Electronic Journal of Qualitative Theory of Differential Equations 2005, No. 12, 1-22; http://www.math.u-szeged.hu/ejqtde/

Existence Results for Impulsive Dynamic Inclusions on Time Scales

Abdelkader Belarbi, Mouffak Benchohra^{1} and Abdelghani Ouahab

Laboratoire de Mathématiques, Université de Sidi Bel Abbès BP 89, 22000, Sidi Bel Abbès, Algérie e-mail: aek_belarbi@yahoo.fr, benchohra@univ-sba.dz, ouahab@univ-sba.dz

Dedicated to the memory of Professor Bernd Aulbach

Abstract

In this paper, we investigate the existence of solutions and extremal solutions for a first order impulsive dynamic inclusion on time scales. By using suitable fixed point theorems, we study the case when the right hand side has convex as well as nonconvex values.

Key words and phrases: Impulsive dynamic inclusions, delta derivative, contraction, extremal solutions, fixed point, time scales. AMS (MOS) Subject Classifications: 34A37, 34A60, 34B37.

1 Introduction

This paper is concerned with the existence of solutions and extremal solutions for a class of initial value problem for impulsive dynamic inclusions on time scales. More precisely, in Section 3, we consider the following problem:

y^{Δ}(t) + p(t)y^{σ}(t) ∈ F (t, y(t)), t ∈ J := [0, b] ∩ 𝕋, t ≠ t_{k}, k = 1, …, m, (1)
y(t_{k}^{+}) − y(t_{k}^{−}) = I_{k}(y(t_{k}^{−})), k = 1, …, m,  (2)
y(0) = η,  (3)

where 𝕋 is a time scale, F : [0, b] × IR → P(IR) is a compact valued multi-valued map, P(IR) is the family of all nonempty subsets of IR, I_{k} ∈ C(IR, IR), η ∈ IR, 0 = t_{0} < t_{1} < ... < t_{m} < t_{m+1} = b, and for each k = 1, …, m,

y(t_{k}^{+}) = lim_{h→0^{+}} y(t_{k} + h) and y(t_{k}^{−}) = lim_{h→0^{−}} y(t_{k} + h) represent the right and left limits

of y(t) at t = t_{k} in the sense of time scales, that is, t_{k} + h ∈ [0, b] ∩ 𝕋 for each h in a neighborhood of 0 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}), σ is a function that will be defined later and y^{σ}(t) = y(σ(t)).

1Corresponding author

Impulsive differential equations have become important in recent years in mathematical models of real processes and they arise in 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 Lakshmikantham et al [22], Samoilenko and Perestyuk [25] and the references therein. In recent years dynamic equations on time scales have received much attention. We refer to the books by Bohner and Peterson [9, 10], Lakshmikantham et al [23] and to the references cited therein. The time scales calculus has tremendous potential for applications in mathematical models of real processes and phenomena, for example in physics, chemical technology, population dynamics, biotechnology and economics, neural networks, social sciences, see the monographs of Aulbach and Hilger [2], Bohner and Peterson [9, 10], Lakshmikantham et al [23] and to the references therein. Recently Henderson [18] and Benchohra et al [1, 7, 8] have initiated the study of impulsive dynamic equations on time scales. The first paper for impulsive dynamic inclusions was proposed by Belarbi, Benchohra and Ouahab [4]. In this paper, we continue this study by considering more general classes of impulsive dynamic inclusions on time scales. We shall provide existence results for the problem (1)-(3). The first one relies on the nonlinear alternative of Leray-Schauder type [16] when the right hand side is convex valued, the second and the third rely also on the nonlinear alternative of Leray-Schauder type but under weaker conditions on the functions I_{k} (k = 1, ..., m) and the mixed generalized Lipschitz and Carathéodory’s conditions and the last one on the fixed point theorem for contraction multi-valued maps due to Covitz and Nadler [13] when the right hand side is not necessarily convex valued. The last section is concerned with the existence of extremal solutions of the above mentioned problem by using a recent fixed point theorem due to Dhage [14] for the sum of a contraction multivalued map and a completely continuous one defined on ordered Banach spaces. These results complement the few existence results devoted to dynamic inclusions on time scales.

2 Preliminaries

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

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

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

(supplemented by inf ∅ := sup 𝕋 and sup ∅ := inf 𝕋) are well defined. The point t ∈ 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 𝕋 has a right-scattered minimum m, define 𝕋_{k} := 𝕋 − {m}; otherwise, set 𝕋_{k} = 𝕋. If 𝕋 has a left-scattered maximum M, define 𝕋^{k} := 𝕋 − {M}; otherwise, set 𝕋^{k} = 𝕋. The notations [0, b], [0, b), and so on, will denote time scales intervals

[0, b] = {t ∈ 𝕋 : a ≤ t ≤ b},

where 0, b ∈ 𝕋 with 0 < ρ(b).

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

Definition 2.2 Let t ∈ 𝕋^{k}, the Δ derivative of f at t, denoted f^{Δ}(t), be the number(provided it exists) if 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, at fix t.

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

F^{Δ}(t) = f(t) for each t ∈ 𝕋^{k}.

Remark 2.3 (i) If f is continuous, then f rd−continuous.

(ii) If f is delta differentiable at t then f is continuous at t.

A function p : 𝕋 → IR is called regressive if

1 + μ(t)p(t) ≠ 0 for all t ∈ 𝕋,

where μ(t) = σ(t) − t, which is called the graininess function. We denote by R^{+} the set of the regressive functions. The generalized exponential function e_{p} is defined as the unique solution of the initial value problem y^{Δ} = p(t)y, y(0) = 1, where p is a regressive function. An explicit formula for e_{p}(t, 0) is given by

e_{p}(t, s) = exp \{∫_{s}^{t} ξ_{μ(τ)}(p(τ))Δτ\} with ξ_{h}(z) = \{ \begin{matrix}\frac{Log(1 + hz)}{h} & if h ≠ 0, \\ z & if h = 0. \end{matrix}

For more details, see [9]. Clearly, e_{p}(t, s) never vanishes. We now give some fundamental properties of the exponential function. Let p, q : 𝕋 → IR two regressive functions. We define

p ⊕ q = p + q + μpq, ⊖p := −\frac{p}{1 + μp}, p ⊖ q := p ⊕ (⊖q).

