Multiple positive solutions for first-order impulsive integral boundary value problems on time scales

In this paper, we first present a class of first-order nonlinear impulsive integral boundary value problems on time scales. Then, using the well-known Guo-Krasnoselskii fixed point theorem and Legget-Williams fixed point theorem, some criteria for the existence of at least one, two, and three positive solutions are established for the problem under consideration, respectively. Finally, examples are presented to illustrate the main results.

MSC: 34B10; 34B37; 34N05.

In fact, continuous and discrete systems are very important in implementing and applications. It is well known that the theory of time scales has received a lot of attention, which was introduced by Stefan Hilger in order to unify continuous and discrete analyses. Therefore, it is meaningful to study dynamic systems on time scales, which can unify differential and difference systems.

In recent years, a great deal of work has been done in the study of the existence of solutions for boundary value problems on time scales. For the background and results, we refer the reader to some recent contributions [1-5] and references therein. At the same time, boundary value problems for impulsive differential equations and impulsive difference equations have received much attention [6-12], since such equations may exhibit several real-world phenomena in physics, biology, engineering, etc. see [13-15] and the references therein.

In paper [16], Sun studied the first-order boundary value problem on time scales

where 0 < β < 1. By means of the twin fixed point theorem due to Avery and Henderson, some existence criteria for at least two positive solutions were established.

Tian and Ge [17] studied the first-order three-point boundary value problem on time scales

Using several fixed point theorems, the existence of at least one positive solution and multiple positive solutions is obtained.

However, except BVP of differential and difference equations, that is, for particular time scales ( or ), there are few papers dealing with multi-point boundary value problems more than three-point for first-order systems on time scales. In addition, problems with integral boundary conditions arise naturally in thermal conduction problems [18], semiconductor problems [19], hydrodynamic problems [20]. In continuous case, since integral boundary value problems include two-point, three-point,..., n-point boundary value problems, such boundary value problems for continuous systems have received more and more attention and many results have worked out during the past ten years, see Refs. [21-27] for more details. To the best of authors' knowledge, up to the present, there is no paper concerning the boundary value problem with integral boundary conditions on time scales. This paper is to fill the gap in the literature.

In this paper, we are concerned with the following first-order nonlinear impulsive integral boundary value problem on time scales:

where is a time scale which is a nonempty closed subset of ℝ with the topology and ordering inherited from ℝ, 0, and T are points in , an interval which has finite right-scattered points, , and p is regressive, ℝ⁺), I_i(1 ≤ i ≤ m) ∈ C([0, +∞), [0, +∞)), g is a nonnegative integrable function on and , e_p(0,σ(T)) is the exponential function on time scale , which will be introduced in the next section, , 0 < t₁ < · · · < t_m < T, and for each and represent the right and left limits of x(t) at .

Remark 1.1. Let denote the set of right-scattered points in interval , 0 ≤ θ₁ < · · · < θ_q ≤ T, σ(θ₀) = 0, θ_q+1 = T. By some basic concepts and time scale calculus formulae in the book by Bohner and Peterson [28], we have

The main purpose of this paper is to establish some sufficient conditions for the existence of at least one, two, or three positive solutions for BVP (1.3) using Guo-Krasnoselskii and Legget-Williams fixed point theorem, respectively.

For convenience, we introduce the following notation:

where i = 1, 2,..., m.

This paper is organized as follows. In Section 2, some basic definitions and lemmas on time scales are introduced without proofs. In Section 3, some useful lemmas are established. In particular, Green's function for BVP (1.3) is established. We prove the main results in Sections 4-6.

In this section, we shall first recall some basic definitions, lemmas that are used in what follows. For the details of the calculus on time scales, we refer to books by Bohner and Peterson [28,29].

Definition 2.1. [28]A time scale is an arbitrary nonempty closed subset of the real set ℝ with the topology and ordering inherited from ℝ. The forward and backward jump operators and the graininess are defined, respectively, by

In this definition, we put (i.e., σ(t) = t if has a maximum t) and (i.e., ρ(t) = t if has a minimum t). The point is called left-dense, left-scattered, right-dense, or right-scattered if ρ(t) = t, ρ(t) < t, σ(t) = t, or σ(t) > t, respectively. Points that are right-dense and left-dense at the same time are called dense. If has a left-scattered maximum m₁, defined ; otherwise, set . If has a right-scattered minimum m₂, defined , otherwise, set .

