Processing math: 74%
Research article

Nonlocal coupled system for ψ-Hilfer fractional order Langevin equations

  • In the present work a coupled system consisting by ψ-Hilfer fractional order Langevin equations supplemented with nonlocal integral boundary conditions is studied. Existence and uniqueness results are obtained by using standard fixed point theorems. The obtained results are well illustrated by numerical examples.

    Citation: Weerawat Sudsutad, Sotiris K. Ntouyas, Chatthai Thaiprayoon. Nonlocal coupled system for ψ-Hilfer fractional order Langevin equations[J]. AIMS Mathematics, 2021, 6(9): 9731-9756. doi: 10.3934/math.2021566

    Related Papers:

    [1] Weerawat Sudsutad, Chatthai Thaiprayoon, Sotiris K. Ntouyas . Existence and stability results for ψ-Hilfer fractional integro-differential equation with mixed nonlocal boundary conditions. AIMS Mathematics, 2021, 6(4): 4119-4141. doi: 10.3934/math.2021244
    [2] Subramanian Muthaiah, Dumitru Baleanu, Nandha Gopal Thangaraj . Existence and Hyers-Ulam type stability results for nonlinear coupled system of Caputo-Hadamard type fractional differential equations. AIMS Mathematics, 2021, 6(1): 168-194. doi: 10.3934/math.2021012
    [3] Subramanian Muthaiah, Manigandan Murugesan, Muath Awadalla, Bundit Unyong, Ria H. Egami . Ulam-Hyers stability and existence results for a coupled sequential Hilfer-Hadamard-type integrodifferential system. AIMS Mathematics, 2024, 9(6): 16203-16233. doi: 10.3934/math.2024784
    [4] Najla Alghamdi, Bashir Ahmad, Esraa Abed Alharbi, Wafa Shammakh . Investigation of multi-term delay fractional differential equations with integro-multipoint boundary conditions. AIMS Mathematics, 2024, 9(5): 12964-12981. doi: 10.3934/math.2024632
    [5] Arjumand Seemab, Mujeeb ur Rehman, Jehad Alzabut, Yassine Adjabi, Mohammed S. Abdo . Langevin equation with nonlocal boundary conditions involving a ψ-Caputo fractional operators of different orders. AIMS Mathematics, 2021, 6(7): 6749-6780. doi: 10.3934/math.2021397
    [6] Murugesan Manigandan, Kannan Manikandan, Hasanen A. Hammad, Manuel De la Sen . Applying fixed point techniques to solve fractional differential inclusions under new boundary conditions. AIMS Mathematics, 2024, 9(6): 15505-15542. doi: 10.3934/math.2024750
    [7] Nichaphat Patanarapeelert, Jiraporn Reunsumrit, Thanin Sitthiwirattham . On nonlinear fractional Hahn integrodifference equations via nonlocal fractional Hahn integral boundary conditions. AIMS Mathematics, 2024, 9(12): 35016-35037. doi: 10.3934/math.20241667
    [8] Ugyen Samdrup Tshering, Ekkarath Thailert, Sotiris K. Ntouyas . Existence and stability results for a coupled system of Hilfer-Hadamard sequential fractional differential equations with multi-point fractional integral boundary conditions. AIMS Mathematics, 2024, 9(9): 25849-25878. doi: 10.3934/math.20241263
    [9] Ayub Samadi, Chaiyod Kamthorncharoen, Sotiris K. Ntouyas, Jessada Tariboon . Mixed Erdélyi-Kober and Caputo fractional differential equations with nonlocal non-separated boundary conditions. AIMS Mathematics, 2024, 9(11): 32904-32920. doi: 10.3934/math.20241574
    [10] Mona Alsulami . Existence theory for a third-order ordinary differential equation with non-separated multi-point and nonlocal Stieltjes boundary conditions. AIMS Mathematics, 2023, 8(6): 13572-13592. doi: 10.3934/math.2023689
  • In the present work a coupled system consisting by ψ-Hilfer fractional order Langevin equations supplemented with nonlocal integral boundary conditions is studied. Existence and uniqueness results are obtained by using standard fixed point theorems. The obtained results are well illustrated by numerical examples.



    Fractional calculus is a natural generalization of ordinary calculus when the order of the derivative is non-integer. Many fractional operators are defined like, Riemman-Liouville, Hadamard, Caputo and Grunwald-Letnikov [12,20,21]. The choice of the operator depends upon the considered system. Due to the variations in the physical systems, many researchers are defining a number of fractional operators, for details, we refer the readers to the article [24].

    Many of the complex physical problems may be better understood in the framework of fractional differential equations. One can find applications of fractional calculus in diverse fields like biology, chemistry, physics, fluid mechanics, economics and social sciences, etc. [1,16,17,21,26].

    For theoretical details of fractional differential equations see [16,18] and the references cited therein. The existence results for non-integer order differential equations are discussed in many articles, for example, see [10,11,13,14]. Stability analysis of fractional differential equations is an important aspect of the qualitative theory of fractional differential equations form numerical and optimization point of view. Ulam in [25], raised the question ``Under what conditions there exist an additive mapping near to an approximate additive mapping?". This question initiated the study of stability of differential equations and the answer of Ulam's question was given by Hyers in [15]. Later on Rassias [22], developed a technique for Hyers-Ulam stability of linear and nonlinear mappings. The theory of fractional equations involving different kinds of boundary conditions has always been remained a field of interest in physical sciences. Non-local and integral boundary conditions are widely used where classical boundary conditions fail to examine many physical properties of the models. Many researchers have already been involved in the existence theory of boundary value problems involving non-local and integral boundary conditions, for example [2,3,4,6,7,8,9,19,23]. Recently Alsaedi et al. [5], discussed the existence theory of the following second order boundary value problem with non-local non-separated type integral multi-point boundary conditions on an arbitrary domain.

