Research article

Existence, continuous dependence and finite time stability for Riemann-Liouville fractional differential equations with a constant delay

  • Received: 24 November 2019 Accepted: 31 March 2020 Published: 21 April 2020
  • MSC : 34A08, 34A37

  • Non-linear scalar Riemann-Liouville fractional differential equation with a constant delay is studied on a finite interval. An initial value problem is set up in appropriate way combining the idea of the initial time interval in ordinary differential equations with delays and the properties of Riemann-Liouville fractional derivatives. The mild solution of the studied initial value problem is defined. The existence, uniqueness, continuous dependence on the initial functions, finite time stability of the mild solutions are investigated.

    Citation: Snezhana Hristova, Antonia Dobreva. Existence, continuous dependence and finite time stability for Riemann-Liouville fractional differential equations with a constant delay[J]. AIMS Mathematics, 2020, 5(4): 3809-3824. doi: 10.3934/math.2020247

    Related Papers:

    [1] Ravi Agarwal, Snezhana Hristova, Donal O'Regan . Integral presentations of the solution of a boundary value problem for impulsive fractional integro-differential equations with Riemann-Liouville derivatives. AIMS Mathematics, 2022, 7(2): 2973-2988. doi: 10.3934/math.2022164
    [2] Ravi P. Agarwal, Snezhana Hristova . Stability of delay Hopfield neural networks with generalized proportional Riemann-Liouville fractional derivative. AIMS Mathematics, 2023, 8(11): 26801-26820. doi: 10.3934/math.20231372
    [3] Wei Liu, Qinghua Zuo, Chen Xu . Finite-time and global Mittag-Leffler stability of fractional-order neural networks with S-type distributed delays. AIMS Mathematics, 2024, 9(4): 8339-8352. doi: 10.3934/math.2024405
    [4] Khalid K. Ali, K. R. Raslan, Amira Abd-Elall Ibrahim, Mohamed S. Mohamed . On study the fractional Caputo-Fabrizio integro differential equation including the fractional q-integral of the Riemann-Liouville type. AIMS Mathematics, 2023, 8(8): 18206-18222. doi: 10.3934/math.2023925
    [5] Md. Asaduzzaman, Md. Zulfikar Ali . Existence of positive solution to the boundary value problems for coupled system of nonlinear fractional differential equations. AIMS Mathematics, 2019, 4(3): 880-895. doi: 10.3934/math.2019.3.880
    [6] Hasanen A. Hammad, Hassen Aydi, Manuel De la Sen . The existence and stability results of multi-order boundary value problems involving Riemann-Liouville fractional operators. AIMS Mathematics, 2023, 8(5): 11325-11349. doi: 10.3934/math.2023574
    [7] Iman Ben Othmane, Lamine Nisse, Thabet Abdeljawad . On Cauchy-type problems with weighted R-L fractional derivatives of a function with respect to another function and comparison theorems. AIMS Mathematics, 2024, 9(6): 14106-14129. doi: 10.3934/math.2024686
    [8] Yucai Ding, Hui Liu . A new fixed-time stability criterion for fractional-order systems. AIMS Mathematics, 2022, 7(4): 6173-6181. doi: 10.3934/math.2022343
    [9] Thabet Abdeljawad, Sabri T. M. Thabet, Imed Kedim, Miguel Vivas-Cortez . On a new structure of multi-term Hilfer fractional impulsive neutral Levin-Nohel integrodifferential system with variable time delay. AIMS Mathematics, 2024, 9(3): 7372-7395. doi: 10.3934/math.2024357
    [10] Lakhlifa Sadek, Tania A Lazǎr . On Hilfer cotangent fractional derivative and a particular class of fractional problems. AIMS Mathematics, 2023, 8(12): 28334-28352. doi: 10.3934/math.20231450
  • Non-linear scalar Riemann-Liouville fractional differential equation with a constant delay is studied on a finite interval. An initial value problem is set up in appropriate way combining the idea of the initial time interval in ordinary differential equations with delays and the properties of Riemann-Liouville fractional derivatives. The mild solution of the studied initial value problem is defined. The existence, uniqueness, continuous dependence on the initial functions, finite time stability of the mild solutions are investigated.


    Differential equations with various types of fractional derivatives such as Caputo fractional derivative, Riemann-Liouville fractional derivative, are intensively studied theoretically and applied to varies models in the last decades. For example, they are successfully applied to study various types of neural networks (see, for example, [1,2,3]). Fractional differential equations with delays are rapidly developed. One of the main studied qualitative questions about fractional delay differential equation is the one about stability. In 1961, Dorato [4] introduced a concept of finite time stability (FTS). FTS is different from asymptotic stability. However, it is regarded as one of the core problems in delay systems from practical considerations. Later this type of stability has been applied to different types of differential equations. Recently, it is applied for Caputo delta fractional difference equations [5,6], for Caputo fractional differential equations [7] for -Hilfer fractional differential equation [8].

