Dynamic Systems and Applications 15 (2006) 43-52

IMPULSIVE FUNCTIONAL DYNAMIC EQUATIONS ON TIME SCALES

M. BENCHOHRA^{1}, J. HENDERSON^{2}, S. K. NTOUYAS^{3}, AND A. OUAHAB^{1}

1 Laboratoire de Mathématiques, Université de Sidi Bel Abbès, BP 89, 22000 Sidi Bel Abbès, Algérie. E-mail: benchohra@univ-sba.dz

2 Department of Mathematics, Baylor University, Waco, Texas 76798-7328 USA. E-mail: Johnny Henderson@baylor.edu

3 Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece. E-mail: sntouyas@cc.uoi.gr

Dedicated to John Baxley on the occasion of his retirement from Wake Forest University.

ABSTRACT. In this paper, the nonlinear alternative of Leray-Schauder type is used to investigate the existence of solutions for first order impulsive functional dynamic equations on time scales with infinite delay. AMS (MOS) Subject Classification. 39A12.

1. INTRODUCTION

This paper is concerned with the existence of solutions of functional impulsive dynamic equations on time scales with infinite delay. First, in Section 3, we consider the impulsive problem,

(1.1)  y^{Δ}(t) = f(t, y_{t}), t ∈ J := [0, b], t ≠ t_{k}, k = 1, …, m,
(1.2)  y(t_{k}^{+}) − y(t_{k}^{−}) = I_{k}(y(t_{k}^{−})), k = 1, …, m,
(1.3)  y_{0} = ϕ ∈ B,

where I T is a time scale which has at least finitely many right-dense points, [0, b] ⊂ (−∞, b] ⊂ I T, f : I T × B → ℝ is a given function, I_{k} ∈ C(ℝ, ℝ), t_{k} ∈ I T, 0 < t_{1} < … < t_{m} < t_{m+1} = b, ϕ ∈ B, and B is called the phase space that will be defined later. Here, y(t_{k}^{+}) and y(t_{k}^{−}) represent right and left limits with respect to the time scale, and in addition, if t_{k} is right scattered, then y(t_{k}^{+}) = y(t_{k}), whereas, if t_{k} is left scattered, then y(t_{k}^{−}) = y(t_{k}).

Received November 1, 2004               1056-2176 $03.50 \begin{matrix}c \\ ◯ \end{matrix}Dynamic Publishers, Inc.

In Section 4, we consider first order impulsive neutral functional dynamic equations on time scales of the form,

(1.4)  [y(t) − g(t, y_{t})]^{Δ} = f(t, y_{t}), t ∈ [0, b], t ≠ t_{k}, k = 1, …, m,

(1.5)  y(t_{k}^{+}) − y(t_{k}^{−}) = I_{k}(y(t_{k}^{−})), k = 1, …, m,

(1.6)  y_{0} = ϕ ∈ B,

where f, ϕ, I_{k} are as in problem (1.1)–(1.3) and g : J × B → ℝ.

The notion of the phase space B plays an important role in the study of both qualitative and quantitative theory. A usual choice is a semi-normed space satisfying suitable axioms, which was introduced by Hale and Kato [11] (see also Kappel and Schappacher [20] and Schumacher [21]). For a detailed discussion on this topic we refer the reader to the book by Hino et al. [14]. In the case where the impulses are absent (i.e., I_{k} = 0, k = 1, …, m), an extensive theory is developed for the problem (1.1)-(1.3). We refer to the monographs of Hale and Lunel [12], Hino et al. [14], and Lakshmikantham et al. [16], and the paper of Corduneanu and Lakshmikantham [9].

Impulsive differential equations have become more important in recent years in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics. There has been a significant development in impulse theory, in recent years, especially in the area of impulsive differential equations with fixed moments; see the monographs of Bainov and Simeonov [4], Lakshmikantham et al. [15], and Samoilenko and Perestyuk [18], and the references therein.

In recent years dynamic equations on times scales have received much attention. We refer to the books by Agarwal and O’Regan [2], Bohner and Peterson [7, 8], Lakshmikantham et al. [17].

