Existence of solutions for impulsive wave equations

1.   Introduction

Mechanical systems with impact, heart beats, blood flows, population dynamics, industrial robotics, biotechnology, economics, etc are real world and applied science phenomena which are abruptly changed in their states at some time instants due to short time perturbations whose duration is negligible in comparison with the duration of these phenomena, please see ^[18]. They are called impulsive phenomena. A natural framework for mathematical simulation of such phenomena are impulsive differential equations or impulsive partial differential equations when more factors are taking into account.

Whereas impulsive differential equations are well studied, see for example the books ^[4,8,39,42] and the references therein, the literature concerning impulsive partial differential equations does not seem to be very rich. The history of impulsive partial differential equations starts at the end of the 20th century with the pioneering work ^[17], in which, impulsive partial differential systems have been showed to be a natural framework for the mathematical modeling of processes in ecology and biology, like population growth, see also ^{[10,11,13,31]}. We can find studies of first order partial differential equations with impulses in ^[5,23,32,40]. Higher-order linear and nonlinear impulsive partial parabolic equations were considered in ^[22]. An initial boundary value problem for a nonlinear parabolic partial differential equation was discussed in ^[9]. The approximate controllability of an impulsive semilinear heat equation was proved in ^[1]. A class of impulsive wave equations was investigated in ^[21]. In ^[29] a class of impulsive semilinear evolution equations with delays is investigated for existence and uniqueness of solutions. The investigations in ^[29] includes several important partial differential equations such as the Burgers equation and the Benjamin-Bona-Mohany equation with impulses, delays and nonlocal conditions. A class of semilinear neutral evolution equations with impulses and nonlocal conditions in a Banach space is investigated in ^[2] for existence and uniqueness of solutions. To prove the main results in ^[2] the authors use a Karakostas fixed point theorem. In ^[2] an example involving Burger's equation is provided to illustrate the application of the main results. Some studies concerning impulsive Burgers equation can be found in ^[16,27,34].

Many classical methods have been successfully applied for solving impulsive partial differential equations. By using variational method, the existence of solutions for a fourth-order impulsive partial differential equations with periodic boundary conditions was obtained in ^[30]. The Krasnoselskⅱ theorem is used to prove existence and uniqueness of solutions for impulsive Hamilton-Jacobi equation in ^[35]. Some other references on impulsive partial differential equations are: ^{[3,6,12,19,20,24,25,26,28,33,36,38,41]}.

In this paper, we investigate the following impulsive wave equation

where

(H1) $T = t_{m+1} > t_m > \ldots > t_1 > t_{0} = 0$, $J = [0, T]$, $J_0 = J\backslash \{t_k\}_{k = 1}^m$, $m\in \mathbb{N}$, $\lambda\ne 1$.

(H2) $J_k\in \mathcal{C}([0, T]\times \mathbb{R}^n)$, $|J_k(v)|\leq A|v|^{r_k}$, $v\in \mathbb{R}$, $r_k > 0$, $k\in \{1, \ldots, m\}$, $A > 0$ is a constant.

(H3) $u_0\in \mathcal{C}^1(\mathbb{R}^n)$, $|u_0|\leq B$ on $\mathbb{R}^n$.

(H4) $f\in \mathcal{C}([0, T]\times \mathbb{R}^n\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}^n)$,

$t \in [0, T]$, $x\in \mathbb{R}^n$, $u, v, w\in \mathbb{R}$, $a_j, b_j, c_{ji}\in \mathcal{C}([0, T]\times \mathbb{R}^n)$, $0\leq a_j, b_j, c_{ji}\leq C$ on $[0, T]\times \mathbb{R}^n$,

$C > 0$, $p_j, q_j, r_{ji} > 0$, $j\in \{1, \ldots, r\}$, $i\in \{1, \ldots, n\}$, $r\in \mathbb{N}$.

Here

The main aim of this paper is to investigate the problem (1.1) for existence and non-uniqueness of solutions. The main objective in this article is to show and develop some topological methods in order to prove the existence of solutions to the problem from many points of view and angles under different conditions to give a good and useful data and information on the solutions to the implementers to be exploited in the best way.

