Systems of differential equations with implicit impulses and fully nonlinear boundary conditions

We show that systems of second-order ordinary differential equations, x^{″}=f(t,x,x^{\prime }), subject to compatible nonlinear boundary conditions and impulses, have a solution x such that (t,x(t)) lies in an admissible bounding subset of [0,1]\times {\mathbb{R}}^n when f satisfies a Hartman-Nagumo growth bound with respect to x^{\prime }. We reformulate the problem as a system of nonlinear equations and apply Leray-Schauder degree theory. We compute the degree by homotopying to a new system of nonlinear equations based on the simpler system of ordinary differential equations, x^{″}=M_{0}L(x-v), subject to Picard boundary conditions and impulses and using the Leray index theorem. Our proof is simpler than earlier existence proofs involving nonlinear boundary conditions without impulses and requires weak assumptions on f.

MSC:34A37, 34A34, 34B15.

Let q\in \mathbb{N}, the natural numbers,

J_0=[t_0,t_1] and J_k=(t_k,t_{k+1}] for 1\le k\le q. We call Q a division of the interval [0,1].

We consider the system of second-order ordinary differential equations

subject to very general nonlinear boundary conditions of the form

and very general nonlinear implicit impulses of the form

where

satisfies f{|}_{J_k\times {\mathbb{R}}^{2n}} has an extension to f_k\in C({\overline{J}}_k\times {\mathbb{R}}^{2n};{\mathbb{R}}^n) and

for 0\le k\le q. Our fully nonlinear boundary conditions (2) include the Picard, periodic, and Neumann boundary conditions as special cases. We establish a general existence result for solutions lying in an admissible bounding set for the system of ordinary differential equations (1) satisfying boundary conditions (2) and impulses (3).

Our result is closely related to those of Thompson [1] and of Kongson et al. [2]. In [1] and [2], the authors established existence results for systems of second-order ordinary differential equations in more general bounding sets and subject to general boundary conditions (2) but not subject to impulses. Moreover, the proof in [1] is incomplete as it fails to establish the required derivative bounds; these appear to require more assumptions on the Hartman-Nagumo growth bound than we assume here. Although our bounding sets are more restrictive than those in [2], our proof is much simpler than theirs. In particular, the ideas introduced in our proof offer a fresh starting point for further work aimed at identifying the natural and most general concept of a bounding set and with this the natural and most general existence results possible for system (1) subject to nonlinear boundary conditions (2).

Earlier works on boundary value problems homotopies the original problem (1), plus nonlinear boundary conditions (2), to x^{″}(t)=0 plus the Picard boundary conditions; see, for example, [3]. This requires f to be redefined for (t,x) outside the admissible bounding set in such a way that solutions to the associated boundary value problem lie in the admissible bounding set. This in turn imposes restrictive assumptions on f and the associated bounding set. A key to our new idea is the observation that it suffices to homotopy our associated system of nonlinear equations to a new system of nonlinear equations associated with the simpler system x^{″}(t)=M_{0}L[x-v(t)] subject to Picard boundary conditions and impulses. This is uniquely solvable with the solution lying in the admissible bounding set. We use the Leray index theorem and the multiplication theorem to show that the degree of the associated nonlinear equation is not zero. Using our homotopies, we do not need to redefine the system outside the admissible bounding set. In the current work, we require the bounding set to be , where r:[0,1]\times {\mathbb{R}}^n\to \mathbb{R} and r(t,\cdot ) is strongly convex as a function of x (see Remark 1(i)).

A further motivation for our work comes from the paper by Cabada and Thompson [4] for a single equation with impulses. Recently, many papers devoted to the study of boundary value problems for nonlinear differential equations with impulses have appeared because of their wide applicability and associated rich theory. In the literature one can find different kinds of existence results for first-order [5, 6], second-order [7–9], and higher-order [10, 11] ordinary differential equations with periodic boundary conditions and impulses. In addition, some existence results for first-order impulsive differential equation with nonlinear boundary conditions can be found in [12–15]. In the papers [4, 16, 17], the ϕ-Laplacian and φ-Laplacian equations with impulses are considered.

