Research Article | Open Access
On Darboux-Type Differential Inclusions with Uncertainty
In this article we prove the existence results for solutions of the Darboux-type problems in fuzzy partial differential inclusions with local conditions of integral types. We present two problems involving open and closed level sets of a given fuzzy mapping. In the first case fuzzy differential inclusion has been transformed into an equivalent Darboux-type problem for partial differential equations and then using the Tychonoff fixed point theorem we prove the existence result for this crisp case. For the second case we use Nadler’s fixed point theorem and selection theorem of Kuratowski-Ryll-Nardzewski to find the solution of given differential inclusions problem. We furnish an example to validate our results.
The uncertainties that occur in modelling of the physical problems may originate some types of ambiguities. Zadeh, in , introduced the fuzzy sets to deal with ambiguities. For modelling the physical problems in a better way, the formulation of these problems by virtue of fuzzy differential equations plays a significant role. The notion of fuzzy derivatives was familiarized in . Afterwards, this idea was explored by using Zadeh’s extension principle (ZEP), in . The generalized Hukuhara derivative for fuzzy functions was introduced and studied in . For detailed overview of theory of fuzzy differential equations, we refer the readers to consult the monographs [5, 6].
The ambiguous and complex physical models can be demonstrated in a convenient way by means of fuzzy differential equations (FDEs). The basic idea of fuzzy partial differential equations (FPDEs) has been originated in . Many of the physical problems were designed, solved, and analyzed with fuzzy differential equations. Many authors put their efforts to study and extend this theory; see, for example [8–12]. The importance of both the ODEs and the PDEs in fuzzy sense is evident. Up till now the theory of FDEs has been studied in various ways but the analysis of fuzzy partial differential equations still needs to be explored. Most of the results are not concerned for fuzzy partial differential inclusions. In , the authors discuss the existence of solution of a fuzzy differential inclusion problem constructed via level sets of fuzzy mappings. In [14, 15], the authors extended this technique for system of fuzzy differential inclusions. We generalize and extend this study to discuss the existence of solutions of fuzzy partial differential inclusions (FPDIs). In this article we investigate two models for Darboux-type partial fuzzy differential inclusions with local conditions of integral type. We furnish an example to justify our main results.
Let be -dimensional Euclidean space; for , and denote the closure and interior of , respectively. Let be the family of all nonempty, convex, and compact subsets of a given linear normed space . The Hausdorff metric on is defined as is a complete and separable metric space. By we mean the open subset of ; denote by an open ball with center at and radius . For a given nonempty set , let be the family of all nonempty subsets of
Definition 1 (see ). A fuzzy subset of is a function For , denote the open level and closed level sets of by and , respectively, defined as follows: and
Definition 2 (see ). A fuzzy subset of is called
fuzzy normal set if there exists an such that ;
fuzzy convex fuzzy set, if, for and , fuzzy upper semicontinuous (u.s.c), if, for any , is closed set in the usual topology on .
For a fuzzy subset of , the support of is denoted and defined as
Definition 3 (see ). Let and be two metric spaces and let be a set-valued mapping. Then is said to be upper semicontinuous on , if, for each and any open set , with , there exists an open neighborhood of such that .
() is called lower semicontinuous on if, for each and any open set with , there exists an open neighborhood of such that for all .
Definition 4 (see ). Let denote the set of all fuzzy sets of that are fuzzy normal, fuzzy convex, and fuzzy upper and have compact support.
It is well known that, for each and , the closed level sets are compact and convex in the usual topology on .
For any nonempty subset of , the fuzzy map can be presented by a real valued function , where, , we denote by
Definition 5 (see ). A fuzzy mapping , where , is called lower open if is at each
There is extraordinary useful weakening of compactness that is satisfied by many topological spaces that arise in geometry and analysis, called paracompactness. Paracompactness is weaker than compactness and is often adequate for many purposes. For example, with usual topology is not compact but a paracomapct space. Before defining the paracompact spaces, we first recall some notions used in the definition of a paracompact space.
Definition 6 (see ). An open covering of a topological space is the collection of open sets, so that their union covers (contains) An open covering of refines open covering of , if each is contained in some
Definition 7 (see ). An open covering of is locally finite if every admits a neighborhood such that is empty for all but finitely many
For example, the covering of by open intervals for is locally finite, whereas the covering of the interval by intervals for barely fails to be locally finite, as there is a problem at the origin; now we define the paracompact space.