Theorem 2.4 [9] Assume that p, q : 𝕋 → IR are regressive functions, then the following hold:

(i) e_{0}(t, s) ≡ 1 and e_{p}(t, t) ≡ 1;

(ii) e_{p}(σ(t), s) = (1 + μ(t)p(t))e_{p}(t, s);

(iii) \frac{1}{e_{p}(t, s)} = e_{⊖p}(t, s);

(iv) e_{p}(t, s)\frac{1}{e_{p}(s, t)} = e_{⊖p}(s, t);

(v) e_{p}(t, s)e_{p}(s, r) = e_{p}(t, r);

(vi) e_{p}(t, s)e_{q}(t, s) = e_{p⊕q}(t, s);

(vii) \frac{e_{p}(t, s)}{e_{q}(t, s)} = e_{p⊖q}(t, s).

C([0, b], IR) is the Banach space of all continuous functions from [0, b] into IR with the norm

∥y∥_{∞} = sup{|y(t)| : t ∈ [0, b]}.

L^{1}([0, b], IR) denote the space of functions from [0, b] into IR which are Lebesgue integrable in the time scale sense normed by

∥y∥_{L^{1}} = ∫_{0}^{b} |y(t)|Δt for each y ∈ L^{1}([0, b], IR)

AC((0, b), IR) is the space of differentiable functions y : (0, b) → IR whose first delta derivative, y^{Δ}, is absolutely continuous.

Let (X, | · |) be a normed space, P(X) = {Y ⊂ X : Y ≠ ∅}, P_{cl}(X) = {Y ∈ P (X) : Y closed}, P_{b}(X) = {Y ∈ P(X) : Y bounded}, P_{c}(X) = {Y ∈ P (X) : Y convex}, P_{cp}(X) = {Y ∈ P(X) : Y compact}. A multivalued map N : [0, b] → P_{cl}(IR) is said to be measurable, if for every y ∈ IR, the function t ⟼ d(y, N(t)) = inf{|y − z| : z ∈ N(t)} is measurable where d is the metric induced by the Banach space IR. In what follows, we will assume that the function F : [0, b] × IR → P(IR) is Carathéodory, i.e.

(i) t → F (t, x) is measurable for each x ∈ IR,

(ii) x → F (t, x) is upper semicontinuous for almost all t ∈ [0, b],

For each y ∈ C([0, b], IR), let S_{F,y} the set of selections of F defined by

S_{F,y} = {v ∈ L^{1}([0, b], IR) : v(t) ∈ F (t, y(t)), a.e. t ∈ [0, b]}.

The following Lemma is crucial in the proof of our main results when the multivalued map has convex values:

Lemma 2.5 [24]. Let X be a Banach space. Let F : J × X ⟶ P_{cp,c}(X) be a Carathéodory multivalued map and let Γ be a linear continuous mapping from L^{1}(J, X) to C(J, X), then the operator

\begin{matrix}Γ ∘ S_{F} : C(J, X) & ⟶ P_{cp,c}(C(J, X)), \\ y & ⟼ (Γ ∘ S_{F})(y) := Γ(S_{F (y)}) \end{matrix}

is a closed graph operator in C(J, X) × C(J, X).

3 Existence Results

We will assume for the remainder of this paper that, for each k = 1, …, m, the points of impulse t_{k} are right dense. In order to define the solution of (1)–(3), we shall consider the following space:

P C = {y : [0, b] ⟶ IR : y_{k} ∈ C(J_{k}, IR), k = 0, …, m, and there exist y(t_{k}^{−})
and y(t_{k}^{+}) with y(t_{k}^{−}) = y(t_{k}), k = 1, …, m},

which is a Banach space with the norm

∥y∥_{P C} = max{∥y_{k}∥_{J_{k}}, k = 0, …, m},

where y_{k} is the restriction of y to J_{k} = (t_{k}, t_{k+1}] ⊂ [0, b], k = 1, …, m, and J_{0} = [t_{0}, t_{1}].

Let us start by defining what we mean by a solution of problem (1)-(3).

Definition 3.1 A function y ∈ P C ∩ AC(J\{t_{1}, … t_{m}}, IR) is said to be a solution of (1)–(3) if there exists a function v ∈ L^{1}([0, b], IR) such that

y^{Δ}(t) + p(t)y^{σ}(t) = v(t) a.e. on J\{t_{k}}, k = 1, …, m,

and for each k = 1, …, m, the function y satisfies the condition y(t_{k}^{+}) − y(t_{k}^{−}) = I_{k}(y(t_{k}^{−})), and the intial condition y(0) = η.

We need the following auxiliary result (see [7]).

Lemma 3.2 Let p : 𝕋 → IR be rd−continuous and regressive. Suppose f : 𝕋 → IR rd−continuous. Let t_{0} ∈ 𝕋, and y_{0} ∈ IR. Then, y is the unique solution of the initial value problem

y^{Δ}(t) + p(t)y^{σ}(t) = f(t), t ∈ [0, b] ∩ 𝕋, t ≠ t_{k}, k = 1, …, m  (4)
y(t_{k}^{+}) − y(t_{k}^{−}) = I_{k}(y(t_{k}^{−})), k = 1, …, m,  (5)
y(0) = y_{0},  (6)

if and only if

y(t) = e_{⊖p}(t, 0)y_{0} + ∫_{0}^{t} e_{⊖p}(t, s)f(s)Δs + ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y(t_{k}^{−})).  (7)

Let us introduce the following hypotheses which are assumed hereafter:

(H1) The function F : [0, b] × IR → P(IR) is Carathéodory.

(H2) There exist constants c_{k} > 0 such that

|I_{k}(x)| ≤ c_{k} for each k = 1, …, m and for all x ∈ IR.

