Nomenclature

s,distance from the measured position to the bottom along the string;

T,axial tension of the component;

M_t,torque of the string;

N,distributional stress between the string and the hole wall;

N_b,binormal component of N;

N_n,principal normal component of N;

q,linear weight of the string submerged in the drill fluid;

E,Young’s modulus;

I,string inertia moment;

r,string radius;

μ,friction coefficient between the string and the hole wall;

g,unit vector in the gravitational direction;

t,tangential unit vector along the borehole;

n,unit vector in the normal direction of the borehole;

b,unit vector in the binormal direction of the borehole;

k_b,curvature of the borehole;

k_n,torsion of the borehole;

M_b,flexure moment of the borehole. 1 Introduction

Fuzzy differential equations (FDEs) are useful tool to describe the dynamic performance for the systems with uncertainties. Therefore,it is important to study the solutions of the FDEs in practice^{[1, 2]}. In 1983,Puri and Ralescu^[3] introduced the concept of fuzzy valued maps, and employed the Hukuhara derivative (H-derivative) to define the differentiability of fuzzy maps. Thereafter,the theory of FDEs has been extensively developed^{[4, 5, 6]},and the existence and uniqueness of the solutions of FDEs have been widely discussed^{[7, 8, 9]}. However,there still are many open problems. Choudary and Donchev^[10] pointed out that the proof of the celebrated theorem of Nieto^[11] was not true,making it still an open question that under what conditions, the FDE could have a solution. Kaleva^[12] introduced a convergent iteration semigroup of a nonlinear fuzzy-valued function,whose limit function was a solution to an autonomous fuzzy Cauchy problem. Guo et al.^[13] discussed the oscillation properties of a class of second-order FDEs with delay,and provided an oscillation criterion. However,it is hard to get the analytic solutions of FDEs. Therefore,numerical methods have also been extensively discussed^{[14, 15, 16, 17]}. The idea of taking the randomness into consideration has received much attention,which might offer a better model for the uncertain dynamical systems^{[18, 19, 20]}. Malinowski^{[21, 22, 23, 24, 25, 26]} discussed the solutions of the random or stochastic FDEs under different conditions.

From the literatures,we can find that there are about five types of methods to research FDEs. The first approach is the classic one,i.e.,the Hukuhara derivative is generalized to a fuzzy valued function. However,in this framework,Diamond^[27] pointed out that the diameter of some FDEs’ solution was unbounded with the increase in the time t,which was inconsistent with the crisp cases. The second approach was presented by H¨ullermeier^[28]. In his idea,the FDE was replaced by a family of differential inclusions,which overcame the preceding problem. However,the solutions obtained by this method might not be fuzzy valued maps^{[29, 30]}. Combining the idea of Aubin^{[31, 32]},Diamond and Watson^[33] exploited this approach by removing the assumption of fuzzy convexity and the compactness of level sets. The third approach is to generalize the differentiability of a fuzzy valued function. Bede et al.^[2] and Bede and Gal^[34] studied a class of fuzzy initial valued problems (FIVPs) and a class of 2-point boundary value problems with a generalized differentiability,which also allowed to obtain the solutions of FDEs with decreasing diameters. However,there are usually 2 solutions for the FIVPs with respect to this derivative^{[35, 36, 37, 38]}. The fourth approach is to use the parametric representation of fuzzy numbers. Chen et al.^{[39, 40]} proved the existence and uniqueness of the solutions for the fuzzy 2-point boundary value problems based on a redefined differentiability. However,none of the FDEs in this framework possesses a periodic solution. The fifth approach was proposed by Liu^[41], where the FDEs were regarded as a type of differential equations,similar to the stochastic differential equation. Thereafter,this method is introduced into the option pricing for fuzzy financial markets^[42] and the fuzzy optimal control with application to portfolio selections^[43].

