MalayaJ.Mat. 5(2)(2017) 337–345
                        Initial value problems for fractional differential equations involving
                                                             Riemann-Liouvillederivative
                                                                      a∗                     b                                 c
                                                 J.A. Nanware , N.B. Jadhav                      and D.B.Dhaigude
                                                 aDepartment of Mathematics, Shrikrishna Mahavidyalaya, Gunjoti–413 606, Maharashtra, India.
                                                  bDepartment of Mathematics, Yashwantrao Chavan Mahavidyalaya, Tuljapur– 413 605, India.
                                      cDepartment of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad-431 004, Maharashtra, India.
                    Abstract
                        Existence results are obtained for fractional differential equations with Cp continuity of functions.
                    Monotone method for nonlinear initial value problem is developed by introducing the notion of coupled
                    loweranduppersolutions. Asanapplicationofthemethodexistenceanduniquenessresultsareobtained.
                    Keywords:    Fractional derivative, initial value problem, coupled lower and upper solutions, existence and
                    uniqueness.
                                                                                                                             c
                    2010 MSC:34A12,34C60,34A45.                                                                             
2012MJM.Allrightsreserved.
                    1     Introduction
                        The advantages of fractional derivatives become apparent in modeling mechanical and electrical
                    properties of real materials, and in many other fields, like theory of fractals. Analytical as well as numerical
                    methodsareavailableforstudyingfractionaldifferential equations such as compositional method, transform
                    method, Adomain methods and power series method etc. ( see details in [4, 23] and references therein).
                    Monotone method [5] coupled with method of lower and upper solutions is an effective mechanism that
                    offers constructive procedure to obtain existence results in a closed set. Basic theory of fractional differential
                    equations with Riemann-Liouville fractional derivative is well developed in [2, 7, 9]. Lakshamikantham and
                    Vatsala [1, 6, 8] obtained the local and global existence of solution of Riemann-Liouville fractional differential
                    equation and uniqueness of solution.                    In the year 2009, McRae developed monotone method for
                    Riemann-Liouvile fractional differential equation with initial conditions and studied the qualitative
                    properties of solutions of initial value problem [10]. Nanware and Dhaigude [11, 13, 14, 16–22] developed
                    monotone method for system of fractional differential equations with various conditions and successfully
                    applied to study qualitative properties of solutions. Nanware obtained existence results for the solution of
                    fractional differential equations involving Caputo derivative with boundary conditions [12, 15]. In 2012,
                    Yaker and Koksal have studied initial value problem (1.1) − (1.2) for Riemann- Liouville fractional
                    differential equations. They have proved existence results by using concept of lower and upper solutions
                    andlocalexistence results under the strong hypothesis that the functions are locally Holder continuous.
                    In this paper, we develop monotone method without such strong hypothesis for the following nonlinear
                    Riemann-Liouville fractional differential equation with initial condition
                                                            Dqu(t) = f(t,u(t))+g(t,u(t)),                 t ∈ [t0, T]                                    (1.1)
                       ∗Correspondingauthor.
                    E-mail address:  jag skmg91@rediffmail.com (J.A. Nanware), narsingjadhav4@gmail.com (N.B. Jadhav), dnyaraja@gmail.com (D.B.
                    Dhaigude).
                     338               J.A.Nanwareetal. / Initial value problems for fractional differential equations involving R-L derivative
                                                                          u0 = u(t)(t−t0)1−q}t=t0                                                         (1.2)
                     where f,g ∈ C(J ×R,R),J = [t0,T], f(t,u) is nondecreasing in u , g(t,u) is nonincreasing in u for each t and
                     Dq denotes the Riemann-Liouville fractional derivative with respect to t of order q(0 < q < 1). This is called
                     initial value problem(IVP). We develop monotone method coupled with lower and upper solutions for the
                     IVP(1.1)−(1.2). ThemethodisappliedtoobtainexistenceanduniquenessofsolutionoftheIVP(1.1)−(1.2).
                         The paper is organized in the following manner : In section 2, we consider some definitions and lemmas
                     required in next section and obtained result for nonstrict inequalities. In section 3, we improve the existence
                     results due to Yaker and Koksal. In section 4, we develop monotone method and apply it to obtain existence
                     and uniqueness results for Riemann-Liouville fractional differential equation with initial condition when
                     nonlinearfunctionontherighthandsideisconsideredassumofnondecreasingandnonincreasingfunctions.