    {u(t)=f(t,u(t)) , a<t<T, a, TR,α1u(a)+α2u(T)=α3ζau(τ)dτ+mι=1γιu(vι),β1 u(a)+β2 u(T)=β3ζa u(s)ds+mι=1ρι u(vι), (1.1)

    where aζviT. Motivated by [5], we prove the existence results for solution of a fractional boundary value problem involving Caputo fractional derivative of order 1<m2 and nonlocal non-separated type integral multipoint boundary conditions involving Caputo fractional derivative of order 0<p1 on an arbitrary domain. In fact we consider the following fractional differential equation

    cDmu(t)=ϝ(t,u(t)) where 1<m2, (1.2)

    with boundary conditions

    γ1u(a)+γ2u(T)=γ3ζau(τ)dτ+γ4eι=1ϑιu(vι)ϱ1 cDpu(a)+ϱ2 cDpu(T)=ϱ3ζa cDpu(τ)dτ+ϱ4eι=1ρι cDpu(vι), 0<p1. (1.3)

    Clearly for m=2, p=1 and γ4=ϱ4=1, the above problems (1.2)(1.3) reduces to (1.1).

    The remaining part of the article is arranged as follows. Section 2 contains the existence results for the fractional boundary value problems (1.2)(1.3) which are proved by applying Schaefer type and Krasnoselskii's fixed point theorems. In Section 3, we prove an existence and uniqueness result by using Banach contraction principle. Hyers-Ulam stability is discussed in Section 4.

    Let H=C[a,T] be the Banach space of all continuous functions defined on closed interval [a,T]R with norm

    u=supt[a,T]|u(t)| (2.1)

    for uH. Set

    α1=γ1+γ2γ3(ζa)γ4eι=1ϑι (2.2)

    and

    β=ϱ2(Ta)1pΓ(2p)ϱ3(ζa)2pΓ(3p)ϱ4eι=1ρι(vιa)1pΓ(2p). (2.3)

    The following lemma will be crucial for coming existence results.

    Lemma 2.1. Let σH and α1 and β are nonzero. Then the solution of the following linear problem (2.4) with boundary conditions (1.3)

    cDmu(t)=σ(t), for a<t<T (2.4)

    has an integral representation given as

    u(t)=ta(tr)m1Γ(m)σ(r)dr+1Kζa(βγ3(ζr)pΓ(m+1)+ϱ3Γ(mp+1)η(t))(ζr)mpσ(r)dr1KTa(βγ2(Tr)pΓ(m)+ϱ2Γ(mp)η(t))(Tr)mp1σ(r)dr+1Keι=1vιa(βγ4ϑι(vιr)pΓ(m)+ϱ4Γ(mp)ριη(t))(vιr)mp1σ(r)dr (2.5)

    where

    α2=γ2(Ta)γ3(ζa)22γ4eι=1ϑι(vιa), (2.6)
    η(t)=α1(ta)α2 (2.7)

    and

    K=βα1. (2.8)

    Proof. The general solution for linear problem

    cDmu(t)=σ(t), a<t<T and 1<m2

    as given in [16], is

    u(t)=ta(tr)m1Γ(m)σ(r)drd1d2(ta) (2.9)

    for some real constants d1 and d2. Making use of (1.3),(2.2) and (2.6) in (2.9), we have

    d1α1+d2α2=γ2Ta(Tr)m1Γ(m)σ(r)drγ3ζa(ζr)mΓ(m+1)σ(r)drγ4eι=1ϑιvιa(vιr)m1Γ(m)σ(r)dr.
    d1=1α1(d2α2+γ2Ta(Tr)m1Γ(m)σ(r)drγ3ζa(ζr)mΓ(m+1)σ(r)drγ4eι=1ϑιvιa(vιr)m1Γ(m)σ(r)dr). (2.10)

    From (2.9)

    u(t)=ta(tr)m1Γ(m)σ(r)drd1d2t

    taking Caputo fractional derivative of order p(0,1) on both sides and using its properties, we get

    cDpu(t)=ta(tr)mp1Γ(mp)σ(r)drd2(ta)1pΓ(2p). (2.11)

    Now using the second boundary condition from (1.3) in (2.11) we get

    ϱ2Ta(Tr)mp1Γ(mp)σ(r)drϱ2d2(Ta)1pΓ(2p)=ϱ3ζaτa(τr)mp1Γ(mp)σ(r)drdτϱ3d2(ζa)2p(2p)Γ(2p)+ϱ4eι=1ριvιa(vιr)mp1Γ(mp)σ(r)drϱ4eι=1ριd2(vιa)1pΓ(2p),

    simplifying, we obtain

    d2(ϱ2(Ta)1pΓ(2p)ϱ3(ζa)2p(2p)Γ(2p)ϱ4eι=1ρι(vιa)1pΓ(2p))=ϱ2Ta(Tr)mp1Γ(mp)σ(r)drϱ4eι=1ριvιa(vιr)mp1Γ(mp)σ(r)drϱ3ζa(ζr)mp(mp)Γ(mp)σ(r)dr.

