Abstract
In this paper we present existence and uniqueness results for a class of fuzzy fractional integral equations. To prove the existence result, we give a variant of the Schauder fixed point theorem in semilinear Banach spaces.
MSC:34A07, 34A08.
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1 Introduction
The topic of fuzzy differential equations has been extensively developed in recent years as a fundamental tool in the description of uncertain models that arise naturally in the real world. Fuzzy differential equations have become an important branch of differential equations with many applications in modeling real world phenomena in quantum optics, robotics, gravity, artificial intelligence, medicine, engineering and many other fields of science. The fundamental notions and results in the fuzzy differential equations can be found in the monographs [1] and [2].
The concept of fuzzy fractional differential equations has been recently introduced in some papers [3–10]. In [7], the authors established the existence and uniqueness of the solution for a class of fuzzy fractional differential equations, where a fuzzy derivative is used in the sense of Seikkala. In [5], the authors proposed the concept of Riemann-Liouville H-differentiability which is a direct extension of strongly generalized H-differentiability (see Bede and Gal [11]) to the fractional literature. They derived explicit solutions to fuzzy fractional differential equations under Riemann-Liouville H-differentiability. In [6], the authors established an existence result for fuzzy fractional integral equations using a compactness-type condition. In this paper, we present an existence result for a class of fuzzy fractional integral equations without a Lipschitz condition. For this we use a variant of the Schauder fixed point theorem. Since the space of continuous fuzzy functions is a semilinear Banach space, we prove a variant of the Schauder fixed point theorem in semilinear Banach spaces.
The paper is organized as follows. Section 2 includes the properties and results which we will use in the rest of the paper. We present an example which shows that a fuzzy fractional differential equation is generally not equivalent to a fuzzy fractional integral equation. In Section 3, we establish the Schauder fixed point theorem in semilinear Banach spaces. In Section 4, we prove an existence result for a class of fuzzy fractional integral equations without a Lipschitz condition. Finally, using Weissinger’s fixed point theorem, we give an existence and uniqueness result.
2 Preliminaries
In the sequel, will denote the n-dimensional Euclidean space with the norm . Let denote the family of all nonempty, compact and convex subsets of . A semilinear structure in is defined by
-
(i)
,
-
(ii)
,
for all , .
The distance between A and B is defined by the Hausdorff-Pompeiu metric
is a complete and separable metric space with respect to the Hausdorff-Pompeiu metric [12].
In the following, we give some basic notions and results on fuzzy set theory. We denote by the space of all fuzzy sets in , that is, is the space of all functions with the following properties:
-
(i)
y is normal, i.e., there exists such that ;
-
(ii)
is compact;
-
(iii)
y is a convex fuzzy function, i.e., for all , and for all , we have
-
(iv)
y is an upper semi-continuous function.
The fuzzy null set is defined by
If , then the set
is called the α-level set of y. Then from (i)-(iv) it follows that the set for all .
The following operations, based on a generalization of Zadeh’s extension principle, define a semilinear structure on :
where and . The α-level set of fuzzy sets satisfy the following properties (see [2]):
-
(i)
;
-
(ii)
for all , and .
We define a metric d on by
where is the Hausdorff-Pompeiu metric. Then is a complete metric space (see [13]).
Proposition 2.1 [2]
If , then
-
(i)
,
-
(ii)
for all ,
-
(iii)
.
Define as the space of fuzzy sets with the property that the function is continuous with respect to the Hausdorff-Pompeiu metric on .
Let be an interval. We denote by the space of all continuous fuzzy functions on T.
It is known that is a complete metric space (see [14]). Therefore, is a complete metric space where
A subset is said to be compact-supported if there exists a compact set such that for all .
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 2.2 [14]
Let A be a compact-supported subset of . Then the following are equivalent:
-
(a)
A is a relatively compact subset of ;
-
(b)
A is level-equicontinuous on .
Remark 2.3 [14]
Let K be a compact subset of and
Then is relatively compact in .
A continuous function is said to be compact if and is bounded imply that is relatively compact in .
Let denote the space of Lebesgue integrable functions from to . Let . The fractional integral of order of y is given by
provided the expression on the right-hand side is defined.
We denote by the set of all Lebesgue integrable selections of , that is,
The Aumann integral of F is defined by
A function is called measurable (see [15]) if for all closed set , where ℬ denotes the Borel algebra of . A function is called integrably bounded if there exists a function such that for a.e. . If such F has measurable selectors, then they are also integrable and is nonempty.
The fractional integral of the function of order is defined by (see [16])
A fuzzy function is measurable if, for all , the set-valued function , defined by
is measurable.
A fuzzy function is integrably bounded if there exists a function such that for all . A measurable and integrably bounded fuzzy function is said to be integrable on if there exists such that for all .
Lemma 2.4 [6]
Let , and let be an integrable fuzzy function. Then for each there exists a unique fuzzy set such that
Let be an integrable fuzzy function. The fuzzy fractional integral of order of the function y,
is defined by (see [6])
Its level sets are given by
that is, we have
Let . If the fuzzy function is Hukuhara differentiable on , then we define the fractional derivative of order of y by
provided that the equation defines a fuzzy number . It is easy to see that , .
Lemma 2.5 [6]
Let and be integrable. Then
Remark 2.6 Let . The equality
is not true in the fuzzy case. Indeed, let be a fuzzy function defined by
Then it is easy to see that
define the α-level intervals of .
Now take . Then
Since
then
which is a fuzzy number for . However, it is not a fuzzy number for . Thus does not satisfy equation (2.1).