The times scales calculus has a tremendous potential for applications in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics, neural networks, and social sciences; see the monographs of Aulbach and Hilger [3], Bohner and Peterson [7, 8], Lakshmikantham et al. [17]. We also mention the recent magazine article [19].

Very recently, impulsive dynamic equations were studied by Agarwal et al. [1], Benchohra et al. [5, 6], and Henderson [13]. We shall prove our existence results for the problems (1.1)–(1.3) and (1.4)–(1.6) by using the nonlinear alternative of Leray-Schauder type [10].

2. PRELIMINARIES

We will briefly recall some basic definitions and facts from the times scales calculus that we will use in the sequel.

A time scale I T is a closed nonempty subset of ℝ. It follows that the jump operators σ, ρ : I T → I T defined by

σ(t) = inf{s ∈ I T : s > t} and ρ(t) = sup{s ∈ I T : s < t}

(supplemented by inf ∅ := sup I T and sup ∅ := inf I T) are well defined. The point t ∈ I T is left-dense, left-scattered, right-dense, right-scattered if ρ(t) = t, ρ(t) < t, σ(t) = t, σ(t) > t, respectively. If I T has a right-scattered minimum m, define I T_{k} := I T − {m}; otherwise, set I T_{k} = I T. If I T has a left-scattered maximum M, define I T^{k} := I T − {M}; otherwise, set I T^{k} = I T. The notations [c, d], [c, d), and so on, will denote time scales intervals such as

[c, d] = {t ∈ I T : c ≤ t ≤ d},

where c, d ∈ I T with c < ρ(d).

Definition 2.1. Let X be a Banach space. The function f : I T → X is called rdcontinuous provided it is continuous at each right-dense point and has a left-sided limit at each point; write f ∈ C_{rd}( I T) = C_{rd}( I T, X).

For t ∈ I T^{k}, let the Δ derivative of f at t, denoted f^{Δ}(t), be the number (provided it exists), such that for all ε > 0 there exists a neighborhood U of t such that

|f(σ(t)) − f(s) − f^{Δ}(t)[σ(t) − s]| ≤ ε|σ(t) − s|

for all s ∈ U.

A function F is called an antiderivative of f : I T → X provided

F^{Δ}(t) = f(t) for each t ∈ I T^{k}.

Let C([a, b], ℝ) denote the Banach space of all continuous functions from [a, b] into ℝ where [a, b] ⊂ I T with the norm

∥y∥_{∞} = sup{|y(t)| : t ∈ [a, b]}.

Remark 2.2. (i) If f is continuous, then f is rd−continuous. (ii) If f is delta differentiable at t, then f is continuous at t.

A function p : I T → ℝ is called regressive if

1 + μ(t)p(t) ≠ 0 for all t ∈ I T^{k},

where μ(t) = σ(t) − t, which is called the graininess function.

3. IMPULSIVE FUNCTIONAL DYNAMIC EQUATIONS

In order to define the phase space and the solution of (1.1)–(1.3), we shall consider the space

B_{b} = {y : (−∞, b] → ℝ^{n} | there exist t_{0} < t_{1} < … < t_{m} < b such that y(t_{k}^{−}) and

y(t_{k}^{+}) exist, with y(t_{k}) = y(t_{k}^{−}), 0 ≤ k ≤ m, and y(t) = ϕ(t), t ≤ 0, y_{k} ∈ C(J_{k}, ℝ^{n})},

where y_{k} is the restriction of y to J_{k} = (t_{k}, t_{k+1}], k = 0, …, m. Let ∥ · ∥_{b} be the seminorm in B_{b} defined by

∥y∥_{b} = ∥y_{0}∥_{B} + sup{|y(s)| : 0 ≤ s ≤ b}, y ∈ B_{b}.