    Using (2.3) in above equation, we have

    d2=1β(ϱ2Ta(Tr)mp1Γ(mp)σ(r)drϱ3ζa(ζr)mpΓ(mp+1)σ(r)drϱ4eι=1ριvιa(vιr)mp1Γ(mp)σ(r)dr), (2.12)

    substituting (2.12) in (2.10) gives

    d1=α2βα1(ϱ2Ta(Tr)mp1Γ(mp)σ(r)drϱ3ζa(ζr)mpΓ(mp+1)σ(r)drϱ4eι=1ριvιa(vιr)mp1Γ(mp)σ(r)dr)+1α1(γ2Ta(Tr)m1Γ(m)σ(r)drγ3ζa(ζr)mΓ(m+1)σ(r)drγ4eι=1ϑιvιa(vιr)m1Γ(m)σ(r)dr). (2.13)

    Making use of (2.7),(2.8),(2.12) and (2.13) in (2.9), we get the required solution

    u(t)=ta(tr)m1Γ(m)σ(r)dr+1Kζa(βγ3(ζr)pΓ(m+1)+ϱ3Γ(mp+1)η(t))(ζr)mpσ(r)dr1KTa(βγ2(Tr)pΓ(m)+ϱ2Γ(mp)η(t))(Tr)mp1σ(r)dr+1Keι=1vιa(βγ4ϑι(vιr)pΓ(m)+ϱ4ριΓ(mp)η(t))(vιr)mp1σ(r)dr.

    The solution of boundary value problems (1.2)(1.3) exists if and only if the following operator Q:HH defined by

    Q(u(t))=ta(tr)m1Γ(m)ϝ(r,u(r))dr+1Kζa(βγ3(ζr)pΓ(m+1)+ϱ3Γ(mp+1)η(t))(ζr)mpϝ(r,u(r))dr1KTa(βγ2(Tr)pΓ(m)+ϱ2Γ(mp)η(t))(Tr)mp1ϝ(r,u(r))dr+1Keι=1vιa(βγ4ϑι(vιr)pΓ(m)+ϱ4ριΓ(mp)η(t))(vιr)mp1ϝ(r,u(r))dr, (2.14)

    has a fixed point.

    To reduce computational complexities we set

    c=(Ta)mΓ(m+1)+1|α1|(|γ3|(ζa)m+1Γ(m+2)+|γ2|(Ta)mΓ(m+1)+eι=1|γ4|ϑι(vιa)mΓ(m+1))
    +|α1(Ta)α2|K(|ϱ3|(ζa)mp+1Γ(mp+2)+|ϱ2|(Ta)mpΓ(mp+1)+eι=1|ϱ4|ρι(vιa)mpΓ(mp+1)). (2.15)
    d=|1α1|(|γ3|(ζa)m+1Γ(m+2)+|γ2|(Ta)mΓ(m+1)+|γ4|eι=1ϑι(vιa)mΓ(m+1))
    +|α1(Ta)α2K|(|ϱ3|(ζa)mp+1Γ(mp+2)+|ϱ2|(Ta)mpΓ(mp+1)+|ϱ4|eι=1ρι(vιa)mpΓ(mp+1)). (2.16)

    Theorem 2.2 (Schaefer).[16] Let H be a Banach space and Q:HH be a completely continuous mapping. Then either the equation y=λQy has a solution for λ=1, or the set {yH:y=λQy for some λ(0,1)} is unbounded.

    The following is the existence results by applying above fixed point theorem.

    Theorem 2.3. Let ϝ:[a,T]×RR be a continuous function. If there exists L1>0 such that |ϝ(t,u(t))|L1 for all t[a,T], uR, then there exists a solution of the boundary value problems (1.2)(1.3).

    Proof. First it will be verified that the operator Q:HH is completely continuous. Since ϝ is continuous, this implies the continuity of Q. For a positive constant ϵ, define Bϵ={uH:uϵ} a bounded set in H. We prove that the operator Q maps bounded sets into bounded sets of H. For uBϵ, t[a,T], we consider

       (Qu)=supt[a,T]|ta(tr)m1Γ(m)ϝ(r,u(r))dr                    +1Kζa(βγ3(ζr)pΓ(m+1)+ϱ3Γ(mp+1)η(t))(ζr)mpϝ(r,u(r))dr                    1KTa(βγ2(Tr)pΓ(m)+ϱ2Γ(mp)η(t))(Tr)mp1ϝ(r,u(r))dr                    +1Keι=1vιa(βγ4ϑι(vιr)pΓ(m)+ϱ4ριΓ(mp)η(t))(vιr)mp1ϝ(r,u(r))dr|             |ϝ(r,u(r))|[(ta)mΓ(m+1)+1|α1|(|γ3|(ζa)m+1Γ(m+2)+|γ2|(Ta)mΓ(m+1)+eι=1|γ4|ϑι(vιa)mΓ(m+1))                  +|α1(ta)α2|k(|ϱ3|(ζa)mp+1Γ(mp+2)+|ϱ2|(Ta)mpΓ(mp+1)+|ϱ4|ρι(vιa)mpΓ(mp+1))]                      L1c, (2.17)

    where c is defined by (2.15).

