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EXISTENCE OF POSITIVE SOLUTIONS FOR SINGULAR IMPULSIVE DIFFERENTIAL EQUATIONS WITH INTEGRAL BOUNDARY CONDITIONS
EXISTENCE OF POSITIVE SOLUTIONS FOR SINGULAR IMPULSIVE DIFFERENTIAL EQUATIONS WITH INTEGRAL BOUNDARY CONDITIONS
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2014. Jul, 21(3): 147-163
Copyright © 2014, Korean Society of Mathematical Education
  • Received : December 08, 2013
  • Accepted : June 28, 2014
  • Published : July 28, 2014
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About the Authors
MIAO CHUNMEI
aSCHOOL OF MATHEMATICS AND STATISTICS, NORTHEAST NORMAL UNIVESITY, 5268 RENMIN STREET, CHANGCHUN, JILIN, 130024, P.R. CHINA.Email address:mathchunmei2012@aliyun.com
GE WEIGAO
bSCHOOL OF MATHEMATICS, BEIJING INSTITUTE OF TECHNOLOGY, BEIJING 100081, P.R. CHINA.Email address:gew@bit.edu.cn
ZHANG ZHAOJUN
cSCHOOL OF SCIENCE, CHANGCHUN UNIVERSITY, CHANGCHUN, JILIN, 130022, P.R. CHINA.Email address:1448494218@qq.com

Abstract
In this paper, we study the existence of positive solutions for singular impulsive differential equations with integral boundary conditions where the nonlinearity f ( t, u, v ) may be singular at v = 0. The proof is based on the theory of Leray-Schauder degree, together with a truncation technique. Some recent results in the literature are generalized and improved.
Keywords
1. Introduction
Impulsive differential equations are basic instruments to study the dynamics of processes that are subjected to abrupt changes in their states. Recent development in this field has been focused by many applied problems, such as control theory [8 , 9] , population dynamics [19] and medicine [4 , 5] . For the general aspects of impulsive diffrential equations, we refer the reader to the classical monograph [14] .
During the last two decades, impulsive differential equations have been studied by many authors [1 - 3 , 10 , 13 , 15 - 16 , 20 - 25] . Many of them are on impulsive di®erential equation boundary value problems (BVPs for short). In recent years, there have been many studies related to impulsive multi-point boundary value problems [6 - 7 , 11 - 12 , 17 , 26] . They include three, four, multi-point impulsive BVPs and impulsive BVPs with integral boundary conditions. However, very few papers consider singular impulsive differential equations with integral boundary conditions.
In [2] , using the Schauder’s fixed point theorem, Agarwal et al. investigated the existence of at least one positive solution for singular BVPs for first and second order impulsive differential equations. In [18] , using the Schauder’s fixed point theorem, Miao et al. studied a singular BVP with integral boundary condition for a first-order impulsive differential equation. Motivated by [2] , we extend the results in [18] to a second order singular impulsive differential equation. In this paper, we consider the following singular impulsive BVP
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where 0 < t 1 < t 2 < ⋯ < t p < 1, 𝕁' = 𝕁 \ { t 1 , t 2 , ⋯ , tp }, 𝕁 = [0, 1], 𝚫 u ( tk ) denotes the jump of u ( t ) at t = tk , i.e., 𝚫 u ( tk ) = u ( tk +0)− u ( tk −0), u ( tk +0) and u ( tk −0)) represent the right and left limits of u ( t ) at t = tk , 𝚫 u ′( tk ) denotes the jump of u ′( t ) at t = tk , i.e., 𝚫 u ′( tk ) = u ′( tk + 0) − u ′( tk − 0), u ′( tk + 0) and u ′( tk − 0)) represent the right and left derivative of u ( t ) at t = tk . We are mainly interested in the case that f ( t, u, v ) may be singular at v = 0.
The method used in this paper mainly depends on the theory of Leray-Schauder degree. We first consider the existence of positive solutions for a constructed nonsingular BVP. Then, using Arzelà-Ascoli theorem, we obtain positive solutions for the singular problem that is approximated by the family of solutions to the nonsingular BVPs.