    The investigations of the properties of the solutions of Riemann-Liouville (RL) fractional differential equations with delays are still at his initial stage. The asymptotic stability of the zero solution of the linear homogeneous differential system with Riemann-Liouville fractional derivative is studied in [9]. Li and Wang introduced the concept of a delayed Mittag-Leffler type matrix function, and then they presented the finite-time stability results by virtue of a delayed Mittag-Leffler type matrix in [10,11,12]. In connection with the presence of the bounded delay the initial condition is given on a whole finite interval called initial interval. In the above mentioned papers ([10,11,12]) the authors study the case when the lower limit of the RL fractional derivative coincides with the left side end of the initial interval. It changes the meaning of the initial condition in differential equations. In connection with this in the paper we set up an initial condition satisfying two main properties: first, it is similar to the initial condition in differential equations with ordinary derivatives and, second, the RL fractional condition is defined at the right side end of the initial interval which is connected with the presence of RL fractional derivative.

    In this paper we study initial value problems for scalar nonlinear RL fractional differential equations with constant delays. Similarly to the case of ordinary derivative, the differential equation is given to the right of the initial time interval. It requires the lower bound of the RL fractional derivative to coincides with the right side end of the initial time interval. We present an integral representation of of the studied initial value problem. By the help of fractional generalization of Gronwall inequality we study the existence, continuous dependence and finite time stability of the scalar nonlinear RL fractional differential equations.

    The main contributions of the current paper include:

    (ⅰ) An appropriate initial value problem for nonlinear RL fractional differential equations is set up based on the idea of the initial time interval for delay differential equations with ordinary derivatives.

    (ⅱ) A mild solution of the considered initial value problem is defined based on an appropriate integral representation of the solution.

    (ⅲ) The existence, continuous dependence and finite time stability of the scalar nonlinear RL fractional differential equations is studied by the help of fractional generalization of Gronwall inequality.

    The rest of this paper is organized as follows. In Section 2, some notations and preliminary lemmas are presented. In Section 3, main results are obtained. In Section 3.1. mild solution of the studied initial value problem is defined and some sufficient conditions by Banach contraction principle are obtained. In Section 3.2. continuous dependence on the initial functions is investigated based on the fractional extension of Gronwall inequality. In Section 3.3. some sufficient conditions for finite time stability are given.

    Let J=[τ,T], I=[0,T] where τ>0 is a constant, T<. Without loss of generality we can assume there exists a natural number N such that T=(N+1)τ. Let Lloc1(I,R) be the linear space of all locally Lebesgue integrable functions m:IR, PC(J)=C([τ,0),R)C((0,T],R).

    Let xPC(J,R). Denote ||x||J=suptJ|x(t)|.

    In this paper we will use the following definitions for fractional derivatives and integrals:

    Riemann - Liouville fractional integral of order q(0,1) ([13,14])

    0Iqtm(t)=1Γ(q)t0m(s)(ts)1qds,     tI,

    where Γ(.) is the Gamma function.

    Riemann - Liouville fractional derivative of order q(0,1) ([13,14])

    RL0Dqtm(t)=ddt( 0I1qtm(t))=1Γ(1q)ddtt0(ts)qm(s)ds,     tI.

    We will give fractional integrals and RL fractional derivatives of some elementary functions which will be used later:

    Proposition 1. The following equalities are true:

    RL0Dqttβ=Γ(1+β)Γ(1+βq)tβq,       0I1qtβ1=Γ(β)Γ(1+βq)tβq,
    0I1qtq1=Γ(q),       RL0Dqttq1=0.

    The definitions of the initial condition for fractional differential equations with RL-derivatives are based on the following result:

    Proposition 2. (Lemma 3.2 [15]). Let q(0,1), and mLloc1([0,T],R).

    (a) If there exists a.e. a limit limt0+[tq1m(t)]=c, then there also exists a limit

    0I1qtm(t)|t=0:=limt0+ 0I1qtm(t)=cΓ(q).

    (b) If there exists a.e. a limit 0I1qtm(t)|t=0=b and if there exists the limit limt0+[t1qm(t)] then

    limt0+[t1qm(t)]=bΓ(q).

    We will use the Mittag - Leffler functions with one and with two parameters, respectively, (see, for example, [14]) given by Ep(z)=j=0zjΓ(jp+1) and Ep,q(z)=j=0zjΓ(jp+q).

    Proposition 3. [16] (Gronwall fractional inequality) Suppose a(t) is a nonnegative function locally integrable on [0,T) (some T) and b(t) is a nonnegative, nondecreasing continuous function defined on [0,T), b(t)M (constant), and suppose u(t) is nonnegative and locally integrable on [0,T) with

    u(t)a(t)+b(t)t0(ts)q1u(s)ds,   t[0,T).

    Then

    u(t)a(t)+t0(n=1(b(t)Γ(q))nΓ(nq)(ts)nq1a(s))ds, t[0,T).