    Next we show that operator Q:HH maps bounded set into equicontinuous set of H. For at1t2T and uBϵ, we consider

    |(Qu)(t2)(Qu)(t1)||t2a(t2r)m1Γ(m)ϝ(r,u(r))dr|                               
    +|1Kζa(βγ3(ζr)pΓ(m+1)+ϱ3Γ(mp+1)η(t2))(ζr)mpϝ(r,u(r))dr|
    +|1KTa(βγ2(Tr)pΓ(m)+ϱ2Γ(mp)η(t2))(Tr)mp1ϝ(r,u(r))dr|
    +|1Keι=1vιa(βγ4ϑι(vιr)pΓ(m)+ϱ4ριΓ(mp)η(t2))(vιr)mp1ϝ(r,u(r))dr|
    +|t1a(t1r)m1Γ(m)ϝ(r,u(r))dr|
    +|1Kζa(βγ3(ζr)pΓ(m+1)+ϱ3Γ(mp+1)η(t1))(ζr)mpϝ(r,u(r))dr|
    +|1KTa(βγ2(Tr)pΓ(m)+ϱ2Γ(mp)η(t1))(Tr)mp1ϝ(r,u(r))dr|
    +|1Keι=1vιa(βγ4ϑι(vιr)pΓ(m)+ϱ4ριΓ(mp)η(t1))(vιr)mp1ϝ(r,u(r))dr|
    |ϝ(r,u(r))|[(t2t1)mΓ(m+1)+(t1a)m(t2a)mΓ(m+1)+(t2t1)mΓ(m+1)                   +|α1(t2t1)K|(|ϱ3|(ζa)mp+1Γ(mp+2)+|ϱ2|(Ta)mpΓ(mp+1)+eι=1|ϱ4|ρι(vιa)mpΓ(mp+1))]L1[(t2t1)mΓ(m+1)+(t1a)m(t2a)mΓ(m+1)+(t2t1)mΓ(m+1)                               +|α1(t2t1)K|(|ϱ3|(ζa)mp+1Γ(mp+2)+|ϱ2|(Ta)mpΓ(mp+1)+eι=1|ϱ4|ρι(vιa)mpΓ(mp+1)).

    As t1t2, the above expression approaches to zero independent of uBϵ. Hence, by the Arzelá Ascoli theorem, the operator Q:HH is completely continuous. Finally, we show that the set V={uH:u=λQu, 0λ1} is bounded. For uV and t[a,T], and using inequalities (2.17) and (2.15), we have

    u=supt[a,T]|λ(Qu)t|L1c.

    Hence by Theorem 2.2, Q has a fixed point in H.

    Theorem 2.4 (Krasnoselskii).[16] Let Ω be a closed bounded, convex and nonempty subset of a Banach space X. Let g1,g2 be the operators such that

    (i) g1y1+g2y2Ω, whenever y1,y2Ω.

    (ii) g1 is compact and continuous;

    (iii) g2 is a contraction.

    Then there exist y3Ω such that y3= g1y3+g2y3.

    We apply the above theorem to prove the following existence result.

    Theorem 2.5. Let ϝ:[a,T]×RR be a continuous function satisfying:

    (H1) |ϝ(t,u)ϝ(t,v)|L|uv|, for all t[a,T], L>0, and u,vR,

    (H2) there exists a function μC([a,T],R+) with |ϝ(t,u)|μ(t),  (t,u)[a,T]×R.

    If d in (2.16) satisfies

    d<1L, (2.18)

    then there exists a solution of problems (1.2)(1.3).

    Proof. Consider a set Br={uH:ur} with rcu, clearly Br is a closed subset of H, where c is given in (2.15). We decompose the operator Q defined in (2.14) into sum of two operators Q1 and Q2 on Br as follows:

    (Q1u)(t)=ta(tr)m1Γ(m)ϝ(r,u(r))dr

    and

    (Q2u)(t)=1Kζa(βγ3(ζr)pΓ(m+1)+ϱ3Γ(mp+1)η(t))(ζr)mpϝ(r,u(r))dr1KTa(βγ2(Tr)pΓ(m)+ϱ2Γ(mp)η(t))(Tr)mp1ϝ(r,u(r))dr+1Keι=1vιa(βγ4ϑι(vιr)pΓ(m)+ϱ4ριΓ(mp)η(t))(vιr)mp1ϝ(r,u(r))dr.

    For u,vBr, consider

    Q1u+Q2v                                                                                            =supt[a,T]|ta(tr)m1Γ(m)ϝ(r,u(r))dr                                                                +1Kζa(βγ3(ζr)pΓ(m+1)+ϱ3Γ(mp+1)η(t))(ζr)mpϝ(r,v(r))dr1KTa(βγ2(Tr)pΓ(m)+ϱ2Γ(mp)η(t))(Tr)mp1ϝ(r,v(r))dr+1Keι=1vιa(βγ4ϑι(vιr)pΓ(m)+ϱ4ριΓ(mp)η(t))(vιr)mp1ϝ(r,v(r))dr|
                μsupt[a,T][(ta)mΓ(m+1)+|βγ3k|(ζa)m+1Γ(m+2)+|βγ2k|(Ta)mΓ(m+1)+eι=1|βγ4k|ϑι(vιa)mΓ(m+1)         +|ϱ3η(t)k|(ζa)mp+1Γ(mp+2)+|ϱ2η(t)k|(Ta)mpΓ(mp+1)+|ϱ4η(t)k|ρι(vιa)mpΓ(mp+1)]
                  μ[(Ta)mΓ(m+1)+1|α1|(|γ3|(ζa)m+1Γ(m+2)+|γ2|(Ta)mΓ(m+1)+eι=1|γ4|ϑι(vιa)mΓ(m+1))        +|α1(Ta)α2|k(|ϱ3|(ζa)mp+1Γ(mp+2)+|ϱ2|(Ta)mpΓ(mp+1)+|ϱ4|ρι(vιa)mpΓ(mp+1))]            μcr.                                                                                     