Definition 2.2. [28]A function is rd continuous provided it is continuous at each right-dense point in and has a left-sided limit at each left-dense point in . The set of rd-continuous functions will be denoted by .

Definition 2.3. [28]If is a function and , then the delta derivative of f at the point t is defined to be the number f^Δ(t) (provided it exists) with the property that for each ε > 0 there is a neighborhood U of t such that

Definition 2.4. [28]For a function (the range ℝ of f may be actually replaced by Banach space), the (delta) derivative is defined by

if f is continuous at t and t is right-scattered. If t is not right-scattered, then the derivative is defined by

provided this limit exists.

Definition 2.5. [28]If F^Δ(t) = f(t), then we define the delta integral by

Definition 2.6. [28]A function is said to be regressive provided 1 + μ(t)p(t) ≠ 0 for all , where μ(t) = σ(t) - t is the graininess function. The set of all regressive rd-continuous functions is denoted by , while the set is given by for all. Let . The exponential function is defined by

where ξ_h(z) is the so-called cylinder transformation.

Lemma 2.1. [28]Let p, . Then

(1) e₀(t, s) ≡ 1 and e_p(t, t) ≡ 1;

(2) e_p(σ(t), s) = (1 + μ(t)p(t))e_p(t, s);

(3) , where ;

(4) e_p(t, s)e_p(s, r) = e_p(t, r),

(5) .

Lemma 2.2. [28]Assume that are delta differentiable at . Then

Lemma 2.3. [28]Let , , and assume that is continuous at (t, t), where with t > a. Also, assume that f^Δ(t, ·) is rd-continuous on [a, σ(t)]. Suppose that for each ε > 0 there exists a neighborhood U of t, independent of τ ∈ [a, σ(t)], such that

where f^Δ denotes the derivative of f with respect to the first variable. Then

(1) ;

(2) .

In this section, we first introduce some background definitions, fixed point theorems in Banach space, then present basic lemmas that are very crucial in the proof of the main results.

We define is a piecewise continuous map with first-class discontinuous points in and at each discontinuous point it is continuous on the left} with the norm , then PC is a Banach Space.

Definition 3.1. A function x is said to be a positive solution of problem (1.3) if x ∈ PC satisfying problem (1.3) and x(t) > 0 for all .

Definition 3.2. Let X be a real Banach space, the nonempty set K ⊂ X is called a cone of X, if it satisfies the following conditions.

(1) x ∈ K and λ ≥ 0 implies λx ∈ K;

(2) x ∈ K and -x ∈ K implies x = 0.

Every cone K ⊂ X induces an ordering in X, which is given by x ≤ y if and only if y - x ∈ K.

Definition 3.3. An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.

Lemma 3.1. (Guo-Krasnoselskii [30]) Let X be a Banach space and K ⊂ X be a cone in X. Assume that Ω₁, Ω₂ are bounded open subsets of X with and is a completely continuous operator such that, either

(1) ||Φx|| ≤ ||x||, x ∈ K ∩ ∂Ω₁, and ||Φx|| ≥ ||x||, x ∈ K ∩ ∂Ω₂; or

(2) ||Φx|| ≥ ||x||, x ∈ K ∩ ∂Ω₁, and ||Φx|| ≤ ||x||, x ∈ K ∩ ∂Ω₂.

Then Φ has at least one fixed point in .

Lemma 3.2. Suppose , ν_i ∈ ℝ, then x is a solution of

where

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

Proof. Assume that x(t) is a solution of (3.2). By the first equation in (3.2), we have

If t ∈ [0, t₁], integrating (3.3) from 0 to t, we get

while t → t₁, we have

then

Now, let t ∈ (t₁, t₂], integrating (3.3) from t₁ to t, we obtain

For t ∈ (t_k, t_k+1], repeating the above process, we can get

that is

It follows from that

where . Then

This means that if x is a solution of (3.2) then x satisfies (3.1).

On the other hand, if x satisfies (3.1), we have

Then

where

Notice that

Similarly,

Hence, we get from (3.5) that

that is

Finally, we can obtain from (3.1) that

and

So the proof of this lemma is completed.

