Existence, uniqueness and exponential boundedness of global solutions to delay fractional differential equations
aa r X i v : . [ m a t h . C A ] J a n Existence, uniqueness and exponentialboundedness of global solutions to delayfractional differential equations
N.D. Cong ∗ and H.T. Tuan † January 3, 2017
Abstract
Under a mild Lipschitz condition we prove a theorem on the existenceand uniqueness of global solutions to delay fractional differential equa-tions. Then, we establish a result on the exponential boundedness forthese solutions.
Key words:
Fractional differential equations; Delay differential equationswith fractional derivatives; Existence and uniqueness; Growth and boundedness.2010 Mathematics Subject Classification:
Recently, delay fractional differential equations (DFDEs) have received con-siderable attentions because they provide mathematical models of real-worldproblems in which the fractional rate of change depends on the influence oftheir hereditary effects, see e.g., [L08, BHNO08, K11, YC13, CHK16] and thereferences therein. The simplest form of DFDEs is ( C D α x ( t ) = f ( t, x ( t ) , x ( t − r )) , t ∈ [0 , T ] ,x ( t ) = φ ( t ) , ∀ t ∈ [ − r, , (1)where α > C D α , and theinitial condition φ is a continuous function on the interval [ − r, r, T > ∗ [email protected], Institute of Mathematics, Vietnam Academy of Science and Tech-nology, 18 Hoang Quoc Viet, 10307 Ha Noi, Viet Nam † [email protected], Institute of Mathematics, Vietnam Academy of Science and Tech-nology, 18 Hoang Quoc Viet, 10307 Ha Noi, Viet Nam α is an integer), under some Lipschitz conditions a delay equation has an uniquelocal solution (see Hale and Lunel [HL93, Section 2.2]); furthermore, by usingcontinuation property (see [HL93, Section 2.3]) one can derive global solutionsas well. However, in the fractional case (non-integer α ) the problem of existenceand uniqueness of (local and global) solutions is more complex because of the fractional order feature of the equation which implies history dependence of thesolutions, hence, among others, the continuation property is not applicable.Abbas [A11] has discussed the existence of solutions to the DFDE (1) and usedKrasnoselskii’s fixed point theorem to show the existence of at least one localsolution to (1). Y. Jalilian and R. Jalilian [JJ13] have proved the existence of aglobal solution on a finite interval to (1) for a class of DFDEs by using a fixedpoint theorem of Leray–Schauder type. Note that in the two papers [A11] and[JJ13] the authors did not derive uniqueness of the solutions. Yang and Cao[YC13] have dealt with the problem of existence and uniqueness of solution ofa general DFDE, they presented theorems on existence and uniqueness of solu-tions to the initial values problems for DFDE, however the Lipschitz conditionthey assume is restrictive and hard to verify because it is a Lipschitz conditionwith respect to an infinite dimensional (functional) variable which varies in a(big) functional space B . Recently, Wang et al. [WCZW16, Theorem 3.2] haveformulated and proved uniqueness of global solutions to the equation similar to(1) by using the generalized Gronwall inequality. However their proof contain aflaw which make the proof incomplete (see Remark 3.3).In the investigation of long term behavior of the DFDEs, as in the classicaltheory of dynamical systems, the understanding of growth rate of the solutions isof basic importance. One needs to know whether the solutions are exponentiallybounded so that the theory of Lyapunov exponents as well as the tools of theLaplace transform are applicable to the study of the qualitative behavior of thesystems. Wang et al. [WCZW16, Theorem 3.3] have formulated and proved atheorem on exponential boundedness of solutions of DFDEs; however there areflaws in the proof and the statement of their theorem is false (see Remak 4.2for details).This paper is devoted to the investigation of the existence, uniqueness andgrowth rate of global solutions of DFDEs. Namely, we