In a complex system,it is necessary to treat the qualitative properties. Most of the qualitative problems for FDEs are based on the idea of differential inclusions. Therefore,discussing the solutions of fuzzy differential inclusions (FDIs) is important in the qualitative theory of FDEs. FDIs were first presented by Baidosov^[44]. Aubin^[32] and Dordan^[45] discussed the viability of the FDIs in an equivalent form with toll sets. Lakshmikantham and Mohapatra^[30] presented a theorem to show that under what conditions the attainable sets of FDIs are the level sets of a fuzzy map. Zhu and Rao^[46] presented two types of FDIs,i.e.,

and where the existence of the solutions was proved for the open situation and the closed situation, respectively. It is easy to see that the FDI is actually a type of parameterized differential inclusions. Therefore,the results of Zhu and Rao^[46] can extend the theory of FDIs to timevarying cases by turning α into α(x(t)).

Since the implicit differential equations play an important role in the differential equation theory,it is interesting to generalize the results of Zhu and Rao^[46] to the implicit fuzzy differential inclusions (IFDIs) as follows:

and where F(t,x) can be regarded as a fuzzy disturbance to the system. In this paper,we will discuss the existence of the solutions of the above IFDIs under different conditions. Moreover, the viable trajectory on some set K is proved to exist. We also give an application to a implicit fuzzy differential equation which comes from the drilling dynamics in petroleum engineering. 2 Preliminaries

Definition 1 Given a function μ_A : Rⁿ → [0, 1],a fuzzy set A can be written as {(x,μ_A) : x ∈ Rⁿ},where μ_A is called the membership function. For any α ∈ (0,1],we denote the α-level set of A by

and define [A]₀ by Given α ∈ [0,1),(A)_α = {x : μ_A(x) > α} is called the α-open level set of A.

A fuzzy set A and its membership function μ_A are usually taken as one,i.e.,we often simply write the membership of x ∈ Rⁿ in A as A(x). The family of the fuzzy sets on a space X is denoted by F(X).

The space of n-dimensional fuzzy numbers is a set Eⁿ of the fuzzy sets {u : Rⁿ → [0, 1]}, where u satisfies the following conditions:

(i) There must exist an x ∈ Rⁿ such that u(x₀) = 1,i.e.,u is normal.

(ii) u is a fuzzy convex set,which means that for any x,y ∈ Rⁿ and λ ∈ [0, 1],

(iii) u is an upper semicontinuous function,which means that [u]_α is closed for any α[0, 1].

(iv) [u]_{0 = {x ∈ Rⁿ : u(x) > 0}is a compact set.}

It is obvious that,for each u ∈ Eⁿ and any α ∈ [0, 1],[u]_α is a convex compact subset of Rⁿ.

The extension principle^[47] is presented to extend a crisp function f: X → Y to its fuzzy situation,i.e.,a fuzzy function : F(X) → F(Y) is defined as follows:

where u ∈ F(X) and y ∈ Y .

Therefore,we can define the addition operator between a crisp vector a ∈ Rⁿ and a fuzzy set u ∈ Eⁿ as a ⊕ u,that is,for any x ∈ Rⁿ,

It is obvious that [a⊕u]_α = a+ [u]_α,where the addition on the right-hand side of the equation is the Minkowski sum. Throughout this paper,we denote the sum between a vector and a fuzzy set,a ⊕ u,by a + u without confusion.

Let B_X be the unit ball of a metric space X. B_X(x,η) denotes the ball in X centered at x with the radius η. A function f : X → R is said to be Lipschitzean with a constant L > 0 if for any x,y ∈ X,

The following definition is the set-valued scenario of this concept.

Definition 2 If F : X → Y,where X and Y are both metric spaces,F is said to be Lipschitzean if and only if there exists a constant L ≥ 0 such that

where L is called the Lipschitz constant.

Moreover,a fuzzy function F : X → Eⁿ is said to be Lipschitzean if there exists a constant L such that for any α ∈ [0, 1],the set-valued map [F]_α is L-Lipschitzean.

Remark 1 We notice that a fuzzy map F : X → Eⁿ can generate a real valued function : X × Rⁿ → [0, 1],where for any x ∈ X and y ∈ Rⁿ,(x,y) = F(x)(y). For convenience,we do not distinguish F and in the following discussion,and denote them both by F.

