EXISTENCE OF SOLUTION FOR IMPULSIVE FRACTIONAL DYNAMIC EQUATIONS WITH DELAY ON TIME SCALES

Abstract

This paper is mainly concerned with the existence of solution for nonlinear impulsive fractional dynamic equations on a special time scale.We introduce the new concept and propositions of fractional q-integral, q-derivative, and α-Lipschitz in the paper. By using a new fixed point theorem, we obtain some new existence results of solutions via some generalized singular Gronwall inequalities on time scales. Further, an interesting example is presented to illustrate the theory.

1. Introduction

The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his Ph.D. thesis in 1988, in order to unify continuous and discrete analysis. A time scale is an arbitrary nonempty closed subset of the real numbers. In recent years, there has been much research activity concerning some different equations on time scales. We refer the reader to the paper [3].

In the last few decades, fractional differential equations have gained considerable importance and attention due to their applications in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, mechanics, chemistry, aerodynamics, and the electrodynamics of complex mediums. See the monographs of Kilbas, Miller and Ross [10], Podlubny and the papers of Daftardar-Gejji and Jafari [6], Diethelm [5], Lakshmikantham.

The concept of fractional q-calculus is not new. Recently and after the appearance of time scale calculus(see for example [4]), some authors started to pay attention and apply the techniques of time scale to discrete fractional calculus [1,2] benefitting from the results announced before in [7].

In paper [11], JinRong Wang discussed the impulsive fractional differential equations with order q ∈ (1, 2) as follows:

A unique solution u of (1.1) is given by

Motivated by the above result, we reconsider the existence of solution for impulsive fractional dynamic equations with delay on time scales

where a ∈ ℝ+,ν ∈ (1, 2), q ∈ (0, 1), f : × ℝ × ℝ → ℝ is jointly continuous, = {t : t = aqn , n ∈ N0} ∪ {0}, N0 = {0, 1, 2, · · ·}, Ik , Jk : ℝ → ℝ, tk satisfies 0 = t0 < t1 < · · · < tm < tm+1 = a, and α(t), β(t) ≤ t, u0 , ū0 are fixed real numbers. For t ∈ , we define the forward jump operator σ : → by σ(t) := inf{s ∈ : s > t}. For any function υ, we define

The rest of this paper is organized as follows. In Section 2, we give some notations, recall some concepts and preparation results. In Section 3, we give four main results. At last, we give an example to demonstrate the application of our main results.

 

2. Preliminaries

In this section, we introduce notations, definitions, and preliminaries. Throughout this paper, let C(,ℝ) be the Banach space of all continuous functions from to ℝ with the norm ∥u∥C := sup{|u(t)| : t ∈ } for u ∈ C(,ℝ)We also introduce the Banach space PC(,ℝ) = {u :→ℝ: u ∈ C((tk , tk+1 ], ℝ), k = 0, 1, 2, . . . , m with the norm ∥u∥ PC := sup{|u(t)| : t ∈}}.

Let us recall the following known definitions. For more details see [2,13].

Definition 2.1. For a function f : →ℝ, the nabla q-derivative of f is

for all t ∈╲{0}, q ∈ (0, 1).

The q-factorial function is defined in the following way.

Definition 2.2. If n is a positive integer, t, s ∈ ╲{0}, q ∈ (0, 1), then

If ν is not a positive integer, then

We state several properties of the q-factorial function.

Proposition 2.3. (i)

where β, γ ∈ℝ.

Definition 2.4. The q-Gamma function is defined by

where α ∈ ℝ╲{. . . , −2, −1, 0}, q ∈ (0, 1), eq (t) = (1 − qnt), eq (0) = 1.

The q-Beta function is defined by

Proposition 2.5. (i) Γq (α + 1) = Γq (α), Γq (1) = 1, where α ∈ ℝ+ .

Definition 2.6. The fractional q-integral is defined by

where q ∈ (0, 1).

Definition 2.7. If X is a Banach space and B is a family of all its bounded sets, the function α : B → ℝ+ defined by α(B) = inf{d > 0 : B admits a finite cover by sets of diameter ≤ d}, B ∈ B, is called the Kuratowski measure of noncompactness.

Consider Ω ⊂ X and F : Ω → X is a continuous bounded map. We say that F is α-Lipschitz, if there exists κ ≥ 0 such that α(F(B)) ≤ κα(B) for all B ⊂ Ω bounded. If, in addition, κ < 1, then we say that F is strict α contraction.

We say that F is α-condensing if α(F(B)) < α(B) for all B ⊂ Ω bounded with α(B) > 0. In other words, α(F(B)) ≥ α(B) implies α(B) = 0.

