Existence results for the system of partial differential inclusions with uncertainty

Abstract

It is well known while dealing with uncertainty, fuzzy sets are assumed to be more efficient than ordinary. In this article, the existence results for a certain types of the system of fuzzy differential inclusions with integral types of local conditions have been obtained. Two systems of fuzzy differential inclusions are considered, the existence result for first system is obtained by using a well-known topological fixed point result, while for the other, a multivalued fixed point result is used. This study has generalized many results present in the literature. To validate the study, non-trivial examples are provided.

Keywords

Fuzzy partial differential inclusions solutions existence results

1 Introduction

Most of the phenomena existing in the nature shows uncertain behavior and Zadeh [21] introduced an effective notion of fuzzy sets to handle the vagueness of the data. Due to the flexible nature of fuzzy set theory, acquired an important place in almost every field of science (social as well as natural). Differential equations (DEs) are most important tool amongst the scientists in case of the mathematical modeling for physical real life problems. Modeling with uncertainty by ordinary differential equations may loose the accuracy, while using fuzzy differential equations (FDEs) could be more suitable option. In this regard many researchers have been attracted and motivated to model the physical uncertain problems using FDEs.

Chang [6] introduced the concept of fuzzy derivatives. In [9], the authors exploit these notions by using Zadeh’s extension principle. Seikkala [19] used the generalized Hukuhara derivative as a fuzzy derivative, which enhanced the modeling techniques of incomplete and uncertain systems. For a detailed study in the theory of fuzzy differential equations we refer the readers to the monographs [10, 13].

The theory of fuzzy partial differential equations was first investigated in [5]. Afterwards many authors explored this field in various directions, (for example) [1 , 18], and the references therein. The theory of fuzzy ordinary and partial differential equations is although very young but still evident and has attraction for the researchers, dealing with ambiguous results. The theory of differential inclusion is also very innovative in the sense of modeling the physical problems. In [22], Zhu and Rao discussed the existence of solution of a problem involving fuzzy differential inclusion after that Min et.al, in [16], generalized this method for a system of fuzzy differential inclusions. Recently, Mehmood and Azam [15] presented a model for hyperbolic type partial fuzzy differential inclusion with integral local conditions and also discussed the existence of solution of fuzzy partial differential inclusions.

In this article, we extend the study related to the existence of solutions to the system of fuzzy partial differential inclusions. These results will be useful in application point of view and generalized known results in the literature, for instant [11 , 22]. We also provide an example to justify our main result.

2 Preliminaries

Let $Γ (ℝ^{n})$ be the family of all nonempty, convex and compact subsets of $ℝ^{n} .$ The Hausdorff distance in $Γ (ℝ^{n})$ is defined as: $H (G, J) = max {sup inf_{a \in G} | | a - b | |, sup inf_{b \in J} | | a - b | |} .$ Then $(H, Γ (ℝ^{n}))$ is a complete and separable space.

2.1 Definition [16]

A fuzzy subset G of $ℝ^{n}$ is a function $G : ℝ^{n} \to [0, 1] .$ The α open level and α closed level sets of G are (G) _α and [G] _α are defined as: ${(G)}_{α} = {x : G (x) > α} for α \in [0, 1),$ and ${[G]}_{α} = {x : G (x) \geq α} for α \in [0, 1]$ respectively.

2.2 Definition [16]

Denote by $E^{n},$ the set of all fuzzy sets of $ℝ^{n},$ such that for $w \in E^{n}$ , the following properties are satisfied:

w is normal, that is, there exists an $x \in ℝ^{n}$ such that w (x) = 1 ;

w is fuzzy convex, that is for $x, y \in ℝ^{n}$ and λ ∈ (0, 1) , w (λx+ (1 - λ) x) ≥ min { w (x) , w (y) } ;

w is upper semicontinuous, that is for any α, [w] _α is closed set;

[w] ₀ (the closure of the set ${x \in ℝ^{n} : w (x) \geq 0}$ ) is compact.

It is obvious that for each $w \in ℝ^{n},$ the α closed level sets are compact and convex in $ℝ^{n} .$

2.3 Lemma [16]

If $w \in E^{n},$ then

[w] _α is compact and convex for all 0 ≤ α ≤ 1 .

For all α ≤ β, [w] _β ⊂ [w] _α .

If {α_n} is a non-decreasing sequence converges to some α in [0, 1] , then ${[w]}_{α} = \underset{n \in ℕ}{\cap} {[w]}_{α_{n}} .$

If {G_α : 0≤ α ≤ 1 } is a family of subsets of $ℝ^{n}$ satisfying above conditions of Lemma 2.3, then there exists a $w \in E^{n}$ such that [w] _α = G_α and ${[w]}_{α} = \underset{0 \leq α \leq 1}{\cup} G_{α} \subset G_{0} .$

2.4 Definition [22]

Let (X, d) and (Y, d₁) be two metric spaces and N : X → 2^Y be a set-valued mapping, N is said to be upper semicontinuous at x₀ ∈ X if and only if for any neighborhood U of N (x₀) , there exists a neighborhood V of x₀, such that N (x) ⊂ U for all x ∈ V .

Every fuzzy map $N : X \to E^{n}$ can generate a real valued function $\tilde{N} : X \times ℝ^{n} \to [0, 1],$ where for $x \in X, y \in ℝ^{n},$ $\tilde{N} (x, y) = N (x) (y) : = N_{(x) (y)} .$

2.5 Definition [16]

A fuzzy mapping $N : Ω \to E^{n},$ where $Ω \subset [0, a] \times [0, b] \times ℝ^{n},$ is called lower open if N_(x,y,u) (w) is lower semicontinuous in (x, y, u) ∈ Ω .

3 Main results

In this section first we present few results from literature, these results will be useful to prove our main results. We proved existence results for two types of fuzzy differential inclusions, first for a given fuzzy mapping, the differential term belongs to the closed level sets of fuzzy mapping and in second it belongs to the open level sets of a given fuzzy mapping. Let us start with a very useful proposition.

3.1 Proposition [20]

Let D be a paracompact Hausdorff topological space, Y be a topological vector space and N : D → 2^Y be a multivalued nonempty convex valued function. If N has open lower sections, that is; for any y ∈ Y, N^-1 (y) is open in D, then there exists a continuous selection f : D → Y such that f (x) ∈ N (x) for all x ∈ D .