The work is organized as follows. In the next section, we give some preliminary results and tools. In Section 3, we prove existence of at least one solution for the problem (1.1). In Section 4, we prove existence of at least one nonnegative solution of the problem (1.1). In Section 5, we prove existence of at least two nonnegative solutions of the problem (1.1). In Section 6, we give an example that illustrates our main results.

2.   Tools

Some preliminary tools are needed to prove the main results. The following fixed point theorem for a sum of two operators is used to prove existence of at least one solution to the problem (1.1).

Theorem 2.1. Let $\epsilon \in (0, 1)$, $B > 0$, $E$ be a Banach space and $X = \{x\in E$: $\Vert x\Vert\leq B\}$. Let also, $Tx = -\epsilon x$, $x\in X$, $S$: $X\to E$ is continuous, $(I-S)(X)$ resides in a compact subset of $E$ and

for any $\lambda\in \left(0, \frac{1}{\epsilon}\right)$. Then there exists a $x^*\in X$ so that

Here $\mu X = \{\mu x$: $x\in X\}$ for any $\mu\in \mathbb{R}$.

Proof. Define

Then $r\left(-\frac{1}{\epsilon}(I-S)\right)$: $X\to X$ is continuous and compact. Now, we apply the Schauder fixed point theorem and we conclude that there exists $x^*\in X$ so that

Assume that $-\frac{1}{\epsilon}(I-S)x^*\not\in X$. Then

and

and hence, $\Vert x^*\Vert = B$. This contradicts with (2.1). Therefore $-\frac{1}{\epsilon}(I-S)x^*\in X$ and

or

or

This completes the proof.

Let $X$ be a real Banach space.

Definition 2.1. A mapping $K$: $X\to X$ is said to be completely continuous if it is continuous and maps bounded sets into relatively compact sets.

The concept for $l$-set contraction is related to that of the Kuratowski measure of noncompactness which we recall for completeness.

Definition 2.2. Let $\Omega_X$ be the class of all bounded sets of $X$. The Kuratowski measure of noncompactness $\alpha$: $\Omega_X\to [0, \infty)$ is defined by

where $diam\; (Y_j) = \sup\{\Vert x-y\Vert_X$: $x, y\in Y_j\}$ is the diameter of $Y_j$, $j\in \{1, \ldots, m\}$.

For the main properties of measure of noncompactness we refer the reader to ^[7].

Definition 2.3. A mapping $K$: $X\to X$ is said to be $l$-set contraction if it is continuous, bounded and there exists a constant $l\geq 0$ such that

for any bounded set $Y\subset X$. The mapping $K$ is said to be a strict set contraction if $l < 1$.

Obviously, if $K$: $X\to X$ is a completely continuous mapping, then $K$ is $0$-set contraction (see ^[15]).

Definition 2.4. Let $X$ and $Y$ be real Banach spaces. A mapping $K$: $X\to Y$ is said to be expansive if there exists a constant $h > 1$ such that

for any $x, y\in X$.

Definition 2.5. A closed, convex set $\mathcal{P}$ in $X$ is said to be cone if

(1) $\alpha x\in \mathcal{P}$ for any $\alpha\geq 0$ and for any $x\in \mathcal{P}$,

(2) $x, -x\in \mathcal{P}$ implies $x = 0$.

Denote $\mathcal{P}^* = \mathcal{P}\backslash \{0\}$.

Lemma 2.1. Let $X$ be a closed convex subset of a Banach space $E$ and $U\subset X$ a bounded open subset with $0\in U.$ Assume there exists $\varepsilon > 0$ small enough and that $K$: $\overline{U}\rightarrow X$ is a strict $k$-set contraction that satisfies the boundary condition:

Then $\, i\, (K, U, X) = 1.$

Proof. Consider the homotopic deformation $H$: $[0, 1]\times \overline{U}\rightarrow X$ defined by

