Applied Mathematics E-Notes, 1(2001), 65-72 \begin{matrix}c \\ ◯ \end{matrix} Available free at mirror sites of http://math2.math.nthu.edu.tw/∼amen/

On Nonresonance Impulsive Functional Differential Equations with Periodic Boundary Conditions ^{∗}

Mouffak Benchohra^{†}, Paul W. Eloe^{‡}

Received 27 March 2001

Abstract

In this paper a fixed point theorem due to Schaefer is used to investigate the existence of solutions for first order nonresonance impulsive functional differential equations in Banach spaces with periodic boundary conditions.

1 Introduction

This paper is concerned with the existence of solutions for the nonresonance boundary value problem for functional differential equations with impulsive effects

y′(t) − λy(t) = f(t, y_{t}), t ∈ J = [0, T], t ≠ t_{k}, k = 1, …, m,  (1)
Δy|_{t=t_{k}} = I_{k}(y(t_{k}^{−})), k = 1, …, m,  (2)
y(t) = y(0), t ∈ J_{0}, My(0) − Ny(T) = 0,  (3)

where λ ∈ R, f : J × C(J_{0}, E) → E is a given function, J_{0} = [−r, 0], 0 < r < ∞, 0 = t_{0} < t_{1} < … < t_{m} < t_{m+1} = T, I_{1}, ..., I_{m} ∈ C(E, E) are bounded, Δy|_{t=t_{k}} = y(t_{k}^{+}) − y(t_{k}^{−}), y(t_{k}^{−}) and y(t_{k}^{+}) represent the left and right limits of y(t) at t = t_{k}, respectively, E a real Banach space with norm | · |, and M and N are constant. Note that if M = N = 1, then (3) represents periodic boundary conditions. For notational purposes, let t_{−1} = −r.

For any continuous function y defined on [−r, T] − {t_{1}, …, t_{m}} and any t ∈ J, we denote by y_{t} the element of C(J_{0}, E) defined by

y_{t}(θ) = y(t + θ), θ ∈ J_{0}.

Here y_{t}(·) represents the history of the state from time t − r, up to the present time t.

Impulsive differential equations have become more important in recent years in some mathematical models of real world phenomena, especially in the biological or medical domain see; the monographs of Bainov and Simeonov [2], Lakshmikantham, Bainov and Simeonov [10], and Samoilenko and Perestyuk [13], and the papers of Agur et al. [1], Goldbeter et al. [6].

∗Mathematics Subject Classifications: 34A37, 34G20, 34K25.

†Department of Mathematics, University of Sidi Bel Abbès, BP 89, 22000 Sidi Bel Abbès, Algérie

‡Department of Mathematics, University of Dayton, Dayton, OH 45469-2316, U. S. A.

Recently an extension to functional differential equations with impulsive effects has been done in [17] by using the coincidence degree theory. For other results on functional differential equations we refer the interested reader to the monograph of Erbe, Kong and Zhang [5], Hale [7], Henderson [8], and the survey paper of Ntouyas [12].

The fundamental tools used in the existence proofs of all above mentioned works are essentially fixed point arguments, nonlinear alternative, topological transversality [3], degree theory [11] or the monotone method combined with upper and lower solutions [4], [9].

This paper will be divided into three sections. In Section 2 we will recall briefly some basic definitions and preliminary facts which will be used throughout Section 3. In Section 3 we shall establish an existence theorem for (1)–(3). We consider the case when λ ≠ 0. Note that when the impulses are absent (i.e. for I_{k} ≡ 0, k = 1, …, m), then the problem (1)–(3) is a nonresonance problem since the linear part in equation (1) is invertible. Our approach is based on a fixed point theorem due to Schaefer [14] (see also, Smart [15]).

2 Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. C(J_{0}, E) is the Banach space of all continuous functions from J_{0} into E with the norm

∥ϕ∥ = sup{|ϕ(θ)| : −r ≤ θ ≤ 0}.

By C(J, E) we denote the Banach space of all continuous functions from J into E with the norm

∥y∥_{J} = sup{|y(t)| : t ∈ J}.

A measurable function y : J → E is Bochner integrable if, and only if, |y| is Lebesgue integrable. (For properties of the Bochner integral, see for instance, Yosida [16]). L^{1}(J, E) denotes the Banach space of functions y : J → E which are Bochner integrable normed by

∥y∥_{L^{1}} = ∫_{0}^{T} |y(t)|dt for all y ∈ L^{1}(J, E).