3.2 Lemma [3]

Let $Ω \subset ℝ \times ℝ^{n} \times ℝ^{m}$ be an open set, (t, x, y) ∈ Ω and $T : Ω \to K (ℝ^{n} \times ℝ^{m})$ an upper semicontinuous set-valued operator. Then there exists an open interval J of $ℝ,$ for a > 0, M > 0, such that;

$J \times S_{aM}^{(x, y)} \subset Ω .$

||T (t, x, y) || ≤ M on $J \times S_{aM}^{(x, y)} .$

Where K (X) denotes the collection of compact subsets of X .

3.3 Lemma [3]

Let X be a Banach space and S, T : J → K (X) be two multivalued measurable operators. If v (t) ∈ S (t) , is measurable selection, then there exists a measurable selection u (t) ∈ T (t) such that; $| | u (t) - v (t) | | \leq H (S (t), T (t))$ for all t ∈ J .

Now we present our first problem in which for a given pair of fuzzy mappings, the inclusions holds for open level sets of the fuzzy maps, the differential inclusion problem is given as;

3.4 Problem

Let $ℝ^{n}, ℝ^{m}$ be two Banach spaces with norm || · ||_∞ and Ω be an open subset of $[0, a] \times [0, b] \times ℝ^{n} \times ℝ^{m} .$ Let $F : Ω \to E^{n}, G : Ω \to E^{m}$ be any two fuzzy mappings and $α, β : ℝ^{n} \times ℝ^{m} \to [0, 1)$ are upper semicontinuous functions. Consider the following system of fuzzy partial differential inclusions: $\begin{matrix} F_{(w, x, u (w, x), v (w, x))} \\ (\begin{matrix} \frac{\partial^{2} u}{\partial w \partial x} + \frac{\partial (ξ_{1} (w, x) u (w, x))}{\partial w} \\ + \frac{\partial (ξ_{2} (w, x) u (w, x))}{\partial x} \end{matrix}) > α (u, v), \end{matrix}$ and $\begin{matrix} G_{(w, x, u (w, x), v (w, x))} \\ (\begin{matrix} \frac{\partial^{2} v}{\partial w \partial x} + \frac{\partial (μ_{1} (w, x) v (w, x))}{\partial w} \\ + \frac{\partial (μ_{2} (w, x) v (w, x))}{\partial x} \end{matrix}) > β (u, v), \end{matrix}$ for (w, x, u, v) ∈ Ω, with local conditions of integral type $\begin{matrix} u (w, 0) + \underset{0}{\int^{b}} k_{1} (w) u (w, x) dx & = & η_{1} (w), \\ u (0, 0) = 0, \\ u (0, x) + \underset{0}{\int^{a}} k_{2} (x) u (w, x) dw & = & η_{2} (x), \\ v (w, 0) + \underset{0}{\int^{b}} h_{1} (w) v (w, x) dx & = & γ_{1} (w), \\ v (0, 0) = 0, \\ v (0, x) + \underset{0}{\int^{a}} h_{2} (x) v (w, x) dw & = & γ_{2} (x), (C 1) \\ for x & \in & [0, b], \\ w \in [0, a] . \end{matrix}$ This problem is equivalent to $\frac{\partial^{2} u}{\partial w \partial x} + \frac{\partial (ξ_{1} (w, x) u (w, x))}{\partial w}$ $+ \frac{\partial (ξ_{2} (w, x) u (w, x))}{\partial x} \in {(F_{(w, x, u (w, x), v (w, x))})}_{α (u, v)}$ and $\frac{\partial^{2} v}{\partial w \partial x} + \frac{\partial (μ_{1} (w, x) v (w, x))}{\partial w}$ $+ \frac{\partial (μ_{2} (w, x) v (w, x))}{\partial x} \in {(G_{(w, x, u (w, x), v (w, x))})}_{β (u, v)}$ with integral type conditions (C1) .

In the next Theorem 3.5, the existence of a common solution of the above system of fuzzy partial differential inclusion will be obtained.

3.5 Theorem

Suppose that $F : Ω \to F (ℝ^{n}), G : Ω \to F (ℝ^{m})$ are bounded convex and lower open fuzzy surjections and $α, β : ℝ^{n} \times ℝ^{m} \to [0, 1)$ are upper semicontinuous functions such that (F_{(w,x,u(w,x),v(w,x))}) _α(u,v) and (G_{(w,x,u(w,x),v(w,x))}) _β(u,v) are nonempty for each (w, x, u, v) ∈ Ω . Then there exist continuous selections $\tilde{f} : Ω \to ℝ^{n}, \tilde{g} : Ω \to ℝ^{m}$ with $\tilde{f} (w, x, u, v) \in {(F_{(w, x, u (w, x), v (w, x))})}_{α (u, v)}$ and $\tilde{g} (w, x, u, v) \in {(G_{(w, x, u (w, x), v (w, x))})}_{β (u, v)},$ such that $\frac{\partial^{2} u}{\partial w \partial x} + \frac{\partial (ξ_{1} (w, x) u (w, x))}{\partial w}$ $+ \frac{\partial (ξ_{2} (w, x) u (w, x))}{\partial x} = \tilde{f} (w, x, u (w, x), v (w, x)),$ and $\frac{\partial^{2} v}{\partial w \partial x} + \frac{\partial (μ_{1} (w, x) v (w, x))}{\partial w}$ $\begin{matrix} + \frac{\partial (μ_{2} (w, x) v (w, x))}{\partial x} & = \tilde{g} (w, x, u (w, x), v (w, x)), \\ for all (w, x, u, v) & \in & Ω . \end{matrix}$

Proof. We define two set-valued functions $\tilde{F} : Ω \to 2^{ℝ^{n}} \tilde{G} : Ω \to 2^{ℝ^{m}}$ as; $\begin{matrix} \tilde{F} (w, x, u (w, x), v (w, x)) \\ = {(F_{(w, x, u (w, x), v (w, x))})}_{α (u, v)}, \\ \tilde{G} (w, x, u (w, x), v (w, x)) \\ = {(G_{(w, x, u (w, x), v (w, x))})}_{β (u, v)}, \\ for each (w, x, u, v) \in Ω . \end{matrix}$ Clearly $\tilde{F} (w, x, u, v)$ and $\tilde{G} (w, x, u, v)$ are nonempty for each (w, x, u, v) ∈ Ω. Consider for $w_{1}, w_{2} \in \tilde{F} (w, x, u, v)$ , $z_{1}, z_{2} \in \tilde{G} (w, x, u, v)$ and θ, σ∈ [0, 1] ; $\begin{matrix} F_{(w, x, u (w, x), v (w, x))} (θ w_{1} + (1 - θ) w_{2}) \\ \geq & min {F_{(w, x, u, v)} (w_{1}), F_{(w, x, u, v)} (w_{2})} \\ > & α (u, v), \\ G_{(w, x, u (w, x), v (w, x))} (σ z_{1} + (1 - σ) z_{2}) \\ \geq & min {G_{(w, x, u, v)} (z_{1}), G_{(w, x, u, v)} (z_{2})} \\ > & β (u, v), \end{matrix}$ the convexity of F and G implies $θ w_{1} + (1 - θ) w_{2} \in \tilde{F} (w, x, u, v)$ and $σ z_{1} + (1 - σ) z_{2} \in \tilde{G} (w, x, u, v),$ thus $\tilde{F} (w, x, u, v)$ and $\tilde{G} (w, x, u, v)$ are convex on Ω . Now to show that $\tilde{F}$ and $\tilde{G}$ have open lower sections. Consider for any $δ \in ℝ^{n}, ω \in ℝ^{m}$

