133x Filetype PDF File size 0.35 MB Source: www.malayajournal.org
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]
no reviews yet
Please Login to review.