Abstract
In this paper, we first present a class of firstorder nonlinear impulsive integral boundary value problems on time scales. Then, using the wellknown GuoKrasnoselskii fixed point theorem and LeggetWilliams 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.
Keywords:
integral boundary value problem; fixed point; multiple solutions; time scale1 Introduction
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 [15] and references therein. At the same time, boundary value problems for impulsive differential equations and impulsive difference equations have received much attention [612], since such equations may exhibit several realworld phenomena in physics, biology, engineering, etc. see [1315] and the references therein.
In paper [16], Sun studied the firstorder 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 firstorder threepoint 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 multipoint boundary value problems more than threepoint for firstorder 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 twopoint, threepoint,..., npoint 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. [2127] 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 firstorder 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 rightscattered 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 < t_{1 }< · · · < 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 rightscattered points in interval , 0 ≤ θ_{1 }< · · · < θ_{q }≤ T, σ(θ_{0}) = 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 GuoKrasnoselskii and LeggetWilliams 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 46.
2 Preliminaries
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 leftdense, leftscattered, rightdense, or rightscattered if ρ(t) = t, ρ(t) < t, σ(t) = t, or σ(t) > t, respectively. Points that are rightdense and leftdense at the same time are called dense. If has a leftscattered maximum m_{1}, defined ; otherwise, set . If has a rightscattered minimum m_{2}, defined , otherwise, set .
Definition 2.2. [28]A function is rd continuous provided it is continuous at each rightdense point in and has a leftsided limit at each leftdense point in . The set of rdcontinuous 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 rightscattered. If t is not rightscattered, 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 rdcontinuous 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 socalled cylinder transformation.
Lemma 2.1. [28]Let p, . Then
(1) e_{0}(t, s) ≡ 1 and e_{p}(t, t) ≡ 1;
(2) e_{p}(σ(t), s) = (1 + μ(t)p(t))e_{p}(t, s);
(4) e_{p}(t, s)e_{p}(s, r) = e_{p}(t, r),
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 rdcontinuous 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
3 Foundational lemmas
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 firstclass 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. (GuoKrasnoselskii [30]) Let X be a Banach space and K ⊂ X be a cone in X. Assume that Ω_{1}, Ω_{2 }are bounded open subsets of X with and is a completely continuous operator such that, either
(1) Φx ≤ x, x ∈ K ∩ ∂Ω_{1}, and Φx ≥ x, x ∈ K ∩ ∂Ω_{2}; or
(2) Φx ≥ x, x ∈ K ∩ ∂Ω_{1}, and Φx ≤ x, x ∈ K ∩ ∂Ω_{2}.
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, t_{1}], integrating (3.3) from 0 to t, we get
while t → t_{1}, we have
then
Now, let t ∈ (t_{1}, t_{2}], integrating (3.3) from t_{1 }to t, we obtain
For t ∈ (t_{k}, t_{k+1}], repeating the above process, we can get
that is
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.
(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 lefthand side of (2) holds. And it is easy to show that the righthand 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 righthand side tends to zero uniformly as x  y → 0. Consequently, {Φx : x ∈ U} is the set of equicontinuous functions.
By ArzelaAscoli 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. ■
4 Existence of at least one positive solution
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:
(H_{1}) max f_{0 }= 0, min f_{∞ }= ∞, and I_{i0 }= 0, i = 1, 2,..., m; or
(H_{2}) max f_{∞ }= 0, min f_{0 }= ∞, and I_{i∞ }= 0, i = 1, 2,..., m.
Then, problem (1.3) has at least one positive solution.
Proof. Firstly, we assume that (H_{1}) holds. In this case, since max f_{0 }= 0 and I_{i0 }= 0, i = 1, 2,..., m, for ε ≤ (Bσ(T) + Bm)^{1}, there exists a positive constant r_{1 }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 ∩ ∂Ω_{1}, we have
which yields
On the other hand, if x ∈ K ∩ ∂Ω_{2}, 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 (H_{2}) 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 r_{3 }such that
In view of min f_{∞ }= ∞, we have that for M' ≥ (Aσ(T)δ)^{1}, there exists a positive constant r_{4 }< δr_{3 }such that
Let Ω_{i }= {x ∈ PC : x < r_{i}}, i = 3, 4.