$\begin{matrix} {\tilde{F}}^{- 1} (δ) \\ = & {(w, x, u, v) \in Ω : δ \in \tilde{F} (w, x, u (w, x), v (w, x))} \\ = & {(w, x, u, v) \in Ω : (F_{(w, x, u (w, x), v (w, x))}) (δ) > α (u, v)} \end{matrix}$ and $\begin{matrix} {\tilde{G}}^{- 1} (ω) \\ = & {(w, x, u, v) \in Ω : ω \in \tilde{G} (w, x, u (w, x), v (w, x))} \\ = & {(w, x, u, v) \in Ω : (G_{(w, x, u (w, x), v (w, x))}) (ω) > β (u, v)} . \end{matrix}$ It is enough to prove that the complements of ${\tilde{F}}^{- 1} (δ)$ and ${\tilde{G}}^{- 1} (ω),$ that is, the sets; ${(w, x, u, v) \in Ω : (F_{(w, x, u, v)}) (δ) \leq α (u, v)},$ and ${(w, x, u, v) \in Ω : (F_{(w, x, u, v)}) (ω) \leq β (u, v)}$ respectively are closed. Let (w_n, x_n, u_n, v_n) be a sequence in the complement of ${\tilde{F}}^{- 1} (δ)$ i.e., ${({\tilde{F}}^{- 1} (δ))}^{c},$ such that (w_n, x_n, u_n, v_n) → (w, x, u, v) . Since F is lower open and α is upper semicontinuous we have; $F_{(w_{n}, x_{n}, u_{n}, v_{n})} (δ) \leq α (u_{n}, v_{n}),$ which implies $\begin{matrix} F_{(w, x, u, v)} (δ) & \leq & lim_{n \to \infty} inf F_{(w_{n}, x_{n}, u_{n}, v_{n})} (δ) \\ \leq & lim_{n \to \infty} sup α (u_{n}, v_{n}) \\ \leq & α (u, v) . \end{matrix}$ Thus $(w, x, u, v) \in {({\tilde{F}}^{- 1} (δ))}^{c}$ and hence $\tilde{F}$ has open lower sections. Similarly we can prove the closeness of ${\tilde{G}}^{- 1} (ω)$ . Thus by Proposition 3.1 there exist continuous selections $\tilde{f} : Ω \to ℝ^{n}, \tilde{g} : Ω \to ℝ^{m}$ such that $\tilde{f} (w, x, u, v) \in {(F_{(w, x, u (w, x), v (w, x))})}_{α (u, v)}$ and $\tilde{g} (w, x, u, v) \in$ (G_{(w,x,u(w,x),v(w,x))}) _β(u,v) for each (w, x, u, v)∈ Ω . As F and G are surjections and $\tilde{F} (w, x, u (w, x)),$ (v (w, x)) and $\tilde{G} (w, x, u (w, x), v (w, x))$ are bounded we get the problem $\begin{matrix} \frac{\partial^{2} u}{\partial w \partial x} + \frac{\partial (ξ_{1} (w, x) u (w, x))}{\partial w} \\ + \frac{\partial (ξ_{2} (w, x) u (w, x))}{\partial x} \\ = & \tilde{f} (w, x, u (w, x), v (w, x)), \\ \frac{\partial^{2} v}{\partial w \partial x} + \frac{\partial (μ_{1} (w, x) v (w, x))}{\partial w} \\ + \frac{\partial (μ_{2} (w, x) v (w, x))}{\partial x} \\ = & \tilde{g} (w, x, u (w, x), v (w, x)), \\ for all (w, x, u, v) \in Ω . \end{matrix}$ with local conditions (C1) .■

3.6 Problem

Now consider the following system of fuzzy partial differential inclusion: $F_{(w, x, u (w, x), v (w, x))} (\begin{matrix} \frac{\partial^{2} u}{\partial w \partial x} + \frac{\partial (ξ_{1} (w, x) u (w, x))}{\partial w} \\ + \frac{\partial (ξ_{2} (w, x) u (w, x))}{\partial x} \end{matrix}) \geq α (u, v),$ and $G_{(w, x, u (w, x), v (w, x))} (\begin{matrix} \frac{\partial^{2} v}{\partial w \partial x} + \frac{\partial (μ_{1} (w, x) v (w, x))}{\partial w} \\ + \frac{\partial (μ_{2} (w, x) v (w, x))}{\partial x} \end{matrix}) \geq β (u, v),$ for (w, x, u, v) ∈ Ω, with local conditions of integral type conditions (C1) . The problem is equivalent to $\begin{matrix} \frac{\partial^{2} u}{\partial w \partial x} + \frac{\partial (ξ_{1} (w, x) u (w, x))}{\partial w} \\ + \frac{\partial (ξ_{2} (w, x) u (w, x))}{\partial x} \\ \in & {[F_{(w, x, u (w, x), v (w, x))}]}_{α (u, v)} \end{matrix}$ and $\begin{matrix} \frac{\partial^{2} v}{\partial w \partial x} + \frac{\partial (μ_{1} (w, x) v (w, x))}{\partial w} \\ + \frac{\partial (μ_{2} (w, x) v (w, x))}{\partial x} \\ \in & {[G_{(w, x, u (w, x), v (w, x))}]}_{β (u, v)} \end{matrix}$ with integral type conditions (C1) .

In the next Theorem 3.7, we find the existence result for the solution of above system of fuzzy partial differential inclusions.

