Abstract
We study the existence of solution for nonlinear fuzzy differential equations of fractional order involving the Riemann-Liouville derivative.
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1 Introduction
The Schauder Fixed-Point Theorem is one of the most celebrated results in Fixed-Point Theory and it states that any compact convex nonempty subset of a normed space has the fixed-point property (Schauder, 1930; Theorem 2.3.7 in [1]). It is also valid in locally convex spaces (Tychonoff, 1935; Theorem 2.3.8 in [1]). Recently, this Schauder fixed-point theorem has been generalized to semilinear spaces [2].
As explained in detail in many works in the field of fuzzy differential equations, uncertainty has to be considered in the formulation of a mathematical model for a better adequacy, due to the imprecision inherent to the information available or the behavior of the dynamical system itself (see, for instance, [3]). For this reason, the construction of models which try to be faithful to a certain real process involves, in many occasions, the consideration of fuzzy differential equations.
On the other hand, the subject of fractional calculus is not a recent topic, since many interesting questions concerning its main concepts and properties have been discussed since the end of the seventeenth century, with the contribution of mathematicians such as Leibnitz, Euler, Laplace, Lacroix, Fourier, Liouville or Riemann, among others (for details, see the introduction of [4] and other monographs on fractional calculus [5, 6]). The main references on fractional calculus also point out the power and usefulness of this topic to the modeling of phenomena in a wide range of scientific fields. The complexity of some processes in the physical world can be reproduced more faithfully with the help of fractional order models better than using classical integer order models, for instance, in electromagnetism, astrophysics, diffusion, material theory, chemistry, control theory, wave propagation, signal theory, electricity and thermodynamics [5]. Some theoretical aspects on the existence and uniqueness results for fractional differential equations have been considered by some authors [7–9].
Agarwal et al. have proposed the concept of the solution of fuzzy fractional differential equations in [10]. Arshad and Lupulescu [11] have deduced some existence and uniqueness results for fuzzy fractional differential equations under Riemann-Liouville derivative. Allahviranloo et al. have presented the explicit solutions of fuzzy fractional differential equations and some related results in [12, 13]. Some existence and uniqueness results for fuzzy fractional integral equations and fuzzy fractional integro-differential equations have been proposed in [14, 15].
In this paper, we consider nonlinear fuzzy fractional differential equations of the form
where and is the Riemann-Liouville fractional derivative and is a fuzzy real number for each , . We present some conditions to obtain a solution.
The paper is organized as follows. In Section 2, we recall the definitions of fuzzy fractional integral and derivative and related properties used in the paper. In Section 3, we present sufficient conditions to have at least a solution.
2 Preliminaries
In this section, we give some definitions and introduce the necessary notation which will be used throughout the paper, see for example [16].
Let us denote by the class of fuzzy subsets of the real axis, that is, maps satisfying the following properties:
-
(i)
u is normal, i.e., there exists such that ,
-
(ii)
u is a convex fuzzy set (i.e. , , ),
-
(iii)
u is upper semicontinuous on ℝ,
-
(iv)
is compact where cl denotes the closure of a subset.
Then is called the space of fuzzy numbers. For denote and . Then from (i)-(iv), it follows that the α-level set is a nonempty compact interval for all and any . The notation
denotes explicitly the α-level set of u. We refer to and as the lower and upper branches of u, respectively.
For and , the sum and the product λu are defined by , , where means the usual addition of two intervals (subsets) of ℝ and means the usual product between a scalar and a subset of ℝ. This is a consequence of Zadeh’s Extension Principle [16].
The metric structure is given by the Hausdorff distance
by
The following properties are well known:
and is a complete metric space.
The concept of a semi-linear space and similar concepts were already considered, for instance, in [17]. A semilinear metric space is a semilinear space S with a metric which is translation invariant and positively homogeneous, that is,
-
,
-
, for all ,
for all and . In this case, we can define a norm on S by , where is the zero element in S. If S is a semilinear metric space, then addition and scalar multiplication on S are continuous. If S is a complete metric space, then we say that S is a semilinear Banach space. For example, the set of fuzzy real numbers is not a vector space and hence it cannot be Banach space. The set of continuous functions from the real compact interval into the set of fuzzy real numbers is a semilinear Banach space. We say semilinear space S has cancellation property if implies for .
Let . We denote by the space of all continuous fuzzy functions defined on . Now, let . We define
where , . Obviously, is a complete metric space with respect to the metric
We denote by , which is not a norm in the classical sense, since is not a vector space. We point out that . We define as the space of fuzzy sets with the property that the function is continuous with respect to the Hausdorff metric on . It is well known that is a complete metric space [18]. If the functions take values in , we get the sets , .
Definition 2.1 ([18])
A subset is said to be compact-supported if there exists a compact set such that for all .