prove a general theoremon the existence and uniqueness of global solutions to the equation (1) under amild Lipschitz condition on f (see Theorem 3.1). An interesting feature of ourresult is the fact that for the existence and uniqueness of the global solutionsof (1) we do not need to require Lipschitz property of f with respect to thethird (delay) variable of f , but only the Lipschitz property of f with respectto the second (non-delay) variable. As concerns the growth rate of solutions ofDFDEs, we derive a result on the exponential boundedness of solutions to the2quation (1) (see Theorem 4.1).The rest of this paper is organized as follows. In Section 2, we recall some basicnotations of fractional calculus and a lemma concerning the equivalence betweena DFDE and a Volterra integral equation. In Section 3, we show the existenceand uniqueness of global solutions to DFDEs (Theorem 3.1). Finally, in Sec-tion 4 we establish a result on the exponential boundedness of these solutions(Theorem 4.1). This section is devoted to recalling briefly a framework of DFDEs. We firstintroduce some notations which are used throughout this paper. Let R ≥ bethe set of all non-negative real numbers and R d be the d -dimensional Euclideanspace endowed with a norm k · k . For any [ a, b ] ⊂ [ a, ∞ ), let C ([ a, b ]; R d ) be thespace of continuous functions ξ : [ a, b ] → R d with the sup norm k · k ∞ , i.e., k ξ k ∞ := sup a ≤ t ≤ b k ξ ( t ) k , ∀ ξ ∈ C ([ a, b ]; R d ) . For α >
0, [ a, b ] ⊂ R and a measurable function x : [ a, b ] → R such that R ba | x ( τ ) | dτ < ∞ , the Riemann–Liouville integral operator of order α is definedby ( I αa + x )( t ) := 1Γ( α ) Z ta ( t − τ ) α − x ( τ ) dτ, t ∈ ( a, b ] , where Γ( · ) is the Gamma function. The Caputo fractional derivative C D αa + x ofa function x ∈ AC m ([ a, b ]; R ) is defined by( C D αa + x )( t ) := ( I m − αa + D m x )( t ) , t ∈ ( a, b ] , where AC m ([ a, b ]; R ) denotes the space of real functions x which has continuousderivatives up to order m − a, b ] and the ( m − th -order deriva-tive x ( m − is absolutely continuous, D m = d m dt m is the usual m th -order derivativeand m := ⌈ α ⌉ is the smallest integer larger or equal to α . The Caputo frac-tional derivative of a d -dimensional vector function x ( t ) = ( x ( t ) , · · · , x d ( t )) T isdefined component-wise as( C D αa + x )( t ) := ( C D αa + x ( t ) , · · · , C D αa + x d ( t )) T . From now on, we consider only the case α ∈ (0 , r be an arbitrary positiveconstant, and φ ∈ C ([ − r, R d ) be a given continuous function, we study thedelay Caputo fractional differential equations C D α x ( t ) = f ( t, x ( t ) , x ( t − r )) , t ∈ [0 , T ] , (2)3ith the initial condition x ( t ) = φ ( t ) , ∀ t ∈ [ − r, , (3)where x ∈ R d , T > f : [0 , T ] × R d × R d → R d is continuous.We also consider the initial condition problem (2)-(3) on the infinite time interval[ − r, ∞ ) as well with the obvious change from finite T to ∞ .A function ϕ ( · , φ ) ∈ C ([ − r, T ]; R d ) is called a solution of the initial conditionproblem (2)-(3) over the interval [ − r, T ] if ( C D α ϕ ( t, φ ) = f ( t, ϕ ( t, φ ) , ϕ ( t − r, φ )) , t ∈ [0 , T ] ,ϕ ( t, φ ) = φ ( t ) , ∀ t ∈ [ − r, . To prove the existence of solutions to the initial condition problem (2)-(3) andto investigate the asymptotic behavior of solutions to this problem we need toconvert it into an equivalent delay integral equation with the initial condition(3). This is stated in the following lemma.
Lemma 2.1.
The function ϕ ∈ C ([ − r, T ]; R d ) is a solution of the initial condi-tion problem (2)-(3) on the interval [ − r, T ] if and only if it is a solution of thedelay integral equation x ( t ) = φ (0) + 1Γ( α ) Z t ( t − τ ) α − f ( τ, x ( τ ) , x ( τ − r )) dτ, ∀ t ∈ [0 , T ] (4)with the initial condition x ( t ) = φ ( t ) , ∀ t ∈ [ − r, . (5) Proof.
Using the same arguments as in the proof of [D10, Lemma 6.2, p. 86].
We show that under a mild Lipschitz condition a DFDE has unique globalsolution.