The following hypotheses are adopted throughout this paper:
(H 1 ) q C [𝕁], q ( t ) > 0, t ∈ (0, 1), f : 𝕁 × [0,∞) × (0,∞) → (0,∞) is continuous, Ik , Lk : [0,∞) × [0,∞) → [0,∞)( k = 1, 2, ⋯ , p ) are continuous, g L 1 [𝕁], g ( t ) ≥ 0, t ∈ 𝕁 and 0 ≤ 𝜎 :=
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g ( t ) dt < 1.
(H 2 ) f ( t, u, v ) ≤ h ( u )[ f 1 ( v )+ f 2 ( v )], ( t, u, v ) ∈ 𝕁×[0,∞)×(0,∞), where f 1 ( u ) > 0 is continuous, nonincreasing on (0,∞); h ( u ), f 2 ( u ) ≥ 0 are continuous on [0,∞).
(H 3 ) For any given constants K > 0, N > 0, there is a constant 𝛾 ∈ [0, 1) and a continuous function 𝜓 K,N : 𝕁 → (0,∞) such that f ( t, u, v ) ≥ 𝜓 K,N ( t ) u 𝛾 , ( t, u, v ) ∈ 𝕁 × [0, K ] × (0, N ].
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2. Preliminaries
For convenience, we first give some notations:
(1) 𝕁 0 = [0, t 1 ], 𝕁 k = ( tk , t k+1 ], k = 1, 2, ⋯ , p − 1, 𝕁 p = ( tp , 1].
(2) PC 1 [𝕁] = { u : 𝕁 → ℝ | u ′( t ) is continuous in 𝕁' and there exist u ′( tk − 0) = u ′( tk ), u ′( tk + 0) < ∞( k = 1, 2, ⋯ , p )}.
Obviously, ( PC 1 [𝕁], ∥ u PC1 ) is a Banach space with the norm ∥ u PC1 = max{∥ u ∥, ∥ u ′∥}, here
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( PC 1 [𝕁], ∥ u PC1 ) is abbreviated as PC 1 [𝕁].
Definition 2.1. We say a function u PC 1 [𝕁] is a positive solution to problem (1.1) if u satisfies (1.1) and u ( t ) > 0, t ∈ (0, 1).
Definition 2.2 ( [14] ). A set S PC 1 [𝕁] is said to be quasiequicontinuous if for all u S and ε > 0, there exists 𝛿 > 0 such that s , t ∈ 𝕁 k ( k = 1, 2, ⋯ , p ) and | s t | < 𝛿 implies
|u(s) − u(t)| < ε and |u′(s) − u′(t)| < ε.
We present the following result about relatively compact sets in PC 1 [
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] which is a consequence of the Arzelà-Ascoli Theorem.
Lemma 2.3 ( [14] ). A set S PC 1 [𝕁] is relatively compact in PC 1 [𝕁] if and only if S is bounded and quasiequicontinuous.
Lemma 2.4. Suppose that e L 1 [𝕁], e ( t ) > 0, t ∈ (0, 1), ak , bk ≥ 0 ( k = 1, 2, ⋯ , p ), a ≥ 0 are constants. Then, BVP
PPT Slide
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has a unique solution. Moreover, this solution can be expressed by
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where
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Proof . It is easy to verify that (2.2) is a solution of (2.1). On the other hand if u is a solution of (2.1), then
  • u′′(t) = −e(t),t∈ 𝕁'.
For any t ∈ 𝕁 k , k = 0, 1, 2, ⋯ , p , integrating on the both sides of the above equation from 0 to t , one obtains
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Using the boundary condition u ′(1) = a , we have
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and then
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Integrate on the both sides of (2.3) from 0 to t , and one obtains
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Multiplying (2.4) with g ( t ) and integrating it from 0 to 1, we have
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and substituting (2.5) into (2.4) yields
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that is,
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The proof is complete.
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In order to solve (1.1), we consider the following BVP
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where F : 𝕁 × ℝ 2 → (0,∞) is continuous,
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,
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: ℝ 2 → [0,∞)( k = 1, 2, ⋯ , p ) are continuous, q , g are the same as in (H 1 ), and a ≥ 0 is a constant.
Let u PC 1 [𝕁]. We define an operator T : PC 1 [𝕁] → PC 1 [𝕁] by
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We have the following result:
Lemma 2.5. T : PC 1 [𝕁] → PC 1 [𝕁] is completely continuous .
Proof . It is easy to prove that T : PC 1 [𝕁] → PC 1 [𝕁] is well defined.
