Electronic Journal of Qualitative Theory of Differential Equations Spec. Ed. I, 2009 No. 23, 1–23; http://www.math.u-szeged.hu/ejqtde/

FILIPPOV’S THEOREM FOR IMPULSIVE DIFFERENTIAL INCLUSIONS WITH FRACTIONAL

ORDER

Abdelghani Ouahab

Laboratoire de Mathématiques, Université de Sidi Bel Abbès BP 89, 22000, Sidi Bel Abbès, Algérie e-mail: agh ouahab@yahoo.fr

Honoring the Career of John Graef on the Occasion of His Sixty-Seventh Birthday

Abstract In this paper, we present an impulsive version of Filippov’s Theorem for

fractional differential inclusions of the form:

D_{∗}^{α}y(t) ∈ F (t, y(t)),  a.e. t ∈ J\{t_{1}, …, t_{m}}, α ∈ (1, 2],
y(t_{k}^{+}) − y(t_{k}^{−}) = I_{k}(y(t_{k}^{−})),  k = 1, …, m,
y′(t_{k}^{+}) − y′(t_{k}^{−}) = \overline{I}_{k}(y′(t_{k}^{−})),  k = 1, …, m,
y(0) = a, y′(0) = c,

where J = [0, b], D_{∗}^{α} denotes the Caputo fractional derivative and F is a setvalued map. The functions I_{k}, \overline{I}_{k} characterize the jump of the solutions at impulse points t_{k} (k = 1, …, m).

Key words and phrases: Fractional differential inclusions, fractional derivative, fractional integral. AMS (MOS) Subject Classifications: 34A60, 34A37.

1 Introduction

Differential equations with impulses were considered for the first time in the 1960’s by Milman and Myshkis [47, 46]. A period of active research, primarily in Eastern Europe from 1960-1970, culminated with the monograph by Halanay and Wexler [32].

The dynamics of many evolving processes are subject to abrupt changes, such as shocks, harvesting and natural disasters. These phenomena involve short-term perturbations from continuous and smooth dynamics, whose duration is negligible in comparison with the duration of an entire evolution. In models involving such perturbations, it is natural to assume these perturbations act instantaneously or in the form of “impulses”. As a consequence, impulsive differential equations have been developed in modeling impulsive problems in physics, population dynamics, ecology, biotechnology, industrial robotics, pharmcokinetics, optimal control, and so forth. Again, associated with this development, a theory of impulsive differential equations has been given extensive attention. Works recognized as landmark contributions include [7, 42, 53, 57]. There are also many different studies in biology and medicine for which impulsive differential equation are a good model (see, for example, [2, 39, 40] and the references therein).

In recent years, many examples of differential equations with impulses with fixed moments have flourished in several contexts. In the periodic treatment of some diseases, impulses correspond to administration of a drug treatment or a missing product. In environmental sciences, impulses correspond to seasonal changes of the water level of artificial reservoirs.

During the last ten years, impulsive ordinary differential inclusions and functional differential inclusions with different conditions have been intensely studied by many mathematicians. At present the foundations of the general theory are already laid, and many of them are investigated in detail in the books of Aubin [3] and Benchohra et al [9], and in the papers of Graef et al [26, 27, 31], Graef and Ouahab [28, 29, 30] and the references therein.

Differential equations with fractional order have recently proved valuable tools in the modeling of many physical phenomena [19, 23, 24, 43, 44]. There has been a significant theoretical development in fractional differential equations in recent years; see the monographs of Kilbas et al [36], Miller and Ross [45], Podlubny [54], Samko et al [56], and the papers of Bai and Lu [6], Diethelm et al [19, 18, 20], El-Sayed and Ibrahim [21], Kilbas and Trujillo [37], Mainardi [43], Momani and Hadid, [48], Momani et al [49], Nakhushev [50], Podlubny et al [55], and Yu and Gao [59].

Very recently, some basic theory for initial value problems for fractional differential equations and inclusions involving the Riemann-Liouville differential operator was discussed by Benchohra et al [10] and Lakshmikantham [41]. El-Sayed and Ibrahim [21] initiated the study of fractional multivalued differential inclusions.

Applied problems require definitions of fractional derivatives allowing a utilization that is physically interpretable for initial conditions containing y(0), y′(0), etc. The same requirements are true for boundary conditions. Caputo’s fractional derivative satisfies these demands. For more details on the geometric and physical interpretation for fractional derivatives of both the Riemann-Liouville and Caputo types, see Podlubny [54].

Recently fractional functional differential equations and inclusions with standard Riemann-Liouville and Caputo derivatives with difference conditions were studied by Benchohra et al [8, 10, 11], Henderson and Ouahab [34], and Ouahab [52].

When α ∈ (1, 2], the impulsive differential equations with Captuo fractional derivatives were studied by Agarwal et al [1].

In this paper, we shall be concerned with Filippov’s theorem and global existence of solutions for impulsive fractional differential inclusions with fractional order. More precisely, we will consider the following problem,

D_{∗}^{α}y(t) ∈ F (t, y(t)), a.e. t ∈ J = [0, b], 1 < α ≤ 2,  (1)
Δy|_{t=t_{k}} = I_{k}(y(t_{k}^{−})), k = 1, …, m,  (2)
Δy′|_{t=t_{k}} = \overline{I}_{k}(y(t_{k}^{−})), k = 1, …, m,  (3)

y(0) = a, y′(0) = c,  (4)

where D_{∗}^{α} is the Caputo fractional derivative, F : J × ℝ → P(ℝ) is a multivalued map with compact values (P(ℝ) is the family of all nonempty subsets of ℝ), 0 = t_{0} < t_{1} < ⋯ < t_{m} < t_{m+1} = b, I_{k}, Ī_{k} ∈ C(ℝ, ℝ) (k = 1, …, m), Δy|_{t=t_{k}} = y(t_{k}^{+}) − y(t_{k}^{−}),

Δy′|_{t=t_{k}} = y′(t_{k}^{+}) − y′(t_{k}^{−}), y(t_{k}^{+}) = lim_{h→0^{+}} y(t_{k} + h) and y(t_{k}^{−}) = lim_{h→0^{+}} y(t_{k} − h) stand for

the right and the left limits of y(t) at t = t_{k}, respectively.

