Existence results for ϕ-Laplacian impulsive differential equations with periodic conditions

1.   Introduction

Differential equations with impulses were considered for the first time by Milman and Myshkis ^[19] and then followed by a period of active research which culminated with the monograph by Halanay and Wexler ^[14]. Many phenomena and evolution processes in the field of physics, chemical technology, population dynamics, and natural sciences may change state abruptly or be subject to short-term perturbations, (see, for instance ^[1,16] and the references therein). These short perturbations may be seen as impulses. Impulsive problems arise also in various applications in communications, mechanics (jump discontinuities in velocity), electrical engineering, medicine, and biology. For example, in the periodic treatment of some diseases, impulses correspond to the administration of a drug treatment. In environmental sciences, impulses correspond to seasonal changes of the water level of artificial reservoirs. Their models are described by impulsive differential equations and inclusions. To date, a spectrum of mathematical results (such as existence, asymptotic behavior, $\ldots$) have been obtained (see ^{[2,3,8,9,13,16,28,29]} and the references therein).

There have been many approaches to the study of the existence of solutions of impulsive differential equations, such as fixed point theory, topological degree theory (including continuation methods and coincidence degree theory), comparison methods and monotone iterative methods (see ^{[15,23,27,31,33]}).

Recently in ^[20,30], the authors studied the existence and multiplicity of solutions of some classes of second order impulsive problems by variational methods.

In the last few years, $\phi-$Laplacian problems for differential equations such as

have been investigated by several authors. Existence and multiplicity results for the nonresonance and resonance cases have been presented in ^[4,5,24,26]. These types of problems with impulse effects have been considered by ^[25], in which they are based on the lower and upper functions.

Coincidence degree, introduced by Mawhin in 1972, is a topological tool for the investigation of the semilinear equation $Lu+Nu = f$, where $L$ is a linear Fredholm operator of index zero (not necessarily invertible) and $N$ is a nonlinear perturbation. Continuation theorems involving these kinds of mappings $(L, N)$ became an effective procedure in proving the existence of solutions of a large variety of boundary value problems. This method is extended in ^[18] to the case when $L$ is a quasi-linear operator in view of its application to problems involving $p-$Laplacian-like operators.

In this paper we use the continuation methods to prove the existence of solutions to the $\phi-$Laplacian problem for differential, equations with impulse effects and periodic boundary conditions

where $0 < t_{1} < t_{2} < \cdots < t_{m} < b, \ f: [0, b]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}$ is a Carathéodory function, $I_{k}, \overline{I}_{k}\in C(\mathbb{R}^{n}, \mathbb{R}^{n})$ and $\phi:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is a suitable monotone homeomorphism.

Because the $\phi-$Laplacian is nonlinear, the continuation theorem of Mawhin ^[11] is not applicable, which leads to difficulty for solving the problems (1.1)–(1.4). By using a Manasevich and Mawhin continuation theorem ^[18] and some analysis techniques, we establish some sufficient conditions for the existence of periodic solutions of the problems (1.1)–(1.4).

2.   Preliminaries

Let $\phi:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a continuous function which satisfies the following two conditions:

($H_1$) For any $x_1, x_2\in\mathbb{R}^{n}$, $x_1\neq x_2$,

($H_2$) There exists a function $\alpha:[0, \infty)\rightarrow[0, \infty)$, $\alpha(s)\rightarrow+\infty$ as $s\rightarrow+\infty$, such that

It is well-known that under these two conditions $\phi$ is an homeomorphism from $\mathbb{R}^n$ onto $\mathbb{R}^n$, and that $\|\phi^{-1}(y)\|\rightarrow+\infty$ as $\|y\|\rightarrow+\infty$.

Definition 2.1. A map $f:[p, q]\times\mathbb{R}^n\to \mathbb{R}^n$ is said to be $L^{1}$-Carathéodory if

(ⅰ) $t\to f(t, y)$ is measurable for all $y\in \mathbb{R}^n$,

(ⅱ) $y\to f(t, y)$ is continuous for almost each $t\in[p, q]$,

(ⅲ) for each $r > 0$, there exists $h_{r}\in L^{1}([p, q], \mathbb{R}_{+})$ such that

Let $X$ and $Z$ be two real Banach spaces with norms $\|\cdot\|_X$ and $\|\cdot\|_Z,$ respectively. A continuous operator $M:X\cap domM\rightarrow Z$ is said to be quasi-linear if

(ⅰ) $dim KerM = dim M^{-1}(0) = n < \infty$;

(ⅱ) $ImM = M(X\cap domM)$ is a closed subset of $Z$.

