Elsevier

Fuzzy Sets and Systems

Volume 265, 15 April 2015, Pages 39-62
Fuzzy Sets and Systems

Random fuzzy fractional integral equations – theoretical foundations

https://doi.org/10.1016/j.fss.2014.09.019Get rights and content

Abstract

This paper presents mathematical foundations for studies of random fuzzy fractional integral equations which involve a fuzzy integral of fractional order. We consider two different kinds of such equations. Their solutions have different geometrical properties. The equations of the first kind possess solutions with trajectories of nondecreasing diameter of their consecutive values. On the other hand, the solutions to equations of the second kind have trajectories with nonincreasing diameter of their consecutive values. Firstly, the existence and uniqueness of solutions is investigated. This is showed by using a method of successive approximations. An estimation of error of nth approximation is given. Also a boundedness of the solution is indicated. To show well-posedness of the considered theory, we prove that solutions depend continuously on the data of the equations. Some concrete examples of random fuzzy fractional integral equations are solved explicitly.

Introduction

The theory of fractional calculus initiated in XVII century has gained more and more attention since the derivatives and integrals of a non-integer order became good tools e.g. in description of mechanical and electrical properties of some real materials or rheological properties of rocks. The fractional calculus has been used in the theory of fractals, in physics, chemistry, engineering, seismology, in problems of viscoelasticity. Although the fractional calculus was available for a long time and applicable to different areas of research, the investigation of the theory of fractional differential equations has been developed quite recently. Currently it focuses a lot of interest because of the challenges it offers compared to the study of ordinary differential equations. The current state of this area of nonlinear analysis is given e.g. in the monographs [27], [29], [52]. Many theoretical aspects on the results for fractional differential equations have been considered e.g. in [11], [17], [19], [20], [26], [31].

Applications of the ordinary differential equations and the fractional differential equations in modeling dynamical systems are evident. On the other hand the realistic information available on a considered dynamical system is often incomplete, imprecise, imperfect, vague (in a word – uncertain) and this causes a dilemma in applications of single-valued differential equations which are suitable in the case of a perfect, precise knowledge about the considered system. To handle the systems with the uncertain initial values and the uncertain relationships between parameters, it has been proposed the theory of fuzzy differential equations (see e.g. [3], [4], [7], [8], [9], [12], [14], [15], [18], [24], [25], [30], [32], [33], [34], [48] and references therein). In these equations the fuzzy sets (see [59], [60]), which replace (and generalize) the real numbers, are used to model the uncertain values. The fuzzy differential equation has been extended to the random fuzzy differential equations [21], [36], [37], [39], [51] and stochastic fuzzy differential equations [40], [41], [44], [45], [46], [49] which incorporate into the equations two different sources of uncertainties, i.e. fuzziness and randomness. The latter equations are more appropriate in modeling uncertain dynamical systems subjected to random forces or stochastic noises. They constitute the directions of extensions to the fuzzy context of the studies on single-valued random differential/integral equations [13], [55], [57] and single-valued stochastic differential equations [6], [22], [50].

Recently, some studies of fuzzy fractional differential and integral equations have been proposed by several papers, see e.g. [1], [2], [5], [10], [47], [54], [58]. They involve the notions of the fuzzy Riemann–Liouville differential operators, fuzzy Caputo derivatives and fuzzy fractional integrals and offer a more comprehensive apparatus to process the dynamical phenomena with fuzziness. However, since the influence of some random factors on the evolution of dynamical system takes place very often, this apparatus would be more and more comprehensive in the case when it allowed for consideration both the fuzziness and randomness. Some investigations without fuzziness can be found in [23], where a polynomial chaos method is applied to solve three concrete examples of crisp random fractional differential equations.

In this paper we initiate some studies on the random fuzzy fractional integral equations where we use a concept of a fuzzy fractional integral. We consider two forms of the random fuzzy fractional integral equations similarly as we have done in [37], [39] for the random fuzzy differential equations. The first one leads to the solutions which possess the trajectories with nondecreasing diameter of their values. In the second form the solutions have the trajectories with nonincreasing diameter of the values. We prove the existence and uniqueness of a global solution to the first kind of the equations, but in some cases for the second kind of the equations we are able to prove this only for a local solution. Hence, here remains an open problem of establishing a method allowing to obtain the existence and uniqueness of the global solution to the equations with nonincreasing diameter of values. For the both kinds of the random fuzzy fractional integral equations we estimate a distance between the approximate solutions and the exact solution, we show that the solutions depend continuously on the initial value and nonlinearity involved in the formulation of the equations. A boundedness result of the solutions is also investigated. Some concrete examples of random fuzzy fractional integral equations are solved explicitly. The presented theory of random fuzzy fractional integral equations can be viewed as an extension in analysis of crisp random fractional differential equations and deterministic crisp fractional differential equations. The latter were never studied in general infinite dimensional metric spaces. They were studied in Banach spaces, but these have linear structure. The space of fuzzy sets does not have the linear structure. The presented theory involves special features of fuzzy sets. In fuzzy case we consider two different equations that reduce to the same crisp equation when we allow the coefficients to be single-valued. In fuzzy case they are completely different. Their solutions have different geometrical properties. This does not appear in the crips case. Moreover the second equation considered in the paper involves, in fact, the Hukuhara differences of fuzzy sets. This causes some more delicate studies in the fuzzy sets setting, because Hukuhara differences may not exist. Such a problem does not appear in crisp case, because there the crisp differences exist always.

All the results established in this paper apply immediately to random set-valued fractional integral equations.

Section snippets

Preliminaries

Let K(Rd) denote the family of all nonempty, compact and convex subsets of Rd. The addition and scalar multiplication in K(Rd) are defined as usual, i.e. for A,BK(Rd) and λRA+B:={a+b|aA,bB},λA:={λa|aA}. In K(Rd) we consider the Hausdorff metric H which for A,BK(Rd) is defined byH(A,B):=max{supaAinfbBab,supbBinfaAab}. It is known (see [28]) that K(Rd) is a complete, separable and locally compact metric space with respect to H. Also it becomes a semilinear metric space with

The main results

Let β>0. In this section we introduce studies concerning the random fuzzy integral equations of the fractional order. The investigations of the fuzzy fractional integral equations in the deterministic case are also new and can be found in [1], [10], [58].

Definition 3.1

The fuzzy fractional β-order integral of the measurable and integrably bounded fuzzy mapping F:[a,b]F(Rd) at t[a,b] is the fuzzy set (IaβF)(t)F(Rd) defined by(IaβF)(t):=1Γ(β)at1(ts)1βF(s)ds, where Γ is the well-known gamma function.

If β=1

Concluding remarks

In this paper, for the first time, the random fuzzy fractional integral equations are studied. They can be viewed as some extensions of the random fuzzy differential equations [36], [37], [39] and the deterministic fuzzy fractional integral equations investigated in [1], [10], [58]. They are appropriate to model dynamic systems operating in fuzzy and random environment.

A main purpose of the paper is to present a framework within which such equations can be investigated and to justify the

References (61)

  • D. Delbosco et al.

    Existence and uniqueness for a nonlinear fractional differential equation

    J. Math. Anal. Appl.

    (1996)
  • K. Diethelm et al.

    Analysis of fractional differential equations

    J. Math. Anal. Appl.

    (2002)
  • Y. Feng

    Fuzzy stochastic differential systems

    Fuzzy Sets Syst.

    (2000)
  • G. González-Parra et al.

    Polynomial Chaos for random fractional order differential equations

    Appl. Math. Comput.

    (2014)
  • O. Kaleva

    Fuzzy differential equations

    Fuzzy Sets Syst.

    (1987)
  • O. Kaleva

    A note on fuzzy differential equations

    Nonlinear Anal.

    (2006)
  • V. Lakshmikantham et al.

    Basic theory of fractional differential equations

    Nonlinear Anal. TMA

    (2008)
  • J. Li et al.

    The Cauchy problem of fuzzy differential equations under generalized differentiability

    Fuzzy Sets Syst.

    (2012)
  • V. Lupulescu

    Initial value problem for fuzzy differential equations under dissipative conditions

    Inf. Sci.

    (2008)
  • V. Lupulescu

    On a class of fuzzy functional differential equations

    Fuzzy Sets Syst.

    (2009)
  • V. Lupulescu

    Fractional calculus for interval-valued functions

    Fuzzy Sets Syst.

    (2015)
  • M.T. Malinowski

    On random fuzzy differential equations

    Fuzzy Sets Syst.

    (2009)
  • M.T. Malinowski

    Existence theorems for solutions to random fuzzy differential equations

    Nonlinear Anal. TMA

    (2010)
  • M.T. Malinowski

    Interval differential equations with a second type Hukuhara derivative

    Appl. Math. Lett.

    (2011)
  • M.T. Malinowski

    Random fuzzy differential equations under generalized Lipschitz condition

    Nonlinear Anal., Real World Appl.

    (2012)
  • M.T. Malinowski

    Strong solutions to stochastic fuzzy differential equations of Itô type

    Math. Comput. Model.

    (2012)
  • M.T. Malinowski

    Itô type stochastic fuzzy differential equations with delay

    Syst. Control Lett.

    (2012)
  • M.T. Malinowski

    Interval Cauchy problem with a second type Hukuhara derivative

    Inf. Sci.

    (2012)
  • M.T. Malinowski

    On set differential equations in Banach spaces – a second type Hukuhara differentiability approach

    Appl. Math. Comput.

    (2012)
  • M.T. Malinowski

    Some properties of strong solutions to stochastic fuzzy differential equations

    Inf. Sci.

    (2013)
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