The paper is organized as follows. We first collect some background material and basic results from multi-valued analysis and fractional calculus in Sections 2 and 3, respectively. Then, we shall be concerned with Filippov’s theorem for impulsive differential inclusions with fractional order in Section 4.

2 Preliminaries

In this section, we introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper. Let AC^{i}([0, b], ℝ^{n}) be the space of functions y : [0, b] → ℝ^{n}, i-differentiable, and whose i^{th} derivative, y^{(i)}, is absolutely continuous.

We take C(J, ℝ) to be the Banach space of all continuous functions from J into ℝ with the norm

∥y∥_{∞} = sup{|y(t)| : 0 ≤ t ≤ b}.

L^{1}(J, ℝ) refers to the Banach space of measurable functions y : J ⟶ ℝ which are Lebesgue integrable; it is normed by

|y|_{1} = ∫_{0}^{b} |y(s)|ds.

Let (X, ∥ · ∥) be a Banach space, and denote:

P(X) = {Y ⊂ X : Y ≠ ∅},
P_{cl}(X) = {Y ∈ P(X) : Y closed},
P_{b}(X) = {Y ∈ P(X) : Y bounded},
P_{cp}(X) = {Y ∈ P(X) : Y compact}.

We say that a multivalued mapping G : X → P(X) has a fixed point if there exists x ∈ X such that x ∈ G(x).

A multi-valued map G : J ⟶ P(ℝ) is said to be measurable if for each x ∈ ℝ the function Y : J ⟶ ℝ defined by

Y (t) = d(x, G(t)) = inf{|x − z| : z ∈ G(t)},

is measurable.

Lemma 2.1 (see [25], Theorem 19.7) Let E be a separable metric space and G a multi-valued map with nonempty closed values. Then G has a measurable selection.

Lemma 2.2 (see [60], Lemma 3.2) Let G : [0, b] → P(E) be a measurable multifunction and u : [0, b] → E a measurable function. Then for any measurable v : [0, b] → ℝ^{+} there exists a measurable selection g of G such that for a.e. t ∈ [0, b],

|u(t) − g(t)| ≤ d(u(t), G(t)) + v(t).

Let (X, d) be a metric space induced from the normed space (X, | · |). Consider H_{d} : P(X) × P(X) ⟶ ℝ_{+} ∪ {∞} 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 [38].

Definition 2.1 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.

For more details on multi-valued maps we refer to the books by Aubin et al [5, 4], Deimling [16], Gorniewicz [25], Hu and Papageorgiou [35], Kisielewicz [38] and Tolstonogov [58].

3 Fractional Calculus

According to the Riemann-Liouville approach to fractional calculus, the notation of fractional integral of order α (α > 0) is a natural consequence of the well known formula (usually attributed to Cauchy), that reduces the calculation of the n−fold primitive of a function f(t) to a single integral of convolution type. In our notation, the Cauchy formula reads

J^{n}f(t) := \frac{1}{(n − 1)!} ∫_{0}^{t}(t − s)^{n−1}f(s)ds, t > 0 , n ∈ ℕ.

Definition 3.1 The fractional integral of order α > 0 of a function f ∈ L^{1}([a, b], ℝ) is defined by

J_{a^{+}}^{α}f(t) = ∫_{a}^{t} \frac{(t − s)^{α−1}}{Γ(α)} f(s)ds,

where Γ is the gamma function. When a = 0, we write J^{α}f(t) = f(t) ∗ ϕ_{α}(t), where

ϕ_{α}(t) = \frac{t^{α−1}}{Γ(α)} for t > 0, and ϕ_{α}(t) = 0 for t ≤ 0, and ϕ_{α} → δ(t) as α → 0, where δ is

the delta function and Γ is the Euler gamma function defined by

Γ(α) = ∫_{0}^{∞} t^{α−1}e^{−t}dt, α > 0.

Also J^{0} = I (Identity operator), i.e. J^{0}f(t) = f(t). Furtheremore, by J^{α}f(0^{+}) we mean the limit (if it exists) of J^{α}f(t) for t → 0^{+}; this limit may be infinite.)

After the notion of fractional integral, that of fractional derivative of order α (α > 0) becomes a natural requirement and one is attempted to substitute α with −α in the above formulas. However, this generalization needs some care in order to guarantee the convergence of the integral and preserve the well known properties of the ordinary derivative of integer order. Denoting by D^{n} with n ∈ ℕ, the operator of the derivative of order n, we first note that

D^{n}J^{n} = I, J^{n}D^{n} ≠ I, n ∈ ℕ,

i.e., D^{n} is the left-inverse (and not the right-inverse) to the corresponding integral operator J^{n}. We can easily prove that

J^{n}D^{n}f(t) = f(t) − ∑_{k=0}^{n−1} f^{(k)}(a^{+})\frac{(t − a)^{k}}{k!} , t > 0.

As consequence, we expect that D^{α} is defined as the left-inverse to J^{α}. For this purpose, introducing the positive integer n such that n − 1 < α ≤ n, one defines the fractional derivative of order α > 0:

Definition 3.2 For a function f given on interval [a, b], the αth Riemann-Liouville fractional-order derivative of f is defined by

D^{α}f(t) = \frac{1}{Γ(n − α)} (\frac{d}{dt})^{n} ∫_{a}^{t}(t − s)^{−α+n−1}f(s)ds,

where n = [α] + 1 and [α] is the integer part of α.

Also, we define D^{0} = J^{0} = I. Then we easily recognize that

D^{α}J^{α} = I, α ≥ 0,  (5)

and

D^{α}t^{γ} = \frac{Γ(γ + 1)}{Γ(γ + 1 − α)}t^{γ−α}, α > 0, γ − 1, t > 0.  (6)

Of course, the properties (5) and (6) are a natural generalization of those known when the order is a positive integer.

Note the remarkable fact that the fractional derivative D^{α}f is not zero for the constant function f(t) = 1 if α ∉ ℕ. In fact, (6) with γ = 0 teaches us that

D^{α}1 = \frac{(t − a)^{−α}}{Γ(1 − α)} , α > 0, t > 0.  (7)

It is clear that D^{α}1 = 0 for α ∈ ℕ, due to the poles of the gamma function at the points 0, −1, −2, ….

We now observe an alternative definition of fractional derivative, originally introduced by Caputo [12, 13] in the late sixties and adopted by Caputo and Mainardi [14] in the framework of the theory of “linear viscoelasticity” (see a review in [43]).

Definition 3.3 Let f ∈ AC^{n}([a, b]). The Caputo fractional-order derivative of f is defined by

(D_{∗}^{α}f)(t) := \frac{1}{Γ(n − α)} ∫_{a}^{t}(t − s)^{n−α−1}f^{n}(s)ds.

This definition is of course more restrictive than Riemann-Liouville definition, in that it requires the absolute integrability of the derivative of order m. Whenever we use the operator D_{∗}^{α} we (tacitly) assume that this condition is met. We easily recognize that in general

D^{α}f(t) := D^{m}J^{m−α}f(t) ≠ J^{m−α}D^{m}f(t) := D_{∗}^{α}f(t),  (8)

unless the function f(t), along with its first m − 1 derivatives, vanishes at t = a^{+}. In fact, assuming that the passage of the m−derivative under the integral is legitimate, one recognizes that, m − 1 < α < m and t > 0,

D^{α}f(t) = D_{∗}^{α}f(t) + ∑_{k=0}^{m−1} \frac{(t − a)^{k−α}}{Γ(k − α + 1)}f^{(k)}(a^{+}),  (9)

and therefore, recalling the fractional derivative of the power function (6),

D^{α} (f(t) − ∑_{k=0}^{m−1} \frac{(t − a)^{k−α}}{Γ(k − α + 1)}f^{(k)}(a^{+})) = D_{∗}^{α}f(t).  (10)

The alternative definition, that is, Definition 3.3, for the fractional derivative thus incorporates the initial values of the function and of order lower than α. The subtraction of the Taylor polynomial of degree m − 1 at t = a^{+} from f(t) means a sort of regularization of the fractional derivative. In particular, according to this definition, a relevant property is that the fractional derivative of a constant is sill zero, i.e.,

D_{∗}^{α}1 = 0, α > 0.  (11)

We now explore the most relevant differences between Definition 3.2 and Definition 3.3 for the two fractional derivatives. From the Riemann-Liouville fractional derivative, we have

D^{α}(t − a)^{α−j} = 0, for j = 1, 2, …, [α] + 1.  (12)

From (11) and (12) we thus recognize the following statements about functions, which for t > 0 admit the same fractional derivative of order α, with n − 1 < α ≤ n, n ∈ ℕ,

D^{α}f(t) = D^{α}g(t) ⇔ f(t) = g(t) + ∑_{j=1}^{m} c_{j}(t − a)^{α−j},  (13)

and

D_{∗}^{α}f(t) = D_{∗}^{α}g(t) ⇔ f(t) = g(t) + ∑_{j=1}^{m} c_{j}(t − a)^{n−j}.  (14)

In these formulas the coefficients c_{j} are arbitrary constants. For proving all mains results we present the following auxiliary lemmas.

Lemma 3.1 [36] Let α > 0 and let y ∈ L^{∞}(a, b) or C([a, b]). Then

(D_{∗}^{α}J^{α}y)(t) = y(t).

Lemma 3.2 [36] Let α > 0 and n = [α] + 1. If y ∈ AC^{n}[a, b] or y ∈ C^{n}[a, b], then

(J^{α}D_{∗}^{α}y)(t) = y(t) − ∑_{k=0}^{n−1} \frac{y^{(k)}(a)}{k!}(t − a)^{k}.

For further readings and details on fractional calculus we refer to the books and papers by Kilbas [36], Podlubny [54], Samko [56], Captuo [12, 13, 14].

4 Filippov’s Theorem

Let J_{k} = (t_{k}, t_{k+1}], k = 0, …, m, and let y_{k} be the restriction of a function y to J_{k}. In order to define mild solutions for problem (1)-(4), consider the space

P C = {y: J → ℝ | y_{k} ∈ C(J_{k}, ℝ), k = 0, …, m, and y(t_{k}^{−})
and y(t_{k}^{+}) exist and satisfy y(t_{k}^{−}) = y(t_{k}) for k = 1, …, m}.

Endowed with the norm

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

this is a Banach space.

Definition 4.1 A function y ∈ P C is said to be a solution of (1)-(4) if there exists v ∈ L^{1}(J, ℝ) with v(t) ∈ F (t, y(t)) for a.e. t ∈ J such that y satisfies the fractional differential equation D_{∗}^{α}y(t) = v(t) a.e. on J, and the conditions (2)-(4).

Let \overline{a}, \overline{c} ∈ ℝ, g ∈ L^{1}(J, ℝ) and let x ∈ P C be a solution of the impulsive differential problem with fractional order:

\{ \begin{matrix}D_{∗}^{α}x(t) & = & g(t), & a.e. t ∈ J\\{t_{1}, …, t_{m}\}, α ∈ (1, 2], \\ Δx_{t=t_{k}} & = & I_{k}(x(t_{k}^{−})), & k = 1, …, m, \\ Δx′_{t=t_{k}} & = & \overline{I}_{k}(x(t_{k}^{−})), & k = 1, …, m, \\ x(0) & = & \overline{a}, x′(0) = \overline{c}. \end{matrix}  (15)

We will need the following two assumptions:

(H_{1}) The function F: J × ℝ → P_{cl}(ℝ) is such that (a) for all y ∈ ℝ, the map t ↦ F (t, y) is measurable,

(b) the map γ : t ↦ d(g(t), F (t, x(t)) is integrable.

(H_{2}) There exists a function p ∈ L^{1}(J, ℝ^{+}) such that

H_{d}(F (t, z_{1}), F (t, z_{2})) ≤ p(t)|z_{1} − z_{2}|, for all z_{1}, z_{2} ∈ ℝ.

Remark 4.1 From Assumptions (H_{1}(a)) and (H_{2}), it follows that the multi-function t ↦ F (t, x_{t}) is measurable, and by Lemmas 1.4 and 1.5 from [22], we deduce that γ(t) = d(g(t), F (t, x(t)) is measurable (see also the Remark on p. 400 in [4]).

Let P (t) = ∫_{0}^{t} p(s)ds. Define the functions η_{0} and H_{0} by

η_{0}(t) = Mδ_{0} + M ∫_{0}^{t} [H_{0}(s)p(s) + γ(s)] ds, t ∈ [0, t_{1}],

where H_{0}(t) = δ_{0}M exp (Me^{P (t)}) + M ∫_{0}^{t} γ(s) exp (Me^{P (t)−P (s)}) ds,

where M = max (1, b, \frac{b^{α−1}}{Γ(α)}) and δ_{0} = |a − \overline{a}| + |c − \overline{c}|.

Theorem 4.1 Suppose that hypotheses (H_{1}) − (H_{2}) are satisfied. Problem (1)-(4) has at least one solution y satisfying, for a.e. t ∈ [0, b], the estimates

|y(t) − x(t)| ≤ M ∑_{0≤k<i} δ_{k} + M ∑_{0≤t_{k}<t} η_{k}(t),

and

|D_{∗}^{α}y(t) − g(t)| ≤ Mp(t) ∑_{0<t_{k}<t} H_{k}(t) + ∑_{0<t_{k}<t} γ_{k}(t),

and for k = 1, …, m, where

η_{k}(t) = M ∫_{t_{k}}^{t} [H_{k}(s)p(s) + γ(s)] ds, t ∈ (t_{k}, t_{k+1}],

and

where H_{k}(t) = δ_{k} exp (Me^{P_{k}(t)}) + ∫_{t_{k}}^{t} γ(s) exp (Me^{P_{k}(t)−P_{k}(s)}) ds,

where

δ_{k} := |x(t_{k}) − y(t_{k})| + |I_{1}(y(t_{k})) − I_{k}(x(t_{k}))| + |x′(t_{k}) − y′(t_{k})| + |I_{1}(y′(t_{k})) − I_{1}(x′(t_{k}))|.

Before proving the theorem, we present a lemma.

Lemma 4.1 Let G : [0, b] → P_{cl}(ℝ) be a measurable multifunction and u : [0, b] → ℝ a measurable function. Assume that there exist p ∈ L^{1}(J, ℝ) such that G(t) ⊆ p(t)B(0, 1), where B(0, 1) denotes the closed ball in ℝ. Then there exists a measurable selection g of G such that for a.e. t ∈ [0, b],

|u(t) − g(t)| ≤ d(u(t), G(t)).

Proof. Let v_{ϵ} : [0, b] → ℝ_{+} define by v_{ϵ}(t) = ϵ > 0. Then from Lemma 2.2, there exists a measurable selection g_{ϵ} of G such that

|u(t) − g_{ϵ}(t)| ≤ d(u(t), G(t)) + ϵ.

Let ϵ = 1, \frac{1}{2}, …, \frac{1}{n}, …, n ∈ ℕ, hence

|u(t) − g_{n}(t)| ≤ d(u(t), G(t)) + \frac{1}{n}.

Using the fact that G is integrable bounded, then

g_{n}(t) ∈ p(t)B(0, 1), t ∈ J,

and we may pass to a subsequence if necessary to get that g_{n} converges to a measurable function g. Then

|u(t) − g(t)| ≤ d(u(t), G(t)).

Proof of Theorem 4.1. We are going to study Problem (1)-(4) in the respective intervals [0, t_{1}], (t_{1}, t_{2}], …, (t_{m}, b]. The proof will be given in three steps and then continued by induction, and then summarized in a fourth step. Step 1. In this first step, we construct a sequence of functions (y_{n})_{n∈ℕ} which will be shown to converge to some solution of Problem (1)-(4) on the interval [0, t_{1}], namely to

\{ \begin{matrix}D_{∗}^{α}y(t) & ∈ & F (t, y(t)), & t ∈ J_{0} = [0, t_{1}], α ∈ (1, 2], \\ y(0) & = & a, & y′(0) = c. \end{matrix}  (16)

Let f_{0} = g on [0, t_{1}] and y_{0}(t) = x(t), t ∈ [0, t_{1}], i.e.,

y_{0}(t) = \overline{a} + t\overline{c} + \frac{1}{Γ(α)} ∫_{0}^{t}(t − s)^{α−1}f_{0}(s)ds, t ∈ [0, t_{1}].

Then define the multi-valued map U_{1}: [0, t_{1}] → P(ℝ) by U_{1}(t) = F (t, y_{0}(t)) ∩ (g(t) + γ(t)B(0, 1)). Since g and γ are measurable, Theorem III.4.1 in [15] tells us that the ball (g(t)+γ(t)B(0, 1)) is measurable. Moreover F (t, y_{0}(t)) is measurable (see Remark 4.1) and U_{1} is nonempty. It is clear that

d(0, F (t, 0)) ≤ d(0, g(t)) + d(g(t), F (t, y_{0}(t))) + H_{d}(F (t, y_{0}(t)), F (t, 0))
≤ |g(t)| + γ(t) + p(t)|y_{0}(t)|, a.e. t ∈ [0, t_{1}].

Hence for all w ∈ F (t, y_{0}(t)) we have

|w| ≤ d(0, F (t, 0)) + H_{d}(F (t, 0), F (t, y_{0}(t)))
≤ |g(t)| + γ(t) + 2p(t)|y_{0}(t)| := M(t), a.e.t ∈ [0, t_{1}].

This implies that

F (t, y_{0}(t)) ⊆ M(t)B(0, 1), t ∈ [0, t_{1}].

From Lemma 4.1, there exists a function u which is a measurable selection of F (t, y_{0}(t)) such that

|u(t) − g(t)| ≤ d(g(t), F (t, y_{0}(t))) = γ(t).

Then u ∈ U_{1}(t), proving our claim. We deduce that the intersection multivalued operator U_{1}(t) is measurable (see [4, 15, 25]). By Lemma 2.1 (Kuratowski-Ryll-Nardzewski selection theorem), there exists a function t → f_{1}(t) which is a measurable selection for U_{1}. Consider

y_{1}(t) = a + tc + \frac{1}{Γ(α)} ∫_{0}^{t}(t − s)^{α−1}f_{1}(s)ds, t ∈ [0, t_{1}].

For each t ∈ [0, t_{1}], we have

\begin{aligned}|y_{1}(t) − y_{0}(t)| & ≤ & |a − \overline{a}| + t|c − \overline{c}| \\ & + & \frac{1}{Γ(α)} ∫_{0}^{t}(t − s)^{α−1}|f_{0}(s) − f_{1}(s)| ds \\ & ≤ & |a − \overline{a}| + b|c − \overline{c}| \\ & + & \frac{1}{Γ(α)} ∫_{0}^{t} |f_{0}(s) − f_{1}(s)|ds. \end{aligned}  (17)

Hence

|y_{1}(t) − y_{0}(t)| ≤ δ + \frac{t_{1}^{α−1}}{Γ(α)} ∫_{0}^{t} γ(s)ds, t ∈ [0, t_{1}].

Now Lemma 1.4 in [22] tells us that F (t, y_{1}(t)) is measurable. The ball (f_{1}(t) + p(t)|y_{1}(t) − y_{0}(t)|B(0, 1)) is also measurable by Theorem III.4.1 in [15]. The set U_{2}(t) = F (t, y_{1}(t)) ∩ (f_{1}(t) + p(t)|y_{1}(t) − y_{0}(t)|B(0, 1)) is nonempty. Indeed, since f_{1} is a measurable function, Lemma 4.1 yields a measurable selection u of F (t, y_{1}(t)) such that

|u(t) − f_{1}(t)| ≤ d(f_{1}(t), F (t, y_{1}(t))).

Then using (H_{2}), we get

|u(t) − f_{1}(t)| ≤ d(f_{1}(t), F (t, y_{1}(t)))
≤ H_{d}(F (t, y_{0}(t)), F (t, y_{1}(t)))
≤ p(t)|y_{0}(t) − y_{1}(t)|,

i.e. u ∈ U_{2}(t), proving our claim. Now, since the intersection multi-valued operator U_{2} defined above is measurable (see [4, 15, 25]), there exists a measurable selection f_{2}(t) ∈ U_{2}(t). Hence

|f_{1}(t) − f_{2}(t)| ≤ p(t)|y_{1}(t) − y_{0}(t)|.  (18)

Define

y_{2}(t) = a + ct + \frac{1}{Γ(α)} ∫_{0}^{t}(t − s)^{α−1}f_{2}(s)ds, t ∈ (0, t_{1}].

Using (17) and (18), a simple integration by parts yields the following estimates, valid for every t ∈ [0, t_{1}],

|y_{2}(t) − y_{1}(t)| ≤ \frac{t_{1}^{α−1}}{Γ(α)} ∫_{0}^{t} |f_{2}(s) − f_{1}(s)|ds
≤ ∫_{0}^{t} p(s) (δ + ∫_{0}^{s} γ(u)du) ds

= M^{2} (δ ∫_{0}^{t} p(s)ds + ∫_{0}^{t} p(s)ds ∫_{0}^{s} γ(u)du)
≤ M^{2} (δ ∫_{0}^{t} p(s)e^{P (s)}ds + ∫_{0}^{t} p(s)e^{P (s)}ds ∫_{0}^{s} e^{−P (u)}γ(u)du)
≤ M^{2} (δe^{P (t)} + ∫_{0}^{t} γ(s)e^{P (t)−P (s)}ds) , t ∈ [0, t_{1}].

Let U_{3}(t) = F (t, y_{2}(t)) ∩ (f_{2}(t) + p(t)|y_{2}(t) − y_{1}(t)|B(0, 1)). Arguing as for U_{2}, we can prove that U_{3} is a measurable multi-valued map with nonempty values; so there exists a measurable selection f_{3}(t) ∈ U_{3}(t). This allows us to define

y_{3}(t) = a + tc + \frac{1}{Γ(α)} ∫_{0}^{t}(t − s)^{α−1}f_{3}(s)ds, t ∈ [0, t_{1}].

For t ∈ [0, t_{1}], we have

|y_{3}(t) − y_{2}(t)| ≤ M ∫_{0}^{t} |f_{2}(s) − f_{3}(s)| ds
≤ M ∫_{0}^{t} p(s)|y_{2}(s) − y_{1}(s)| ds.

Then

|y_{3}(s) − y_{2}(s)| ≤ M^{3} (δe^{P (s)} + ∫_{0}^{s} γ(u)e^{P (s)−P (u)}du)
≤ M^{3} (δe^{P (s)} + ∫_{0}^{s} γ(u)e^{P (s)−P (u)}du) .

Performing an integration by parts, we obtain, since P is a nondecreasing function, the following estimates

|y_{3}(t) − y_{2}(t)| ≤ \frac{M^{3}}{2} ∫_{0}^{t} 2p(s) (δe^{2P (s)} + ∫_{0}^{s} γ(u)e^{P (s)−P (u)}du) ds
≤ \frac{M^{3}}{2} (δe^{2P (t)} + ∫_{0}^{t} 2p(s)ds ∫_{0}^{s} γ(u)e^{2(P (s)−P (u))}du)
≤ \frac{M^{3}}{2} (δe^{2P (t)} + ∫_{0}^{t} (e^{2P (s)})′ ds ∫_{0}^{s} γ(u)e^{−2P (u))}du)
≤ \frac{M^{3}}{2} (δe^{2P (t)} + e^{2P (t)} ∫_{0}^{t} γ(s)e^{−2P (s)}ds − ∫_{0}^{t} γ(s)ds)
≤ \frac{M^{3}}{2} (δe^{2P (t)} + ∫_{0}^{t} γ(s)e^{2(P (t)−P (s))}ds) , t ∈ [0, t_{1}].

Repeating the process for n = 1, 2, 3, …, we arrive at the following bound

\begin{aligned}|y_{n}(t) − y_{n−1}(t)| & ≤ & \frac{M^{n}}{(n − 1)!} ∫_{0}^{t} γ(s)e^{(n−1)(P (t)−P (s))}ds \\ & + & \frac{M^{n}}{(n − 1)!}δe^{(n−1)P (t)}, t ∈ [0, t_{1}]. \end{aligned}  (19)

By induction, suppose that (19) holds for some n and check (19) for n+1. Let U_{n+1}(t) = F (t, y^{n}(t)) ∩ (f_{n} + p(t)|y_{n}(t) − y_{n−1}(t)|B(0, 1)). Since U_{n+1} is a nonempty measurable set, there exists a measurable selection f_{n+1}(t) ∈ U_{n+1}(t), which allows us to define for n ∈ ℕ

y_{n+1}(t) = a + tc + \frac{1}{Γ(α)} ∫_{0}^{t}(t − s)^{α−1}f_{n+1}(s)ds, t ∈ [0, t_{1}].  (20)

Therefore, for a.e. t ∈ [0, t_{1}], we have

|y_{n+1}(t) − y_{n}(t)| ≤ M ∫_{0}^{t} |f_{n+1}(s) − f_{n}(s)| ds
≤ \frac{M^{n+1}}{(n − 1)!} ∫_{0}^{t} p(s)|y_{n}(s) − y_{n−1}(s)| ds
≤ \frac{M^{n+1}}{(n − 1)!} ∫_{0}^{t} p(s)ds (δe^{(n−1)P (s)} + ∫_{0}^{s} γ(u)e^{(n−1)(P (s)−P (u))}du)
≤ \frac{M^{n+1}}{n!} ∫_{0}^{t} δnp(s)e^{nP (s)}ds
+ \frac{M^{n+1}}{n!} ∫_{0}^{t} np(s)e^{nP (s)}ds ∫_{0}^{s} γ(u)e^{−nP (u)}du.

Again, an integration by parts leads to

\begin{aligned}|y_{n+1}(t) − y_{n}(t)| & ≤ & \frac{M^{(n+1)}}{n!} ∫_{0}^{t} γ(s)e^{n(P (t)−P (s))}ds \\ & + & \frac{M^{(n+1)}}{n!} δe^{nP (t)}. \end{aligned}

Consequently, (19) holds true for all n ∈ ℕ. We infer that {y_{n}} is a Cauchy sequence in P C_{1}, converging uniformly to a limit function y ∈ P C_{1}, where

P C_{1} = C([0, t_{1}], ℝ).

Moreover, from the definition of {U_{n}}, we have

|f_{n+1}(t) − f_{n}(t)| ≤ p(t)|y_{n}(t) − y_{n−1}(t)|, a.e t ∈ [0, t_{1}].

Hence, for almost every t ∈ [0, t_{1}], {f_{n}(t)} is also a Cauchy sequence in ℝ and then converges almost everywhere to some measurable function f(·) in ℝ. In addition, since f_{0} = g, we have for a.e. t ∈ [0, t_{1}]

|f_{n}(t)| ≤ ∑_{i=1}^{n} p(t)|f_{i}(t) − f_{i−1}(t)| + |f_{0}(t)|

≤ ∑_{i=2}^{n} p(t)|y_{i−1}(t) − y_{i−2}(t)| + |g(t)|

≤ p(t) ∑_{i=2}^{∞} |y_{i}(t) − y_{i−1}(t)| + γ(t) + |g(t)|.

Hence

|f_{n}(t)| ≤ H_{0}(t)p(t) + γ(t) + |g(t)|,

where

H_{0}(t) := δM exp (e^{P (t)}) + M ∫_{0}^{t} γ(s) exp (e^{P (t)−P (s)}) ds.  (21)

From the Lebesgue dominated convergence theorem, we deduce that {f_{n}} converges to f in L^{1}([0, t_{1}], ℝ). Passing to the limit in (20), we find that the function

y(t) = a + tc + \frac{1}{Γ(α)} ∫_{0}^{t}(t − s)^{α−1}f(s)ds, t ∈ (0, t_{1}]

is solution to Problem (1)-(4) on [0, t_{1}]; thus y ∈ S_{[0,t_{1}]}(a, c). Moreover, for a.e. t ∈ [0, t_{1}], we have

|x(t) − y(t)| = |a + tc + \frac{1}{Γ(α)} ∫_{0}^{t}(t − s)^{α−1}g(s)ds

−\overline{a} − t\overline{c} − \frac{1}{Γ(α)} ∫_{0}^{t}(t − s)^{α−1}f(s)ds|

≤ Mδ + M ∫_{0}^{t} |f(s) − f_{0}(s)|ds
≤ Mδ + M ∫_{0}^{t} |f(s) − f_{n}(s)|ds
+ M ∫_{0}^{t} |f_{n}(s) − f_{0}(s)|ds
≤ Mδ + M ∫_{0}^{t} |f(s) − f_{n}(s)|ds
+ M ∫_{0}^{t} (H(s)p(s) + γ(s)) ds.

Passing to the limit as n → ∞, we get

|x(t) − y(t)| ≤ η_{0}(t), a.e. t ∈ [0, t_{1}],  (22)

with

η_{0}(t) := Mδ + M ∫_{0}^{t} (H_{0}(s)p(s) + γ(s)) ds.

Next, we give an estimate for |D^{α}y(t) − g(t)| for t ∈ [0, t_{1}]. We have

|D_{∗}^{α}y(t) − g(t)| = |f(t) − f_{0}(t)|
≤ |f_{n}(t) − f_{0}(t)| + |f_{n}(t) − f(t)|
≤ p(t) ∑_{i=1}^{∞} |y_{i+1}(t) − y_{i}(t)| + γ(t) + |f_{n}(t) − f(t)|.

Arguing as in (21) and passing to the limit as n → +∞, we deduce that

|D_{∗}^{α}y(t) − g(t)| ≤ H_{0}(t)p(t) + γ(t), t ∈ [0, t_{1}].

The obtained solution is denoted by y_{1} := y_{|[0,t_{1}]}.

Step 2. Consider now Problem (1)-(4) on the second interval (t_{1}, t_{2}], i.e.,

\{ \begin{matrix}D_{∗}^{α}y(t) & ∈ & F (t, y(t)), & a.e. t ∈ (t_{1}, t_{2}], \\ y(t_{1}^{+}) & = & y_{1}(t_{1}) + I_{1}(y_{1}(t_{1})), \\ y′(t_{1}^{+}) & = & y′_{1}(t_{1}) + \overline{I}_{1}(y_{1}(t_{1})). \end{matrix}  (23)

Let f_{0} = g and set

\begin{aligned}y^{0}(t) & = & x(t_{1}) + I_{1}(x(t_{1})) + (t − t_{1})[x(t_{1}) + \overline{I}_{1}(x(t_{1}))] \\ & + & \frac{1}{Γ(α)} ∫_{t_{1}}^{t}(t − s)^{α−1}f_{0}(s)ds, t ∈ (t_{1}, t_{2}]. \end{aligned}

Notice that (22) allows us to use Assumption (H_{2}), apply again Lemma 1.4 in [22] and argue as in Step 1 to prove that the multi-valued map U_{1}: [t_{1}, t_{2}] → P(ℝ) defined by U_{1}(t) = F (t, y^{0}(t)) ∩ (g(t) + γ(t)B(0, 1)) is U_{1}(t) is measurable. Hence, there exists a function t ↦ f_{1}(t) which is a measurable selection for U_{1}. Define

\begin{aligned}y^{1}(t) & = & y_{1}(t_{1}) + I_{1}(y_{1}(t_{1})) + (t − t_{1})[y_{1}(t_{1}) + \overline{I}_{1}(y_{1}(t_{1}))] \\ & + & \frac{1}{Γ(α)} ∫_{t_{1}}^{t}(t − s)^{α−1}f_{1}(s)ds, t ∈ (t_{1}, t_{2}]. \end{aligned}

Next define the measurable multi-valued map U_{2}(t) = F (t, y^{1}(t)) ∩ (f_{1}(t) + p(t)|y^{1}(t) − y^{0}(t)|B(0, 1)). It has a measurable selection f_{2}(t) ∈ U_{2}(t) by the Kuratowski-Ryll-Nardzewski selection theorem. Repeating the process of selection as in Step 1, we can define by induction a sequence of multi-valued maps U_{n}(t) = F (t, y^{n−1}(t)) ∩ (f_{n−1}(t) + p(t)|y^{n−1}(t) − y^{n−2}(t)|B(0, 1)) where {f_{n}} ∈ U_{n} and (y^{n})_{n∈ℕ} is as defined by

\begin{aligned}y^{n}(t) & = & y_{1}(t_{1}) + I_{1}(y_{1}(t_{1})) + (t − t_{1})[y_{1}(t_{1}) + \overline{I}_{1}(y_{1}(t_{1}))] \\ & + & \frac{1}{Γ(α)} ∫_{t_{1}}^{t}(t − s)^{α−1}f_{n}(s)ds, t ∈ (t_{1}, t_{2}], \end{aligned}

and we can easily prove that

\begin{aligned}|y_{n+1}(t) − y_{n}(t)| & ≤ & \frac{M^{(n+1)}}{n!} ∫_{t_{1}}^{t} γ(s)e^{n(P_{1}(t)−P_{1}(s))}ds \\ & + & \frac{M^{(n+1)}}{n!} δ_{1}e^{nP_{1}(t)}, t ∈ (t_{1}, t_{2}]. \end{aligned}

Let

P C_{2} = {y: y ∈ C(t_{1}, t_{2}] and y(t_{1}^{+}) exists}.

As in Step 1, we can prove that the sequence {y^{n}} converges to some y ∈ P C_{2} solution to Problem (23) such that, for a.e. t ∈ (t_{1}, t_{2}], we have

|x(t) − y(t)| ≤ M|x_{1}(t_{1}) − y_{1}(t_{1})| + M|x′(t_{1}) − y′_{1}(t_{1})|
+ M|I_{1}(x(t_{1})) − I_{1}(y_{1}(t_{1}))| + M|\overline{I}_{1}(x(t_{1})) − \overline{I}_{1}(y_{1}(t_{1}))|
+ M ∫_{t_{1}}^{t} (H_{1}(s)p(s) + γ(s)) ds
≤ Mδ_{1} + M ∫_{t_{1}}^{t} (H_{1}(s)p(s) + γ(s)) ds,

and

|D_{∗}^{α}y(t) − g(t)| := |f(t) − f_{0}(t)| ≤ H_{1}(t)p(t) + γ(t).

Denote the restriction y_{|(t_{1},t_{2}]} by y_{2}.

Step 3. We continue this process until we arrive at the function y_{m+1} := y|_{(t_{m},b]}

as a solution of the problem

\{ \begin{matrix}D_{∗}^{α}y(t) & ∈ & F (t, y(t)), & a.e. t ∈ (t_{m}, b], \\ y(t_{m}^{+}) & = & y_{m−1}(t_{m}) + I_{m}(y_{m−1}(t_{m})), \\ y′(t_{m}^{+}) & = & y′_{m−1}(t_{m}) + \overline{I}_{m}(y(t_{m})). \end{matrix}

Then, for a.e. t ∈ (t_{m}, b], the following estimates are easily derived:

|x(t) − y(t)| ≤ |y_{m}(t_{m}) − x(t_{m})| + |y′_{m}(t_{m}) − x′(t_{m})|
+ M[|I_{m}(x(t_{m})) − I_{m}(y(t_{m}))| + |\overline{I}_{m}(x(t_{m})) − \overline{I}_{m}(y(t_{m}))|
+ M ∫_{t_{m}}^{t} (H_{m}(s)p(s) + γ(s)) ds
≤ Mδ_{m} + M ∫_{t_{m}}^{t} (H_{m}(s)p(s) + γ(s)) ds

and

|D_{∗}^{α}y(t) − g(t)| ≤ H_{m}(t)p(t) + γ(t)).