Let $X = X_1\oplus X_2$ and $Z = Z_1\oplus Z_2$, where $X_1 = KerM$, $Z_2 = ImM$ and $X_2, Z_1$ are respectively the complementary spaces of $X_1$ in $X$, $Z_2$ in $Z$. Assume that $dimX_1 = dimZ_1;$ then we can define $P:X\rightarrow X_1$ and $Q:Z\rightarrow Z_1$ as the corresponding orthogonal projections such that

Denote by $J:Z_1\rightarrow X_1$ a homeomorphism with $J(0) = 0.$

Let $\Omega$ be a bounded open subset of $X$, with $0\in\Omega$ such that $domM \cap\Omega\neq\emptyset$, and consider a parameter family of nonlinear perturbations (generally) $N_\lambda:[0, 1]\times \bar\Omega\rightarrow Z$ with $N_1 = N$. The continuous operator $N_\lambda$ is said to be $M-$compact in $\bar\Omega$ with respect to $M$ if there is an operator $K:ImM\rightarrow X_2$ with $K(0) = 0$ such that for $\lambda\in[0, 1]$,

We introduce the intermediate map

which is clearly compact under the above assumptions.

Consider the abstract equation

Now, we consider the continuation theorem for a quasilinear operator due to Ge and Ren ^[12].

Lemma 2.1. ^[12] Let $X$ and $Z$ be Banach spaces, $\Omega \subset X$ be an open and bounded nonempty set, $M$ be a quasi-linear operator and $N_\lambda$ be a $M-$compact operator in $\bar\Omega$. Then (2.2) has a solution $x\in\bar\Omega$ [resp $x \in\partial\Omega$] if and only if $x\in\bar\Omega$ [resp $x \in\partial\Omega$] is a fixed point of $F(\lambda, \cdot)$ defined in (2.1).

Theorem 2.2. ^[12] Let $X$ and $Z$ be two Banach spaces and $\Omega\subset X$ be an open and bounded nonempty set. Suppose $M: X\cap dom M\to Z$ is a quasilinear operator and $N_\lambda: \overline{\Omega}\to Z, \ \lambda\in[0, 1],$ is $M-$ compact. In addition, if

i) $Mx\not = N_\lambda x, \quad \lambda\in(0, 1), \quad x\in\partial\Omega,$

ii) $deg(JQN_1, \Omega\cap Ker M, 0)\not = 0,$

then the abstract equation $Mx = N_1x$ has at least one solution in $\overline{\Omega}.$

3.   Fixed point formulation

In order to define a solution for Problems (1.1)–(1.2), consider the space of piece-wise continuous functions:

endowed with the norm

and let

be a Banach space with the same norm of $PC.$

Also, consider

endowed with the norm

where $y_{k} = y|_{J_{k}}$ and $J_k = (t_{k}, t_{k+1}],$ and let

be a Banach space with the same norm of $PC^1.$ Define $X = PC^1_b$, $Z = L^{1}([0, b], \mathbb{R}^{n})\times\mathbb{R}^{nm}\times\mathbb{R}^{nm}\times\mathbb{R}^n$, and for $z = (x, c_{1}, c_{2}, c_3)\in Z$, we have

Let

Let $h\in PC(J, \mathbb{R}^{n})$. Then we define $y$ by:

where

We will prove that $M$ is a quasi-linear operator.

For $y\in KerM,$ we have

and then

The boundary conditions imply that

Thus

Next, $ImM = \{ (x,{a_1}, \ldots ,{a_m},{\bar a_1}, \ldots ,{\bar a_m},d) \in Z:(\phi {(y'(t))^\prime } = x(t)$ , a. e. $t \in [0,b],y({t_k}) = \;{a_k}$, $y'({t_k}) = {\bar a_k},k = 1$, $\ldots ,m,x(t) \in domM\}$.