Definition 2.2 ([18])
A subset is said to be level-equicontinuous at if for all , there exists such that
A is level-equicontinuous on if A is level-equicontinuous at α for all .
Theorem 2.3 ([18])
Let A be a compact-supported subset of . Then the following assertions are equivalent:
-
A is a relatively compact subset of ,
-
A is level-equicontinuous on .
In fact, if A is relatively compact in , then A is compact-supported and also level-equicontinuous on . Conversely, if A is compact-supported in and level-equicontinuous on , then A is relatively compact in .
Definition 2.4 ([18])
A continuous function is said to be compact if for every subinterval and every bounded subset , then is relatively compact in .
Let denote the family of all nonempty compact convex subsets of ℝ. is endowed with the topology generated by the Hausdorff metric .
Definition 2.5 A mapping is strongly measurable if, for all , the set-valued mapping defined by the following:
is Lebesgue measurable.
Definition 2.6 Let . The integral of F over I, denoted by , is defined level-wise by the following expression:
for all .
A function is called integrably bounded if there exists an integrable function such that , for all . A strongly measurable and integrably bounded mapping is said to be integrable over I if .
Corollary 2.7 If is continuous, then it is integrable.
We denote by the space of Lebesgue integrable functions from I to .
Theorem 2.8 ([19])
Let be integrable and . Then
-
(i)
,
-
(ii)
,
-
(iii)
is integrable on I,
-
(iv)
.
Theorem 2.9 ([2], Schauder Fixed-Point Theorem for Semilinear Spaces)
Let B be a nonempty, closed, bounded and convex subset of a semilinear Banach space S having the cancellation property and suppose is a compact operator. Then P has at least one fixed point in B.
Definition 2.10 Let . The fuzzy fractional integral of order of u is defined as
provided the integral in the right-hand side is defined for a.e. . For we obtain ; that is, the classical integral operator.
Remark 2.11 ([11])
Let . If with , then and . If , then is bounded at , whereas if with , then we may expect to be unbounded at . This is similar to the crisp case [7].
Proposition 2.12 ([11])
If and , then
Example 2.13 ([11])
Let be a constant fuzzy function, , for . Then
Example 2.14 ([11])
Let be a fuzzy function given by , where and . Then
Definition 2.15 ([2])
Let and . If the fuzzy function is Hukuhara differentiable on , then we define the fuzzy fractional derivative of order q of u at t by
which defines a fuzzy number .
Remark 2.16 Obviously for . Also we have
and
Proposition 2.17 ([11])
If and , then
Example 2.18 ([11])
Let be a constant fuzzy function, for . Then
Example 2.19 ([11])
Let be a fuzzy function given by where and , . Then
We note that .
According to Definition 2.15 and Example 2.19, we obtain the following lemma.
Lemma 2.20 Let and . Then the solutions of the fuzzy fractional differential equation
are , .
3 Fuzzy fractional differential equations
Consider the fuzzy fractional differential equation
where and is a continuous fuzzy function on .
Definition 3.1 A fuzzy function with continuous fractional derivative on is a solution of the fuzzy fractional differential equation (1) if
Remark 3.2 We may apply the results in Section 2 to consider a fuzzy integral equation which allows to obtain a solution to Eq. (1). Indeed, if is a solution to the fuzzy integral equation
and , then u is also a solution to Eq. (1).
Lemma 3.3 If is continuous, then u is bounded.
Proof If u is continuous, the function and are continuous functions on , and bounded. Then is bounded. □
In the sequel, if and , we define the operator by
Lemma 3.4 The operator is well-defined and continuous on .
Proof First we show that is well defined, i.e., for fixed , we check that . In fact we prove that is uniformly continuous on . Let , , and M such that
Then
Therefore when .
Next, we prove continuity of . Let as in , i.e., as . Then we have
Therefore as in . □
Remark 3.5 If is bounded, then is bounded in . Indeed, for we have
Lemma 3.6 If is bounded, then is equicontinuous in .
Proof Let , for all and , . Then from the first part of the proof of the Lemma 3.4, we have, for all ,
which tends to 0 as uniformly in . Hence is equicontinuous in . □
Remark 3.7 If is such that is compact-supported in , then G is bounded. Indeed, there exists a compact set K in ℝ such that . On the other hand, for ,
Then is bounded.
Lemma 3.8 If is such that
is compact-supported and level-equicontinuous, then is relatively compact in .
Proof By the Arzelà-Ascoli Theorem, we show that is an equicontinuous subset of and is relatively compact in for each . Since by Remark 3.7, G is bounded, using Lemma 3.6, it is sufficient to show is relatively compact for each in . By Theorem 2.3, it is equivalent to showing that is a compact-supported subset of and level-equicontinuous on for each . Since is compact-supported, there exists a compact set such that for all and . Then, for all and ,
Then is compact-supported for each .
Now, to prove level-equicontinuity, take fixed and . If , then , for some so that
Therefore
Since is level-equicontinuous, then for given , there exists such that , then
Hence
Then is level-equicontinuous in on for every . □
In the following, we consider
where
The operator is well-defined if is a continuous function in with values in .
We define as , . Now, let . Then we have the following result.
Lemma 3.9 Suppose that is uniformly continuous and bounded in . Then the operator is continuous and bounded in .
Proof Let the sequence , as in where . Then for a given by the uniform continuity of in , there exists such that for ,
Now, given , since as , there exists such that, for , we have , i.e., . Then , , , so that , , , and
This proves that in . On the other hand, if B is bounded in Ω, then , , i.e., . Then , . Since is bounded in , then there exists a such that , . Then is bounded. □
Lemma 3.10 If is compact-supported and level-equicontinuous, then
is relatively compact.
Proof Since
is compact-supported and level-equicontinuous, it is relatively compact. □
Lemma 3.11 Let be a continuous mapping on . Then it is compact on if and only if the set is compact-supported and level-equicontinuous.
Proof First, let be continuous and compact on . Then by Theorem 2.3, is compact-supported and level-equicontinuous.
Now let is compact-supported and level-equicontinuous. Again by Theorem 2.3, it is relatively compact. Hence is compact on . Since for any bounded set , the set is relatively compact. □
Definition 3.12 is a bounded operator if for every bounded B in , is bounded in .
In the following, we present a local existence theorem for the fuzzy fractional differential equation (1). For simplicity, in the rest of the paper, we shall often limit arguments to the choice .
Theorem 3.13 Let and let be a given continuous fuzzy function in . If is compact and uniformly continuous on , then the fuzzy integral equation has at least one continuous solution defined on , for a suitable .
Proof According to Remark 3.2, we need only consider the following fuzzy integral equation:
Define the set
It is easy to see that Ω is a closed, bounded and convex subset of the semilinear Banach space . On the set Ω, we define the operator by
We claim that the operator is continuous and compact. Indeed, the operator is the composition of two continuous and bounded operators , where
and
The operator is well defined since it is the composition of and . Since is continuous and compact, by Lemma 3.11, is compact-supported and level-equicontinuous. Then by Lemma 3.10, is compact-supported and level-equicontinuous. Therefore, by Lemma 3.8, is relatively compact in . Then operator is compact on Ω.
Moreover, from Example 2.14, we have, for ,
Therefore, we have
where we may assume to be as small as we want by shrinking . Now, fix as a domain of the operator , where , which is a convex, bounded, and closed subset of the complete metric space .
For sufficiently small, we have
Theorem 2.9 ensures that operator has at least one fixed point. In consequence, Eq. (1) has at least one continuous solution u defined on , where and . □
Corollary 3.14 Under the conditions of Theorem 3.13 and assuming that , for every , then the fuzzy fractional differential equation (1) has at least a continuous solution defined on , for a suitable .
Proof If is a solution to the fuzzy integral equation
using that is continuous and , for every , then it is clear that and u is a solution to Eq. (1) in . □
Remark 3.15 If f is Lipchitz continuous in the second variable u, then in Theorem 3.13, one has uniqueness of the solution by using the classical Banach contraction fixed-point theorem. Note that Lipchitz continuity implies uniform continuity.
Theorem 3.16 Let be a given continuous fuzzy function in . If f is a Lipschitz continuous function in the second variable on , that is,
then the fuzzy fractional integral equation has a unique solution defined on , for a suitable .
Proof Similar to proof of Theorem 3.13, we define . Then is Lipschitz continuous and for small, is a contraction map. □
Corollary 3.17 Under the conditions of Theorem 3.16 and assuming that , for every , the fuzzy fractional differential equation (1) has at least a continuous solution defined on , for a suitable .
Remark 3.18 As indicated in results of [11], we cannot expect uniqueness for such solutions in general. Consider the equation
with , which admits two solutions and
where
with as we see from Example 2.19. It is easy to check that is also a solution to this problem.
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Acknowledgements
This work has been partially supported by Ministerio de Economía y Competitividad (Spain), project MTM2010-15314, and co-financed by the European Community fund FEDER. This research was completed during the visit of the first author to the USC.
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Each of the authors, AK, JJN, and RRL, contributed to each part of this study equally and read and approved the final version of the manuscript.
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Khastan, A., Nieto, J.J. & Rodríguez-López, R. Schauder fixed-point theorem in semilinear spaces and its application to fractional differential equations with uncertainty. Fixed Point Theory Appl 2014, 21 (2014). https://doi.org/10.1186/1687-1812-2014-21
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DOI: https://doi.org/10.1186/1687-1812-2014-21