Abstract

We present some results for the asymptotic stability of solutions for nonlinear fractional difference equations involvingRiemann-Liouville-likedifference operator. The results are obtained by using Krasnoselskii's fixed point theorem and discrete Arzela-Ascoli's theorem. Three examples are also provided to illustrate our main results.

1. Introduction

In this paper we consider the asymptotic stability of solutions for nonlinear fractional difference equations:Δ𝛼π‘₯(𝑑)=𝑓(𝑑+𝛼,π‘₯(𝑑+𝛼)),π‘‘βˆˆπ‘0Ξ”,0<𝛼≀1,π›Όβˆ’1π‘₯(𝑑)|𝑑=0=π‘₯0,(1.1) where Δ𝛼 is aRiemann-Liouville-likediscrete fractional difference, π‘“βˆΆ[0,+∞)×𝑅→𝑅 is continuous with respect to 𝑑 and π‘₯, π‘π‘Ž={π‘Ž,π‘Ž+1,π‘Ž+2,…}.

Fractional differential equations have received increasing attention during recent years since these equations have been proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Most of the present works were focused on fractional differential equations, see [1–12] and the references therein. However, very little progress has been made to develop the theory of the analogous fractional finite difference equation [13–19].

Due to the lack of geometry interpretation of the fractional derivatives, it is difficult to find a valid tool to analyze the stability of fractional difference equations. In the case that it is difficult to employ Liapunov’s direct method, fixed point theorems are usually considered in stability [20–25]. Motivated by this idea, in this paper, we discuss asymptotic stability of nonlinear fractional difference equations by using Krasnoselskii’s fixed point theorem and discrete Arzela-Ascoli’s theorem. Different from our previous work [18], in this paper, the sufficient conditions of attractivity are irrelevant to the initial value π‘₯0.

2. Preliminaries

In this section, we introduce preliminary facts of discrete fractional calculus. For more details, see [14].

Definition 2.1 (see [14]). Let 𝜈>0. The 𝜈-th fractional sum π‘₯ is defined by Ξ”βˆ’πœˆ1𝑓(𝑑)=Ξ“(𝜈)π‘‘βˆ’πœˆξ“π‘ =π‘Ž(π‘‘βˆ’π‘ βˆ’1)(πœˆβˆ’1)𝑓(𝑠),(2.1) where 𝑓 is defined for 𝑠=π‘Žβ€‰mod (1) and Ξ”βˆ’πœˆπ‘“ is defined for 𝑑=(π‘Ž+𝜈) mod (1), and 𝑑(𝜈)=Ξ“(𝑑+1)/Ξ“(π‘‘βˆ’πœˆ+1). The fractional sum Ξ”βˆ’πœˆ maps functions defined on π‘π‘Ž to functions defined on π‘π‘Ž+𝜈.

Definition 2.2 (see [14]). Let πœ‡>0 and π‘šβˆ’1<πœ‡<π‘š, where π‘š denotes a positive integer, π‘š=βŒˆπœ‡βŒ‰, βŒˆβ‹…βŒ‰ ceiling of number. Set 𝜈=π‘šβˆ’πœ‡. The πœ‡-th fractional difference is defined as Ξ”πœ‡π‘“(𝑑)=Ξ”π‘šβˆ’πœˆπ‘“(𝑑)=Ξ”π‘š(Ξ”βˆ’πœˆπ‘“(𝑑)).(2.2)

Theorem 2.3 (see [15]). Let 𝑓 be a real-value function defined on π‘π‘Ž and πœ‡,𝜈>0, then the following equalities hold:(i)Ξ”βˆ’πœˆ[Ξ”βˆ’πœ‡π‘“(𝑑)]=Ξ”βˆ’(πœ‡+𝜈)𝑓(𝑑)=Ξ”βˆ’πœ‡[Ξ”βˆ’πœˆπ‘“(𝑑)]; (ii)Ξ”βˆ’πœˆΞ”π‘“(𝑑)=Ξ”Ξ”βˆ’πœˆπ‘“(𝑑)βˆ’(π‘‘βˆ’π‘Ž)(πœˆβˆ’1)Ξ“(𝜈)𝑓(π‘Ž).

Lemma 2.4 (see [15]). Let πœ‡β‰ 1 and assume πœ‡+𝜈+1 is not a nonpositive integer, then Ξ”βˆ’πœˆπ‘‘(πœ‡)=Ξ“(πœ‡+1)𝑑Γ(πœ‡+𝜈+1)(πœ‡+𝜈).(2.3)

Lemma 2.5 (see [15]). Assume that the following factorial functions are well defined:(i)If 0<𝛼<1, then 𝑑(𝛼𝛾)β‰₯(𝑑(𝛾))𝛼;(ii)𝑑(𝛽+𝛾)=(π‘‘βˆ’π›Ύ)(𝛽)𝑑(𝛾).