A set-valued map F : X → Y is strict if its domain

By the definition of Eⁿ,we can see that,for any fixed x ∈ X,the function F(x,·) : Rⁿ → [0, 1] is strict,convex,and upper semicontinuous. Moreover,the closed set {y∈ Rⁿ : F(x,y) ≥ α} is compact for any α ∈ [0, 1].

Definition 3 (i) For a metric space X,we say that a set-valued map F : X → Y is upper semicontinuous at x ∈ X if and only if for any neighbourhood U of F(x),there exists an η > 0 such that for all x'∈ B_X(x,η),F(x') ⊂ U. Moreover,if F is upper semicontinuous at any point of X,we say that F is upper semicontinuous.

(ii) We say that a set-valued map F is lower semicontinuous at x ∈ Dom(F) if and only if for any y ∈ F(x) and any sequence {x_n} ⊂ Dom(F) converging to x,there exists a sequence consisting of the elements y_n ∈ F(x_n) converging to y. Moreover,if F is lower semicontinuous at any point x ∈ Dom(F),we say that F is lower semicontinuous.

Contingent cone and contingent derivative are useful tools in the study of the solutions and the viability of the differential inclusions. Let K be a non-empty set in a Hilbert space,and the contingent cone T_K(x) to K at x be defined as

where d_K(x+hv) is the distance from x+hv to K. Let F : X → Y be a strict set-valued map, and (x₀,y₀) belong to the graph of F. For a set-valued map,denoted by DF(x₀,y₀),if the graph of DF(x₀,y₀) is the contingent cone T_graph(F)(x₀,y₀) to the graph of F at (x₀,y₀),i.e., DF(x₀,y₀) is called the contingent derivative of F at x₀ ∈ K and y₀ ∈ F(x₀). 3 Mainresults In this section,we mainly discuss the existence of the solutions of a class of IFDIs in different conditions.

Lemma 1^[48] For a metric space X,let F be a Lipschizean compact convex set-valued map from X into Rⁿ. If for some M,each x ∈ X,and F(x) ⊂ MB,then F has a Lipschizean selection f : X → Rⁿ with a Lipschiz constant K.

Now,we present the following existence theorem by Lemma 1 under some strong conditions. The cases under weakened conditions will be discussed later.

Theorem 1 For an open subset Ω ⊂ R×Rⁿ and (t₀,x₀) ∈ Ω,assume that the fuzzy map F : Ω → Eⁿ is Lipschitzean,and the Lipschitz constant is denoted by L > 0. If there exists some constant M such that [F(x)]₀ ⊂ MB,and the function g : Ω × Rⁿ → Rⁿ satisfies the Lipschitz condition that there is a neighborhood V = {(t,x,y) ∈ Ω×Rⁿ,|| t−t₀||≤a, ||x−x₀≤b},then,

for some constants L₁ > 0 and 0 < L₂ < 1. Moreover,suppose that α : Rⁿ → [0, 1] is continuous. Then,on some interval I = [t₀,t₀ +T],the following IFDI has a continuous differentiable solution x : I → Rⁿ:

Proof First of all,we define the set-valued map : Ω → Rⁿ by

Because of the properties of Eⁿ,one can easily see that (t,x) is nonempty. By the convexity of F,we have that,for any u,v ∈(t,x) and λ ∈ [0, 1], Thus,(·,·) is nonempty and convex.

As F(t,x) ∈ Eⁿ and the elements in En are upper semicontinuous,[F(t,x)]α(_x) is a closed subset of the compact set [F(t,x)]₀. Therefore,[F(t,x)]α(_x) is also compact. Overall,we can see that is a nonempty compact convex valued map from Ω into Rⁿ. Moreover,it is also Lipschitzean with the Lipschitz constant L.

