Existence Results for Fuzzy Partial Differential Inclusions

Abstract

We discuss the existence of solution of a certain type of fuzzy partial differential inclusions with local conditions of integral types.

1. Introduction

The complexities of physical phenomena depict some kind of uncertainty, and dealing with uncertainty by fuzzy sets was introduced by Zadeh in [1]. For the betterment of mathematical modelling for physical problems, formulation by fuzzy differential equations plays an important role. The notion of fuzzy derivatives was introduced in [2]. In [3], the authors define these concepts by using Zadeh’s extension principle, while, in [4], the fuzzy derivative was defined by generalized Hukuhara derivative. For more details about the theory of fuzzy differential equations, we refer the readers to the monograph [5].

The incomplete and uncertain systems can be modelled in a better way by virtue of fuzzy differential equations. The theory of fuzzy partial differential equations was initiated in [6]. After that many authors work out in this branch of analysis in a productive way, for example, [7–11] and the references therein. The importance of ordinary differential equations and partial differential equations in fuzzy sense is obvious. This field is in its growing age, and it interacts with many researchers. And up till now the literature is mainly concerned with fuzzy ordinary differential equations, inclusions, and fuzzy partial differential equations, but in this paper we present a model for hyperbolic type partial fuzzy differential inclusion with integral local conditions. In [12], the authors discuss the existence of solution of a fuzzy differential inclusion problem and, in [13], the authors extend the technique for a system of fuzzy differential inclusions. We extend this study to discuss the existence of solution of fuzzy partial differential inclusion. We also present an example to justify our main result.

2. Preliminaries

Let be the family of all nonempty, convex, and compact subsets of The Hausdorff metric in is defined as follows: Then is a complete and separable space.

Definition 1 (see [13]). A fuzzy subset of is a function The open level and closed level sets of are and and are defined as follows: respectively.

Definition 2 (see [13]). Denote by the set of all fuzzy sets of , such that satisfy the following properties:(i)is normal; that is, there exists such that .(ii) is fuzzy convex; that is, for and , .(iii) is upper semicontinuous; that is, for any , is closed set.(iv) is compact, where bar denotes the closure of the set.

It is obvious that, for each , the closed level sets are compact and convex in .

Lemma 3 (see [13]). If , then(i) is compact and convex for all ,(ii)for all ,(iii)if is a nondecreasing sequence converging to some in , then .

If is a family of subsets of satisfying the above conditions of the lemma, then there exists such that and .

Definition 4 (see [12]). Let and be two metric spaces and let be a set-valued mapping; is said to be upper semicontinuous at if and only if, for any neighborhood of , there exists a neighborhood of , such that for all .

Every fuzzy map can generate a real valued function , where, for , .

Definition 5 (see [13]). A fuzzy mapping , where , is called lower open if is lower semicontinuous in .

3. Main Results

Proposition 6 (see [14]). Let be a paracompact Hausdorff topological space, let be a topological vector space, and let be a multivalued nonempty complex valued function. If has open lower sections, that is, for any , is open in , then there exists a continuous selection such that for any .

Lemma 7 (see [15]). Let be an open set and and an upper semicontinuous set-valued operator. Then there exists an open interval of , for , , such that(i),(ii) on .

Lemma 8 (see [15]). Let be a Banach space and let be two multivalued measurable operators, where is the collection of compact subsets of Then if is measurable selection, then there exists a measurable selection such thatfor all .

Problem 9. Let be a Banach space with norm and let be an open subset of . Let be a fuzzy mapping and is an upper semicontinuous function. Consider the following partial fuzzy differential inclusion: for with local conditions of integral type The problem is equivalent to with integral type conditions .

In the next theorem we prove the existence of solution of the above partial differential inclusion.

Theorem 10. Suppose that is bounded convex and lower open fuzzy surjection and is an upper semicontinuous function such that is nonempty for each Then there exists a continuous selection such that

Proof. We can define a set-valued function as follows: Clearly is nonempty for each , and consider for and by convexity of . Hence ; thus is convex on .
Now we will show that has open lower sections. For any , It is enough to prove that the complement of , that is, the set is closed. Let be a sequence in such that . Since is lower open and is upper semicontinuous we have and we have Thus and hence has open lower sections. Thus by Proposition 6 there exists a continuous selection for each . As is a surjection and is bounded we get the problem with local conditions .

The following Tychonoff theorem will be used to prove the existence of solution of problem with local conditions .

Theorem 11 (see [16] [Tychonoff]). Let be a complete locally convex linear space and let be a closed and convex subset of Let be continuous and If is compact then has a fixed point.

Let be an -dimensional Euclidean space and let be set of positive integers. Let The topology on is induced by families of seminorms where is the bounded region in , , and It is well known that [17] this topology on is complete and locally convex linear space. We use similar technique to that used in [18].

Observe that problem with is equivalent to finding fixed point of the following mapping defined by

Condition 1. Consider and , , and is subadditive in , for all .

Condition 2. Consider , and

Theorem 12. If Conditions 1 and 2 hold then problem with local conditions has a solution in .

Proof. For existence of the solution we need to prove that has a fixed point in . Clearly is continuous and compact in the topology of It is sufficient to prove that, for any closed bounded and convex subset of , we have For this, consider The above equation implies The subadditivity of implies the following inequality [18]: using this inequality in the above inequality, we get Let By applying Schwartz’s inequality and using the above values of and , we get Using the definition of given seminorm, we obtainLet using this we have this can be written as by choosing and , we have, for any , that is, Hence, by Tychonoff Theorem, there exists such that . This completes the theorem.

Problem 13. Now consider the following partial fuzzy differential inclusion: for , with local conditions of the same integral type .
The problem is equivalent to with integral local conditions .

In the next theorem we find the existence result for a solution of above fuzzy partial differential inclusion.

Theorem 14. Let be -level uniformly continuous and fuzzy integrably bounded; is uniformly continuous. If, for every , in , there exists such that then there exists a solution of Problem 13.

Proof. Define by . We show that is upper semicontinuous. For a given , we can write the neighborhood of as follows: For and , we have Since is uniformly continuous for each and is also uniformly continuous and, using the above inequality, we can find a small enough neighborhood of in , such that for all and thus This means that is upper semicontinuous. So we find, from Lemma 7, that there exists a real constant such that Let with a metric defined by Then is a complete generalized metric space[13]. Define by where Here is multivalued double integral of Aumann [19], and we defined it here as follows: for each .
Clearly for all . Since the operator is upper semicontinuous with compact values, by well known selection theorem of Kuratowski-Ryll-Nardzewski [20], has measurable selection for all , and is Lebesgue integrable [4].
Let ; thus .
Next we show that is closed for all . Let be a sequence in that converges to . Since and is closed [19], we have .
We claim that is multivalued contraction; for this let , which implies that there exists such that By virtue of Lemma 8 [21], there exists a measurable selection such that Let Now consider Similarly we will get Thus we have for all .
Since , by Nadler’s Theorem [22], there exists a fixed point . Thus

Example 15. Consider the hyperbolic PDI with boundary conditions and , where is a bounded rectangle in plane. Suppose that is bounded convex and lower open fuzzy surjection and is an upper semicontinuous function such that is nonempty for each , where is the characteristic function, as Then, by Theorem 10, there exists a continuous selection for each , such that which implies Hence the solution of is thus

Competing Interests

The authors have declared that they have no competing interests.