We will assume that B satisfies the following axioms: (A) If y : (−∞, b] → ℝ, b > 0, is such that y|_{[0,b]} ∈ B_{b} and y_{0} ∈ B, then for every t

in [0, b) the following conditions hold:

(i) y_{t} is in B; (ii) ∥y_{t}∥_{B} ≤ K(t) sup{|y(s)| : 0 ≤ s ≤ t} + M(t)∥y_{0}∥_{B}, where H ≥ 0 is a constant, K : [0, ∞) → [0, ∞) is continuous, M : [0, ∞) → [0, ∞) is locally bounded, and H, K, M are independent of y(·).

(A-1) For the function y(·) in (A), y_{t} is a B-valued continuous function on [0, b). (A-2) The space B is complete.

Definition 3.1. A function y ∈ B_{b}, is said to be a solution of (1.1)–(1.3), if y satisfies the dynamic equation

y^{Δ}(t) = f(t, y_{t}) everywhere on J\{t_{k}}, k = 1, …, m,

and for each k = 1, …, m, the function y satisfies the equations y(t_{k}^{+}) − y(t_{k}^{−}) = I_{k}(y(t_{k})), and y_{0} = ϕ ∈ B.

The following hypotheses will be assumed hereafter: (H1) The function f : [0, b] × B → ℝ is continuous; (H2) There exist constants c_{k}, such that

|I_{k}(y)| ≤ c_{k}, for each k = 1, …, m, and for all y ∈ ℝ;

(H3) There exist a continuous non-decreasing function ψ : [0, ∞) ⟶ (0, ∞), a function p ∈ L^{1}([0, b], ℝ_{+}), and a constant M > 0 such that

|f(t, u)| ≤ p(t)ψ(|u|_{B}) for each (t, u) ∈ [0, b] × B,

and

\frac{M}{K_{b} [∫_{0}^{b} p(s)ψ(M)Δs + ∑_{k=1}^{m} c_{k}] + K_{b}|ϕ(0)| + M_{b}∥ϕ∥_{B}} > 1,

where K_{b} = sup{K(t) : t ∈ [0, b]} and M_{b} = sup{M(t) : t ∈ [0, b]}.

Theorem 3.2. Suppose that hypotheses (H1)–(H3) are satisfied. Then the impulsive IVP (1.1)–(1.3) has at least one solution.

Proof. Transform the problem (1.1)–(1.3) into a fixed point problem. We consider the operator N : B_{b} → B_{b} defined by,

(Ny)(t) = \{ \begin{matrix}ϕ(t), & if t ∈ (−∞, 0], \\ ϕ(0) + ∫_{0}^{t} f(s, y_{s})Δs + ∑_{0<t_{k}<t} I_{k}(y(t_{k})), & if t ∈ [0, b]. \end{matrix}

Clearly, the fixed points of N are solutions to (1.1)–(1.3). We will prove that N has a fixed point.

Let x(·) : (−∞, b) → ℝ be the function defined by

x(t) = \{ \begin{matrix}ϕ(0), & if t ∈ [0, b], \\ ϕ(t), & if t ∈ (−∞, 0]; \end{matrix}

then x_{0} = ϕ. For each z ∈ C([0, b], E) with z_{0} = 0, we denote by z̄ the function defined by

z̄(t) = \{ \begin{matrix}z(t), & if t ∈ [0, b]; \\ 0, & if t ∈ (−∞, 0]. \end{matrix}

If y(·) satisfies

(3.1)  y(t) = ϕ(0) + ∫_{0}^{t} f(s, y_{s})Δs + ∑_{0<t_{k}<t} I_{k}(y(t_{k}^{−})),

we can decompose it as y(t) = z̄(t) + x(t), 0 ≤ t ≤ b, which implies y_{t} = z̄_{t} + x_{t}, for every 0 ≤ t ≤ b and the function z(·) satisfies

(3.2)  z(t) = ∫_{0}^{t} f(s, z̄_{s} + x_{s})Δs + ∑_{0<t_{k}<t} I_{k}(z(t_{k}^{−}) + x(t_{k}^{−})).

Set

B_{b}^{0} = {z ∈ B_{b} : z_{0} = 0}.