Lemma 2.6 (see [13]). Let πœ‡>0 be noninteger, π‘š=βŒˆπœ‡βŒ‰, βŒˆβ‹…βŒ‰, 𝜈=π‘šβˆ’πœ‡, thus one hasπ‘‘βˆ’πœ‡ξ“π‘ =π‘Ž+𝜈(π‘‘βˆ’π‘ βˆ’1)(πœ‡βˆ’1)=(π‘‘βˆ’π‘Žβˆ’πœˆ)(πœ‡)πœ‡.(2.4)

Lemma 2.7. The equivalent fractional Taylor’s difference formula of (1.1) is π‘₯π‘₯(𝑑)=0𝑑Γ(𝛼)(π›Όβˆ’1)+1Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)𝑓(𝑠+𝛼,π‘₯(𝑠+𝛼)),π‘‘βˆˆπ‘π›Ό.(2.5)

Proof. Apply the Ξ”βˆ’π›Ό operator to each side of the first formula of (1.1) to obtain Ξ”βˆ’π›ΌΞ”π›Όπ‘₯(𝑑)=Ξ”βˆ’π›Όπ‘“(𝑑+𝛼,π‘₯(𝑑+𝛼)),π‘‘βˆˆπ‘π›Ό.(2.6)
Apply Theorem 2.3 to the left-hand side of (2.6) to obtain Ξ”βˆ’π›ΌΞ”π›Όπ‘₯(𝑑)=Ξ”βˆ’π›ΌΞ”Ξ”βˆ’(1βˆ’π›Ό)π‘₯(𝑑)=Ξ”Ξ”βˆ’π›ΌΞ”βˆ’(1βˆ’π›Ό)𝑑π‘₯(𝑑)βˆ’(π›Όβˆ’1)π‘₯Ξ“(𝛼)π‘₯(π›Όβˆ’1)=π‘₯(𝑑)βˆ’0𝑑Γ(𝛼)(π›Όβˆ’1).(2.7)
So, applying Definition 2.1 to the right-hand side of (2.6), for π‘‘βˆˆπ‘π›Ό we obtain (2.5). The recursive iteration to this Taylor’s difference formula implies that (2.5) represents the unique solution of the IVP (1.1). This completes the proof.

Lemma 2.8 (see [4, (1.5.15)]). The quotient expansion of two gamma functions at infinityisΞ“(𝑧+π‘Ž)Ξ“(𝑧+𝑏)=π‘§π‘Žβˆ’π‘ξ‚ƒξ‚€11+𝑂𝑧,ξ€·||||ξ€Έ.arg(𝑧+π‘Ž)<πœ‹,|𝑧|⟢∞(2.8)

Corollary 2.9. One has 𝑑(βˆ’π›½)>(𝑑+𝛼)(βˆ’π›½)for𝛼,𝛽,𝑑>0.(2.9)

Proof. According to Lemma 2.8, 𝑑(βˆ’π›½)(𝑑+𝛼)(βˆ’π›½)=Ξ“(𝑑+1)β‹…Ξ“(𝑑+𝛽+1)Ξ“(𝑑+𝛼+𝛽+1)=Ξ“(𝑑+𝛼+1)Ξ“(𝑑+1)⋅ΓΓ(𝑑+𝛼+1)(𝑑+𝛼+𝛽+1)Ξ“(𝑑+𝛽+1)=π‘‘βˆ’π›Όξ‚ƒξ‚€11+𝑂𝑑⋅(𝑑+𝛽)𝛼11+𝑂=𝛽𝑑+𝛽1+𝑑𝛼11+𝑂𝑑11+𝑂𝑑+𝛽>1.(2.10) Then, 𝑑(βˆ’π›½)>(𝑑+𝛼)(βˆ’π›½) for 𝛼,𝛽,𝑑>0. This completes the proof.

Definition 2.10. The solution π‘₯=πœ‘(𝑑) of the IVP (1.1) is said to be(i) stable if for any πœ€>0 and 𝑑0βˆˆπ‘…+, there exists a 𝛿=𝛿(𝑑0,πœ€)>0 such that||π‘₯𝑑,π‘₯0,𝑑0ξ€Έ||βˆ’πœ‘(𝑑)<πœ€(2.11) for |π‘₯0βˆ’πœ‘(𝑑0)|≀𝛿(𝑑0,πœ€) and all 𝑑β‰₯𝑑0;(ii) attractive if there exists πœ‚(𝑑0)>0 such that β€–π‘₯0β€–β‰€πœ‚ implieslimπ‘‘β†’βˆžπ‘₯𝑑,π‘₯0,𝑑0ξ€Έ=0;(2.12)(iii) asymptotically stable if it is stable and attractive.
The space π‘™βˆžπ‘›0 is the set of real sequences defined on the set of positive integers where any individual sequence is bounded with respect to the usual supremum norm. It is well known that under the supremum norm π‘™βˆžπ‘›0 is a Banach space [26].

Definition 2.11 (see [27]). A set Ξ© of sequences in π‘™βˆžπ‘›0 is uniformly Cauchy (or equi-Cauchy), if for every πœ€>0, there exists an integer 𝑁 such that |π‘₯(𝑖)βˆ’π‘₯(𝑗)|<πœ€, whenever 𝑖,𝑗>𝑁 for any π‘₯={π‘₯(𝑛)} in Ξ©.

Theorem 2.12 (see [27, (discrete Arzela-Ascoli’s theorem)]). A bounded, uniformly Cauchy subset Ξ© of π‘™βˆžπ‘›0 is relatively compact.

Theorem 2.13 (see [20, (Krasnoselskii’s fixed point theorem)]). Let 𝑆 be a nonempty, closed, convex, and bounded subset of the Banach space 𝑋 and let π΄βˆΆπ‘‹β†’π‘‹ and π΅βˆΆπ‘†β†’π‘‹ be two operators such that(a)A is a contraction with constant 𝐿<1,(b)B is continuous, 𝐡𝑆 resides in a compact subset of 𝑋,(c)[π‘₯=𝐴π‘₯+𝐡𝑦,π‘¦βˆˆπ‘†]β‡’π‘₯βˆˆπ‘†.Then the operator equation 𝐴π‘₯+𝐡π‘₯=π‘₯ has a solution in 𝑆.