The operator $H$ is continuous and uniformly continuous in $t$ for each $x,$ and the mapping $H(t, .)$ is a strict set contraction for each $t\in[0, 1]$. In addition, $H(t, .)$ has no fixed point on $\partial U$. On the contrary,

$\bullet$ If $t = 0$, there exists some $x_0\in \partial U$ such that $x_0 = 0$, contradicting $x_0\in U.$

$\bullet$ If $t\in(0, 1]$, there exists some $x_0 \in {\cal P}\cap\partial U$ such that $\frac{1}{\varepsilon+1}\, tKx_0 = x_0$; then $Kx_0 = \frac{1+\varepsilon}{t}x_0$ with $\frac{1+\varepsilon}{t}\geq 1+\varepsilon,$ contradicting the assumption. From the invariance under homotopy and the normalization properties of the index, we deduce

Now, we show that

We have

Then there exists $\gamma > 0$ such that

In other hand, we have $\frac{1}{\epsilon+1}Kx \rightarrow Kx$ as $\epsilon\rightarrow 0,$ for $x\in \overline{U}.$ So for $\varepsilon$ small enough

Define the convex deformation $G$: $[0, 1]\times\overline{U}\rightarrow X$ by

The operator $G$ is continuous and uniformly continuous in $t$ for each $x,$ and the mapping $G(t, .)$ is a strict set contraction for each $t\in[0, 1]$ (since $t+\frac{1}{\varepsilon+1}(1-t) < t+1-t = 1$). In addition, $G(t, .)$ has no fixed point on $\partial U$. In fact, for all $x\in \partial U$, we have

Then our claim follows from the invariance property by homotopy of the index.

Proposition 2.1. Let $\mathcal{P}$ be a cone in a Banach space $E$. Let also, $U$ be a bounded open subset of $\mathcal{P}$ with $0\in U.$ Assume that $T$: $\Omega\subset\mathcal{P}\rightarrow E$ is an expansive mapping with constant $h > 1,$ $S$: $\overline{U}\rightarrow E$ is a $l$-set contraction with $0\le l < h-1$, and $S(\overline{U})\subset(I-T)(\Omega).$ If {{there exists $\varepsilon\ge0$ such that }}

then the fixed point index $i_*\, (T+S, U\cap \Omega, {\mathcal P}) = 1.$

Proof. The mapping $(I-T)^{-1}S$: $\overline{U}\rightarrow \mathcal{P}$ is a strict set contraction and it is readily seen that the following condition is satisfied

Our claim then follows from the definition of $i_*$ and the following Lemma 2.1.

The following result will be used to prove existence of at least two nonnegative solutions to the problem (1.1).

Theorem 2.2. Let $\mathcal{P}$ be a cone of a Banach space $E$; $\Omega$ a subset of $\mathcal{P}$ and $U_1, U_2\; and\; U_3$ three open bounded subsets of $\mathcal{P}$ such that $\overline{U}_1\subset \overline{U}_2\subset U_3$ and $0\in U_1.$ Assume that $T$: $\Omega\rightarrow \mathcal{P}$ is an expansive mapping with constant $h > 1,$ $S$: $\overline{U}_3\rightarrow E$ is a $k$-set contraction with $0\le k < h-1$ and $S(\overline{U}_3)\subset(I-T)(\Omega).$ Suppose that $({U}_2\setminus \overline{U}_1)\cap\Omega\neq\emptyset, \, ({U}_3\setminus \overline{U}_2)\cap\Omega\neq\emptyset,$ and there exists $u_{0}\in\mathcal{P}^*$ such that the following conditions hold:

(ⅰ) $Sx\neq (I-T)(x-\lambda u_{0}), \;$ for all $\lambda > 0$ and $x\in \partial U_1\cap(\Omega+\lambda u_0).$

(ⅱ) There exists $\epsilon\geq 0$ such that $Sx\neq (I-T)(\lambda x), \;$ for all $\, \lambda\geq 1+\epsilon, \, x\in\partial U_2$ and $\lambda x\in \Omega$.

