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