Variational-like inequalities for n-dimensional fuzzy-vector-valued functions and fuzzy optimization

Example 2.1

Let u ∈ E² is defined by

then [u]^r = {(x, y): x² + y² ≤ 1 − r²}, r ∈ [0, 1].

Theorem 2.2

[32] If u ∈ Eⁿ, then

1. [u]^r is a nonempty compact convex subset of Rⁿ for any r ∈ (0, 1],

2. [u]^r₁ ⊆ [u]^r₂, whenever 0 ≤ r₂ ≤ r₁ ≤ 1,

3. if r_n > 0 and r_n converging to r ∈ [0, 1] is nondecreasing, then ⋂n=1∞[u]rn=[u]r.

   Conversely, suppose for any r ∈ [0, 1], there exists an A^r ⊆ Rⁿ which satisfies the above (i)-(iii), then there exists a unique u ∈ Eⁿ such that [u]^r = A^r, r ∈ (0, 1], [u]⁰ = ⋃_r∈(0,1][u]^r ⊆ A⁰.

   Let u, v ∈ Eⁿ, k ∈ R. For any x ∈ Rⁿ, the addition and scalar multiplication can be defined, respectively, as:

   (u+v)(x)=sups+t=xmin{u(s),v(t)}, (2.3)

   (ku)(x)=u(xk),k≠0, (2.4)

   (0u)(x)=1,x=0,0,x≠0. (2.5)

   It is well known that for any u, v ∈ Eⁿ and k ∈ R, the addition u + v and the scalar multiplication ku have the level sets

   [u+v]r=[u]r+[v]r={x+y:x∈[u]r,y∈[v]r}, (2.6)

   [ku]r=k[u]r={kx:x∈[u]r}. (2.7)

Proposition 2.3

[33] If u, v ∈ Eⁿ, k, k₁, k₂ ∈ R, then

1. k(u + v) = ku + kv,

2. k₁(k₂ u) = (k₁ k₂)u,

3. (k₁+k₂) u = k₁ u + k₂u when k₁ ≥ 0 and k₂ ≥ 0.

   Give two subsets A, B ⊆ Rⁿ and k ∈ R, the Minkowski difference is given by A − B = A + (−1)B = {a − b : a ∈ A, b ∈ B}. However, in general, A + (−A) ≠ 0, i.e. the opposite of A is not the inverse of A in Minkowski addition (unless A = {a} is a singleton). The spaces 𝓚ⁿ and KCn are not linear spaces since they do not contain inverse elements and therefore subtraction is not defined. To partially overcome this situation, Hukuhara [36] introduced the following H-difference A ⊖ B = C ⟺ A = B + C and an important property of ⊖ is that A ⊖ A = {0}, ∀ A ∈ Rⁿ and (A + B) ⊖ B = A, ∀ A, B ∈ Rⁿ. The H-difference is unique, but a necessary condition for A ⊖_HB to exist is that A contains a translation {c} + B of B. In order to overcome this situation, Stefanini [37] defined the generalized Hukuhara difference of two sets A, B ∈ 𝓚ⁿ as follows

   A⊖gHB=C⟺(1)A=B+C,or(2)B=A+(−1)C. (2.8)

   The generalized Hukuhara difference has been extended to the fuzzy case in [38]. For any u, v ∈ Eⁿ, the generalized Hukuhara difference (gH-difference for short) is the fuzzy number w, if it exists, such that

   u⊖gHv=w⟺(1)u=v+w,or(2)v=u+(−1)w. (2.9)

   It is possible that the gH-difference of two fuzzy numbers does not exist. To solve this shortcoming, in [39] a new difference between fuzzy numbers was proposed. Using the convex hull (conv) the new difference was defined as follows.

Definition 2.4

[39, 40] The generalized difference (g-difference for short) of two fuzzy numbers u, v ∈ Eⁿ is given by its level sets as

where the gH-difference ⊝_gH is with interval operands [u]^β and [v]^β.

A necessary condition for u ⊝_g v to exist is that either [u]^r contains a translation of [v]^r or [v]^r contains a translation of [u]^r for any r ∈ [0, 1].