For any z ∈ B_{b}^{0}, we have

∥z∥_{B_{b}^{0}} = ∥z_{0}∥_{B} + sup{|z(s)| : 0 ≤ s ≤ b} = sup{|z(s)| : 0 ≤ s ≤ b}.

Thus, (B_{b}^{0}, ∥ · ∥_{B_{b}^{0}}) is a Banach space. Let the operator P : B_{b}^{0} → B_{b}^{0} be defined by,

(P z)(t) = \{ \begin{matrix}0, & t ≤ 0, \\ ∫_{0}^{t} f(s, z̄_{s} + x_{s})Δs + ∑_{0<t_{k}<t} I_{k}(x(t_{k}^{−}) + z(t_{k}^{−})), & t ∈ [0, b]. \end{matrix}

48 M. BENCHOHRA, J. HENDERSON, S. K. NTOUYAS, AND A. OUAHAB

Obviously the operator N has a fixed point is equivalent to P has one, so we turn to prove that P has a fixed point. We shall use the Leray-Schauder alternative to prove that P has fixed point.

Step 1: P is continuous.

Let {z_{n}} be a sequence such that z_{n} → z in B_{b}^{0}. Then

|P (z_{n})(t) − P (z)(t)| ≤ ∫_{0}^{b} |f(s, z̄_{n_{s}} + x_{s}) − f(s, z̄_{s} + x_{s})|Δs.

+ ∑_{k=1}^{m} |I_{k}(z_{n}(t_{k}) + x(t_{k})) − I_{k}(z(t_{k}) + x(t_{k}))|.

Hence,

∥P (z_{n}) − P (z)∥_{B_{b}^{0}} ≤ ∥f(·, z̄_{n}(·) + x(·)) − f(·, z̄(·) + x(·))∥_{L^{1}}

+ ∑_{k=1}^{m} |I_{k}(z_{n}(t_{k}) + x(t_{k})) − I_{k}(z(t_{k}) + x(t_{k}))|.

Thus,

∥P (z_{n}) − P (z)∥_{B_{b}^{0}} → 0 as n → ∞.

Step 2: P sends bounded sets into bounded sets.

We will show that for any q > 0 there exists a positive constant ℓ such that for each z ∈ B_{q} = {z ∈ B_{b}^{0} : ∥z∥_{B_{b}} ≤ q} one has ∥P ∥_{B_{b}} ≤ ℓ. For every x ∈ B_{q}, we have

∥x_{t} + z̄_{t}∥_{B} ≤ ∥x_{t}∥_{B} + ∥z̄_{t}∥_{B}
≤ K(t) sup{|x(s)| : 0 ≤ s ≤ t} + M(t)∥x_{0}∥_{B}
+K(t) sup{|z̄(s)| : 0 ≤ s ≤ t} + M(t)∥z_{0}∥_{B}
≤ K_{b}q + K_{b}|ϕ(0)| + M_{b}∥ϕ∥_{B} := q^{∗}.

By (H2)–(H3), for each t ∈ J, we have that

|(P z)(t)| ≤ ∫_{0}^{t} p(s)ψ(∥x_{s} + z̄_{s}∥_{B})Δs + ∑_{k=1}^{m} c_{k}
≤ ψ(q^{∗}) ∫_{0}^{b} p(s)Δs + ∑_{k=1}^{m} c_{k}.

Then, we have

∥P ∥_{B_{b}} ≤ ψ(q^{∗}) ∫_{0}^{b} p(s)Δs + ∑_{k=1}^{m} c_{k} := ℓ.

Step 3: P sends bounded sets into equicontinuous sets.

IFDE’S ON TIME SCALES 49

Let τ_{1}, τ_{2} ∈ J, 0 < τ_{1} < τ_{2}. Then we have

|(P z)(τ_{2}) − (P z)(τ_{1})| ≤ ψ(r^{∗}) ∫_{τ_{1}}^{τ_{2}} p(s)Δs + ∑_{τ_{1}≤t_{k}<τ_{2}} c_{k}.

The right-hand side tends to zero as τ_{2} → τ_{1}.