From the following problem

we have

Since $y(0) = y(b)$, we also have

so that

and then

Using the projections,

we obtain that

Thus

and moreover, $ImM$ is a closed subspace of $Z$. Therefore $M$ is a quasi-linear.

For any $\Omega\subset domM,$ define the family

The problems (1.1)–(1.4) is equivalent to the operator equation

We will prove that $N_\lambda$ is $M-$compact in $\bar\Omega$. It is easy to show that

Let

and the homeomorphism

is given by

We supposed that $f(t, y, y')$ is a Carathéodory function and $\phi^{-1}$ is continuous. Then we have the following result.

Lemma 3.1. Suppose $\Omega\subset PC^1([0, b], \mathbb{R}^n)$ is a bounded open set. Then $N_\lambda$ is $M$-compact in $\bar\Omega$.

Proof. We will prove that

● $QN_\lambda: X\rightarrow Z$ is continuous and sends bounded sets into bounded sets.

● $K(I-Q)N_\lambda: \bar\Omega\rightarrow X$ is completely continuous.

Step 1: $QN_\lambda$ sends bounded sets into bounded sets and it is continuous.

First,

$\circ$ $QN_\lambda$ sends bounded sets into bounded sets in $PC^1([0, b], \mathbb{R}^n)$.

Let $y\in\bar\Omega = \{y\in PC^1([0, b], \mathbb{R}^n): \|y\|_1\leq r \}.$

Clearly $\|y\|_{1}\leq r,$ and then, there exists $r_* > 0$ such that

Since $I_{k}$ and $\bar I_{k}$ are continuous and $\bar B(0, r_*)$ is compact in $\mathbb{R}^n,$ we have

Using the fact that $f$ is a Carathédory function, we obtain

Since $\phi^{-1}$ is continuous, we also have

Thus

$\circ$ $QN_\lambda$ is continuous.

Let $(y_{\alpha})_{\alpha\in\mathbb{N}}$ be a sequence such that $y_n\rightarrow y$ in $PC^1([0, b], \mathbb{R}^n)$. Then there exists $r > 0$ such that

Hence there exists $h_r\in L^1([0, b], \mathbb{R}^n)$ such that

From the definition of $QN_\lambda$, we have

By the dominated convergence theorem, and since $I_k, \ \bar I_k, \ \phi$ and $\phi^{-1}$ are continuous functions, we get

Thus

Hence, $QN_\lambda$ is continuous.

Step 2: $K(I-Q)N_\lambda$ is completely continuous.

For this we prove that $(I-Q)N_\lambda$ sends bounded sets into bounded sets and $KN_\lambda$ is completely continuous. The first part is immediate. For the second part, as in Step 1, we can prove that $KN_\lambda$ is bounded and continuous.

It remains to show that $KN_\lambda$ is equicontinuous, then by using the Arzelà-Ascoli theorem $KN_\lambda$is compact. Indeed Let $l_1, \ l_2\in[0, b]$, $l_1 < l_2$ and $\Omega$ be a bounded set in $PC^1([0, b], \mathbb{R}^n)$, and let $y\in\Omega$. Then

where $l$ is defined in Step 1. By the mean value theorem, we obtain

As $l_2\rightarrow l_1$ the right hand side of the above inequality tends to zero. Also we have

Since $\phi^{-1}$ is continuous function, we conclude that $\overline{K(\Omega)}$ is compact.

● Using the fact that $f$ is a Carathéodory function and that $I_k, \ \bar I_k, \ \phi$ and $\phi^{-1}$ are continuous functions, we can easily prove that $KN_\lambda$ is continuous.

Let us now consider the abstract differential periodic problem

and define

We obtain that $F$ is a completely continuous operator. Furthermore, the operator equation (3.4) is equivalent to the following fixed point equation:

Let us now consider the simple periodic boundary value problem

where $h\in L^1$ is such that $\int_{0}^{b}h(s)ds = 0$, and let us recall that $y$ is a continuous solution to (3.5)–(3.8) defined by

where

and

The boundary conditions imply that

with

For fixed $l\in PC([0, b], \mathbb{R}^n)$ and $c\in\mathbb{R}^n$, let us define

Proposition 3.2. If $\phi$ satisfies conditions ($H_1$) and ($H_2$), then the function $G_{c, l}$ has the following properties:

(i) For any fixed $l\in PC([0, b], \mathbb{R}^n)$ and $c\in\mathbb{R}^n$, the equation

has unique solution $\widetilde{a}(l)$.