We introduce some notation in order to define the solution of (1)–(3). Suppose y : [−r, T] → E and each y(t_{k}^{−}) and y(t_{k}^{+}) exist, k = 1, …, m. By convention, set y(t_{k}^{−}) = y(t_{k}) for k = 1, …, m. Let y_{k} denote the restriction of y to J_{k} = [t_{k−1}, t_{k}] in the following sense. If t ∈ (t_{k−1}, t_{k}], then y_{k}(t) = y(t). If t = t_{k−1}, then y_{k}(t_{k−1}) = y(t_{k−1}^{+}). Define

Ψ = {y : [−r, T] → E| y_{k} ∈ C(J_{k}, E), 0 ≤ k ≤ m + 1, and y(t) = y(0), t ∈ J_{0}}.

Ψ is a Banach space with the norm

∥y∥_{Ψ} = max{∥y_{k}∥_{k}| k = 0, …, m + 1},

where || · ||_{k} denotes the supremum norm on J_{k}, k = 0, …, m + 1.

We shall also consider the set

Ψ^{1} = {y : [−r, T] → E| y_{k} ∈ W^{1,1}(J_{k}, E), 1 ≤ k ≤ m + 1, and y(t) = y(0), t ∈ J_{0}}

The set Ψ^{1} is a Banach space with the norm

∥y∥_{Ψ^{1}} = max{∥y_{k}∥_{W^{1,1}(J_{k},E)}| k = 1, …, m + 1}.

A map f : J × C(J_{0}, E) ⟶ E is said to be L^{1}-Carathéodory if (i) t ⟼ f(t, u) is measurable for each u ∈ C(J_{0}, E); (ii) u ⟼ f(t, u) is continuous for almost all t ∈ J; and (iii) for each k > 0, there exists g_{k} ∈ L^{1}(J, R_{+}) such that |f(t, u)| ≤ g_{k}(t) for all ∥u∥ ≤ k and almost all t ∈ J.

We now define a solution of problem (1)–(3). A function y ∈ Ψ ∩ Ψ^{1} is said to be a solution of (1)–(3) if y satisfies the equation y′(t)−λy(t) = f(t, y_{t}) a.e. on J −{t_{1}, ...t_{m}} and the conditions Δy|_{t=t_{k}} = I_{k}(y(t_{k}^{−})), k = 1, …, m, y(t) = y(0) for all t ∈ J_{0}, and My(0) − Ny(T) = 0.

Our main result is based on the following:

LEMMA 1 (See also [15], p. 29). Let S be a convex subset of a normed linear space X and assume 0 ∈ S. Let K : S → S be a completely continuous operator, and let

Φ(K) = {y ∈ S : y = μK(y) for some 0 < μ < 1}.

Then either Φ(K) is unbounded or K has a fixed point.

We now consider the following “linear problem” (4), (2), (3), where (4) is the equation

y′(t) − λy(t) = g(t), t ≠ t_{k}, k = 1, …, m,  (4)

where g ∈ L^{1}(J_{k}, E), k = 1, …, m. For short, we shall refer to (4), (2), (3) as (LP). Note that (LP) is not really a linear problem since the impulsive functions are not necessarily linear. However, if I_{k}, k = 1, ..., m, are linear, then (LP) is a linear impulsive problem.

We state and prove the following auxiliary result. Eloe and Henderson [4] have constructed the analogous Green’s function for the problem (1), (2), (3) in the case of n-dimensional systems. The proof here gives an alternate development. In the development we require that N be a nonzero constant, although the conclusion of the lemma is valid in the case N = 0.

LEMMA 2. y ∈ Ψ^{1} is a solution of (LP), if and only if y ∈ Ψ is a solution of the following impulsive integral equation

y(t) = \{ \begin{matrix}y(0) & t ∈ J_{0} \\ ∫_{0}^{T} H(t, s)g(s)ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(y(t_{k})) & t ∈ J \end{matrix} ,  (5)

where

H(t, s) = (M − Ne^{λT})^{−1} \{ \begin{matrix}Me^{−λ(s−t)} & 0 ≤ s ≤ t ≤ T \\ Ne^{λT} e^{−λ(s−t)} & 0 ≤ t < s ≤ T \end{matrix} .  (6)

68 Impulsive Functional Differential Equations

PROOF. We prove only one of the implications. Suppose that y ∈ Ψ^{1} is a solution of (LP). Then

y′ − λy = g(t), t ≠ t_{k},

i.e.,

(e^{−λt}y(t))′ = e^{−λt}g(t), t ≠ t_{k}.  (7)

Assume that t_{k} < t ≤ t_{k+1}, k = 0, ..., m. By integration of (7) we obtain

e^{−λt_{i+1}}y(t_{i+1}) − e^{−λt_{i}^{+}}y(t_{i}^{+}) = ∫_{t_{i}}^{t_{i+1}} e^{−λs}g(s)ds, i = 0, …, k − 1.