    Recently, in [17] the non-homogeneous scalar linear Riemann-Liouville fractional differential equations with a constant delay :

     RL0Dqtx(t)=Ax(t)+Bx(tτ)+f(t) for t>0. (2.1)

    with the initial conditions

    x(t)=g(t),   t[τ,0], (2.2)
    limt0+(t1qx(t))=g(0)Γ(q) (2.3)

    where fC(R+,R), gC([τ,0],R) was studied. It was proved the solution is given by the function

     Λq(t)={g(t)t(τ,0]g(0)Eq,q(Atq)tq1+t0(ts)q1Eq,q(A(ts)q)(Bg(sτ)+f(s))ds t(0,τ]g(0)Eq,q(Atq)tq1+t0(ts)q1Eq,q(A(ts)q)f(s)ds        +Bn1i=0(i+1)τiτ(ts)q1Eq,q(A(ts)q)Λq(sτ)ds        +Btnτ(ts)q1Eq,q(A(ts)q)Λq(sτ)ds                                             for   t(nτ,(n+1)τ],n=1,2, (2.4)

    where Eq,q(z)=i=0ziΓ(iq+q) and Eq(z)=i=0ziΓ(iq+1) are Mittag-Leffler functions with two and one parameter, respectively.

    Now, we will study the following nonlinear fractional delay differential equations

     RL0Dqtx(t)=Ax(t)+Bx(tτ)+f(t,x(t)) for tI. (2.5)

    with the initial conditions (2.2), (2.3) where A,BR are given constants, f:I×RR.

    Remark 1. Note that in the case of the linear equation (2.1) we have formula (2.4) for the explicit solution since in the case of nonlinear equation (2.5) we are not able to obtain an explicit formula, we could provide only an integral presentation of the solution (see Example 1 and Example 2).

    Example 1. Consider the special case of (2.1):

     RL0D0.5tx(t)=x(t1)+t for t>0x(t)=t,   t[1,0],limt0+(t0.5x(t))=0. (2.6)

    Then applying Eq,q(0)=1Γ(q) we obtain the solution of (2.6):

     x(t)={t,         t(1,0]1πt0(ts)0.5(s1+s)ds=2tπ(43t1),      t(0,1]1πt0(ts)0.5sds+1π10(ts)0.5(s1)ds+1πtτ(ts)0.5x(s1)ds    =43πt1.543π(t1)1.5+43πt(t1.5)+t(t1)+43πt 2F1[0.5,1.5,2.5,1t]                1615πt 2F1[0.5,2.5,3.5,1t],     t(1,2]. (2.7)

    In connection with Remark 1 we will define a mild solution:

    Definition 1. A function xPC(J,R) is called a mild solution of the IVP (2.5), (2.2), (2.3) if it satisfies the following integral equation

     x(t)={g(t)   for   t[τ,0],g(0)Eq,q(Atq)tq1+Bt0(ts)q1Eq,q(A(ts)q)g(sτ)ds        +t0(ts)q1Eq,q(A(ts)q)f(s,x(s))ds   for   t(0,τ],g(0)Eq,q(Atq)tq1+t0(ts)q1Eq,q(A(ts)q)f(s,x(s))ds        +Bn1i=0(i+1)τiτ(ts)q1Eq,q(A(ts)q)x(sτ)ds        +Btnτ(ts)q1Eq,q(A(ts)q)x(sτ)ds                                             for   t(nτ,(n+1)τ],n=1,2,,N

    Example 2. Consider the partial case of (2.5) (compare with (2.6):

     RL0D0.5tx(t)=x(t1)+sin(x(t)) for t>0x(t)=t,   t[1,0],limt0+(t0.5x(t))=0. (3.1)

    Now similarly to Example 1 we are not able to obtain the exact solution of (2.6). But using Definition 1 we can consider the mild solution x(t) of the IVP (3.1) satisfying:

     x(t)={t,         t(1,0]1πt0(ts)0.5(s1)ds+1πt0(ts)0.5sin(x(s))ds      =2tπ(23t1)+1πt0(ts)0.5sin(x(s))ds,      t(0,1]1πt0(ts)0.5sin(x(s))ds+1π10(ts)0.5(s1)ds+1πtτ(ts)0.5x(s1)ds    =1πt0(ts)0.5sin(x(s))ds43π(t1)1.5+43πt(t1.5)        +1πtτ(ts)0.5x(s1)ds,     t(1,2]. (3.2)

    Examples 1 and 2 show the main difference between the linear RL fractional differential equations and the nonlinear RL fractional differential equations with a linear part.

    We will introduce the following conditions:

    (A1). The function fC([0,T]×R,R) and there exists a function wC(I,R+) such that |f(t,x)|w(t) for all tI,xR.

    (A2). The function fC([0,T]×R,R) and there exists a constant L>0 such that |f(t,x)f(t,y)|L|xy| for all tI,x,yR. First, we will consider the case of Lipschitz nonlinear function.

    Theorem 1. Let A0, the condition (A2) be satisfied and

    1. The function gC([τ,0],R and |g(0)|<.

    2. ρ=Lh1+|B|h2|A|<1 where h1=maxt[0,T]|Eq(Atq)1|,

    h2=maxn=0,1,2,,N{maxt(nτ,(n+1)τ](|Eq(A(tnτ)q)1|+n1j=0|Eq(A(tjτ)q)Eq(A(t(j+1)τ)q)|)}.

    Then the the IVP (2.5), (2.2), (2.3) has a unique mild solution xPC(J,R).