This paper is organized as follows. In Section 2, we introduce the notation and definitions that we use in this paper. We give the definition of compatible boundary conditions and introduce our definition of compatible impulses in Section 3. In Section 4, we present the Nagumo-type condition that we use in our existence result to a priori bound the derivative of solutions. Section 5 is principally devoted to our main result where we prove that there are solutions to (1), (2), and (3) lying in an admissible bounding set. In Section 6, we present an example.

In this section, we present the notation, definitions, and assumptions that we use to obtain a priori bounds on solutions.

Let H denote finite or infinite dimensional Hilbert spaces. For a bounded subset V of H, let V^{\circ } denote its interior, ∂V its boundary and \overline{V} its closure. For a bounded subset U of [0,1]\times {\mathbb{R}}^n and t\in [0,1], let U(t) denote its t-cross section and \partial U(t) denote the boundary of U(t) in {\mathbb{R}}^n. Thus U(t)=\{x\in {\mathbb{R}}^n:(t,x)\in U\}. Let {\partial }_{C}U denote the curved boundary of U, so {\partial }_{C}U={\bigcup }_{t\in [0,1]}\partial U(t) excludes the sets \{0\}\times U^{\circ }(0) and \{1\}\times U^{\circ }(1) from ∂U. For x\in \mathbb{R}, |x| denotes the absolute value of x. For x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n and y=(y_1,\dots ,y_n)\in {\mathbb{R}}^n, x^T denotes the transpose of x while x\cdot y denotes the scalar product of x and y. Let ℐ denote the identity on H so \mathcal{I}(x)=x for all x. If X is a Banach space and A\subset H, then C^m(A;X) denotes the space of m-times continuously differentiable functions from A to X with a finite norm. In the case of continuous functions, we omit the m, while in the case of real-valued functions, we omit the X.

Let J\subset \mathbb{R} be an interval. For r\in C^2(J\times {\mathbb{R}}^n), let r_t(t,x) denote the partial derivative with respect to t, r_x(t,x) denote the gradient, and r_{xx}(t,x) denote the matrix of second-order partial derivatives of r with respect to x.

The norm on C^m(J;{\mathbb{R}}^n) is given by

where u^{(k)} denotes the k th derivative of u. By abuse of notation, we abbreviated C^m(J;{\mathbb{R}}^n) to C^m(J). Further we will abbreviate {\parallel u\parallel }_{C^m(J)} to \parallel u\parallel when the meaning is clear from the context.

For \tau \in [0,1), let u^{(l)}({\tau }^{+})={lim}_{t\to {\tau }^{+}}{u}^{(l)}(t) and for \tau \in (0,1], let u^{(l)}({\tau }^{-})={lim}_{t\to \tau -}{u}^{(l)}(t) for 0\le l\le m. To simplify statements of results, set u^{(l)}(1^{+})=u^{(l)}(1) and u^{(l)}(0^{-})=u^{(l)}(0) for 0\le l\le m, where u^{(l)}(1) and u^{(l)}(0) are the appropriate one-sided derivatives.

In order to define the concept of solution for our problem, we consider the following sets. Let

All our limits are assumed to be {\mathbb{R}}^n-valued when they exist. Thus, for u\in C_Q^m, u^{(l)}(t_k^{\pm }) exists for k=0,\dots ,q+1, l=0,\dots ,m. Note that C_Q^0 is defined in the obvious way. Thus we may identify x\in C_Q^m with \tilde{x}=(x_0,\dots ,x_q)\in {\prod }_{k=0}^q{C}^m({\overline{J}}_k), where \tilde{x}(t)=x_k(t), for all t\in J_k. By abuse of notation, we will denote \tilde{x} by x where the meaning is clear from the context. Further we define a norm on C_Q^m by

If A is a bounded open subset of H, \mathcal{G}(x)=x+\mathcal{K}(x), where \mathcal{K}\in C(\overline{A},H), \mathcal{K}(\overline{A}) has compact closure and p\in H\setminus \mathcal{G}(\partial A), then d(\mathcal{G},A,p) denotes the Leray-Schauder degree of \mathcal{G} on A at p. In the special case that H={\mathbb{R}}^n and \mathcal{G}\in C(\overline{A},{\mathbb{R}}^n), p\in {\mathbb{R}}^n\setminus \mathcal{G}(\partial A), d(\mathcal{G},A,p) is the Brouwer degree.

