On fractional Langevin equation involving two fractional orders in different intervals

https://doi.org/10.15388/NA.2019.6.3 Nonlinear Analysis: Modelling and Control, Vol. 24, No. 6, 884–897 eISSN: 2335-8963 ISSN: 1392-5113 On fractional Langevin equation involving two fractional orders in different intervals Hamid Baghania,1, Juan J. Nietob aDepartment of Mathematics, Faculty of Mathematics, Statistics and Computer Science, University of Sistan and Baluchestan, Zahedan, Iran [email protected];[email protected].ac.ir bDepartment of Mathematical Analysis, Institute of Mathematics, University of Santiago de Compostela, Santiago de Compostela, 15782 Spain [email protected] Received: October 24, 2018 / Revised: April 5, 2019 / Published online: November 8, 2019 Abstract. In this paper, we study a nonlinear Langevin equation involving two fractional orders α∈(0,1] and β∈(1,2] with initial conditions. By means of an interesting fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for the fractional equations. Some illustrative numerical examples are also discussed. Keywords: Caputo fractional derivative, initial boundary conditions, existence and uniqueness, fractional Langevin equation. 1 Introduction Fractional differential equations have proven strong connection to many physical and engineering applications such as viscoelasticity, ground water flows, boundary layer theory, granular flows, dynamics of cold atoms in optical lattices, plasma turbulence, dynamics of polymeric materials. On the other hand, the fractional differential equations involving Riemann–Liouville and Caputo operators of fractional orders have gained intensive interest in the last years. It has been shown that these types of equations have also numerous applications in diverse fields and thus have evolved into multidisciplinary subjects. An excellent account in the study of fractional differential equations can be found in [10, 11, 14, 16, 21, 26, 30, 33]. For more details and recent examples, see [3, 5–7, 9, 20] and the references therein. Some new and recent aspects on fractional calculus can be seen in [1,2,15,22,25,28,29, 31]. 1Corresponding author. c 2019 Authors. Published by Vilnius University Press This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. On fractional Langevin equation 885 The Langevin differential equation was first proposed by Paul Langevin in 1908. He gave a mathematical description of Brownian motion x00 +λx0=f, where xrepresents the position of a unit mass. The force acting on that particle is written as a sum −λx0+f of a viscous force proportional to the particle’s velocity −λDβx,β= 1 and a nonlinear noisy term f. The parameter λrepresents a viscosity related to the velocity or the fractional derivative. Indeed, the equation has been widely used in different disciplines to describe the dynamical processes of various fluctuating environments [8]. However, it has been realized that the ordinary Langevin differential equation comes up with insufficient illustrations for some systems existing in disordered or fractal mediums. Such occurrence was observed in anomalous transport phenomena [17]. To overcome this deficiency, Kubo introduced the generalized Langevin differential equation in which a fractional memory kernel was incorporated to interpret the fractal and memory properties [19]. The nonlinear Langevin equation with two fractional orders was introduced and studied in [32]. Most of the nonlinear fractional equations cannot be analytically solved, so, firstly, the existence and uniqueness of solutions should be verified. Recently, the existence of solutions of the initial and boundary value problems for nonlinear fractional equations are extensively studied. For example, see Kosmatov [18], Li et al. [24], Deng et al. [9], Gorka et al. [12], Lakshmikantham et al. [23], Zhou et al. [34] and Guo et al. [13]. The authors in [11] have investigated the existence and uniqueness of solutions for nonlinear Langevin equation involving two fractional orders with anti-periodic boundary conditions based on coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. Motivated by the work, we study, with a new and different method, the existence and uniqueness of solutions for an initial value problem of Langevin equation involving two fractional orders in different intervals as follows: DβDα+λx(t) = ft, x(t),0⩽t⩽1,0< α ⩽1,1< β ⩽2, x(0) = x(1) = 0,D2αx(1) + λDαx(1) = 0,(1) where Dαis the Caputo fractional derivative of order α,f: [0,1] ×R→Ris a given continuous function, and λis a real number. Furthermore, D2αis the sequential fractional derivative presented by Miller and Ross [27] as follows: Dαu=Dαu, Dkαu=DαD(k−1)αu, k = 2,3, . . . . 2 Preliminaries In this section, we recall some notations and preliminaries for fractional calculus and a new fixed point theorem which are needed prior to addressing our results. Nonlinear Anal. Model. Control, 24(6):884–897 886 H. Baghani, J.J. Nieto Definition 1. (See [9,18].) The Riemann–Liouville integral of order α > 0for a function x: (0,∞)→Ris defined as Iαx(t) = t Z 0 (t−s)α−1 Γ(α)x(s) ds, 0⩽t⩽1,(2) where Γis the Gamma function, provided the right-hand side integral exists and is finite. Definition 2. (See [9,18].) The Caputo fractional derivative of order α > 0for a continuous function x: (0,∞)→Ris given by Dαx(t) = 1 Γ(n−α) t Z 0 (t−s)n−α−1x(n)(s) ds, 0⩽t⩽1,(3) where nis the smallest integer greater than or equal to α, provided the right-hand side integral exists and is finite. We note that the Caputo derivative reduces to the conventional nth derivative of a given function as α→n, and the initial conditions keep the same form as that of the classical Langevin equation. The relationship between (2) and (3) and other properties of these notions are stated in the