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

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

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 $\{(t,x)\in [0,1]\times {\mathbb{R}}^n:r(t,x)<0\}$, 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.

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.

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).

We assume that f satisfies the following conditions.

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].

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.

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].

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].

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 $\parallel x^{\prime }(t)\parallel <N$.

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 $\parallel x^{\prime }(t)\parallel <N_k(r,M_1,M_2,K,\mathrm{\Phi })$ for $t\in {\overline{J}}_k$. Thus $\parallel x^{\prime }(t)\parallel <N$ 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 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.

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

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 $\parallel x^{\prime }(t)\parallel <N$ 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 $r_{f_{\lambda }}^{\prime \prime }(t,x,p)>0$.

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

 □

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

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

where $R_1$ is given in (10).

If x is a solution of (21) with $(t,x)\in \overline{\mathrm{\Omega }}$, then $\parallel x^{\prime }(t)\parallel <N$, 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 $r_{f_{1,\lambda }}^{\prime \prime }(t,x,p)>0$.

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

 □

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 $t\in J_k$ 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, $\parallel V^{\prime }(t)\parallel <N$, where N is given in Lemma  1.