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

∥F (t, x)∥_{P} = sup{|v| : v ∈ F (t, x)} ≤ p(t)ψ(|x|) for each (t, x) ∈ [0, b] × IR,

and

\frac{M}{|η| sup_{t∈[0,b]} e_{⊖p}(t, 0) + ∑_{k=1}^{m} c_{k} sup_{t∈[0,b]} e_{⊖p}(t, t_{k}) + sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s)ψ(M) ∫_{0}^{b} p(s)Δs} > 1.

Theorem 3.3 Suppose that hypotheses (H1)–(H3) hold. Then the impulsive dynamic inclusions (1)–(3) has at least one solution on [0, b].

Proof. Transform the problem (1)–(3) into a fixed point problem. Consider the operator N : P C ⟶ P(P C) defined by

\begin{aligned}N(y) & = \{h ∈ P C : h(t) = e_{⊖p}(t, 0)η + ∫_{0}^{t} e_{⊖p}(t, s)v(s)Δs \\ & + ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y(t_{k}^{−})), v ∈ S_{F,y}\}. \end{aligned}

Remark 3.4 Clearly, from Lemma 3.2, the fixed points of N are solutions to (1)–(3).

We shall show that N satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof will be given in several steps.

Step 1: N(y) is convex for each y ∈ P C.

Indeed, if h_{1}, h_{2} belong to N(y), then there exist v_{1}, v_{2} ∈ S_{F,y} such that for each t ∈ [0, b] we have

h_{i}(t) = e_{⊖p}(t, 0)η + ∫_{0}^{t} e_{⊖p}(t, s)v_{i}(s)Δs + ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y(t_{k}^{−})) (i = 1, 2).

Let 0 ≤ d ≤ 1. Then, for each t ∈ [0, b] we have

\begin{aligned}(dh_{1} + (1 − d)h_{2})(t) & = e_{⊖p}(t, 0)η + ∫_{0}^{t} e_{⊖p}(t, s)[dv_{1}(s) + (1 − d)v_{2}(s)]Δs \\ & + ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y(t_{k}^{−})). \end{aligned}

Since S_{F,y} is convex (because F has convex values), then

dh_{1} + (1 − d)h_{2} ∈ N(y).

Step 2: N maps bounded sets into bounded sets in P C.

Let B_{q} = {y ∈ P C : ∥y∥_{P C} ≤ q} be a bounded set in P C and y ∈ B_{q}, then for each h ∈ N(y), there exists v ∈ S_{F,y} such that for each t ∈ [0, b],

h(t) = e_{⊖p}(t, 0)η + ∫_{0}^{t} e_{⊖p}(t, s)v(s)Δs + ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y(t_{k}^{−})).

From (H2) and (H3) we have

|h(t)| ≤ |η| sup_{t∈[0,b]} e_{⊖p}(t, 0) + sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s) ∫_{0}^{b} |v(s)|Δs

+ ∑_{k=0}^{m} e_{⊖p}(t, t_{k})c_{k}

≤ |η| sup_{t∈[0,b]} e_{⊖p}(t, 0) + sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s) ∫_{0}^{b} ψ(q)p(s)Δs

+ ∑_{k=0}^{m} sup_{t∈[0,b]} e_{⊖p}(t, t_{k})c_{k}

≤ |η| sup_{t∈[0,b]} e_{⊖p}(t, 0) + sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s)ψ(q)∥p∥_{L^{1}}

+ ∑_{k=0}^{m} sup_{t∈[0,b]} e_{⊖p}(t, t_{k})c_{k}.

Step 3: N maps bounded sets into equicontinuous sets of P C.

Let u_{1}, u_{2} ∈ J, u_{1} < u_{2} and B_{q} be a bounded set of P C as in Step 2 and y ∈ B_{q}. For each h ∈ N(y), there exists v ∈ S_{F,y} such that for each t ∈ [0, b],

h(t) = e_{⊖p}(t, 0)η + ∫_{0}^{t} e_{⊖p}(t, s)v(s)Δs + ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y(t_{k}^{−})).

EJQTDE, 2005 No. 12, p. 7 Then, we have

|h(u_{2}) − h(u_{1})| ≤ |e_{⊖p}(u_{2}, 0) − e_{⊖p}(u_{1}, 0)||η|

+ψ(q)∥p∥_{L^{1}} ∫_{0}^{u_{1}} |e_{⊖p}(u_{2}, s) − e_{⊖p}(u_{1}, s)|Δs

+ψ(q)∥p∥_{L^{1}} ∫_{u_{1}}^{u_{2}} e_{⊖p}(u_{2}, s)Δs

+ ∑_{0≤t_{k}<u_{1}} |e_{⊖p}(u_{2}, t_{k}) − e_{⊖p}(u_{1}, t_{k})|c_{k}

+ ∑_{u_{1}≤t_{k}<u_{2}} e_{⊖p}(u_{2}, t_{k})c_{k}.

The right hand side tends to zero as u_{2} − u_{1} → 0. As a consequence of Steps 1 to 3 together with the Arzelá-Ascoli Theorem, we can conclude that N : P C ⟶ P(P C) is completely continuous.

Step 4: N has a closed graph.

Let y_{n} → y_{∗}, h_{n} ∈ N(y_{n}) and h_{n} → h_{∗}. We need to show that h_{∗} ∈ N(y_{∗}). h_{n} ∈ N(y_{n}) means that there exists v_{n} ∈ S_{F,y_{n}} such that for each t ∈ [0, b],

h_{n}(t) = e_{⊖p}(t, 0)η + ∫_{0}^{t} e_{⊖p}(t, s)v_{n}(s)Δs + ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y_{n}(t_{k}^{−})).

We must show that there exists h_{∗} ∈ S_{F,y_{∗}} such that for each t ∈ [0, b],

h_{∗}(t) = e_{⊖p}(t, 0)η + ∫_{0}^{t} e_{⊖p}(t, s)v_{∗}(s)Δs + ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y_{∗}(t_{k}^{−})).

Clearly, since I_{k}, k = 1, …, m, are continuous, we have

∥(h_{n}− ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y_{n}(t_{k}^{−})))−(h_{∗}− ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y_{∗}(t_{k}^{−})))∥_{P C} ⟶ 0, as n → ∞.