On the one hand, if x ∈ K ∩ ∂Ω_{3}, we have
which yields
On the other hand, if x ∈ K ∩ ∂Ω_{4}, 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. ■
5 Existence of at least two positive solutions
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.
(H_{3}) min f_{0 }= +∞, min f_{∞ }= +∞.
(H_{4}) There exists a positive constant R such that for all 0 < x ≤ R.
(H_{5}) , 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 (H_{4}) and (H_{5}), for x ∈ K ∩ ∂Ω_{R}, we get
So
Since min f_{0 }= +∞, for M ≥ (Aσ(T)δ)^{1}, there exists a positive constant R_{1 }< δ_{R }such that
Hence,
Similarly, since min f_{∞ }= +∞, for M' ≥ (Aσ(T)δ)^{1}, there exists a positive constant such that
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 x_{1 }and x_{2 }satisfying 0 < R_{1 }≤ x_{1} < R < x_{2} ≤ R_{2}. The proof is complete. ■
Theorem 5.2. Assume that the following conditions hold.
(H_{6}) max f_{0 }= 0, max f_{∞ }= 0, I_{i0 }= 0, I_{i∞ }= 0, i = 1, 2,..., m.
(H_{7}) 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 (H_{7}), for x ∈ K ∩ ∂Ω_{r}, we get
So
Since max f_{0 }= 0 and I_{i0 }= 0, i = 1, 2,..., m, for ε ≤ (Bσ(T) + Bm)^{1}, there exists a positive constant r_{1 }< δ_{r }such that
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
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 x_{1 }and x_{2 }satisfying 0 < r_{1 }≤ x_{1} < r < x_{2} ≤ r_{2}. The proof is complete. ■
Similar to Theorems 5.1 and 5.2, one can easily obtain the following corollary:
Corollary 5.1. Assume that (H_{7}) and the following conditions hold.
(H_{8}) max f_{0 }= 0, max f_{∞ }= 0, I_{i0 }= 0, i = 1, 2,..., m.
(H_{9}) 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.
6 Existence of at least three positive solutions
In this section, we will state and prove our multiplicity result of positive solutions to problem (1.3) via LeggetWilliams fixed point theorem. For readers' convenience, we first illustrate LeggetWilliams 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. (LeggetWilliams 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 x_{1}, x_{2}, x_{3 }in satisfying x_{1} < d, a < α(x_{2}), x_{3} > d, and α(x_{3}) < 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 .
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
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 x_{1}, x_{2}, x_{3 }such that
The proof is complete. ■
7 Examples
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) = x^{3}, α = 1, , and
From (1.4), system (7.1) reduces to
By calculating, we get Γ = 0.3033 > 0, max f_{0 }= 0, min f_{∞ }= ∞, and I_{0 }= 0. Therefore, (H_{1}) 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 f_{0 }= ∞, 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) = 4e^{2 }and . Take R = 4976, then we can choose that
By calculating, it is easy to see that , I(x) ∈ C(ℝ^{0}, ℝ^{0}) 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) = 2e^{2}. 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.
8 Conclusion
In this paper, we first present a class of integral boundary value problems on time scales. Using the time scales calculus theory, the wellknown GuoKrasnoselskii fixed point theorem, and LeggetWilliams 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 secondorder integral boundary problems and higherorder integral boundary problems.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
In this paper, the authors first presented a class of firstorder nonlinear impulsive integral boundary value problems on time scales. Then, by using the wellknown GuoKrasnoselskii fixed point theorem and LeggetWilliams 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.
Acknowledgements
This work is supported by the National Natural Sciences Foundation of People's Republic of China under Grant 10971183.
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