Proposition 2.5

[41] Let u, v ∈ Eⁿ. Then

1. if the g-difference exists, it is unique,

2. u ⊖_g u = 0,

3. (u + v) ⊖_g v = u, (u + v) ⊖_g u = v,

4. u ⊖_g v = −(v ⊖_g u).

   Given u, v ∈ Eⁿ, the distance D : Eⁿ × Eⁿ → [0, +∞) between u and v is defined by the equation

   D(u,v)=supr∈[0,1]d([u]r,[v]r), (2.11)

   where d is the Hausdorff metric given by

   d([u]r,[v]r)=inf{ε:[u]r⊂N([v]r,ε),[v]r⊂N([u]r,ε)}=max{supa∈[u]rinfb∈[v]r∥a−b∥,supb∈[v]rinfa∈[u]r∥a−b∥}.

   N([u]^r, ε) = {x ∈ Rⁿ : d(x, [u]^r) = inf_y∈[u]^rd(x, y) ≤ ε} is the ε-neighborhood of [u]^r. Then, (Eⁿ, D) is a complete metric space, and satisfies D(u + w, v + w) = D(u, v), D(ku, kv) = ∣k∣D(u, v) for any u, v, w ∈ Eⁿ and k ∈ R.

   Let Sⁿ⁻¹ = {x ∈ Rⁿ: ∥x∥ = 1} be the unit sphere of Rⁿ and 〈⋅, ⋅〉 be the inner product in Rⁿ, i.e. 〈 x, y 〉 = ∑i=1n x_iy_i, where x = (x₁, x₂, ⋯, x_n) ∈ Rⁿ, y = (y₁, y₂, ⋯, y_n) ∈ Rⁿ. Suppose u ∈ Eⁿ, r ∈ [0, 1] and x ∈ Sⁿ⁻¹, the support function of u is defined by

   u∗(r,x)=supa∈[u]r〈a,x〉. (2.12)

Theorem 2.6

[42] Suppose u ∈ Eⁿ, r ∈ [0, 1], then

For u ∈ Eⁿ, we denote the centroid of [u]^r, r ∈ [0, 1] as

where ∫ ⋯ ∫_[u]^r1dx₁dx₂ ⋯ dx_n is the solidity of [u]^r, r ∈ [0, 1] and ∫ ⋯ ∫_[u]^rx_idx₁dx₂ ⋯ dx_n (i = 1, 2, ⋯, n) is the multiple integral of x_i on measurable sets [u]^r, r ∈ [0, 1]. Next we define an order ⪯_C for Eⁿ.

Let τ : Eⁿ → Rⁿ be a real vector-valued function defined by ([29])

where ∫01r∫⋯∫[u]rxidx1dx2⋯dxn∫⋯∫[u]r1dx1dx2⋯dxndr(i=1,2,⋯,n) is the Lebesgue integral of r∫⋯∫[u]rxidx1dx2⋯dxn∫⋯∫[u]r1dx1dx2⋯dxn(i=1,2,⋯,n) on [0, 1]. The vector-valued function τ is called a ranking value function defined on Eⁿ.

Definition 2.7

[29] Let u, v ∈ Eⁿ, C ⊆ Rⁿ be a closed convex cone with 0 ∈ C and C ≠ Rⁿ. We say that u ⪯_c v (u precedes v) if

We say that u ≺_c v if u ⪯_c v and τ(u) ≠ τ(v). Sometimes we may write v ⪰_c u (resp. v ≻_c u) instead of u ⪯_c v (resp. u ≺_c v). In addition, ε͠ ∈ Eⁿ is said to be an arbitrary positive fuzzy-number if ε͠≻_c0 (0 ∈ Rⁿ) and D(ε͠, 0) < ε, where ε is an arbitrary positive real number.

Example 2.8

If u, v ∈ E¹, then τ(u) = ∫01r(ur−+ur+)dr,τ(v)=∫01r(vr−+vr+)dr. Suppose C = R⁺ = [0, +∞) ⊆ R, u ⪯_c v if and only if τ(u) ≤ τ(v), i.e., τ(v) ∈ τ(u) + [0, +∞). Therefore, when u, v ∈ E¹, Definition 2.7 coincides with the definition of ordering of u, v proposed by Goetschel ([27]).