Consider the continuous linear operator

Γ : L^{1}([0, b], IR) → C([0, b], IR)

given by

v ⟼ (Γv)(t) = ∫_{0}^{t} e_{⊖p}(t, s)v(s)ds.

>From Lemma 2.5, it follows that Γ ∘ S_{F} is a closed graph operator. Moreover, we have

(h_{n}(t) − ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y_{n}(t_{k}^{−}))) ∈ Γ(S_{F,y_{n}}).

Since y_{n} → y_{∗}, it follows from Lemma 2.5 that for each t ∈ [0, b],

h_{∗}(t) = e_{⊖p}(t, 0)η + ∫_{0}^{t} e_{⊖p}(t, s)v_{∗}(s)Δs + ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y_{∗}(t_{k}^{−})),

for some v_{∗} ∈ S_{F,v_{∗}}.

Step 5: A priori bounds on solutions.

Let y be such that y ∈ λN(y) for some λ ∈ (0, 1). Then, there exists v ∈ S_{F,y} such that for each t ∈ [0, b],

y(t) = λe_{⊖p}(t, 0)η + λ ∫_{0}^{t} e_{⊖p}(t, s)v(s)Δs + λ ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y(t_{k}^{−})).

This implies by (H2) and (H3) that, for each t ∈ [0, b],

|y(t)| ≤ |η| sup_{t∈[0,b]} e_{⊖p}(t, 0) + ∑_{k=1}^{m} c_{k} sup_{t∈[0,b]} e_{⊖p}(t, t_{k})

+ sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s) ∫_{0}^{b} p(s)ψ(|y(s)|)Δs

≤ |η| sup_{t∈[0,b]} e_{⊖p}(t, 0) + ∑_{k=1}^{m} c_{k} sup_{t∈[0,b]} e_{⊖p}(t, t_{k})

+ sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s)ψ(∥y∥_{P C}) ∫_{0}^{b} p(s)Δs.

Consequently

\frac{∥y∥_{P C}}{|η| sup_{t∈[0,b]} e_{⊖p}(t, 0) + ∑_{k=1}^{m} c_{k} sup_{t∈[0,b]} e_{⊖p}(t, t_{k}) + sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s)ψ(∥y∥_{P C}) ∫_{0}^{b} p(s)Δs} ≤ 1.

Then by (H3), there exists M such that ∥y∥_{P C} ≠ M. Let

U = {y ∈ P C : ∥y∥_{P C} < M}.

The operator N : \overline{U} → P(P C) is upper semicontinuous and completely continuous. >From the choice of U, there is no y ∈ ∂U such that y ∈ λN(y) for some λ ∈ (0, 1). As a consequence of the nonlinear alternative of Leray-Schauder type [16], we deduce that N has a fixed point y in U which is a solution of the problem (1)–(3).

We now present two other existence results for the problem (1)–(3) when the right hand side has convex values under weaker conditions on the functions I_{k} (k = 1, ..., m) (as used in [11] for impulsive differential inclusions).

Theorem 3.5 In addition to (H1), assume that the following conditions hold: (H4) There exist constants c_{k} > 0 such that

|I_{k}(x)| ≤ c_{k}|x| for each k = 1, ..., m and all x ∈ IR.

(H5) H_{d}(F (t, y), F (t, \overline{y})) ≤ l(t)|y − \overline{y}| for each t ∈ [0, b] and all y, \overline{y} ∈ IR where l ∈

L^{1}([0, b], IR_{+}) ∩ R^{+} and d(0, F (t, 0)) ≤ l(t) a.e t ∈ [0, b]. If

sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s)∥l∥_{L^{1}} + ∑_{k=1}^{m} sup_{t∈[0,b]} e_{⊖p}(t, t_{k})c_{k} < 1,

then the problem (1)-(3) has at least one solution on [0, b].

Proof. Let y be such that y ∈ λN(y) for some λ ∈ (0, 1). Then, there exist v ∈ S_{F,y} such that for each t ∈ [0, b],

y(t) = λe_{⊖p}(t, 0)η + λ ∫_{0}^{t} e_{⊖p}(t, s)v(s)Δs + λ ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y(t_{k}^{−})).

This implies by (H4) and (H5) that for each t ∈ [0, b],

|y(t)| ≤ |η| sup_{t∈[0,b]} e_{⊖p}(t, 0) + ∑_{k=1}^{m} sup_{t∈[0,b]} e_{⊖p}(t, t_{k})c_{k}|y(t_{k}^{−})|

+ sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s) ∫_{0}^{b} |v(s)|Δs.

≤ |η| sup_{t∈[0,b]} e_{⊖p}(t, 0) + ∑_{k=1}^{m} sup_{t∈[0,b]} e_{⊖p}(t, t_{k})c_{k}|y(t_{k}^{−})|

+ sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s) ∫_{0}^{b} |l(s)y(s) + l(s)|Δs

≤ |η| sup_{t∈[0,b]} e_{⊖p}(t, 0) + ∑_{k=1}^{m} sup_{t∈[0,b]} e_{⊖p}(t, t_{k})c_{k}∥y∥_{P C}

+ sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s)∥y∥_{P C}∥l∥_{L^{1}}

+ sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s)∥l∥_{L^{1}}.

Consequently

∥y∥_{P C} ≤ \frac{|η| sup_{t∈[0,b]} e_{⊖p}(t, 0) + sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s)∥l∥_{L^{1}}}{1 − sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s)∥l∥_{L^{1}} − ∑_{k=1}^{m} sup_{t∈[0,b]} e_{⊖p}(t, t_{k})c_{k}} := M.

Let

U = {y ∈ P C : ∥y∥_{P C} < M + 1}.

The operator N : \overline{U} → P(P C) is upper semicontinuous and completely continuous. From the choice of U, there is no y ∈ ∂U such that y ∈ λN(y) for some λ ∈ (0, 1). As a consequence of the nonlinear alternative of Leray-Schauder type [16], we deduce that N has a fixed point y in U which is a solution of the problem (1)–(3).

Theorem 3.6 In addition to (H1) and (H5), assume that the following conditions hold

(H6) lim_{|x|→+∞} \frac{I_{k}(x)}{x} = 0 for each k = 1, ..., m.