By the continuity of
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,
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( k = 1, 2, ⋯ , p ) and F , we have T is continuous.
Next we shall show that T is compact. Suppose B = { u PC 1 [𝕁]| ∥ u PC1 r } ⊂ PC 1 [𝕁] is a bounded set. For any u B , which implies ∥ u ∥ ≤ r , ∥ u ′∥ ≤ r , we have
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In addition,
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This implies that T ( B ) is uniformly bounded.
For any given ε > 0, t , s ∈ 𝕁 k ( k = 0, 1, ⋯ , p ) (without loss of generality, let s < t ), when t s , we obtain
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Additional,
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This implies that T ( B ) is quasiequicontinuous. By Lemma 2.3, T ( B ) is relatively compact. Therefore, T is completely continuous.
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Now we state a existence principle, which plays an important role in our proof of main results.
Lemma 2.6 (Existence Principle). Assume that there exists a constant
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independent of λ, such that for λ ∈ (0, 1), ∥ u PC1 R , where u ( t ) satisfies
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Then (2.8) 1 has at least one solution u ( t ) such that u PC1 R .
Proof . For any λ ∈ 𝕁, u PC 1 [𝕁], define one operator
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By Lemma 2.5, N λ : PC 1 [𝕁] → PC 1 [𝕁] is completely continuous. It can be verified that a solution of BVP (2.8) λ equivalent to a fixed point of N λ in PC 1 [𝕁]. Let Ω = { u PC 1 [𝕁]| ∥ u PC1 < R }, then Ω is an open set in PC 1 [𝕁]. If there exists u ∈ 𝟃Ω such that N 1 u = u , then u ( t ) is a solution of (2.8) 1 with ∥ u PC1 R . Thus the conclusion is true. Otherwise, for any u ∈ 𝟃Ω, N 1 u u . If λ = 0, for u ∈ 𝟃Ω, ( I N 0 ) u ( t ) = u ( t ) − N 0 u ( t ) = u ( t ) −
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a ≢ 0 since ∥ u PC1 = R >
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a , so N 0 u u for any u ∈ 𝟃Ω. For λ ∈ (0, 1), if there is a solution u ( t ) to BVP (2.8) λ by the assumption, one gets ∥ u PC1 R , which is a contradiction to u ∈ 𝟃Ω.
In a word, for any u ∈ 𝟃Ω­ and λ ∈ 𝕁, N λ u u . Homotopy invariance of Leray-Schauder degree deduce that
Deg{IN1,­Ω,0} = Deg{IN0,­Ω,0} = 1.
Hence, N 1 has a fixed point u in Ω. That is, BVP (2.8) 1 has a solution u ( t ) with ∥ u PC1 R . The proof is completed.
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Lemma 2.7. Suppose (H 1 ) holds. If u is a solution to problem (2.6), then
  • (i)u(t)is concave on𝕁k(k= 0, 1, ⋯ ,p);
  • (ii)u′(t) ≥a, t∈ 𝕁',u′(tk− 0) ≥u′(tk+ 0) ≥a, 𝚫u(tk) ¸ 0,k= 1, 2, ⋯ ,p;
  • (iii)u(t) ≥a and u(t) ≥t∥u∥,t∈ 𝕁.
Proof . (i) Because u ( t ) is a solution of problem (2.6), we have
u′′(t) = −q(t)F(t, u(t), u′(t)) < 0, t ∈ 𝕁'.
Therefore u ′ is nonincreasing on 𝕁', which implies u ( t ) is concave on 𝕁 k ( k = 0, 1, ⋯ , p ).
(ii) Since u ′ is nonincreasing on 𝕁', and u ′(1) = a , therefore u ′( t ) ≥ a , t ∈ 𝕁', u ′( tk − 0) ≥ u ′( tk + 0) ≥ a , 𝚫 u ( tk ) ≥ 0, k = 1, 2, ⋯ , p .
(iii) From Lemma 2.4, we have for t ∈ 𝕁,
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Because u ( t ) is concave, we have
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thus u ( t ) ≥ t u ∥, t ∈ 𝕁.
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3. Existence Results
In this section, we give the main results for BVP (1.1) in this paper.
Theorem 3.1. Suppose (H 1 )-(H 5 ) hold, then BVP (1.1) has at least one positive solution .
