]
subject to the two-point boundary conditions
i
i
y(∆∇) (a) = 0 = y(∆∇) (b), 0 ≤ i ≤ n − 1,
where n ∈ N, [a, b] is a time scale interval and f : Rn → [0, ∞) is continuous. We establish the existence of
at least three symmetric positive solutions by using the well-known Avery generalization of the LeggettWilliams fixed point theorem. And then, we establish at least 2m − 1 symmetric positive solutions for an
arbitrary positive integer m.
1 Introduction
Recently, there have been considerable activities in developing the theory of dynamic equations on time
scales as this theory unifies the theories of differential equations and finite difference equations. The new
methods developed are not only significant in the theoretical study of differential equations and difference
equations, but also potentially important to nonlinear numerical analysis, as well as to its applications.
The primary purpose of this investigation is to study the symmetry properties of the solutions of even
order dynamic equation on time scales. For recent discussions on symmetry properties of differential equations or finite difference equations or time scales, to mention a few, we list the papers of Erbe and Wang
[15], Eloe and Henderson [12, 13], Eloe, Henderson and Sheng [14], Henderson and Thompson [19], Avery
0
2008 Mathematics Subject Classification: 39A10; 34B15; 34A40.
Keywords: Boundary value problem; cone; symmetric positive solution; symmetric time scale.
2
J. Henderson, P. Murali, K.R. Prasad
and Henderson [4, 5, 6], Avery, Davis and Henderson [7], Davis and Henderson [10], Davis, Henderson and
Wong [11], Anderson [2] and Henderson and Wong [18].
In this setting it is natural to assume the time scale is symmetric. One of the primary results is that symmetry of solutions is not to be expected even under the aforementioned assumptions. Instead, the dynamic
equation requires consideration as forward and backward operator at a time. Hence our study continues into
forward and backward operators for boundary value problems.
In this paper, we are dealing with symmetric time scales. By an interval time scale, we mean the intersection of a real interval with a given time scale.
An interval time scale T = [a, b] is said to be a symmetric interval time scale if
t ∈ T ⇔ a + b − t ∈ T.
If T = R or T = hZ, (h > 0) then the symmetry definition is always satisfied. In addition, the interval
time scale T = [1, 2] ∪ {3, 4, 5} ∪ [6, 7] ∪ {8} ∪ [9, 10] ∪ {11, 12, 13} ∪ [14, 15] has the symmetry property. But
the time scale T = {0} ∪ { 1n : n ∈ N} is not a symmetric time scale. By a symmetric solution y(t) of the BVP
(1.1)-(1.2), we mean y(t) is a solution of the BVP (1.1)-(1.2) and satisfies
y(t) = y(b + a − t) for t ∈ [a, b].
In this paper, we address the question of the existence of multiple symmetric positive solutions for the
even order boundary value problem on a symmetric interval time scale,
n
i
n−1
(−1)n y(∆∇) (t) = f (y(t), y∆∇ (t), ..., y(∆∇) (t), ..., y(∆∇)
(t)), t ∈ [a, b]
(1.1)
satisfying the two-point boundary conditions
i
i
y(∆∇) (a) = 0 = y(∆∇) (b), 0 ≤ i ≤ n − 1,
(1.2)
n
where n ∈ N, f : Rn → [0, ∞) is continuous, a ∈ Tkn , b ∈ Tk for a time scale T, and σn (a) < ρn (b).
This paper is organized as follows. In Section 2, we briefly describe some salient features of time scales.
In Section 3, we estimate the bounds of the Green’s function. In Section 4, we establish the existence of
at least three symmetric positive solutions for two-point boundary value problem (1.1)-(1.2). And then, we
establish the existence of at least 2m − 1 symmetric positive solutions for two-point boundary value problem
(1.1)-(1.2) for an arbitrary positive integer m.