Step 4. Summarizing, a solution y of Problem (1)-(4) can be defined as follows

y(t) = \{ \begin{matrix}y_{1}(t), & if t ∈ [0, t_{1}], \\ y_{2}(t), & if t ∈ (t_{1}, t_{2}], \\ … & … \\ y_{m+1}(t), & if t ∈ (t_{m}, b]. \end{matrix}

From Steps 1 to 3, we have that, for a.e. t ∈ [0, t_{1}],

|x(t) − y(t)| ≤ η_{0}(t), and |D_{∗}^{α}y(t) − g(t)| ≤ H_{0}(t)p(t) + γ(t),

as well as the following estimates, valid for t ∈ (t_{1}, b]

|x(t) − y(t)| ≤ M ∑_{k=0}^{m} δ_{k} + ∑_{k=0}^{m} η_{k}(t).

Similarly

|D_{∗}^{α}y(t) − g(t)| ≤ p(t) ∑_{0<t_{k}<t} H_{k}(t) + ∑_{0<t_{k}<t} γ_{k}(t),

where γ_{k} := γ_{|}_{J_{k}}. The proof of Theorem 4.1 is complete.

4.1 Filippov’s Theorem on the Half-Line

We may consider Filippov’s Problem on the half-line as given by,

\{ \begin{matrix}D_{∗}^{α}y(t) & ∈ & F (t, y(t)), & a.e. t ∈ J̃\\{t_{1}, …\}, \\ Δy_{t=t_{k}} & = & I_{k}(y(t_{k}^{−})), & k = 1, …, \\ Δy′_{t=t_{k}} & = & \overline{I}_{k}(y(t_{k}^{−})), & k = 1, …, \\ y(t) & = & a, & y′(0) = c, \end{matrix}  (24)

where J̃ = [0, ∞), 0 = t_{0} < t_{1} < ⋯ < t_{m} < ⋯, lim_{m→∞} t_{m} = +∞, F: J̃× ℝ → P(ℝ) is

a multifunction, and a, c ∈ ℝ. Let x be the solution of Problem (15) on the half-line. We will need the following assumptions:

(H̃_{1}) The function F: J̃× ℝ → P_{cl}(ℝ) is such that (a) for all y ∈ ℝ, the map t ↦ F (t, y) is measurable,

(b) the map t ↦ γ(t) = d(g(t), F (t, x(t)) ∈ L^{1}([0, ∞), ℝ_{+})

(H̃_{2}) There exist a function p ∈ L^{1}([0, ∞), ℝ^{+}) such that

H_{d}(F (t, z_{1}), F (t, z_{2})) ≤ p(t)|z_{1} − z_{2}|, for all z_{1}, z_{2} ∈ ℝ,

(H̃_{3}) For every x ∈ ℝ, we have

∑_{k=1}^{∞} |I_{k}(x)| < ∞, ∑_{k=1}^{∞} |\overline{I}_{k}(x)| < ∞.

Then we can extend Filippov’s Theorem to the half-line.

Theorem 4.2 Let γ_{k} := γ_{|}_{J_{k}} and assume (H̃_{1}) − (H̃_{3}) hold. Then, Problem (24) has at least one solution y satisfying, for t ∈ [0, ∞), the estimates

|y(t) − x(t)| ≤ M ∑_{0<t_{k}<t} δ_{k} + M ∑_{0<t_{k}<t} η_{k}(t),

and

|D_{∗}^{α}y(t) − g(t)| ≤ p(t) ∑_{0<t_{k}<t} H_{k}(t) + ∑_{0<t_{k}<t} γ_{k}(t).

Proof. The solution will be sought in the space

P̃ C = {y: [0, ∞) → ℝ, y_{k} ∈ C(J_{k}, ℝ), k = 0, …, such that
y(t_{k}^{−}) and y(t_{k}^{+}) exist and satisfy y(t_{k}^{−}) = y(t_{k}) for k = 1, …},

where y_{k} is the restriction of y to J_{k} = (t_{k}, t_{k+1}], k ≥ 0. Theorem 4.1 yields estimates of y_{k} on each one of the bounded intervals J_{0} = [0, t_{1}], and J_{k} = (t_{k−1}, t_{k}], k = 2, …. Let y_{0} be solution of Problem (1)-(4) on J_{0}.

Then, consider the following problem

\{ \begin{matrix}D_{∗}^{α}y(t) & ∈ & F (t, y(t)), & a.e. t ∈ (t_{1}, t_{2}], \\ y(t_{1}^{+}) & = & y_{0}(t_{1}) + I_{1}(y_{0}(t_{1})), \\ y′(t_{1}^{+}) & = & y′_{0}(t_{1}) + \overline{I_{1}}(y_{0}(t_{1})). \end{matrix}

From Theorem 4.1, this problem has a solution y_{1}. We continue this process taking into account that y_{m} := y|_{(t_{m},t_{m+1}]} is a solution to the problem

\{ \begin{matrix}D_{∗}^{α}y(t) & ∈ & F (t, y(t)), & a.e. t ∈ (t_{m}, t_{m+1}], \\ y(t_{m}^{+}) & = & y_{m−1}(t_{m}) + I_{m}(y_{m−1}(t_{m}^{−})), \\ y′(t_{m}^{+}) & = & y′_{m−1}(t_{m}) + \overline{I}_{m}(y_{m−1}(t_{m}^{−})). \end{matrix}

Then a solution y of Problem (24) may be rewritten as

y(t) = \{ \begin{matrix}y_{1}(t), & if t ∈ [0, t_{1}], \\ y_{2}(t), & if t ∈ (t_{1}, t_{2}], \\ … & … \\ y_{m}(t), & if t ∈ (t_{m}, t_{m+1}], \\ … & … \end{matrix}

Acknowledgements

The author wishes to express his gratitude to referee for his/her many corrections and constructive criticism.

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