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EXISTENCE OF SOLUTION FOR IMPULSIVE FRACTIONAL DYNAMIC EQUATIONS WITH DELAY ON TIME SCALES†
EXISTENCE OF SOLUTION FOR IMPULSIVE FRACTIONAL DYNAMIC EQUATIONS WITH DELAY ON TIME SCALES†
Journal of Applied Mathematics & Informatics. 2015. May, 33(3_4): 275-292
Copyright © 2015, Korean Society of Computational and Applied Mathematics
  • Received : August 25, 2014
  • Accepted : November 05, 2014
  • Published : May 30, 2015
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ZHI-JUAN GAO
XU-YANG FU
QIAO-LUAN LI

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. AMS Mathematics Subject Classification : 34K11, 39A10.
Keywords
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:
PPT Slide
Lager Image
A unique solution u of (1.1) is given by
PPT Slide
Lager Image
Motivated by the above result, we reconsider the existence of solution for impulsive fractional dynamic equations with delay on time scales
PPT Slide
Lager Image
where a ∈ ℝ + , ν ∈ (1, 2), q ∈ (0, 1), f :
PPT Slide
Lager Image
× ℝ × ℝ → ℝ is jointly continuous,
PPT Slide
Lager Image
= { t : t = aqn , n N 0 } ∪ {0}, N 0 = {0, 1, 2, · · ·}, Ik , Jk : ℝ → ℝ, tk satisfies 0 = t 0 < t 1 < · · · < tm < tm +1 = a , and α ( t ), β ( t ) ≤ t , u 0 , ū 0 are fixed real numbers. For t
PPT Slide
Lager Image
, we define the forward jump operator σ :
PPT Slide
Lager Image
PPT Slide
Lager Image
by σ ( t ) := inf{ s
PPT Slide
Lager Image
: s > t }. For any function υ , we define
PPT Slide
Lager Image
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 (
PPT Slide
Lager Image
,ℝ) be the Banach space of all continuous functions from
PPT Slide
Lager Image
to ℝ with the norm ∥ u C := sup{| u ( t )| : t
PPT Slide
Lager Image
} for u C (
PPT Slide
Lager Image
,ℝ)We also introduce the Banach space PC (
PPT Slide
Lager Image
,ℝ) = { u :
PPT Slide
Lager Image
→ℝ: u C (( tk , tk +1 ], ℝ), k = 0, 1, 2, . . . , m with the norm ∥ u PC := sup{| u ( t )| : t
PPT Slide
Lager Image
}}.
Let us recall the following known definitions. For more details see [2 , 13] .
Definition 2.1. For a function f :
PPT Slide
Lager Image
→ℝ, the nabla q-derivative of f is
PPT Slide
Lager Image
for all t
PPT Slide
Lager Image
╲{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
PPT Slide
Lager Image
╲{0}, q ∈ (0, 1), then
PPT Slide
Lager Image
If ν is not a positive integer, then
PPT Slide
Lager Image
We state several properties of the q -factorial function.
Proposition 2.3. ( i )
PPT Slide
Lager Image
  • (ii)
  • (iii)The nabla q-derivative of the q-factorial function with respect to t is
PPT Slide
Lager Image
  • (iv)The nabla q-derivative of the q-factorial function with respect to s is
PPT Slide
Lager Image
where β, γ ∈ℝ.
Definition 2.4. The q -Gamma function is defined by
PPT Slide
Lager Image
where α ∈ ℝ╲{. . . , −2, −1, 0}, q ∈ (0, 1), eq ( t ) =
PPT Slide
Lager Image
(1 − qnt ), eq (0) = 1.
The q -Beta function is defined by
PPT Slide
Lager Image
Proposition 2.5. ( i ) Γ q ( α + 1) =
PPT Slide
Lager Image
Γ q ( α ), Γ q (1) = 1, where α ∈ ℝ + .