Lemma 3.3. Let G(t, s) be defined the same as that in Lemma 3.2, then the following properties hold.

(1) G(t, s) > 0 for all ;

(2) A ≤ G(t, s) ≤ B for all , where

Proof. Since , then it is clear that (1) holds. Now we will show that (2) holds.

Hence, the left-hand side of (2) holds. And it is easy to show that the right-hand side of (2) also holds. The proof is complete. ■

Define an operator Φ : PC → PC by

By Lemma 3.2, the fixed points of Φ are solutions of problem (1.3).

Lemma 3.4. The operator Φ : PC → PC is completely continuous.

Proof. The first step we will show that Φ : PC → PC is continuous. Let be a sequence such that in PC. Then

Since f(t, x) and I_i(x)(1 ≤ i ≤ m) are continuous in x, we have |(Φx_n)(t) - (Φx)(t)| → 0, which leads to ||Φx_n - Φx||_PC → 0, as n → ∞. That is, Φ : PC → PC is continuous.

Next, we will show that Φ : PC → PC is a compact operator by two steps.

Let U ⊂ PC be a bounded set.

Firstly, we will show that {Φx : x ∈ U}is bounded. For any x ∈ U, we have

In virtue of the continuity of f(t, x) and I_i(x)(1 ≤ i ≤ m), we can conclude that {Φx : x ∈ U} is bounded from above inequality.

Secondly, we will show that {Φx : x ∈ U} is the set of equicontinuous functions. For any x, y ∈ U, then

In virtue of the continuity of f(t, x) and I_i(x)(1 ≤ i ≤ m), the right-hand side tends to zero uniformly as |x - y| → 0. Consequently, {Φx : x ∈ U} is the set of equicontinuous functions.

By Arzela-Ascoli theorem on time scales [31], {Φx : x ∈ U} is a relatively compact set. So Φ maps a bounded set into a relatively compact set, and Φ is a compact operator.

From above three steps, it is easy to see that Φ : PC → PC is completely continuous. The proof is complete. ■

Let , where . It is not difficult to verify that K is a cone in PC.

Lemma 3.5. Φ maps K into K.

Proof. Obviously, Φ(K) ⊂ PC. ∀x ∈ K, we have

which implies

Therefore,

Hence, Φ(K) ⊂ K. The proof is complete. ■

In this section, we will state and prove our main result about the existence of at least one positive solution of problem (1.3).

Theorem 4.1. Assume that one of the following conditions is satisfied:

(H₁) max f₀ = 0, min f_∞ = ∞, and I_i0 = 0, i = 1, 2,..., m; or

(H₂) max f_∞ = 0, min f₀ = ∞, and I_i∞ = 0, i = 1, 2,..., m.

Then, problem (1.3) has at least one positive solution.

Proof. Firstly, we assume that (H₁) holds. In this case, since max f₀ = 0 and I_i0 = 0, i = 1, 2,..., m, for ε ≤ (Bσ(T) + Bm)^-1, there exists a positive constant r₁ such that

In view of min f_∞ = ∞, we have that for M ≥ (Aσ(T)δ)^-1, there exists a constant such that

Let Ω_i = {x ∈ PC : ||x|| < r_i}, i = 1, 2.

On the one hand, if x ∈ K ∩ ∂Ω₁, we have

which yields

On the other hand, if x ∈ K ∩ ∂Ω₂, we have

which implies

Therefore, by (4.1), (4.2), and Lemma 3.1, it follows that Φ has a fixed point in .

Next, we assume that (H₂) holds. In this case, since max f_∞ = 0 and I_i∞ = 0, i = 1, 2,..., m, for ε' ≤ (Bσ(T) + Bm)^-1, there exists a positive constant r₃ such that

In view of min f_∞ = ∞, we have that for M' ≥ (Aσ(T)δ)^-1, there exists a positive constant r₄ < δr₃ such that

Let Ω_i = {x ∈ PC : ||x|| < r_i}, i = 3, 4.