By a solution x we mean a function x\in C_Q^2 satisfying (1) for all t\in [0,1]\setminus Q, (2) and (3).

We look for solutions to problem (1) together with the fully nonlinear boundary conditions (2) and impulses (3) in the following admissible bounding set which provides a priori bounds on solutions to (1).

Definition 1 Let \mathrm{\Omega }\subset [0,1]\times {\mathbb{R}}^n be a bounded set and v\in C_Q^2. We call (\mathrm{\Omega },v) an admissible bounding set for (1) if it has the following properties:

1. (i)

   There is r:[0,1]\times {\mathbb{R}}^n\to \mathbb{R} such that

   1. (a)

      r{|}_{J_k\times {\mathbb{R}}^n} can be uniquely extended to r_k\in C^2({\overline{J}}_k\times {\mathbb{R}}^n) for all 0\le k\le q;

   2. (b)

      ;

   3. (c)

      {\sum }_{i,j=1}^n{r}_{x_i{x}_j}(t,x){\xi }_i{\xi }_j\ge \mathrm{\Theta }{\parallel \xi \parallel }^2 for some constants \mathrm{\Theta }>0, all \xi \in {\mathbb{R}}^n and (t,x)\in \mathrm{\Omega };

2. (ii)

   There is \epsilon >0 such that B_{\epsilon }(v_k(t))\subseteq {\mathrm{\Omega }}_k(t), where for all 0\le k\le q;

3. (iii)

   If t\in (0,1)\setminus Q, p\in {\mathbb{R}}^n, r(t,u)=0 and r^{\prime }(t,u,p)=0, then

   r_f^{\prime \prime }(t,u,p)>0,

   where

   r^{\prime }(t,u,p)=r_t(t,u)+r_x^T(t,u)p,

   (4)
   r_f^{\prime \prime }(t,u,p)=r_{tt}(t,u)+2r_{tx}^T(t,u)p+p^T{r}_{xx}(t,u)p+r_x^T(t,u)f(t,u,p);

   (5)
4. (iv)

   \parallel r_x(t,x)\parallel \ge c>0 for all (t,x)\in {\partial }_C\mathrm{\Omega } and some constant c>0.

Remark 1

1. (i)

   A function r\in C^2({\mathbb{R}}^n) is strongly convex iff for some constants \mathrm{\Theta }>0,

   \sum _{i,j=1}^n{r}_{xx}(x){\xi }_i{\xi }_j\ge \mathrm{\Theta }{\parallel \xi \parallel }^2

   (6)

   for x,\xi \in {\mathbb{R}}^n (see Part 4 in [18]). If r\in C^2({\mathbb{R}}^n) satisfies (6), then r is uniformly convex, see Appendix B.1. in [19]. Moreover, r\in C^2({\mathbb{R}}^n) satisfies (6) when \mathrm{\Theta }=0 iff r is convex (see Appendix B.1. in [19]). From the definition of convex function, it is easy to see that

   is a convex set for t\in [0,1].

   It follows from Definition 1(i)(c) that for (t,x,p)\in \overline{\mathrm{\Omega }}\times {\mathbb{R}}^n,

   \begin{array}{rcl}{r}_0^{\prime \prime }(t,x,p)& =& p^T{r}_{xx}(t,x)p+2p^T{r}_{tx}(t,x)+r_{tt}(t,x)\\ \ge & \mathrm{\Theta }{\parallel p\parallel }^2-2D\parallel p\parallel -D\\ >& -K_1,\end{array}

   (7)

   where D={sup}_{(t,x)\in \mathrm{\Omega }}\{\parallel r_{tx}(t,x)\parallel ,\parallel r_{tt}(t,x)\parallel \} and K_1>0.

2. (ii)

   It follows from Definition 1(i)(a), (ii) and (iv) that

   \begin{array}{r}\parallel r_x(t,x)\parallel \ge c>0,\\ (x-v(t))\cdot r_x(t,x)>\eta \parallel r_x(t,x)\parallel \text{and hence}\\ (x-v(t))\cdot r_x(t,x)>\eta c>0\end{array}

   (8)

   for all (t,x)\in {\bigcup }_k{\partial }_C{\mathrm{\Omega }}_k and some \eta >0.