    P r o o f: Existence. Define the operator Ξ:PC(J,R)PC(J,R) by the equality

     Ξ(x(t))={g(t)            t[τ,0]g(0)Eq,q(Atq)tq1+Bt0(ts)q1Eq,q(A(ts)q)g(sτ)ds        +t0(ts)q1Eq,q(A(ts)q)f(s,x(s))ds   for   t(0,τ],g(0)Eq,q(Atq)tq1+t0(ts)q1Eq,q(A(ts)q)f(s,x(s))ds        +Bn1i=0(i+1)τiτ(ts)q1Eq,q(A(ts)q)x(sτ)ds        +Btnτ(ts)q1Eq,q(A(ts)q)x(sτ)ds                                             for   t(nτ,(n+1)τ],n=1,2,,N

    Let z,yPC(J,R). We will prove that

    |Ξ(z(t))Ξ(y(t))|(L|Eq(Atq)1||A|+|B|n1j=0|Eq(A(tjτ)q)Eq(A(t(j+1)τ)q)||A|        +|B||Eq(A(tnτ)q)1||A|)||zy||J for   t(nτ,(n+1)τ], n=0,1,2,,N (3.3)

    Let t(0,τ]. Then applying Definition 1 and the equality

    t0(ts)q1Eq,q(A(ts)q)ds=i=0AiΓ((i+1)q)t0(ts)(i+1)q1ds=i=0Ai(1+i)qΓ((i+1)q)t(i+1)q=Eq(Atq)1A,   t(0,τ], (3.4)

    we obtain

    |Ξ(z(t))Ξ(y(t))||t0(ts)q1Eq,q(A(ts)q)|f(s,z(s))f(s,y(s))|ds|L|t0(ts)q1Eq,q(A(ts)q)|z(s)y(s)|ds|L||zy||J|t0(ts)q1Eq,q(A(ts)q)ds|=L|Eq(Atq)1||A|||zy||J (3.5)

    Let t(τ,2τ]. Then according to Definition 1 and the equality

    tτ(ts)q1Eq,q(A(ts)q)ds=i=0AiΓ((i+1)q)tτ(ts)(i+1)q1ds=i=0Ai(1+i)qΓ((i+1)q)(tτ)(i+1)q=Eq(A(tτ)q)1A,    t(τ,2τ], (3.6)

    we have

    |Ξ(z(t))Ξ(y(t))||t0(ts)q1Eq,q(A(ts)q)|f(s,z(s))f(s,t(s))|ds|        +|B| |τ0(ts)q1Eq,q(A(ts)q)|z(sτ)y(sτ)|ds|        +|B| |tτ(ts)q1Eq,q(A(ts)q)|z(sτ)y(sτ)|ds|(L|Eq(Atq)1|A+|B| |tτ(ts)q1Eq,q(A(ts)q)ds|)||zy||J(L|Eq(Atq)1||A|+|B||Eq(A(tτ)q)1||A|)||zy||J. (3.7)

    Let t(2τ,3τ]. Then according to Definition 1 and the equalities

    t2τ(ts)q1Eq,q(A(ts)q)ds=i=0AiΓ((i+1)q)t2τ(ts)(i+1)q1ds=i=0Ai(1+i)qΓ((i+1)q)(tτ)(i+1)q=Eq(A(t2τ)q)1A,    t(2τ,3τ], (3.8)

    and

    2ττ(ts)q1Eq,q(A(ts)q)ds=i=0AiΓ((i+1)q)2ττ(ts)(i+1)q1ds=i=0Ai(1+i)qΓ((i+1)q)(tτ)(i+1)qi=0Ai(1+i)qΓ((i+1)q)(t2τ)(i+1)q=Eq(A(tτ)q)Eq(A(t2τ)q)A, (3.9)

    we have

    |Ξ(z(t))Ξ(y(t))||t0(ts)q1Eq,q(A(ts)q)|f(s,z(s))f(s,t(s))|ds|        +|B| |τ0(ts)q1Eq,q(A(ts)q)|z(sτ)y(sτ)|ds|        +|B| |2ττ(ts)q1Eq,q(A(ts)q)|z(sτ)y(sτ)|ds|        +|B| |t2τ(ts)q1Eq,q(A(ts)q)|z(sτ)y(sτ)|ds|(L|Eq(Atq)1||A|+|B| |2ττ(ts)q1Eq,q(A(ts)q)ds|         +|B| |t2τ(ts)q1Eq,q(A(ts)q)ds|)||zy||J(L|Eq(Atq)1||A|+|B||Eq(A(tτ)q)Eq(A(t2τ)q)||A|       +|B||Eq(A(t2τ)q)1||A|)||zy||J. (3.10)

    Following the induction process and the definition of ρ we obtain that ||Ξ(z(t))Ξ(y(t))||Jρ||zy||J. Therefore, the operator Ξ:PC(J,R)PC(J,R) is a contraction.

    Uniqueness. Let z(t),y(t) be two mild solutions of the IVP (2.5), (2.2), (2.3). Applying induction process w.r.t. the intervals and from condition 2 we obtain that ||zy||(kτ,(k+1)τ]<ρ||zy||(kτ,(k+1)τ] for k=0,1,,N which proves the uniqueness.