On the one hand, if x ∈ K ∩ ∂Ω₃, we have

which yields

On the other hand, if x ∈ K ∩ ∂Ω₄, we have

which implies

Hence, from (4.3) and (4.4) and Lemma 3.1, we conclude that Φ has a fixed point in , that is, problem (1.3) has at least one positive solution. The proof is complete. ■

In this section, we will state and prove our main results about the existence of at least two positive solutions to problem (1.3).

Theorem 5.1. Assume that the following conditions hold.

(H₃) min f₀ = +∞, min f_∞ = +∞.

(H₄) There exists a positive constant R such that for all 0 < x ≤ R.

(H₅) , x ∈ (0, ∞), i = 1, 2,..., m.

Then, problem (1.3) has at least two positive solutions.

Proof. Let Ω_R = {x ∈ PC : ||x|| < R}. From (H₄) and (H₅), for x ∈ K ∩ ∂Ω_R, we get

So

Since min f₀ = +∞, for M ≥ (Aσ(T)δ)^-1, there exists a positive constant R₁ < δ_R such that

Let . For any , we have

Hence,

Similarly, since min f_∞ = +∞, for M' ≥ (Aσ(T)δ)^-1, there exists a positive constant such that

Let . For any , we have

Hence,

Equations 5.1 and 5.2 imply that Φ has at least one fixed point in , which is a positive solution of problem (1.3). Besides, (5.1) and (5.3) imply that Φ has at least one fixed point in , which is a positive solution of problem (1.3). Therefore, problem (1.3) has at least two positive solutions x₁ and x₂ satisfying 0 < R₁ ≤ ||x₁|| < R < ||x₂|| ≤ R₂. The proof is complete. ■

Theorem 5.2. Assume that the following conditions hold.

(H₆) max f₀ = 0, max f_∞ = 0, I_i0 = 0, I_i∞ = 0, i = 1, 2,..., m.

(H₇) There exists a positive constant r such that for all 0 < x ≤ r.

Then problem (1.3) has at least two positive solutions.

Proof. Let Ω_r = {x ∈ PC : ||x|| < r}. From (H₇), for x ∈ K ∩ ∂Ω_r, we get

So

Since max f₀ = 0 and I_i0 = 0, i = 1, 2,..., m, for ε ≤ (Bσ(T) + Bm)^-1, there exists a positive constant r₁ < δ_r such that

Let . For any , we have

Hence,

Similarly, since max f_∞ = 0 and I_i∞ = 0, i = 1, 2,..., m, for ε' ≤ (Bσ(T) + Bm)^-1, there exists a positive constant such that

Let . For any , we have

Hence,

Equations 5.4 and 5.5 imply that Φ has at least one fixed point in , which is a positive solution of problem (1.3). Besides, (5.4) and (5.6) imply that Φ has at least one fixed point in , which is a positive solution of problem (1.3). Therefore, problem (1.3) has at least two positive solutions x₁ and x₂ satisfying 0 < r₁ ≤ ||x₁|| < r < ||x₂|| ≤ r₂. The proof is complete. ■

Similar to Theorems 5.1 and 5.2, one can easily obtain the following corollary:

Corollary 5.1. Assume that (H₇) and the following conditions hold.

(H₈) max f₀ = 0, max f_∞ = 0, I_i0 = 0, i = 1, 2,..., m.

(H₉) There exists a positive constant d such that for all x ≥ d, i = 1, 2,..., m.

Then, problem (1.3) has at least two positive solutions.

In this section, we will state and prove our multiplicity result of positive solutions to problem (1.3) via Legget-Williams fixed point theorem. For readers' convenience, we first illustrate Legget-Williams fixed point theorem.

Let be a real Banach space with cone K. A map α : K → [0, +∞) is said to be a continuous concave functional on K if α is continuous and

for all x, y ∈ K and t ∈ [0, 1]. Let a, b be two numbers such that 0 < a < b and α be a nonnegative continuous concave functional on K. We define the following convex sets:

Lemma 6.1. (Legget-Williams fixed point theorem [32]). Let be completely continuous and α be a nonnegative continuous concave functional on K such that α(x) ≤ ||x|| for all x ∈ K_c. Suppose that there exist 0 < d < a < b ≤ c such that

(1) {x ∈ K(α, a, b) : α(x) > a} ≠ ∅, and α(Φ(x)) > a for all x ∈ K(α, a, b);

(2) ||Φx|| < d for all ||x|| ≤ d;

(3) α(Φ(x)) > a for all x ∈ K(α, a, c) with ||Φ(x)|| > b.