2 Preliminaries about Time Scales
A time scale T is a nonempty closed subset of R. For an excellent introduction to the overall area of dynamic
equation on time scales, we suggest the recent texts by Bohner and Peterson [8, 9] and Lakshmikantham,
Sivasundaram and Kaymakcalan [20], from which we cull the following definitions. The functions σ, ρ :
T → T are jump operators given by
Multiple Symmetric Positive Solutions for Two-Point Even Order Boundary Value Problems on Time Scales
3
σ(t) = inf{s ∈ T : s > t} and ρ(t) = sup{s ∈ T : s < t}
(supplemented by inf 0/ = sup T and sup 0/ = inf T). The point t ∈ T is left-dense, left-scattered, right-dense,
right-scattered if ρ(t) = t, ρ(t) < t, σ(t) = t, σ(t) > t, respectively. If T has a right- scattered minimum m,
define Tk = T − {m}; otherwise, set Tk = T. If T has a left-scattered maximum M, define Tk = T − {M};
otherwise, set Tk = T.
For f : T → R and t ∈ Tk , the delta derivative of f at t, denoted f ∆ (t), is the number (provided it exists)
with the property the given any ε > 0, there is a neighborhood U of t such that
| f (σ(t)) − f (s) − f ∆ (t)[σ(t) − s] |≤ ε | σ(t) − s |
for all s ∈ U.
For f : T → R and t ∈ Tk , the nabla derivative of f at t, denoted f ∇ (t), is the number (provided it exists)
with the property the given any ε > 0, there is a neighborhood U of t such that
| f (ρ(t)) − f (s) − f ∇ (t)[ρ(t) − s] |≤ ε | ρ(t) − s |
for all s ∈ U.
A function f : T → R is left-dense continuous or ld-continuous on [a, b], denoted f ∈ Cld [a, b], provided
it is continuous at left-dense points in T and its right-sided limits exist (finite) at right-dense points in T. It is
known that if f is ld-continuous, then there is a function F(t) such that F ∇ (t) = f (t). In this case, we define
Z b
a
f (t)∇t = F(b) − F(a).
3 Green’s Function and Bounds
In this section, we construct the Green’s function for the BVP (1.1)-(1.2). We estimate bounds on the Green’s
function, and some lemmas which are needed in our main result are provided.
Let G1 (t, s) be the Green’s function of the BVP,
−y∆∇ (t) = 0, t ∈ [a, b],
y(a) = 0 = y(b),
which is given by
( (b−s)(t−a)
G1 (t, s) =
(b−a) ,
(b−t)(s−a)
(b−a) ,
t≤s
s≤t
for all t, s ∈ [a, b]. Then for 2 ≤ j ≤ n we can recursively define
G j (t, s) =
Z b
a
G j−1 (t, r)G1 (r, s)∇r, for all t, s ∈ [a, b].
(3.1)
4
J. Henderson, P. Murali, K.R. Prasad
As a result, G j (t, s) is the Green’s function for the BVP
j
(−1) j y(∆∇) (t) = 0, t ∈ [a, b],
i
i
y(∆∇) (a) = y(∆∇) (b) = 0, 0 ≤ i ≤ j − 1,
(3.2)
(3.3)
and G j (t, s) ≥ 0 for all t, s ∈ [a, b].
Let D = {v|v : C[a, b] → R}. For each 1 ≤ j ≤ n − 1, define the operator T j : D → D by
(T j v)(t) =
Z b
a
G j (t, s)v(s)∇s, t ∈ [a, b].
By the construction of T j and properties of G j (t, s), it is clear that
j
(−1) j (T j v)(∆∇) (t) = v(t), t ∈ [a, b],
i
i
(T j v)(∆∇) (a) = (T j v)(∆∇) (b) = 0, 0 ≤ i ≤ j − 1.
Hence, we see that the BVP (1.1)-(1.2) has a solution if and only if the following BVP has a solution
v∆∇ (t) + f (Tn−1 v(t), Tn−2 v(t), ..., T1 v(t), v(t)) = 0, t ∈ [a, b],
(3.4)
v(a) = 0 = v(b).
(3.5)
n−1
Indeed, if y is a solution of the BVP (1.1)-(1.2), then v(t) = y(∆∇) (t) is a solution of the BVP (3.4)-(3.5).
Conversely, if v is a solution of the BVP (3.4)-(3.5), then y(t) = Tn−1 v(t) is a solution of the BVP (1.1)-(1.2).
