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aimsmathematics 6 8 8654 8666 doi 10 3934 math 2021503 received 15 january 2021 accepted 27 may 2021 http www aimspress com journal math published 07 june 2021 research article ...

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                                                        AIMSMathematics,6(8): 8654–8666.
                                                        DOI:10.3934/math.2021503
                                                        Received: 15 January 2021
                                                        Accepted: 27 May 2021
         http://www.aimspress.com/journal/Math          Published: 07 June 2021
        Research article
        Dynamicsandstability for Katugampola random fractional differential
        equations
                   1   ¨     2,∗               3             4
        Fouzia Bekada , Saıd Abbas , Mouffak Benchohra and Juan J. Nieto
         1                               ¨
          Laboratory of Mathematics, University of Saıda–Dr. Moulay Tahar, P. O. Box 138, EN-Nasr, 20000
            ¨
          Saıda, Algeria
         2                                ¨
          Department of Mathematics, University of Saıda–Dr. Moulay Tahar, P. O. Box 138, EN-Nasr,
                ¨
          20000Saıda, Algeria
         3                                                 `
          Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes, P. O. Box 89, Sidi
                `
          Bel-Abbes 22000, Algeria
         4                       ´       ´            ´               ´
          Departamento de Estatistica, Analise Matematica e Optimizacion, Instituto de Matematicas,
          Universidade de Santiago de Compostela, Santiago de Compostela, Spain
        * Correspondence: Email: abbasmsaid@yahoo.fr, said.abbas@univ-saida.dz.
        Abstract: This paper deals with some existence of random solutions and the Ulam stability for a
        class of Katugampola random fractional differential equations in Banach spaces. A random fixed point
        theorem is used for the existence of random solutions, and we prove that our problem is generalized
        Ulam-Hyers-Rassias stable. An illustrative example is presented in the last section.
        Keywords: differential equation; Katugampola fractional integral; Katugampola fractional
        derivative; random solution; Banach space; Ulam stability; fixed point
        MathematicsSubjectClassification: 26A33, 34A37, 34G20
        1. Introduction
           The history of fractional calculus dates back to the 17th century. So many mathematicians define
        the most used fractional derivatives, Riemann-Liouville in 1832, Hadamard in 1891 and Caputo in
        1997 [24,28,34]. Fractional calculus plays a very important role in several fields such as physics,
        chemicaltechnology,economics,biology;see[2,24]andthereferencestherein. In2011,Katugampola
        introduced a derivative that is a generalization of the Riemann-Liouville fractional operators and the
        fractional integral of Hadamard in a single form [21,22].
           There are several articles dealing with different types of fractional operators; see [1,3,9–13,16,32].
        Variousresults about existence of solutions as well as Ulam stability are provided in [6–8,14,15,17,19,
                                                                                                                                 8655
              20,23,25–31,33]. In this article we investigate the following class of Katugampola random fractional
              differential equation
                                           ρ   ς
                                          ( D0x)(ξ,w) = f(ξ, x(ξ,w),w); ξ ∈ I = [0,T], w ∈ Ω,                                    (1.1)
              with the terminal condition
                                                            x(T,w) = xT(w); w ∈ Ω,                                               (1.2)
                                                                                                                          ρ   ς
              where x : Ω → E is a measurable function, ς ∈ (0,1], T > 0, f : I × E × Ω → E, D is the
                       T                                                                                                      0
              Katugampola operator of order ς, and Ω is the sample space in a probability space, and (E,k · k) is a
              Banach space.
              2. Preliminaries
                 By C(I) := C(I,E) we denote the Banach space of all continuous functions x : I → E with the
              norm
                                                               kxk∞ = supkx(ξ)k,
                                                                         t∈I
              and L1(I,E) denotes the Banach space of measurable function x : I → E with are Bochner integrable,
              equipped with the norm                                    Z
                                                              kxkL1 =      kx(ξ)kdξ.
                                                                          I
              Let Cς,ρ(I) be the weighted space of continuous functions defined by
                                               Cς,ρ(I) = {x : (0,T] → E : ξρ(1−ς)x(ξ) ∈ C(I)},
              with the norm
                                                           kxk := supkξρ(1−ς)x(ξ)k.
