EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR THE SYSTEMS OF HIGHER ORDER BOUNDARY VALUE PROBLEMS ON TIME SCALES

Abstract

This paper is concerned with boundary value problems for systems of n-th order dynamic equations on time scales. Under the suitable conditions, the existence and multiplicity of positive solutions are established by using abstract fixed-point theorems.

1. Introduction

Let be a time scale with Given an interval J of ℝ, we will use the interval notation

In this paper we are concerned with the existence and multiplicity of positive solutions for dynamic equation on time scales

satisfying the boundary conditions

where ƒ(t, 0) ≡ 0, g(t, 0) ≡ 0.

Recently, existence and multiplicity of solutions for boundary value problems of dynamic equations have been of great interest in mathematics and its applications to engineering sciences. To our knowledge, most existing results on this topic are concerned with the single equation and simple boundary conditions.

It should be pointed out that Eloe and Henderson [6] discussed the boundary value problem as follows

By using a Krasnosel’skii fixed point theorem, the existence of solutions are obtained in the case when, either ƒ is superlinear, or ƒ is sublinear. Yang and Sun [15] considered the boundary value problem of the system of differential equations

By appealing to the degree theory, the existence of solutions are established. Note, that, there is only one differential equation in [4] and BVP in [15] contains the system of second order differential equations.

The arguments for establishing the existence of solutions of the BVP (1)-(2) involve properties of Green’s function that play a key role in defining some cones. A fixed point theorem due to Krasnosel’skii [10] is applied to yield the existence of positive solutions of the BVP (1)-(2). Another fixed point theorem about multiplicity is applied to obtain the multiplicity of positive solutions of BVP (1)-(2).

The rest of this paper is organized as follows. In Section 2, we shall provide some properties of certain Green’s functions and preliminaries which are needed later. For the sake of convenience, we also state Krasnosel’skii fixed point theorem in a cone. In Section 3, we establish the existence and multiplicity of positive solutions of the BVP (1)-(2). In Section 4, some examples are given to illustrate our main results.

 

2. Preliminaries

In this section, we will give some lemmas which are useful in proving our main results.

To obtain solutions of the BVP (1)-(2), we let G(t, s) be the Green’s function for the boundary value problem

Using the Cauchy function concept G(t, s) is given by

Lemma 1. For we have

Proof. For t ≤ s, we have

Similarly, for σ(s) ≤ t and σi(s) ≤ t, i = 1, · · · , n − 1, we have

Thus, we have

□

Lemma 2. Let we have

Proof. The Green’s function for the BVP (3)-(4) is given in (5) shows that

For t ≤ s, t ∈ I and σi−1(a) ≤ σn−2(a), i = 1, 2, · · · , n − 1, we have

For σ(s) ≤ t, t ∈ I and σi−1(a) ≤ σn−2(a), i = 1, 2, · · · , n − 1, we have

Therefore

□

We note that a pair (u(t), v(t)) is a solution of the BVP (1)-(2) if and only if

Assume throughout that is such that

both exist and satisfy

Next, let be defined by

Finally, we define

and let

For our construction, let with supremum norm

Then (E, ∥·∥) is a Banach space. Define

It is obivious that P is a positive cone in E. Define an integral operator T : P → E by

Lemma 3. If the operator T is defined as (10), then T : P → P is completely continuous.

Proof. From the continuity of ƒ and g, and (8) that, for u ∈ P, Tu(t) ≥ 0 on Also, for u ∈ P, we have from (6) that

so that

Next, if u ∈ P, we have from (7), (9), and (10) that

Therefore T : P → P. Since G(t, s), ƒ(t, u) and g(t, u) are continuous, it is easily known that T : P → P is completely continuous. The proof is complete. □

From above arguments, we know that the existence of positive solutions of (1)-(2) can be transferred to the existence of positive fixed points of the operator T.