In fact, we have the representation
y(t) =
where
v(s) =
Z b
a
Z b
a
Gn−1 (t, s)v(s)∇s,
G1 (s, τ) f (Tn−1 v(τ), Tn−2 v(τ), ..., T1 v(τ), v(τ))∇τ.
It is also noted that a symmetric solution v of the BVP (3.4)-(3.5);
i.e. v(t) = v(b + a − t), t ∈ [a, b],
gives rise to a symmetric solution y of the BVP (1.1)-(1.2).
The following lemmas are needed for the main result.
£
¤
3b+a
Lemma 1. Let I = b+3a
. For (t, s) ∈ I × [a, b], we have
4 , 4
G j (t, s) ≥
where φ =
Rb
a
G1 (s, s)∇s > 0.
1 j−1
φ G1 (s, s), j ≥ 1
4j
(3.6)
Multiple Symmetric Positive Solutions for Two-Point Even Order Boundary Value Problems on Time Scales
5
Proof. The proof is by induction. First, for j = 1 the inequality (3.6) is obvious. Next, for fixed j, assuming
that (3.6) is true, from (3.1) we have for (t, s) ∈ I × [a, b],
G j+1 (t, s) =
≥
≥
Z b
a
Z b
a
G j (t, r)G1 (r, s)∇r for t, s ∈ [a, b]
G j (t, r)G1 (r, s)∇r for (t, s) ∈ I × [a, b]
Z b j−1
φ
=
a
4j
1
G1 (r, r) × G1 (s, s)∇r
4
φj
G1 (s, s).
4 j+1
Hence, by induction the proof is complete.
u
t
Lemma 2. For t, s ∈ [a, b], we have
G j (t, s) ≤ φ j−1 G1 (s, s), j ≥ 1,
where φ =
Rb
a
(3.7)
G1 (s, s)∇s > 0.
Proof. Again, the proof is by induction. For j = 1 the inequality (3.7) is obvious. Next, for fixed j, assume
that (3.7) is true, then from (3.1) we have
G j+1 (t, s) =
≤
Z b
a
Z b
a
j
G j (t, r)G1 (r, s)∇r
φ j−1 G1 (r, r) × G1 (s, s)∇r
= φ G1 (s, s).
Hence, by induction the proof is complete.
u
t
Lemma 3. For t, s ∈ [a, b], the Green’s function G j (t, s) satisfies the symmetric property,
G j (t, s) = G j (b + a − t, b + a − s).
(3.8)
Proof. By the definition of G j (t, s), (2 ≤ j ≤ n)
G j (t, s) =
Z b
a
G j−1 (t, r)G1 (r, s)∇r, for all t, s ∈ [a, b].
Clearly, we can see; G1 (t, s) = G1 (a+b−t, a+b−s). Now, the proof is by induction. For j = 2 the inequality
(3.8) is obvious. Next, assume that (3.8) is true, for fixed j ≥ 2, then from (3.1) we have
6
J. Henderson, P. Murali, K.R. Prasad
G j+1 (t, s) =
=
=
Z b
a
Z b
a
Z b
a
G j (t, r)G1 (r, s)∇r
G j (a + b − t, a + b − r)G1 (a + b − r, a + b − s)∇r
G j (a + b − t, r1 )G1 (r1 , a + b − s)∇r1
(by using a transformation r1 = a + b − r)
= G j+1 (a + b − t, a + b − s).
u
t
Lemma 4. For t ∈ [a, b], the operator T j satisfies the symmetric property
T j y(t) = T j y(b + a − t).
Proof. By definition of T j , and using the transformation s1 = b + a − s,
T j y(t) =
=
=
Z b
a
Z b
a
Z b
a
G j (t, s)v(s)∇s
G j (a + b − t, a + b − s)v(s)∇s (from Lemma 3 )
G j (a + b − t, s1 )v(s1 )∇s1
= T j y(b + a − t).
u
t
4 Multiple Symmetric Positive Solutions
In this section, we establish the existence of multiple symmetric positive solutions for the even order twopoint BVP (1.1)-(1.2), by using Avery generalization of the Leggett-Williams fixed point theorem. And then,
we establish 2m − 1 symmetric positive solutions for an arbitrary positive integer m.