next theorems. For their proofs, we refer to [18,30]. Theorem 1. Let α, β ⩾0. If xis continuous, then IαIβx=IβIαx=Iα+βx. Theorem 2. Let α⩾0. If xis continuous, then DαIαx=x. Theorem 3. (See [16, 30].) Let nbe a positive integer and α∈(n−1, n). Then the general solution of the fractional differential equation Dαx= 0 is given by x(t) = c0+c1t+c2t2+· · · +cn−1tn−1, where ci∈R,i= 0,1, . . . , n −1. Now, we present an interesting fixed point theorem (Amini–Emami’s fixed point theorem) which plays main role in our discussion. Definition 3. (See [4].) Let (X, 4)be a partially ordered set and f:X→Xbe a self mapping. Then fis called increasing (decreasing) if fx 4fy (fy 4fx) whenever x4y. Also, we say that elements x, y ∈Xare comparable either x4yor y4x. Moreover, {xn}is called an increasing sequence (a decreasing sequence) if xn4xn+1 for all n∈N(xn+1 4xnfor all n∈N). Theorem 4. (See [4].) Let (X, 4)be a partially ordered set and suppose that there exists a metric din Xsuch that (X, d)is a complete metric space. Let f:X→Xbe an http://www.journals.vu.lt/nonlinear-analysis On fractional Langevin equation 887 increasing mapping such that there exists an element x0∈Xwith x04f(x0). Suppose that there exists 0⩽α < 1such that df(x), f(y)⩽α d(x, y) for all comparable x, y ∈X. Assume that either fis continuous or Xis such that if an incrasing sequence {xn} → xin X, then xn4xfor all n∈N. Beside, if for each x, y ∈X, there exists z∈Xwhich is comparable to xand y. Then fhas a unique fixed point x∗. 3 Existence and uniqueness In this section, we intend to give an existence and uniqueness result for problem (1). First, we establish the equivalence of this problem and a mixed Fredholm–Volterra integral equation. Then we break our discussion up into two subsections. Two cases of interest are discussed separately in Section 3.1 for the case λ⩾0and in Section 3.2 for the case λ < 0. The reason for discussing them separately is that the definition of coupled lower and upper solution is different in these two cases. Also, in the end of this section, we conduct a comparison between our results and the main results of [11]. Lemma 1. xis a solution of problem (1) if and only if it is a solution of the nonlinear integral equation x(t) = 1 Γ(α+β) t Z 0 (t−τ)α+β−1fτ, x(τ)dτ−λ Γ(α) t Z 0 (t−τ)α−1x(τ) dτ +tα"− 1 Z 0 (1 −τ)α+β−1 Γ(α+β)fτ, x(τ)dτ+λ 1 Z 0 (1 −τ)α−1 Γ(α)x(τ) dτ +Γ(2 −α) Γ(α+ 2) 1 Z 0 (1 −τ)β−α−1 Γ(β−α)fτ, x(τ)dτ# −tα+1 Γ(α+ 2)"Γ(2 −α) 1 Z 0 (1 −τ)β−α−1 Γ(β−α)fτ, x(τ)dτ#(4) for all t∈[0,1]. Proof. Let xbe a solution of the problem (1). Then from Theorem 2 we obtain DβDα+λx(t)−Iβf·, x(·)(t)= 0. Nonlinear Anal. Model. Control, 24(6):884–897 888 H. Baghani, J.J. Nieto Now by applying Theorem 3 we deduce Dα+λx(t)−Iβf·, x(·)(t) = c1+c2t or, equivalently, Dαx(t) + λIαx(·)(t)−Iα+βf·, x(·)(t)−c1 tα Γ(α+ 1) −c2 tα+1 Γ(α+ 2)= 0. Applying Theorem 3 again, the general form of problem (1) can be written as x(t) = Iα+βf·, x(·)(t)−λIαx(·)(t) + c0+c1 tα Γ(α+ 1) +c2 tα+1 Γ(α+ 2).