3. Main Results

Let π‘™βˆžπ›Ό be the set ofall real sequencesπ‘₯={π‘₯(𝑑)}βˆžπ‘‘=𝛼 with norm β€–π‘₯β€–=supπ‘‘βˆˆπ‘π›Ό|π‘₯(𝑑)|, then π‘™βˆžπ›Ό is a Banach space.

Define the operatorπ‘₯𝑃π‘₯(𝑑)=0𝑑Γ(𝛼)(π›Όβˆ’1)+1Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)π‘₯𝑓(𝑠+𝛼,π‘₯(𝑠+𝛼)),𝐴π‘₯(𝑑)=0𝑑Γ(𝛼)(π›Όβˆ’1),1𝐡π‘₯(𝑑)=Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)𝑓(𝑠+𝛼,π‘₯(𝑠+𝛼)),π‘‘βˆˆπ‘π›Ό.(3.1) Obviously, 𝑃π‘₯=𝐴π‘₯+𝐡π‘₯, the operator 𝐴 is a contraction with the constant 0, which implies that condition (a) of Theorem 2.13 holds, and π‘₯(𝑑) is a solution of (1.1) if it is a fixed point of the operator 𝑃.

Lemma 3.1. Assume that the following condition is satisfied:
(𝐻1) there exist constants 𝛽1∈(𝛼,1) and 𝐿1β‰₯0 such that ||||𝑓(𝑑,π‘₯(𝑑))≀𝐿1𝑑(βˆ’π›½1)forπ‘‘βˆˆπ‘π›Ό.(3.2) Then the operator 𝐡 is continuous and 𝐡𝑆1 is a compact subset of 𝑅 for π‘‘βˆˆπ‘π›Ό+𝑛1, where 𝑆1=ξ€½||||π‘₯(𝑑)∢π‘₯(𝑑)≀𝑑(βˆ’π›Ύ1)forπ‘‘βˆˆπ‘π›Ό+𝑛1ξ€Ύ,(3.3)𝛾1=(βˆ’1/2)(π›Όβˆ’π›½1), and 𝑛1βˆˆπ‘ satisfies that ||π‘₯0||ξ€·Ξ“(𝛼)𝛼+𝑛1+𝛾1ξ€Έ((1/2)(𝛼+𝛽1)βˆ’1)+𝐿1Ξ“ξ€·1βˆ’π›½1ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½1𝛼+𝑛1+𝛾1ξ€Έ(βˆ’π›Ύ1)≀1.(3.4)