If

sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s)∥l∥_{L^{1}} + ∑_{k=1}^{m} sup_{t∈[0,b]} e_{⊖p}(t, t_{k})ε_{k} < 1,

where ε_{k}, k = 1, …, m are positive constants that will specified later, then the problem (1)–(3) has at least one solution on [0, b].

Proof. Let y be such that y ∈ λN(y) for some λ ∈ (0, 1). Then, there exist v ∈ S_{F,y} such that for each t ∈ [0, b],

y(t) = λe_{⊖p}(t, 0)η + λ ∫_{0}^{t} e_{⊖p}(t, s)v(s)Δs + λ ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y(t_{k}^{−})).

(H6) implies that for each ε_{k} > 0, there exists a constant A > 0 such that

|x| ≥ A ⇒ |I_{k}(x)| ≤ ε_{k}|x|.

Let

E_{1} = {t; t ∈ [0, b] : |x(t)| < A},
E_{2} = {t; t ∈ [0, b] : |x(t)| ≥ A}

and

C_{1} = max{|I_{k}(x(t))|, t ∈ E_{1}}.

By (H5) and (H6), for each t ∈ [0, b],

|y(t)| ≤ |η| sup_{t∈[0,b]} e_{⊖p}(t, 0) + ∑_{t_{k}∈E_{1}} e_{⊖p}(t, t_{k})|I_{k}(y(t_{k}^{−}))|

+ ∑_{t_{k}∈E_{2}} e_{⊖p}(t, t_{k})|I_{k}(y(t_{k}^{−}))|

+ sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s)∥y∥_{P C}∥l∥_{L^{1}}

+ sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s)∥l∥_{L^{1}}

≤ |η| sup_{t∈[0,b]} e_{⊖p}(t, 0) + C_{1} ∑_{k=1}^{m} sup_{t∈E_{1}} e_{⊖p}(t, t_{k})

+ ∑_{k=1}^{m} sup_{t∈E_{2}} e_{⊖p}(t, t_{k})ε_{k}∥y∥_{P C}

+ sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s)∥y∥_{P C}∥l∥_{L^{1}}

+ sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s)∥l∥_{L^{1}}.

Consequently

∥y∥_{P C} ≤ \frac{|η| sup_{t∈[0,b]} e_{⊖p}(t, 0) + C_{1} ∑_{k=1}^{m} sup_{t∈E_{1}} e_{⊖p}(t, t_{k}) + sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s)∥l∥_{L^{1}}}{1 − sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s)∥l∥_{L^{1}} − ∑_{k=1}^{m} sup_{t∈E_{2}} e_{⊖p}(t, t_{k})ε_{k}} := M̄.

Let

U = {y ∈ P C : ∥y∥_{P C} < M̄ + 1}.

The operator N : \overline{U} → P(P C) is upper semicontinuous and completely continuous. >From the choice of U, there is no y ∈ ∂U such that y ∈ λN(y) for some λ ∈ (0, 1). As a consequence of the nonlinear alternative of Leray-Schauder type [16], we deduce that N has a fixed point y in U which is a solution of the problem (1)–(3).

We present now a result for the problem (1)-(3) with a nonconvex valued right hand side. Let (X, d) be a metric space induced from the normed space (X, | · |). Consider H_{d} : P(X) × P(X) ⟶ IR_{+} ∪ {∞} given by

H_{d}(A, B) = max {sup_{a∈A} d(a, B), sup_{b∈B} d(A, b)} ,

where d(A, b) = inf_{a∈A} d(a, b), d(a, B) = inf_{b∈B} d(a, b). Then (P_{b,cl}(X), H_{d}) is a metric space

and (P_{cl}(X), H_{d}) is a generalized metric space (see [21]).

Definition 3.7 A multivalued operator N : X → P_{cl}(X) is called

a) γ-Lipschitz if and only if there exists γ > 0 such that

H_{d}(N(x), N(y)) ≤ γd(x, y) for each x, y ∈ X,

b) a contraction if and only if it is γ-Lipschitz with γ < 1.

N has a fixed point if there is x ∈ X such that x ∈ N(x). The fixed point set of the multivalued operator N will be denoted by F ixN. For more details on multivalued maps we refer to the books of Deimling [15], Gorniewicz [17], Hu and Papageorgiou [20] and Tolstonogov [26].

Our considerations are based on the following fixed point theorem for contraction multivalued operators given by Covitz and Nadler in 1970 [13] (see also Deimling, [15] Theorem 11.1).

Lemma 3.8 Let (X, d) be a complete metric space. If N : X → P_{cl}(X) is a contraction, then F ixN ≠ ∅.

Let us introduce hypotheses which are assumed hereafter

(H8) F : [0, b] × IR ⟶ P_{cp}(IR) has the property that F (·, y) : [0, b] → P_{cp}(IR) is measurable for each y ∈ IR;

(H9) There exist constants d_{k} ≥ 0 such that

|I_{k}(y) − I_{k}(\overline{y})| ≤ d_{k}|y − \overline{y}| for each y, \overline{y} ∈ IR.

Theorem 3.9 Assume that (H5), (H8)-(H9) are satisfied. Let τ > 1. If

\frac{1}{τ} + ∑_{k=1}^{m} G_{∗}d_{k} < 1, where G_{∗} = sup_{(t,s)∈[0,b]×[0,b]} e_{⊖p}(t, s),

then the IVP (1)-(3) has at least one solution on [0, b].

Remark 3.10 For each y ∈ P C, the set S_{F,y} is nonempty since by (H8), F has a measurable selection (see [12], Theorem III.6).

Proof. We shall show that N satisfies the assumptions of Lemma 3.8. The proof will be given in two steps.

Step 1: N(y) ∈ P_{cl}(P C) for each y ∈ P C.

Indeed, let (y_{n})_{n≥0} ∈ N(y) such that y_{n} ⟶ ỹ in P C. Then ỹ ∈ P C and there exists v_{n} ∈ S_{F,y} such that for each t ∈ [0, b]

y_{n}(t) = e_{⊖p}(t, 0)η + ∫_{0}^{t} e_{⊖p}(t, s)v_{n}(s)Δs + ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y(t_{k}^{−})).

Using the fact that F has compact values and from (H5), we may pass to a subsequence if necessary to get that v_{n} converges to v in L^{1}([0, b], IR) and hence v ∈ S_{F,y}. Then, for each t ∈ [0, b],

y_{n}(t) ⟶ ỹ(t) = e_{⊖p}(t, 0)η + ∫_{0}^{t} e_{⊖p}(t, s)v(s)Δs + ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y(t_{k}^{−})).

So, ỹ ∈ N(y).

Step 2: There exists γ < 1 such that

H_{d}(N(y), N(\overline{y})) ≤ γ∥y − \overline{y}∥ for each y, \overline{y} ∈ P C.

Let y, \overline{y} ∈ P C and h_{1} ∈ N(y). Then, there exists v_{1}(t) ∈ F (t, y(t)) such that for each t ∈ [0, b],

h_{1}(t) = e_{⊖p}(t, 0)η + ∫_{0}^{t} e_{⊖p}(t, s)v_{1}(s)Δs + ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y(t_{k}^{−})).

From (H5) it follows that for each t ∈ [0, b],

H_{d}(F (t, y(t)), F (t, \overline{y}(t))) ≤ l(t)|y(t) − \overline{y}(t)|.

Hence, there exists w ∈ F (t, \overline{y}(t)) such that for each t ∈ [0, b],

∥v_{1}(t) − w∥ ≤ l(t)|y(t) − \overline{y}(t)|.

Consider U : [0, b] → P(IR) given by

U(t) = \{w ∈ IR : ∥v_{1}(t) − w∥ ≤ l(t)|y(t) − \overline{y}(t)|\}.

Since the multivalued operator V (t) = U(t) ∩ F (t, \overline{y}(t)) is measurable (see Proposition III.4 in [12]), there exists a function v_{2}(t) which is a measurable selection for V. So, v_{2}(t) ∈ F (t, \overline{y}(t)) and for each t ∈ [0, b],

∥v_{1}(t) − v_{2}(t)∥ ≤ l(t)|y(t) − \overline{y}(t)|.

Let us define for each t ∈ [0, b],

h_{2}(t) = e_{⊖p}(t, 0)η + ∫_{0}^{t} e_{⊖p}(t, s)v_{2}(s)Δs + ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(\overline{y}(t_{k}^{−}))

We define on P C an equivalent norm by

∥y∥_{∗} = sup_{t∈J} e_{⊖(τG_{∗}l)}(t, 0)|y(t)| for all y ∈ P C,

where e_{⊖(τG_{∗}l})(t, 0) is the unique solution of the problem

y^{Δ}(t) = τG_{∗}l(t)y(t), y(0) = 1,

where τG_{∗}l is a regressive function. >From (H5) and (H9), for each t ∈ [0, b],

|h_{1}(t) − h_{2}(t)| ≤ ∫_{0}^{1} e_{⊖p}(t, s)∥v_{1}(s) − v_{2}(s)∥Δs

+ ∑_{k=1}^{m} e_{⊖p}(t, s)|I_{k}(y(t_{k}^{−})) − I_{k}(\overline{y}(t_{k}^{−}))|

≤ ∫_{0}^{t} G_{∗}l(s)|y(s) − \overline{y}(s)|Δs

+ ∑_{k=1}^{m} G_{∗}d_{k}|y(t_{k}^{−}) − \overline{y}(t_{k}^{−})|

≤ \frac{1}{τ} ∫_{0}^{t} τG_{∗}l(s)|y(s) − \overline{y}(s)|Δs

+ ∑_{k=1}^{m} d_{k}G_{∗}|y(t_{k}^{−}) − \overline{y}(t_{k}^{−})|

≤ \frac{1}{τ} ∫_{0}^{t} τG_{∗}l(s)e_{τG_{∗}l}(s, 0)e_{τG_{∗}l}(0, s)|y(s) − \overline{y}(s)|Δs

+\frac{τ}{τ} ∑_{k=1}^{m} d_{k}G_{∗}|y(t_{k}^{−}) − \overline{y}(t_{k}^{−})|

≤ \frac{1}{τ} ∫_{0}^{t}[e_{τG_{∗}l}(s, 0)]^{Δ}e_{⊖(τG_{∗}l)}(s, 0)|y(s) − \overline{y}(s)|Δs

+\frac{τ}{τ} ∑_{k=1}^{m} d_{k}G_{∗}|y(t_{k}^{−}) − \overline{y}(t_{k}^{−})|

≤ \frac{1}{τ}e_{τG_{∗}l}(t, 0)∥y − \overline{y}∥_{∗} + ∑_{k=1}^{m} d_{k}G_{∗}e_{τG_{∗}l}(t, 0)∥y − \overline{y}∥_{∗}.

Thus

∥h_{1} − h_{2}∥_{∗} ≤ (\frac{1}{τ} + ∑_{k=1}^{m} G_{∗}d_{k}) ∥y − \overline{y}∥_{∗}.

By an analogous relation, obtained by interchanging the roles of y and \overline{y}, it follows that

H_{d}(N(y), N(\overline{y})) ≤ (\frac{1}{τ} + ∑_{k=1}^{m} G_{∗}d_{k}) ∥y − \overline{y}∥_{∗}.

So, N is a contraction and thus, by Lemma 3.8, N has a fixed point y which is solution to (1)-(3).

4 Extremal Solutions

We equip the space P C with the order relation ” ≤ ” defined by the cone K in P C, that is

K = {y ∈ P C : y(t) ≥ 0, t ∈ [0, b]}.

It is known that the cone K is normal in P C. The details of cones and their properties may be found in Heikkila and Lakshmikantham [19]. Let α, β ∈ P C such that α ≤ β. then by an order interval [α, β] we mean a set of points in P C given by

[α, β] = {y ∈ P C : α ≤ y ≤ β}.

Let D, Q ∈ P_{cl}(P C). Then by D ≤ Q we mean α ≤ β for all α ∈ D and β ∈ Q. Thus α ≤ D implies that α ≤ y for all y ∈ Q in particular, if D ≤ D, then it follows that D is a singleton set.

