Periodic boundary value problems for nonlinear first-order impulsive dynamic equations on time scales

By using the classical fixed point theorem for operators on cone, in this article, some results of one and two positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales are obtained. Two examples are given to illustrate the main results in this article.

Mathematics Subject Classification: 39A10; 34B15.

Let T be a time scale, i.e., T is a nonempty closed subset of R. Let 0, T be points in T, an interval (0, T)_T denoting time scales interval, that is, (0, T)_T: = (0, T) ⋂ T. Other types of intervals are defined similarly.

The theory of impulsive differential equations is emerging as an important area of investigation, since it is a lot richer than the corresponding theory of differential equations without impulse effects. Moreover, such equations may exhibit several real world phenomena in physics, biology, engineering, etc. (see [1-3]). At the same time, the boundary value problems for impulsive differential equations and impulsive difference equations have received much attention [4-18]. On the other hand, recently, the theory of dynamic equations on time scales has become a new important branch (see, for example, [19-21]). Naturally, some authors have focused their attention on the boundary value problems of impulsive dynamic equations on time scales [22-36]. However, to the best of our knowledge, few papers concerning PBVPs of impulsive dynamic equations on time scales with semi-position condition.

In this article, we are concerned with the existence of positive solutions for the following PBVPs of impulsive dynamic equations on time scales with semi-position condition

where T is an arbitrary time scale, T > 0 is fixed, 0, T ∈ T, f ∈ C (J × [0, ∞), (-∞, ∞)), I_k ∈ C([0, ∞), [0, ∞)), t_k ∈ (0, T)_T, 0 < t₁ < ⋯ < t_m < T, and for each k = 1, 2,..., m, and represent the right and left limits of x(t) at t = t_k. We always assume the following hypothesis holds (semi-position condition):

(H) There exists a positive number M such that

By using a fixed point theorem for operators on cone [37], some existence criteria of positive solution to the problem (1.1) are established. We note that for the case T = R and I_k(x) ≡ 0, k = 1, 2,..., m, the problem (1.1) reduces to the problem studied by [38] and for the case I_k(x) ≡ 0, k = 1, 2,..., m, the problem (1.1) reduces to the problem (in the one-dimension case) studied by [39].

In the remainder of this section, we state the following fixed point theorem [37].

Theorem 1.1. Let X be a Banach space and K ⊂ X be a cone in X. Assume Ω₁, Ω₂ are bounded open subsets of X with and Φ: is a completely continuous operator. If

(i) There exists u₀ ∈ K\{0} such that u - Φu ≠ λu₀, u ∈ K ⋂ ∂ Ω₂, λ≥ 0; Φu ≠ τu, u ∈ K ⋂ ∂Ω₁, τ ≥ 1, or

(ii) There exists u₀ ∈ K\{0} such that u - Φu ≠ λu₀, u ∈ K ⋂ ∂Ω₁, λ≥ 0; Φu ≠ τu, u ∈ K ⋂ ∂Ω₂, τ ≥ 1.

Then Φ has at least one fixed point in .

Throughout the rest of this article, we always assume that the points of impulse t_k are right-dense for each k = 1, 2,...,m.

We define

where x_k is the restriction of x to J_k = (t_k, t_k+1]_T ⊂ (0, σ(T)]_T, k = 1, 2,..., m and J₀ = [0, t₁]_T, t_{m +1} = σ(T).

Let

with the norm , then X is a Banach space.

Lemma 2.1. Suppose M > 0 and h: [0, T]_T → R is rd-continuous, then x is a solution of

where

if and only if x is a solution of the boundary value problem

Proof. Since the proof similar to that of [34, Lemma 3.1], we omit it here.

Lemma 2.2. Let G(t, s) be defined as in Lemma 2.1, then

Proof. It is obviously, so we omit it here.

Remark 2.1. Let G(t, s) be defined as in Lemma 2.1, then .

For u ∈ X, we consider the following problem:

It follows from Lemma 2.1 that the problem (2.1) has a unique solution:

where h_u(s) = Mu(σ(s)) - f(s, u(σ(s))), s ∈ [0, T]_T.

We define an operator Φ: X → X by

It is obvious that fixed points of Φ are solutions of the problem (1.1).

Lemma 2.3. Φ: X → X is completely continuous.

Proof. The proof is divided into three steps.

Step 1: To show that Φ: X → X is continuous.

Let be a sequence such that u_n → u (n → ∞) in X. Since f(t, u) and I_k(u) are continuous in x, we have

So

which leads to ||Φu_n - Φu|| → 0 (n → ∞). That is, Φ: X → X is continuous.

Step 2: To show that Φ maps bounded sets into bounded sets in X.