3.7 Theorem

Let $F : Ω \to E^{n}$ and $G : Ω \to E^{m}$ are both uniformly continuous and fuzzy integrably bounded, $α, β : ℝ^{n} \times ℝ^{m} \to [0, 1)$ are uniformly continuous. If for every (w, x, u, v) and $(w, x, \bar{u}, \bar{v})$ in Ω, there exist λ₁, λ₂, λ₃, λ₄ ∈ (0, ab) such that (i) $\begin{matrix} H (F (w, x, u, v), F (w, x, \bar{u}, \bar{v})) \\ \leq & \frac{1}{2 {(ab)}^{2}} {λ_{1} | | u - \bar{u} | | + λ_{2} | | v - \bar{v} | |} \end{matrix}$ and $\begin{matrix} H (G (w, x, u, v), G (w, x, \bar{u}, \bar{v})) \\ \leq & \frac{1}{2 {(ab)}^{2}} {λ_{3} | | u - \bar{u} | | + λ_{4} | | v - \bar{v} | |} . \end{matrix}$ Then there exists a solution of above Problem 3.6 .

Proof. Define two set-valued functions $\tilde{F} : Ω \to 2^{ℝ^{n}}$ and $\tilde{G} : Ω \to 2^{ℝ^{m}}$ by $\begin{matrix} \tilde{F} (w, x, u (w, x), v (w, x)) \\ = & {[F (w, x, u (w, x), v (w, x))]}_{α (u, v)}, \\ \tilde{G} (w, x, u (w, x), v (w, x)) \\ = & {[G (w, x, u (w, x), v (w, x))]}_{β (u, v)}, \end{matrix}$ for each (w, x, u, v) ∈ Ω. To show, both of the mappings $\tilde{F}$ and $\tilde{G}$ are upper semicontinuous. Here only the case for $\tilde{F}$ has been considered, similar arguments will be assumed for $\tilde{G} .$ Now for a given $(\hat{w}, \hat{x}, \hat{u}, \hat{v}) \in Ω,$ consider the neighborhood of $\tilde{F} (\hat{w}, \hat{x}, \hat{u}, \hat{v})$ as follows; $S_{r}^{\tilde{F} (\hat{w}, \hat{x}, \hat{u}, \hat{v})} = {w \in ℝ^{n} : d (w, \tilde{F} (\hat{w}, \hat{x}, \hat{u}, \hat{v})) < r} .$ For (w, x, u, v) ∈ Ω and $δ \in \tilde{F} (w, x, u, v),$ we have $\begin{matrix} d (δ, \tilde{F} (\hat{w}, \hat{x}, \hat{u}, \hat{v})) \\ \leq & H (\tilde{F} (w, x, u, v), \tilde{F} (\hat{w}, \hat{x}, \hat{h}, \hat{v})) \end{matrix}$ $\begin{matrix} = H ({[F (w, x, u, v)]}_{α (u, v)}, {[F (\hat{w}, \hat{x}, \hat{u}, \hat{v})]}_{α (\hat{u}, \hat{v})}) \\ \leq H ({[F (\hat{w}, \hat{x}, \hat{u}, \hat{v})]}_{α (\hat{u}, \hat{v})}, {[F (w, x, u, v)]}_{α (\hat{u}, \hat{v})}) \\ + H ({[F (w, x, u, v)]}_{α (\hat{u}, \hat{v})}, {[F (w, x, u, v)]}_{α (u, v)}) \end{matrix}$ $\begin{matrix} \leq & H (\tilde{F} (\hat{w}, \hat{x}, \hat{u}, \hat{v}), \tilde{F} (w, x, u, v)) + \\ H ({[F (w, x, u, v)]}_{α (\hat{u}, \hat{v})}, {[F (w, x, u, v)]}_{α (u, v)}) . \end{matrix}$ Since [F (w, x, u, v)] _α(u,v) is uniformly continuous for each α (u, v) and α is also uniformly continuous, and using above inequality, we can find a small enough neighborhood W of $(\hat{w}, \hat{x}, \hat{u}, \hat{v})$ in Ω, such that for all (w, x, u, v) ∈ W and $δ \in \tilde{F} (w, x, u, v)$ $d (δ, \tilde{F} (\hat{w}, \hat{x}, \hat{u}, \hat{v})) < r,$ thus $\tilde{F} (W) \subset S_{r}^{\tilde{F} (\hat{w}, \hat{x}, \hat{u}, \hat{v})},$ which shows $\tilde{F} (w, x, u, v)$ is upper semicontinuous. So by using the Lemma 3.2, there exists a real constant ρ > 0 such that $max_{(w, x, u, v) \in Ω} | | \tilde{F} (w, x, u, v) | | \leq ρ .$ Let $U = {\begin{matrix} u \in C (I, ℝ^{n}) : | | u (w, x) - u_{0} (w, x) | | \leq ρ_{1}, \\ for all (w, x) \in I = [0, a] \times [0, b], \\ where u (w_{0}, x_{0}) = u_{0} \end{matrix}}$

with a metric $d_{U} : U \times U \to ℝ \cup {+ \infty}$ defined by; $d_{U} (u_{1}, u_{2}) = sup_{(w, x) \in I} {| | u_{1} (w, x) - u_{2} (w, x) | |},$ and let

$V = {\begin{matrix} v \in C (I, ℝ^{m}) : | | v (w, x) - v_{0} (w, x) | | \leq ρ_{2}, \\ for all (w, x) \in I = [0, a] \times [0, b], \\ where v (w_{0}, x_{0}) = v_{0} \end{matrix}}$

with a metric $d_{V} : V \times V \to ℝ \cup {+ \infty}$ defined by; $d_{V} (v_{1}, v_{2}) = sup_{(w, x) \in I} {| | v_{1} (w, x) - v_{2} (w, x) | |} .$ Then (U, d_U) and (V, d_V) both are complete generalized metric spaces. Now we consider