PPT Slide
Lager Image
Definition 2.6. The fractional q -integral is defined by
PPT Slide
Lager Image
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
PPT Slide
Lager Image
and the class of all α -condensing maps F : Ω → X is denoted by
PPT Slide
Lager Image
. We remark that
PPT Slide
Lager Image
PPT Slide
Lager Image
and every F
PPT Slide
Lager Image
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 t 0 = 0, tm +1 = T ;
(iii) W ( t ) = { u ( t )| u W,t J ╲{ t 1 , · · · , tm }}, W (
PPT Slide
Lager Image
) = { u (
PPT Slide
Lager Image
)| u W } and W (
PPT Slide
Lager Image
) = { u (
PPT Slide
Lager Image
)| 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 :
PPT Slide
Lager Image
→ ℝ, define the norm
PPT Slide
Lager Image
where µ (
PPT Slide
Lager Image
) is the Lebesgue measure on
PPT Slide
Lager Image
. Let Lp (
PPT Slide
Lager Image
,ℝ) be the Banach space of all Lebesgue measurable functions m :
PPT Slide
Lager Image
→ ℝ with ∥ m L p (
PPT Slide
Lager Image
) < ∞.
Theorem 2.13 (Hölder’s inequality). Assume that 1 ≤ p, q ≤ ∞ and
PPT Slide
Lager Image
= 1. For any l Lp (
PPT Slide
Lager Image
,ℝ) and m Lq (
PPT Slide
Lager Image
,ℝ), lm L 1 (
PPT Slide
Lager Image
,ℝ) with lm L1 (
PPT Slide
Lager Image
) ≤ ∥ l Lp(
PPT Slide
Lager Image
) m Lq (
PPT Slide
Lager Image
) .
Lemma 2.14. Suppose u ( t ), b ( t ), g ( t ), f ( t ) are nonnegative on
PPT Slide
Lager Image
, b ( t ) is nondecreasing and locally integrable on
PPT Slide
Lager Image
, g ( t ), f ( t ) are nondecreasing and continuous on
PPT Slide
Lager Image
, a ∈ ℝ + , and h ( t ) := g ( t ) + f ( t ) ≤ M 0 <
PPT Slide
Lager Image
for any ν > 1.
If
PPT Slide
Lager Image
for any α , β :
PPT Slide
Lager Image
PPT Slide
Lager Image
with α ( t ) ≤ t,β (t) ≤ t, then
PPT Slide
Lager Image
Proof . Let
PPT Slide
Lager Image
Since α ( t ) ≤ t, β ( t ) ≤ t , we have
PPT Slide
Lager Image
then
PPT Slide
Lager Image
where h ( t ) := g ( t ) + f ( t ), h ( t ) is a nondecreasing continuous function.
Let
PPT Slide
Lager Image
for locally integrable functions ϕ . Then
PPT Slide
Lager Image
implies
PPT Slide
Lager Image
Let us prove that
PPT Slide
Lager Image
and Dnz ( t ) → 0 as n → +∞ for each t
PPT Slide
Lager Image
.