                     2     Preliminaries
                         In this section, we discuss some basic definitions and results which are required for the development of
                     monotone method for fractional differential equation with initial condition involving Riemann-Liouville
                     derivative when nonlinear function on the right hand side is considered as sum of nondecreasing and
                     nonincreasing functions.
                         TheRiemann-Liouvillefractional derivative of order q(0 < q < 1) [23] is defined as
                                                Dqu(t) =          1     dnZ t(t−τ)n−q−1u(τ)dτ,                  for a ≤ t ≤ b.
                                                  a          Γ(n−q) dt             a
                     Lemma2.1. [2] Let m ∈ Cp([t0,T],R) and for any t1 ∈ (t0,T] we have m(t1) = 0 and m(t) < 0 for t0 ≤ t ≤ t1.
                     Thenit follows that Dqm(t1) ≥ 0.
                     Lemma 2.2. [6] Let {uǫ(t)} be a family of continuous functions on [t0,T], for each ǫ > 0 where Dquǫ(t) =
                     f (t, uǫ(t)),   uǫ(t0) = uǫ(t)(t − t0)1−q}t=t0 and |f(t,uǫ(t))| ≤ M for t0 ≤ t ≤ T. Then the family {uǫ(t)} is
                     equicontinuous on [t0,T].
                         Now,weintroducethenotionofloweranduppersolutionsfortheinitialvalueproblem(1.1)−(1.2).
                     Definition 2.1. A pair of functions v(t) and w(t) in Cp(J,R) are said to be lower and upper solutions of the IVP
                     (1.1) −(1.2) if
                                                             Dqv(t) ≤ f(t,v(t))+g(t,v(t)),                 v0 ≤ u0
                                                             Dqw(t) ≥ f(t,w(t))+g(t,w(t)),                  w0 ≥ u0.
                     Definition2.2. Apairoffunctions v(t) and w(t) in Cp(J,R) are said to be lower and upper solutions of type I of IVP
                     (1.1) −(1.2) if
                                                             Dqv(t) ≤ f(t,v(t))+g(t,w(t)),                  v0 ≤ u0
                                                             Dqw(t) ≥ f(t,w(t))+g(t,v(t)),                  w0 ≥ u0.
                     Definition2.3. Apairoffunctions v(t) and w(t) in Cp(J,R) are said to be lower and upper solutions of type II of IVP
                     (1.1) −(1.2) if
                                                             Dqv(t) ≤ f(t,w(t))+g(t,v(t)),                  v0 ≤ u0
                                                             Dqw(t) ≥ f(t,v(t))+g(t,w(t)),                  w0 ≥ u0.
                     Definition 2.4. A pair of functions v(t) and w(t) in Cp(J,R) are said to be lower and upper solutions of type III of
                     IVP(1.1)−(1.2) if
                                                              Dqv(t) ≤ f(t,w(t))+g(t,w(t)),                  v0 ≤ u0
                                                             Dqw(t) ≥ f(t,v(t))+g(t,v(t)),                  w0 ≥ u0.
                                  J.A. Nanware et al. / Initial value problems for fractional differential equations involving R-L derivative               339
                    3     Existence Results
                        In this section, we improve the existence results due to Yaker and Koksal [24] for IVP (1.1)−(1.2). We now
                    state and prove the following existence results.
                    Theorem3.1. Supposethat:
                    (i) v(t) and w(t) in Cp(J,R) are coupled lower and upper solutions of type I of IVP (1.1)-(1.2) with v(t) ≤ w(t) on J.
                    (ii) f(t,u),g(t,u) ∈ C[Ω,R] and g(t,u(t)) is nonincreasing in u for each t on J.
                    Thenthere exist a solution u(t) of IVP (1.1)-(1.2) satisfying v(t) ≤ u ≤ w(t) on J.
                    Proof. Let P : J ×R → R be defined by
                                                                   P(t,u) = min{w(t),max(u(t),v(t))}
                    Then f(t,P(t,u(t))+g(t,P(t,u(t))) defines a continuous extension of f + g to J ×R which is bounded, since
                    f + g is uniformly bounded on Ω. By Lemma 2.2, it follows that the family P (t,u(t)) is equicontinuous on J.