$U \times V = {\begin{matrix} (u, v) \in C (I, ℝ^{n}) \times C (I, ℝ^{m}) : \\ | | u - u_{0} | | \leq ρ_{1}, | | v - v_{0} | | \leq ρ_{2} \\ for all (w, x) \in I = [0, a] \times [0, b], \end{matrix}}, ∥$ and let $d_{U \times V} : (U \times V) \times (U \times V) \to ℝ \cup {+ \infty}$ be a metric on U × V, defined by $d_{U \times V} ((u_{1}, v_{1}), (u_{2}, v_{2})) = d_{U} (u_{1}, u_{2}) + d_{V} (v_{1}, v_{2})$ then (U × V, d_U×V) is a complete generalized metric space. Define an operator S : U × V → 2^U×V by $\begin{matrix} S (u, v) = \\ {\begin{matrix} (\bar{u}, \bar{v}) : \bar{u} (w, x) \in u_{0} (w, x) + \\ \underset{x_{0} w_{0}}{\iint^{x w}} {[F (s, t, u (s, t), v (s, t))]}_{α (u, v)} dsdt a . e I, \\ \bar{v} (w, x) \in v_{0} (w, x) + \\ \underset{x_{0} w_{0}}{\iint^{x w}} {[G (s, t, u (s, t), v (s, t))]}_{β (u, v)} dsdt a . e I \end{matrix}} . \end{matrix}$ Where $u_{0} (w, x) =$ $\begin{matrix} η_{1} (w) + η_{2} (x) - \underset{0}{\int^{b}} k_{1} (w) u (w, x) dx \\ - \underset{0}{\int^{a}} k_{2} (x) u (w, x) dw - \overset{x}{\int_{0}} ξ_{1} (w, t) u (w, t) dt \end{matrix}$ $\begin{matrix} - \underset{0}{\int^{w}} ξ_{2} (s, x) u (s, x) ds + \overset{x}{\int_{0}} ξ_{1} (0, t) η_{2} (t) dt \\ + \underset{0}{\int^{w}} ξ_{2} (s, 0) η_{1} (s) ds \end{matrix}$ $\begin{matrix} - \underset{0 0}{\iint^{w b}} ξ_{2} (s, 0) k_{1} (w) u (w, x) dsdx \\ - \underset{0 0}{\iint^{x a}} ξ_{1} (0, t) k_{2} (x) u (w, x) dxdt \end{matrix}$ and $v_{0} (w, x) =$ $\begin{matrix} γ_{1} (w) + γ_{2} (x) - \underset{0}{\int^{b}} h_{1} (w) u (w, x) dx \\ - \underset{0}{\int^{a}} h_{2} (x) u (w, x) dw - \overset{x}{\int_{0}} μ_{1} (w, t) u (w, t) dt \\ - \underset{0}{\int^{w}} μ_{2} (s, x) u (s, x) ds + \overset{x}{\int_{0}} μ_{1} (0, t) γ_{2} (t) dt \\ + \underset{0}{\int^{w}} μ_{2} (s, 0) γ_{1} (s) ds \\ - \underset{0 0}{\iint^{w b}} μ_{2} (s, 0) h_{1} (w) u (w, x) dsdx \\ - \underset{0 0}{\iint^{x a}} ξ_{1} (0, t) h_{2} (x) u (w, x) dwdt . \end{matrix}$ Also $\underset{x_{0} w_{0}}{\iint^{x w}} {[F (s, t, u (s, t), v (s, t))]}_{α (u, v)} dsdt$ and $\underset{x_{0} w_{0}}{\iint^{x w}} {[G (s, t, u (s, t), v (s, t))]}_{β (u, v)} dsdt$ are multivalued double integral of Aumann type [4], defined by $\begin{matrix} \underset{x_{0} w_{0}}{\iint^{x w}} \tilde{F} (s, t, u (s, t), v (s, t)) dsdt \\ = & \underset{x_{0} w_{0}}{\iint^{x w}} {[F (s, t, u (s, t), v (s, t))]}_{α (u, v)} dsdt \end{matrix}$ $= {\begin{matrix} \underset{x_{0} w_{0}}{\iint^{x w}} f (s, t, u (s, t), v (s, t)) dsdt, \\ where f : Ω \to ℝ^{n} is a measurable selection for \\ {[F (s, t, u (s, t), v (s, t))]}_{α (u, v)} \end{matrix}},$ and $\begin{matrix} \underset{x_{0} w_{0}}{\iint^{x w}} \tilde{G} (s, t, u (s, t), v (s, t)) dsdt \\ = & \underset{x_{0} w_{0}}{\iint^{x w}} {[G (s, t, u (s, t), v (s, t))]}_{β (u, v)} dsdt \end{matrix}$ $= {\begin{matrix} \underset{x_{0} w_{0}}{\iint^{x w}} g (s, t, u (s, t), v (s, t)) dsdt, \\ where g : Ω \to ℝ^{m} is a measurable selection for \\ {[G (s, t, u (s, t), v (s, t))]}_{β (u, v)} \end{matrix}},$ for each α, β ∈ (0, 1] . Clearly S (u, v) = φ for all (u, v) ∈ U × V . Since the multivalued operators $\tilde{F} (w, x, u, v) = {[F (w, x, u, v)]}_{α (u, v)}$ and $\tilde{G} (w, x, u, v) = {[G (w, x, u, v)]}_{β (u, v)}$ are both upper semicontinuous with compact values, therefore by using the well known selection Theorem of Kuratowski-Ryll-Nardzewski [12], $\tilde{F} (w, x, u, v)$ and $\tilde{G} (w, x, u, v)$ have measurable selections $f (w, x, u, v) \in \tilde{F} (w, x, u, v)$ and $g (w, x, u, v) \in \tilde{G} (w, x, u, v)$ for all (w, x) ∈ I, are Lebesgue integrable by given conditions. Let $\tilde{u} (w, x) = u_{0} + \underset{x_{0} w_{0}}{\iint^{x w}} f (s, t, u (s, t), v (s, t)) dsdt$ and $\tilde{v} (w, x) = v_{0} + \underset{x_{0} w_{0}}{\iint^{x w}} g (s, t, u (s, t), v (s, t)) dsdt,$ so that $(\tilde{u}, \tilde{v}) \in S (u, v)$ , thus S (u, v) = φ . Next we claim that S (u, v) is closed for all (u, v) ∈ U × V . Let (u_n, v_n) be a sequence in S (u, v) converges to $(\hat{u}, \hat{v}) \in U \times V .$ Since $u_{n} \in u_{0} + \underset{x_{0} w_{0}}{\iint^{x w}} \tilde{F} (s, t, u (s, t), v (s, t)) dsdt a . e I,$ and $v_{n} \in v_{0} + \underset{x_{0} w_{0}}{\iint^{x w}} \tilde{G} (s, t, u (s, t), v (s, t)) dsdt a . e I,$ and the sets [4] $u_{0} + \underset{x_{0} w_{0}}{\iint^{x w}} \tilde{F} (s, t, u (s, t), v (s, t)) dsdt$ and $v_{0} + \underset{x_{0} w_{0}}{\iint^{x w}} \tilde{G} (s, t, u (s, t), v (s, t)) dsdt$ are closed, which implies $(\hat{u}, \hat{v}) \in S (u, v) .$ Now we prove that S is a multivalued contraction. For this let (u₂, v₂) ∈ S (u₁, v₁) , which implies there exists $f (w, x, u_{1}, v_{1}) \in \tilde{F} (w, x, u_{1}, v_{1})$ and $g (w, x, u_{1}, v_{1}) \in \tilde{G} (w, x, u_{1}, v_{1})$ such that $u_{2} (w, x) = u_{0} + \underset{x_{0} w_{0}}{\iint^{x w}} f (s, t, u_{1} (s, t), v_{1} (s, t)) dsdt,$ and $v_{2} (w, x) = v_{0} + \underset{x_{0} w_{0}}{\iint^{x w}} g (s, t, u_{1} (s, t), v_{1} (s, t)) dsdt .$ Then by applying Lemma 3.3, we get a pair of measurable selections $f (w, x, u_{2}, v_{2}) \in \tilde{F} (w, x, u_{2}, v_{2})$ and $g (w, x, u_{2}, v_{2}) \in \tilde{G} (w, x, u_{2}, v_{2})$ which gives $\begin{matrix} | | f (w, x, u_{2}, v_{2}) - f (w, x, u_{1}, v_{1}) | | \\ \leq & H (\tilde{F} (w, x, u_{1}, v_{1}), \tilde{F} (w, x, u_{2}, v_{2})) \\ = & H (\begin{matrix} {[F (w, x, u_{1}, v_{1})]}_{α (u_{1}, v_{1})}, \\ {[F (w, x, u_{2}, v_{2})]}_{α (u_{2}, v_{2})} \end{matrix}) \\ \leq & H (F (w, x, u_{1}, v_{1}), F (w, x, u_{2}, v_{2})) \end{matrix}$ and $\begin{matrix} | | g (w, x, u_{2}, v_{2}) - g (w, x, u_{1}, v_{1}) | | \\ \leq & H (G (w, x, u_{1}, v_{1}), G (w, x, u_{2}, v_{2})) . \end{matrix}$ Let (u₃, v₃) ∈ S (u₂, v₂) , then $u_{3} (w, x) = u_{0} + \underset{x_{0} w_{0}}{\iint^{x w}} f (s, t, u_{2} (s, t), v_{2} (s, t)) dsdt,$ and $v_{3} (w, x) = v_{0} + \underset{x_{0} w_{0}}{\iint^{x w}} g (s, t, u_{2} (s, t), v_{2} (s, t)) dsdt .$ Now consider $\begin{matrix} | | u_{3} (w, x) - u_{2} (w, x) | | \\ = & | | \begin{matrix} u_{0} + \underset{x_{0} w_{0}}{\iint^{x w}} f (s, t, u_{2} (s, t), v_{2} (s, t)) dsdt - \\ u_{0} - \underset{x_{0} w_{0}}{\iint^{x w}} f (s, t, u_{1} (s, t), v_{1} (s, t)) dsdt \end{matrix} | | \\ = & | | \begin{matrix} \underset{x_{0} w_{0}}{\iint^{x w}} f (s, t, u_{2} (s, t), v_{2} (s, t)) dsdt - \\ \underset{x_{0} w_{0}}{\iint^{x w}} f (s, t, u_{1} (s, t), v_{1} (s, t)) dsdt \end{matrix} | | \end{matrix}$ $\begin{matrix} \leq & \underset{x_{0} w_{0}}{\iint^{x w}} | | \begin{matrix} f (s, t, u_{2} (s, t), v_{2} (s, t)) - \\ f (s, t, u_{1} (s, t), v_{1} (s, t)) \end{matrix} | | dsdt \\ \leq & \underset{x_{0} w_{0}}{\iint^{x w}} H (F (w, x, u_{1}, v_{1}), F (w, x, u_{2}, v_{2})) dsdt \\ \leq & \underset{x_{0} w_{0}}{\iint^{x w}} \frac{1}{2 {(ab)}^{2}} {λ_{1} | | u_{1} - u_{2} | | + λ_{2} | | v_{1} - v_{2} | |} dsdt \\ \leq & \frac{max {λ_{1}, λ_{2}}}{2 {(ab)}^{2}} \underset{x_{0} w_{0}}{\iint^{x w}} {d_{U} (u_{1}, u_{2}) + d_{V} (v_{1}, v_{2})} dsdt \end{matrix}$ $\begin{matrix} \leq & \frac{max {λ_{1}, λ_{2}}}{2 {(ab)}^{2}} d_{U \times V} ((u_{1}, v_{1}), (u_{2}, v_{2})) \underset{x_{0} w_{0}}{\iint^{x w}} dsdt \\ \leq & \frac{max {λ_{1}, λ_{2}}}{2 {(ab)}^{2}} d_{U \times V} ((u_{1}, v_{1}), (u_{2}, v_{2})) \\ (x - x_{0}) (w - w_{0}), \end{matrix}$ so we have, $d_{U} (u_{2}, u_{3}) \leq \frac{max {λ_{1}, λ_{2}}}{2 ab} d_{U \times V} ((u_{1}, v_{1}), (u_{2}, v_{2})) .$ Also consider $\begin{matrix} | | v_{3} (w, x) - v_{2} (w, x) | | \\ = & | | \begin{matrix} v_{0} + \underset{x_{0} w_{0}}{\iint^{x w}} g (s, t, u_{2} (s, t), v_{2} (s, t)) dsdt - \\ v_{0} - \underset{x_{0} w_{0}}{\iint^{x w}} g (s, t, u_{1} (s, t), v_{1} (s, t)) dsdt \end{matrix} | | \\ = & | | \begin{matrix} \underset{x_{0} w_{0}}{\iint^{x w}} g (s, t, u_{2} (s, t), v_{2} (s, t)) dsdt - \\ \underset{x_{0} w_{0}}{\iint^{x w}} g (s, t, u_{1} (s, t), v_{1} (s, t)) dsdt \end{matrix} | | \end{matrix}$ $\begin{matrix} \leq & \underset{x_{0} w_{0}}{\iint^{x w}} | | \begin{matrix} g (s, t, u_{2} (s, t), v_{2} (s, t)) - \\ g (s, t, u_{1} (s, t), v_{1} (s, t)) \end{matrix} | | dsdt \\ \leq & \underset{x_{0} w_{0}}{\iint^{x w}} H (G (w, x, u_{1}, v_{1}), G (w, x, u_{2}, v_{2})) dsdt \\ \leq & \underset{x_{0} w_{0}}{\iint^{x w}} \frac{1}{2 {(ab)}^{2}} {λ_{3} | | u_{1} - u_{2} | | + λ_{4} | | v_{1} - v_{2} | |} dsdt \end{matrix}$ $\begin{matrix} \leq & \frac{max {λ_{3}, λ_{4}}}{2 {(ab)}^{2}} {d_{U} (u_{1}, u_{2}) + d_{V} (v_{1}, v_{2})} \underset{x_{0} w_{0}}{\iint^{x w}} dsdt \\ \leq & \frac{max {λ_{3}, λ_{4}}}{2 {(ab)}^{2}} d_{U \times V} ((u_{1}, v_{1}), (u_{2}, v_{2})) (ab), \end{matrix}$ which implies $d_{V} (v_{2}, v_{3}) \leq \frac{max {λ_{3}, λ_{4}}}{ab} d_{U \times V} ((u_{1}, v_{1}), (u_{2}, v_{2})) .$ Thus we have $d_{U \times V} ((u_{2}, v_{2}), (u_{3}, v_{3})) \leq \frac{Θ}{2 ab} d_{U \times V} ((u_{1}, v_{1}), (u_{2}, v_{2})),$ where Θ = max { λ₁, λ₂ } + max { λ₃, λ₄ } . By interchanging (u₁, v₁) and (u₂, v₂) , we get similar inequalities and thus $H (S (u_{1}, v_{1}), S (u_{2}, v_{2})) \leq \frac{Θ}{2 ab} d_{U \times V} ((u_{1}, v_{1}), (u_{2}, v_{2})),$ for all (u₁, v₁) , (u₂, v₂) ∈ U × V . Since $\frac{Θ}{2 ab} \in (0, 1)$ so by Nadler’s Theorem [17], there exists a fixed point (u, v) ∈ S (u, v). This proves the result.■