    Remark 2. It is obvious that the condition A0 in Theorem 1 is not a restriction because the nonzero term Ax could be added to the nonlinear part without losing the Lipschitz property.

    Example 3. Consider the IVP (3.1) In this case A=0.1,f(t,x)=sin(x)0.1x,B=1. Then the condition (A2) is satisfied with L=1.1 but the condition 2 of Theorem 1 is not satisfied.

    Now, we change the equation in the IVP (3.1) to  RL0D0.5tx(t)=0.1x(t1)+0.01sin(x(t)). In this case A=0.1,f(t,x)=0.01sin(x)0.1x,B=0.1, h1=h2=0.43581 and ρ=(0.11+0.1)0.435810.1<1. According to Theorem the IVP (3.1) has unique mild solution which is satisfying the integral presentation given in Definition 1.

    In the case of a bounded nonlinear function we have the following result:

    Theorem 2. Let the condition (A1) be satisfied and

    1. The function gC([τ,0],R and |g(0)|<.

    2. ρ=2||w||Ih1+|B|h2|A|<1 where h1 and h2 are defined in Theorem 1.

    Then the the IVP (2.5), (2.2), (2.3) has a unique solution xPC(J,R).

    The proof of Theorem 2 is similar to the one of Theorem 1 and we omit it.

    We will study the continuous dependence of mild solutions of the IVP (2.5), (2.2), (2.3) on the initial functions.

    Consider IVP (2.5), (2.2), (2.3) and the RL fractional equation (2.5) with initial conditions

    x(t)=p(t),   t[τ,0], (3.11)
    limt0+(t1qx(t))=p(0)Γ(q) (3.12)

    Theorem 3. Let the following conditions be satisfied:

    1. The functions g,pC([τ,0],R, |g(0)|<, |p(0)|<.

    2. The function fC([0,T]×R,R) and it is Lipschitz with a constant L>0 on [0,T]×R.

    Then for any number δ>0 there exist numbers Kk,Ck>0, k=0,1,2,,N such that the inequality ||gp||[τ,0]<δ implies

    |x(t)y(t)|δ(Kk(tkτ)q1+Ck) for  t(kτ,(k+1)τ],  k=0,1,2,,N (3.13)

    where x(t),y(t) are mild solutions of the IVPs (2.5), (2.2), (2.3) and (2.5), (3.11), (3.12) respectively.

    P r o o f: We will use the induction w.r.t. the intervals to prove the claim.

    Let M=suptJ|Eq,q(Atq)|.

    Let t(0,τ]. Then from Definition 1 and Eq. (3.4) we get

    |x(t)y(t)|δMtq1+|B|Mδt0(ts)q1ds+LMt0(ts)q1|x(s)y(s)|dsδMtq1+δM|B|τqq+LMt0(ts)q1|x(s)y(s)|dsδMtq1+δP0+LMt0(ts)q1|x(s)y(s)|ds (3.14)

    where P0=M|B|τqq.

    According to Proposition 3, the inequality t0(ts)nq1sq1ds=tnq+q1Γ(q)Γ(nq)Γ(nq+q) we obtain

    |x(t)y(t)|δMtq1+δP0+δP0t0n=1(MLΓ(q))n(ts)nq1Γ(nq)ds+δMt0n=1(MLΓ(q))n(ts)nq1Γ(nq)sq1dsδP0n=0(MLΓ(q))ntnqΓ(nq+1)+δMtq1Γ(q)n=0(MLΓ(q))n tnqΓ(nq+q)=δ(K0tq1+C0),   t(0,τ] (3.15)

    where K0=MΓ(q)Eq,q(MLΓ(q)τq), C0=M|B|τqqEq(MLΓ(q)τq).

    Let t(τ,2τ]. Then applying Definition 1, (3.15), the inequalities τ0(τs)q1sq1ds=τ2q1Γ(q)Γ(q)Γ(2q), qΓ2(q)Γ(2q)2 we get

    |x(t)y(t)|δMtq1+LMt0(ts)q1|x(s)y(s)|ds+|B|δMτ0(ts)q1ds+|B|Mtτ(ts)q1(δK0(sτ)q1+δC0)dsδMτq1+LMδK0(tτ)2q1Γ2(q)Γ(2q)+LMδC0(2τ)qq+|B|δMτqq+δ|B|MK0(tτ)2q1Γ2(q)Γ(2q)+δ|B|MC0(tτ)qq+LMtτ(ts)q1|x(s)y(s)|dsδMτq1+2δ(L+|B|)MK0(tτ)q1(τ)qq+δ|B|MC0τqq+|B|δM(τ)qq+δLMK0τ2q1Γ(q)Γ(q)Γ(2q)+LMtτ(ts)q1|x(s)y(s)|ds2δ|B|MK0τqq(tτ)q1+δP1+LMtτ(ts)q1|x(s)y(s)|ds (3.16)

    where P1=Mτq1+|B|MC0τqq+|B|M(τ)qq+LMK0τ2q1Γ(q)Γ(q)Γ(2q).