3 Schauder fixed point theorem for semilinear spaces
In this section, we prove the Schauder fixed point theorem for semilinear Banach spaces. First, we recall the Schauder fixed point theorem.
Theorem 3.1 ([17], Schauder fixed point theorem)
Let Y be a nonempty, closed, bounded and convex subset of a Banach space X, and suppose that is a compact operator. Then P has at least one fixed point in Y.
We recall that a semilinear metric space is a semilinear space S with a metric which is translation invariant and positively homogeneous, that is,
-
(i)
,
-
(ii)
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.
Let S be a semilinear space having the cancelation property. Define an equivalence relation ∼ on by
for all , and let denote the equivalence class containing . Let G denote the collection of all equivalence classes of . On G define addition and scalar multiplication as follows:
and
for all , and . Further define a map by
for all . Let S be a semilinear metric space. On G, define a norm by
for all .
Theorem 3.2 [18]
Suppose that S is a semilinear space having the cancelation property. Then G is a vector space satisfying and j is an injection such that
-
(i)
;
-
(ii)
for all and .
Theorem 3.3 [18]
Suppose that S is a semilinear metric space. Then the set all equivalence classes G, constructed above, is a metric vector space and j is an isometry.
Now, we are able to prove a variant of the Schauder fixed point theorem in semilinear Banach spaces.
Theorem 3.4 (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 cancelation property, and suppose that is a compact operator. Then P has at least one fixed point in B.
Proof By Theorem 3.3, there exists an embedding . Let B be a nonempty, closed, bounded and convex subset of S. Since j is isometry, it follows that is also a closed and bounded subset of G. For convexity, let and . Then there exist such that and . By Theorem 3.2, we obtain
Since B is convex, we have , which implies . Hence is convex. Let be defined by , that is, . First we show that is a compact operator. Note that is a continuous operator because P, j and are continuous. Further, we have
Since is relatively compact, it follows that is relatively compact. Hence, by the Schauder fixed point theorem, has a fixed point , that is, . Let . Then
Thus is a fixed point of P. □
Remark 3.5 The space of fuzzy sets is a semilinear Banach space S having the cancelation property. Therefore, the Schauder fixed point theorem holds true for fuzzy metric spaces.
4 Existence and uniqueness
Consider the following fuzzy fractional integral equation:
where , and is continuous on .
A function is called a solution for (4.1) if
holds for all .
Remark 4.1 Let . Consider the following fuzzy fractional differential equation:
If is a solution of the integral equation
then by Lemma 2.5 is a solution of (4.2), but the converse is not true.
In [19], the authors showed that the space can be embedded in , the Banach space of continuous real-valued functions defined on , where is the unit ball. In [14], an Ascoli-Arzelá-type theorem was proved. We use this theorem to establish an existence theorem for fuzzy fractional integral equations. Let be the zero function in .
Theorem 4.2 Let , and . Define
Suppose that is a compact function and . Let such that is compact-supported and . Then integral equation (4.1) has at least one solution , where is chosen such that
Proof Define the set
It is evident that Ω is a closed, bounded and convex subset of the Banach space . On the set Ω, we define the operator by
In order to prove our desired existence result, we show that T has a fixed point. First we show that the operator T is continuous on Ω. For this, let in Ω. Then we have
This implies that T is a continuous operator on Ω. For and , we have
It follows that
Thus, T maps the set Ω to itself. Now we will prove that is relatively compact in . Using the Arzela-Ascoli theorem, we just need to prove:
-
(i)
is an equicontinuous subset of ;
-
(ii)
is relatively compact in for each .
Let , and , we obtain
so when for all . This implies that is equicontinuous on . Now we show that is relatively compact in and by Theorem 2.2 this is equivalent to proving that is a level-equicontinuous and compact-supported subset of .
Fixing , we see that and if , then
Since is relatively compact in , Theorem 2.2 implies that is level-equicontinuous. Then for each there exists such that
Also, implies
Hence, we obtain
Therefore is level-equicontinuous in . Finally, due to the relative compactness of and , we have that there exist compact sets such that for all and for all . Thus, we have
Since is bounded on , hence there exists a compact set such that
which proves that is compact-supported. Thus, T is a compact operator. Hence, by Theorem 3.4, it follows that T has a fixed point in Ω, which is a solution of integral equation (4.1). □
The following Weissinger fixed point theorem will be used to prove an existence and uniqueness result.
Theorem 4.3 [20]
Let be a nonempty complete metric space, and let for all be such that converges. Moreover, let the mapping satisfy the inequality
for all and for all . Then the operator T has a unique fixed point . Furthermore, for any , the sequence converges to the above fixed point .
Theorem 4.4 Let . Suppose that is continuous and satisfies a Lipschitz condition, that is, there exists such that
for all , where . Then there exists a unique solution to integral equation (4.1).
Proof From Theorem 4.2, we have that the integral equation has a solution. In order to prove uniqueness of this solution, we prove that the operator T has a unique fixed point. For this, we shall first prove that, for all , and , the following inequality holds:
For , this statement is trivially true. Suppose that (4.5) is true for some . Then from inequality (4.4) we have
Taking the supremum over , we get
The series with is a convergent series (see Theorem 4.1 in [21]). Thus by Theorem 4.3 we deduce the uniqueness of the solution of our integral equation. □
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The second author acknowledges the financial support of Higher Education Commission (HEC) of Pakistan.
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Agarwal, R.P., Arshad, S., O’Regan, D. et al. A Schauder fixed point theorem in semilinear spaces and applications. Fixed Point Theory Appl 2013, 306 (2013). https://doi.org/10.1186/1687-1812-2013-306
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DOI: https://doi.org/10.1186/1687-1812-2013-306