Definition 8 (see ). A topological space is called paracompact if every open covering has a locally finite refinement.
Definition 9 (see ). A Hausdorff space is a topological space in which every two distinct points can be separated by disjoint open sets.
Definition 10 (see ). Let and be two topological spaces and let be a set valued mapping. is said to have open lower sections if is open in for any .
Definition 11 (see ). For a given complete -finite measure space and a separable Banach space a compact set-valued operator is called an integrably bounded if and only if there exits a such that for all , a.e.
For , define the set being the set of selections of and the closed subset of . is nonempty if and only if is integrably bounded . A fuzzy mapping is integrably bounded if is integrably bounded for any
Definition 12 (see ). A fuzzy mapping is called -level uniformly continuous, if for a given there exists such that for any and for such that implies
3. Main Results
In this section, first we recall some important results that will be useful in the sequel; then we state and prove our main results.
Proposition 13 (see ). Let and be the paracompact Hausdorff topological space and a topological vector space, respectively. Let be a convex valued mapping. If possesses an open lower sections, then there exists a continuous selection of ; that is, is continuous and for all
Lemma 14 (see ). Consider an open subset and let be an upper semicontinuous operator. Then there exists an open interval of , and , such that(i) for (ii) on .
Lemma 15 ( see ). Let be a Banach space with borel measure space . Let be two measurable compact operators. Then if is a measurable selection of , then there exists a measurable selection of such that for all , where is Pompeiu Hausdorff metric on .
Let us propose our first Darboux type differential inclusion problem, which involves open level sets of a fuzzy mapping defined on an open subset of given space. We state the first problem as follows.
Problem 16. Let be a Banach space and let be an open subset of . Let be a continuous function and let be a fuzzy mapping with an function. Consider the following FPDI: for , withThe problem is equivalent towith .
In the next theorem we discuss the conditions, under which the above problem can be transformed into a nonfuzzy problem of partial differential equation, known as Darboux problem.
Theorem 17. Let be a bounded convex and lower open fuzzy surjection. If there exists an function such that is nonempty for each then there exists a continuous such that satisfying
Proof. Define a set-valued map by for each , where is open level set of the fuzzy mapping . By assumptions is nonempty for each ; consider for and by convexity of . Hence ; thus is convex on
Clearly has open lower sections, since for any We show that is closed. Let be a sequence in such that As is lower open and is therefore which implies Thus which shows that has open lower sections. Thus, by Proposition 13, there exists a continuous for each As is a surjection and is bounded we get the problemwith local conditions .
Theorem 18 (see  [Tychonoff]). Let be a complete locally convex vector space. Let be a closed and convex subset of and let be continuous compact operator such that . Then admits a fixed point.
Let be a -dimensional Euclidean space and we denote The topology on is induced by the families of seminormswhere is the bounded region in , and . is complete and locally convex linear space . We use the similar technique used in .
Define the topology on induced by the families of seminormsthen with this topology, is complete and locally convex linear space.
Condition 1. and is subadditive in , for all .
Condition 2. ,
Proof. Clearly is continuous and compact in the topology of We need to use Tychonoff Theorem to find the fixed point of . It is sufficient to prove for a closed set ConsiderThis impliesThe subadditivity of implies the following inequality, as given in : Using the above inequality, we getLet and . By applying Schwartz’s inequality and using above values of and , we getUsing the definition of given seminorm, we obtainLet Using the above inequality, we have and equivalently Choose and set , a closed and bounded subset of ; we have for any that is Hence, by the above Tychonoff theorem, there exists such that ; that is, is the required solution.
Now we state our second Darboux type fuzzy differential inclusion, involving closed level sets of a fuzzy mapping defined on an open subset of given space. This problem is stated as follows.
The next theorem describes the conditions under which the solution of above problem exists.
Theorem 21. Let be a fuzzy integrably bounded and - level uniformly continuous mapping. Let be uniformly continuous. If, for every , , there exists satisfying Then Problem 20 possesses a solution.
Proof. Define by We show that is . For a given , we can write the neighborhood of as For and , we have Since is -level uniformly continuous and is uniformly continuous, using the above inequality, we can find a small enough neighborhood of in , such that for all and thus which means that is . So by Lemma 14 there exists a real constant such that Let , with a metric defined by Then is a complete generalized metric space . Define by where and Here is multivalued triple integral of Aumann , defined by for each
Clearly for all Since the operator is compact so by well-known selection theorem of Kuratowski-Ryll-Nardzewski ,