    Thus Q1u+Q2vBr, which verifies assumption (i) in Theorem 2.4. For u,vBr, consider

       Q2uQ2v=supt[a,T]|[1Kζa(βγ3(ζr)pΓ(m+1)+ϱ3Γ(mp+1)η(t))(ζr)mpϝ(r,u(r))dr1KTa(βγ2(Tr)pΓ(m)+ϱ2Γ(mp)η(t))(Tr)mp1ϝ(r,u(r))dr+1Keι=1vιa(βγ4ϑι(vιr)pΓ(m)+ϱ4ριΓ(mp)η(t))(vιr)mp1ϝ(r,u(r))dr][1Kζa(βγ3(ζr)pΓ(m+1)+ϱ3Γ(mp+1)η(t))(ζr)mpϝ(r,v(r))dr1KTa(βγ2(Tr)pΓ(m)+ϱ2Γ(mp)η(t))(Tr)mp1ϝ(r,v(r))dr+1Keι=1vιa(βγ4ϑι(vιr)pΓ(m)+ϱ4ριΓ(mp)η(t))(vιr)mp1ϝ(r,v(r))dr]|

    using (H1)

           L[|1α1|(|γ3|(ζa)m+1Γ(m+2)+|γ2|(Ta)mΓ(m+1)+|γ4|eι=1ϑι(vιa)mΓ(m+1))                                +|α1(Ta)α2K|(|ϱ3|(ζa)mp+1Γ(mp+2)+|ϱ2|(Ta)mpΓ(mp+1)+|ϱ4|eι=1ρι(vιa)mpΓ(mp+1))]uv.          =Lduv.                                                                                       

    From (2.18), we have Ld<1, so Q2 is a contraction.

    Next, we show that Q1 is compact and continuous. The continuity of ϝ implies the continuity of Q1 and since Q1uμ(Ta)mΓ(m+1), therefore Q1 is uniformly bounded on Br.

    Set supt[a,T]×βr|ϝ(t,u)|=ˆϝ and for at1t2T, consider

    Q1u(t2)Q1u(t1)=t2a(t2r)m1Γ(m)ϝ(r,u(r))drt1a(t1r)m1Γ(m)ϝ(r,u(r))drˆϝt1a(t2r)m1Γ(m)dr+t2t1(t2r)m1Γ(m)drt1a(t1r)m1Γ(m)dr=[(t1a)m(t2a)mΓ(m+1)]ˆϝ.

    As t1t2, the above expression tends to zero independent of uBr. This implies that Q1 is relatively compact on Br. Hence, it follows by the Arzelá Ascoli theorem that the operator Q1 is compact on Br. Thus all the hypothesis of the Theorem 2.4, are satisfied. Therefore the problems (2.1)(1.3) has at least one solution.

    Theorem 3.1 (Banach). Let (M,ρ) be a complete metric space and T:MM be a self mapping. If there exists δ(0,1) such that

    ρ(Tx,Ty)δρ(x,y)

    for all x,yM. Then T has a unique fixed point.

    Now we state and prove our result regarding uniqueness of solution.

    Theorem 3.2. Suppose ϝ:[a.T]×RR is continuous and satisfying (H1). If L<c1, where c is defined in (2.15), then there exists a unique solution of the problems (1.2)(1.3).

    Proof. Set

    supt[a,T]ϝ(t,0)=L2 (3.1)

    and choose ϵcL21Lc>0. We show that the mapping Q defined in (2.14) satisfies Q(Bϵ)Bϵ. For uBϵ and t[a,T], we consider

    |ϝ(t,u(t))|=|[ϝ(t,u(t))ϝ(t,0)]+ϝ(t,0)||ϝ(t,u(t))ϝ(t,0)|+|ϝ(t,0)|Lu+L2.

    Now for uBϵ, consider

    Q(u)=supt[a,T]|ta(tr)m1Γ(m)ϝ(r,u(r))dr+1Kζa(βγ3(ζr)pΓ(m+1)+ϱ3Γ(mp+1)η(t))(ζr)mpϝ(r,u(r))dr1KTa(βγ2(Tr)pΓ(m)+ϱ2Γ(mp)η(t))(Tr)mp1ϝ(r,u(r))dr+1Keι=1vιa(βγ4ϑι(vιr)pΓ(m)+ϱ4ριΓ(mp)η(t))(vιr)mp1ϝ(r,u(r))dr|.(Lϵ+L2)cϵ.

    Thus Q(Bϵ)Bϵ.

    Now we show that Q is a contraction. For u,vH, we have

    QuQv=supt[a,T]|ta(tr)m1Γ(m)(ϝ(r,u(r))ϝ(r,v(r)))dr+1Kζa(βγ3(ζr)pΓ(m+1)+ϱ3Γ(mp+1)η(t))(ζr)mp(ϝ(r,u(r))ϝ(r,v(r)))dr1KTa(βγ2(Tr)pΓ(m)+ϱ2Γ(mp)η(t))(Tr)mp1(ϝ(r,u(r))ϝ(r,v(r)))dr+1Keι=1vιa(βγ4ϑι(vιr)pΓ(m)+ϱ4ριΓ(mp)η(t))(vιr)mp1(ϝ(r,u(r))ϝ(r,v(r)))dr|.Lcuv, (3.2)

    Since Lc<1, therefore Q is contraction, so there exists a unique solution w of Q which is a unique solution of (1.2)(1.3).