Proof. For π‘‘βˆˆπ‘π›Ό, apply Lemma 2.8 and 𝛾1>0, 𝑑(βˆ’π›Ύ1)=Ξ“(𝑑+1)Γ𝑑+𝛾1ξ€Έ+1=π‘‘βˆ’π›Ύ111+𝑂𝑑,(3.5) and we have that 𝑑(βˆ’π›Ύ1)β†’0 as π‘‘β†’βˆž, then there exists a 𝑛1βˆˆπ‘ such that inequality (3.4) holds, which implies that the set 𝑆1 exists.
We firstly show that 𝐡 maps 𝑆1 in 𝑆1.
It is easy to know that 𝑆1 is a closed, bounded, and convex subset of 𝑅.
Apply condition (𝐻1), Lemma 2.5, Corollary 2.9 and (3.4), for π‘‘βˆˆπ‘π›Ό+𝑛1, we have ||||≀1𝐡π‘₯(𝑑)Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)||||≀1𝑓(𝑠+𝛼,π‘₯(𝑠+𝛼))Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)𝐿1(𝑠+𝛼)(βˆ’π›½1)=𝐿1Ξ”βˆ’π›Ό(𝑑+𝛼)(βˆ’π›½1)=𝐿1Ξ“ξ€·1βˆ’π›½1ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½1ξ€Έ(𝑑+𝛼)(π›Όβˆ’π›½1)<𝐿1Ξ“ξ€·1βˆ’π›½1ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½1𝑑(π›Όβˆ’π›½1)=𝐿1Ξ“ξ€·1βˆ’π›½1ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½1𝑑+𝛾1ξ€Έ(βˆ’π›Ύ1)𝑑(βˆ’π›Ύ1)≀𝐿1Ξ“ξ€·1βˆ’π›½1ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½1𝛼+𝑛1+𝛾1ξ€Έ(βˆ’π›Ύ1)𝑑(βˆ’π›Ύ1)≀𝑑(βˆ’π›Ύ1),(3.6) which implies that 𝐡𝑆1βŠ‚π‘†1 for π‘‘βˆˆπ‘π›Ό+𝑛1.
Nextly, we show that 𝐡 is continuous on 𝑆1.
Let πœ€>0 be given then there exist 𝑇1βˆˆπ‘ and 𝑇1β‰₯𝑛1 such that π‘‘βˆˆπ‘π›Ό+𝑇1 implies that 𝐿1Ξ“ξ€·1βˆ’π›½1ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½1𝑑(π›Όβˆ’π›½1)<πœ€2.(3.7)
Let {π‘₯𝑛} be a sequence such that π‘₯𝑛→π‘₯. For π‘‘βˆˆ{𝛼+𝑛1,𝛼+𝑛1+1,…,𝛼+𝑇1βˆ’1}, applying the continuity of 𝑓 and Lemma 2.6, we have ||𝐡π‘₯𝑛||≀1(𝑑)βˆ’π΅π‘₯(𝑑)Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)||𝑓𝑠+𝛼,π‘₯𝑛||(𝑠+𝛼)βˆ’π‘“(𝑠+𝛼,π‘₯(𝑠+𝛼))≀maxξ€½π‘ βˆˆ0,1,…,𝑇1ξ€Ύβˆ’1||𝑓𝑠+𝛼,π‘₯𝑛||Γ—1(𝑠+𝛼)βˆ’π‘“(𝑠+𝛼,π‘₯(𝑠+𝛼))Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)=𝑑(𝛼)Ξ“(𝛼+1)maxξ€½π‘ βˆˆ0,1,…,𝑇1ξ€Ύβˆ’1||𝑓𝑠+𝛼,π‘₯𝑛||≀(𝑠+𝛼)βˆ’π‘“(𝑠+𝛼,π‘₯(𝑠+𝛼))𝛼+𝑇1ξ€Έβˆ’1(𝛼)Ξ“(𝛼+1)maxξ€½π‘ βˆˆ0,1,…,𝑇1ξ€Ύβˆ’1||𝑓𝑠+𝛼,π‘₯𝑛||(𝑠+𝛼)βˆ’π‘“(𝑠+𝛼,π‘₯(𝑠+𝛼))⟢0asπ‘›βŸΆβˆž.(3.8)
For π‘‘βˆˆπ‘π›Ό+𝑇1, ||𝐡π‘₯𝑛||≀1(𝑑)βˆ’π΅π‘₯(𝑑)Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)ξ€Ί||𝑓𝑠+𝛼,π‘₯𝑛||+||||≀(𝑠+𝛼)𝑓(𝑠+𝛼,π‘₯(𝑠+𝛼))2𝐿1Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)(𝑠+𝛼)(βˆ’π›½1)=2𝐿1Ξ”βˆ’π›Ό(𝑑+𝛼)(βˆ’π›½1)=2𝐿1Ξ“ξ€·1βˆ’π›½1ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½1ξ€Έ(𝑑+𝛼)(π›Όβˆ’π›½1)<2𝐿1Ξ“ξ€·1βˆ’π›½1ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½1𝑑(π›Όβˆ’π›½1)<πœ€.(3.9)
Thus, for all π‘‘βˆˆπ‘π›Ό+𝑛1, we have ||𝐡π‘₯𝑛||(𝑑)βˆ’π΅π‘₯(𝑑)⟢0asπ‘›β†’βˆž.(3.10) which implies that 𝐡 is continuous.
Lastly, we show that 𝐡𝑆1 is relatively compact.
Let 𝑑1,𝑑2βˆˆπ‘π›Ό+𝑇1 and 𝑑2>𝑑1, thus we have ||𝑑𝐡π‘₯2ξ€Έξ€·π‘‘βˆ’π΅π‘₯1ξ€Έ||=|||||1Ξ“(𝛼)𝑑2βˆ’π›Όξ“π‘ =0𝑑2ξ€Έβˆ’π‘ βˆ’1(π›Όβˆ’1)βˆ’1𝑓(𝑠+𝛼,π‘₯(𝑠+𝛼))Ξ“(𝛼)𝑑1βˆ’π›Όξ“π‘ =0𝑑1ξ€Έβˆ’π‘ βˆ’1(π›Όβˆ’1)|||||≀1𝑓(𝑠+𝛼,π‘₯(𝑠+𝛼))Ξ“(𝛼)𝑑2βˆ’π›Όξ“π‘ =0𝑑2ξ€Έβˆ’π‘ βˆ’1(π›Όβˆ’1)||||+1𝑓(𝑠+𝛼,π‘₯(𝑠+𝛼))Ξ“(𝛼)𝑑1βˆ’π›Όξ“π‘ =0𝑑1ξ€Έβˆ’π‘ βˆ’1(π›Όβˆ’1)||||≀𝐿𝑓(𝑠+𝛼,π‘₯(𝑠+𝛼))1Ξ“ξ€·1βˆ’π›½1ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½1𝑑2ξ€Έ+𝛼(π›Όβˆ’π›½1)+𝐿1Ξ“ξ€·1βˆ’π›½1ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½1𝑑1ξ€Έ+𝛼(π›Όβˆ’π›½1)<𝐿1Ξ“ξ€·1βˆ’π›½1ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½1𝑑2(π›Όβˆ’π›½1)+𝐿1Ξ“ξ€·1βˆ’π›½1ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½1𝑑1(π›Όβˆ’π›½1)<πœ€.(3.11) Thus, {𝐡π‘₯∢π‘₯βˆˆπ‘†1} is a bounded and uniformly Cauchy subset by Definition 2.11, and 𝐡𝑆1 is relatively compact by means of Theorem 2.12. This completes the proof.

Lemma 3.2. Assume that condition (𝐻1) holds, then a solution of (1.1) is in 𝑆1 for π‘‘βˆˆπ‘π›Ό+𝑛1.