It is very difficult to find the solutions of differential inclusions, but in the next examples we consider the systems of differential inclusions under some conditions to obtain their solutions by using our main results. In the next example a system of fuzzy differential inclusions is solved by using Theorem 3.5.

3.8 Example

Consider the hyperbolic PDI $\begin{matrix} \frac{\partial^{2} u}{\partial w \partial x} + \frac{\partial (wu)}{\partial w} + \frac{\partial (xu)}{\partial x} \\ \in & {[χ_{{(w + x) cos (w + x) + sin (w + x)}}]}_{α (u, v)}, \end{matrix}$ and $\begin{matrix} \frac{\partial^{2} v}{\partial w \partial x} + \frac{\partial (wv)}{\partial w} + \frac{\partial (xv)}{\partial x} \\ \in & {[χ_{{(w + x) cos (w + x) + sin (w + x)}}]}_{α (u, v)}, \end{matrix}$ with boundary conditions; $\begin{matrix} v (w, 0) & = & u (w, 0) = sin (w), \\ v (0, x) & = & u (0, x) = sin (x) \end{matrix}$ and w, x ∈ Δ, where Δ is a bounded rectangle in plane. Suppose that $F, G : Ω \to F (ℝ^{n})$ be bounded convex and lower open fuzzy surjections and $α, β : ℝ^{n} \to [0, 1)$ be the upper semicontinuous functions such that $\begin{matrix} {[F_{(w, x, u (w, x), v (w, x))}]}_{α (u (w, x), v (w, x))} \\ : & = {[χ_{{(w + x) cos (w + x) + sin (w + x)}}]}_{α (u (w, x), v (w, x))} \end{matrix}$ and $\begin{matrix} {[G_{(w, x, u (w, x), v (w, x))}]}_{β (u (w, x), v (w, x))} \\ : & = {[χ_{{(w + x) cos (w + x) + sin (w + x)}}]}_{α (u (w, x), v (w, x))} \end{matrix}$ is nonempty for each (w, x, u, v) ∈ Ω, where χ is the characteristic function. As $H ({[F_{(w, x, u, v)}]}_{α (u, v)}, {[G_{(w, x, u, v)}]}_{β (u, v)})$ is zero. Then by Theorem 3.5, there exist two continuous selections $f (w, x, u, v) \in {[F_{(w, x, u (w, x), v (w, x))}]}_{α (u (w, x), v (w, x))}$ and $g (w, x, u) \in {[G_{(w, x, u (w, x), v (w, x))}]}_{α (u (w, x), v (w, x))}$ for each (w, x, u, v) ∈ Ω, such that $(w + x) cos (w + x) + sin (w + x) = f (w, x, u, v)$ and $(w + x) cos (w + x) + sin (w + x) = g (w, x, u, v)$ Which implies $\begin{matrix} (w + x) cos (w + x) + sin (w + x) \\ = & \frac{\partial^{2} u}{\partial w \partial x} + \frac{\partial (wu)}{\partial w} + \frac{\partial (xu)}{\partial x}, \end{matrix}$ and $\begin{matrix} (w + x) cos (w + x) + sin (w + x) \\ = & \frac{\partial^{2} v}{\partial w \partial x} + \frac{\partial (wv)}{\partial w} + \frac{\partial (xv)}{\partial x} . \end{matrix}$ Hence the common solution of the system $\frac{\partial^{2} u}{\partial w \partial x} + \frac{\partial (wu)}{\partial w} + \frac{\partial (xu)}{\partial x}$ $- (w + x) cos (w + x) - sin (w + x) = 0,$ $\frac{\partial^{2} v}{\partial w \partial x} + \frac{\partial (wv)}{\partial w} + \frac{\partial (xv)}{\partial x}$ $- (w + x) cos (w + x) - sin (w + x) = 0$ is thus $v (w, x) = u (w, x) = sin (w + x) .$

In the next example a system of fuzzy differential inclusions is solved by using Theorem 3.7.