    According to Proposition 3 we obtain

    |x(t)y(t)|2|B|MK0τqq(tτ)q1+P1++tτ[n=1(MLΓ(q))n(ts)nq1Γ(nq)(2|B|MK0τqq(sτ)q1+P1]ds2|B|MK0τqq(tτ)q1+P1Eq(MLΓ(q)(tτ)q)+2|B|MK0τqqn=1(MLΓ(q))n(tτ)1+q+nqΓ(q)Γ(nq+q)P1Eq(MLΓ(q)(tτ)q)+2|B|MK0Γ(q)τqq(tτ)q1Eq,q(MLΓ(q)(tτ)q)P1Eq(MLΓ(q)τq)+2|B|MK0Γ(q)τqq(tτ)q1Eq,q(MLΓ(q)τq)=δ(K1(tτ)q1+C1),  t(τ,2τ] (3.17)

    where K1=2|B|M2Γ2(q)E2q,q(MLΓ(q)τq)τqq and C1=P1Eq(MLΓ(q)τq).

    Let t(2τ,3τ]. Then applying Definition 1 and (3.6) we get

    |x(t)y(t)|δMtq1+LMδτ0(ts)q1(K0sq1+C0)ds+LMδ2ττ(ts)q1(K1(sτ)q1+C1)ds+LMt2τ(ts)q1|x(s)y(s)|ds+|B|Mδτ0(ts)q1ds+|B|Mδ2ττ(ts)q1(K0(sτ)q1+C0)ds+|B|Mδt2τ(ts)q1(K1(s2τ)q1+C1)dsδM(2τ)q1+δLMK0(3τ)2q1Beta[1/3,q,q]+δLMC0(3τ)qq+δLMK1(2τ)2q1Beta[1/2,q,q]+δLMC1(2q)qq+LMt2τ(ts)q1|x(s)y(s)|ds+|B|Mδ(3τ)qq+|B|MδK0(2τ)2q1Beta[1/2,q,q]+|B|MδC0(2τ)qq+δ|B|MK1τqΓ2(q)Γ(2q)(t2τ)q1+δ|B|MC1τqqδ|B|MK1τqΓ2(q)Γ(2q)(t2τ)q1+δP2+LMt2τ(ts)q1|x(s)y(s)|ds

    where Beta[x,q,q] is the incomplete beta function.

    According to Proposition 3 we obtain

    |x(t)y(t)|δ|B|MK1τqΓ2(q)Γ(2q)(t2τ)q1+δP2+δP2t2τ[n=1(MLΓ(q))n(ts)nq1Γ(nq)]ds+δ|B|MK1τqΓ2(q)Γ(2q)t2τ[n=1(MLΓ(q))n(ts)nq1Γ(nq)(s2τ)q1]dsδP2n=0(MLΓ(q))n(t2τ)nqΓ(nq+1)+δ|B|MK1τqΓ3(q)Γ(2q)(t2τ)q1n=0(MLΓ(q))n(t2τ)nqΓ(q+nq)=δ|B|MK1τqΓ3(q)Γ(2q)Eq,q(MLΓ(q)(t2τ)q)(t2τ)q1+δP2Eq(MLΓ(q)(t2τ)q)=K2(t2τ)q1+C2,   t(2τ,3τ],

    where K2=δ|B|MK1τqΓ3(q)Γ(2q)Eq,q(MLΓ(q)(t2τ)q) and C2=δP2Eq(MLΓ(q)(τ)q).

    Continue the induction process we prove the claim.

    Corollary 1. Let the conditions of Theorem 3 be satisfied and q>0.5.

    Then for any positive numbers δ,ε: ε<τ there exists a number K,C>0 such that the inequality ||gp||[τ,0]<δ implies

    |x(t)y(t)|δK for  t(ε,T], (3.18)

    where x(t),y(t) are mild solutions of the IVPs (2.5), (2.2), (2.3) and (2.5), (3.11), (3.12) respectively.

    P r o o f: The proof is similar to the one of Theorem 3 applying the inequality (tkτ)2q1qτqq for t(kτ,(k+1)τ], k=0,1,,N.

    In this section we will define and study the finite time stability of mild solutions of the IVP for Riemann-Liouville (2.5), (2.2), (2.3).

    Note that because of the singularity of tq1 at 0, we could prove the FTS on an interval which does not contain 0.

    Theorem 4. Let the function gC([τ,0],R, |g(0)|<, q>0.5 and the condition (A1) be satisfied.

    Then for any real positive numbers δ,ε: ε<τ there exists a number K depending on δ and ε such that the inequality ||g||[τ,0]<δ implies |x(t)|<K for t(ε,T] where x(t) is the mild solution of the IVP (2.5), (2.2), (2.3).

    P r o o f: Let ||g||[τ,0]<δ and M=suptJ|Eq,q(Atq)|.

