Solvability of boundary value problem for second order impulsive differential equations with one-dimensional p-Laplacian on whole line ∗

The motivation for the present work stems from both practical and theoretical aspects. In fact, boundary value problems on the whole line arise quite naturally in the study of radially symmetric solutions of nonlinear elliptic equations modelling various physical phenomena such as unsteady flow of gas though a whole, porous media, and the theory of drain flows. The asymptotic theory of ordinary differential equations is an area in which there is great activity among a large number of investigators. In this theory, it is of great interest to investigate, in particular, the existence of solutions with prescribed asymptotic behavior, which are global in the sense that they are solutions on the whole line (half line). The existence of global solutions with prescribed asymptotic behavior is usually formulated as the existence of solutions of boundary value problems on the whole line (half line). In recent years, the existence of solutions of boundary value problems of the differential equations governed by nonlinear differential operator [Φ(u′)]′ = [|u′|p−2u′]′ has been studied by many authors, see [5, 7, 8, 9, 10, 11, 12, 13, 15, 17].


Introduction
The motivation for the present work stems from both practical and theoretical aspects.In fact, boundary value problems on the whole line arise quite naturally in the study of radially symmetric solutions of nonlinear elliptic equations modelling various physical phenomena such as unsteady flow of gas though a whole, porous media, and the theory of drain flows.
The asymptotic theory of ordinary differential equations is an area in which there is great activity among a large number of investigators.In this theory, it is of great interest to investigate, in particular, the existence of solutions with prescribed asymptotic behavior, which are global in the sense that they are solutions on the whole line (half line).The existence of global solutions with prescribed asymptotic behavior is usually formulated as the existence of solutions of boundary value problems on the whole line (half line).
Impulsive differential equation is one of the main tools to study the dynamics of processes in which sudden changes occur.The theory of impulsive differential equation has recently received considerable attention.However, the study on existence of positive solutions of nonlocal boundary value problems for impulsive differential equations on whole real line has not been sufficiently developed [1,2,3,4,5,6,16].
In all above mentioned papers, the boundary conditions are subjected to the two end points 0 and +∞ (or −∞ and +∞) and the obtained solutions are defined on [0, +∞) (or R).An interesting question occurs: when one subjects the boundary conditions on two intermediate points ξ, η, how can we get solutions defined on R of a boundary value problem of differential equations on whole line?On the other hand, in known papers [7,8,9,10,11,12,13,14], concerning the differential equations [Φ(ρ(t)x (t))] + p(t)f (t, x(t), ρ(t)x (t)) = 0, it is supposed that (t, u, v) → p(t)f (t, u, v) is a Carathéodory function.To the best of our knowledge, there has been no paper concerning the solvability of [Φ(ρ(t)x (t))] + p(t)f (t, x(t), ρ(t)x (t)) = 0 with (t, u, v) → p(t)f (t, u, v) being a non-Carathéodory function.
Motivated by mentioned papers, to fill this gap, we consider the following boundary value problem for the impulsive singular differential equation on the whole line: where ds|, and the following cases will be discussed: The homogeneous boundary conditions x(ξ) = 0, x(η) = 0 of special case in (1) come from the four-point boundary conditions a lim t→−∞ x(t)−bx(ξ) = c lim t→+∞ (t)− dx(η) = 0 (if a = c = 0 and b = d = 1), which arise in the study of heat flow problems involving a bar of unit length with two controllers at t = −∞ and t = +∞ adding or removing heat according to the temperatures detected by two sensors at t = ξ and t = η.It is well known that Consider the problem (tx One can get from ( |t|x (t)) = 1, a.e.t ∈ R, that x(t) = (2/3)t 3/2 + 2c 1 √ t + c 2 for t > 0 and x(t) = −(2/3)|t| 3/2 − 2c 1 |t| + c 2 for t < 0. Thus the mentioned problem has infinitely many continuous solutions Here c 1 , c 2 , c 3 ∈ R.So this kind of problem is interesting.Our purpose is to establish sufficient conditions for the existence of solutions of BVP (1) in Cases 1-4, respectively.The remainder of this paper is organized as follows: the first result are given in Section 2 in Case 1, the existence result of solutions of BVP (1) in Cases 2 is given in Section 3, and similarly, we can establish existence results in Cases 3 and 4, respectively, we omit the details.Finally, in Section 4, two examples are given to illustrate the main results.
Definition 3. Let X be a real Banach space.An operator T : X → X is completely continuous if it is continuous and maps bounded sets into relatively compact sets.