3.9 Example

Consider a system of fuzzy differential inclusions $\frac{\partial^{2} (u + e^{- 2 (x + w)})}{\partial w \partial x} + \frac{\partial (\frac{1}{12} sin (x + w) u)}{\partial w}$ $+ \frac{\partial (\frac{1}{4} cos (x + w) u)}{\partial x} \in {[F_{(w, x, u (w, x), v (w, x))}]}_{α (u, v)}$ $\frac{\partial^{2} (v + e^{- 2 (x + w)})}{\partial w \partial x} + \frac{\partial (\frac{1}{12} sin (x + w) v)}{\partial w}$ $+ \frac{\partial (\frac{1}{4} cos (x + w) v)}{\partial x} \in {[G_{(w, x, u (w, x), v (w, x))}]}_{α (u, v)}$ and $F, G : {[0, 1]}^{2} \times ℝ \times ℝ \to E^{1}$ defined by $\begin{matrix} F_{(x, w, u, v)} \\ = & G_{(x, w, u, v)} \\ = & \frac{5}{12} (sin (x + w) + cos (x + w)) \\ {(e^{- 2 (x + w)}; 2 e^{- 2 (x + w)})}_{S} \end{matrix}$ where (e^-2(x+w) ; 2e^-2(x+w)) _S is a symmetric triangular fuzzy number with compact support [e^-2(x+w), 2e^-2(x+w)]. Let $q (x, w) = \frac{5}{12} (sin (x + w) + cos (x + w)),$ then for any α ∈ [0, 1] , the α-level sets are $\begin{matrix} {[F_{(x, w, u, v)}]}_{α} \\ = & q (x, w) [(1 + 0.5 α) e^{- 2 (x + w)}, (2 - 0.5 α) e^{- 2 (x + w)}] . \end{matrix}$ Clearly F is convex and bounded. Now for α, β ∈ [0, 1] , consider $\begin{matrix} H ({[F_{(x, w, u, v)}]}_{α}, {[F_{(x, w, \bar{u}, \bar{v})}]}_{β}) \\ = & | q (x, w) e^{- 2 (x + w)} | \times \\ H ([1 + 0.5 α, 2 - 0.5 α], [1 + 0.5 β, 2 - 0.5 β]) \\ \leq & | \frac{5}{6} e^{- 2 (x + w)} | | α - β | \\ \leq & | \frac{5}{6} e^{- 2 (x + w)} | [α | | u - \bar{u} | | + β | | v - \bar{v} | |] \end{matrix}$ All the calculations will remain same for the case of G_(x,w,u,v), and thanks to Theorem 3.7 which allows us to obtain f (x, w, u, v) ∈ [F_(x,w,u,v)] _α and g (x, w, u, v) ∈ [G_(x,w,u,v)] _α such that $\frac{\partial^{2} (u + e^{- 2 (x + w)})}{\partial w \partial x} + \frac{\partial (\frac{1}{12} sin (x + w) u)}{\partial w}$ $+ \frac{\partial (\frac{1}{4} cos (x + w) u)}{\partial x} = f (x, w, u, v)$ and $\frac{\partial^{2} (v + e^{- 2 (x + w)})}{\partial w \partial x} + \frac{\partial (\frac{1}{12} sin (x + w) v)}{\partial w}$ $+ \frac{\partial (\frac{1}{4} cos (x + w) v)}{\partial x} = g (x, w, u, v);$ By simple investigation here we find f (x, w, u, v) = g (x, w, u, v) = q (x, w) e^-2(x+w), therefore we get the system $\begin{matrix} \frac{\partial^{2} (u + e^{- 2 (x + w)})}{\partial w \partial x} + \frac{\partial (\frac{1}{12} sin (x + w) u)}{\partial w} \\ + \frac{\partial (\frac{1}{4} cos (x + w) u)}{\partial x} \\ = & \frac{5}{12} (sin (x + w) + cos (x + w)) e^{- 2 (x + w)} \end{matrix}$ $\begin{matrix} \frac{\partial^{2} (v + e^{- 2 (x + w)})}{\partial w \partial x} + \frac{\partial (\frac{1}{12} sin (x + w) v)}{\partial w} \\ + \frac{\partial (\frac{1}{4} cos (x + w) v)}{\partial x} \\ = & \frac{5}{12} (sin (x + w) + cos (x + w)) e^{- 2 (x + w)}, \end{matrix}$ with common solution $u (x, w) = v (x, w) = u (x, w) = - e^{2 (x + w)} .$

4 Competing Interests

The authors are grateful to Higher Education Commission (HEC) of Pakistan, for providing the grant of project no: 5766 under National Research Program for Universities.

Footnotes

Acknowledgments

We are grateful to referees and editor for their useful suggestions and comments to improve this article.

References

Arara

and Benchohra

, Fuzzy solutions for boundary value problems with integral boundary conditions, Acta Mathematica Universitatis Comenianae 75(1) (2006), 119–126.

Arara

, Benchohra

, Ntouyas

S.K.

and Ouahab

, Fuzzy solutions for hyperbolic partial differential equations, International Journal of Applied Mathematical Sciences 2(2) (2005), 181–195.

J.P Aubin and A. Cellina, Differential inclusions: Set-valued maps and viability theory, Springer Science & Business Media, 264, 2005.

Aumann

R.J.

, Integrals of set-valued functions, J Math Anal Appl 12(1) (1965), 1–12.

Buckley

and Feuring

, Introduce to fuzzy partial differential equations, Fuzzy Sets and Systems 105 (1999), 241–248.

Chang

S.L.

and Zadeh

L.A.

, On fuzzy mapping and control, IEEE Transactions on Systems, Man, and Cybernetics 2 (1972), 30–34.

S.S Chang, Fixed Point Theory with Applications, Chongqing Publishing House, Chongqing, 1984.

Chen

Y.Y.

, Chang

Y.T.

and Chen

B.S.

, Fuzzy solutions to partial differential equations: Adaptive Approach, IEEE Transactions on Fuzzy Systems 17(1) (2009), 116–127.

Dubois

and Prade

, Towards fuzzy differential calculus part 3: Differentiation, Fuzzy Sets and Systems 8 (1982), 225–233.

10.

Gomes

L.T.

and de

L.C.

, Barros and B. Bede, Fuzzy differential equations in various approaches, Berlin: Springer, 2015.

11.

Hüllermeier

, An approach to modeling and simulation of uncertain dynamical systems, Internat J Uncertain Fuzziness Knowledge-Based Systems 5 (1997), 117–137.

12.

Kuratowski

and Ryll-Nardzewski

, A general theorem on selectors, Bull Polish Acad Sci 12 (1965), 397–403.

13.

Lakshmikantham

and Mohapatra

, Theory of fuzzy differential equations and inclusions, London: Taylor and Francis Publishers, 2003.

14.

Long

H.V.

, Son

N.T.K.

and Ha

N.T.M.

, The existence and uniqueness of fuzzy solutions for hyperbolic partial differential equations, Fuzzy Optimization and Decision Making 13(4) (2014), 435–462.

15.

Mehmood

and Azam

, Existence results for fuzzy partial differential inclusions, Journal of Function Spaces (2016) 8. Article ID 6759294. 10.1155/2016/6759294.

16.

Min

, Liu

Z.B.

, Zhang

L.H.

and Huang

N.J.

, On a system of fuzzy differential inclusions, Filomat 29(6) (2015), 1231–1244.

17.

Nadler

S.B.

, Jr, Multi-valued contraction mappings, Pacific J Math 30(2) (1969), 475–488.

18.

Nikravesh

, Zadeh

L.A.

and Korotkikh

, Fuzzy partial differential equations and relational equations, Berlin, Springer, 2004.

19.

Seikkala

, On the fuzzy initial value problem, Fuzzy Sets and Systems 24 (1987), 319–330.

20.

Yannelis

N.C.

and Prabhakar

N.D.

, Existence of minimal elements and equilibria in linear topological spaces, J Math Econom 12 (1983), 233–245.

21.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

22.

Zhu

Y.G.

and Rao

, Differential inclusions for fuzzy maps, Fuzzy Sets and Systems 112 (2000), 257–261.