Generalized Fractional Operators on Time Scales with Application to ...

arXiv:1804.02536v1 [math.CA] 7 Apr 2018

Generalized Fractional Operators on Time Scales with Application to Dynamic Equations∗ Kheira Mekhalfi1 [email protected] 1

2

Delfim F. M. Torres2,† [email protected]

Institute of Sciences, Department of Mathematics, University of Ain Temouchent, BP 284, 46000 Ain Temouchent, Algeria

Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Abstract We introduce more general concepts of Riemann–Liouville fractional integral and derivative on time scales, of a function with respect to another function. Sufficient conditions for existence and uniqueness of solution to an initial value problem described by generalized fractional order differential equations on time scales are proved. Keywords: fractional derivatives and integrals, initial value problem, dynamic equations, fixed point, existence and uniqueness of solution. MSC 2010: 26A33, 34K37, 34N05.

1

Introduction

The theory of fractional differential equations, specifically the question of existence and uniqueness of solutions, is a research topic of great importance [1,8,24]. Another important area of study is dynamic equations on time scales, which goes back to 1988 and the work of Aulbach and Hilger, and has been used with success to unify differential and difference equations [4, 6, 18]. Starting with a linear dynamic equation, Bastos et al. have introduced the notion of fractional-order derivative on time scales, involving time-scale analogues of Riemann–Liouville operators [9–11]. Another approach originate from ∗ Preprint whose final and definite form is with The European Physical Journal Special Topics (EPJ ST), ISSN 1951-6355 (Print), ISSN 1951-6401 (Online). Submitted 20-June-2017; revised 26-Oct-2017; accepted for publication 06-April-2018. † Corresponding author.

1

the inverse Laplace transform on time scales [12]. After such pioneer work, the study of fractional calculus on time scales developed in a popular research subject: see [13–15, 17, 29] and references therein. Recent results, since 2015, cover fractional q-symmetric systems on time scales [33]; existence of solutions for delta Riemann–Liouville fractional differential equations [35]; existence and uniqueness of solutions for boundary-value problems of fractional-order dynamic equations on time scales in Caputo sense [34]; existence of solutions for impulsive fractional dynamic equations with delay on time scales [21]; and existence of solutions for Cauchy problems with Caputo nabla fractional derivatives [36]. As a real application, we mention the study of calcium ion channels that are retarded with injection of calcium-chelator Ethylene Glycol Tetraacetic Acid [20]. In fact, physical applications of fractional initial value problems in different time scales abound [25,32]. For example, fractional differential equations that govern the behaviors of viscoelastic materials with memory and the creep phenomenon have been proposed in [19] for the continuous time scale T = R; fractional difference equations in discrete time scales T = hZ, h > 0, or T = q Z , with relevance in the description of several physical phenomena, are studied in [2,27]; nonlocal thermally sensitive resistors on arbitrary time scales T are investigated in [31]. Here, we extend available results in the literature by introducing more general concepts of fractional operators on time scales of a function with respect to another function. Then, we investigate corresponding generalized fractional order dynamic equations on time scales. The paper is organized as follows. In Section 2, we briefly recall necessary definitions and results. Our own results are then given in Section 3: we define a fractional integral operator of Riemann–Liouville type on time scales (Definition 3.1) and a generalized fractional derivative (Definition 3.2). Such generalizations on time scales help us to study relations between fractional difference equations and fractional differential equations. On the other hand, they also provide a background to study boundary value problems including fractional order difference and differential equations. In this direction, we generalize the recent results of [16] by proving some sufficient conditions for uniqueness and existence of solutions of a fractional initial value problem on an arbitrary time scale (see Theorems 3.5 and 3.6). We end with Section 4 of illustrative examples.

2

Preliminaries

In this section, we recall some definitions and results that are used in the sequel. We use C(J , R) to denote the Banach space of continuous functions with the norm kyk∞ = sup {|y(t)| : t ∈ J } , where J is an interval. A time scale T is an arbitrary nonempty closed subset of the real numbers. The calculus on time scales was initiated by Aulbach and Hilger [7, 26] in order to create a theory that can unify and extend discrete and continuous analysis. The real numbers R, the integers Z, the natural numbers N, the nonnegative 2

integers N0 , the h-numbers (hZ = {hk : k ∈ Z}, where h > 0 is a fixed real number), and the q-numbers (q Z ∪ {0} = {q k : k ∈ Z} ∪ {0}, where q > 1 is a fixed real number), are examples of time scales, as are [0, 1]∪[2, 3], [0, 1]∪N, and the Cantor set, where [0, 1] and [2, 3] are intervals of real numbers. Any time scale T, being a closed subset of the real numbers, has a topology inherited from the real numbers with the standard topology. It is a complete metric space with the metric (distance) d : T2 → R, d(t, s) = |t − s| for t, s ∈ T. Consequently, according to the well-known theory of general metric spaces, we have for T the fundamental concepts such as open balls (intervals), neighborhoods of points, open sets, closed sets, compact sets, and so on. In particular, for a given number N > 0, the N -neighborhood Uδ (t) of a given point t ∈ T is the set of all points s ∈ T such that d(t, s) < N . By a neighborhood of a point t ∈ T it is meant an arbitrary set in T containing a N -neighborhood of the point t. Also, we have for functions f : T → R the concepts of limit, continuity, and the properties of continuous functions on general complete metric spaces (note that, in particular, any function f : Z → R is continuous at each point of Z). The main task is to introduce and investigate the concept of derivative for functions f : T → R. This proves to be possible due to the special structure of the metric space T. In the definition of derivative, an important role is played by the so-called forward and backward jump operators. Definition 2.1 (See [18]). For t ∈ T, one defines the forward jump operator σ : T → T by σ(t) = inf{s ∈ T : s > t},

while the backward jump operator ρ : T → T is defined by ρ(t) = sup{s ∈ T : s < t}.

In addition, we put σ(max T) = max T if there exists a finite max T, and ρ(min T) = min T if there exists a finite min T. Obviously, both σ(t) and ρ(t) are in T when t ∈ T. This is because of our assumption that T is a closed subset of R. Let t ∈ T. If σ(t) > t, then we say that t is right-scattered, while if ρ(t) < t, then we say that t is left-scattered. Also, if t < max T and σ(t) = t, then t is called right-dense, and if t > min T and ρ(t) = t, then t is called left-dense. The derivative makes use of the set Tκ , which is derived from the time scale T as follows: if T has a left-scattered maximum M , then Tκ := T\{M }; otherwise, Tκ := T. Definition 2.2 (Delta derivative [3]). Assume f : T → R and let t ∈ Tκ . One defines f (σ(s)) − f (t) f ∆ (t) := lim , t 6= σ(s), s→t σ(s) − t

provided the limit exists. We call f ∆ (t) the delta derivative (or Hilger derivative) of f at t. Moreover, we say that f is delta differentiable on Tκ provided f ∆ (t) exists for all t ∈ Tκ . The function f ∆ : Tκ → R is then called the (delta) derivative of f on Tκ . 3

Definition 2.3 (Delta Integral [3,18]). Let [a, b] denote a closed bounded interval in T. A function F : [a, b] → R is called a delta antiderivative of function f : [a, b) → R provided F is continuous on [a, b], delta differentiable on [a, b), and F ∆ (t) = f (t) for all t ∈ [a, b). Then, we define the ∆-integral of f from a to b by Z b f (t)∆t := F (b) − F (a). a

Definition 2.4 (See [3, 18]). A function f : T → R is called rd-continuous, provided it is continuous at right-dense points in T and its left-sided limits exist (finite) at left-dense points in T. The set of rd-continuous functions f : T → R is denoted by Crd . Similarly, a function f : T → R is called ld-continuous provided it is continuous at left-dense points in T and its right-sided limits exist (finite) at right-dense points in T. The set of ld-continuous functions f : T → R is denoted by Cld . All rd-continuous bounded functions on [a, b) are delta integrable from a to b. For a more general treatment of the delta integral on time scales (Riemann, Lebesgue, and Riemann-Stieltjes integration on time scales), see [18, 23, 28]. Proposition 2.5 (See [18]). Suppose a, b ∈ T, a < b, and f (t) is continuous on [a, b]. Then, Z

a

b

f (t)∆t = [σ(a) − a]f (a) +

Z

b

f (t)∆t. σ(a)

Proposition 2.6 (See [5]). Suppose T is a time scale and f is an increasing continuous function on [a, b]. If F is the extension of f to the real interval [a, b] given by ( f (s) if s ∈ T, F (s) := f (t) if s ∈ (t, σ(t)) 6⊂ T, then Z

b

a

f (t)∆t ≤

Z

b

F (t)dt.

a

Definition 2.7 (Fractional integral on time scales [16]). Suppose T is a time scale, [a, b] is an interval of T, and h is an integrable function on [a, b]. Let 0 < α < 1. Then the (left) fractional integral of order α of h is defined by T α a I t h(t)

:=

Z

a

t

(t − s)α−1 h(s)∆s, Γ(α)

where Γ is the gamma function. Using Definition 2.7, one defines the Riemann–Liouville fractional derivative on time scales.

4

Definition 2.8 (Riemann–Liouville derivative on time scales [16]). Let T be a time scale, t ∈ T, 0 < α < 1, and h : T → R. The (left) Riemann–Liouville fractional derivative of order α of h is defined by T α a D t h(t)

3

1 := Γ(1 − α)

Z

t −α

a

(t − s)

h(s)∆s

∆

.

Main Results

We begin by generalizing the concepts of fractional integral and fractional derivative given by Definitions 2.7 and 2.8, by introducing the concept of fractional integration and fractional differentiation on time scales of a function with respect to another function. Definition 3.1 (Generalized fractional integral on time scales). Suppose T is a time scale, [a, b] is an interval of T, h is an integrable function on [a, b], and g is monotone having a delta derivative g ∆ with g ∆ (t) 6= 0 for any t ∈ [a, b]. Let 0 < α < 1. Then, the (left) generalized fractional integral of order α of h with respect to g is defined by Z t 1 α T I h(t) = (g(t) − g(s))α−1 g ∆ (s)h(s)∆s. a;g t a Γ(α) Definition 3.2 (Generalized fractional derivative on time scales). Suppose T is a time scale, [a, b] is an interval of T, h is an integrable function on [a, b], and g is monotone having a delta derivative g ∆ with g ∆ (t) 6= 0 for any t ∈ [a, b]. Let 0 < α < 1. Then, the (left) generalized fractional derivative of order α of h with respect to g is defined by α T a;g D t h(t)

=

1 1 ∆ Γ(1 − α) g (t)

Z

t

a

(g(t) − g(s))−α g ∆ (s)h(s)∆s

∆

.

Remark 3.3. If T = R, then Definitions 3.1 and 3.2 give, respectively, the wellknown generalized fractional integral and derivative of Riemann–Liouville [30, Section 18.2]. Let z be a monotone function having a delta derivative z ∆ with z ∆ (t) 6= 0 for any t ∈ J . We consider the following initial value problem: T α t0 ;z Dt y(t)

= f (t, y(t)),

t ∈ [t0 , t0 + a] = J ⊆ T,

T α t0 ;z It y(t0 )

0 < α < 1,

= 0,

(3.1)

where Tt0 ;z Dtα and Tt0 ;z It1−α is the (left) Riemann–Liouville generalized fractional derivative and integral with respect to function z, and f : J × R → R is a rightdense continuous function. Our main goal is to obtain sufficient conditions for the existence and uniqueness of solution to problem (3.1). In what follows, T is a given time scale and J = [t0 , t0 + a] ⊆ T. 5

Lemma 3.4. Let 0 < α < 1, J ⊆ T, and f : J × R → R. Function y ∈ C(J , R) is a solution of problem (3.1) if and only if is a solution of the following integral equation: Z z ∆ (t) t y(t) = (z(t) − z(s))α−1 z ∆ (s)f (s, y(s))∆s. Γ(α) t0 Proof. By Definitions 3.1 and 3.2, we have Z t 1 1−α T I y(t) = (z(t) − z(s))−α z ∆ (s)y(s)∆s t0 ;z t Γ(1 − α) t0 and α T t0 ;z D t y(t)

Z t ∆ 1 1 −α ∆ (z(t) − z(s)) z (s)y(s)∆s z ∆ (t) Γ(1 − α) t0  ∆ 1 T 1−α 1  1−α y(t). = ∆ ∆ ◦ Tt0 ;z I t y(t) = ∆ t0 ;z I t z (t) z (t)

=

Moreover, α T α T t0 ;z I t [t0 ;z D t y(t)]

∆ 1 T α T 1−α I ( I y(t)) t ;z t ;z z ∆ (t) 0 t 0 t ∆ 1 T 1 = ∆ t0 ;z I t y(t) z (t) 1 = ∆ y(t). z (t)

=

(3.2)

It follows from (3.2), (3.1) and Definition 3.1 that h i α α y(t) = z ∆ (t)Tt0 ;z I t Tt0 ;z Dt y(t) α

= z ∆ (t)Tt0 ;z I t f (t, y(t)) Z z ∆ (t) t = (z(t) − z(s))α−1 z ∆ (s)f (s, y(s))∆s. Γ(α) t0

The proof is complete. Our first main result is based on the Banach fixed point theorem [22]. Theorem 3.5. Let f : J × R → R be continuous and assume there exists a constant L > 0 such that |f (t, u) − f (t, v)| ≤ Lku − vk∞ for t ∈ J and u, v ∈ C(J , R). If Lz ∆ (t) Mα (t)

(t − t0 )Γ(α) < 1, 6

t ∈ J,

(3.3)

where

Z

t

t0

Mα (t) :=

(z(t) − z(s))α−1 z ∆ (s)ds t − t0

,

(3.4)

then problem (3.1) has a unique solution on J . Proof. We transform problem (3.1) into a fixed point problem. Consider the operator F : C(J , R) → C(J , R) defined by z ∆ (t) F (y)(t) = Γ(α)

Z

t

t0

(z(t) − z(s))α−1 z ∆ (s)f (s, y(s))∆s.

(3.5)

We need to prove that F has a fixed point, which is a unique solution of (3.1) on J . For that, we show that F is a contraction. Let x, y ∈ C(J , R). For t ∈ J , we have |F (x)(t) − F (y)(t)| ∆ Z t z (t) = (z(t) − z(s))α−1 z ∆ (s)[f (s, x(s)) − f (s, y(s))]∆s Γ(α) t0 Z z ∆ (t) t (z(t) − z(s))α−1 z ∆ (s)|f (s, x(s)) − f (s, y(s))|∆s ≤ Γ(α) t0 Z Lz ∆ (t)kx − yk∞ t (z(t) − z(s))α−1 z ∆ (s)∆s ≤ Γ(α) t0 Z Lz ∆ (t)kx − yk∞ t (z(t) − z(s))α−1 z ∆ (s)ds. ≤ Γ(α) t0 Then, it follows from (3.4) that |F (x)(t) − F (y)(t)| ≤

Lz ∆ (t)Mα (t)(t − t0 ) kx − yk∞ . Γ(α)

By (3.3), F is a contraction and thus, by the contraction mapping theorem, we deduce that F has a unique fixed point. This fixed point is the unique solution of (3.1). Now, we give our second main result, which is an existence result based upon the nonlinear alternative of Leray–Schauder, applied to completely continuous operators [22]. Theorem 3.6. Suppose f : J × R → R is a rd-continuous bounded function such that there exists N > 0 with |f (t, y)| ≤ N for all t ∈ J , y ∈ R. Then problem (3.1) has a solution on J . Proof. We use Schauder’s fixed point theorem [22] to prove that F defined by (3.5) has a fixed point. The proof is given in four steps.

7

Step 1: F is continuous. Let yn be a sequence such that yn → y in C(J , R). Then, for each t ∈ J , |F (yn )(t)

− F (y)(t)| Z z ∆ (t) t ≤ (z(t) − z(s))α−1 z ∆ (s) |f (s, yn (s)) − f (s, y(s))| ∆s Γ(α) t0 Z z ∆ (t) t ≤ (z(t) − z(s))α−1 z ∆ (s) sup |f (s, yn (s)) − f (s, y(s))| ∆s Γ(α) t0 s∈J Z z ∆ (t) kf (·, yn (·)) − f (·, y(·))k∞ t = (z(t) − z(s))α−1 z ∆ (s)∆s Γ(α) t0 Z z ∆ (t)kf (·, yn (·)) − f (·, y(·))k∞ t ≤ (z(t) − z(s))α−1 z ∆ (s)ds Γ(α) t0 ≤

z ∆ (t)Mα (t)(t − t0 ) kf (·, yn (·)) − f (·, y(·))k∞ . Γ(α)

Since f is a continuous function, we have |F (yn )(t) − F (y)(t)| ≤

z ∆ (t)Mα (t)(t − t0 ) kf (·, yn (·)) − f (·, y(·))k∞ −→ 0 as n → ∞. Γ(α)

Step 2: the map F sends bounded sets into bounded sets in C(J , R). It is enough to show that there exists a positive constant l such that F (y) ∈ Bl = {F (y) ∈ C(J , R) : kF(y)k∞ ≤ l}. By hypothesis, for each t ∈ J , one has Z z ∆ (t) t (z(t) − z(s))α−1 z ∆ (s)|f (s, y(s))|∆s |F(y)(t)| ≤ Γ(α) t0 Z z ∆ (t)N t ≤ (z(t) − z(s))α−1 z ∆ (s)∆s Γ(α) t0 Z z ∆ (t)N t ≤ (z(t) − z(s))α−1 z ∆ (s)ds Γ(α) t0 ≤

z ∆ (t)N Mα (t)(t − t0 ) = l. Γ(α)

Step 3: F sends bounded sets into equicontinuous sets of C(J , R). Let t1 , t2 ∈ J , t1 < t2 . Then, |F (y)(t2 )

− F (y)(t1 )| Z z ∆ (t) t1 (z(t1 ) − z(s))α−1 z ∆ (s)f (s, y(s))∆s ≤ Γ(α) t0

8

−

≤

≤

≤

Z

t2

t

z (s)f (s, y(s))∆s

α−1 ∆

(z(t2 ) − z(s))

Z 0 z ∆ (t) t1 (z(t1 ) − z(s))α−1 − (z(t2 ) − z(s))α−1 Γ(α) t0
+(z(t2 ) − z(s))α−1 z ∆ (s)f (s, y(s))∆s Z t2 (z(t2 ) − z(s))α−1 z ∆ (s)f (s, y(s))∆s − t0 Z z ∆ (t)N t1 ((z(t1 ) − z(s))α−1 − (z(t2 ) − z(s))α−1 )z ∆ (s)∆s Γ(α) t0 Z t2 α−1 ∆ (z(t2 ) − z(s)) z (s)∆s + t1 Z ∆ z (t)N t1 ((z(t1 ) − z(s))α−1 − (z(t2 ) − z(s))α−1 )z ∆ (s)ds Γ(α) t0 Z t2 α−1 ∆ (z(t2 ) − z(s)) z (s)ds . + t1

As t1 → t2 , the right-hand side of the above inequality tends to zero. As a consequence of Steps 1 to 3, together with the Arzela–Ascoli theorem, we conclude that F : C(J , R) → C(J , R) is completely continuous. Step 4: a priori boundedness of solutions. It remains to show that the set E = {y ∈ C(J , R) : y = λF (y), 0 < λ < 1} is bounded. Let y ∈ E be any element. Then, for each t ∈ J , Z z ∆ (t) t (z(t) − z(s))α−1 z ∆ (s)f (s, y(s))∆s. y(t) = λF (y)(t) = λ Γ(α) t0 It follows that |y(t)| ≤ ≤ ≤ ≤ ≤

∆ Z t z (t) α−1 ∆ (z(t) − z(s)) z (s)f (s, y(s))∆s Γ(α) t0 Z z ∆ (t) t (z(t) − z(s))α−1 z ∆ (s)|f (s, y(s))|∆s Γ(α) t0 Z z ∆ (t)N t (z(t) − z(s))α−1 z ∆ (s)∆s Γ(α) t0 Z z ∆ (t)N t (z(t) − z(s))α−1 z ∆ (s)ds Γ(α) t0 z ∆ (t)N Mα (t)(t − t0 ) . Γ(α)

Hence, the set E is bounded. As a consequence of Schauder’s fixed point theorem, we conclude that F has a fixed point, which is solution of (3.1). 9

4

Examples

α When function z is the identity id, we have Ia;id = Iaα , that is, we get the ordinary left Riemann–Liouville fractional integral. In this very particular case, existence of solution to fractional problem (3.1) on time scales has been recently investigated in [16]. Here we cover the general situation: existence of solution to the fractional initial value problem (3.1), for a general function z. For illustrative purposes, let z(t) = t2 , T = 2N , α = 21 , and t0 = 0. Then, σ(t) = 2t and α (I0;t 2 f )(t)

(t2 )∆ = Γ(α)

Z

(t2 )∆ = √ π

Z

t

(t2 − s2 )α−1 (s2 )∆ f (s)∆s

0 t 0

1

(t2 − s2 )− 2 (s2 )∆ f (s)∆s.

In this case, (t2 )∆ F (y)(t) = √ π

Z

t

0

1

(t2 − s2 )− 2 (s2 )∆ f (s, y(s))∆s

and we have |F (x)(t) − F (y)(t)|

≤ = ≤ ≤

Z (t + σ(t))Lkx − yk∞ t 2 1 √ (t − s2 )− 2 (s + σ(s))∆s π 0 Z (t + 2t)Lkx − yk∞ t 2 1 √ (t − s2 )− 2 (s + 2s)∆s π 0 Z t 1 3tLkx − yk∞ √ (t2 − s2 )− 2 (3s)ds π 0 9t2 L √ kx − yk∞ . π

In this example, Mα (t) =

Z

0

t

1

(t2 − s2 )− 2 (3s)ds

t and (3.4) is reduced to Mα (t) ≡ 3. Choose b such that 9t2 L b = √ < 1. π Then the conditions of Theorem 3.5 are satisfied, and we conclude that there is a function y ∈ C(J , R) solution of (3.1).

Acknowledgments This research was initiated while Mekhalfi was visiting the Department of Mathematics of University of Aveiro, Portugal, February 2017. The hospitality of the 10

host institution and the financial support of University of Ain Temouchent, Algeria, are here gratefully acknowledged. Torres was supported by Portuguese funds through CIDMA and FCT, within project UID/MAT/04106/2013. The authors are very grateful to two anonymous referees, for their suggestions and invaluable comments.

Author Contribution Statement Both authors contributed equally to the paper.

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