Proof . Step 1. From (H 5 ), we choose M > 0 and 0 <
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ε < M such that
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Furthermore, we have
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Let n 0 ∈ {1, 2, ⋯ } satisfy that
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< ε , and set ℕ 0 = { n 0 , n 0 + 1, n 0 + 2, ⋯ }.
In what follows, we show that the following BVP
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has a positive solution for each m ∈ ℕ 0 .
To this end, we consider the following BVP
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where
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then f * : 𝕁 × [0,∞) × ℝ → (0,∞),
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,
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: [0,∞) × ℝ → [0,∞), ( k = 1, 2, ⋯ , p ).
To obtain a solution of BVP (3.4) for each m ∈ ℕ 0 , by applying Lemma 2.6, we consider the family of BVPs
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where λ ∈ 𝕁. Let u ( t ) be a solution of (3.5). From Lemma 2.7, we observe that u ( t ) is concave on 𝕁 k ( k = 0, 1, ⋯ , p ), u ′( t ) ≥
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, t ∈ 𝕁', u ′( tk − 0) ¸ u ′( tk +0) ¸
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, k = 1, 2, ⋯ , p and u ( t ) ≥
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, u ( t ) ≥ t u PC1 , t ∈ 𝕁.
For any x ∈ 𝕁', by (H 2 ), we have
u′′(x) = λq(x)f*(x, u(x), u′(x)) = λq(x)f(x, u(x), u′(x)) ≤ q(x)h(u(x))[f1(u′(x)) + f2(u′(x))].
Multiply the above inequality by
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and integrate it from t ( t
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) to 1 yield that
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For any t ∈ 𝕁, we have
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and
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Integrate (3.6) from 0 to 1, and one obtains
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If u ′(0) ≥ u (1), then ∥ u PC1 = max{ u (1), u ′(0)} = u ′(0). By (3.7) one obtains
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which together with (3.2) implies
uPC1 = u′(0) ≠ M.
If u ′(0) < u (1), then ∥ u PC1 = max{ u (1), u ′(0)} = u (1). By (3.8), we obtain
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which together with (3.1) implies
uPC1 = u(1) ≠ M.
By Lemma 2.6, for any fixed m ∈ ℕ 0 , BVP (3.4) has at last one positive solution, denoted by um ( t ), and ∥ um PC1 M . From Lemma 2.7, we note that um ( t ) ≥
PPT Slide
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PPT Slide
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, t ∈ 𝕁,
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( t ) ≥
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, t ∈ 𝕁', u ′( tk − 0) ≥ u ′( tk + 0) ≥
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. So f *( t , um ( t ),
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( t )) = f ( t , um ( t ),
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( t )),
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( um ( t ),
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( t )) = Ik ( um ( t ),
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( t )),
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( um ( t ),
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( t )) = Lk ( um ( t ),
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( t ))( k = 1, 2, ⋯ , p ). Therefore, um ( t ) is the solution to BVP (3.3).
Step 2. By
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we conclude that
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where L > 0 is a constant.
In fact, (H 3 ) guarantees the existence of a function 𝜓 M,M which is continuous on 𝕁 and positive on (0, 1) with
f(t, um(t), (t)) ≥ 𝜓M,M (t)u𝛾, t ∈ 𝕁, 𝛾 ∈ [0, 1).
By Lemma 2.4 and Lemma 2.7, one has
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where L 1 :=
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s 𝛾+1 q ( s )𝜓 M,M ( s ) ds , L 0 :=
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s 𝛾+1 q ( s )𝜓 M,M ( s ) ds . Furthermore, we have
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Because um ( t ) is a solution of (3.3), for s ∈ 𝕁',
PPT Slide
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and integrate it from t ( t ∈ 𝕁) to 1, one obtains
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and then, we have
(t) ≥ 𝜑(t), tJ0, (tk − 0) ≥ (tk + 0) ≥ 𝜑(tk), k = 1, 2, ⋯ , p,
Thus, (3.10) holds.
Step 3. It remains to show that
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( j = 0, 1) are both uniformly bounded and quasiequicontinuous on 𝕁. By (3.9), we have
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( j = 0, 1) are both uniformly bounded on 𝕁.