Theorem 3.1 (Existence and uniqueness of global solutions to DFDEs) . As-sume that f : [0 , T ] × R d × R d → R d is continuous and satisfies the followingLipschitz condition with respect to the second variable: there exists a non-negative continuous function L : [0 , T ] × R d → R ≥ such that k f ( t, x, y ) − f ( t, ˆ x, y ) k ≤ L ( t, y ) k x − ˆ x k (6)for all t ∈ [0 , T ], x, y, ˆ x ∈ R d . Then, the initial condition problem (2)-(3) has aunique global solution ϕ ( · , φ ) on the interval [ − r, T ].4 roof. According to Lemma 2.1, the equation (2) with the initial condition (3)is equivalent to the initial condition problem (4)-(5).First we consider the case 0 < T ≤ r . In this case, the equation (4) has theform x ( t ) = φ (0) + 1Γ( α ) Z t ( t − τ ) α − f ( τ, x ( τ ) , φ ( τ − r )) dτ, ∀ t ∈ [0 , T ] . For this integral equation, by Tisdell [T12, Theorem 6.4, p. 310], there exists aunique solution on the interval [0 , T ]. Denote that solution by ξ ∗ r and put ϕ T ( t, φ ) := ( φ ( t ) , ∀ t ∈ [ − r, ,ξ ∗ r ( t ) , ∀ t ∈ [0 , T ] . Then ϕ T ( t, φ ) is the unique solution of the problem (4)-(5) on [ − r, T ].For the case T > r , we divide the interval [0 , T ] into [0 , r ] ∪ · · · ∪ [( k − r, k r ] ∪ [ k r, T ], where k ∈ N and 0 ≤ T − k r < r . On the interval [ − r, r ], using thesame arguments as above, we can find a unique solution of the initial conditionproblem (4)-(5) which is denoted by ϕ r ( · , φ ). We will prove the existence anduniqueness of solution on the interval [ − r, k r ] by induction. Assume that theproblem (4)-(5) has a unique solution on the interval [ − r, kr ] for some 1 ≤ k
Suppose that the assumptions of the corollary are satisfied. Note thatif T > T > − r, T ] and [ − r, T ] implying that we findunique global solutions ϕ on [ − r, T ] and ϕ on [ − r, T ]. Due to uniqueness thefunction ϕ coincides with the function ϕ on [ − r, T ]. To complete the proofwe establish a new function ϕ ( · , φ ) on [ − r, ∞ ) as below ϕ ( t, φ ) := ( φ ( t ) , if t ∈ [ − r, ,ϕ t ( t, φ ) , if t > , where ϕ t ( · , φ ) is the function defined as in the proof of Theorem 3.1. Then,this function is the unique solution to the initial condition problem (2)-(3) on[ − r, ∞ ). Remark . Wang et al. [WCZW16, Theorem 3.2, p. 48] have proved a resulton the uniqueness of solutions to a DFDE like the problem (2)-(3) under theLipschitz assumption k f ( t, x, y ) − f ( t, ˆ x, ˆ y ) k ≤ L ( k x − ˆ x k + k y − ˆ y k ) , ∀ t ∈ R ≥ , ∀ x, y, ˆ x, ˆ y ∈ R . Their approach is based on the generalized Gronwall inequality, and the keypoint in their proof is the inequality (18) of [WCZW16, p. 49]. They deducethis inequality from the inequality (17) of [WCZW16, p. 49]. However, thisdeduction is incorrect due to the fractional nature of the equations. Thus, theproof of Wang et al. is incomplete. 7
Exponential boundedness of solutions to delayfractional differential equations
For the qualitative theory of DFDEs the study of growth rate of solutions isof basic importance. In this section we show that solutions of DFDEs areexponentially bounded.Let φ ∈ C ([ − r, R d ) be an arbitrary continuous function. We consider theinitial condition problem (2)-(3) on the semi real axis [ − r, ∞ ). A solution ϕ ( · , φ ) of the initial condition problem (2)-(3) is called exponentially bounded ifthere exist positive constants C and λ such that k ϕ ( t, φ ) k ≤ C exp ( λt ) , ∀ t ≥ . The main result in this section is the following theorem on exponential bound-edness of solutions of the initial condition problem (2)-(3).