    The following is the example that illustrate the above theorem to ensure the existence of a unique solution.

    Example 3.3. Consider the following non-separated multi-point fractional boundary value problem

    cD1.75u(t)=1t+14tan1(u(t)0.5+etcost)+sint, 2<t<3,γ1u(2)+γ2u(3)=γ3ζau(τ)dτ+γ44ι=1ϑιu(vι),ϱ1 cD0.5u(2)+ϱ2 cD0.5u(3)=ϱ3ζa cD0.5u(τ)dτ+ϱ4eι=1ρι cD0.5u(vι),

    where γ1=1,γ2=12,γ3=1,γ4=13,ϱ1=1=ϱ2=ϱ3,ϱ4=13, ϑ1=110,ϑ2=320,ϑ3=14,ϑ4=1, v1=115,v2=125, v3=135,v4=145, ρ1=ρ2=1719,ρ3=ρ4=911. Note that |ϝ(t,u(t))|π4+1=L and |ϝ(t,u)ϝ(t,v)|L|uv|, by assuming L=14. Since Lc<1, so by Theorem 3.1, it has a unique solution.

    In this section we discuss the criteria for Ulam stability of the problems (1.2)(1.3). The following remarks and definitions will be crucial for proof of our result.

    Remark 4.1. From Lemma 2.1 we can write the solution uH of fractional BVP (1.2)(1.3) as;

    u(t)=TaG(t,r)ϝ(r,u(r))dr, (4.1)

    where

    aζvιT,

    and

    G(t,r)={(tr)m1Γ(m)+1K(βγ3(ζr)pΓ(m+1)+ϱ3Γ(mp+1)η(t))(ζr)mp1K(βγ2(Tr)pΓ(m)+ϱ2Γ(mp)η(t))(Tr)mp1,   art1K(βγ3(ζr)pΓ(m+1)+ϱ3Γ(mp+1)η(t))(ζr)mp1K(βγ2(Tr)pΓ(m)+ϱ2Γ(mp)η(t))(Tr)mp1+1Keι=1(βγ4ϑι(vιr)pΓ(m)+ϱ4Γ(mp)ριη(t))(vιr)mp1,  trζ1K(βγ2(Tr)pΓ(m)+ϱ2Γ(mp)η(t))(Tr)mp1+1Keι=1(βγ4ϑι(vιr)pΓ(m)+ϱ4Γ(mp)ριη(t))(vιr)mp1,  ζrvι1K(βγ2(Tr)pΓ(m)+ϱ2Γ(mp)η(t))(Tr)mp1,  vιrT.. (4.2)

    Moreover

    \begin{equation*} \left \vert G(t, r)\right \vert \leq \left \vert \frac{T^{m}}{\Gamma (m)} \right \vert +6\left \vert \frac{T^{m}}{K}\right \vert \left \vert \beta \max \left \{ \gamma _{3}, \gamma _{2}\right \} \frac{\left( \zeta \right) ^{p}}{ \Gamma \left( m+1\right) }\right \vert +\left \vert \frac{\max \left \{ \varrho _{3}, \varrho _{2}\right \} }{\Gamma \left( m-p+1\right) }\eta ^{\ast }\right \vert \end{equation*}
    \begin{equation} +2\frac{1}{K}\sum \limits_{\iota = 1}^{e}\left \vert \beta \gamma _{4}\vartheta _{\iota }\frac{(v_{\iota })^{p}}{\Gamma \left( m\right) }+\frac{\varrho _{4} }{\Gamma (m-p)}\rho _{\iota }\eta ^{\ast }\right \vert : = \Lambda , \ \ \ \ \ \ \ \ \end{equation} (4.3)

    where

    \begin{equation} \eta ^{\ast } = \underset{t\in \left[ a, T\right] }{\max }\left \vert \eta \left( t\right) \right \vert . \end{equation} (4.4)

    Definition 4.2. The fractional BVP \left(1.2\right) \left(1.3\right) is said to be Hyers-Ulam stable if there exist constants \lambda > 0 , such that for each \epsilon > 0 and for each solution v\in H of

    \begin{equation} \left \vert ^{c}D^{m}v(t)-\digamma \left( t, v\left( t\right) \right) \right \vert \leq \epsilon , \text{ }t\in \left[ a, T\right] \end{equation} (4.5)

    there exists a solution u\in H of \left(1.2\right) \left(1.3\right) such that

    \begin{equation} \left \vert v\left( t\right) -u\left( t\right) \right \vert \leq \lambda \epsilon , \text{ }t\in \left[ a, T\right] . \end{equation} (4.6)

    Remark 4.3. A function v\in H is a solution of inequality \left(4.5\right) if and only if there exists a function \omega \in H such that

    \left(i\right) \left \vert \omega \left(t\right) \right \vert \leq \epsilon, t\in \left[a, T\right],

    \left(ii\right) ^{c}D^{m}v(t) = \digamma \left(t, v\left(t\right) \right) +\omega \left(t\right), t\in \left[a, T\right].

    In the following result we obtain the criteria under which the problem \left(1.2\right) \left(1.3\right) is Hyers-Ulam stable.