If u, v ∈ E², in Definition 2.7, let C be the set of nonnegative orthant of R², i.e., C = R²⁺ = {(x₁, x₂) ∈ R² : x₁ ⩾ 0, x₂ ⩾ 0)} ⊆ R².

Example 2.9

A special kind of n-dimension fuzzy numbers is the fuzzy n-cell numbers proposed in [43]. Let u ∈ L(Eⁿ), i.e., [u]^r = ∏i=1n[ui−(r),ui+(r)]=[u1−(r),u1+(r)]×[u2−(r),u2+(r)]×⋯×[un−(r),un+(r)] for any r ∈ [0, 1], where the left endpoint function and the right endpoint function ui−(r),ui+(r)∈R with ui−(r)≤ui+(r) (i = 1, 2, ⋯, n), then we have

For u, v ∈ L(Eⁿ), suppose C = Rⁿ⁺ = {(x₁, x₂, ⋯, x_n) ∈ Rⁿ : x₁ ⩾ 0, x₂ ⩾ 0, ⋯, x_n ⩾ 0)} ⊆ Rⁿ, then we have u ⪰_c v ⟺ τ(u) ∈ τ(v) + c. Furthermore, for k₁, k₂ ∈ R, we obtain

Let M be a convex set of m-dimensional Euclidean space R^m and F be an n-dimensional fuzzy-number-valued function (fuzzy-number-valued function for short) from M into Eⁿ.

Example 2.10

The following function is a 2-dimensional fuzzy-number-valued function. For constants s, t ∈ R, F:[−−ln⁡15,−ln⁡15]2→E2 is defined as

Example 2.11

The following function is a fuzzy 1-cell number function. Furthermore, for constants s ∈ R, F(s)(x) = f(s)u(x).

where f(s) = e^s, and

The epigraph of F, denoted by epi(F), is defined as

For u, v ∈ Eⁿ, we say that u and v are comparable, if either u ⪯_c v or v ⪯_c u,; otherwise, they are non-comparable. F is said to be a comparable fuzzy-number-valued function if for each pair t₁, t₂ ∈ M and t₁ ≠ t₂, F(t₁) and F(t₂) are comparable; otherwise, F is said to be a non-comparable fuzzy-number-valued function.

F is said to be lower semicontinuous (l.c.) at t₀ ∈ M, if for any ε͠≻_c0, there exists a neighborhood U of t₀, when t ∈ U, we have F(t₀)≺_cF(t) + ε͠; F is said to be upper semicontinuous (u.c.) at t₀ ∈ M, if for any ε͠≻_c0, there exists a neighborhood U of t₀, when t ∈ U, we have F(t)≺_cF(t₀)+ε͠. F is continuous at t₀ ∈ M, if it is both l.c. and u.c. at t₀, and that it is continuous if and only if it is continuous at every point of M ([29]).

Definition 2.12

([29]) Let F : M → Eⁿ be a fuzzy-number-valued function.

1. An element t₀ ∈ M is called a local minimum point of F if there exists a neighborhood U of t₀, F(t₀)⪯_c F(t) for any t ∈ U.

2. An element t₀ ∈ M is called a global minimum point of F if F(t₀)⪯_c F(t) for any t ∈ M.

3. An element t₀ ∈ M is called a strictly local minimum point of F if there exists a neighborhood U of t₀, F(t₀)≺_c F(t) for any t ∈ U and t ≠ t₀.

4. An element t₀ ∈ M is called a strictly global minimum point of F if F(t₀)≺_c F(t) for any t ∈ M and t ≠ t₀.

Definition 2.13

Let A = (u₁, u₂, ⋯, u_n) ∈ (Eⁿ)ⁿ, u_i ∈ Eⁿ, i = 1, 2, ⋯, n, and T = (t₁, t₂, ⋯, t_n) ∈ Rⁿ be an n-dimensional fuzzy vector and an n-dimensional real vector, respectively. We define the product of a fuzzy vector with a real vector as TA = ∑i=1n t_iu_i, which is an n-dimensional fuzzy number. In addition, if TA = 0, then we say A is fuzzy orthogonal to T.