Theorem 4.1 (Exponential boundedness of solutions to delay fractional dif-ferential equations) . Assume that f is continuous and satisfies the followingconditions:(H1) there exists a positive constant L such that k f ( t, x, y ) − f ( t, ˆ x, ˆ y ) k ≤ L ( k x − ˆ x k + k y − ˆ y k ) , ∀ t ∈ R ≥ , x, y, ˆ x, ˆ y ∈ R d ;(H2) there exists a constant β > L such thatsup t ≥ R t ( t − τ ) α − k f ( τ, , k dτE α ( βt α ) < ∞ . Then the global solution ϕ ( · , φ ) on the interval [ − r, ∞ ) of the initial conditionproblem (2)-(3) is exponentially bounded. More precisely, there exists a constant C > k ϕ ( t, φ ) k ≤ CE α ( βt α ) , ∀ t ≥ . Proof.
We denote by C β ([0 , ∞ ); R d ) the set of all continuous functions ξ ∈ C ([0 , ∞ ); R d ) satisfying the condition k ξ k β := sup t ≥ k ξ ( t ) k E α ( βt α ) < ∞ . It is easily seen that k ·k β is a norm in C β ([0 , ∞ ); R d ) and ( C β ([0 , ∞ ); R d ) , k ·k β )is a Banach space. 8or any φ ∈ C ([ − r, R d ), we construct a operator T φ on ( C β ([0 , ∞ ); R d ) , k ·k β )as follows:( T φ ξ )( t ) := φ (0) + 1Γ( α ) Z t ( t − τ ) α − f ( τ, ξ ( τ ) , φ ( τ − r )) dτ, ∀ t ∈ [0 , r ] , ( T φ ξ )( t ) := φ (0) + 1Γ( α ) Z r ( t − τ ) α − f ( τ, ξ ( τ ) , φ ( τ − r )) dτ + 1Γ( α ) Z tr ( t − τ ) α − f ( τ, ξ ( τ ) , ξ ( τ − r )) dτ, ∀ t > r. First we consider the case t ∈ (0 , r ]. In this case we have k ( T φ ξ )( t ) k ≤ k φ (0) k + L Γ( α ) Z t ( t − τ ) α − (cid:16) k ξ ( τ ) k + k φ ( τ − r ) k (cid:17) dτ + 1Γ( α ) Z t ( t − τ ) α − k f ( τ, , k dτ ≤ (cid:18) Lr α Γ( α + 1) (cid:19) k φ k ∞ + L Γ( α ) Z t ( t − τ ) α − E α ( βτ α ) k ξ ( τ ) k E α ( βτ α ) dτ + 1Γ( α ) Z t ( t − τ ) α − k f ( τ, , k dτ, where k φ k ∞ := sup − r ≤ s ≤ k φ ( s ) k < ∞ . This implies thatsup t ∈ [0 ,r ] k ( T φ ξ )( t ) k E α ( βt α ) ≤ (cid:18) Lr α Γ( α + 1) (cid:19) k φ k ∞ + Lβ k ξ k β + 1Γ( α ) sup t ≥ R t ( t − τ ) α − k f ( τ, , k dτE α ( βt α ) < ∞ . Next, consider the case t ≥ r . In this case we have k ( T φ ξ )( t ) k ≤ k φ (0) k + L k φ k ∞ Γ( α ) Z r ( t − τ ) α − dτ + 1Γ( α ) Z t ( t − τ ) α − k f ( τ, , k dτ + L Γ( α ) Z t ( t − τ ) α − E α ( βτ α ) k ξ ( τ ) k E α ( βτ α ) dτ + L Γ( α ) Z tr ( t − τ ) α − k ξ ( τ − r ) k dτ. k ( T φ ξ )( t ) k ≤ k φ k ∞ (cid:18) Lt α Γ( α + 1) (cid:19) + 1Γ( α ) Z t ( t − τ ) α − k f ( τ, , k dτ + L k ξ k β Γ( α ) Z t ( t − τ ) α − E α ( βτ α ) dτ + L Γ( α ) Z tr ( t − τ ) α − E α ( β ( τ − r ) α ) k ξ ( τ − r ) k E α ( β ( τ − r ) α ) dτ. Since E α ( · ) is a monotone increasing function on real line, this implies that k ( T φ ξ )( t ) k E α ( βt α ) ≤ k φ k ∞ (cid:18) t ≥ r Lt α Γ( α + 1) E α ( βt α ) (cid:19) + sup t ≥ r R t ( t − τ ) α − k f ( τ, , k dτ Γ( α ) E α ( βt α ) + 2 Lβ k ξ k β < ∞ , ∀ t ≥ r. To