(5) Now, by (5) we obtain Dαx(t) = Iβf·, x(·)(t)−λx(t) + c1+c2t, and D2αx(t) = Iβ−αf·, x(·)(t)−λDαx(t) + c2 t1−α Γ(2 −α). Hence, by using the boundary conditions for problem (1) we have c2=−Γ(2 −α) 1 Z 0 (1 −τ)β−α−1 Γ(β−α)fτ, x(τ)dτ, c1=−Γ(α+ 1) 1 Z 0 (1 −τ)α+β−1 Γ(α+β)fτ, x(τ)dτ +λΓ(α+ 1) 1 Z 0 (1 −τ)α−1 Γ(α)x(τ) dτ +Γ(α+ 1)Γ(2 −α) Γ(α+ 2) 1 Z 0 (1 −τ)β−α−1 Γ(β−α)fτ, x(τ)dτ, c0= 0. Substituting the values of c0,c1and c2in (5), we obtain solution (4). On the other hand, it is easy to prove that, if xis a solution of the integral equation (4), then xis also a solution of problem (1). http://www.journals.vu.lt/nonlinear-analysis On fractional Langevin equation 889 3.1 Consideration in the case λ⩾ ⩾ ⩾0 Definition 4. An element (x0, y0)∈C[0,1]×C[0,1] is called a coupled lower and upper solution of (1) if, for all t∈[0,1], x0(t)⩽1 Γ(α+β) t Z 0 (t−τ)α+β−1fτ, x0(τ)dτ−λ Γ(α) t Z 0 (t−τ)α−1y0(τ) dτ +tα"− 1 Z 0 (1 −τ)α+β−1 Γ(α+β)fτ, y0(τ)dτ+λ Γ(α) 1 Z 0 (1 −τ)α−1x0(τ) dτ +Γ(2 −α) Γ(α+ 2) 1 Z 0 (1 −τ)β−α−1 Γ(β−α)fτ, x0(τ)dτ# −tα+1 Γ(α+ 2)"Γ(2 −α) 1 Z 0 (1 −τ)β−α−1 Γ(β−α)fτ, y0(τ)dτ# and y0(t)⩾1 Γ(α+β) t Z 0 (t−τ)α+β−1fτ, y0(τ)dτ−λ Γ(α) t Z 0 (t−τ)α−1x0(τ) dτ +tα"− 1 Z 0 (1 −τ)α+β−1 Γ(α+β)fτ, x0(τ)dτ+λ Γ(α) 1 Z 0 (1 −τ)α−1y0(τ) dτ +Γ(2 −α) Γ(α+ 2) 1 Z 0 (1 −τ)β−α−1 Γ(β−α)fτ, y0(τ)dτ# −tα+1 Γ(α+ 2)"Γ(2 −α) 1 Z 0 (1 −τ)β−α−1 Γ(β−α)fτ, x0(τ)dτ#. To prove the main results, we need the following assumptions: (H1) f: [0,1] ×R→Rbe a function such that f(·, x(·)) ∈C[0,1] for each x∈ C[0,1]. (H2) There exists L > 0such that 0⩽f(t, x)−f(t, y)⩽L(x−y)for all x, y ∈R with x⩾y. Theorem 5. With assumptions (H1)–(H2), if problem (1) has a coupled lower and upper solution and Λ < 1, where Λ=2L Γ(α+β+ 1) +2λ Γ(α+ 1) +2LΓ(2 −α) Γ(α+ 2)Γ(β−α+ 1), then it has a unique solution in C[0,1]. Nonlinear Anal. Model. Control, 24(6):884–897 890 H. Baghani, J.J. Nieto Proof. It is easy to see that X:= C[0,1] is a partially ordered set with the following order relation in X: x4y x, y ∈X⇐⇒ x(t)⩽y(t)∀t∈[0,1]. Also, (X, d)is a complete metric space with metric d(x, y) = maxt∈[0,1] |x(t)−y(t)|. Obviously, if {xn}is an increasing sequence in Xthat converges to x∈Xand {yn}is a decreasing sequence in Xthat converges to y∈X, then xn4xand y4ynfor all n. Also, for any x, y ∈X, the functions max{x, y}and min{x, y}are the upper and lower bounds of x, y, respectively. Also, X×Xis a partially ordered set if we define the following order relation in X×X: (x, y)ˆ 4(u, v)⇐⇒ x4u, v 4y. Furthermore, for every (x, y),(u, v)∈X×X, there exists a (max{x, u},min{y, v})∈ X×Xthat is comparable to (x, y)and (u, v). Moreover, (X×X, ˆ d)is a complete metric space, where ˆ d((x, y),(u, v)) = d(x, u) + d(y, v). Also, if {(xn, yn)}is an increasing sequence in X×Xthat converges to (x, y)then (xn, yn)ˆ 4(x, y)for each n. Now we define g1, g2:X→Xand S:X×X→X×Xas follows: g1(x)(t) = 1 Γ(α+β) t Z 0 (t−τ)α+β−1fτ, x(τ)dτ+tα"λ Γ(α) 1 Z 0 (1 −τ)α−1x(τ) dτ +Γ(2 −α) Γ(α+ 2)Γ(β−α) 1 Z 0 (1 −τ)β−α−1fτ, x(τ)dτ#, t ∈[0,1], g2(y)(t) = −λ Γ(α) t Z 0 (t−τ)α−1y(τ) dτ−tα Γ(α+β) 1 Z 0 (1 −τ)α+β−1fτ, y(τ)dτ −tα+1 Γ(2 −α) Γ(α+ 2)Γ(β−α) 1 Z 0 (1 −τ)β−α−1fτ, y(τ)dτ, t ∈[0,1], and S(x, y) = g1(x) + g2(y), g1(y) + g2(x). It is easy to see that (x0, y0)ˆ 4S(x0, y0),g1is an increasing mapping and g2is a decreasing mapping. Hence, Sis an increasing mapping in X×X. Now, for (x, y),(u, v)∈X×Xwith (x, y)ˆ 4(u, v)and t∈[0,1], we have S(x, y)(t)− S(u, v)(t) =g1x(t) + g2y(t), g1y(t) + g2x(t)−g1u(t) + g2v(t), g1v(t) + g2u(t) ⩽g1x(t)−g1u(t)+g2y(t)−g2v(t)+g1y(t)−g1v(t)+g2x(t)−g2u(t). http://www.journals.vu.lt/nonlinear-analysis On fractional Langevin equation 891 On the other hand g1x(t)−g1u(t) ⩽1 Γ(α+β) t Z 0 (t−τ)α+β−1fτ, x(τ)−fτ, u(τ)dτ +tα"λ Γ(α) 1 Z 0 (1 −τ)α−1x(τ)−u(τ)dτ +Γ(2 −α) Γ(α+ 2)Γ(β−α) 1 Z 0 (1 −τ)β−α−1fτ, x(τ)−fτ, u(τ)dτ# ⩽L Γ(α+β+ 1) +λ Γ(α+ 1) +LΓ(2 −α) Γ(α+ 2)Γ(β−α+ 1)!d(x, u), g1y(t)−g1v(t) ⩽1 Γ(α+β) t Z 0 (t−τ)α+β−1fτ, y(τ)−fτ, v(τ)dτ +tα"λ Γ(α) 1 Z 0 (1 −τ)α−1y(τ)−v(τ)dτ +Γ(2 −α) Γ(α+ 2)Γ(β−α) 1 Z 0 (1 −τ)β−α−1fτ, y(τ)−fτ, v(τ)dτ# ⩽L Γ(α+β+ 1) +λ Γ(α+ 1) +LΓ(2 −α) Γ(α+ 2)Γ(β−α+ 1)!d(y, v), g2x(t)−g2u(t) ⩽λ Γ(α) t Z 0 (t−τ)α−1x(τ)−u(τ)dτ +1 Γ(α+β) 1 Z 0 (1 −τ)α+β−1fτ, x(τ)−fτ, u(τ)dτ +Γ(2 −α) Γ(α+ 2)Γ(β−α) 1 Z 0 (1 −τ)β−α−1fτ, x(τ)−fτ, u(τ)dτ ⩽L Γ(α+β+ 1) +λ Γ(α+ 1) +LΓ(2 −α) Γ(α+ 2)Γ(β−α+ 1)d(x, u) Nonlinear Anal. Model. Control, 24(6):884–897 892 H. Baghani, J.J. Nieto and g2y(t)−g2v(t) ⩽λ Γ(α) t Z 0 (t−τ)α−1y(τ)−v(τ)dτ +1 Γ(α+β) 1 Z 0 (1 −τ)α+β−1fτ, y(τ)−fτ, v(τ)dτ +Γ(2 −α) Γ(α+ 2)Γ(β−α) 1 Z 0 (1 −τ)β−α−1fτ, y(τ)−fτ, v(τ)dτ ⩽L Γ(α+β+ 1) +λ Γ(α+ 1) +LΓ(2 −α) Γ(α+ 2)Γ(β−α+ 1)d(y, v) for all (x, y),(u, v)∈X×Xwith (x, y)ˆ 4(u, v)and t∈[0,1]. Then we conclude that S(x, y)(t)− S(u, v)(t) ⩽2L Γ(α+β+ 1) +2λ Γ(α+ 1) +2LΓ(2 −α) Γ(α+ 2)Γ(β−α+ 1)d(x, u) +2L Γ(α+β+ 1) +2λ Γ(α+ 1) +2LΓ(2 −α) Γ(α+ 2)Γ(β−α+ 1)d(y, v) =Λˆ d(x, y),(u, v) for all (x, y),(u, v)∈X×Xwith (x, y)ˆ 4(u, v)and t∈[0,1]. Hence, for each (x, y), (u, v)∈X×Xwith (x, y)ˆ 4(u, v), we obtain ˆ dS(x, y),S(u, v)⩽Λˆ d(x, y),(u, v) Thus, according to Theorem 4, there is a unique element (x∗, y∗)∈X×Xsuch that (x∗, y∗) = S(x∗, y∗). On the other hand, since (y∗, x∗)is another fixed point of S, then y∗=x∗. Hence, x∗=g1(x∗) + g2(x∗), i.e., x∗is a unique solution of (1). 3.2 Consideration in the case λ < 0 For consideration the existence and uniqueness result in this case, we can use the same technique as in previous subsection. The main difference is the structure of the function S and consequently the definition of coupled lower and upper solution of problem (1). We could carry out a similar argument to prove the existence and uniqueness result. For this reason, we omit the full details of the processes. http://www.journals.vu.lt/nonlinear-analysis