By Lemma 1, has a Lipschitzean selection f : Ω → Rⁿ such that f(t,x) ∈ (t,x). We should notice that the Lipschitz constant denoted by K of f might be different with L. Thus, it suffices to verify the existence of the solution of the following implicit initial value problem (IIVP):

Denote the ball in Rⁿ by

For x ∈ B(x₀,b),we set Px = g(t₀,x₀,x) + f(t₀,x₀). From the Lipschitz condition,we have for any u,v ∈ Rⁿ. Since L₂ < 1,P : Rⁿ → Rⁿ is obviously a contraction mapping. Thus,the Banach fixed point theorem shows that there exits a unique y₀ ∈ Rⁿ such that Let Then,U is closed in C¹([t₀,t₀ + a],Rⁿ). Define a distance on U as follows: where Then,it is easy to show that U is a complete metric space. Define an operator T on U as follows: Then,one can directly see that,for any φ ∈ U, This shows that T is a map from U to U.

Next,we will prove that T is a contractive map. Actually,∀φψ ∈ U and ∀t ∈ [t₀,t₀ + a],

Similarly,we have and When a on the right-side of the above inequality is small enough,we have and It follows that Therefore,T is a contractive map on U. By the Banach fixed point theorem,there exists a unique φ^* ∈ U such that φ^* = Tφ^*. Thus,φ^* is the required solution,and this completes the proof.

Remark 2 From the proof of the above theorem,we can see that the Lipschitz constant L₂ of g with respect to y is necessarily less than 1. From the routine of the proof,we only need the selection f of to be Lipschitzean on some neighborhood. Thus,the condition of F can be weakened.

Next,we will consider the IFDI in an open case,i.e.,

where

Definition 4 ^[46] If there exists a constant L > 0 for a fuzzy map F : Ω → Eⁿ such that for any (t,x₁),(t,x₂) ∈ Ω,x₁ ≠ x₂,y₁,y₂ ∈ Rⁿ,F(t,x₁,y1) > 0,and F(t,x₂,y₂) > 0,we have

then F is called F-Lipschitzean.

Lemma 2 ^[49] For a paracompact Hausdorff topological space D and a topological vector space Y,F : D → 2^Y is a nonempty convex valued map. If for any y ∈ Y,F⁻¹(y) = {x ∈ D| y ∈ F(x)} is open in D,i.e.,F has open lower sections,then F has a continuous selection f : D → Y such that ∀x ∈ D,f(x) ∈ F(x).

Theorem 2 For an open subset Ω ⊂ R × Rⁿ,(t₀,x₀) ∈ Ω,and an F-Lipschitzean fuzzy map F : Ω → Eⁿ with the constant K,suppose that the corresponding function F(t,x,y) : Ω × Rⁿ → [0, 1] is lower semicontinuous at (t,x) ∈ Ω. Suppose that the continuous function g : Ω × Rⁿ → Rⁿ satisfies the same Lipschitz condition (L) in Theorem 1. Let α : Rⁿ → [0,1) be an upper semicontinuous function. Then,we have that on some interval I = [t₀,t₀ + T], there exists a continuous differentiable function x : I → Rⁿ,which is the solution of (2).

Proof Similar to the proof of Theorem 1,we first define the set-valued map : Ω → Rⁿ as

Because of the convexity and normality of Eⁿ,for any (t,x) ∈ Ω,(t,x) is nonempty and convex.

For any y ∈ Rⁿ,we assert that ⁻¹(y) = {(t,x) ∈ Ω,F(t,x,y) > α(x)} is open in Rⁿ. It is sufficient to verify that ( ⁻¹(y))^c = {(t,x) ∈ Ω,F(t,x,y) ≤α(x)} is closed. In fact,for any {(t_n,x_n)} ⊂ ( ⁻¹(y))^c,we have F(t_n,x_n,y) ≤ α(x_n). Let (t_n,x_n,y) → (t,x) as n→∞. By the lower semicontinuity of F(·,·,y) and the upper semicontinuity of α(·),we have

Thus,( ⁻¹(y))^c is closed,and is a nonempty convex set-valued map with open lower sections. By Lemma 2,there exists a continuous selection f : Ω → Rⁿ of such that,for each (t,x) ∈ Ω, f(t,x) ∈ (t,x). By the F-Lipschitzean condition of F,f is obviously Lipschitzean with the Lipschitz constant K. Thus,the solution of the following IIVP is also the solution of (2):

The process is similar to the routine in Theorem 1,and thus is omitted here. This completes the proof.