Proof. Notice if that π‘₯(𝑑) is a fixed point of 𝑃, then it is a solution of (1.1). To prove this, it remains to show that, for fixed π‘¦βˆˆπ‘†1, π‘₯=𝐴π‘₯+𝐡𝑦⇒π‘₯βˆˆπ‘†1 holds.
If π‘₯=𝐴π‘₯+𝐡𝑦, applying condition (𝐻1) and (3.4), for π‘‘βˆˆπ‘π›Ό+𝑛1, we have ||||≀||||+||||≀||π‘₯π‘₯(𝑑)𝐴π‘₯(𝑑)𝐡𝑦(𝑑)0||𝑑Γ(𝛼)(π›Όβˆ’1)+1Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)||||≀||π‘₯𝑓(𝑠+𝛼,𝑦(𝑠+𝛼))0||𝑑Γ(𝛼)(π›Όβˆ’1)+𝐿1Ξ“ξ€·1βˆ’π›½1ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½1ξ€Έ(𝑑+𝛼)(π›Όβˆ’π›½1)<||π‘₯0||Γ𝑑(𝛼)(π›Όβˆ’1)+𝐿1Ξ“ξ€·1βˆ’π›½1ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½1𝑑(π›Όβˆ’π›½1)=||π‘₯0||ξ€·Ξ“(𝛼)𝑑+𝛾1ξ€Έ((1/2)(𝛼+𝛽1)βˆ’1)+𝐿1Ξ“ξ€·1βˆ’π›½1ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½1𝑑+𝛾1ξ€Έ(βˆ’π›Ύ1)𝑑(βˆ’π›Ύ1)≀||π‘₯0||Ξ“ξ€·(𝛼)𝛼+𝑛1+𝛾1ξ€Έ((1/2)(𝛼+𝛽1)βˆ’1)+𝐿1Ξ“ξ€·1βˆ’π›½1ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½1𝛼+𝑛1+𝛾1ξ€Έ(βˆ’π›Ύ1)𝑑(βˆ’π›Ύ1)≀𝑑(βˆ’π›Ύ1).(3.12) Thus, π‘₯(𝑑)βˆˆπ‘†1 for π‘‘βˆˆπ‘π›Ό+𝑛1. According to Theorem 2.13 and Lemma 3.1, there exists a π‘₯βˆˆπ‘†1 such that π‘₯=𝐴π‘₯+𝐡π‘₯, that is, 𝑃 has a fixed point in 𝑆1 which is a solution of (1.1) for π‘‘βˆˆπ‘π›Ό+𝑛1. This completes the proof.

Theorem 3.3. Assume that condition (𝐻1) holds, then the solutions of (1.1) is attractive.

Proof. By Lemma 3.2, the solutions of (1.1) exist and are in 𝑆1. All functions π‘₯(𝑑) in 𝑆1 tend to 0 as π‘‘β†’βˆž. Then the solutions of (1.1) tend to zero as π‘‘β†’βˆž. This completes the proof.

Theorem 3.4. Assume that the following condition is satisfied:
(𝐻2) there exist constants 𝛽2∈(𝛼,1) and 𝐿2β‰₯0 such that ||||𝑓(𝑑,π‘₯(𝑑))βˆ’π‘“(𝑑,𝑦(𝑑))≀𝐿2𝑑(βˆ’π›½2)β€–π‘₯βˆ’π‘¦β€–forπ‘‘βˆˆπ‘π›Ό.(3.13) Then the solutions of (1.1) are stable provided that πΏπ‘βˆΆ=2Ξ“ξ€·(1+𝛼)Ξ“1βˆ’π›½2ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½2ξ€ΈΞ“ξ€·1+𝛽2ξ€Έ<1.(3.14)

Proof. Let π‘₯(𝑑) be a solution of (1.1), and let Μƒπ‘₯(𝑑) be a solution of (1.1) satisfying the initial value condition Μƒπ‘₯(0)=Μƒπ‘₯0. For π‘‘βˆˆπ‘π›Ό, applying condition (𝐻2), we have ||||≀𝑑π‘₯(𝑑)βˆ’Μƒπ‘₯(𝑑)(π›Όβˆ’1)||π‘₯Ξ“(𝛼)0βˆ’Μƒπ‘₯0||+1Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)Γ—||||≀𝑑𝑓(𝑠+𝛼,π‘₯(𝑠+𝛼))βˆ’π‘“(𝑠+𝛼,Μƒπ‘₯(𝑠+𝛼))(π›Όβˆ’1)||π‘₯Ξ“(𝛼)0βˆ’Μƒπ‘₯0||+𝐿2Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)(𝑠+𝛼)(βˆ’π›½2)β€–=𝑑π‘₯βˆ’Μƒπ‘₯β€–(π›Όβˆ’1)Ξ“||π‘₯(𝛼)0βˆ’Μƒπ‘₯0||+𝐿2Ξ”βˆ’π›Ό(𝑑+𝛼)(βˆ’π›½2)=𝑑‖π‘₯βˆ’Μƒπ‘₯β€–(π›Όβˆ’1)||π‘₯Ξ“(𝛼)0βˆ’Μƒπ‘₯0||+𝐿2Ξ“ξ€·1βˆ’π›½2ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½2ξ€Έ(𝑑+𝛼)(π›Όβˆ’π›½2)≀𝛼‖π‘₯βˆ’Μƒπ‘₯β€–(π›Όβˆ’1)||π‘₯Ξ“(𝛼)0βˆ’Μƒπ‘₯0||+𝐿2Ξ“ξ€·1βˆ’π›½2ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½2𝛼(π›Όβˆ’π›½2)||π‘₯β€–π‘₯βˆ’Μƒπ‘₯β€–=𝛼0βˆ’Μƒπ‘₯0||+𝐿2Ξ“ξ€·(1+𝛼)Ξ“1βˆ’π›½2ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½2ξ€ΈΞ“ξ€·1+𝛽2ξ€Έ||π‘₯β€–π‘₯βˆ’Μƒπ‘₯β€–=𝛼0βˆ’Μƒπ‘₯0||+𝑐‖π‘₯βˆ’Μƒπ‘₯β€–,(3.15) which implies that 𝛼‖π‘₯βˆ’Μƒπ‘₯‖≀||π‘₯1βˆ’π‘0βˆ’Μƒπ‘₯0||.(3.16)
For any given πœ€>0, let 𝛿=((1βˆ’π‘)/𝛼)πœ€, |π‘₯0βˆ’Μƒπ‘₯0|<𝛿 follows that β€–π‘₯βˆ’Μƒπ‘₯β€–<πœ€, which yields that the solutions of (1.1) are stable. This completes the proof.