Set

for 1\le k\le q, where \mathrm{\Omega }(t_k^{+})={\mathrm{\Omega }}_k(t_k) and \mathrm{\Omega }(t_k^{-})={\mathrm{\Omega }}_{k-1}(t_k). Let

We assume that f satisfies the following conditions.

Definition 2 Let (\mathrm{\Omega },v) be an admissible bounding set for (1). We say that f satisfies the Hartman-Nagumo condition on Ω if:

1. (i)

   f{|}_{J_k\times {\mathbb{R}}^{2n}} has an extension to f_k\in C({\overline{J}}_k\times {\mathbb{R}}^{2n});

2. (ii)

   \parallel f(t,x,p)\parallel \le \mathrm{\Phi }(\parallel p\parallel ) for all (t,x,p)\in \mathrm{\Omega }\times {\mathbb{R}}^n, where

   {\int }^{\mathrm{\infty }}\frac{s}{\mathrm{\Phi }(s)}ds=\mathrm{\infty };
3. (iii)

   \parallel f(t,x,p)\parallel \le M{r}_f^{\prime \prime }(t,x,p)+K for all (t,x,p)\in \mathrm{\Omega }\times {\mathbb{R}}^n, where M and K are nonnegative constants and r_f^{″} is given by (5).

Remark 2 If conditions (ii) and (iii) above are satisfied, a solution x of (1) with (t,x(t))\in \mathrm{\Omega } satisfies the Hartman-Nagumo inequality (see the second paragraph on p.702 in [20]).

Following [1], we give the definition of compatible boundary conditions and introduce the definition of compatible impulses. These are simple, degree-based relationships between the boundary conditions, the impulses, and the associated admissible bounding set. For more information on compatibility of boundary conditions, we refer the reader to [1, 21], and [[4], Definition 14].

Definition 3 For 1\le k\le q, we call the vector field {\mathrm{\Psi }}_k=({\psi }_k^0,{\psi }_k^1)\in C({\overline{\mathrm{\Delta }}}_k;{\mathbb{R}}^{2n}) strongly inwardly pointing on {\mathrm{\Delta }}_k if for all (C_k,D_k)\in {\overline{\mathrm{\Delta }}}_k,

where {\mathrm{\Delta }}_k, {\mathrm{\Omega }}_k(t_k), {\mathrm{\Omega }}_{k-1}(t_k) are given in (9). We call the vector field {\mathrm{\Psi }}_0=({\psi }_0^0,{\psi }_0^1)\in C({\overline{\mathrm{\Delta }}}_0;{\mathbb{R}}^{2n}) strongly inwardly pointing on {\mathrm{\Delta }}_0:={\mathrm{\Omega }}_0(0)\times {\mathrm{\Omega }}_q(1) if for all (C_0,D_0)\in {\overline{\mathrm{\Delta }}}_0,

where r_k (k=0,\dots ,q) is the extension to {\overline{J}}_k of r{|}_{J_k^{\circ }} and r_k^{\prime } is given by (4). From (9), {\mathrm{\Omega }}_0(0)=\mathrm{\Omega }(0); {\mathrm{\Omega }}_q(1)=\mathrm{\Omega }(1). For k=0,\dots ,q, we call {\mathrm{\Psi }}_k inwardly pointing on {\mathrm{\Delta }}_k if the above inequalities are weak.

In what follows, where there is a strongly inwardly pointing vector field {\mathrm{\Psi }}_k on {\overline{\mathrm{\Delta }}}_k for all 0\le k\le q, then {\mathcal{G}}_k is defined by

for all (C_k,D_k)\in {\overline{\mathrm{\Delta }}}_k, 0\le k\le q.

The following definition is a variant of Definition 2.5 given in [1].

Definition 4 Let 0\le k\le q and g_k\in C({\overline{\mathrm{\Delta }}}_k\times {\mathbb{R}}^{2n};{\mathbb{R}}^{2n}). We say g_k is strongly compatible with Ω if

for all (C_k,D_k,u_k,v_k)\in {\overline{\mathrm{\Delta }}}_k\times {\mathbb{R}}^{2n} such that

and/or

and

for any strongly inwardly pointing vector field {\mathrm{\Psi }}_k on {\overline{\mathrm{\Delta }}}_k.