Let B be a real Banach space with cone P. A map α : P → [0, ∞) is said to be a nonnegative continuous
concave functional on P if α is continuous and
α(λx + (1 − λ)y) ≥ λα(x) + (1 − λ)α(y),
for all x, y ∈ P and λ ∈ [0, 1]. Similarly, we say that a map β : P → [0, ∞) is said to be a nonnegative
continuous convex functional on P if β is continuous and
β(λx + (1 − λ)y) ≤ λβ(x) + (1 − λ)β(y),
Multiple Symmetric Positive Solutions for Two-Point Even Order Boundary Value Problems on Time Scales
7
for all x, y ∈ P and λ ∈ [0, 1]. Let γ, β, θ be nonnegative continuous convex functional on P and α, ψ be
nonnegative continuous concave functionals on P, then for nonnegative numbers h0 , a0 , b0 , d 0 and c0 , we
define the following convex sets
P(γ, c0 ) = {y ∈ P|γ(y) < c0 },
P(γ, α, a0 , c0 ) = {y ∈ P|a0 ≤ α(y), γ(y) ≤ c0 },
Q(γ, β, d 0 , c0 ) = {y ∈ P|β(y) ≤ d 0 , γ(y) ≤ c0 },
P(γ, θ, α, a0 , b0 , c0 ) = {y ∈ P|a0 ≤ α(y), θ(y) ≤ b0 , γ(y) ≤ c0 },
Q(γ, β, ψ, h0 , d 0 , c0 ) = {y ∈ P|h0 ≤ ψ(y), β(y) ≤ d 0 , γ(y) ≤ c0 }.
In obtaining multiple symmetric positive solutions of the BVP (1.1)-(1.2), the following so called Five
Functionals Fixed Point Theorem will be fundamental.
Theorem 1. [3] Let P be a cone in a real Banach space B. Suppose α and ψ are nonnegative continuous
concave functionals on P and γ, β and θ are nonnegative continuous convex functionals on P such that, for
some positive numbers c0 and k,
α(y) ≤ β(y) and k y k≤ kγ(y) f or all y ∈ P(γ, c0 ).
Suppose further that T : P(γ, c0 ) → P(γ, c0 ) is completely continuous and there exist constants h0 , d 0 , a0 , b0 ≥
0 with 0 < d 0 < a0 such that each of the following is satisfied.
(B1) {y ∈ P(γ, θ, α, a0 , b0 , c0 )|α(y) > a0 } 6= 0/ and
α(Ty) > a0 for y ∈ P(γ, θ, α, a0 , b0 , c0 ),
(B2) {y ∈ Q(γ, β, ψ, h0 , d 0 , c0 )|β(y) < d 0 } 6= 0/ and
β(Ty) < d 0 for y ∈ Q(γ, β, ψ, h0 , d 0 , c0 ),
(B3) α(Ty) > a0 provided y ∈ P(γ, α, a0 , c0 ) with θ(Ty) > b0 ,
(B4) β(Ty) < d 0 provided y ∈ Q(γ, β, d 0 , c0 ) with ψ(Ty) < h0 .
Then T has at least three fixed points y1 , y2 , y3 ∈ P(γ, c0 ) such that
β(y1 ) < d 0 , a < α(y2 ), and d 0 < β(y3 ) with α(y3 ) < a0 .
Let B = {v|v : C[a, b] → R} be the Banach space equipped with the norm
k v k= max |v(t)|.
t∈[a,b]
Define the cone P ⊂ B by
½
y ∈ B : v(t) ≥ 0 and v∆∇ (t) ≤ 0 on [a, b];
P=
v(t) = v(b + a − t) ∀ t ∈ [a, b]; and mint∈I v(t) ≥
¾
1
4
kvk
.
£
¤
a+7b
Now, let I1 = 7a+b
and define the nonnegative continuous concave functionals α, ψ and the nonneg8 , 8
ative continuous convex functionals β, θ, γ on P by
8
J. Henderson, P. Murali, K.R. Prasad
γ(v) =
max
a+3b
t∈[a, 3a+b
4 ]∪[ 4 ,b]
|v(t)|, ψ(v) = min |v(t)|, β(v) = max |v(t)|,
t∈I1
t∈I1
α(v) = min |v(t)|, θ(v) = max |v(t)|.
t∈I
t∈I
We observe that for any v ∈ P,
α(v) = min |v(t)| ≤ max |v(t)| = β(v),
t∈I
k v k≤ 4 min v(t) ≤ 4
t∈I
t∈I1
max
a+3b
t∈[a, 3a+b
4 ]∪[ 4 ,b]
|v(t)| = 4γ(v).