Next we need only to show that
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( j = 0, 1) are quasiequicontinuous on 𝕁. By um ( t ) is the solution (3.3), for s ∈ 𝕁', we have
PPT Slide
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and by integrate it from t ( t ∈ 𝕁) to 1, one obtains
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For any t, s ∈ 𝕁 k ( k = 0, 1, ⋯ , p ),
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By the conditions (H 2 ) and (H 4 ), one gets
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Therefore,
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( j = 0, 1) are quasiequicontinuous on 𝕁.
The Arzelà-Ascoli theorem guarantees that there is a subsequence ℕ* of ℕ 0 (without loss of generality, we assume ℕ* = ℕ 0 ) and functions
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( t )( j = 0, 1) with
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( t ) →
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( t )( j = 0, 1) uniformly on 𝕁 as m → +∞ through ℕ*. So u (0) =
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g ( t ) u ( t ) dt , u ′(1) = 0, ∥ u PC1 M ,
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um ( tk + 0) = u ( tk + 0),
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( tk + 0) = u ′( tk + 0)( k = 1, 2, ⋯ , p ), and
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u ′( t ) ≥ ϕ ( t ), t ∈ 𝕁.
For t ∈ ( tp , 1), by um ( t )( m ∈ ℕ*) is the solution of (3.3) and Lemma 2.4, we have
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Let m → +∞ through ℕ* in (3.13), one has
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and furthermore, we have u ′′( t ) + q ( t ) f ( t , u ( t ), u ′( t )) = 0, t ∈ ( tp , 1). Similarly, for any t ∈ 𝕁 k ( k = 1, ⋯ , p − 1), t ∈ (0, t 1 ), one has u ′′( t ) + q ( t ) f ( t , u ( t ), u ′( t )) = 0.
Thus, we have
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i.e. u ( t ) is positive solution of BVP (1.1), and ∥ u PC1 M ,
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u ′( t ) ≥ 𝜑( t ), t ∈ 𝕁. The proof of Theorem 3.1 is complete.
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4. An Example
In this section, we give an example to illustrate our results.
Example 4.1. Consider the following BVP
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where 0 < t 1 < t 2 < ⋯ < tp < 1, ck , dk ≥ 0, k = 1, 2, ⋯ , p are constants and
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Conclusion. BVP (4.1) has at least one positive solution.
Proof . Obviously, q ( t ) = t, f ( t, u, v ) =
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g ( t ) = t , Ik = ck ≥ 0, Lk = dk ≥ 0, k = 1, 2, ⋯ , p . 𝜎 =
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tdt =
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∈ [0, 1), so (H 1 ) holds.
Let
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then (H 2 ) holds. For any K, N > 0, choose 𝜓 K,N ( t ) =
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and 𝛾 =
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such that
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( t, u, v ) ∈
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× [0, K ] × (0, N ],
thus (H 3 ) holds.
By a direct calculation, we have
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and
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which implies that condition (H 4 ) holds.
Next, we show that the conditions (H 5 ) holds. In fact, because
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then (H 5 ) holds. Therefore, by Theorem 3.1, we can obtain that (4.1) has at least one positive solution.
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Acknowledgements
Supported by NNSF of China (11001032, 11071014).
References
Agarwal R.P. , O’Regan D. 2000 Multiple nonnegative solutions for second-order impulsive differential equations Appl. Math. Comput. 114 51 - 59    DOI : 10.1016/S0096-3003(99)00074-0
Agarwal R.P. , Franco D. , O’Regan D. 2005 Singular boundary value problems for first and second order impulsive differential equations Aequationes Math. 69 83 - 96    DOI : 10.1007/s00010-004-2735-9
Agarwal R.P. , Franco D. , O’Regan D. 2005 Singular boundary value problems for first and second order impulsive differential equations Aequationes Math. 69 83 - 96    DOI : 10.1007/s00010-004-2735-9
Choisy M. , Guéga J.F. , Rohani P. 2006 Dynamics of infectious deseases and pulse vaccination: teasing apart the embedded resonance effects Phys. D. 223 26 - 35    DOI : 10.1016/j.physd.2006.08.006
D’Onofrio A. 2005 On pulse vaccination strategy in the SIR epidemic model with vertical transmission Appl. Math. Lett. 18 729 - 732    DOI : 10.1016/j.aml.2004.05.012
Feng M.Q. , Xie D.X. 2009 Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations J. Comput. Appl. Math. 223 438 - 448    DOI : 10.1016/j.cam.2008.01.024
Feng M.Q. , Du B. , Ge W.G. 2009 Impulsive boundary value problems with integral boundary conditions and one-dimensional p-Laplacian Nonlinear Anal. 70 3119 - 3126    DOI : 10.1016/j.na.2008.04.015
George R.K. , Nandakumaran A.K. , Arapostathis A. 2000 A note on contrability of impulsive systems J. Math. Anal. Appl. 241 276 - 283    DOI : 10.1006/jmaa.1999.6632
Jiang G. , Lu Q. 2007 Impulsive state feedback control of a predator-prey model J. Comput. Appl. Math. 200 193 - 207    DOI : 10.1016/j.cam.2005.12.013
Jiang D.Q. , Chu J.F. , He Y. 2007 Multiple positive solutions of Sturm-Liouville problems for second order impulsive differential equations Dynam. Systems Appl. 16 611 - 624
Jankowski T. 2008 Positive solutions of three-point boundary value problems for second order impulsive differential equations with advanced arguments Appl. Math. Comput. 197 179 - 189    DOI : 10.1016/j.amc.2007.07.081
Jankowski T. 2008 Positive solutions to second order four-point boundary value problems for impulsive differential equations Appl. Math. Comput. 202 550 - 561    DOI : 10.1016/j.amc.2008.02.040
Lin X. , Jiang D. 2006 Multiple positive solutions of Dirichlet boundary value problems for second-order impulsive differential equations J. Math. Anal. Appl. 321 501 - 514    DOI : 10.1016/j.jmaa.2005.07.076
Lakshmikantham V. , Bainov D.D. , Simeonov P. S. 1989 Theory of Impulsive Differential Equations World Scientific Singapore
Lee E.K. , Lee Y.H. 2004 Multiple positive solutions of singular two point boundary value problems for second-order impulsive differential equations Appl. Math. Comput. 158 745 - 759    DOI : 10.1016/j.amc.2003.10.013
Lee Y.H. , Liu X.Z. 2007 Study of singular boundary value problems for second order impulsive differential equations J. Math. Anal. Appl. 331 159 - 176    DOI : 10.1016/j.jmaa.2006.07.106
Liang S.H. , Zhang J.H. 2009 The existence of countably many positive solutions for some nonlinear singular three-point impulsive boundary value problems Nonlinear Anal. 71 4588 - 4597    DOI : 10.1016/j.na.2009.03.016
Miao C.M. , Ge W.G , Zhang J. 2009 A singular boundary value problem with integral boundary condition for a first-order impulsive differential equation (Chinese) Mathematics in Practice and Theory 39 (15) 182 - 186
Nenov S. 1999 Impulsive controllability and optimization problems in population dynamics Nonlinear Anal. 36 881 - 890    DOI : 10.1016/S0362-546X(97)00627-5
Nieto J.J. 1997 Basic theory for nonresonance impulsive periodic problems of first order J. Math. Anal. Appl. 205 (2) 423 - 433    DOI : 10.1006/jmaa.1997.5207
Nieto J.J. , O’Regan D. 2009 Variational approach to impulsive differential equations Nonlinear Anal. RWA 10 680 - 690    DOI : 10.1016/j.nonrwa.2007.10.022
Nieto J.J. 2010 Variational formulation of a damped Dirichlet impulsive problem Appl. Math. Lett. 23 940 - 942    DOI : 10.1016/j.aml.2010.04.015
Rachunková I. , Tomecek J. 2006 Singular Dirichlet problem for ordinary differential equation with impulses Nonlinear Anal. 65 210 - 229    DOI : 10.1016/j.na.2005.09.016
Tian Y. , Ge W.G. 2010 Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations Nonlinear Anal. 72 277 - 287    DOI : 10.1016/j.na.2009.06.051
Tian Y. , Ge W.G. 2012 Multiple solutions of impulsive Sturm-Liouville boundary value problem via lower and upper solutions and variational methods J. Math. Anal. Appl. 387 475 - 489    DOI : 10.1016/j.jmaa.2011.08.042
Xu J.F. , O’Regan D. , Yang Z.L. Positive Solutions for a nth-Order Impulsive Differential Equation with Integral Boundary Conditions Differ Equ Dyn Syst.    DOI : 10.1007/s12591-013-0176-4