Choose
the following limits exist and are finite: Lemma 1. X is a Banach space with • defined.
Proof.It is easy to see that X is a normed linear space.Let {x u } be a Cauchy sequence in X.Then x u − x v → 0, u, v → +∞.We will prove that there x 0 ∈ X such that x u → x 0 as u → +∞.Since x u ∈ X, we have http://www.mii.lt/NAThen there exists functions x s,0 , y s,0 ∈ C 0 [t s , t s+1 ] such that lim u→+∞ x u (t)/σ(t) = x s,0 (t) and lim u→+∞ ρ(t)x u (t)/Φ −1 (τ (t)) = y s,0 (t) uniformly on [t s , t s+1 ].Define x 0 (t) = x s,0 (t), y 0 (t) = y s,0 (t) for all t ∈ (t s , t s+1 ](s ∈ Z).Then x 0 , y 0 : R → R is well defined on R, and It follows that Now we do the following three steps: Step The details are omitted.It follows that x u → x 0 as u → +∞.So X is a Banach space.
Lemma 2. Let M be a subset of X.Then M is relatively compact if and only if the following conditions are satisfied: Proof.(⇐) From Lemma 1 we know X is a Banach space.In order to prove that the subset M is relatively compact in X, we only need to show M is totally bounded in X, that is, for all > 0, M has a finite -net.
. By (i), (ii), and Ascoli-Arzela theorem, we can know that M | (−ts 0 ,ts 0 ] is relatively compact.Thus there exist x 1 , x 2 , . . ., x k ∈ M such that, for any x ∈ M , we have that there exists some i = 1, 2, . . ., k such that Therefore, for x ∈ M , we can get x − x i X .So, for any > 0, M has a finite -net (⇒) Assume that M is relatively compact, then for any > 0, there exists a finite -net of M .Let the finite -net be {U x1 , U x2 , . . ., U x k } with x i ⊂ M .Then for any x ∈ M , there exists U xi such that x ∈ U xi and x x − Furthermore, there exists t −s0 < 0 and t s0 > 0 such that |x i (w 1 )−x i (w 2 )| < for all w 1 , w 2 t s0 and all w 1 , w 2 t −s0 and i = 1, 2, . . ., k.Then we have for w 1 , w 2 t s0 , all w 1 , w 2 t −s0 and x ∈ M that http://www.mii.lt/NASimilarly, for w 1 , w 2 t s0 , all w 1 , w 2 t −s0 and x ∈ M , we have that Thus (iii) is valid.Similarly, we can prove that (ii) holds.Consequently, the lemma is proved.
For ease expression, denote G x (t) = G(t, x(t), ρ(t)x (t)) for a function G : R 3 → R and x ∈ X. Lemma 3. Suppose that x ∈ X.Then there exists a unique and A x satisfies . Proof.Denote .
We define a ts<b k s := − b ts<a k s for a > b.For x ∈ X, define (T x)(t) by where A x is defined by (3).
Lemma 4. Suppose that f is a strong Carathéodory function, I, J are discrete Carathéodory functions, and for each r > 0, f (t, σ(t)u, Φ −1 (τ (t))v) converges uniformly as t → ±∞ on [−r, r] × [−r, r].Then (i) T : X → X is well defined; (ii) x ∈ X is a solution of (1) if and only if x ∈ X is a fixed point of T in X; (iii) T is completely continuous.
(ii) By direct computation, we can get Thus it follows that x ∈ X is a solution of (1) if and only if x ∈ X is a fixed point of T in X.
(iii) Now we prove that T is completely continuous.The following five steps are needed (Steps 1-2 imply that T : X → X is continuous, and Steps 3-5 imply that T maps bounded sets into relatively compact sets).We omit the details of the proofs.

Y. Liu
Step 1.We prove that the function A x : X → R is continuous in x.
Step 2. We show that T is continuous on X.Since A x is continuous, f, φ, ψ are strong Carathéodory functions, I, J are discrete Carathédory functions, then the result follows.
Step 3. We show that T is maps bounded subsets into bounded sets.
From Steps 3-5 and Lemma 2 we see that T maps bounded sets into relatively compact sets.
Therefore, the operator T : X → X is completely continuous.The proof of (iii) is complete.The proof is complete.Now, we address the first result of this paper.We need the following assumption.
Assumption A. There exist nonnegative constants A j , a ij , b ij 0 (i = 1, 2, j = 0, 1, 2, . . ., m), φ s 0, ψ s 0 (s ∈ Z) and k j , l j 0 (j = 1, 2, . . ., m) with k j + l j > 0 and We denote σ = max{k j + l j : j = 1, 2, . . ., m} and Theorem 1. Suppose that Assumption A holds, f is a strong Carathéodory function, I, J are discrete Carathéodory functions, and for each r > 0, f (t, σ(t)u, Φ −1 (τ (t))v) converges uniformly as t → ±∞ on [−r, r] × [−r, r].Then BVP (1) has at least one solution if Proof.Let X and T be defined above.From Lemma 3, T : X → X is well defined and is a completely continuous operator.We prove that T has a fixed point in X to get a solution of BVP (1).For x ∈ X, we have x r < +∞.Then Assumption A implies that By the definition of T , we get by using (4) that . Then A j r kj +lj .

It follows that
Then On the other hand, we have Nonlinear Anal.Model.Control, 21(5):651-672
(iii) σ > 1 and B(A Then we get T x A + B + Br σ 0 r 0 .So T Ω 0 ⊂ Ω 0 .Thus Schauder's fixed point theorem implies that the operator T has at least one fixed point in Ω 0 .So BVP(1) has at least one solution.
The proof of Theorem 1 is completed.
3 Solvability of (1) (1) (1) in Case 2 In this section, we present existence result of BVP (1) in Case the following limits exist and are finite: http://www.mii.lt/NALemma 5. X is a Banach space with • defined.
Proof.It is similar to the proof of Lemma 1 and is omitted.
Lemma 6.Let M be a subset of X.Then M is relatively compact if and only if the following conditions are satisfied: Proof.It is similar to the proof of Lemma 2 and is omitted.
For x ∈ X, let (T x)(t) be defined by where A x is defined by (3).
Lemma 7. Suppose that f is a strong Carathéodory function and I, J are discrete Carathéodory functions, and for each r > 0, f (t, u, Φ −1 (τ (t))v) converges uniformly as t → ±∞ on [−r, r] × [−r, r].Then T : X → X is well defined and is completely continuous, x ∈ X is a solution of (1) if and only if x ∈ X is a fixed point of T in X.
Proof.It is similar to the proof of Lemma 4 and is omitted.
To state and prove Theorem 2, we need the following assumption.