(ⅲ) $Sx\neq (I-T)(x-\lambda u_{0}), \;$ for all $\lambda > 0$ and $x\in \partial U_3\cap(\Omega+\lambda u_0).$

Then $T+S$ has at least two non-zero fixed points $x_1, x_2\in \mathcal{P}$ such that

or

Proof. If

for $x\in \partial U_2\cap \Omega$, then we get a fixed point $x_1\in \partial U_2\cap\Omega$ of the operator $T+S$.

Suppose that

for any $x\in \partial U_2\cap\Omega$. Without loss of generality, assume that

Then, by Propositions 2.11 and 2.16 in ^[14] and Proposition 2.1, we have

The additivity property of the index yields

Consequently, by the existence property of the index, $T+S$ has at least two fixed points $x_1\in (U_2\setminus {U}_1)\cap\Omega {\mbox{ and }} x_2\in (\overline{U}_3\setminus \overline{U}_2)\cap\Omega.$ For $l, s\in \mathbb{N}\cup\{0\}$, define

and it is continuous on $(t_k, t_{k+1}) \exists \lim\limits_{t\to t_k+}g^{(l)}(t)$ and $\lim\limits_{t\to t_k}g^{(l)}(t)\}$,

and

In $E = PC^2([0, T], \mathcal{C}^2(\mathbb{R}^n))$, define a norm

provided it exists. Note that $E$ is a Banach space. For $u\in E$, set

$t \in [0, T]$, $x\in \mathbb{R}^n$.

Lemma 2.2. Suppose $(H1)$ and $(H4)$. Then for any $u\in E$ with $\Vert u\Vert \leq B$, we have

and

for $t \in [0, T]$ and $x\in \mathbb{R}^n$.

Proof. We have

for $t \in [0, T]$, $x\in \mathbb{R}^n$. Next,

for $t \in [0, T]$, $x\in \mathbb{R}^n$. This completes the proof.

For $u\in E$, define

Lemma 2.3. Suppose $(H1)$–$(H4)$. If $u\in E$ satisfies the equation $S_1u = 0$ on $[0, T]\times \mathbb{R}^n$, then $u$ satisfies the problem (1.1).

Proof. We have

Then

and

and

By the last two equations, we find

Next,

and

for $t \in [0, T]$, $x\in \mathbb{R}^n$. Hence,

Therefore

This completes the proof.

Lemma 2.4. Suppose $(H1)$–$(H4)$. For $u\in E$, $\Vert u\Vert \leq B$, we have

Proof. We have, using Lemma 2.2,

This completes the proof.

$\it{(H5)}$ Now, suppose that $D$ is a positive constant and $g$ is a nonnegative continuous function on $\mathbb{R}^n$. Moreover $g > 0$ on $\mathbb{R}^n\backslash \{\cup_{i = 1}^n\{x_i = 0\}\}$,

and

where

In the last section, we will give an example for the constant $D$ and the function $g$ that satisfy $(H5)$. For $u\in E$, define the operator

Lemma 2.5. Suppose $(H1)$–$(H4)$. If $u\in E$ satisfies the equation

for some constant $a$, then $u$ satisfies the problem (1.1).

Proof. We differentiate four times in $t$ and four times in $x_1$, $\ldots$, $x_n$ the Eq (2.3) and we get

whereupon

Since $S_1$ is continuous, we get

Thus,

Hence and Lemma 2.3, we conclude that $u$ satisfies the problem (1.1). This completes the proof.

Lemma 2.6. Suppose $(H1)$–$(H5)$. If $u\in E$ and $\Vert u\Vert \leq B$, then

Proof. We have

$t \in [0, T]$, $x\in \mathbb{R}^n$, and

$t \in [0, T]$, $x\in \mathbb{R}^n$, and

$t \in [0, T]$, $x\in \mathbb{R}^n$, and

$t \in [0, T]$, $x\in \mathbb{R}^n$, $l\in \{1, \ldots, n\}$, and

$t \in [0, T]$, $x\in \mathbb{R}^n$, $l\in \{1, \ldots, n\}$. Consequently

This completes the proof.