Theorem 3.5. Assume that conditions (𝐻1) and (𝐻2) hold, then the solutions of (1.1) are asymptotically stable provided that (3.14) holds.
Theorem 3.5 is the simple consequence of Theorems 3.3 and 3.4.

Theorem 3.6. Assume that the following condition is satisfied:
(𝐻3) there exist constants 𝛽3∈(𝛼,(1/2)(1+𝛼)), 𝛾2=(1/2)(1βˆ’π›Ό), and 𝐿3β‰₯0 such that ||||𝑓(𝑑,π‘₯(𝑑))≀𝐿3𝑑+𝛾2ξ€Έ(βˆ’π›½3)||||π‘₯(𝑑)forπ‘‘βˆˆN𝛼.(3.17) Then the solutions of (1.1) is attractive.

Proof. Set 𝑆2=ξ€½||||π‘₯(𝑑)∢π‘₯(𝑑)≀𝑑(βˆ’π›Ύ2)forπ‘‘βˆˆπ‘π›Ό+𝑛2ξ€Ύ,(3.18) where 𝑛2βˆˆπ‘ satisfies that ||π‘₯0||ξ€·Ξ“(𝛼)𝛼+𝑛2+𝛾2ξ€Έ(βˆ’π›Ύ2)+𝐿3Ξ“ξ€·1βˆ’π›½3βˆ’π›Ύ2ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½3βˆ’π›Ύ2𝛼+𝑛2+𝛾2ξ€Έ(π›Όβˆ’π›½3)≀1.(3.19)
We first prove condition (c) of Theorem 2.13, that is, for fixed π‘¦βˆˆπ‘†2 and for all π‘₯βˆˆπ‘…, π‘₯=𝐴π‘₯+𝐡𝑦⇒π‘₯βˆˆπ‘†2 holds.
If π‘₯=𝐴π‘₯+𝐡𝑦, applying condition (𝐻3) and (3.19), for π‘‘βˆˆπ‘π›Ό+𝑛2, we have ||||≀||||+||||≀||π‘₯π‘₯(𝑑)𝐴π‘₯(𝑑)𝐡𝑦(𝑑)0||𝑑Γ(𝛼)(π›Όβˆ’1)+1Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)||||≀||π‘₯𝑓(𝑠+𝛼,𝑦(𝑠+𝛼))0||𝑑Γ(𝛼)(π›Όβˆ’1)+1Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)𝐿3𝑠+𝛼+𝛾2ξ€Έ(βˆ’π›½3)||||≀||π‘₯𝑦(𝑠+𝛼)0||Γ𝑑(𝛼)(π›Όβˆ’1)+𝐿3Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)𝑠+𝛼+𝛾2ξ€Έ(βˆ’π›½3)(𝑠+𝛼)(βˆ’π›Ύ2)≀||π‘₯0||𝑑Γ(𝛼)(π›Όβˆ’1)+𝐿3Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)(𝑠+𝛼)(βˆ’π›½3βˆ’π›Ύ2)≀||π‘₯0||𝑑Γ(𝛼)(π›Όβˆ’1)+𝐿3Ξ“ξ€·1βˆ’π›½3βˆ’π›Ύ2ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½3βˆ’π›Ύ2ξ€Έ(𝑑+𝛼)(π›Όβˆ’π›½3βˆ’π›Ύ2)<||π‘₯0||𝑑Γ(𝛼)(π›Όβˆ’1)+𝐿3Ξ“ξ€·1βˆ’π›½3βˆ’π›Ύ2ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½3βˆ’π›Ύ2𝑑(π›Όβˆ’π›½3βˆ’π›Ύ2)=||π‘₯0||ξ€·Ξ“(𝛼)𝑑+𝛾2ξ€Έ(βˆ’π›Ύ2)+𝐿3Ξ“ξ€·1βˆ’π›½3βˆ’π›Ύ2ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½3βˆ’π›Ύ2𝑑+𝛾2ξ€Έ(π›Όβˆ’π›½3)𝑑(βˆ’π›Ύ2)≀||π‘₯0||ξ€·Ξ“(𝛼)𝛼+𝑛2+𝛾2ξ€Έ(βˆ’π›Ύ2)+𝐿3Ξ“ξ€·1βˆ’π›½3βˆ’π›Ύ2ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½3βˆ’π›Ύ2𝛼+𝑛2+𝛾2ξ€Έ(π›Όβˆ’π›½3)𝑑(βˆ’π›Ύ2)≀𝑑(βˆ’π›Ύ2).(3.20) Thus, condition (c) of Theorem 2.13 holds.
The proof of condition (b) of Theorem 2.13 is similar to that of Lemma 3.1, and we omit it. Therefore, 𝑃 has a fixed point in 𝑆2 by using Theorem 2.13, that is, the IVP (1.1) has a solution in 𝑆2. Moreover, all functions in 𝑆2 tend to 0 as π‘‘β†’βˆž, then the solution of (1.1) tends to zero as π‘‘β†’βˆž, which shows that the zero solution of (1.1) is attractive. This completes the proof.