We denote the fuzzy vector 0 by 0={0,0,⋯,0⏟n}, where 0 ∈ Eⁿ. If A = (u₁, u₂, ⋯, u_n) ∈ (E)ⁿ, u_i ∈ E¹, i = 1, 2, ⋯, n, then Definition 2.13 coincides with Definition 2.4 proposed in [34]. It is not difficult to obtain

For any n-dimensional fuzzy vectors X and Y, let X = {X₁, X₂, ⋯, X_n}, Y = {Y₁, Y₂, ⋯, Y_n}, we use the following convention for equalities and inequalities throughout the paper:

1. X ≤ Y ⟺ x_i⪯_c y_i, i = 1, 2, ⋯ n, with strict inequality holding for at least one i;

2. X ≤ q Y ⟺ x_i⪯_c y_i, i = 1, 2, ⋯ n;

3. X = Y ⟺ x_i =_c y_i, i = 1, 2, ⋯ n;

4. X < Y ⟺ x_i≺_c y_i, i = 1, 2, ⋯ n.

   In the following, we assume that the fuzzy-number-valued function F : M → Eⁿ and fuzzy-vector-valued function F: M → (Eⁿ)ⁿ are comparable, respectively.

Definition 2.14

Let F: M → (Eⁿ)ⁿ be an n-dimensional fuzzy-vector-valued function, denoted by F(t) = (u₁(t), u₂(t), ⋯, u_n(t)), where u_i(t) (i = 1, 2, ⋯, n) is a fuzzy-number-valued function on M. For the sake of brevity, F is called a fuzzy-vector-valued function.

1. F is said to be a comparable fuzzy-vector-valued function if any u_i(t) (i = 1, 2, ⋯, n) is a comparable fuzzy-number-valued function.

2. For s, t ∈ M, we define g-difference of fuzzy-vector-valued functions as

   F(s)⊝gF(t)=(u1(s)⊝gu1(t),u2(s)⊝gu2(t)⋯,un(s)⊝gun(t)). (2.22)

Example 2.15

Let f(t) = (t₁, t₂, t₃) = (et,et(et−1)3,et3) ∈ R³, t ∈ R, be a 3-dimensional real-vector-valued function, and u = (u₁, u₂, u₃) ∈ (E)³ (u_i ∈ E, i = 1, 2, 3) be a 3-dimensional fuzzy-vector-valued function, where

and

Then according to Definition 2.13, we have the following 1-dimensional fuzzy-number-valued function F : (0, ∞)² → E and

and

Theorem 2.16

Let F, G ∈ (Eⁿ)ⁿ. Then

1. if the g-difference exists, it is unique,

2. F ⊖_gF = 0,

3. (F + G) ⊖_gG = F, (F + G) ⊖_gF = G,

4. F ⊖_gG = −(G ⊖_gF).

Proof

It is not difficult to obtain from Proposition 2.5 and Definition 2.14.

Definition 2.17

Let F: M → (Eⁿ)ⁿ be a fuzzy-vector-valued function, denoted by F(t) = (u₁(t), u₂(t), ⋯, u_n(t)), where u_i(t) (i = 1, 2, ⋯, n) is a fuzzy-number-valued function on M.

1. F is said to be lower semicontinuous (l.c.) at t₀ ∈ M if there exists a neighborhood U of t₀, any u_i(t) (i = 1, 2, ⋯, n) is l.c. at t₀.

2. F is said to be upper semicontinuous (u.c.) at t₀ ∈ M if there exists a neighborhood U of t₀, any u_i(t) (i = 1, 2, ⋯, n) is u.c. at t₀.

   A fuzzy-vector-valued function F : M → (Eⁿ)ⁿ is continuous at t₀ ∈ M, if it is both l.c. and u.c. at t₀, and that it is continuous if and only if it is continuous at every point of M.

Definition 2.18

Let F : M → Eⁿ be a fuzzy-number-valued function, t0=(t10,t20,⋯,tm0)∈intM. If g-difference F(t10,⋯,tj0+h,⋯,tm0)⊖gF(t10,⋯,tj0,⋯,tm0) exists and there exists u_j ∈ Eⁿ (j = 1, 2, ⋯, m), such that

then we say that F has the jth partial generalized derivative (g-derivative for short) at t₀, denoted by uj=∂F/∂tj0.