(ii) If $\langle\sum_{k = 1}^{m}I_k(x), \widetilde{a}(l)\rangle > 0$, $x\in\mathbb{R}^n$, where the function $\widetilde{a}: PC([0, b], \mathbb{R}^n)\rightarrow\mathbb{R}^n$ is defined in (i), then $\widetilde{a}$ is continuous and sends bounded sets into bounded sets.

Proof. (ⅰ) By ($H_1$), it is immediate that

and hence if (3.11) has a solution, it is unique. To prove its existence we will show that $\langle G_{c, l}(a), a\rangle > 0$ for $\|a\|$ sufficiently large. Indeed we have

and thus

From ($H_2$), $\forall y\in \mathbb{R}^n$, we have that

Thus from (3.12) and (3.13),

Since $\|a\|\rightarrow\infty$ implies that $\|\phi^{-1}(a+l(t))\|\rightarrow\infty$, uniformly for $t\in [0, b]$, we find from (3.14) that there exists an $r > 0$ such that

It follows by an elementary topological degree argument that the equation $G_{c, l}(a) = 0$ has a solution for each $l\in PC$, which by our previous argument is unique. In this way we define a function $\widetilde{a}: PC\rightarrow\mathbb{R}^n$ which satisfies

To prove (ⅱ) let $B$ be a bounded subset of $PC$ and let $l\in B$. From (3.15)

Then

and hence

Assume that $\{\widetilde{a}(l), l\in B\}$ is not bounded. Then, for an arbitrary $A > 0,$ there is an $l\in B$, with $\|l\|_0$ sufficiently large, so that

Hence by using (3.16) and (3.13), we find that

Thus $A\leq\|l\|_0$, which is a contradiction. Therefore $\widetilde{a}$ sends bounded sets in $PC$ into bounded sets in $\mathbb{R}^n$.

Finally, to show the continuity of $\widetilde{a}$, let $(l_\alpha)$ be a convergent sequence in $PC$, say $l_\alpha \rightarrow l$, as $\alpha\rightarrow \infty$. Since $(\widetilde{a}(l_\alpha))$ is a bounded sequence, any subsequence of it contains a convergent subsequence, denoted by $(a(l_{\alpha_{j}}))$. Let $a(l_{\alpha_{j}})\rightarrow \widehat{a}$, as $j\rightarrow\infty$. By letting $j\rightarrow\infty$ in

we find that

and hence $\widetilde{a}(l) = \widehat{a}$, which shows the continuity of $\widetilde{a}$.

4.   Existence results

We will assume in this section that $\phi: \mathbb{R}^n\rightarrow\mathbb{R}^n$ is continuous and satisfies conditions $(H_1)-(H_2)$ of Section 2. Our aim in this part is to apply the Manasevich and Mawhin continuation theorem ^[18] for quasilinear equations to the quasilinear problem with impulse effects (1.1)–(1.4).

Theorem 4.1. Assume that $\Omega$ is an open bounded set in $PC([0, b], \mathbb{R}^n)$ such that the following conditions hold:

$(H_3)$ For each $\lambda\in (0, 1)$ the problem

has no solution on $\partial\Omega.$

$(H_4)$ The equation

where $h = f(t, a, 0)$ and $L_0(h)(t) = c+\int_0^t\phi^{-1}[(\phi(d)+\int_0^sf(\tau, a, 0)d\tau]ds$, $t\in[0, t_1]$ has no solution on $\partial\Omega\cap\mathbb{R}^{n}$.

$(H_5)$ The Brouwer degree

Then the problems (1.1)–(1.4) has a solution in $\Omega$.

Proof. We transform the problems (1.1)–(1.4) into the one parameter family of problems

For $\lambda\in(0, 1],$ observe that in both cases, $y$ is a solution of problem (4.1) or $y$ is a solution of problem (4.3). We have necessarily

It follows that, for $\lambda\in(0, 1]$, problems (4.1) and (4.3) have the same solutions. Furthermore it is easy to see that $f$ Carathéodory implies that $N: PC^1([0, b], \mathbb{R}^n)\times[0, 1]\rightarrow Z$ defined by

where

is continuous and sends bounded sets into bounded sets. Problem (4.3) can be written in the equivalent form

with

We assume that for $\lambda = 1$, (4.4) does not have a solution on $\partial\Omega$. By hypothesis $(H_3),$ it follows that (4.4) has no solution for $(y, \lambda)\in\partial\Omega\times(0, 1]$. For $\lambda = 0$, (4.3) is equivalent to the problem

And thus if $y$ is a solution of this problem, we must have

Hence

which implies that

where $c\in\mathbb{R}^{n}$ is a constant. Integrating this last equation over $[0, b],$ we obtain $\phi^{-1}(c) = 0$, and thus $y(t) = e$, a constant, and by (4.6)

which, together with hypothesis $(H_4),$ implies that $y = e \notin \partial\Omega$. Thus we prove that (4.4) has no solution $(y, \lambda)\in\partial\Omega\times[0, 1]$. Then we have that for each $\lambda\in[0, 1]$ the Leray-Schauder degree $d_{LS}[I-F_{f}(., \lambda), \Omega, 0]$ is well defined and by properties of that degree, that

Now it is clear that the problem

is equivalent to the problem (1.1)-(1.4), and will have a solution, if we can prove that

We have that

Thus we obtain

Hence by the properties of Leray-Schauder degree (proved in ^[7]), we have that

where the function $G$ is defined in Proposition 3.2, and $d_B$ denotes the Brouwer degree. Since by the Hypothesis $(H_5)$ this last degree is different from zero, the theorem is proved.

Our next theorem is a consequence of Theorem 4.1. We need first the following definition.

Let $f = (f_1, \ldots, f_n):J\times\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be a Carathéodory function. We will say that $f$ satisfies a generalized Villari condition if there is an $\rho_0 > 0$ such that for all $y\in PC_{b}^{1}$, $y = (y_1, \ldots, y_n)$, with

for some $j\in\{1, \ldots, n\}$, it holds that

for some $i\in\{1, \ldots, n\}$.

Let $B(R)$ denote the open ball in $\mathbb{R}^{n}$ with center zero and radius $R$.

Theorem 4.2. Assume that the following conditions hold:

$(K_1)$ There exist $\nu\in C^{1}(\mathbb{R}^{n}, \mathbb{R}^{n})$ and $h\in L^{1}(J, \mathbb{R}_+)$ such that

and

for all $x, y\in \mathbb{R}^n$ and a.e $t\in J$.

$(K_2)$ f satisfies a generalized Villari condition.

$(K_3)$ There exist positive constants $c_{k}$, such that for each $k = 1, \ldots, m,$ we have

and

for any $x, y, z\in \mathbb{R}^n.$

$(K_4)$ There is an $R_0 > 0$, such that all the possible solutions to the equation

belong $B(0, R_0)$.

$(K_5)$ The Brouwer degree

Then the problems (1.1)–(1.4) has at least one solution.

Proof. Let $(y, \lambda)$, $y\in PC^{1}_{b}$, $\lambda\in (0, 1)$, be a solution to (4.1), then using (4.2), we have

Then, from hypothesis $(K_1), \ (K_2)$ and $(K_3)$, we get

Let us set $\phi(y'(t)) = \widetilde{b}(t)+\overline{b}$, with $\int_{0}^{b}\widetilde{b}(t)dt = 0$, and $\overline{b} = \frac{1}{b}\int_{0}^{b}\phi(y'(t))dt$. From (4.11) and (4.2), we get

Then,

which yields

We next find an a priori bound for $\overline{b}$. We have that

Hence, by integrating on each $J_{k} = (t_{k-1}, t_k]$ and using the boundary conditions, we obtain

Then,

By Proposition 3.2, it follows that $\overline{b} = \widetilde{a}(\widetilde{b})$, where the function $\widetilde{a}$ is defined in that proposition. Recalling that $\widetilde{a}$ sends bounded sets into bounded sets, we have that there is a positive constant $C_1$ such that

Hence, from (4.13) and the fact that $\phi^{-1}$ is continuous, we obtain a positive constant $C_2$ such that

