A Schauder fixed point theorem in semilinear spaces and applications
© Agarwal et al.; licensee Springer. 2013
Received: 24 May 2013
Accepted: 10 October 2013
Published: 22 November 2013
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.
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  and .
The concept of fuzzy fractional differential equations has been recently introduced in some papers [3–10]. In , 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 , the authors proposed the concept of Riemann-Liouville H-differentiability which is a direct extension of strongly generalized H-differentiability (see Bede and Gal ) to the fractional literature. They derived explicit solutions to fuzzy fractional differential equations under Riemann-Liouville H-differentiability. In , 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.
for all , .
is a complete and separable metric space with respect to the Hausdorff-Pompeiu metric .
y is normal, i.e., there exists such that ;
- (iii)y is a convex fuzzy function, i.e., for all , and for all , we have
y is an upper semi-continuous function.
is called the α-level set of y. Then from (i)-(iv) it follows that the set for all .
for all , and .
where is the Hausdorff-Pompeiu metric. Then is a complete metric space (see ).
Proposition 2.1 
for all ,
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.
A subset is said to be compact-supported if there exists a compact set such that for all .
A is level-equicontinuous on if A is level-equicontinuous at α for all .
Theorem 2.2 
A is a relatively compact subset of ;
A is level-equicontinuous on .
Remark 2.3 
Then is relatively compact in .
A continuous function is said to be compact if and is bounded imply that is relatively compact in .
provided the expression on the right-hand side is defined.
A function is called measurable (see ) 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.
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 
provided that the equation defines a fuzzy number . It is easy to see that , .
Lemma 2.5 
define the α-level intervals of .
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 (, 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.
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 all .
Theorem 3.2 
for all and .
Theorem 3.3 
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.
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
where , and is continuous on .
holds for all .
then by Lemma 2.5 is a solution of (4.2), but the converse is not true.
In , 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 , 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 .
is an equicontinuous subset of ;
is relatively compact in for each .
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 .
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 
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 .
for all , where . Then there exists a unique solution to integral equation (4.1).
The series with is a convergent series (see Theorem 4.1 in ). Thus by Theorem 4.3 we deduce the uniqueness of the solution of our integral equation. □
The second author acknowledges the financial support of Higher Education Commission (HEC) of Pakistan.
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