Adding appropriate terms, we obtain

e^{−λt}y(t) − y(0) = ∑_{0<t_{k}<t} e^{−λt_{k}}(y(t_{k}^{+}) − y(t_{k})) + ∫_{0}^{t} e^{−λs}g(s)ds.  (8)

Thus,

y(T) = e^{λT} [y(0) + ∑_{k=1}^{m} e^{−λt_{k}}I_{k}(y(t_{k})) + ∫_{0}^{T} e^{−λs}g(s)ds].

Substitute this expression into (3) to obtain

y(0) = (M − Ne^{λT})^{−1}Ne^{λT} [∑_{k=1}^{m} e^{−λt_{k}}I_{k}(y(t_{k})) + ∫_{0}^{T} e^{−λs}g(s)ds] .  (9)

Substitute (9) into (8) to obtain

e^{−λt}y(t) = (M − Ne^{λT})^{−1}Ne^{λT} [∑_{k=1}^{m} e^{−λt_{k}}I_{k}(y(t_{k})) + ∫_{0}^{T} e^{−λs}g(s)ds]
+ ∑_{0<t_{k}<t} e^{−λt_{k}}I_{k}(y(t_{k})) + ∫_{0}^{t} e^{−λs}g(s)ds.  (10)

Now employ (10) to obtain

e^{−λt}y(t) = (M − Ne^{λT})^{−1}Ne^{λT} [ ∑_{0<t_{k}<t} e^{−λt_{k}}I_{k}(y(t_{k})) + ∑_{t≤t_{k}<T} e^{−λt_{k}}I_{k}(y(t_{k}))

+ ∫_{0}^{T} e^{−λs}g(s)ds + (M − Ne^{λT})(Ne^{λT})^{−1} ∑_{0<t_{k}<t} e^{−λt_{k}}I_{k}(y(t_{k}))

+(M − Ne^{λT})(Ne^{λT})^{−1} ∫_{0}^{t} e^{−λs}g(s)ds]

= (M − Ne^{λT})^{−1}Ne^{λT} [MN^{−1}e^{−λT} ∑_{0<t_{k}<t} e^{−λt_{k}}I_{k}(y(t_{k}))

+ ∑_{t≤t_{k}<T} e^{−λt_{k}}I_{k}(y(t_{k})) + MN^{−1}e^{−λT} ∫_{0}^{t} e^{−λs}g(s)ds + ∫_{t}^{T} e^{−λs}g(s)ds].

Thus

y(t) = (M − Ne^{−λT})^{−1}[M ∫_{0}^{t} e^{−λ(s−t)}g(s)ds + N ∫_{t}^{T} e^{−λ(s−t−T)}g(s)ds

+M ∑_{0<t_{k}<t} e^{−λ(t_{k}−t)}I_{k}(y(t_{k})) + N ∑_{t≤t_{k}<T} e^{−λ(t_{k}−t−T)}I_{k}(y(t_{k}))]

= ∫_{0}^{T} H(t, s)g(s)ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(y(t_{k})).

3 Main Result

We are now in a position to state and prove our existence result for the problem (1)–(3). For the study of this problem we first list the following hypotheses:

(H1) f : J × C(J_{0}, E) ⟶ E is an L^{1}−Carathéodory map;

(H2) there exist constants c_{k} such that |I_{k}(y)| ≤ c_{k}, k = 1, ..., m for each y ∈ E;

(H3) there exists m ∈ L^{1}(J, R) such that

|f(t, y_{t})| ≤ m(t) for almost all t ∈ J and all y ∈ Ψ;

(H4) for each bounded B ⊂ Ψ and t ∈ J the set

{∫_{0}^{T} H(t, s)f(s, y_{s})ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(y(t_{k})) : y ∈ B}

is relatively compact in E.

REMARK. (i) If the dimension of E is finite then (H4) is trivially satisfied. (ii) Condition (H4) is satisfied if for each t ∈ J the map C(J_{0}, E) → E : u ⟼ f(t, u) sends bounded sets into relatively compact sets.

THEOREM 1. Assume that hypotheses (H1)-(H4) hold. Then the problem (1)–(3) has at least one solution on J_{1}.

PROOF. Transform the problem into a fixed point problem. Consider the operator, K : Ψ → Ψ defined by:

(Ky)(t) = \{ \begin{matrix}y(0) & t ∈ J_{0} \\ ∫_{0}^{T} H(t, s)f(s, y_{s})ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(y(t_{k})) & t ∈ J \end{matrix} .

Then clearly from Lemma 2 the fixed points of K are solutions to (1)–(3). We shall show that K satisfies the assumptions of Lemma 1. The proof will be given in several steps.

Step 1: K maps bounded sets into bounded sets in Ψ. Indeed, it is enough to show that there exists a positive constant ℓ such that for each y ∈ B_{q} = {y ∈ Ψ : ∥y∥_{Ψ} ≤ q} one has ∥Ky∥_{Ψ} ≤ ℓ. Let y ∈ B_{q}, then for each t ∈ J, we have

(Ky)(t) = ∫_{0}^{T} H(t, s)f(s, y_{s})ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(y(t_{k})).

By (H1) we have for each t ∈ J,

|(Ky)(t)| ≤ ∫_{0}^{T} |H(t, s)||f(s, y_{s})|ds + ∑_{k=1}^{m} |H(t, t_{k})||I_{k}(y(t_{k}))|

≤ ∫_{0}^{T} |H(t, s)||g_{q}(s)|ds + ∑_{k=1}^{m} |H(t, t_{k})| sup{|I_{k}(y)| : ∥y∥_{Ψ} ≤ q}.

Then for each h ∈ K(B_{q}) we have

∥h∥_{Ψ} ≤ sup_{(t,s)∈J×J} |H(t, s)| ∫_{0}^{T} |g_{q}(s)|ds + ∑_{k=1}^{m} sup_{t∈J} |H(t, t_{k})| sup{|I_{k}(y)| : ∥y∥_{Ψ} ≤ q}
= ℓ.

Step 2: K maps bounded sets into equicontinuous sets of Ψ. Indeed, let τ_{1}, τ_{2} ∈ J, τ_{1} < τ_{2} and B_{q} be a bounded set of Ψ as in Step 1. Let y ∈ B_{q}. Then

|(Ky)(τ_{2}) − (Ky)(τ_{1})|
≤ ∫_{0}^{T} |H(τ_{2}, s) − H(τ_{1}, s)||g_{q}(s)|ds + ∑_{k=1}^{m} |H(τ_{2}, t_{k}) − H(τ_{1}, t_{k})|c_{k}.

As τ_{2} → τ_{1} the right-hand side of the above inequality tends to zero.

Step 3: K : Ψ → Ψ is continuous. Indeed, let {y_{n}} be a sequence such that y_{n} → y in Ψ. Then there is an integer q such that ∥y_{n}∥_{Ψ} ≤ q for all n = 0, 1, 2, ... and ∥y∥_{Ψ} ≤ q, so y_{n} ∈ B_{q} and y ∈ B_{q}. We have then by the dominated convergence theorem

∥Ky_{n} − Ky∥_{Ψ} ≤ sup_{t∈J} [ ∫_{0}^{T} |H(t, s)||f(s, y_{ns}) − f(s, y_{s})|ds

+ ∑_{k=1}^{m} |H(t, t_{k})||I_{k}(y_{n}(t_{k})) − I_{k}(y(t_{k}))|]

→ 0.

Thus K is continuous.

As a consequence of Steps 1 to 3 and (H4) together with the Arzela-Ascoli theorem we can conclude that K : Ψ → Ψ is completely continuous.

Step 4: The set

Φ(K) := {y ∈ Ψ : y = μK(y), for some 0 < μ < 1}

is bounded. Indeed, let y ∈ Φ(K). Then y = μK(y) for some 0 < μ < 1. Thus for each t ∈ J

y(t) = μ ∫_{0}^{T} H(t, s)f(s, y_{s})ds + μ ∑_{k=1}^{m} H(t, t_{k})I_{k}(y(t_{k})).

This implies by (H2)-(H3) that for each t ∈ J we have

|y(t)| ≤ ∫_{0}^{T} |H(t, s)f(s, y_{s})|ds + ∑_{k=1}^{m} |H(t, t_{k})I_{k}(y(t_{k}))|

≤ sup_{(t,s)∈J×J} |H(t, s)| ∫_{0}^{T} m(s)ds + ∑_{k=1}^{m} sup_{t∈J} |H(t, t_{k})| c_{k}
= b,

where b is independent of y. This shows that Φ(K) is bounded.

Set X := Ψ. As a consequence of Lemma 1 we deduce that K has a fixed point which is a solution of (1)–(3). The proof is complete.

Clearly, hypothesis (H3) is a strong hypothesis. Now that the alternative method due to Schaefer [14] has been established, standard hypotheses to obtain a priori bounds on solutions can be applied. For example, since H(t, s) ≤ Me^{|μ|T} for some positive constant M, if |f(t, x)| ≤ g(t)|x| on [0, T] × R, then a standard Gronwall inequality can be applied to obtain a priori bounds on solutions.

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