As a consequence of Steps 2-3 together with the Arzelá-Ascoli Theorem, it suffices to show that P maps B_{q} into precompact sets.

Step 4 A priori bounds on solutions. Let z be a solution of the integral equation,

z(t) = ∫_{0}^{t} f(s, z̄_{s} + x_{s})Δs + ∑_{0<t_{k}<t} I_{k}(x(t_{k}^{−}) + z(t_{k}^{−})).

By (H2), we have that

(3.3)  |z(t)| ≤ ∫_{0}^{t} p(s)ψ(∥x_{s} + z̄_{s}∥_{B})Δs + ∑_{0<t_{k}<t} c_{k}.

But

∥x_{t} + z̄_{t}∥_{B} ≤ ∥x_{t}∥_{B} + ∥z̄_{t}∥_{B}
≤ K(t) sup{|x(s)| : 0 ≤ s ≤ t} + M(t)∥x_{0}∥_{B}
+K(t) sup{|z(s)| : 0 ≤ s ≤ t} + M(t)∥z̄_{0}∥_{B}
≤ K_{b} sup{|z(s)| : 0 ≤ s ≤ t} + K_{b}|ϕ(0)| + M_{b}∥ϕ∥_{B}.

If we let w(t) denote the right hand side of the above inequality, we have that

∥x_{t} + z̄_{t}∥_{B} ≤ w(t),

and therefore (3.3) becomes

(3.4)  |z(t)| ≤ ∫_{0}^{t} p(s)ψ(w(s))Δs + ∑_{0<t_{k}<t} c_{k}.

Using (3.4) in the definition of w, we have that

w(t) ≤ K_{b} [∫_{0}^{t} p(s)ψ(w(s))Δs + ∑_{0<t_{k}<t} c_{k}] + K_{b}|ϕ(0)| + M_{b}∥ϕ∥_{B}.

Consequently,

\frac{∥w∥_{∞}}{K_{b} [∫_{0}^{b} p(s)ψ(∥w∥_{∞})Δs + ∑_{0<t_{k}<t} c_{k}] + K_{b}|ϕ(0)| + M_{b}∥ϕ∥_{B}} ≤ 1.

50 M. BENCHOHRA, J. HENDERSON, S. K. NTOUYAS, AND A. OUAHAB

Then, by (H3), there exists M such that ∥w∥_{∞} ≠ M. Set

U = {z ∈ B_{b}^{0} : ∥z∥_{B_{b}^{0}} < M + 1}.

The operator P : \overline{U} → B_{b}^{0} is completely continuous. From the choice of U, there is no z ∈ ∂U such that z = λP (z), for some λ ∈ (0, 1). As a consequence of the nonlinear alternative of Leray-Schauder type [10], we deduce that P has a fixed point z in U. Then the problem (1.1)–(1.3) has at least one solution. □

4. NEUTRAL FUNCTIONAL DYNAMIC EQUATIONS

Definition 4.1. A function y ∈ B_{b}, is said to be a solution of (1.4)–(1.6), if y satisfies the dynamic equation

[y(t) − g(t, y_{t})]^{Δ} = f(t, y_{t}) everywhere on J\{t_{k}}, k = 1, …, m,

and for each k = 1, …, m, the function y satisfies the equations y(t_{k}^{+}) − y(t_{k}^{−}) = I_{k}(y(t_{k})), and y_{0} = ϕ ∈ B.

Theorem 4.2. Let f : J × B → ℝ be an continuous function. Assume (H2) and the conditions:

(B1) The function g is continuous and completely continuous, and for any bounded set Q ⊆ C((−∞, b], ℝ), the set {t → g(t, x_{t}) : x ∈ Q} is equicontinuous in C([0, b], ℝ^{n}), and there exist constants 0 ≤ c_{1} < 1, c_{2} ≥ 0 such that

|g(t, u)| ≤ c_{1}∥u∥_{B} + c_{2}, t ∈ [0, b], u ∈ B;