For 0\le k\le q, we say that g_k is compatible with Ω if there is a sequence g_{k_i}\in C({\overline{\mathrm{\Delta }}}_k\times {\mathbb{R}}^{2n};{\mathbb{R}}^{2n}) strongly compatible with Ω and converging uniformly to g_k on compact subsets of {\overline{\mathrm{\Delta }}}_k\times {\mathbb{R}}^{2n}.

In the literature, there are many variants of the ‘Nagumo condition’ which are used to establish a priori bounds on the derivative of bounded solutions.

We use the following variant of Lemma 4.1 in [2].

Lemma 1 Let \mathrm{\Phi }\in C([0,\mathrm{\infty });[0,\mathrm{\infty })) satisfy

and r be given in Definition  1(i). Let x be a solution of(1) satisfying r(t,x(t))\le 0. Assume that

for (t,x) such that r(t,x)\le 0 and p\in {\mathbb{R}}^n, where M_1, M_2, and K are nonnegative constants and r_f^{\prime \prime } is given by (5). Then there exists N=N(r,M_1,M_2,K,\mathrm{\Phi })>0 such that .

Proof Since r is given in Definition 1(i), then \parallel x\parallel \le R_2 when r(t,x)\le 0, where R_2 is given in (10). Thus the proof of Lemma 5.2 of Hartman [22] carries over to our case on {\overline{\mathrm{\Omega }}}_k, and it follows that for t\in {\overline{J}}_k. Thus for t\in [0,1], where N={max}_{0\le k\le q}{N}_k(r,M_1,M_2,K,\mathrm{\Phi }). □

Remark 3 The function \mathrm{\Phi }\equiv 1 satisfies (13).

In this section, we present the main result of this paper. We prove the existence of at least one solution to nonlinear problem (1), (2), and (3) lying in an admissible bounding set. To achieve this, we turn our impulsive boundary value problem into an equivalent nonlinear equation and use Leray-Schauder degree theory. We compute the degree using three homotopies, the Leray index theorem and the multiplication theorem.

The first homotopy involves \mathcal{S}(x,C,D,\lambda )=({\mathcal{S}}_0,\dots ,{\mathcal{S}}_q), where

The second and third homotopies are constructed using one-parameter families of systems of ordinary differential equations.

We construct our first family of systems of differential equations using f_0 defined below.

Let Φ, K_1, M, and K be given in Lemma 1, Remark 1, and Definition 2, respectively. Let

for (t,x,p)\in [0,1]\times {\mathbb{R}}^{2n}, where

and ε is given below. Firstly, we consider M_2>M>0 where M is given in Definition 2. For case M=0, see Remark 5. Let

Remark 4

1. (i)

   It follows that f_0{|}_{J_k\times {\mathbb{R}}^{2n}} has a continuous extension to {\overline{J}}_k\times {\mathbb{R}}^{2n}.

2. (ii)

   It follows from Remark 1 that M_0, K_2, and L are well defined when M>0 where M is given in Definition 2.

For \lambda \in [0,1], we define f_{\lambda }:[0,1]\times {\mathbb{R}}^{2n}\to {\mathbb{R}}^n by

where f_0 and f are given in (14) and (1), respectively.

We consider the system

Lemma 2 Let (\mathrm{\Omega },v) be an admissible bounding set for (1) and assume that f satisfies the Hartman-Nagumo condition and that f_{\lambda } is given by (18). Then for (t,x,p)\in \overline{\mathrm{\Omega }}\times {\mathbb{R}}^n,

where Φ is given in Lemma  1, K_2 is given in (17) and r_f^{″} is given by (5) and M_1, M_2 are nonnegative numbers.

If x is a solution of (19) with (t,x)\in \overline{\mathrm{\Omega }}, then where N is given in Lemma  1.

Moreover, if t\in (0,1)\setminus Q, p\in {\mathbb{R}}^n, r(t,x)=0, and r^{\prime }(t,x,p)=0, then r_{f_{\lambda }}^{\prime \prime }(t,x,p)>0.