(4.1)
(4.2)
We are now ready to present the main result of this section. We denote
Z
L=
Z
s∈I
G1 (s, s)∇s and L1 =
s∈I1
G1 (s, s)∇s.
Theorem 2. Suppose there exist 0 < a0 < b0 < 4b0 ≤ c0 such that f satisfies the following conditions:
0
(A1) f (un−1 , un−2 , ..., u1 , u0 ) < aφ for all (|un−1 |, |un−2 |, ..., |u1 |, |u0 |) in
0 j−1
0
Π1j=n−1 [ a φ4 j+1L1 , 4c0 φ j ] × [ a4 , a0 ],
0
(A2) f (un−1 , un−2 , ..., u1 , u0 ) > 4bφ for all (|un−1 |, |un−2 |, ..., |u1 |, |u0 |) in
0 j−1
Π1j=n−1 [ a φ4 j L , 4c0 φ j ] × [b0 , 4b0 ],
0
(A3) f (un−1 , un−2 , ..., u1 , u0 ) < cφ for all (|un−1 |, |un−2 |, ..., |u1 |, |u0 |) in
Π1j=n−1 [0, 4c0 φ j ] × [0, c0 ].
Then the BVP (1.1)-(1.2) has at least three symmetric positive solutions.
Proof. Define the completely continuous operator T : P → B by
T v(t) =
Z b
a
G1 (t, s) f (Tn−1 v(s), Tn−2 v(s), ..., T1 v(s), v(s))∇s.
(4.3)
It is obvious that a fixed point of T is a solution of the BVP (3.4)-(3.5). We seek three fixed points
v1 , v2 , v3 ∈ P of T . First, we show that T : P → P. Let v ∈ P. Clearly, T v(t) ≥ 0 and T ∆∇ v(t) ≤ 0 for
t ∈ [a, b]. Further, since G j (t, s) = G j (b + a −t, b + a − s), we see that T j v(t) = T j v(b + a −t), 1 ≤ j ≤ n − 1,
for t ∈ [a, b]. Hence, it follows that T v(t) = T v(b + a −t), for t ∈ [a, b]. Also, noting that T v(a) = 0 = T v(b),
and we have
Z
min T v(t) = min
t∈I
t∈I
≥
Z
1 b
b
a
G1 (t, s) f (Tn−1 v(s), Tn−2 v(s), ..., T1 v(s), v(s))∇s
G1 (s, s) f (Tn−1 v(s), Tn−2 v(s), ..., T1 v(s), v(s))∇s
4 a
1
= k Tv k .
4
Thus, T : P → P. Next, for all v ∈ P, by the equations (4.1)-(4.2), respectively, we have α(v) ≤ β(v) and
k v k≤ 4γ(v). To show that T : P(γ, c0 ) → P(γ, c0 ), let v ∈ P(γ, c0 ). This implies k v k≤ 4c0 . Using Lemma 2,
we find for 1 ≤ j ≤ n − 1 and t ∈ [a, b],
Multiple Symmetric Positive Solutions for Two-Point Even Order Boundary Value Problems on Time Scales
T j v(t) =
Z b
a
≤ 4c0
G j (t, s)v(s)∇s
Z b
a
G j (t, s)∇s
0 j−1
≤ 4c φ
Z b
a
G1 (s, s)∇s
= 4c0 φ j .
We may now use condition (A3) to obtain
γ(T v) =
Z b
a+3b
a
t∈[a, 3a+b
4 ]∪[ 4 ,b]
c0
φ
= c0 .
≤
max
Z b
a
G1 (t, s) f (Tn−1 v(s), Tn−2 v(s), ..., T1 v(s), v(s))∇s
G1 (s, s)∇s
Therefore, T : P(γ, c0 ) → P(γ, c0 ).