                                                               C
                                                                      ξ∈I
              Definition2.1. [2]. The Riemann-Liouville fractional integral operator of the function h ∈ L1(I,E) of
              order ς ∈ R is defined by
                            +                                           Z
                                                                    1      ξ
                                                     RL ς                           r−1
                                                       I h(ξ) =             (ξ − s)    h(s)ds.
                                                        0         Γ(r)    0
              Definition 2.2. [2]. The Riemann-Liouville fractional operator of order ς ∈ R is defined by
                                                                                                           +
                                                                           ! Z
                                                               1        d n     ξ
                                             RL ς                                        n−ς−1
                                               D0h(ξ) = Γ(n−ς) dς              0 (ξ − s)       h(s)ds.
              Definition 2.3. (Hadamard fractional integral) [4]. The Hadamard fractional integral of order r is
              defined as                                          Z
                                                Iςh(ξ) =     1      ξ log ξς−1 h(s)ds, ς > 0,
                                                 0         Γ(ς)    1       s            s
              provided that the left-hand side is well defined for almost every ξ ∈ (0,T).
              AIMSMathematics                                                                          Volume6,Issue 8, 8654–8666.
                                                                                                                                                                 8656
                 Definition 2.4. (Hadamard fractional derivative ) [4]. The Hadamard fractional derivative of order r
                 is defined as                                                        ! Z
                                                                                      n     ξ         
                                                 Dςh(ξ) =            1         ξ d              log ξ n−ς−1 h(s)ds, ς > 0,
                                                    0           Γ(n−ς)           dξ        1         s                  s
                 provided that the left-hand side is well defined for almost every ξ ∈ (0,T).
                 Definition2.5. (Katugampolafractionalintegral)[21]. TheKatugampolafractionalintegralsoforder
                 (ς > 0) is defined by                                                  Z
                                                                                  1−ς      ξ        ρ−1
                                                                ρIςx(ξ) = ρ                        s          x(s)ds                                            (2.1)
                                                                   0                            ρ      ρ 1−ς
                                                                               Γ(ς)      0   (ξ − s )
                 for ρ > 0 and ξ ∈ I, provided that the left-hand side is well defined for almost every ξ ∈ (0,T).
                 Definition 2.6. (Katugampola fractional derivative) [21]. The Katugampola fractional derivative of
                 order ς > 0 is defined by:
                                                                                 !
                                                                              d n
                                                  ρ   r                  1−ρ          ρ n−r
                                                   D0u(ξ) =            ξ             ( I     u)(ξ)
                                                                             dξ          0
                                                                                               ! Z
                                                                          r−n+1                 n     ξ          ρ−1
                                                                =       ρ           ξ1−ρ d                     s            u(s)ds,
                                                                                                           ρ      ρ r−n+1
                                                                      Γ(n−r)               dξ       0   (ξ − s )
                 provided that the left-hand side is well defined for almost every ξ ∈ (0,T).
                      We present in the following theorem some properties of Katugampola fractional integrals and
                 derivatives.
                 Theorem2.7. [21]Let0 < Re(ς) < 1 and 0 < Re(η) < 1 and ρ > 0, for a > 0:
                     • Index property:
                                                                           ρ    ς  ρ   η                ρ   ς+η
                                                                          ( Da)( Dah)(t)           = Da h(t)
                                                                              ρ r    ρ η                ρ r+η
                                                                             ( I )( I h)(t)        = I h(t)
                                                                                 a     a                   a
                     • Linearity property:
                                                                       ρ   r                   ρ   r         ρ    r
                                                                        D (h+g) =                D h(t) + D g(t)
                                                                           a                       a              a
                                                                        ρ r                    ρ r          ρ r
                                                                          I (h + g)       = I h(t)+ I g(t)
                                                                           a                      a            a
                 and we have                                                       d
                                                                            (t1−ρ     )Ir(I1−r)u(s)ds.
                                                                                   dt    0   0
                 Theorem2.8. [21]Letr be a complex number, Re(r) ≥ 0, n = [Re(r)] and ρ > 0. Then, for t > a;
                                    ρ r              1 R t           r−1
                   (1) limρ→1( I h)(t) =                    (t − τ)      h(τ)dτ.
                                       a           Γ(r)   a
                   (2) lim        +(ρIrh)(t) = 1 R t(log t)r−1h(τ)dτ.