Then, Φ has at least three fixed points x₁, x₂, x₃ in satisfying ||x₁|| < d, a < α(x₂), ||x₃|| > d, and α(x₃) < a.

Theorem 6.1. Assume that there exist numbers d, a, and c with such that

Then, problem (1.3) has at least three positive solutions.

Proof. For x ∈ K, we define

It is easy to verify that α is a nonnegative continuous concave functional on K with α(x) < ||x|| for all x ∈ K.

We first claim that if there exists a positive constant r such that , , i = 1, 2,..., m, for x ∈ (0, r], then .

Indeed, if ,

Thus, ||Φx|| < r, that is Φx ∈ K_r. Hence, we have shown that (6.1) or (6.2) hold, then Φ maps into K_d or into K_c, respectively. So condition (2) of Lemma 6.1 holds.

Let . Next, we will show that {x ∈ K(α, a, b) : α(x) > a} ≠ ∅, and α(Φ(x)) > a for x ∈ K(α, a, b). In fact, , then the constant function .

Since (6.3) holds, for x ∈ K(α, a, b), we obtain

So α(Φ(x)(t)) > a for all x ∈ K(α, a, b), then condition (1) of Lemma 6.1 holds.

Finally, suppose x ∈ K(α, a, c) and , then we have

for all . Thus,

To sum up, all the conditions of Theorem 6.1 are satisfied. Hence, Φ has at least three fixed points, that is, problem (1.3) has at least three positive solutions x₁, x₂, x₃ such that

The proof is complete. ■

In this section, we give some examples to illustrate our main results.

Example 7.1. Take . We consider the following IBVP on :

where T = 3, p(t) = t, f(t, x(σ(t))) = (t + 1)(x(σ(t)))², I(x) = x³, α = 1, , and

From (1.4), system (7.1) reduces to

By calculating, we get Γ = 0.3033 > 0, max f₀ = 0, min f_∞ = ∞, and I₀ = 0. Therefore, (H₁) holds. From Theorem 4.1w, it follows that the IBVP (7.1) has at least one solution.

Example 7.2. Take . We consider the following IBVP on :

where p(t) = t, , , α = 1, , and

By calculating, we get Γ = 0.5732 > 0, max f_∞ = 0, min f₀ = ∞, and I_∞ = 0. Therefore, by Theorem 4.1, it follows that the IBVP (7.2) has at least one solution.

Example 7.3. Take . We consider the following IBVP on :

Since p(t) = 1, T = 3, and σ(T) = 4, we know that e_p(σ(T), 0) = 4e² and . Take R = 4976, then we can choose that

By calculating, it is easy to see that , I(x) ∈ C(ℝ⁰, ℝ⁰) and

Therefore, all the conditions of Theorem 5.1 are fulfilled. So system (7.3) has at least two positive solutions.

Example 7.4. Take . We consider the following IBVP on :

where

Since p(t) = 1, T = 3, and σ(T) = 3, we know that e_p(σ(T), 0) = 2e². Then, we can get

Thus, if we choose , , and c is sufficiently large, then all the conditions of Theorem 6.1 are satisfied. So system (7.4) has at least three positive solutions.

In this paper, we first present a class of integral boundary value problems on time scales. Using the time scales calculus theory, the well-known Guo-Krasnoselskii fixed point theorem, and Legget-Williams fixed point theorem, we establish the existence of at least one, two, and three positive solutions for the problems. In addition, the methods in this paper may be applied to some other systems such as second-order integral boundary problems and higher-order integral boundary problems.

The authors declare that they have no competing interests.

In this paper, the authors first presented a class of first-order nonlinear impulsive integral boundary value problems on time scales. Then, by using the well-known Guo-Krasnoselskii fixed point theorem and Legget-Williams fixed point theorem, they established some criteria for the existence of at least one, two, and three positive solutions to the problem under consideration, respectively. All authors typed, read and approved the final manuscript.

This work is supported by the National Natural Sciences Foundation of People's Republic of China under Grant 10971183.

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