Here the limit is taken in the metric space (Eⁿ, D). If all the partial g-derivatives at t₀ exist, then we say F is said to be generalized differentiable (g-differentiable for short) on t₀. If F is g-differentiable at any interior point of M, then F is said to be g-differentiable on M. The fuzzy vector(u₁, u₂, ⋯, u_m) ∈ (Eⁿ)^m is said to be the gradient of F at t₀, denoted by ∇ F(t₀), that is,

In addition, t₀ ∈ M is said to be a stationary point of F if ∇ F(t₀) = 0.

Note that if M = [a, b], then Definition 2.18 coincides with the definition of F is g-differentiable on [a, b] proposed by Gong and Hai ([41]).

We call F : [a, b] → (L(Eⁿ))ⁿ, denoted by F = (F₁, F₂, ⋯, F_n), is an n-dimensional fuzzy n-cell vector-valued function (fuzzy n-cell vector-valued function for short). If F = (f₁(t)u₁, f₂(t)u₂, ⋯, f_n(t)u_n), where f_i : [a, b] → R, u_i ∈ L(Eⁿ), i = 1, 2, ⋯ n, then the gradient of F at t₀ is defined as

and it is not difficult to obtain ∇Fi=(ui∂fi/∂t10,ui∂fi/∂t20,⋯,ui∂fi/∂tm0),i=1,2,⋯n.

Definition 2.19

The function η : M × M → Rⁿ is said to be a skew function if

Definition 2.20

[35] An n-dimensional fuzzy set u is a fuzzy cone if u (γ x) = u (x) for all γ > 0 and x ∈ Rⁿ.

Definition 2.21

Let A = (u₁, u₂, ⋯, u_n) ∈ (Eⁿ)ⁿ (u_i ∈ Eⁿ, i = 1, 2, ⋯, n) be an n-dimensional fuzzy vector. A fuzzy dual cone of A is the n-dimensional fuzzy vector A^∗ given by

for nonzero y ∈ Rⁿ, and A∗(0)=(1,1,⋯,1⏟n).

Notice that if A = (u) ∈ Eⁿ is a 1-dimensional fuzzy vector, i.e., an n-dimensional fuzzy number, then Definition 2.21 reduces to Definition 8 proposed in [35].

Definition 3.1

The mapping F : M → (Eⁿ)ⁿ is said to be

1. fuzzy monotone over M if

   (y−x)(F(y)⊝gF(x))⪰c0,∀x,y∈M. (3.1)

2. fuzzy invex monotone over M if there exists a continuous map η: M × M → Rⁿ such that

   η(y,x)(F(y)⊝gF(x))⪰c0,∀x,y∈M. (3.2)

   Note that this definition reduces to the definition of monotone functions if η(y, x) = y − x.

3. fuzzy strictly invex monotone over M if there exists a continuous map η : M × M → Rⁿ such that

   η(y,x)(F(y)⊝gF(x))⪰c0,∀x,y∈M,x≠y. (3.3)

Definition 3.2

A g-differentiable fuzzy mapping F : M → Eⁿ is called

1. fuzzy invex (FIX) with respect to a function η : M × M → Rⁿ, if for all x, y ∈ M

   F(x)⊝gF(y)⪰cη(x,y)∇F(y). (3.4)

2. fuzzy strictly invex (FSIX) with respect to a function η : M × M → Rⁿ, if for all x, y ∈ M

   F(x)⊝gF(y)≻cη(x,y)∇F(y),∀x≠y. (3.5)

3. fuzzy incave (FIC) with respect to a function η : M × M → Rⁿ, if for all x, y ∈ M

   F(x)⊝gF(y)⪯cη(x,y)∇F(y). (3.6)

4. fuzzy strictly incave (FSIC) with respect to a function η : M × M → Rⁿ, if for all x, y ∈ M

   F(x)⊝gF(y)≺cη(x,y)∇F(y),∀x≠y. (3.7)

Theorem 3.3

The function F : M → Eⁿ will be a fuzzy invex fuzzy-number-valued function with respect to some function η if and only if each stationary point of F is a global minimum point.

Proof

Necessity. Let F be a fuzzy invex fuzzy-number-valued function with respect to some function η. If x₀ is a stationary point of F, then ∇ F(x₀) = 0. Since F is fuzzy invex, using (3.4), we have F(x)⊝_gF(x₀)⪰_c η(x, y)∇ F(x₀) = 0, ∀ x ∈ M. Thus, we obtain F(x₀)⪯_cF(x), ∀ x ∈ M. Therefore, x₀ is a global minimum point.

Sufficiency. If y is a stationary point of F, i.e., ∇ F = 0, and also a global minimum point of F, then for a function η : M × M → Rⁿ, we have

Therefore, F is a fuzzy invex fuzzy-number-valued function.□

Theorem 3.4

If a g-differentiable fuzzy mapping F : M → Eⁿ is fuzzy invex on M with respect to η : M × M → Rⁿ and η is a skew function. Then, ∇ F : M → (Eⁿ)ⁿ is fuzzy invex monotone with respect to the same η.

Proof

By the fuzzy invexity of F, there exists η(x, y) ∈ Rⁿ, such that

By changing x for y,

Adding the above two formulas, we obtain

Since η is a skew function, η(y, x) = −η(x, y), thus, we have

that is,

Therefore, ∇ F is fuzzy invex monotone.□

Theorem 3.5

If a g-differentiable fuzzy mapping F : M → Eⁿ is fuzzy strictly invex on M with respect to η : M × M → Rⁿ and η is a skew function. Then, ∇ F: M → (Eⁿ)ⁿ is fuzzy strictly invex monotone with respect to the same η.

Proof

By the fuzzy invexity of F, there exists η(x, y) ∈ Rⁿ, such that

By changing x for y,

Adding the above two formulas, we obtain

Since η is a skew function, η(y, x) = −η(x, y), thus, we have

that is,

Therefore, ∇ F is fuzzy strictly invex monotone.□

Theorem 4.1

(Decomposition theorem)[39] If u ∈ Eⁿ, then

Let f : M → Eⁿ be a fuzzy-number-valued function, then ∀ x ∈ M, [f(x)]^α = f_α(x) = f(x)(α) = f(x, {α}) = {x ∈ Rⁿ : f(x) ⩾ α}, α ∈ [0, 1], denotes the α-cut set of f. According to Theorem 2.2, ∀ x ∈ M, f_α(x) ⊆ KCn ⊆ 2^Rn, where 2^Rn is the family of all nonempty subsets of Rⁿ.

Definition 4.2

Let M be a closed and convex set in R^m. Given a continuous mapping η : M × M → Rⁿ.

1. The variational-like inequality problem for n-dimensional fuzzy mappings (fuzzy variational-like inequality problem for short), denoted by FVLIP(M, F, η), is to find x ∈ M such that

   η(x,y)F(x)⪰c0,∀y∈M, (4.2)

   where F : M → (Eⁿ)ⁿ is a continuous fuzzy-vector-valued mapping.

2. The variational inequality problem for n-dimensional fuzzy mappings (fuzzy variational inequality problem for short), denoted by FVIP(M, F), is to find x ∈ M such that

   (y−x)F(x)⪰c0,∀y∈M, (4.3)

   where F : M → (Eⁿ)ⁿ is a continuous fuzzy-vector-valued mapping.

3. The generalized variational-like inequality problem for n-dimensional fuzzy mappings (generalized fuzzy variational-like inequality problem for short), denoted by GFVLIP(M, F,η), is to find x ∈ M with x^∗ ∈ F(x) such that

   η(x,y)x∗⪰c0,∀y∈M. (4.4)

   where F : M → 2^(En)n is a continuous set-valued fuzzy vector mapping, and 2^(En)n is the family of all nonempty subsets of (Eⁿ)ⁿ.