summarize, the following estimate is truesup t ≥ k ( T φ ξ )( t ) k E α ( βt α ) < ∞ , ∀ ξ ∈ ( C β ([0 , ∞ ); R d ) , k · k β ) . Thus, T φ (( C β ([0 , ∞ ); R d ) , k · k β )) ⊂ ( C β ([0 , ∞ ); R d ) , k · k β ). We now show thatthe operator T φ is contractive on ( C β ([0 , ∞ ); R d ) , k · k β ). Indeed, for any ξ, ˆ ξ ∈ C β ([0 , ∞ ); R d ) , k · k β ), on [0 , r ] we have the estimate kT φ ξ ( t ) − T φ ˆ ξ ( t ) k ≤ L Γ( α ) Z t ( t − τ ) α − k ξ ( τ ) − ˆ ξ ( τ )) k dτ ≤ L k ξ − ˆ ξ k β Γ( α ) Z t ( t − τ ) α − E α ( βτ α ) dτ, ∀ t ∈ (0 , r ] . This implies that sup t ∈ [0 ,r ] kT φ ξ ( t ) − T φ ˆ ξ ( t ) k E α ( βt α ) ≤ Lβ k ξ − ˆ ξ k β . (8)Furthermore, for all t ≥ r , we have kT φ ξ ( t ) − T φ ˆ ξ ( t ) k ≤ L Γ( α ) Z t ( t − τ ) α − k ξ ( τ ) − ˆ ξ ( τ )) k dτ + L Γ( α ) Z tr ( t − τ ) α − k ξ ( τ − r ) − ˆ ξ ( τ − r ) k dτ. By using the same arguments as above, we obtainsup t ≥ r kT φ ξ ( t ) − T φ ˆ ξ ( t ) k E α ( βt α ) ≤ Lβ k ξ − ˆ ξ k β (9)10or all t ≥ r . Combining (8) and (9), we get kT φ ξ − T φ ˆ ξ k β ≤ Lβ k ξ ( τ ) − ˆ ξ ( τ ) k β for all ξ, ˆ ξ ∈ ( C β ([0 , ∞ ); R d ) , k · k β ). Since Lβ <
1, according to the Banachfixed point theorem, there exists a unique fixed point ξ ∗ of T φ in the space( C β ([0 , ∞ ); R d ) , k · k β ). Put ϕ ( t, φ ) := ( φ ( t ) , ∀ t ∈ [ − r, ,ξ ∗ ( t ) , ∀ t ∈ [0 , ∞ ) . It is obvious that ϕ ( · , φ ) is the unique global solution of the initial condi-tion problem (2)-(3) on the interval [ − r, ∞ ). From the definition of the space( C β ([0 , ∞ ); R d ) , k · k β ), we can find a constant C > k ϕ ( t, φ ) k = k ξ ∗ ( t ) k ≤ CE α ( βt α ) , ∀ t ≥ . Due to the asymptotic growth rate of the Mittag-Leffler function E α ( βt α ), thesolution ϕ ( · , φ ) is exponentially bounded. The proof is complete. Remark . Wang et al. [WCZW16] stated a theorem on exponential bounded-ness of solutions of delay fractional differential equations. In particular, they as-serted that under the condition (H1), solutions of (2) are exponentially boundedfor any initial condition φ ∈ C ([ − r, R d ) (see [WCZW16, Theorem 3.3, p. 49]).However, their statement is false. For an easy counterexample let us considerthe equation ( C D α x ( t ) = exp( t ) , t > ,x ( t ) = x ∈ R ≥ for t ∈ [ − r, , (10)where the fractional order α ∈ (0 , t ) satisfiesthe condition (H1) above as well as the condition (H1) in the statement of[WCZW16, Theorem 3.3, p. 49]. By Theorem 3.1, the equation (10) has aunique global solution on [0 , ∞ ), which can be computed explicitly as ϕ ( t, x ) = x + 1Γ( α ) Z t ( t − τ ) α − exp( τ ) dτ. It is easily seen that the solution ϕ ( · , x ) is not exponential bounded. Acknowledgement
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