We first verify that conditions (B1), (B2) of Theorem 1 are satisfied. It is obvious that
/
{v ∈ P(γ, θ, α, b0 , 4b0 , c0 )|α(v) > b0 } 6= 0,
and
{v ∈ Q(γ, β, ψ,
a0 0 0
/
, a , c )|β(v) < a0 } 6= 0.
4
0
Next, let v ∈ P(γ, θ, α, b0 , 4b0 , c0 ) or v ∈ Q(γ, β, ψ, a4 , a0 , c0 ).
Then, for 1 ≤ j ≤ n − 1,
T j v(t) =
Z b
a
G j (t, τ)v(τ)∇τ
0 j−1
≤ 4c φ
Z b
a
G1 (τ, τ)∇τ,
≤ 4c0 φ j−1 φ = 4c0 φ j
and for v ∈ P(γ, θ, α, b0 , 4b0 , c0 ),
Z
φ j−1 b
G1 (τ, τ)v(τ)∇τ
4j a
Z
φ j−1 b0
G1 (τ, τ)v(τ)∇τ
≥
4j
τ∈I
Z
φ j−1 b0
≥
G1 (τ, τ)∇τ
4j
τ∈I
φ j−1 b0 L
=
.
4j
T j v(t) ≥
9
10
J. Henderson, P. Murali, K.R. Prasad
0
and also for v ∈ Q(γ, β, ψ, a4 , a0 , c0 ),
Z
φ j−1 b
G1 (τ, τ)v(τ)∇τ
4j a
Z
φ j−1 a0
G1 (τ, τ)v(τ)∇τ
≥ j+1
4
τ∈I1
Z
φ j−1 a0
≥ j+1
G1 (τ, τ)∇τ
4
τ∈I1
φ j−1 a0 L1
=
.
4 j+1
T j v(t) ≥
Now, we may apply condition (A2) to get
α(T v) = min
Z b
t∈I
≥
Z
1 b
a
G1 (t, s) f (Tn−1 v(s), Tn−2 v(s), ..., T1 v(s), v(s))∇s
G1 (s, s) f (Tn−1 v(s), Tn−2 v(s), ..., T1 v(s), v(s))∇s
4 a
Z
b0 b
G1 (s, s)∇s
>
φ a
= b0 .
Clearly, by condition (A1) we obtain
β(T v) = max
≤
Z b
t∈I1
a0
a
max
G1 (t, s) f (Tn−1 v(s), Tn−2 v(s), ..., T1 v(s), v(s))∇s
Z b
G1 (t, s)∇s
φ t∈I1 a
Z
a0 b
≤
G1 (s, s)∇s = a0 .
φ a
To see that (B3) is satisfied, let v ∈ P(γ, α, b0 , c0 ) with θ(T v) > 4b0 . Using Lemma 3, we get
α(T v) = min
t∈I
≥
Z b
Z
1 b
a
G1 (t, s) f (Tn−1 v(s), Tn−2 v(s), ..., T1 v(s), v(s))∇s
G1 (s, s) f (Tn−1 v(s), Tn−2 v(s), ..., T1 v(s), v(s))∇s
4 a
Z b
1
≥ max
G1 (t, s) f (Tn−1 v(s), Tn−2 v(s), ..., T1 v(s), v(s))∇s
4 t∈[a,b] a
Z b
1
≥ max
G1 (t, s) f (Tn−1 v(s), Tn−2 v(s), ..., T1 v(s), v(s))∇s
4 t∈I a
1
= θ(T v) > b0 .
4
Multiple Symmetric Positive Solutions for Two-Point Even Order Boundary Value Problems on Time Scales
Finally, we show that (B4) holds. Let v ∈ Q(γ, β, a0 , c0 ) with ψ(T v) <
β(T v) = max
Z b
t∈I1
a
≤
t∈[a,b] a
Z b
=4
a
4.