Remark 3 Theorem 2 extends Theorem 4 of Zhu and Rao^[46] to an implicit case.

Remark 4 In the first part of the proof above,we should notice that,for any open subset U ⊂ Rⁿ,

is still an open set,which means that the set-valued map is lower semicontinuous. However, if is not closed valued,Michael’s selection theorem cannot be applied here.

Lemma 3 ^[50] (Michael’s selection theorem) For a metric space X and a Banach space Y, if F : X → Y is a lower semicontinuous closed convex valued map,then F has a continuous selection f : X → Y .

Theorem 3 For an open subset Ω ⊂ R × Rⁿ and (t₀,x₀) ∈ Ω,suppose that F : Ω → Eⁿ is an F-Lipschitzean fuzzy map with a constant K,whose corresponding function F(t,x,y) : Ω×Rⁿ → [0, 1] is continuous at (t,x) ∈ Ω and differentiable at y ∈ Rⁿ. The continuous function g : Ω × Rⁿ → Rⁿ satisfies the same Lipschitz condition (L) in Theorem 1. Let α : Rⁿ → (0,1] be an upper semicontinuous function. Assume that the transversality condition is satisfied,i.e., for any (t₀,x₀) ∈ Ω and y₀ ∈ Rⁿ,there exist constants c > 0 and η > 0 such that

and for the unit ball of R,B_R⊂ cF_y^' (t,x,B_Rⁿ)−[a(x)−z,+∞). Then,on some I = [t₀,t₀+T], there exists a continuous differentiable function x : I → Rⁿ,which is the solution of (1).

Proof Define the set-valued map G(t,x) : Ω → R by

Obviously,G(t,x) is closed. We will prove that G(t,x) is lower semicontinuous. For any r ∈ G(t,x),we need to find r_n ∈ G(t_n,x_n) such that r_n → r. Since α(x) is upper semicontinuous and x_n → x, It is obvious that there exists a most finite α(x_n),which is greater than r. Denote N the max index of α(x_n) greater than 1 and e_N the difference between the max α(x_n) and r. Let Then,{r_n} is the required sequence.

For any z ∈ G(t,x) and T_G(t,x)(z) = [α(x) − z,+∞),the transversality condition can be rewritten as follows:

∀(t,x) ∈ B((t₀,x₀),η),y ∈ B(y₀,η),and z ∈ B(F(t₀,x₀,y₀)) ∩ G(t,x),the unit ball of R B_R ⊂ cF_y^'(t,x,B_Rⁿ) − T_G(t,x)(z).

From Theorem 1.5.5 in Ref. ^[51],we can get that (t,x) is lower semicontinuous,and

By Lemma 3,there exists a continuous selection f(t,x) ∈ (t,x). Similar to the proof of Theorem 2,we can get the existence of the solution of (1). This completes the proof.

In the following theorem,we will discuss the situation when α(x) is lower semicontinuous.

Theorem 4 For an open subset Ω ⊂ R × Rⁿ and (t₀,x₀) ∈ Ω,suppose that F : Ω → Eⁿ is an F-Lipschitzean fuzzy map,whose corresponding function F(t,x,y) : Ω× Rⁿ → [0, 1] is upper semicontinuous at (t,x) ∈ Ω. The continuous function g : Ω × Rⁿ → Rⁿ satisfies the same Lipschitz condition (L) in Theorem 1. Let α : Rⁿ → [0,1) be a lower semicontinuous function. Moreover,if there exists a neighborhood D of (t₀,x₀) such that [F(t,x)]α(x) is compact in Rⁿ,then on some I = [t₀,t₀ + T],there exists a continuous differentiable function x : I → Rⁿ,which is the solution of (1).