    Theorem 4.4. Suppose \digamma :\left[a.T\right] \times \mathbb{R} \rightarrow \mathbb{R} is continuous and satisfying \left(H1\right) . If 1\not = L\Lambda \left(T-a\right), then the problems \left(1.2\right) \left(1.3\right) is Hyers-Ulam stable.

    Proof. Let \digamma :\left[a.T\right] \times \mathbb{R} \rightarrow \mathbb{R} be continuous and satisfying \left(H1\right) . Let u\in H be any solution of inequality \left(4.5\right) then by Remark 4.3, we have

    \begin{equation*} ^{c}D^{m}u(t) = \digamma \left( t, u\left( t\right) \right) +\omega \left( t\right) , \text{ for all }t\in \left[ a, T\right] . \end{equation*}

    Using Remark 4.1, we can write

    \begin{equation*} u(t) = \int_{a}^{T}G(t, r)\digamma \left( r, u\left( r\right) \right) dr+\int_{a}^{T}G(t, r)\omega \left( r\right) dr, \end{equation*}

    which gives

    \begin{equation} \left \vert u(t)-\int_{a}^{T}G(t, r)\digamma \left( r, u\left( r\right) \right) dr\right \vert \leq \Lambda \left( T-a\right) \epsilon , \end{equation} (4.7)

    where G(t, r) and \Lambda are defined in \left(4.2\right) and \left(4.3\right) . Now let v\in H be a unique solution of fractional BVP \left(1.2\right) \left(1.3\right), consider

    \begin{eqnarray*} \left \vert u\left( t\right) -v\left( t\right) \right \vert & = &\left \vert u\left( t\right) -\int_{a}^{T}G(t, r)\digamma \left( r, v\left( r\right) \right) dr\right \vert \\ &\leq &\left \vert u\left( t\right) -\int_{a}^{T}G(t, r)\digamma \left( r, u\left( r\right) \right) dr\right \vert \\ &&+\left \vert \int_{a}^{T}G(t, r)\digamma \left( r, u\left( r\right) \right) dr-\int_{a}^{T}G(t, r)\digamma \left( r, v\left( r\right) \right) dr\right \vert \end{eqnarray*}

    from \left(4.7\right) we have

    \begin{equation*} \left \vert u\left( t\right) -v\left( t\right) \right \vert \leq \Lambda \left( T-a\right) \epsilon +\left \vert \int_{a}^{T}G(t, r)\digamma \left( r, u\left( r\right) \right) dr-\int_{a}^{T}G(t, r)\digamma \left( r, v\left( r\right) \right) dr\right \vert \end{equation*}

    using \left(H1\right) we get

    \begin{equation*} \left \Vert u-v\right \Vert \leq \Lambda \left( T-a\right) \epsilon +L\Lambda \left( T-a\right) \left \Vert u-v\right \Vert \end{equation*}

    which further implies

    \begin{equation*} \left \Vert u-v\right \Vert \leq \lambda \epsilon , \end{equation*}

    where \lambda = \frac{\Lambda \left(T-a\right) }{1-L\Lambda \left(T-a\right) }. Since 1\not = L\Lambda \left(T-a\right), therefore the problems \left(1.2\right) \left(1.3\right) is Hyers-Ulam stable. This proves the theorem.

    Remark 4.5. In Example 3.3, the system is Hyers-Ulam stable.

    Authors are grateful to the reviewers and editors for their suggestions and comments to improve the manuscript.

    The authors declare no conflict of interest.