(B2) There exist a continuous nondecreasing function ψ : [0, ∞) ⟶ (0, ∞) and p ∈ L^{1}(J, ℝ_{+}) such that

|f(t, x)| ≤ p(t)ψ(∥u∥_{B}), for a.e. t ∈ [0, b] and each u ∈ B,

and there exists M_{∗} > 0 such that

\frac{M_{∗}}{\frac{1}{1 − c_{1}K_{b}} [K_{b}|g(0, ϕ(0))| + c_{2}K_{b} + α + K_{b}ψ(M_{∗}) ∫_{0}^{b} p(s)Δs]} > 1,

where α = K_{b}|ϕ(0)| + M_{b}∥ϕ∥_{B},

are satisfied. Then the IVP (1.4)-(1.6) has at least one solution.

Proof. In analogy to Theorem 3.2 we consider the operator P^{∗} : B_{b}^{0} → B_{b}^{0} defined by

(P^{∗}z)(t) = \{ \begin{matrix}0, & t ≤ 0, \\ g(0, ϕ(0)) − g(t, \overline{z}_{t} + x_{t}) + ∫_{0}^{t} f(s, \overline{z}_{s} + x_{s})Δs, & t ∈ [0, b]. \end{matrix}

IFDE’S ON TIME SCALES 51

As in Theorem 3.2, we can prove that the operator P^{∗} is completely continuous. In order to use the Leray-Schauder alternative, we shall obtain a priori estimates for the solutions of the integral equation,

z(t) = λ [g(0, ϕ(0)) − g(t, \overline{z}_{t} + x_{t}) + ∫_{0}^{t} f(s, \overline{z}_{s} + x_{s})Δs] ,

where z_{0} = λϕ for some λ ∈ (0, 1). Then,

|z(t)| ≤ |g(0, ϕ(0))| + |g(t, z̄_{t} + x_{t})| + ∫_{0}^{t} p(s)ψ(∥z̄_{s} + x_{s}∥_{B})ds
≤ |g(0, ϕ(0))| + c_{1}∥z̄_{t} + x_{t}∥_{B} + c_{2} + ∫_{0}^{t} p(s)ψ(∥\overline{z}_{s} + x_{s}∥_{B})ds.

If we put α = K_{b}|ϕ(0)| + M_{b}∥ϕ∥_{B}, then

∥z̄_{t} + x_{t}∥_{B} ≤ K_{b} sup_{s∈[0,t]} |z(s)| + α := w(t),

and

|z(t)| ≤ |g(0, ϕ(0))| + c_{1}w(t) + c_{2} + ∫_{0}^{t} p(s)ψ(w(s))Δs.

But

w(t) ≤ K_{b}|g(0, ϕ(0))| + c_{1}K_{b}w(t) + c_{2}K_{b} + K_{b} ∫_{0}^{t} p(s)ψ(w(s))Δs + α,

or

w(t) ≤ \frac{1}{1 − c_{1}K_{b}} [K_{b}|g(0, ϕ(0))| + c_{2}K_{b} + α + K_{b} ∫_{0}^{b} p(s)ψ(w(s))Δs] , t ∈ [0, b].

Hence,

\frac{∥w∥_{∞}}{\frac{1}{1 − c_{1}K_{b}} [K_{b}|g(0, ϕ)| + c_{2}K_{b} + α + K_{b} ∫_{0}^{b} p(s)ψ(∥w∥_{∞})Δs]} ≤ 1.

Then, by (B2), there exists M_{∗} such that ∥w∥_{∞} ≠ M_{∗}. Set

U_{∗} = {z ∈ B_{b}^{0} : ∥z∥_{B_{b}^{0}} < M_{∗} + 1}.

The operator P^{∗} : \overline{U}_{∗} → B_{b}^{0} is completely continuous. From the choice of U_{∗}, there is no z ∈ ∂U_{∗} such that z = λP^{∗}(z), for some λ ∈ (0, 1). As a consequence of the nonlinear alternative of Leray-Schauder type [10], we deduce that P^{∗} has a fixed point z in U_{∗}, and so the problem (1.4)–(1.6) has at least one solution. □

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