Definition 4.1 Let X be an ordered Banach space. A mapping N : X → P_{cl}(X) is called isotone increasing if x, y ∈ X with x < y, then we have that N(x) ≤ N(y).

Theorem 4.2 [14] Let [a, b] be an order interval in a Banach space and let A, B : [a, b] → P_{cl}(X) be two multivalued operators satisfying

(i) A is multivalued contraction,

(ii) B is completely continuous,

(iii) A and B are isotone increasing, and

(iv) A(x) + B(x) ⊂ [a, b] for all x ∈ [a, b].

Further if the cone K in X is normal, then the operator inclusion x ∈ A(x) + B(x) has a least fixed point x_{∗} and a greatest fixed point x^{∗} in [a, b]. Moreover x_{∗} = lim_{n→∞} x_{n} and x^{∗} = lim_{n∞} y_{n}, where {x_{n}} and {y_{n}} are the sequences in [a, b] defined by

x_{n+1} ∈ A(x_{n}) + B(x_{n}), x_{0} = a and y_{n+1} ∈ A(y_{n}) + B(y_{n}), y_{0} = b.

The following concept of lower and upper solutions for (1)-(3) has been introduced by Benchohra, Henderson, and Ntouyas [6] for periodic boundary value problems for impulsive differential inclusions at fixed moments (see also [5]). It will be the basic tool in the approach that follows.

Definition 4.3 A function α ∈ P C is said to be a lower solution of (1)-(3) if there exists v_{1} ∈ L^{1}(J, IR) such that v_{1}(t) ∈ F (t, α(t)) a.e. on J, α^{Δ}(t) + p(t)α^{σ}(t) ≤ v_{1}(t) a.e. on J, t ≠ t_{k}, α(t_{k}^{+}) − α(t_{k}^{−}) ≤ I_{k}(α(t_{k}^{−})), t = t_{k}, k = 1, …, m, and α(0) ≤ η. Similarly, a function β ∈ P C is said to be an upper solution of (1)-(3) if there exists v_{2} ∈ L^{1}(J, IR) such that v_{2}(t) ∈ F (t, β(t)) a.e. on J, β^{Δ}(t) + p(t)β^{σ}(t) ≥ v_{2}(t) a.e. on J, t_{k} ≠ t_{k}, β(t_{k}^{+}) − β(t_{k}^{−}) ≥ I_{k}(β(t_{k}^{−})), t = t_{k}, k = 1, …, m, and β(0) ≥ η.

The following hypotheses will be assumed hereafter:

(A1) The multifunction F (t, y) is nondecreasing in y almost everywhere for t ∈ [0, b];

(A2) The functions I_{k}, k = 1, …, m are continuous and nondecreasing.

(A3) There exist α and β ∈ P C, respectively lower and upper solutions for the problem (1)–(3) such that α ≤ β.

Theorem 4.4 Suppose that hypotheses (h5), (H9), (A1)–(A3) are satisfied. Then the impulsive IVP (1)–(3) has minimal and maximal solutions on [0, b].

Proof. Define two multivalued maps A, B : P C → P(P C) by

A(y) = {h ∈ P C : h(t) = e_{⊖p}(t, 0)η + ∫_{0}^{t} e_{⊖p}(t, s)v(s)Δs, v ∈ S_{F,y}},

and

B(y) = {h ∈ P C : h(t) = ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y(t_{k}^{−}))}.

It can be shown, as in the proofs of Theorems 3.3 and 3.9, that A and B define the multi-valued operators A : [α, β] → P_{cl,cv,bd}(P C) and B : [α, β] → P_{cp,cv}(P C). It can be similarly shown that A and B are respectively multi-valued contraction and completely continuous on [α, β]. We shall show that A and B are isotone increasing on [α, β]. Let x, y ∈ [α, β] be such that x < y. Then by (A1), for each t ∈ [0, b],

A(x) = {h ∈ P C : h(t) = e_{⊖p}(t, 0)η + ∫_{0}^{t} e_{⊖p}(t, s)v(s)Δs, v ∈ S_{F,x}}
≤ {h ∈ P C : h(t) = e_{⊖p}(t, 0)η + ∫_{0}^{t} e_{⊖p}(t, s)v(s)Δs, v ∈ S_{F,y}}
= A(y).

Hence A(x) ≤ A(y). Similarly by (A2), for each t ∈ [0, b], B(x) ≤ B(y). Thus A and B are isotone increasing on [α, β]. Finally we prove that A(y) + B(y) ⊂ [α, β] for each y ∈ [α, β]. Let h ∈ A(y) + B(y) be any element. Then, there exists v ∈ S_{v,y} such that for each t ∈ [0, b],

h(t) = e_{⊖p}(t, 0)η + ∫_{0}^{t} e_{⊖p}(t, s)v(s)Δs + ∑_{0<t_{k}<t} e_{⊖p}(t, t_{k})I_{k}(y(t_{k}^{−})).

Let

t_{i} = max{t_{k} : t_{k} < t}.

By definition of the upper solution and the conditions (A1)-(A3), we get for each t ∈ [0, b],

h(t) ≤ e_{⊖p}(t, 0)β(0) + ∫_{0}^{t_{1}} e_{⊖p}(t, s)[β^{Δ}(s) + p(t)β^{σ}(s)]Δs

+ ∫_{t_{1}^{+}}^{t_{2}} e_{⊖p}(t, s)[β^{Δ}(s) + p(t)β^{σ}(s)]Δs

+ … + ∫_{t_{i}}^{t} e_{⊖p}(t, s)[β^{Δ}(s) + p(t)β^{σ}(s))]Δs

+ ∑_{k=1}^{k=i} e_{⊖p}(t, t_{k})I_{k}(β(t_{k}^{−})).