The class of all strict α-contractions F : Ω → X is denoted by and the class of all α-condensing maps F : Ω → X is denoted by . We remark that ⊂ and every F ∈ is α-Lipschitz with constant κ = 1.

Proposition 2.8. If F, G : Ω → X are α-Lipschitz maps with constants κ, respectively κ′ , then F + G : Ω → X are α-Lipschitz with constant κ + κ′ .

Proposition 2.9. If F : Ω → X is compact, then F is α-Lipschitz with constant κ = 0.

Proposition 2.10. If F : Ω → X is Lipschitz with constant κ, then F is α-Lipschitz with the same constant κ.

Theorem 2.11 (PC-type Ascoli-Arzela theorem, Theorem 2.1 of [12]). Let X be a Banach space and W ⊂ PC(J,X). If the following conditions are satisfied:

(i) W is uniformly bounded subset of PC(J, X);

(ii) W is equicontinuous in (tk,tk+1 ), k = 0, 1, · · · , m, where t0 = 0, tm+1 = T ;

(iii) W(t) = {u(t)|u ∈ W,t ∈ J╲{t1 , · · · , tm }}, W() = {u()|u ∈ W} and W() = {u()|u ∈ W} is a relatively compact subsets of X, then W is a relatively compact subset of PC(J,X).

Theorem 2.12 (Theorem 2, [8]). Let F : X → X be α-condensing and S = {x ∈ X : exists λ ∈ [0, 1] such that x = λFx}.

If S is a bounded set in X, so there exists r > 0 such that S ⊂ Br (0), then F has at least one fixed point and the set of the fixed points of F lies in Br (0).

For measurable functions m : → ℝ, define the norm

where µ() is the Lebesgue measure on . Let Lp(,ℝ) be the Banach space of all Lebesgue measurable functions m : → ℝ with ∥m∥ L p() < ∞.

Theorem 2.13 (Hölder’s inequality). Assume that 1 ≤ p, q ≤ ∞ and = 1.For any l ∈ Lp (,ℝ)and m ∈ Lq (,ℝ), lm ∈ L1 (,ℝ) with ∥lm∥ L1() ≤ ∥l∥ Lp()∥m∥Lq() .

Lemma 2.14. Suppose u(t), b(t), g(t), f(t) are nonnegative on , b(t) is nondecreasing and locally integrable on , g(t), f(t) are nondecreasing and continuous on , a ∈ ℝ+ , and h(t) := g(t) + f(t) ≤ M0 < for any ν > 1.

If

for any α, β : → with α(t) ≤ t,β(t) ≤ t, then

Proof. Let

Since α(t) ≤ t, β(t) ≤ t, we have

then

where h(t) := g(t) + f(t), h(t) is a nondecreasing continuous function.

Let for locally integrable functions ϕ. Then

implies

Let us prove that

and Dnz(t) → 0 as n → +∞ for each t ∈ .

We know this relation (2.1) is true for n = 1. Assume that it is true for some n = k, that is

If n = k + 1, then the induction hypothesis implies

since h(t) is nondecreasing, it follows that

By interchanging the order of integration (see [9]), we have

where the integral

since (t−s) is decreasing about s, we have (t−qτ)≤ (t−qντ ), ν > 1,and by Proposition 2.3 and Definition 2.4, we get

The relation (2.1) is proved. By (2.1) and h(t) ≤ M0 , we have

Since M0 < and by Proposition 2.5, we get

as n → ∞, for t ∈ . Then the Lemma 2.14 is proved. ΋

Lemma 2.15. Let ν ∈ (1, 2) and h : → ℝ be jointly continuous. A function u given by

is the unique solution of the following impulsive problem on time scales

where = {t : t = aqn , n ∈ N0 }∪{0}, a ∈ ℝ+ , q ∈ (0, 1), N0 = {0, 1, 2, · · · }, k = 1, 2, 3, · · · , m, Ik , Jk : ℝ → ℝ, tk satisfy 0 = t0 < t1 < · · · < tm < tm+1 = a. σ : → is the forward jump operator σ(t) := inf{s ∈ : s > t}. And for any function υ, we define

Proof. Assume the general solution u of the Eq. (2.4) is given by

where t0 = 0, tm+1 = a. Then, we have

Applying the cauchy conditions of (2.4), we get

Next, using the impulsive condition of (2.4), we find that

which by (2.7) implies

Furthermore, using the impulsive condition of (2.4), we find that

which implies

So by (2.9), (2.11), we have

Thus, we can get (2.3).

Conversely, assume that u satisfies (2.3). By a direct computation, it follows that the solution given by (2.3) satisfies (2.4). This completes the proof. ΋

 

3. Main results

In this section, we deal with the existence and uniqueness of solutions for the problem (1.3).

Before stating and proving the main results, we introduce the following hypotheses:

[H1] f : × ℝ × ℝ → ℝ is jointly continuous.

[H2] For arbitrary (t, u, v) ∈ × ℝ × ℝ, there exist L1 , L2 > 0, such that

[H3] There exist q1 , q2 ∈ (0, 1), real functions h ∈ L( , ℝ), y ∈ L( , ℝ) such that

for all t ∈ and u1 (·), u2 (·) ∈ ℝ.

[H4] There exist constants L3,L4 > 0, such that

for all u,v ∈ ℝ, and k = 1, 2, · · · ,m.

[H5] For arbitrary u ∈ ℝ, there exist constants M1 , M2 > 0, such that

Theorem 3.1. Assume that [H1]-[H5] hold, and if L1 + L2 < ,then the problem (1.3) has at least one solution on .

Proof. By Lemma 2.15, we define an operator : PC(, ℝ) → PC(,ℝ) by

for t ∈ (tk , tk+1], k = 0, 1, 2, · · · , m, where

For the sake of convenience, we subdivide the proof into several steps.

Step 1. is continuous.

Let {un } be a sequence such that un → u in PC( , ℝ). Then for each t ∈ (tk, tk+1], by conditions [H3], [H4], we have

Further, we can obtain

Step 2. maps bounded sets into bounded sets in PC(,ℝ).

For each u ∈ Bη = {u ∈ PC(,ℝ) : ∥u∥ PC ≤ η}, t ∈ (tk, tk+1], by [H2], [H5], we have

Let ℓ := |u0 | + a|ū0| + M2(a − σ(ti)) + mM1 + (1 − q)σ(ti)(|ū0| + iM2) +, we get

Step 3. maps bounded sets into equicontinuous sets of PC(, ℝ). It is easy to know is equicontinuous on interval (tk,tk+1], k = 1, 2, · · · , m. For any 0 ≤ s1 < s2 ≤ t1, u ∈ Bη = {u ∈ PC( , ℝ) : ∥u∥PC ≤ η}, we have

considering s2 → s1, we have

Thus, we find that is equicontinuous on .

Step 4. Now it remains to show that the set

is bounded.

Without loss of generality, for any u ∈ E(), t ∈ (tk,tk+1 ], by [H2], [H5], we have

where k = 0, 1, 2, · · · , m.

By Lemma 2.14, there exists a Mk > 0 such that

Set M =Mk, thus for every t ∈ , we get

This shows that the set E() is bounded.

As a consequence of Schaefer’s fixed point theorem, we know that has a fixed point which is a solution of the problem (1.3). The proof is complete. ΋

Theorem 3.2. Assume that [H1],[H3] and [H4] hold, and if

then the problem (1.3) has an unique solution on .

Proof. Consider the operator : PC( , ℝ) → PC(, ℝ) defined as (3.1), and transform the problem (1.3) into a fixed point problem of .

Step 1. u ∈ PC(, ℝ) for every u ∈ PC(, ℝ).

If t = 0, for any δ > 0, we have

then

Thus, we find that u is continuous at 0. It is easy to see that u ∈ C((tk,tk+1],ℝ), k = 0, 1, · · · ,m.

From the above discussion, we get u ∈ PC(,ℝ) for every u ∈ PC(,ℝ).

Step 2. is a contraction operator on PC(,ℝ).

In fact, for arbitrary u1, u2 ∈ PC(,ℝ), by [H3], [H4] and Theorem 2.13, we obtain

Thus, due to (3.2), we know that is a contraction mapping on PC(,ℝ).

By applying the well-known Banach’s contraction mapping principle, we get that the operator has a unique fixed point on PC(,ℝ). Therefore, the problem (1.3) has a unique solution. ΋

Before proving the next results, we introduce the following hypotheses. [H2] ′ For arbitrary (t, u, v) ∈ ×ℝ×ℝ, there exist and q1 ,q2 ∈ [0, 1) such that

[H3] ′ There exist such that

for each t ∈ , and all u1, u2 ∈ ℝ. [H4] ′ There exist constants such that

for all u,v ∈ ℝ, and k = 1, 2, · · · , m. [H5] ′ For arbitrary u ∈ ℝ, there exist constants CI ,CJ > 0 and q3 ,q4 ∈ [0, 1) such that

Theorem 3.3. Assume that [H1] and [H2] ′ -[H5] ′ hold, and if

where

then the promble (1.3) has at least one solution u ∈ PC(,ℝ) and the set of the solutions of the problem (1.3) is bounded in PC(,ℝ).

Proof. Now, we define the operators as follows:

H : PC(,ℝ) → PC(,ℝ) given by

where 1 (u,t) is defined as in (3.1).

Let F : PC(,ℝ) → PC(,ℝ) given by

Thus, the existence of a solution for the problem (1.3) is equivalent to the existence of a fixed point for operator F.

Step 1. The operator H is Lipschitz with constant κ1 = Nk , by Proposition 2.10, consequently H is α-Lipschitz with the same constant κ1 = Nk . Moreover, the operator H satisfies the following growth condition:

For every t ∈ [0,t1], u,v ∈ PC(,ℝ), it is obvious that

If t ∈ (tk ,tk+1], k = 1, 2, · · · , m, u, v ∈ PC(,ℝ), by [H4] ′ , we have

Let Nk := (a − σ(ti ))CiJ +CiI + (1 − q)σ(ti)(CjJ).

For every u,v ∈ PC(,ℝ), t ∈ (tk,tk+1], k = 1, 2, 3, · · · , m. Using [H4] ′ step by step, and by Proposition 2.8 and Proposition 2.10, we know that H is α- Lipschitz with the constant κ1 = Nk. And by [H5] ′ , we get (3.4).

Step 2. The operator g is compact, by Proposition 2.9, then g is α-Lipschitz with constant κ2 = 0.

In order to prove the compactness of g, we consider a bounded set E ⊆ C((tk,tk+1),ℝ),k = 0, 1, 2, · · · , m, and we will show that g(E) is relatively compact in C((tk,tk+1]) with the help of Theorem 2.11.

(i) For t ∈ [0,t1], let un be a sequence on E ⊆ C([0,t1],ℝ), for every un ∈ E, by [H2] ′ , we have

thus, the set g(E) is bounded in C([0,t1]).

For each (tk,tk+1], k = 1, 2, 3, · · · , m, repeating the above process again, one can obtain that the set g(E) is an uniformly bounded subset of PC(,ℝ).

(ii) For t ∈ (tk,tk+1], k = 1, 2, · · · , m, it is easy to know gun is equicontinuous. For any 0 ≤ s1 < s2 ≤ t1, un ∈ E, we have

then

Thus, we find that g is equicontinuous on . From (i), (ii), we get the compactness of the operator g on PC(,ℝ).

By Proposition 2.9, we know that the operator g is α-Lipschitz with constant 0.

Step 3. The operator g is continuous. Moreover, by [H2] ′ , the operator g satisfies the following growth condition:

Let {un} be a sequence such that un → u in PC(,ℝ). Then for each t ∈ (tk,tk+1], by condition [H3] ′ , we have

Further, we can obtain

We know that the operator g is continuous on (tk,tk+1], k = 0, 1, · · · , m.

Step 4. From Step 1 and Step 2, by Proposition 2.8 and condition (3.3), we ob-tain that the operator F is strict α-contraction with constant κ = Nk. Further, by Definition 2.7, we finally get that the operator F is a α-condensing map.

Step 5. Let E(F) := {u ∈ PC(,ℝ) : ∃ ∈ (0, 1) such that u = Fu}. Consider every u ∈ E(F), by (3.4),(3.5), we have

This inequality, together with qi ∈ [0, 1), i = 1, 2, 3, 4, shows us that E(F) is bounded in PC( , ℝ). If not, we suppose by contradiction ξ := ∥u∥ PC → ∞.

Dividing both sides of (3.6) by ξ, and taking ξ → ∞, we get

this is a contraction.

From above, by Theorem 2.12, we deduce that the operator F has at least one fixed point and the set of the fixed points of F is bounded in PC(, ℝ). ΋

Theorem 3.4. Assume that [H1], [H2] ′ -[H5] ′ and condition (3.3) hold, and if

then the problem (1.3) has a unique solution u ∈ PC(, ℝ).

Proof. By Theorem 3.3, the problem (1.3) has at least one solution. Now, let u(.), v(.) be the solutions of problem (1.3) with the same initial values,

by [H3] ′ , [H4] ′ , then

then

From condition (3.3), (3.8), we obtain

Due to Lemma 2.14, we get

The proof is complete. ΋

Next, we give an example to illustrate the usefulness of our main results.

Example 3.5. Let us consider the following fractional impulsive problem with delay on time scales

where m > 0, is a constant,1 (.),2 (.) are defined as in (1.4).

Let

Obviously, for all u ∈ C(1 , ℝ+ ) and each t ∈ 1 , we have

where

it is easy to know that L1 + L2 = 1 <

For u1,u2 ∈ C(1,ℝ+) and t ∈ 1 , we get

where

Set

we have

and for every u,v ∈ C(1 ,ℝ+ ), we get

Set

we have

and for every u,v ∈ C(1 ,ℝ+ ), we get

Thus, all the assumptions in Theorem 3.1 are satisfied. Eq. (3.10) has at least one solution.