Proof It follows from (17) that

for (t,x,p)\in \overline{\mathrm{\Omega }}\times {\mathbb{R}}^n. Since f satisfies the Hartman-Nagumo condition, thus for (t,x,p)\in \overline{\mathrm{\Omega }}\times {\mathbb{R}}^n, and R_1 is given in (10), it is easy to see that

Since M_2>M>0, it follows from (17) that

If x is a solution of (19) with (t,x)\in \overline{\mathrm{\Omega }}, it follows from Lemma 1 that where N is given in Lemma 1.

From Definition 1(iii), if t\in (0,1)\setminus Q, p\in {\mathbb{R}}^n, r(t,x)=0, and r^{\prime }(t,x,p)=0, then r_f^{\prime \prime }(t,x,p)>0. Since f satisfies the Hartman-Nagumo condition, so \parallel f\parallel \le \mathrm{\Phi }(\parallel p\parallel ). It follows from (15) that

If \mathrm{\Phi }(\parallel p\parallel )\le L from (14), then f_0=M_0\mathrm{\Phi }(\parallel p\parallel )[x-v(t)] and

If L\le \mathrm{\Phi }(\parallel p\parallel ) from (14), then f_0=M_{0}L[x-v(t)]. It follows from (16) and (7) that

Thus

 □

Now we construct the second one-parameter family of systems of ordinary differential equations.

For \lambda \in [0,1], we define f_{1,\lambda }:[0,1]\times {\mathbb{R}}^{2n}\to {\mathbb{R}}^n by

where f_0, M_0, L are given in (14), (15), and (16), respectively.

We consider the system

Lemma 3 Assume that (\mathrm{\Omega },v) is an admissible bounding set for (1) and that f_{1,\lambda } is defined in (20). Then, for (t,x,p)\in \overline{\mathrm{\Omega }}\times {\mathbb{R}}^n,

where R_1 is given in (10).

If x is a solution of (21) with (t,x)\in \overline{\mathrm{\Omega }}, then , where N is given in Lemma  1.

Moreover, if t\in (0,1)\setminus Q, p\in {\mathbb{R}}^n, r(t,x)=0, and r^{\prime }(t,x,p)=0, then r_{f_{1,\lambda }}^{\prime \prime }(t,x,p)>0.

Proof Clearly,

for all (t,x,p)\in \overline{\mathrm{\Omega }}\times {\mathbb{R}}^n, where R_1 is given in (10). From the proof of Lemma 2, \parallel f_0(t,x,p)\parallel \le M_2{r}_{f_0}^{\prime \prime }+K_2 for all (t,x)\in \overline{\mathrm{\Omega }}, p\in {\mathbb{R}}^n. It follows from (17) and (8) that

Thus

If x is a solution of (21) with (t,x)\in \overline{\mathrm{\Omega }}, then it follows from Lemma 1 that , where N is given in Lemma 1.

From the proof of Lemma 2, if t\in (0,1)\setminus Q, p\in {\mathbb{R}}^n, r(t,x)=0, r^{\prime }(t,x,p)=0, then r_{f_0}^{\prime \prime }(t,x,p)>0. It follows from (15) and (16), respectively, that

where K_1 is given in Remark 1(i). Therefore,

 □

Remark 5 If M=0, where M is given in Definition 2, we do not need to choose M_2 and K_2 in (14). We set

for (t,x,p)\in [0,1]\times {\mathbb{R}}^{2n}. Moreover, we do not need the second one-parameter family of systems of ordinary differential equations based on f_{1,\lambda } to construct our homotopy.

For 0\le k\le q and (t,x)\in {\overline{\mathrm{\Omega }}}_k, let G_k:{\overline{J}}_k\times {\overline{J}}_k\to \mathbb{R} be Green’s function for (1) restricted to {\overline{J}}_k together with the homogeneous boundary conditions x_k(t_k)=A=0=B=x_k(t_{k+1}), thus

For 0\le k\le q, let

Using the above two families of systems of ordinary differential equations, we can homotopy the original problem (1), (2), and (3) to the following solvable system of ordinary differential equations subject to Picard boundary conditions and impulses.

where M_0, L are given in (15) and (16), respectively. Then (24) and (25) have a solution V\in C_Q^2=C_Q^2[0,1] of the form

for t\in J_k and 0\le k\le q, where we have identified V with \tilde{V}=(V_0,\dots ,V_q).