Let B ⊂ X be a bounded set, that is, ∃ r > 0 such that ∀ u ∈ B we have ||u|| ≤ r. Then, for any u ∈ B, in virtue of the continuities of f(t, u) and I_k(u), there exist c > 0, c_k > 0 such that

We get

Then we can conclude that Φu is bounded uniformly, and so Φ(B) is a bounded set.

Step 3: To show that Φ maps bounded sets into equicontinuous sets of X.

Let t₁, t₂ ∈ (t_k, t_k+1]_T ⋂ [0, σ(T)]_T, u ∈ B, then

The right-hand side tends to uniformly zero as |t₁ - t₂| → 0.

Consequently, Steps 1-3 together with the Arzela-Ascoli Theorem shows that Φ: X → X is completely continuous.

Let

where . It is not difficult to verify that K is a cone in X.

From condition (H) and Lemma 2.2, it is easy to obtain following result:

Lemma 2.4. Φ maps K into K.

For convenience, we denote

and

Now we state our main results.

Theorem 3.1. Suppose that

(H₁) f₀ > 0, f^∞ < 0, I₀ = 0 for any k; or

(H₂) f_∞ > 0, f⁰ < 0, I_∞ = 0 for any k.

Then the problem (1.1) has at least one positive solutions.

Proof. Firstly, we assume (H₁) holds. Then there exist ε > 0 and β > α > 0 such that

and

Let Ω₁ = {u ∈ X: ||u|| < r₁}, where r₁ = α. Then u ∈ K ⋂ ∂Ω₁, 0 < δα = δ ||u|| ≤ u(t) ≤ α, in view of (3.1) and (3.2) we have

which yields ||Φ(u)|| < ||u||.

Therefore

On the other hand, let Ω₂ = {u ∈ X: ||u|| < r₂}, where .

Choose u₀ = 1, then u₀ ∈ K\{0}. We assert that

Suppose on the contrary that there exist and such that

Let , then , we have from (3.3) that

Therefore,

which is a contradiction.

It follows from (3.4), (3.5) and Theorem 1.1 that Φ has a fixed point , and u* is a desired positive solution of the problem (1.1).

Next, suppose that (H₂) holds. Then we can choose ε' > 0 and β' > α' > 0 such that

and

Let Ω₃ = {u ∈ X: ||u|| < r₃}, where r₃ = α'. Then for any u ∈ K ⋂ ∂Ω₃, 0 < δ ||u|| ≤ u(t) ≤ ||u|| = α'.

It is similar to the proof of (3.5), we have

Let Ω₄ = {u ∈ X: ||u|| < r₄}, where . Then for any u ∈ K ⋂ ∂Ω₄, u(t) ≥ δ ||u|| = δr₄ = β', by (3.6) and (3.7), it is easy to obtain

It follows from (3.9), (3.10) and Theorem 1.1 that Φ has a fixed point , and u* is a desired positive solution of the problem (1.1).

Theorem 3.2. Suppose that

(H₃) f⁰ < 0, f^∞ < 0;

(H₄) there exists ρ > 0 such that

Then the problem (1.1) has at least two positive solutions.

Proof. By (H₃), from the proof of Theorem 3.1, we should know that there exist β" > ρ > α" > 0 such that

where Ω₅ = {u ∈ X: ||u|| < r₅}, Ω₆ = {u ∈ X: ||u|| < r₆},

By (3.11) of (H₄), we can choose ε > 0 such that

Let Ω₇ = {u ∈ X: ||u|| < ρ}, for any u ∈ K ⋂ ∂Ω₇, δρ = δ ||u|| ≤ u(t) ≤ ||u|| = ρ, from (3.12) and (3.15), it is similar to the proof of (3.4), we have

By Theorem 1.1, we conclude that Φ has two fixed points and , and u** and u*** are two positive solution of the problem (1.1).

Similar to Theorem 3.2, we have:

Theorem 3.3. Suppose that

(H₄) f₀ > 0, f_∞ > 0, I₀ = 0, I_∞ = 0;

(H₅) there exists ρ > 0 such that

Then the problem (1.1) has at least two positive solutions.

Example 4.1. Let T = [0, 1] ∪ [2,3]. We consider the following problem on T

where T = 3, f(t, x) = x - (t + 1)x², and I(x) = x²

Let M = 1, then, it is easy to see that

and

Therefore, by Theorem 3.1, it follows that the problem (4.1) has at least one positive solution.

Example 4.2. Let T = [0, 1] ∪ [2,3]. We consider the following problem on T

where , and I(x) = x²e^-x.

Choose M = 1, ρ = 4e², then , it is easy to see that

and

Therefore, together with Theorem 3.3, it follows that the problem (4.2) has at least two positive solutions.

The author declares that they have no competing interests.

The author thankful to the anonymous referee for his/her helpful suggestions for the improvement of this article. This work is supported by the Excellent Young Teacher Training Program of Lanzhou University of Technology (Q200907)

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