11
In view of Lemma 3, we have
G1 (t, s) f (Tn−1 v(s), Tn−2 v(s), ..., T1 v(s), v(s))∇s
Z b
≤ max
a0
G1 (t, s) f (Tn−1 v(s), Tn−2 v(s), ..., T1 v(s), v(s))∇s
G1 (s, s) f (Tn−1 v(s), Tn−2 v(s), ..., T1 v(s), v(s))∇s
Z b
1
a
4
≤ 4 min
t∈I
≤ 4 min
t∈I1
G1 (s, s) f (Tn−1 v(s), Tn−2 v(s), ..., T1 v(s), v(s))∇s
Z b
a
Z b
a
G1 (t, s) f (Tn−1 v(s), Tn−2 v(s), ..., T1 v(s), v(s))∇s
G1 (t, s) f (Tn−1 v(s), Tn−2 v(s), ..., T1 v(s), v(s))∇s
= 4ψ(T v) < a0 .
We have proved that all the conditions of Theorem 1 are satisfied and so there exist at least three symmetric positive solutions v1 , v2 , v3 ∈ P(γ, c0 ) for the BVP (3.4)-(3.5). Therefore the BVP (1.1)-(1.2) has at
least three symmetric positive solutions y1 , y2 , y3 of the form
yi (t) = Tn−1 vi (t) =
Z b
a
Gn−1 (t, s)vi (s)∇s, i = 1, 2, 3.
This completes the proof of the theorem.
u
t
Now we prove the existence of 2m − 1 symmetric positive solutions for the BVP (1.1)-(1.2) by using
induction on m.
Theorem 3. Let m be an arbitrary positive integer. Assume that there exist numbers ai (1 ≤ i ≤ m) and
b j (1 ≤ j ≤ m − 1) with 0 < a1 < b1 < 4b1 < a2 < b2 < 4b2 < ... < am−1 < bm−1 < 4bm−1 < am such that
)
f (un−1 , un−2 , ..., u1 , u0 ) < aφi f or all (|un−1 |, |un−2 |, ..., |u1 |, |u0 |)
(4.4)
j−1
in Π1j=n−1 [ ai φ4 j+1L1 , 4am φ j ] × [ a4i , ai ], 1 ≤ i ≤ m,
)
f (un−1 , un−2 , ..., u1 , u0 ) > 4bφ i f or all (|un−1 |, |un−2 |, ..., |u1 |, |u0 |)
(4.5)
j−1
in Π1j=n−1 [ bl φ4 j L , 4bm−1 φ j ] × [bl , 4bl ], 1 ≤ l ≤ m − 1.
Then the BVP (1.1)-(1.2) has at least 2m − 1 symmetric positive solutions in Pam .
Proof. We use induction on m. First, for m = 1, we know from (4.4) that T : Pa1 → Pa1 , then, it follows
from Schauder fixed point theorem that the BVP (1.1)-(1.2) has at least one symmetric positive solution in
Pa1 . Next, we assume that this conclusion holds for m = k. In order to prove that this conclusion holds for
12
J. Henderson, P. Murali, K.R. Prasad
m = k + 1, we suppose that there exist numbers ai (1 ≤ i ≤ k + 1) and b j (1 ≤ j ≤ k) with 0 < a1 < b1 <
4b1 < a2 < b2 < 4b2 < ... < ak < bk < 4bk < ak+1 such that
)
f (un−1 , un−2 , ..., u1 , u0 ) < aφi for all (|un−1 |, |un−2 |, ..., |u1 |, |u0 |)
(4.6)
j−1
in Π1j=n−1 [ ai φ4 j+1L1 , 4ak+1 φ j ] × [ a4i , ai ], 1 ≤ i ≤ k + 1,
)
f (un−1 , un−2 , ..., u1 , u0 ) > 4bφ i for all (|un−1 |, |un−2 |, ..., |u1 |, |u0 |)
(4.7)
j−1
in Π1j=n−1 [ bl φ4 j L , 4bk φ j ] × [bl , 4bl ], 1 ≤ l ≤ k.
By assumption, the BVP (1.1)-(1.2) has at least 2k − 1 symmetric positive solutions ui (i = 1, 2, ..., 2k − 1)
in Pak . At the same time, it follows from Theorem 2, and (4.6) and (4.7) that the BVP (1.1)-(1.2) has at
least three symmetric positive solutions u, v and w in Pak+1 such that, k u k< ak , bk < mint∈I v(t), k w k>
ak , mint∈I w(t) < bk . Obviously, v and w are different from ui (i = 1, 2, ..., 2k − 1). Therefore, the BVP(1.1)(1.2) has at least 2k + 1 symmetric positive solutions in Pak+1 which shows that this conclusion also holds
for m = k + 1.
u
t
Acknowledgement
One of the authors (P. Murali) is thankful to CSIR, India for an SRF award.