We show that V(t)\in \mathrm{\Omega }(t) for all t\in [0,1].

Lemma 4 Assume that (\mathrm{\Omega },v) is an admissible bounding set for (1) and V(t) is given by (26). Then V(t)\in \mathrm{\Omega }(t) for t\in [0,1]. Moreover, , where N is given in Lemma  1.

Proof Suppose (\tilde{t},V(\tilde{t}))\notin \mathrm{\Omega } for some \tilde{t}\in {\overline{J}}_k. Set q(t)={[V(t)-v(t)]}^2. Since V(t) is a solution of (24) and (25), it follows that q(t_k^{+})=q(t_{k+1}^{-})=0, \mathrm{\forall }0\le k\le q, and so \tilde{t}\ne t_k and \tilde{t}\ne t_{k+1}. Therefore \tilde{t}\in J_k^{\circ }. So q(t) has a local maximum at \tilde{t}\in J_k^{\circ } and (\tilde{t},V(\tilde{t}))\notin \mathrm{\Omega }. Hence q(\tilde{t})\ge {\epsilon }^2, where ε is given below. But it follows from (15) and (16) that M_{0}L\epsilon >\parallel v^{″}(t)\parallel for all t\in [0,1] and hence

a contradiction. Thus (t,V(t))\in \mathrm{\Omega } for t\in [0,1]. Since V(t) is a solution of (24) and (21) is (24) when \lambda =0, it follows from Lemma 3 that , where N is given in Lemma 1. □

Now we present our main result.

Theorem 1 Assume that (\mathrm{\Omega },v) is an admissible bounding set for (1) and that f satisfies the Hartman-Nagumo condition. Suppose that the boundary conditions (2) and impulses (3) are compatible with Ω. Then there is at least one solution x\in C_Q^2 of problem (1), (2), and (3) such that (t,x(t))\in \overline{\mathrm{\Omega }} for t\in [0,1].

Proof Now {\mathrm{\Delta }}_k\ne \mathrm{\varnothing } for 0\le k\le q. First consider the case that all g_k are strongly compatible with Ω.

Choose \epsilon \in (0,1) such that B_{\epsilon }(v(t))\subseteq \mathrm{\Omega }(t) for all t\in [0,1]. It follows from Remark 1(ii) that (x-v(t))\cdot r_x(t,x)>\eta c>0, where \eta >0, c>0, for x\in {\bigcup }_k{\partial }_C{\mathrm{\Omega }}_k(t) and all t\in [0,1]. Let M_0, L, and K_2 be given in (15), (16), and (17), respectively. Let M_2>M where M is given in Definition 2.

Let

and let \mathrm{\Sigma }=\mathrm{\Pi }\times \mathrm{\Delta }, where \mathrm{\Delta }={\prod }_{k=0}^q{\mathrm{\Delta }}_k and {\mathrm{\Delta }}_k is given in (9) and N is given in Lemma 1.

Following [4], we interpret (C,D)=(C_0,\dots ,C_q,D_0,\dots D_q)\in △ to mean (C_k,D_k)\in {\mathrm{\Delta }}_k for k=0,\dots ,q and set D_{q+1}=D_0. Let \mathrm{\Psi }(C,D)=({\mathrm{\Psi }}_0(C_0,D_0),\dots ,{\mathrm{\Psi }}_q(C_q,D_q)), where {\mathrm{\Psi }}_k is a strongly inwardly pointing vector field on {\mathrm{\Delta }}_k for each k. Let \mathcal{G}(C,D)=({\mathcal{G}}_0(C_0,D_0),\dots ,{\mathcal{G}}_q(C_q,D_q)), where {\mathcal{G}}_k(C_k,D_k) is given in (11), for all 0\le k\le q.

Let f_{\lambda } be given in (18). For all t\in {\overline{J}}_k, let

where G_k(t,s) is given in (22). Define

where we identify x and \tilde{x}=(x_0,\dots ,x_q).

Consider the solutions (x,C,D)\in \overline{\mathrm{\Sigma }} of

where