    [1] K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer, New York, 2010.
    [2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of the Fractional Differential Equations, North-Holland Mathematics Studies, 204, 2006.
    [3] V. Lakshmikantham, S. Leela, J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, 2009.
    [4] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, NewYork, 1993.
    [5] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
    [6] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, 1993.
    [7] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.
    [8] B. Ahmad, A. Alsaedi, S.K. Ntouyas, J. Tariboon, Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Springer, Cham, Switzerland, 2017.
    [9] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
    [10] R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, J. Chem. Phys., 284 (2002), 399–408.
    [11] R. Hilfer, Y. Luchko, Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouvill fractional derivatives, Frac. Calc. Appl. Anal., 12 (2009), 299–318.
    [12] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simulat., 44 (2017), 460–481. doi: 10.1016/j.cnsns.2016.09.006
    [13] J. Vanterler da C. Sousa, E. Capelas de Oliveira, On the \psi-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72–91. doi: 10.1016/j.cnsns.2018.01.005
    [14] J. Vanterler da C. Sousa, E. Capelas de Oliveira, Leibniz type rule: \psi-Hilfer fractional operator, Commun. Nonlinear Sci. Numer. Simul., 77 (2019), 305–311. doi: 10.1016/j.cnsns.2019.05.003
    [15] J. Vanterler da C. Sousa, E. Capelas de Oliveira, Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, Appl. Math. Lett., 81 (2018), 50–56. doi: 10.1016/j.aml.2018.01.016
    [16] K. M. Furati, N. D. Kassim, N. E. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64 (2012), 1616–1626. doi: 10.1016/j.camwa.2012.01.009
    [17] H. Gu, J. J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., 257 (2015), 344–354.
    [18] J. Wang, Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput., 266 (2015), 850–859.
    [19] S. Asawasamrit, A. Kijjathanakorn, S. K. Ntouyas, J. Tariboon, Nonlocal boundary value problems for Hilfer fractional differential equations, Bull. Korean Math. Soc., 55 (2018), 1639–1657.
    [20] A. Mali, K. Kucche, Nonlocal boundary value problem for generalized Hilfer implicit fractional differential equations, Math. Meth. Appl. Sci., 43 (2020), 8608–8631. doi: 10.1002/mma.6521
    [21] S. K. Ntouyas, D. Vivek, Existence and uniqueness results for sequential \psi-Hilfer fractional differential equations with multi-point boundary conditions, Acta Mathematica Universitatis Comenianae, 90 (2021), 171–185.
    [22] W. T. Coffey, Yu. P. Kalmykov, J. T. Waldron, The Langevin Equation, second ed., World Scientific, Singapore, 2004.
    [23] A. Alsaedi, S. K. Ntouyas, B. Ahmad, Existence results for Langevin fractional differential inclusions involving two fractional orders with four-point multi-term fractional integral boundary conditions, Abstr. Appl. Anal., 2013 (2013), 1–17.
    [24] J. Tariboon, S. K. Ntouyas, Nonlinear second-order impulsive q-difference Langevin equation with boundary conditions, Bound. Value Probl., 2014 (2014), 85. doi: 10.1186/1687-2770-2014-85
    [25] J. Tariboon, S. K. Ntouyas, C. Thaiprayoon, Nonlinear Langevin equation of Hadamard-Caputo type fractional derivatives with nonlocal fractional integral conditions, Adv. Math. Phys., 2014 (2014), 1–15.
    [26] Ch. Nuchpong, S. K. Ntouyas, D. Vivek, J. Tariboon, Nonlocal boundary value problems for \psi-Hilfer fractional-order Langevin equations, Bound. Value Probl., 2021 (2021), 1–12. doi: 10.1186/s13661-020-01478-2
    [27] C. Thaiprayoon, W. Sudsutad, S. K. Ntouyas, Mixed nonlocal boundary value problem for implicit fractional integro-differential equations via \psi-Hilfer fractional derivative, Adv. Differ. Equ., 2021 (2021), 1–24. doi: 10.1186/s13662-020-03162-2
    [28] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985.
    [29] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2005.
  • This article has been cited by:

    1. Hanadi Zahed, Hoda A. Fouad, Snezhana Hristova, Jamshaid Ahmad, Generalized Fixed Point Results with Application to Nonlinear Fractional Differential Equations, 2020, 8, 2227-7390, 1168, 10.3390/math8071168
    2. Jiabin Xu, Hassan Khan, Rasool Shah, A.A. Alderremy, Shaban Aly, Dumitru Baleanu, The analytical analysis of nonlinear fractional-order dynamical models, 2021, 6, 2473-6988, 6201, 10.3934/math.2021364
    3. Djiab Somia, Nouiri Brahim, A new class of mixed fractional differential equations with integral boundary conditions, 2021, 7, 2351-8227, 227, 10.2478/mjpaa-2021-0016
    4. Min Wang, Naeem Saleem, Shahid Bashir, Mi Zhou, Fixed Point of Modified F-Contraction with an Application, 2022, 11, 2075-1680, 413, 10.3390/axioms11080413
    5. Naeem Saleem, Mi Zhou, Shahid Bashir, Syed Muhammad Husnine, Some new generalizations of F- contraction type mappings that weaken certain conditions on Caputo fractional type differential equations, 2021, 6, 2473-6988, 12718, 10.3934/math.2021734
    6. Vedat Suat Ertürk, Amjad Ali, Kamal Shah, Pushpendra Kumar, Thabet Abdeljawad, Existence and stability results for nonlocal boundary value problems of fractional order, 2022, 2022, 1687-2770, 10.1186/s13661-022-01606-0
    7. Shahid Bashir, Naeem Saleem, Hassen Aydi, Syed Muhammad Husnine, Asma Al Rwaily, Developments of some new results that weaken certain conditions of fractional type differential equations, 2021, 2021, 1687-1847, 10.1186/s13662-021-03519-1
    8. Mi Zhou, Naeem Saleem, Shahid Bashir, Solution of fractional integral equations via fixed point results, 2022, 2022, 1029-242X, 10.1186/s13660-022-02887-w
    9. 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, 2023, 8, 2473-6988, 11325, 10.3934/math.2023574
    10. Hasanen A. Hammad, Hassen Aydi, Doha A. Kattan, New Contributions to Fixed Point Techniques with Applications for Solving Fractional and Differential Equations, 2024, 23, 1575-5460, 10.1007/s12346-023-00932-7
    11. Akbar Azam, Nayyar Mehmood, Niaz Ahmad, Faryad Ali, Reich–Krasnoselskii-type fixed point results with applications in integral equations, 2023, 2023, 1029-242X, 10.1186/s13660-023-03022-z
    12. Elyas Shivanian, On the Solution of Caputo Fractional High-Order Three-Point Boundary Value Problem with Applications to Optimal Control, 2024, 31, 1776-0852, 10.1007/s44198-023-00164-y
    13. Ahsan Abbas, Nayyar Mehmood, Manuel De la Sen, Ahmed Al-Rawashdeh, Mathematical analysis of dynamical systems involving Atangana–Baleanu piecewise derivative, 2025, 120, 11100168, 438, 10.1016/j.aej.2025.02.028
  • Reader Comments
  • © 2021 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(3027) PDF downloads(161) Cited by(7)

Figures and Tables

Figures(6)  /  Tables(4)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog