EXISTENCE OF POSITIVE SOLUTIONS FOR SINGULAR IMPULSIVE DIFFERENTIAL EQUATIONS WITH INTEGRAL BOUNDARY CONDITIONS

Abstract

In this paper, we study the existence of positive solutions for singular impulsive differential equations with integral boundary conditions $$\{u^{{\prime}{\prime}}(t)+q(t)f(t,u(t),u^{\prime}(t))=0,\;t{\in}\mathbb{J}^{\prime},\\{\Delta}u(t_k)=I_k(u(t_k),u^{\prime}(t_k)),\;k=1,2,{\cdots},p,\\{\Delta}u^{\prime}(t_k)=-L_k(u(t_k),u^{\prime}(t_k)),\;k=1,2,{\cdots},p,\\u=(0)={\int}_{0}^{1}g(t)u(t)dt,\;u^{\prime}=0,$$) 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.

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

where 0 < t1 < t2 < ⋯ < tp < 1, 𝕁' = 𝕁 ＼ {t1, t2, ⋯ , tp}, 𝕁 = [0, 1], 𝚫u(tk) denotes the jump of u(t) at t = tk, i.e., 𝚫u(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, 𝚫u′(tk) denotes the jump of u′(t) at t = tk, i.e., 𝚫u′(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:

(H1) 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 ∈ L1[𝕁], g(t) ≥ 0, t ∈ 𝕁 and 0 ≤ 𝜎 := g(t)dt < 1.

(H2) f(t, u, v) ≤ h(u)[f1(v)+f2(v)], (t, u, v) ∈ 𝕁×[0,∞)×(0,∞), where f1(u) > 0 is continuous, nonincreasing on (0,∞); h(u), f2(u) ≥ 0 are continuous on [0,∞).

(H3) For any given constants K > 0, N > 0, there is a constant 𝛾 ∈ [0, 1) and a continuous function 𝜓K,N : 𝕁 → (0,∞) such that f(t, u, v) ≥ 𝜓K,N (t)u𝛾, (t, u, v) ∈ 𝕁 × [0,K] × (0,N].

 

2. Preliminaries

For convenience, we first give some notations:

(1) 𝕁0 = [0, t1], 𝕁k = (tk, tk+1], k = 1, 2, ⋯ , p − 1, 𝕁p = (tp, 1].

(2) PC1[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, (PC1[𝕁], ∥u∥PC1) is a Banach space with the norm ∥u∥PC1 = max{∥u∥, ∥u′∥}, here (PC1[𝕁], ∥u∥PC1) is abbreviated as PC1[𝕁].

Definition 2.1. We say a function u ∈ PC1[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 ⊂ PC1[𝕁] is said to be quasiequicontinuous if for all u ∈ S and ε > 0, there exists 𝛿 > 0 such that s, t ∈ 𝕁k(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 PC1[] which is a consequence of the Arzelà-Ascoli Theorem.

Lemma 2.3 ([14]). A set S ⊂ PC1[𝕁] is relatively compact in PC1[𝕁] if and only if S is bounded and quasiequicontinuous.

Lemma 2.4. Suppose that e ∈ L1[𝕁], e(t) > 0, t ∈ (0, 1), ak, bk ≥ 0 (k = 1, 2, ⋯ , p), a ≥ 0 are constants. Then, BVP

has a unique solution. Moreover, this solution can be expressed by

where

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 ∈ 𝕁k, k = 0, 1, 2, ⋯ , p, integrating on the both sides of the above equation from 0 to t, one obtains

Using the boundary condition u′(1) = a, we have and then

Integrate on the both sides of (2.3) from 0 to t, and one obtains

Multiplying (2.4) with g(t) and integrating it from 0 to 1, we have

and substituting (2.5) into (2.4) yields

that is,

The proof is complete.

In order to solve (1.1), we consider the following BVP

where F : 𝕁 × ℝ2 → (0,∞) is continuous, , : ℝ2 → [0,∞)(k = 1, 2, ⋯ , p) are continuous, q, g are the same as in (H1), and a ≥ 0 is a constant.

Let u ∈ PC1[𝕁]. We define an operator T : PC1[𝕁] → PC1[𝕁] by

We have the following result:

Lemma 2.5. T : PC1[𝕁] → PC1[𝕁] is completely continuous.

Proof. It is easy to prove that T : PC1[𝕁] → PC1[𝕁] is well defined.

By the continuity of , (k = 1, 2, ⋯ , p) and F, we have T is continuous.

Next we shall show that T is compact. Suppose B = {u ∈ PC1[𝕁]| ∥u∥PC1 ≤ r} ⊂ PC1[𝕁] is a bounded set. For any u ∈ B, which implies ∥u∥ ≤ r, ∥u′∥ ≤ r, we have

In addition,

This implies that T(B) is uniformly bounded.

For any given ε > 0, t, s ∈ 𝕁k(k = 0, 1, ⋯ , p) (without loss of generality, let s < t), when t → s, we obtain

Additional,

This implies that T(B) is quasiequicontinuous. By Lemma 2.3, T(B) is relatively compact. Therefore, T is completely continuous.

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

independent of λ, such that for λ ∈ (0, 1), ∥u∥PC1 ≠ R, where u(t) satisfies

Then (2.8)1 has at least one solution u(t) such that ∥u∥PC1 ≤ R.

Proof. For any λ ∈ 𝕁, u ∈ PC1[𝕁], define one operator

By Lemma 2.5, Nλ : PC1[𝕁] → PC1[𝕁] is completely continuous. It can be verified that a solution of BVP (2.8)λ equivalent to a fixed point of Nλ in PC1[𝕁]. Let Ω = {u ∈ PC1[𝕁]| ∥u∥PC1 < R}, then Ω is an open set in PC1[𝕁]. If there exists u ∈ 𝟃Ω such that N1u = 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, N1u ≠ u. If λ = 0, for u ∈ 𝟃Ω, (I − N0)u(t) = u(t) − N0u(t) = u(t) − a ≢ 0 since ∥u∥PC1 = R > a, so N0u ≠ 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, Nλu ≠ u. Homotopy invariance of Leray-Schauder degree deduce that

Deg{I − N1,­Ω,0} = Deg{I − N0,­Ω,0} = 1.

Hence, N1 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.

Lemma 2.7. Suppose (H1) holds. If u is a solution to problem (2.6), then

(i) u(t) is concave on 𝕁k(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 𝕁k(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, 𝚫u(tk) ≥ 0, k = 1, 2, ⋯ , p.

(iii) From Lemma 2.4, we have for t ∈ 𝕁,

Because u(t) is concave, we have

thus u(t) ≥ t∥u∥, t ∈ 𝕁.

 

3. Existence Results

In this section, we give the main results for BVP (1.1) in this paper.

Theorem 3.1. Suppose (H1)-(H5) hold, then BVP (1.1) has at least one positive solution.

Proof. Step 1. From (H5), we choose M > 0 and 0 < ε < M such that

Furthermore, we have

Let n0 ∈ {1, 2, ⋯ } satisfy that < ε, and set ℕ0 = {n0, n0 + 1, n0 + 2, ⋯ }.

In what follows, we show that the following BVP

has a positive solution for each m ∈ ℕ0.

To this end, we consider the following BVP

where

then f* : 𝕁 × [0,∞) × ℝ → (0,∞), , : [0,∞) × ℝ → [0,∞), (k = 1, 2, ⋯ , p).

To obtain a solution of BVP (3.4) for each m ∈ ℕ0, by applying Lemma 2.6, we consider the family of BVPs

where λ ∈ 𝕁. Let u(t) be a solution of (3.5). From Lemma 2.7, we observe that u(t) is concave on 𝕁k(k = 0, 1, ⋯ , p), u′(t) ≥ , t ∈ 𝕁', u′(tk − 0) ¸ u′(tk +0) ¸ , k = 1, 2, ⋯ , p and u(t) ≥ , u(t) ≥ t∥u∥PC1 , t ∈ 𝕁.

For any x ∈ 𝕁', by (H2), 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 and integrate it from t(t ∈ ) to 1 yield that

For any t ∈ 𝕁, we have

and

Integrate (3.6) from 0 to 1, and one obtains

If u′(0) ≥ u(1), then ∥u∥PC1 = max{u(1), u′(0)} = u′(0). By (3.7) one obtains

which together with (3.2) implies

∥u∥PC1 = u′(0) ≠ M.