                              ρ→0       a            Γ(r)  a        τ             τ
                                    ρ   r             d n     1    R t    h(τ)
                   (3) limρ→1( Dah)(t) = (dt) Γ(n−r)                        r−n+1 dτ.
                                                                     a (t−τ)
                                  + ρ    r               1      d n R t         t n−r−1         dτ
                   (4) limρ→0 ( Dah)(t) = Γ(n−r)(tdt)                  a (log τ)         h(τ) τ .
                 AIMSMathematics                                                                                                Volume6,Issue 8, 8654–8666.
                                                                                                                     8657
             Remark2.9.
                          ρ r         RL r
              (1) limρ→1( I h)(t) = (   I h)(t).
                            a            a
              (2) lim    +(ρIrh)(t) = (HIrh)(t).
                      ρ→0    a           a
                          ρ  r         RL r
              (3) limρ→1( D h)(t) = (    D h)(t).
                             a             a
              (4) lim    +(ρDrh)(t) = (HDrh)(t).
                      ρ→0     a             a
                                                                       ρ  r
             Lemma2.10. Let0 < r < 1. The fractional equation ( D v)(t) = 0, has as solution
                                                                          0
                                                            v(t) = ctρ(r−1),                                        (2.2)
             with c ∈ R.
             Lemma2.11. Let0 < r < 1. Then
                                                    ρ r ρ  r                 ρ(r−1)
                                                     I ( D0u)(t) = u(t) + ct     .
             Proof. We have
                                        r  r         1−p d ! r+1   r
                                       I D u(t) = t          I   D u(t)
                                        0  0             dt   0    0
                                         1−ρ d      ρ−r    Z t    sρ−1     ρ  r        !
                                   = (t        )                 ρ    ρ −r( D0u(s))ds
                                            dt! Γ(r +1) 0 (t − s )         "        !          #   !
                                   = t1−ρ d         ρ−r    Z t     sρ−1      s1−ρ d  (I1−ru)(s) ds
                                                                 ρ    ρ −r             0
                                            dt! Γ(r +1) 0 (t − s )"              ds    #   !
                                         1−ρ d      ρ−r    Z t ρ     ρ r  d    1−r
                                   = t                         (t −s )       (I  u)(s) ds .
                                            dt    Γ(r + 1)                ds 0
                                                             0
             Thus, IrDru(t) = I + I , with
                    0  0        1    2
                                                        !     −r                        
                                                                    h                  i
                                                  1−ρ d     ρ          ρ    ρ r 1−r     t
                                           I = t                     (t −s ) I    u(s)    ,
                                            1         dt Γ(r +1)               0        0
             and                                 d !   ρ−r    Z t
                                             1−ρ                      ρ−1 ρ    ρ r−1 1−r
                                      I2 = t                      rρs    (t −s )    I   u(s)ds.
                                                 dt Γ(r +1)                          0
                                                                0
             Hence, we get
                                                             I = ctρ(r−1)
                                                              1
             and
                                                   1−ρ d ! ρ1−r Z t ρ−1 ρ     ρ r−1 1−r
                                         I   = t                   s   (t −s )     I   u(s)ds
                                         2            dt Γ(r)                       0
                                                        !        0
                                             = t1−ρ d Ir(I1−r)u(s)ds
                                                      dt   0 0
                                             = u(t).
             Finally we obtain
                                                      r    r                ρ(r−1)
                                                    (I )(D u)(t) = u(t) + ct     .
                                                      0    0
             AIMSMathematics                                                                 Volume6,Issue 8, 8654–8666.
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...Aimsmathematics doi math received january accepted may http www aimspress com journal published june research article dynamicsandstability for katugampola random fractional dierential equations fouzia bekada sad abbas mouak benchohra and juan j nieto laboratory of mathematics university sada dr moulay tahar p o box en nasr algeria department djillali liabes sidi bel abbes departamento de estatistica analise matematica e optimizacion instituto matematicas universidade santiago compostela spain correspondence email abbasmsaid yahoo fr said univ saida dz abstract this paper deals with some existence solutions the ulam stability a class in banach spaces xed point theorem is used we prove that our problem generalized hyers rassias stable an illustrative example presented last section keywords equation integral derivative solution space mathematicssubjectclassication g introduction history calculus dates back to th century so many mathematicians dene most derivatives riemann liouville hadama...

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