Proof Define the set-valued map (t,x) as follows:

We claim that (t,x) is upper semicontiunous on D. By the definition of Eⁿ,we can clearly find that (t,x) is nonempty,compact,and convex in Rⁿ. Since R × Rⁿ is locally compact, for each compact subset K ⊂ D,the graph of the restriction of ,|_K : K → Rⁿ,is closed. Actually,for any Cauchy sequence (t_n,x_n,y_n) ⊂ graph( |_K) which converges to (t,x,y),by Thus,(t,x,y) ∈ graph(|_K),and graph(|_K) is a closed subset of K× [F(t,x)]_α(x). Therefore,graph(|_K) is also compact,and (t,x) is upper semicontiunous on D. As g(t,x,y) is continuous,g(t,x,y) + (t,x) is also upper semicontinuous with nonempty compact convex values.

By the compactness of [F(t,x)]_α(x) and the continuity of g,for any y ∈ Rⁿ,the minimal selection (t,x) → m(g(t,x,y) + (t,x)) is locally compact. Thus,there exists a compact convex subset K ⊂ Rⁿ and two positive scalars a and b such that

Let T = min(a,b||K||),where ||K|| = max(||z||,z ∈ K). Let f_n be a sequence of the continuous single-valued maps approaching in the sense of the approximate selection theorem 1.12.1 in Ref. ^[52]. Then,we can see that f_n is also Lipschitzean. Let x_n : [t₀,t₀ + T] → Rⁿ be the solutions of the following problem: on Q. Since each ||x'_n|| is bounded by ||K|| and each xn,having the values in the compact set x₀ + (t₀ + T )K,it follows from the Ascoli-Arzel`a theorem that there exists a subsequence x_nk such that x_nk converges uniformly to x(·) on [t₀,t₀ + T] and x'_nk converges weakly to x'(·) in L¹([t₀,t₀ + T]). Fix t₁ and t₂ in I,and let Then, where the integral is in the sense of Aumann^[53]. By the F-Lipschitz condition of F and the Lipschitz condition of g,we have By Lemma 2.1.1 in Ref. ^[52],x'(t) is the solution of (1). This completes the proof.

For a subset K ⊂ Rⁿ,the solution for the differential inclusion

is said to be a viable trajectory on [t₀,T) for any t ∈ [t₀,T) and x(t) ∈ K. Next,we will present a viable theorem for (1). Without loss of generality,we set t₀ = 0 in the following theorem.

Theorem 5 Assume that the graph of K of the set-valued map t ∈ R⁺ :→ K(t) ⊂ Rⁿ is closed,the map g : (t,x,v) ∈ K × Rⁿ :→ g(t,x,v) ∈ Rⁿ is continuous and affine with respect to v,and the corresponding function F(t,x,y) : Ω × Rⁿ → [0, 1] of the fuzzy map F : Ω → Eⁿ is upper semicontinuous at (t,x) ∈ Ω. Let α : Rⁿ → [0,1) be a lower semicontinuous function. We posit the following tangential condition:

There exists a constant c > 0 such that

Then,for all x₀ ∈ K(0),the following implicit differential inclusion has a viable solution x(·) :

Proof Similar to the construction in Theorem 3,let

Then,we know that is an upper semicontinuous nonempty compact convex valued map. Set Since g is continuous and the graph of is closed,we can easily see that the graph of G is also closed. Thus,the set-valued map H : (t,x) → G(t,x) ∩ cB has nonempty values by the tangential condition (TC),and H is also upper semicontinuous (having closed graph and taking values in compact cB). Thus,H is bounded with the compact convex values and satisfies the following condition: Theorem 4.4.1 in Ref. ^[52] implies the existence of the viable solutions to the differential inclusion as follows: which are obviously the viable trajectories to (1^*). This completes the proof.