                                                                                                                      ǫ
                    By Ascoli-Arzela theorem the sequences {P (t,u(t))} has convergent subsequences {P (t,u )} which
                                                                                ǫ                                                          ǫn     1
                    converges       uniformly        to    P(t,u).      Since     f  + g is uniformly                continuous,        we obtain          that
                    f (t, P  (t, u)) + g(t, P (t,u)) tends uniformly to f(t,P(t,u)) + g(t,P(t,u)) as n → ∞. Hence P(t,u(t)) is the
                           ǫn                  ǫn
                    solution of
                                           Dqu(t) = f(t,P(t,u))+g(t,P(t,u)),                 u(t) = u(t0)(t−t0)1−q}t=t0 = u0.                             (3.3)
                    It follows that the equation (3.3) has a solution on the interval J.
                    We wish to prove that v(t) ≤ u(t) ≤ w(t) on J. For ǫ > 0, consider wǫ(t) = w(t) + ǫγ(t) and viǫ(t) =
                    v (t) − ǫγ(t), where γ(t) = (t − t )q−1E               ((t − t )q) Then we have w0 = w0 +ǫγ0,                   v0 = v0 −ǫγ0, where
                     i                                          0       q,q        0                           ǫ                      ǫ
                    γ0 > 0. This shows that v0 < u0 < w0. Next we show that u < wǫ,                            t0 ≤ t ≤ T. On the contrary, suppose
                                                      ǫ              ǫ
                    that vǫ ≥ u ≥ wǫ. Then there exists t1 ∈ (t0,T] such that u(t1) = wǫ(t1) and vǫ > u > wǫ,                             t0 ≤ t < t1. Thus
                    u(t1) > w(t1) and hence P(t1,u(t1)) = w(t1).
                    Set m(t) = u(t)−wǫ(t) we have m(t1) = 0 and m(t) ≤ 0,                          t0 ≤ t ≤ t1. By Lemma 2.1, we have Dqu(t1) ≥
                    Dqwǫ(t1) whichgivesacontradiction
                                                 f (t1, w(t1)) + g(t1,w(t1)) = f(t1,P(t1,u(t1)) + g(t1,P(t1,u(t1)))
                                                                                  =Dqu(t1)
                                                                                  ≥Dqw (t )
                                                                                           ǫ   1
                                                                                  =Dqw(t1)+ǫγ(t1)
                                                                                  >Dqw(t1)
                                                                                  ≥ f(t1,w(t1))+g(t1,v(t1))
                    Similarly, we prove vǫ < u,            t0 ≤ t ≤ T. For this, suppose there exists t1 ∈ (t0,T] such that vǫ(t1) = u(t1)
                    andvǫ(t) > u(t),         t0 ≤ t < t1. Thus u(t1) < v(t1 ≤ w(t1) and hence P(t1,u(t1)) = v(t1).
                    Set m(t) = vǫ(t) − u(t) we have m(t1) = 0 and m(t) ≤ 0,                           t0 ≤ t ≤ t1. Applying Lemma 2.1, we have
                    Dqu(t1) ≥ Dqwǫ(t1). Since g(t,u) is nonincreasing in u for each t and γ(t) > 0, we get a contradiction
                                                  f (t1, v(t1)) + g(t1,v(t1)) = f(t1,P(t1,u(t1)) + g(t1,P(t1,u(t1)))
                                                                                  =Dqu(t1)
                                                                                  ≤Dqv (t )
                                                                                          ǫ  1
                                                                                  =Dqv(t1)−ǫγ(t1)
                                                                                   0,u ≥ u,
                                              g(t,u(t))−g(t,u(t)) ≥ −N(u−u),N > 0,u ≥ u
               Thenthere exist monotone sequences {vn(t)} and {wn(t)} such that
                                           {vn(t)} → v(t)   and  {wn(t)} → w(t)as n → ∞
               and v(t) and w(t)) are minimal and maximal solutions of the IVP (1.1)-(1.2).
               Proof. For any η in C(J,R) such that for v0 ≤ η on J, we consider the following linear fractional differential
               equation
                           Dqu(t) = f(t,η(t))+g(t,η(t))− M(u−η)−N(u−η),           u(t)(t −t0)1−q}t=t0 = u0       (4.4)
               Since the right hand side of equation (4.4) is known, it is clear that for every η there exists a unique solution
               u(t) of IVP (4.4) on J.
                  For each η and µ in C(J,R) such that v0 ≤ η and w0 ≤ µ, define a mapping A by A[η,µ] = u(t) where
               u(t) is the unique solution of IVP (4.4). This mapping defines the sequences {vn(t)} and {wn(t)}. Firstly, we
               prove
               (I) v0 ≤ A[v0,w0],  , w0 ≥ A[w0,v0]