We know this relation (2.1) is true for n = 1. Assume that it is true for some n = k , that is
PPT Slide
Lager Image
If n = k + 1, then the induction hypothesis implies
PPT Slide
Lager Image
since h ( t ) is nondecreasing, it follows that
PPT Slide
Lager Image
By interchanging the order of integration (see [9] ), we have
PPT Slide
Lager Image
where the integral
PPT Slide
Lager Image
since ( t s )
PPT Slide
Lager Image
is decreasing about s , we have ( t )
PPT Slide
Lager Image
≤ ( t qντ )
PPT Slide
Lager Image
, ν > 1,and by Proposition 2.3 and Definition 2.4, we get
PPT Slide
Lager Image
The relation (2.1) is proved. By (2.1) and h ( t ) ≤ M 0 , we have
PPT Slide
Lager Image
Since M 0 <
PPT Slide
Lager Image
and by Proposition 2.5, we get
PPT Slide
Lager Image
as n → ∞, for t
PPT Slide
Lager Image
. Then the Lemma 2.14 is proved. ΋
Lemma 2.15. Let ν ∈ (1, 2) and h :
PPT Slide
Lager Image
→ ℝ be jointly continuous. A function u given by
PPT Slide
Lager Image
is the unique solution of the following impulsive problem on time scales
PPT Slide
Lager Image
where
PPT Slide
Lager Image
= { t : t = aqn , n N 0 }∪{0}, a ∈ ℝ + , q ∈ (0, 1), N 0 = {0, 1, 2, · · · }, k = 1, 2, 3, · · · , m, Ik , Jk : ℝ → ℝ, tk satisfy 0 = t 0 < t 1 < · · · < tm < tm +1 = a . σ :
PPT Slide
Lager Image
PPT Slide
Lager Image
is the forward jump operator σ ( t ) := inf{ s
PPT Slide
Lager Image
: s > t }. And for any function υ, we define
PPT Slide
Lager Image
Proof . Assume the general solution u of the Eq. (2.4) is given by
PPT Slide
Lager Image
where t 0 = 0, tm +1 = a . Then, we have
PPT Slide
Lager Image
Applying the cauchy conditions of (2.4), we get
PPT Slide
Lager Image
Next, using the impulsive condition of (2.4), we find that
PPT Slide
Lager Image
which by (2.7) implies
PPT Slide
Lager Image
Furthermore, using the impulsive condition of (2.4), we find that
PPT Slide
Lager Image
which implies
PPT Slide
Lager Image
So by (2.9), (2.11), we have
PPT Slide
Lager Image
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 :
PPT Slide
Lager Image
× ℝ × ℝ → ℝ is jointly continuous.
[H2] For arbitrary ( t, u, v ) ∈
PPT Slide
Lager Image
× ℝ × ℝ, there exist L 1 , L 2 > 0, such that
PPT Slide
Lager Image
[H3] There exist q 1 , q 2 ∈ (0, 1), real functions h L
PPT Slide
Lager Image
(
PPT Slide
Lager Image
, ℝ), y L
PPT Slide
Lager Image
(
PPT Slide
Lager Image
, ℝ) such that
PPT Slide
Lager Image
for all t
PPT Slide
Lager Image
and u 1 (·), u 2 (·) ∈ ℝ.
[H4] There exist constants L 3 , L 4 > 0, such that
PPT Slide
Lager Image
for all u,v ∈ ℝ, and k = 1, 2, · · · , m .
[H5] For arbitrary u ∈ ℝ, there exist constants M 1 , M 2 > 0, such that
PPT Slide
Lager Image
Theorem 3.1. Assume that [H1]-[H5] hold, and if L 1 + L 2 <
PPT Slide
Lager Image
, then the problem (1.3) has at least one solution on
PPT Slide
Lager Image
.
Proof . By Lemma 2.15, we define an operator
PPT Slide
Lager Image
: PC (
PPT Slide
Lager Image
, ℝ) → PC (
PPT Slide
Lager Image
,ℝ) by
PPT Slide
Lager Image
for t ∈ ( tk , tk +1 ], k = 0, 1, 2, · · · , m , where
PPT Slide
Lager Image
For the sake of convenience, we subdivide the proof into several steps.
Step 1.
PPT Slide
Lager Image
is continuous.
Let { un } be a sequence such that un u in PC (
PPT Slide
Lager Image
, ℝ). Then for each t ∈ ( tk , tk +1 ], by conditions [H3], [H4], we have
PPT Slide
Lager Image
Further, we can obtain
PPT Slide
Lager Image
Step 2.
PPT Slide
Lager Image
maps bounded sets into bounded sets in PC (
PPT Slide
Lager Image
,ℝ).