Remark 5 By transposing g to the left,we can easily see that (1^*) is just a special case of the following IFDI:

where f is a single valued map from Ω × Rⁿ to any vector space R^m. The viable solution of the above IFDI exists under the same assumptions,whose proof is routine. 4 Application

In drilling petroleum engineering dynamics,it is crucial to analyze the torque and the drag of the drill-string. In 1986 and 1988,Ho^{[54, 55]} modeled the dynamic performance of the stiff string under the assumption of large deformation,which took the coupling of the axial tension and the torque into consideration. His model is as follows:

This model is more precise,but it is hard to obtain an analytic solution. One can refer to Ref. ^[56] for the details and the numerical solutions for this model. For simplicity,we rewrite (3) into the following form:

where X is a 2-dimensional vector consisting of the torque of the string M_t and the axial tension of the component T . This is an IIVP that has a unique solution locally under the condition < 1. However,because of the uncertainty underground,the dynamic performance of the drilling system is not that accurate and crisp,which means that the above differential equation can be modified as follows: where F(s,X(s)) is a fuzzy map from R⁺×R² to E². Thus,the dynamics of the drilling system can be represented by an implicit fuzzy initial value problem (IFIVP). To analyze the drilling performance,it is essential to discuss its solution.

Let α(x) be a constant α ∈ [0, 1]. From the existence theorems presented in this paper,we know that there exists a solution set {X(x₀,s)}_α,starting from x₀ ∈ [x₀]_α,for the following IFDI:

Generally,F(s,X(s)) is supposed to be upper semicontinuous. Moreover,if the boundedness assumption holds with the constants b,M,and T ,then,for all X₀ and the inclusion we can get that,when α ranges over [0, 1],by Theorem 6.2.3 in Ref. ^[30],the solution sets {X(x₀,s)}_α are the α-level sets of some fuzzy map X(x₀,s),which are actually the solution sets of the problem (*).

Examples Suppose that the trajectory of the borehole is r(s). Then,with the rudiments of differential geometry,we can get

Suppose that r(s) = (0,s,s₂) with 0 ≤ s ≤ L. Then,we can get k_b = 1/4,which is a constant. Let μ = 0.2,r = 0.118m,q = 1 200 kg·m⁻¹,and the initial condition X0 be a symmetric triangular two-dimensional fuzzy vector with the support ,i.e.,

Suppose that F(s,X(s)) = λX(s),which can be regarded as a fuzzy disturbance to the system. Then,(∗) can be interpreted as a family of the IFDIs as follows:

Since the above IFDIs satisfy all the conditions for the existence of the solutions for any α ∈ [0, 1] and each X_α0 ∈ [X₀]_α. Particularly,since [λX(s)]_α = {λX_α(s)} is singleton,let α ∈ [0, 1] and X_α0 ∈ [X₀]_α,then,the above inclusion becomes which is an IIVP. Let the required error be 0.005,and divide the drill string into the sections [s_i,s_i+1] (i = 0,1,2,· · · ,L/δ),where δ is the length,and s₀ = 0. For the first section of the string,by a similar proof with Theorem 1,we get X'(s₀) = Y₀. Putting φ₀(s) = Y₀s + X₀ into the iteration we can get the value of X(s₁). Taking X(s₁) as the initial value of this IIVP on the section [s₁,s₂] and repeating the same process,we can approximately compute the value of X(s_k) section by section,where the result in the former section is the initial value in the later one.

Let

When α varies on [0, 1],and the initial value X_α0 takes all the values in [X₀]_α,we can get a group of numerical solution sets {X(s)}_α with MATLAB,which are the α-level sets of a fuzzy vector X(s) (the solution of (*). Figure 1 shows {X(s)}₀ on [0,L]. From the figure,we can see that this area is actually the support set of X(s),and for any α,[X(s)]_α is contained in it.

From Fig. 1,we can see that as s increases,X(s) turns out to be crisp. This case comes from the construction of (*),where we simply take F(X(s),s) as λX(s). When we transform this IFIVP into the IFDIs by α-level sets,as [F(X(s),s)]_α is singleton,the IFDIs are actually crisp implicit differential equations. Therefore,the perturbation all over the system just comes from the initial values. Moreover,this system is stable under the fuzzy disturbance F. The situation is similar when we alter λ (see Fig. 2).

Acknowledgements The authors are grateful to the editor and the referees for their valuable comments and suggestions.