Theorem 3.7. Assume that conditions (𝐻2) and (𝐻3) hold, then the solutions of (1.1) are asymptotically stable provided that (3.14) holds.

Theorem 3.8. Assume that the following condition is satisfied:
(𝐻4) there exist constants πœ‚βˆˆ(0,1),𝛽4∈(𝛼,(2+π›Όπœ‚)/(2+πœ‚)), and 𝐿4β‰₯0 such that ||||𝑓(𝑑,π‘₯(𝑑))≀𝐿4(𝑑+1)(βˆ’π›½4)||||π‘₯(𝑑)πœ‚forπ‘‘βˆˆπ‘π›Ό.(3.21) Then the solutions of (1.1) is attractive.

Proof. Set 𝑆3=ξ€½||||π‘₯(𝑑)∢π‘₯(𝑑)≀𝑑(βˆ’π›Ύ3)forπ‘‘βˆˆπ‘π›Ό+𝑛3ξ€Ύ,(3.22) where 𝛾3=(1/2)(𝛽4βˆ’π›Ό), and 𝑛3βˆˆπ‘ satisfies that ||π‘₯0||ξ€·Ξ“(𝛼)𝛼+𝑛3+𝛾3ξ€Έ(π›Όβˆ’1+𝛾3)+𝐿4Ξ“ξ€·1βˆ’π›½4βˆ’π›Ύ3πœ‚ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½4βˆ’π›Ύ3πœ‚ξ€Έξ€·π›Ό+𝑛3+𝛾3ξ€Έβˆ’π›Ύ3≀1.(3.23)
Here we only prove that condition (c) of Theorem 2.13 holds, and the remaining part of the proof is similar to that of Theorem 3.6.
Since πœ‚βˆˆ(0,1),𝛽4∈(𝛼,(2+π›Όπœ‚)/(2+πœ‚)), and 𝛾3=(1/2)(𝛽4βˆ’π›Ό), then 𝛾3,𝛾3πœ‚,𝛼+𝛾3∈(0,1),𝛽4+𝛾3πœ‚βˆˆ(𝛼,1).
If π‘₯=𝐴π‘₯+𝐡𝑦, applying condition (𝐻4), Lemma 2.5 and (3.23), for π‘‘βˆˆπ‘π›Ό+𝑛3, we have ||||≀||||+||||≀||π‘₯π‘₯(𝑑)𝐴π‘₯(𝑑)𝐡𝑦(𝑑)0||𝑑Γ(𝛼)(π›Όβˆ’1)+1Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)||||≀||π‘₯𝑓(𝑠+𝛼,𝑦(𝑠+𝛼))0||𝑑Γ(𝛼)(π›Όβˆ’1)+1Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)𝐿4(𝑠+𝛼+1)(βˆ’π›½4)||||𝑦(𝑠+𝛼)πœ‚β‰€||π‘₯0||𝑑Γ(𝛼)(π›Όβˆ’1)+𝐿4Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)𝑠+𝛼+𝛾3πœ‚ξ€Έ(βˆ’π›½4)ξ€Ί(𝑠+𝛼)(βˆ’π›Ύ3)ξ€»πœ‚β‰€||π‘₯0||Γ𝑑(𝛼)(π›Όβˆ’1)+𝐿4Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)𝑠+𝛼+𝛾3πœ‚ξ€Έ(βˆ’π›½4)(𝑠+𝛼)(βˆ’π›Ύ3πœ‚)=||π‘₯0||𝑑Γ(𝛼)(π›Όβˆ’1)+𝐿4Ξ“(𝛼)π‘‘βˆ’π›Όξ“π‘ =0(π‘‘βˆ’π‘ βˆ’1)(π›Όβˆ’1)(𝑠+𝛼)(βˆ’π›½4βˆ’π›Ύ3πœ‚)≀||π‘₯0||𝑑Γ(𝛼)(π›Όβˆ’1)+𝐿4Ξ“ξ€·1βˆ’π›½4βˆ’π›Ύ3πœ‚ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½4βˆ’π›Ύ3πœ‚ξ€Έ(𝑑+𝛼)(π›Όβˆ’π›½4βˆ’π›Ύ3πœ‚)<||π‘₯0||𝑑Γ(𝛼)(π›Όβˆ’1)+𝐿4Ξ“ξ€·1βˆ’π›½4βˆ’π›Ύ3πœ‚ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½4βˆ’π›Ύ3πœ‚ξ€Έπ‘‘(π›Όβˆ’π›½4βˆ’π›Ύ3πœ‚)≀||π‘₯0||𝑑Γ(𝛼)(π›Όβˆ’1)+𝐿4Ξ“ξ€·1βˆ’π›½4βˆ’π›Ύ3πœ‚ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½4βˆ’π›Ύ3πœ‚ξ€Έπ‘‘(π›Όβˆ’π›½4)=||π‘₯0||ξ€·Ξ“(𝛼)𝑑+𝛾3ξ€Έ(π›Όβˆ’1+𝛾3)+𝐿4Ξ“ξ€·1βˆ’π›½4βˆ’π›Ύ3πœ‚ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½4βˆ’π›Ύ3πœ‚ξ€Έξ€·π‘‘+𝛾3ξ€Έ(βˆ’π›Ύ3)𝑑(βˆ’π›Ύ3)≀||π‘₯0||ξ€·Ξ“(𝛼)𝛼+𝑛3+𝛾3ξ€Έ(π›Όβˆ’1+𝛾3)+𝐿4Ξ“ξ€·1βˆ’π›½4βˆ’π›Ύ3πœ‚ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½4βˆ’π›Ύ3πœ‚ξ€Έξ€·π›Ό+𝑛3+𝛾3ξ€Έ(βˆ’π›Ύ3)𝑑(βˆ’π›Ύ3)≀𝑑(βˆ’π›Ύ3).(3.24) Thus, condition (c) of Theorem 2.13 holds. This completes the proof.