For each u Bη = { u PC (
PPT Slide
Lager Image
,ℝ) : ∥ u PC η }, t ∈ ( tk , tk +1 ], by [H2], [H5], we have
PPT Slide
Lager Image
Let := | u 0 | + a | ū 0 | + M 2
PPT Slide
Lager Image
( a σ ( ti )) + mM 1 + (1 − q )
PPT Slide
Lager Image
σ ( ti )(| ū 0 | + iM 2 ) +
PPT Slide
Lager Image
, we get
PPT Slide
Lager Image
Step 3.
PPT Slide
Lager Image
maps bounded sets into equicontinuous sets of PC (
PPT Slide
Lager Image
, ℝ). It is easy to know
PPT Slide
Lager Image
is equicontinuous on interval ( tk , tk +1 ], k = 1, 2, · · · , m . For any 0 ≤ s 1 < s 2 t 1 , u Bη = { u PC (
PPT Slide
Lager Image
, ℝ) : ∥ u PC η }, we have
PPT Slide
Lager Image
considering s 2 s 1 , we have
PPT Slide
Lager Image
Thus, we find that
PPT Slide
Lager Image
is equicontinuous on
PPT Slide
Lager Image
.
Step 4. Now it remains to show that the set
PPT Slide
Lager Image
is bounded.
Without loss of generality, for any u E (
PPT Slide
Lager Image
), t ∈ ( tk , tk +1 ], by [H2], [H5], we have
PPT Slide
Lager Image
where k = 0, 1, 2, · · · , m .
By Lemma 2.14, there exists a Mk > 0 such that
PPT Slide
Lager Image
Set M =
PPT Slide
Lager Image
Mk , thus for every t
PPT Slide
Lager Image
, we get
PPT Slide
Lager Image
This shows that the set E (
PPT Slide
Lager Image
) is bounded.
As a consequence of Schaefer’s fixed point theorem, we know that
PPT Slide
Lager Image
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
PPT Slide
Lager Image
then the problem (1.3) has an unique solution on
PPT Slide
Lager Image
.
Proof . Consider the operator
PPT Slide
Lager Image
: PC (
PPT Slide
Lager Image
, ℝ) → PC (
PPT Slide
Lager Image
, ℝ) defined as (3.1), and transform the problem (1.3) into a fixed point problem of
PPT Slide
Lager Image
.
Step 1.
PPT Slide
Lager Image
u PC (
PPT Slide
Lager Image
, ℝ) for every u PC (
PPT Slide
Lager Image
, ℝ).
If t = 0, for any δ > 0, we have
PPT Slide
Lager Image
then
PPT Slide
Lager Image
Thus, we find that
PPT Slide
Lager Image
u is continuous at 0. It is easy to see that
PPT Slide
Lager Image
u C (( tk , tk +1 ],ℝ), k = 0, 1, · · · , m .
From the above discussion, we get
PPT Slide
Lager Image
u PC (
PPT Slide
Lager Image
,ℝ) for every u PC (
PPT Slide
Lager Image
,ℝ).
Step 2.
PPT Slide
Lager Image
is a contraction operator on PC (
PPT Slide
Lager Image
,ℝ).
In fact, for arbitrary u 1 , u 2 PC (
PPT Slide
Lager Image
,ℝ), by [H3], [H4] and Theorem 2.13, we obtain
PPT Slide
Lager Image
Thus, due to (3.2), we know that
PPT Slide
Lager Image
is a contraction mapping on PC (
PPT Slide
Lager Image
,ℝ).
By applying the well-known Banach’s contraction mapping principle, we get that the operator
PPT Slide
Lager Image
has a unique fixed point on PC (
PPT Slide
Lager Image
,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 ) ∈
PPT Slide
Lager Image
×ℝ×ℝ, there exist
PPT Slide
Lager Image
and q 1 , q 2 ∈ [0, 1) such that
PPT Slide
Lager Image
[H3] ′ There exist
PPT Slide
Lager Image
such that
PPT Slide
Lager Image
for each t
PPT Slide
Lager Image
, and all u 1 , u 2