Lemma 4 ([3,4,10]). Let (E, ∥ · ∥) be a Banach space, and let P ⊂ E be a cone in E. Assume that Ω1 and Ω2 are open bounded subsets of E such that 0 ∈ Ω1. If

is a completely continuous operator such that either

then T has a fixed point in

Lemma 5 ([3, 4, 10]). Let (E, ∥ · ∥) be a Banach space, and let P ⊂ E be a cone in E. Assume that Ω1, Ω2 and Ω3 are open bounded subsets of E such that If

is a completely continuous operator such that:

then T has at least two fixed points and furthermore

 

3. Main Results

First we give the following assumptions:

(A1)

(A2)

(A3)

(A4)

(A5) ƒ(t, u), g(t, u) are increasing functions with respect to u and, there is a number N > 0, such that

Theorem 1. If (A1) and (A2) are satisfied, then (1)-(2) has at least one positive solution satisfying u(t) > 0, v(t) > 0.

Proof. From (A1) there is a number H1 ∈ (0; 1) such that for each one has

where η > 0 satisfies

For every u ∈ P and ∥u∥ = H1/2, note that

thus

So, ∥Tu∥ ≤ ∥u∥. If we set

then

On the other hand, from (A2) there exist four positive numbers μ, μ′, C1 and C2 such that

where μ and μ′ satisfy

For u ∈ P, we have

where

Therefore

from which it follows that ∥Tu∥ ≥ Tu(τ) ≥ ∥u∥ as ∥u∥ → ∞.

Let Ω2 = {u ∈ E : ∥u∥ < H2}. Then for u ∈ P and ∥u∥ = H2 > 0 sufficient by large, we have

Thus, from (11), (12) and Lemma 4, we know that the operator T has a fixed point in The proof is complete. □

Theorem 2. If (A3) and (A4) are satisfied, then (1)-(2) has at least one positive solution satisfying u(t) > 0, v(t) > 0.

Proof. From (A3) there is a number such that for each one has

where λ > and λ′ satisfy

From g(t, 0) ≡ 0 and the continuity of g(t, u), we know that there exists a number small enough such that

For every u ∈ P and ∥u∥ = H3, note that

thus

So, ∥Tu∥ ≥ ∥u∥. If we set

then

On the other hand, we know from (A4) that there exist three positive numbers η′, C4, and C5 such that for every

where

Thus we have

where

from which it follows that Tu(t) ≤ ∥u∥ as ∥u∥ → ∞. Let Ω4 = {u ∈ E :∥ u ∥< H4}. For each u ∈ P and ∥u∥ = H4 > 0 large enough, we have

From (13), (14) and Lemma 4, we know that the operator T has a fixed point in The proof is complete. □

Theorem 3. If (A2), (A3) and (A5) are satisfied, then (1)-(2) has at least two distinct positive solutions satisfying ui(t) > 0, vi(t) > 0 (i = 1, 2).

Proof. Note that Then from (A5), for every we have

Thus

And from (A2) and (A3) we have

We can choose H2, H3 and N such that H3 ≤ N ≤ H2 and (15)-(17) are satisfied. From Lemma 5, T has at least two fixed points in respectively. The proof is complete. □

 

4. Examples

Some examples are given to illustrate our main results.

Example 1. Consider the following dynamic equations

satisfying the boundary conditions

where f(t, v) = v3, g(t, u) = u2, then conditions of Theorem 1 are satisfied. From Theorem 1, the BVP (18)-(19) has at least one positive solution.

Example 2. Consider the following dynamic equations

satisfying the boundary conditions

where f(t, v) = v2/3, g(t, u) = u3/4, then conditions of Theorem 2 are satisfied. From Theorem 2, the BVP (20)-(21) has at least one positive solution.

Example 3. Consider the following system of boundary value problems

where

We can choose N = 100, then conditions of Theorem 3 are satisfied. From Theorem 3, the BVP (22)-(23) has at least two positive solutions.