If u′(0) < u(1), then ∥u∥PC1 = max{u(1), u′(0)} = u(1). By (3.8), we obtain

which together with (3.1) implies

∥u∥PC1 = u(1) ≠ M.

By Lemma 2.6, for any fixed m ∈ ℕ0, 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) ≥ , t ∈ 𝕁, (t) ≥ , t ∈ 𝕁', u′(tk − 0) ≥ u′(tk + 0) ≥ . So f*(t, um(t), (t)) = f(t, um(t), (t)), (um(t), (t)) = Ik(um(t), (t)), (um(t), (t)) = Lk(um(t),(t))(k = 1, 2, ⋯ , p). Therefore, um(t) is the solution to BVP (3.3).

Step 2. By

we conclude that

where L > 0 is a constant.

In fact, (H3) guarantees the existence of a function 𝜓M,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

where L1 := s𝛾+1q(s)𝜓M,M (s)ds, L0 := s𝛾+1q(s)𝜓M,M (s)ds. Furthermore, we have

Because um(t) is a solution of (3.3), for s ∈ 𝕁',

and integrate it from t(t ∈ 𝕁) to 1, one obtains

and then, we have

(t) ≥ 𝜑(t), t ∈ J0, (tk − 0) ≥ (tk + 0) ≥ 𝜑(tk), k = 1, 2, ⋯ , p,

Thus, (3.10) holds.

Step 3. It remains to show that (j = 0, 1) are both uniformly bounded and quasiequicontinuous on 𝕁. By (3.9), we have (j = 0, 1) are both uniformly bounded on 𝕁.

Next we need only to show that (j = 0, 1) are quasiequicontinuous on 𝕁. By um(t) is the solution (3.3), for s ∈ 𝕁', we have

and by integrate it from t(t ∈ 𝕁) to 1, one obtains

For any t, s ∈ 𝕁k(k = 0, 1, ⋯ , p),

By the conditions (H2) and (H4), one gets

Therefore, (j = 0, 1) are quasiequicontinuous on 𝕁.

The Arzelà-Ascoli theorem guarantees that there is a subsequence ℕ* of ℕ0 (without loss of generality, we assume ℕ* = ℕ0) and functions (t)(j = 0, 1) with (t) → (t)(j = 0, 1) uniformly on 𝕁 as m → +∞ through ℕ*. So u(0) = g(t)u(t)dt, u′(1) = 0, ∥u∥PC1 ≤ M, um(tk + 0) = u(tk + 0), (tk + 0) = u′(tk + 0)(k = 1, 2, ⋯ , p), and 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

Let m → +∞ through ℕ* in (3.13), one has

and furthermore, we have u′′(t) + q(t)f(t, u(t), u′(t)) = 0, t ∈ (tp, 1). Similarly, for any t ∈ 𝕁k(k = 1, ⋯ , p − 1), t ∈ (0, t1), one has u′′(t) + q(t)f(t, u(t), u′(t)) = 0.

Thus, we have

i.e. u(t) is positive solution of BVP (1.1), and ∥u∥PC1 ≤ M, u′(t) ≥ 𝜑(t), t ∈ 𝕁. The proof of Theorem 3.1 is complete.

 

4. An Example

In this section, we give an example to illustrate our results.

Example 4.1. Consider the following BVP

where 0 < t1 < t2 < ⋯ < tp < 1, ck, dk ≥ 0, k = 1, 2, ⋯ , p are constants and

Conclusion. BVP (4.1) has at least one positive solution.

Proof. Obviously, q(t) = t, f(t, u, v) = g(t) = t, Ik = ck ≥ 0, Lk = dk ≥ 0, k = 1, 2, ⋯ , p. 𝜎 = tdt = ∈ [0, 1), so (H1) holds.

Let

then (H2) holds. For any K, N > 0, choose 𝜓K,N (t) = and 𝛾 = such that

(t, u, v) ∈ × [0,K] × (0,N],

thus (H3) holds.

By a direct calculation, we have

and

which implies that condition (H4) holds.

Next, we show that the conditions (H5) holds. In fact, because

then (H5) holds. Therefore, by Theorem 3.1, we can obtain that (4.1) has at least one positive solution.