3.   Existence of at least one solution

Now, suppose that

$\it{(H6)}$ $D\left(2B+ \left(1+\left|\frac{\lambda}{1-\lambda}\right|\right)T^2\left(nB+C\sum\limits_{j = 1}^r \left(B^{p_j}+B^{q_j}+\sum\limits_{i = 1}^n B^{r_{ji}}\right)\right) +A \sum\limits_{k = 1}^m B^{r_k}\right) < B$.

$\it{(H7)}$ $\epsilon\left(B+D\left(2B+ \left(1+\left|\frac{\lambda}{1-\lambda}\right|\right)T^2\left(nB+C\sum\limits_{j = 1}^r \left(B^{p_j}+B^{q_j}+\sum\limits_{i = 1}^n B^{r_{ji}}\right)\right) +A \sum\limits_{k = 1}^m B^{r_k}\right) \right)\leq B$.

Let $X_2$ be the set of all equi-continuous families in $E$, $X_3 = X_2\cup\{u_0\}$, $\widetilde{X} = X_3$. Let also,

For $u\in E$, define the operators

By Lemma 2.5, it follows that any fixed point of the operator $T+S$ is a solution to the problem (1.1).

Lemma 3.1. Suppose that $(H1)$–$(H7)$ hold. For $u\in X$, we have

Proof. By Lemma 2.6, we get

and

This completes the proof.

Our main result in this section is as follows.

Theorem 3.1. Suppose that $(H1)$–$(H7)$ hold. Then the problem (1.1) has at least one solution.

Proof. By Lemma 3.1, it follows that $I-S$: $X\to X$ and it is continuous. Since the continuous map of equi-continuous families are equi-continuous families, we conclude that $(I-S)(X)$ resides in a compact subset of $E$. Now, assume that there is an $u\in \partial X$ and $\lambda\in \left(0, \frac{1}{\epsilon}\right)$ so that

or

or

whereupon

This is a contradiction. Hence and Theorem 2.1, it follows that the operator $T+S$ has a fixed point and the problem (1.1) has at least one solution.

4.   Existence of at least one nonnegative solution

Let

Below, suppose that

$\it{(H8)}$

For $u\in E$, define the operator

Lemma 4.1. Suppose that $(H1)$–$(H5)$ and $(H8)$ hold. If $u\in E$ is a fixed point of the operator $T+\widetilde{S}$, then it satisfies the problem (1.1).

Proof. We have

whereupon

Now, we apply Lemma 2.5 and we get the desired result. This completes the proof.

Lemma 4.2. Suppose that $(H1)$–$(H5)$ and $(H8)$ hold. Then $I-\widetilde{S}: X_1\to X_1$,

Proof. Take $u\in X_1$ arbitrarily. Then

Since

where we have used the first inequality of $(H8)$, we conclude that $(I-\widetilde{S})u\geq 0$. Next, using the second inequality of $(H8)$, we get

Thus, $I-\widetilde{S}$: $X_1\to X_1$. Moreover,

where we have used the third inequality of $(H8)$. This completes the proof.

Our main result in this section is as follows.

Theorem 4.1. Suppose that $(H1)$–$(H5)$ and $(H8)$ hold. Then the problem (1.1) has at least one nonnegative solution.

Proof. By Lemma 4.2, we have that $I-\widetilde{S}$: $X_1\to X_1$ and it is continuous and $(I-\widetilde{S})(X_1)$ resides in a compact subset of $E$. Now, assume that there are an $u\in \partial X_1$ and $\lambda\in \left(0, \frac{1}{\epsilon}\right)$ so that

or

or

Hence, applying Lemma 4.2, we get

This is a contradiction. From here, applying Lemma 4.1 and Theorem 2.1, we get that the problem (1.1) has at least one nonnegative solution. This completes the proof.

5.   Existence of at least two nonnegative solutions

Suppose

$\it{(H9)}$ Let $m > 0$ be large enough and $A$, $B$, $\widetilde{\widetilde{r}}$, $L$, $R_1$ be positive constants that satisfy the following conditions

For $u\in E$, define the operators

Our main result in this section is as follows.