4. The generalized variational inequality problem for n-dimensional fuzzy mappings (generalized fuzzy variational inequality problem for short), denoted by GFVIP(M, F), is to find x ∈ M with x^∗ ∈ F(x) such that

   (y−x)x∗⪰c0,∀y∈M. (4.5)

   where F : M → 2^(En)n is a continuous set-valued fuzzy vector mapping, and 2^(En)n is the family of all nonempty subsets of (Eⁿ)ⁿ.

   Here we would like to point out that (FVLIP) and (FVIP) include many kinds of variational inequality problems as their special cases. For example,

1. If F : M → Rⁿ is a continuous real-vector-valued mapping, and C = R⁺ = [0,∞), then (4.3) reduces to the classical variational inequality problem: to finding x ∈ K such that

   (y−x)F(x)≥0,∀y∈M, (4.6)

   which was considered by Stampacchia [1].

2. If F : M → 2^Rn is a continuous set-valued real-vector mapping, then (4.5) reduces to the classical generalized variational inequality problem: to finding x ∈ M with x^∗ ∈ F(x) such that

   (y−x)x∗≥0,∀y∈M. (4.7)

   This problem was considered and studied by Noor [24].

   Suppose for any r ∈ [0, 1], there exists an A^r ⊆ Rⁿ which satisfies the conditions (i)-(iii) in Theorem 2.2, then there exists a unique F ∈ Eⁿ such that [F]^r = A^r, r ∈ (0, 1], [F]⁰ = ⋃_r∈(0,1][F]^r ⊆ A⁰. We denote A = {A^r:r ∈ [0, 1]}, then A ⊆ KCn ⊆ 2^Rn.

3. Let G : M → A be a set-valued mapping. Now we define a 1-dimensional fuzzy-vector-valued mapping F by

   F:M→F(A),x↦r⋅χA(x)⊆(En)1,

   where χA(x)=1,x∈A,0,x∉A, is the characteristic function of the set A. Then (4.3) is equivalent to the variational inequality for fuzzy mapping, which was considered and studied by Noor [17], i.e., is to find x ∈ M with x^∗ ∈ G(x) such that

   (y−x)x∗≥0,∀y∈M. (4.8)

4. Let G : M → A be a set-valued mapping. Now we define a 1-dimensional fuzzy-vector-valued mapping F by

   F:M→F(A),x↦r⋅χA(x)⊆(En)1,

   where χA(x)=1,x∈A,0,x∉A, is the characteristic function of the set A. Then (4.2) is equivalent to the variational-like inequality for fuzzy mapping, which was considered and studied by Rufián-Lizana [44], i.e., is to find x ∈ M with x^∗ ∈ G(x) such that

   η(x,y)x∗≥0,∀y∈M. (4.9)

   Let a : Rⁿ → [0, 1] be a function, we have f_a(x) = {x ∈ Rⁿ : f(x) ⩾ a(x)}. ∀ x ∈ Rⁿ, suppose for any a(x) ∈ [0, 1], there exists an A^a(x) ⊆ Rⁿ which satisfies the conditions (i)-(iii) in Theorem 2.2, then there exists a unique F ∈ Eⁿ such that [F]^a(x) = A^a(x), a(x) ∈ (0, 1], and [F]⁰ = ⋃_a(x)∈(0,1][F]^a(x) ⊆ A⁰. We denote A = {A^a(x) : a(x) ∈ [0, 1]}, then A ⊆ KCn ⊆ 2^Rn.

   (v) Let G : M → A be a set-valued mapping. Now we define a 1-dimensional fuzzy-vector-valued mapping F by

   F:M→F(A),x↦a(x)⋅χA(x)⊆(En)1,

   where χA(x)=1,x∈A,0,x∉A, is the characteristic function of the set A. Then (4.2) is equivalent to the variational inequality for fuzzy mapping: is to find x ∈ M with x^∗ ∈ F_a(x)(x) such that

   (y−x)x∗⪰c0,∀y∈M. (4.10)

   This problem was studied by Huang [16], where the cut a(x) depends on x. It is slightly different from that of Noor [17], where the cut is a constant. The advantages are that the cuts have more freedom than those of Noor, and the model includes that of Noor as a special case in the viewpoint of mathematics.