4. Examples

Example 4.1. Consider Ξ”0.5π‘₯(𝑑)=0.2(𝑑+0.5)(βˆ’0.75)sin(π‘₯(𝑑+0.5)),π‘‘βˆˆπ‘0,Ξ”βˆ’0.5π‘₯(𝑑)|𝑑=0=π‘₯0,(4.1) where 𝑓(𝑑,π‘₯(𝑑))=0.2𝑑(βˆ’0.75)sin(π‘₯(𝑑)), π‘‘βˆˆπ‘0.5.
Since ||||=||𝑓(𝑑,π‘₯(𝑑))0.2𝑑(βˆ’0.75)||sin(π‘₯(𝑑))≀0.2𝑑(βˆ’0.75),(4.2)thisimplies that condition (𝐻1) holds.
In addition, ||||𝑓(𝑑,π‘₯(𝑑))βˆ’π‘“(𝑑,𝑦(𝑑))≀0.2𝑑(βˆ’0.75)β€–π‘₯βˆ’π‘¦β€–.(4.3) Thus, condition (𝐻2) is satisfied.
Moreover, from 𝐿2=0.2, 𝛼=0.5, and 𝛽2=0.75, we have 𝐿𝑐=2Ξ“ξ€·(1+𝛼)Ξ“1βˆ’π›½2ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½2ξ€ΈΞ“ξ€·1+𝛽2ξ€Έ=0.2Ξ“(1.5)Ξ“(0.25)Ξ“(1.25)Ξ“(1.75)β‰ˆ0.7716<1,(4.4) which implies that inequality (3.14) holds.
Thus the solutions of (4.1) are asymptotically stable by Theorem 3.5.

Example 4.2. Consider Ξ”0.5π‘₯(𝑑)=0.2(𝑑+1.5)(βˆ’0.6)π‘₯(𝑑+0.5),π‘‘βˆˆπ‘0,Ξ”βˆ’0.5π‘₯(𝑑)|𝑑=0=π‘₯0,(4.5) where 𝑓(𝑑,π‘₯(𝑑))=0.2(𝑑+1)(βˆ’0.6)π‘₯(𝑑), π‘‘βˆˆπ‘0.5.
Since 𝛽3=0.6,𝛼=0.5, we have that 𝛽3∈(𝛼,(1/2)(1+𝛼)), 𝛾2=0.25 and ||||=||𝑓(𝑑,π‘₯(𝑑))0.2(𝑑+1)(βˆ’0.6)||π‘₯(𝑑)≀0.2(𝑑+0.25)(βˆ’0.6)||||,π‘₯(𝑑)(4.6) which implies that condition (𝐻3) is satisfied.
Meanwhile, ||||𝑓(𝑑,π‘₯(𝑑))βˆ’π‘“(𝑑,𝑦(𝑑))≀0.2(𝑑+1)(βˆ’0.6)β€–π‘₯βˆ’π‘¦β€–β‰€0.2𝑑(βˆ’0.6)β€–π‘₯βˆ’π‘¦β€–,(4.7) which implies that condition (𝐻2) is satisfied.
From 𝐿2=0.2, 𝛼=0.5, and 𝛽2=0.6, we have 𝐿𝑐=2Ξ“ξ€·(1+𝛼)Ξ“1βˆ’π›½2ξ€ΈΞ“ξ€·1+π›Όβˆ’π›½2ξ€ΈΞ“ξ€·1+𝛽2ξ€Έ=0.2Ξ“(1.5)Ξ“(0.4)Ξ“(0.9)Ξ“(1.6)β‰ˆ0.4120<1,(4.8) which implies that inequality (3.14) holds.
Thus the solutions of (4.5) are asymptotically stable by Theorem 3.7.

Example 4.3. Consider Ξ”0.5π‘₯(𝑑)=(𝑑+1.5)(βˆ’0.6)π‘₯1/3(𝑑+0.5),π‘‘βˆˆπ‘0,Ξ”βˆ’0.5π‘₯(𝑑)|𝑑=0=π‘₯0,(4.9) where 𝑓(𝑑,π‘₯(𝑑))=(𝑑+1)(βˆ’0.6)π‘₯1/3(𝑑), π‘‘βˆˆπ‘0.5.
Since 𝛼=0.5,𝛽4=0.6,πœ‚=1/3, we have that πœ‚βˆˆ(0,1),𝛽4∈(𝛼,(2+π›Όπœ‚)/(2+πœ‚)) and ||||𝑓(𝑑,π‘₯(𝑑))≀(𝑑+1)(βˆ’0.6)||||π‘₯(𝑑)1/3,(4.10) then condition (𝐻4) is satisfied.
The solutions of (4.9) are attractive by Theorem 3.8.

Acknowledgments

Thisresearch was supported by the NSF of Hunan Province (10JJ6007, 2011FJ3013), the Scientific Research Foundation of Hunan Provincial Education Department, and the Construct Program of the Key Discipline in Hunan Province.