Thus, for each t ∈ [0, b]

e_{p}(t, 0)h(t) ≤ β(0) + ∫_{0}^{t_{1}}[e_{p}(s, 0)β(s)]^{Δ}Δs + ∫_{t_{1}^{+}}^{t_{2}}[e_{p}(s, 0)β(s)]^{Δ}Δs

+ … + ∫_{t_{i}^{+}}^{t}[e_{p}(s, 0)β(s)]^{Δ}Δs + ∑_{k=1}^{k=i} e_{p}(t_{k}, 0)I_{k}(β(t_{k}^{−}))

= β(0) + e_{p}(t_{1}^{−}, 0)β(t_{1}^{−}) − e_{p}(0, 0)β(0) + e_{p}(t_{2}^{−}, 0)β(t_{2}^{−})

−e_{p}(t_{1}^{+}, 0)β(t_{1}^{+}) + … + e_{p}(t, 0)β(t) − e_{p}(t_{i}^{+}, 0)β(t_{i}^{+})

+ ∑_{k=1}^{k=i} e_{p}(t_{k}, 0)I_{k}(β(t_{k}^{−}))

= −e_{p}(t_{1}, 0)[β(t_{1}^{+}) − β(t_{1}^{−})] − e_{p}(t_{2}, 0)[β(t_{2}^{+}) − β(t_{2}^{−})]

− … − e_{p}(t_{i}, 0)[β(t_{i}^{+}) − β(t_{i}^{−})] + e_{p}(t, 0)β(t)

+ ∑_{k=1}^{k=i} e_{p}(t_{k}, 0)I_{k}(β(t_{k}^{−}))

≤ −e_{p}(t_{1}, 0)I_{1}(β(t_{1}^{−})) − e_{p}(t_{2}, 0)I_{2}(β(t_{2}^{−}))

− … − e_{p}(t_{i}, 0)I_{i}(β(t_{i}^{−})) + e_{p}(t, 0)β(t)

+ ∑_{k=1}^{k=i} e_{p}(t_{k}, 0)I_{k}(β(t_{k}^{−}))

= e_{p}(t, 0)β(t).

Hence

h(t) ≤ β(t) for each t ∈ [0, b].

Similarly, by replacing β with α and reversing the order, we can prove that

h(t) ≥ α(t), for each t ∈ [0, b].

Then

α ≤ N(y) ≤ β for all y ∈ [α, β].

As a consequence of Theorem 4.2, we deduce that N has least and greatest fixed point in [α, β]. This further implies that the problem (1)-(3) has minimal and maximal solutions on [0, b].

5 Example:

Suppose 𝕋 = [0, 1] ∪ [2, 3] ∪ [4, 5] and p a regressive function. We consider the equation

y^{Δ}(t) = p(t)y(t), y(0) = 1.

We can easily show that the unique solution of the above equation is given by

y(t) = e_{p}(t, 0) = \{ \begin{matrix}e^{∫_{0}^{t} p(s)Δs}, & if t ∈ [0, 1], \\ exp(∫_{0}^{1} p(s)Δs + ∫_{2}^{t} p(s)Δs), & if t ∈ [2, 3], \\ exp(∫_{0}^{1} p(s)Δs + ∫_{2}^{3} p(s)Δs + ∫_{4}^{t} p(s)Δs), & if t ∈ [4, 5]. \end{matrix}

Also we consider the following dynamic inclusion of the form

y^{Δ}(t) + p(t)y^{σ}(t) ∈ F (t, y(t)), t ∈ [0, 1], t ≠ \frac{1}{2},  (8)

y(\frac{1}{2}^{+}) − y(\frac{1}{2}^{−}) = I_{1}(y(\frac{1}{2}^{−})),  (9)
y(0) = 0,  (10)

where F : [0, 1] × IR → P(IR) is the multivalued map defined by

(t, x) → F (t, x) := [\frac{x^{2}}{x^{2} + 2} + t, \frac{x^{2}}{x^{2} + 1} + t + 1].

It is clear that F is a compact convex valued multivalued map and of Carathéodory. Let v ∈ [\frac{x^{2}}{x^{2}+2} + t, \frac{x^{2}}{x^{2}+1} + t + 1], then we have

|v| ≤ max (\frac{x^{2}}{x^{2} + 2} + |t|, \frac{x^{2}}{x^{2} + 1} + |t| + 1) ≤ 3, for each (t, x) ∈ [0, 1] × IR.

Hence

∥F (t, x)∥ := sup\{|v| : v ∈ [\frac{x^{2}}{x^{2} + 1} + t, \frac{x^{2}}{x^{2} + 1} + t + 1]\} ≤ 3 := p(t)ψ(x),

where p(t) = 1 and ψ(x) = 3. Assume that there exists c > 0 such that

|I_{1}(x)| ≤ c, for each x ∈ IR.

We can find M > 0 such that

\frac{M}{c sup_{t∈[0,1]} e_{⊖p}(t, \frac{1}{2}) + 3 sup_{(t,s)∈[0,1]×[0,1]} e_{⊖p}(t, s)} > 1.

Consider the operator N : P C ⟶ P(P C) defined by

N(y) = \{h ∈ P C : h(t) = ∫_{0}^{t} e_{⊖p}(t, s)v(s)Δs + e_{⊖p}(t, \frac{1}{2})I_{1}(y(\frac{1}{2}^{−})), v ∈ S_{F,y}\}.

Let y be such that y ∈ λN(y) for some λ ∈ (0, 1). Then, there exists v ∈ S_{F,y} such that for each t ∈ [0, 1] we have

y(t) = λ ∫_{0}^{t} e_{⊖p}(t, s)v(s)Δs + λe_{⊖p}(t, \frac{1}{2})I_{1}(y(\frac{1}{2}^{−})).

This implies that for each t ∈ [0, 1] we have

|y(t)| ≤ c sup_{t∈[0,1]} e_{⊖p}(t, \frac{1}{2}) + 3 sup_{(t,s)∈[0,1]×[0,1]} e_{⊖p}(t, s)

Thus

∥y∥_{P C} ≤ c sup_{t∈[0,1]} e_{⊖p}(t, \frac{1}{2}) + 3 sup_{(t,s)∈[0,1]×[0,1]} e_{⊖p}(t, s).

Then all the conditions of Theorem 3.3 hold and thus, the problem (8)-(10) has at least one solution on [0,1].

Acknowledgement. The authors thank the referee for his remarks.

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(Received April 8, 2005)