    Let t(0,τ]. Then according to Definition 1 we have

    |x(t)|δEq,q(Atq)tq1+|B|δt0(ts)q1Eq,q(A(ts)q)|ds        +t0(ts)q1Eq,q(A(ts)q)|f(s,x(s))|dsδMtq1+|B|Mδt0(ts)q1ds+M||w||It0(ts)q1dsδMtq1+M(|B|δ+||w||I)τqq,   t(0,τ]. (3.19)

    From (3.19) it follows that

    |x(t)|δMεq1+M(|B|δ+||w||I)τqq,   t(ε,τ]. (3.20)

    Let t(τ,2τ]. Then we have

    |x(t)|δMτq1+Mt0(ts)q1|f(s,x(s))|ds        +|B|Mδτ0(ts)q1ds+|B|Mtτ(ts)q1x(sτ)dsδMτq1+M((||w||I+|B|δ)(2τ)qq+|B|M(δMτ2q12q1+M(||w||I+|B|δ)(τqq)2)=K1.

    Let t(2τ,3τ]. Then we have

    |x(t)|δM(2τ)q1+Mt0(ts)q1|f(s,x(s))|ds+|B|Mδτ0(ts)q1ds+|B|M2ττ(τs)q1(δM(sτ)q1+M(|B|δ+||w||I)τqq)ds+|B|MK1t2τ(ts)q1dsδM(2τ)q1+|M(|w||I+|B|δ)(3τ)qq+|B|M(δMτ2q12q1+M(|B|δ+||w||I)(τqq))2+K1τq)=K2.

    Let t(3τ,4τ]. Then we have

    |x(t)|δM(3τ)q1+|M(|w||I+|B|δ)(4τ)qq+|B|M(δMτ2q12q1+M(|B|δ+||w||I)(τqq)2)+|B|Mτq(K1+K2)=K3.

    Following the induction process we prove the claim with K=δM(Nτ)q1+|M(|w||I+|B|δ)(Nτ)qq+|B|M(δMτ2q12q1+M(|B|δ+||w||I)(τqq)2)+|B|MτqNi=1Ki.

    In the case the nonlinear Lipschitz functions we obtain the following result:

    Theorem 5. Let the function gC([τ,0],R, |g(0)|<, q>0.5 and the condition (A2) be satisfied.

    Then for any real positive numbers δ,ε: ε<τ there exists a number K depending on δ and ε such that the inequality ||g||[τ,0]<δ implies |x(t)|<K for t(ε,T] where x(t) is the mild solution of the IVP (2.5), (2.2), (2.3).

    P r o o f: According to Theorem 1 the the IVP (2.5), (2.2), (2.3) has a unique solution xPC(J,R). Let ||g||[τ,0]<δ and M=suptJ|Eq,q(Atq)|.

    Let t(ε,τ]. Then according to Definition 1 we have

    |x(t)|δEq,q(Atq)tq1+|B|δt0(ts)q1Eq,q(A(ts)q)ds        +Lt0(ts)q1Eq,q(A(ts)q)|x(s)|dsδMtq1+|B|Mδτqq+LMt0(ts)q1|x(s)|ds. (3.21)

    From (3.21) and Proposition 3 it follows that

    |x(t)|δMtq1+|B|Mδτqq+t0(n=1(LMΓ(q))nΓ(nq)(ts)nq1(δMsq1+|B|Mδτqq))dsδMtq1+|B|Mδτqq+δMtq1Γ(q)n=1(LMΓ(q))nΓ(nq+q)tnq+|B|Mδτqqn=1(LMΓ(q))nΓ(nq+1)(t)nqδMtq1+|B|Mδτqq+δMεq1Γ(q)n=1(LMΓ(q))nΓ(nq+q)tnq+|B|Mδτqqn=1(LMΓ(q))nΓ(nq+1)(t)nqδMtq1Γ(q)Eq,q(LMΓ(q))tq)+|B|MδτqqEq(LMΓ(q))tq). (3.22)

    Therefore,

    x(t)δ(Mεq1Γ(q)+|B|Mτqq)Eq(LMΓ(q))τq)=δK0,  t(ε,τ]. (3.23)

    Let t(τ,2τ]. Then from (3.23) we have

    |x(t)|δMtq1+MLt0(ts)q1|x(s)|ds        +|B|Mδτ0(ts)q1ds+|B|Mtτ(ts)q1x(sτ)dsδMtq1+|B|Mδ(1+K0)τqq+MLt0(ts)q1|x(s)|ds.

    From (3.24) and Proposition 3 it follows that

    |x(t)|δMtq1+|B|Mδ(1+K0)τqq+tτ(n=1(MLΓ(q))nΓ(nq)(ts)nq1(δMsq1+|B|Mδ(1+K0)τqq))dsδMtq1Γ(q)Eq,q(LMΓ(q))tq)+|B|Mδ(1+K0)τqq]Eq(LMΓ(q))tq). (3.24)

    Therefore,

    |x(t)|δ(Mεq1Γ(q)+|B|M(1+K0)τqq)Eq(LMΓ(q))τq)=δK1,  t(τ,2τ] (3.25)

    Following the induction process we obtain

    |x(t)|δ(Mεq1Γ(q)+|B|M(1+Kk1)τqq)Eq(LMΓ(q))τq)=δKk,  t(kτ,(k+1)τ],

    where Kk=(Mεq1Γ(q)+|B|M(1+Kk1)τqq)Eq(LMΓ(q))τq), k=1,2,,N.

