Schauder fixed-point theorem in semilinear spaces and its application to fractional differential equations with uncertainty
© Khastan et al.; licensee Springer. 2014
Received: 16 October 2013
Accepted: 5 January 2014
Published: 22 January 2014
We study the existence of solution for nonlinear fuzzy differential equations of fractional order involving the Riemann-Liouville derivative.
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 ). It is also valid in locally convex spaces (Tychonoff, 1935; Theorem 2.3.8 in ). Recently, this Schauder fixed-point theorem has been generalized to semilinear spaces .
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, ). 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  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 . 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 . Arshad and Lupulescu  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].
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.
In this section, we give some definitions and introduce the necessary notation which will be used throughout the paper, see for example .
u is normal, i.e., there exists such that ,
u is a convex fuzzy set (i.e. , , ),
u is upper semicontinuous on ℝ,
is compact where cl denotes the closure of a subset.
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 .
and is a complete metric space.
The concept of a semi-linear space and similar concepts were already considered, for instance, in . 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 .
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 . If the functions take values in , we get the sets , .
Definition 2.1 ()
A subset is said to be compact-supported if there exists a compact set such that for all .
Definition 2.2 ()
A is level-equicontinuous on if A is level-equicontinuous at α for all .
Theorem 2.3 ()
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 ()
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 .
is Lebesgue measurable.
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 ()
is integrable on I,
Theorem 2.9 (, 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.
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 ()
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 .
Proposition 2.12 ()
Example 2.13 ()
Example 2.14 ()
Definition 2.15 ()
which defines a fuzzy number .
Proposition 2.17 ()
Example 2.18 ()
Example 2.19 ()
We note that .
According to Definition 2.15 and Example 2.19, we obtain the following lemma.
are , .
3 Fuzzy fractional differential equations
where and is a continuous fuzzy function on .
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. □
Lemma 3.4 The operator is well-defined and continuous on .
Therefore when .
Therefore as in . □
Lemma 3.6 If is bounded, then is equicontinuous in .
which tends to 0 as uniformly in . Hence is equicontinuous in . □
Then is bounded.
is compact-supported and level-equicontinuous, then is relatively compact in .
Then is compact-supported for each .
Then is level-equicontinuous in on for every . □
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 .
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. □
is relatively compact.
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 .
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.
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 .
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 .
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.
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 .
with as we see from Example 2.19. It is easy to check that is also a solution to this problem.
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.
- Smart DR: Fixed Point Theorems. Cambridge University Press, Cambridge; 1980.MATHGoogle Scholar
- Agarwal RP, Arshad S, O’Regan D, Lupulescu V: A Schauder fixed point theorem in semilinear spaces and applications. Fixed Point Theory Appl. 2013., 2013: Article ID 306 10.1186/1687-1812-2013-306Google Scholar
- Bede B, Tenali GB, Lakshmikantham V: Perspectives of fuzzy initial value problems. Commun. Appl. Anal. 2007, 11: 339–358.MathSciNetMATHGoogle Scholar
- Abbas S, Benchohra M, N’Guerekata GM Developments in Mathematics 27. In Topics in Fractional Differential Equations. Springer, New York; 2012.View ArticleGoogle Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.MATHGoogle Scholar
- Belmekki M, Nieto JJ, Rodríguez-López R: Existence of periodic solutions for a nonlinear fractional differential equation. Bound. Value Probl. 2009., 2009: Article ID 324561Google Scholar
- Bonilla B, Rivero M, Rodríguez-Germà L, Trujillo JJ: Fractional differential equations as alternative models to nonlinear differential equations. Appl. Math. Comput. 2007, 187: 79–88. 10.1016/j.amc.2006.08.105View ArticleMathSciNetMATHGoogle Scholar
- Delbosco D, Rodino L: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 1996, 204: 609–625. 10.1006/jmaa.1996.0456View ArticleMathSciNetMATHGoogle Scholar
- Agarwal RP, Lakshmikantham V, Nieto JJ: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. 2010, 72: 2859–2862. 10.1016/j.na.2009.11.029View ArticleMathSciNetMATHGoogle Scholar
- Arshad S, Lupulescu V: On the fractional differential equations with uncertainty. Nonlinear Anal. 2011, 74: 3685–3693. 10.1016/j.na.2011.02.048View ArticleMathSciNetMATHGoogle Scholar
- Allahviranloo T, Salahshour S, Abbasbandy S: Explicit solutions of fractional differential equations with uncertainty. Soft Comput. 2012, 16: 297–302. 10.1007/s00500-011-0743-yView ArticleMATHGoogle Scholar
- Salahshour S, Allahviranloo T, Abbasbandy S, Baleanu D: Existence and uniqueness results for fractional differential equations with uncertainty. Adv. Differ. Equ. 2012, 112: 1–12.MathSciNetGoogle Scholar
- Agarwal RP, Arshad S, O’Regan D, Lupulescu V: Fuzzy fractional integral equations under compactness type condition. Fract. Calc. Appl. Anal. 2012, 15: 572–590.View ArticleMathSciNetGoogle Scholar
- Alikhani R, Bahrami F: Global solutions for nonlinear fuzzy fractional integral and integrodifferential equations. Commun. Nonlinear Sci. Numer. Simul. 2013, 18: 2007–2017. 10.1016/j.cnsns.2012.12.026View ArticleMathSciNetMATHGoogle Scholar
- Diamond P, Kloeden P: Metric Spaces of Fuzzy Sets. World Scientific, Singapore; 1994.View ArticleMATHGoogle Scholar
- Godini G: A framework for best simultaneous approximation, normed almost linear spaces. J. Approx. Theory 1985, 43: 338–358. 10.1016/0021-9045(85)90110-8View ArticleMathSciNetMATHGoogle Scholar
- Román-Flores H, Rojas-Medar M: Embedding of level-continuous fuzzy sets on Banach spaces. Inf. Sci. 2002, 144: 227–247. 10.1016/S0020-0255(02)00182-2View ArticleMATHGoogle Scholar
- Kaleva O: Fuzzy differential inclusions. Fuzzy Sets Syst. 1987, 24: 301–317. 10.1016/0165-0114(87)90029-7View ArticleMathSciNetMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.