Theorem 5.1. Suppose that $(H1)$–$(H5)$ and $(H9)$ hold. Then the problem (1.1) has at least two nonnegative solutions.

Proof. Define

(1) For $v_1, v_2\in \Omega$, we have

whereupon $T_1$: $\Omega \to E$ is an expansive operator with a a constant $\frac{1}{m\epsilon}$.

(2) For $v\in \overline{\mathcal{P}}_{R_1}$, we get

Therefore $S_3(\overline{\mathcal{P}}_{R_1})$ is uniformly bounded. Since $S_3: \overline{\mathcal{P}}_{R_1}\to E$ is continuous, we have that $S_3(\overline{\mathcal{P}}_{R_1})$ is equi-continuous. Consequently $S_3$: $\overline{\mathcal{P}}_{R_1}\to E$ is a $0$-set contraction.

(3) Let $v_1\in \overline{\mathcal{P}}_{R_1}$. Set

Note that $S_2v_1+ \frac{L}{5}\geq 0$ on $[a, b]$. We have $v_2\geq 0$ on $[a, b]$ and

Therefore $v_2\in \Omega$ and

or

Consequently $S_3(\overline{\mathcal{P}}_{R_1})\subset (I-T_1)(\Omega)$.

(4) Assume that for any $u_0\in \mathcal{P}^*$ there exist $\lambda\geq 0$ and $u\in \partial\mathcal{P}_{r}\cap (\Omega+\lambda u_0)$ or $u\in \partial\mathcal{P}_{R_1}\cap (\Omega+\lambda u_0)$ such that

Then

or

Hence,

This is a contradiction.

(5) Suppose that for any $\epsilon_1\geq 0$ small enough there exist a $u_1\in \partial \mathcal{P}_{L}$ and $\lambda_1\geq 1+\epsilon_1$ such that $\lambda_1u_1\in \overline{\mathcal{P}}_{R_1}$ and

In particular, for $\epsilon_1 > \frac{2}{5m}$, we have $u_1\in \partial \mathcal{P}_{L}$, $\lambda_1u_1\in \overline{\mathcal{P}}_{R_1}$, $\lambda_1\geq 1+\epsilon_1$ and (5.1) holds. Since $u_1\in \partial \mathcal{P}_{L}$ and $\lambda_1u_1\in \overline{\mathcal{P}}_{R_1}$, it follows that

Moreover,

or

From here,

and

which is a contradiction.

Therefore all conditions of Theorem 2.2 hold. Hence, the problem (1.1) has at least two solutions $u_1$ and $u_2$ so that

or

This completes the proof.

6.   An example

Let $T = 1$, $n = 1$, $m = 3$, $t_1 = \frac{1}{4}$, $t_2 = \frac{1}{3}$, $t_3 = \frac{1}{2}$. Consider the problem

Here

Then

Take

Then,

and

Thus, $(H6)$ and $(H7)$ hold. Hence and Theorem 3.1, it follows that the considered problem has at least one solution. Next,

and

and

Therefore $(H8)$ holds. By Theorem 4.1, it follows that the considered problem has at least one nonnegative solution. Moreover,

and

Consequently $(H9)$ holds. By Theorem 5.1, it follows that the considered problem has at least two nonnegative solutions.

Now, we will construct a function $g$ for arbitrary $n$. Let

Then

Therefore

Hence, there exists a positive constant $C_1$ so that

$s\in \mathbb{R}$. Note that by ^[37], we have

Let

and

Then there exists a positive constant $C_2$ so that

Take $g(x) = \frac{D}{C_2}g_1(x)$, $x\in \mathbb{R}^n$. Hence,

7.   Conclusions

In this paper, we investigate initial value problems for impulsive nonlinear wave equations. We reduce the considered problem to a suitable integral equation. Then, we define two operators and show that any fixed point of their sum is a solution of the considered problem. After this, we apply recent fixed point theorems and we show that the considered problem has at least one and at least two classical solutions. The proposed approach can be applied for other classes impulsive partial differential equations.

Acknowledgments

The researchers would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project.

Conflict of interest

The authors declare there are no conflicts of interest.