References
[1] R. P. Agarwal, D. O’Regan, and P. J. Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer
Academic Publishers, Dordrecht, The Netherlands, 1999.
[2] D. R. Anderson, Eigenvalue intervals for even order Sturm-Liouville dynamic equation, Comm. Appl. Nonli. Anali., 12(2005),
no. 4, 1-13.
[3] R. I. Avery, A generalization of the Leggett-Williams xed point theorem, Math.Sci. Res. Hot-Line, 3(1999), 9-14.
[4] R. I. Avery and J. Henderson, Existence of three positive pseudo symmetric solutions for a one dimensional p-Laplacian, J.
Math. Anal. Appl., 10 (2004), no. 6, 529-539.
[5] R. I. Avery and J. Henderson, Existence of three positive pseudo symmetric solutions for a one dimensional discrete pLaplacian, (preprint).
[6] R. I. Avery and J. Henderson, Three symmetric positive solutions for a second order boundary vaue problem, Appl. Math.
Letters., 13 (2003), 1-7.
[7] R. I. Avery, J. M. Davis and J. Henderson, Three symmetric positive solutions for Lidstone problems by a generalization of
the Leggett- Williams theorem, Elec. J. Diff. Equ., 2000 (2000), no 40, 1-15.
[8] M. Bohner and A. C. Peterson, Dynamic Equations on Time scales, An Introduction with Applications, Birkhauser, Boston,
MA, (2001).
[9] M. Bohner and A. C. Peterson, Advances in Dynamic Equations on Time scales, , Birkhauser, Boston, (2003).
[10] J. M. Davis and J. Henderson, Triple positive symmetric solutions for a Lidstone boundary value problem, Diff. Equn. Dyn.
Systm., 7 (1999), 321-330.
[11] J. M. Davis, J. Henderson and P. J. Y. Wong, General Lidstone problem : multiplicity and symmetric of solutions, J. Math.
anal. Appl., 251 (2000), no. 2, 527-548.
[12] P. W. Eloe and J. Henderson, Positive solutions for (n-1,1) conjugate boundary value problems, Nonlinear Anal. 28(1997),
1669-1680.
Multiple Symmetric Positive Solutions for Two-Point Even Order Boundary Value Problems on Time Scales
13
[13] P. W. Eloe and J. Henderson, Positive solutions and nonlinear (k,n-k) conjugate eigenvalue problems, Diff. Equ. Dyna. Syst.,
6(1998), 309-317.
[14] P. W. Eloe, J. Henderson and Q. Sheng, Notes on Crossed symmetry solutions of the two-point boundary value problems on
time scales, J. Difference Equ. Appl., 9 (2003), no. 1, 29-48.
[15] L. H. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math.Soc.
120(1994), 743-748.
[16] D. Guo and V. Lakshmikantam, Nonlinear problems in Abstract Cones, Academic press, San Diego, 1988.
[17] J. Henderson, Multiple symmetric solution for discrete Lidstone boundary value problems, Dynam. Contin. Discrete Impuls.
Systems, 7 (2000), 577-585.
[18] J. Henderson and P. J. Y. Wong, Double symmetric solutions for discrete Lidstone boundary value problems, J. Difference
Equ. and Appl., 7(2001), 811-828.
[19] J. Henderson and H. B. Thompson, Multiple symmetric positive solutiions for a second order boundary value problem, Proc.
Amer. Math. Soc., 128 (2000), 2373-2379.
[20] V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan, Dynanic Systems on Measure Chains, Mathematics and its Applications, 370, Kluwer, Dordrecht, 1996.
[21] R. W. Leggett and L. R. Williams, Multiple positive fixed points of non-linear operations on ordered Banach spaces, Indiana
Univ. Math. J., 28(1979), 673-688.
[22] P. J. Y. Wong, Multiple symmetric solution for discrete Lidstone boundary value problems, J. Difference Equ. and Appl.,
8(2002), no. 9, 765-797.