Remark 4.3

Let M be a convex cone in R^m and F : M → (Eⁿ)ⁿ. The fuzzy variational-like inequality problem is called complementarity-like problem, denoted by NCLP(F). The NCLP(F) is an important special case of FVILP(M, F, η). That is, the NCLP(F) is to find x^⋆ ∈ M such that

where F^⋆ denotes the fuzzy dual cone of F, i.e.,

Remark 4.4

If F : M → (L(Eⁿ))ⁿ, the fuzzy variational-like inequality problem is called the fuzzy box constrained variational-like inequality problem, denoted by FBVLIP(M, F, η).

Example 4.5

If F : M → (L(Eⁿ))ⁿ be fuzzy n-cell vector-valued function, then (FBVLIP) is to find x ∈ M such that

which is equivalent to

where C = Rⁿ⁺ = {(x₁, x₂, ⋯, x_n) ∈ Rⁿ : x₁ ⩾ 0, x₂ ⩾ 0, ⋯, x_n ⩾ 0} ⊆ Rⁿ. Suppose that F = (F₁, F₂, ⋯, F_n), η = (η₁, η₂, ⋯, η_n), then (FVIP) is to find x ∈ M such that

where τ(F_i(x)) = (∫01r(Fi1−(x)(r)+Fi1+(x)(r))dr,∫01r(Fi2−(x)(r)+Fi2+(x)(r))dr,⋯,∫01r(Fin−(x)(r)+Fin+(x)(r))dr), i = 1, 2, ⋯ n.

Theorem 4.6

Let M be a nonempty, compact and convex subset of R^m and let F be a continuous mapping from X into (Eⁿ)ⁿ. Then there exists a solution to the problem FVLIP(M, F, η), that is, there exists x₀ ∈ M such that

Proof

If M is a point, the theorem is trivial. If M is not a point, then it can be supposed that M has interior points for otherwise, without loss of generality, Rⁿ is replaced by a suitable subspace of Rⁿ containing M. Since a translation of the space Rⁿ dose not affect the assumption or assertion, it can be supposed that x = 0 is an interior point of M. We denote a half-space by ∂M = {x ∈ R^m : (x – p)n ≤ 0}, where p is a point in R^m and n is an nonzero vector in R^m.

Let x₀ ∈ ∂M. Then (4.15) holds if and only if there is a hyperplane π through x₀, that is, π = {x ∈ R^m : (x – p)n = 0}, supporting M such that if N ≠ 0 is a fuzzy vector which is fuzzy orthogonal to π and pointing into the half-space not containing M, then F(x₀) = –tN for some t ≥ 0.

1. ∂M is of class C¹. Assume that (4.15) fails to hold for all x₀ ∈ M. We shall show that

   F(x)=0 (4.16)

   has a solution x₀ ∈ M, which satisfies (4.15) trivially.

   Let N(x₀) be the outward, unit normal vector at x₀ ∈ ∂M. Then

   F(x0,t)=(1−t)F(x0)+tN(x0), 0≤t≤1,

   is a deformation of the vector field F(x₀), x₀ ∈ ∂M, into the vector field N(x₀). The assumption that (4.15) dose not hold for x₀ ∈ ∂M implies that F(x₀) ≠ 0 for x₀ ∈ ∂M, 0 ≤ t ≤ 1. Hence the indices of the vector fields F(x₀), N(x₀) with respect to x = 0 are identical.

   There is a deformation D(x₀, s) = (1 – s)N(x₀) + sx₀, 0 ≤ s ≤ 1, of N(x₀) into x₀ and D(x₀, s) ≠ 0 since x = 0 is an interior point of M. Since the vector field x₀, x₀ ∈ ∂M, has index 1 with respect to x = 0, the index of N(x₀) and, hence, of F(x₀) is 1. This proves that (4.16) has a solution in M.

2. ∂M is not of class C¹. By a theorem of Minkowski (see [45], pp. 36-37), there exists a sequence of compact convex sets M₁ ⊆ M₂ ⊆ ⋯ such that M is the closure of the union M₁ ∪ M₂ ∪ ⋯ and ∂M_m, m = 1, 2, ⋯, is of class C¹. By case 1, there exists x_m ∈ M satisfying

   η(y,xm)F(xm)⪰c0,  ∀y∈Mm.