Hence for $t\in[0, t_1]$, we get

For $t\in(t_1, t_2]$

We continue this process and we obtain that, for $t\in(t_m, b],$

Then

Next the solution $y$ satisfies

By the generalized Villari condition, we have, for each $j_1\in\{1, \ldots, n\}$, there exists $t_{j_1}\in [0, t_1]$ such that

Since

we get

For each $j_2\in\{1, \ldots, n\}$, there exists a $t_{j_2}\in (t_1, t_2]$ such that

Since

we get

We continue this process, and we get that, for each $j_m\in\{1, \ldots, n\}$, there exists a $t_{j_m}\in (t_m, b]$ such that

Since

we get

Then for each $j\in\{1, \ldots, n\}$, there exists $t_j\in J$ such that

Thus there is a constant $C_3$ such that $\|y\|_0\leq C_3$. It follows that we can find $R_0 > 0$ such that, if $(y, \lambda)$ is a solution to (4.1), then $\|y\|_1\leq R_0$. We define next the set $\Omega\subset PC_b^1$ that appears in Theorem 4.1 as $\Omega = B(R_0),$ the open ball in $PC_b^1$ of center $0$ and radius $R_0$. Thus condition $(H_3)$ of Theorem 4.1 is satisfied with $\Omega = B(R_0)$, and since the rest of the conditions of Theorem 4.1 are also satisfied, the proof is complete.

Corollary 4.3. Suppose that the following conditions are satisfied.

$(M_1)$ There is a mapping $\nu\in C^{1}(\mathbb{R}^{n}, \mathbb{R}^{n})$ such that conditions $(K_1)$ and $(K_3)$ hold.

$(M_2)$There exist a function $\widetilde{h}\in L^{1}(J, \mathbb{R}_+)$ and continuous function $\eta: [0, \infty)\rightarrow[0, \infty)$, such that

and

for almost all $t\in J$, and all $x, y\in\mathbb{R}^n$.

$(M_3)$ Condition $(K_5)$ holds.

Then the problems (1.1)–(1.4) has at least one solution.

Proof. Let $(y, \lambda)$, $\lambda\in (0, 1)$ be a solution to (4.1). As in the proof of Theorem 4.2, it follows from conditions $(K_1)$ and $(K_3)$, that there is a positive constant $C_2$ such that $\|y'\|_0\leq C_2$. We claim that conditions $(K_3)$ and $(M_2)$ imply that there is a constant $C_3$ such that $\|y\|_0\leq C_3$. Indeed, from (4.12), we have that

Then by (4.18)

Since (4.15) holds by the reasoning of the previous theorem and $\eta(s)\rightarrow\infty$ as $s\rightarrow\infty$, from (4.19) we find the required bound for $\|y\|_0$.

Now let $a\in\mathbb{R}^{n}$ be such that

Then (4.18) implies that $\eta(|a|)\leq C_2$, and hence $|a|\leq C_3$, for some positive constants $C_2$ and $C_3$. Thus there is $R_0 > 0$ such that every solutions of (4.10) belongs to $B(R_0)$, and since condition $(K_5)$ holds, then all condition of Theorem 4.1 are satisfied. Therefore, (1.1)–(1.4) has at least one solution.

Example 4.1. We consider the following

where $h: J\to \mathbb{R}$ defined by

Let $\nu = c\in C^1(\mathbb{R}, \mathbb{R}), \ c\in \mathbb{R}_+$, then

and

for all $x, y\in \mathbb{R}$ and a.e $t\in J$.

$(\bar K_1)$ There exist positive constants $\alpha_{1}, \alpha_2$, such that

and

and

for any $x, y, z\in \mathbb{R}.$

Define

It is clear that $G: \mathbb{R}\to \mathbb{R}$ is a continuous function. We can show that there exists $R_0 > 0$ such that $G(B(0, R_0))\subset B(0, R_0)$. Then $d(G, B(0, R), 0)\not = 0.$ So all the conditions of Theorem 4.2 are satisfied. Hence the above problem has at least one solution.

Acknowledgments

The authors would like to thank the anonymous referees for their careful reading of the manuscript and pertinent comments; their constructive suggestions substantially improved the quality of the work.

Conflict of interest

The authors declare no conflict of interest.