    Example 4. Consider the IVP (3.1) with RL fractional equation  RL0D0.5tx(t)=0.1x(t1)+0.01sin(x(t)). According to Example 3 it has unique mild solution x(t) which is satisfying the integral presentation given in Definition 1. Also, according to Theorem 5 for δ=1, ε=0.001 the inequality |x(t)|<K holds for t[0.001,3] where M=sup[0,3]E0.5,0.5(0.1t0.5)=0.7772, K0=(0.77720.0010.51Γ(0.5)+0.10.777210.5)Eq(0.110.7772Γ(0.5))=52.321, K1=62.0518 and K=K2=63.8615.

    We study scalar nonlinear RL fractional differential equations with constant delays. An appropriate initial value problem for studd equations is set up based on the idea of the initial time interval for delay differential equations with ordinary derivatives. A mild solution is defined based on an appropriate integral representation of the solution. The existence, continuous dependence and finite time stability of the scalar nonlinear RL fractional differential equations is studied by the help of fractional generalization of Gronwall inequality. The obtained integral representations could be successfully applied to study many qualitative investigation of the properties of the solutions of nonlinear RL fractional differential equations.

    Research was partially supported by Fund Scientific Research MU19-FMI-009, Plovdiv University.

    All authors declare no conflicts of interest in this paper.



    [1] H. Zhang, R. Ye, S. Liu, et al. LMI-based approach to stability analysis for fractional-order neural networks with discrete and distributed delays, Int. J. Syst. Sci., 49 (2018), 537-545. doi: 10.1080/00207721.2017.1412534
    [2] W. Zhang, J. Cao, R. Wu, et al. Lag projective synchronization of fractional-order delayed chaotic systems, J. Franklin Institute, 356 (2019), 1522-1534. doi: 10.1016/j.jfranklin.2018.10.024
    [3] W. Zhang, H. Zhang, J. Cao, et al. Synchronization in uncertain fractional-order memristive complex-valued neural networks with multiple time delays, Neural Networks, 110 (2019), 186-198. doi: 10.1016/j.neunet.2018.12.004
    [4] P. Dorato, Short time stability in linear time-varying systems, Proc. IRE Int. Convention Record, 4 (1961), 83-87.
    [5] D. F. Luo, Z. G. Luo, Uniqueness and novel finite-time stability of solutions for a class of nonlinear fractional delay difference systems, Discr. Dynam, Nature Soc., 2018 (2018), 1-7.
    [6] G. C. Wu, D. Baleanu, S. D. Zeng, Finite-time stability of discrete fractional delay systems: Gronwall inequality and stability criterion, Commun. Nonl. Sci. Numer. Simul., 57 (2018), 299-308. doi: 10.1016/j.cnsns.2017.09.001
    [7] V. N. Phat, N. T. Thanh, New criteria for finite-time stability of nonlinear fractional-order delay systems: A Gronwall inequality approach, Appl. Math. Lett., 83 (2018), 169-175. doi: 10.1016/j.aml.2018.03.023
    [8] D. F. Luo, Z. G. Luo, Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses, Adv. Diff. Eq., 155, 2019.
    [9] D. Qian, C. Li, R. P. Agarwal, et al. Stability analysis of fractional differential system with Riemann-Liouville derivative, Math. Comput. Modell., 52 (2010), 862-874. doi: 10.1016/j.mcm.2010.05.016
    [10] M. Li, J. Wang, Finite time stability of fractional delay differential equations, Appl. Math. Lett., 64 (2017), 170-176. doi: 10.1016/j.aml.2016.09.004
    [11] M. Li, J. R. Wang, Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations, Appl. Math. Comput., 324 (2018), 254-265.
    [12] M. Li, J. R. Wang, Finite time stability and relative controllability of Riemann-Liouville fractional delay differential equations, Math. Meth. Appl. Sci., 2019 (2019), 1-17.
    [13] K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, Heidelberg, 2010.
    [14] I. Podlubny, Fractional Differential Equations, Academic Press: San Diego, 1999.
    [15] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
    [16] H. Ye, J. Gao, Y. Ding, A generalized Gronwall inequality and its application to a fracnal differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081. doi: 10.1016/j.jmaa.2006.05.061
    [17] R. Agarwal, S. Hristova, D. O'Regan, Explicit solutions of initial value problems for linear scalar Riemann-Liouville fractional differential equations with a constant delay, Mathematics, 8 (2020), 1-14.
  • This article has been cited by:

    1. S. Hristova, T. Donchev, 2021, 2321, 0094-243X, 030012, 10.1063/5.0040075
    2. Fouzia Bekada, Saïd Abbas, Mouffak Benchohra, Juan J. Nieto, Dynamics and stability for Katugampola random fractional differential equations, 2021, 6, 2473-6988, 8654, 10.3934/math.2021503
    3. Sinan Serkan Bilgici, Müfit ŞAN, Existence and uniqueness results for a nonlinear singular fractional differential equation of order σ(1,2), 2021, 6, 2473-6988, 13041, 10.3934/math.2021754
    4. Chen Chen, Qixiang Dong, Existence and Hyers–Ulam Stability for a Multi-Term Fractional Differential Equation with Infinite Delay, 2022, 10, 2227-7390, 1013, 10.3390/math10071013
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4270) PDF downloads(403) Cited by(4)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog