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Generalized vector complementarity problem with fuzzy mappings Adem Kılıçman a,∗ , Rais Ahmad b , Mijanur Rahaman b a Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia b Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
Received 13 May 2014; received in revised form 11 January 2015; accepted 13 January 2015 Available online 16 January 2015
Abstract In this paper, we introduce and study a generalized vector complementarity problem with fuzzy mappings. Under suitable conditions, we have shown that generalized vector complementarity problem with fuzzy mappings is equivalent to generalized vector variational inequality problem with fuzzy mappings. We derive some existence results for our problem. Results of this paper represent a significant improvement and refinement of the previously known results. © 2015 Elsevier B.V. All rights reserved. Keywords: Vector; Complementarity problem; Variational inequality; Positive homogeneous; Fuzzy mapping
1. Introduction Variational inequality theory provides us a unified frame work for dealing with a wide class of problems arising in elasticity, structural analysis, physical and engineering sciences, etc. (see [1–3,9,15–17] and references therein). Equally important is the area of mathematical programming known as the complementarity theory, which was introduced by Lemke [25] and has been generalized to study a large class of problems occurring in fluid flow through porous media, contact problem in elasticity, economics and transportation equilibria, control optimization, and lubrication problem (see [3,10,31,32] and the references therein). The relationship between a variational inequality problem and a complementarity problem has been noted implicitly by Lions [28] and Mancino and Stampacchia [30]. However, it was Karamardian [22,23], who showed that if the set is involved in a variational inequality problem and complementarity problem is a convex cone, then both the problems are equivalent. The applications of fuzzy set theory [35] can be found in many branches of mathematical and engineering sciences including artificial intelligence, management science, control engineering, computer science, see e.g. [37]. Heilpern [18] introduced the concept of fuzzy mapping and proved a fixed point theorem for fuzzy contraction mapping which is an analogue of Nadler’s fixed point theorem for multi-valued mappings. Chang and Zhu [7] introduced the concept of variational inequalities for fuzzy mappings in abstract spaces. Since then, several types of variational inequalities * Corresponding author.
E-mail addresses:
[email protected] (A. Kılıçman),
[email protected] (R. Ahmad),
[email protected] (M. Rahaman). http://dx.doi.org/10.1016/j.fss.2015.01.008 0165-0114/© 2015 Elsevier B.V. All rights reserved.
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and complementarity problems for fuzzy mappings have been studied by many authors [5,6,19,29] using the approach of Ref. [7]. Let X be a real Banach space with dual space X ∗ and K a closed convex cone of X. In 2001, Yin et al. [34] introduced a class of F -complementarity problems, which consist of finding x ∈ K such that T x, x + F (x) = 0,
T x, y + F (y) ≥ 0,
∀y ∈ K,
X∗
where T : X → and F : K → (−∞, ∞), and they proved that F -complementarity problem is equivalent to the following variational inequality problem i.e., finding x ∈ K such that T x, y − x + F (y) − F (x) ≥ 0,
∀y ∈ K.
After that vector F -complementarity problems were studied by Huang and Fang [20] and Huang and Li [21], Li and Huang [27], Farajzadeh and Zafarani [12] and Farajzadeh et al. [13], etc. in different settings. The purpose of this work is to introduce and study a generalized vector complementarity problem with fuzzy mappings and its corresponding generalized vector variational inequality problem with fuzzy mappings. Further, the equivalence of these problems is shown and some existence results are proved. 2. Preliminaries First we mention some definitions, notations and conclusions needed for achieving our main results. Let Y be a real Banach space. A non-empty subset C of Y is said to be a convex cone if 1. C + C = C; 2. λC ⊆ C, for all λ > 0. C is called pointed cone if C is a convex cone and C ∩ {−C} = {0}. An ordered Banach space (Y, C) is a real Banach space Y with an ordering defined by a closed convex cone C ⊆ Y with an apex at the origin if x≥y
⇔
x − y ∈ C.
Note that this ordering is anti-symmetric if C is pointed. Let X be a real Banach space, K ⊆ X be a non-empty closed convex set and (Y, C) be an ordered Banach space induced by a pointed closed convex cone C. We denote by L(X, Y ), the space of all continuous linear mappings from X into Y , and l, x the value of continuous linear mapping l ∈ L(X, Y ) at x. The following definition of fuzzy set was introduced by Zadeh [35] in 1965 in order to mathematically represent uncertainty and vagueness. Definition 2.1. Let X be a nonempty set. A fuzzy set A in X is characterized by its membership mapping μA : X −→ [0, 1] and μA (x) is interpreted as the degree of membership of element x in the fuzzy set A, for each x ∈ X. We denote the collection of all fuzzy sets of X by F(X) and a mapping F : K → F(X) is called a fuzzy mapping. If F is a fuzzy mapping, then F (x), x ∈ K (denoted by Fx hereafter), is a fuzzy set in F(X) and Fx (y), y ∈ X is the degree of membership of y in Fx . Let A ∈ F(X) and α ∈ (0, 1]. Then the set (A)α = x ∈ X : A(x) ≥ α is called an α-cut set of A. A fuzzy mapping F : K −→ F(X) is said to be closed if, for each x ∈ X, the function y → Fx (y) is upper semicontinuous, i.e., for any given net {yλ } ∈ K satisfying yλ → y0 ∈ X, we have lim sup Fx (yλ ) ≤ Fx (y0 ). λ
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Let F : K −→ F(X) be a closed fuzzy mapping satisfying the following condition: If there exists a mapping α : K −→ [0, 1] such that for each x ∈ K, (Fx )α(x) = {y ∈ K : Fx (y) ≥ α(x)} is a nonempty bounded subset of K. The concept of KKM-mapping was introduced by Knaster et al. [24] which provides the foundation for many well known existence results, such as Ky Fan’s minimax inequality theorem, Ky Fan–Browder’s fixed point theorem and Hartmann–Stampacchia variational inequality theorem, etc., see e.g. [2,4,11]. Definition 2.2. Let K be a non-empty subset of topological vector space X. A multi-valued mapping T : K → 2X is called a KKK-mapping if CoA ⊆ T (x), ∀A ∈ P(K), x∈A
where Co denotes the convex hull and P(K) is the family of all non-empty finite subsets of K. The following definition can be found in [14]. Definition 2.3. Let X be a non-empty set and Y a topological space. The multi-valued mapping G : X → 2Y is called / clG(x ), where clG(x ) denotes the topological transfer closed-valued if y ∈ / G(x), there exists x ∈ X such that y ∈ Y closure of G(x ). It is clear that, G : X → 2 is transfer closed-valued if and only if G(x) = clG(x). x∈K
x∈K
If A ⊆ X and B ⊆ Y , then G : A → 2B is called transfer closed-valued if the multi-valued mapping x → G(x) ∩ B is transfer closed-valued. In the case where X = Y and A = B, G is called transfer closed-valued on A. The following lemmas are needed in the next section. ˆ : K → 2K are the multi-valued Lemma 2.1. (See [14].) Let K be a non-empty convex subset of X. Suppose that G, G mappings such that: (i) (ii) (iii) (iv)
ˆ G(x) ⊆ G(x), for all x ∈ K; ˆ is a KKM-mapping; G for each A ∈ P(K), G is transfer closed-valued on CoA; for each A ∈ P(K), clK G(x) ∩ CoA = G(x) ∩ CoA; x∈CoA
x∈CoA
(v) there is a non-empty compact convex set B ⊆ K such that clK ( x∈B G(x)) is compact. Then,
G(x) = ∅.
x∈K
Lemma 2.2. (See [26].) Let X and Y be two topological spaces and let T : X → 2Y an upper semi-continuous multi-valued mapping with compact values. Suppose {xα } is a net in X such that xα → x0 . If yα ∈ T (xα ) for each α, then there is y0 ∈ T (x0 ) and a subset {yβ } of {yα } such that yβ → y0 . Definition 2.4. (i) A mapping F : K → Y is said to be positive homogeneous if F (αx) = αF (x), for all x ∈ K and α ≥ 0. (ii) A mapping F : K × K → Y is said to be positive homogeneous of order 1 if F (αx, αx) = αF (x, x), for all x ∈ K and α ≥ 0.
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The following definition is useful to solve many variational inequalities and their equivalent problems, which involves a weaker convexity condition. Definition 2.5. (See [33,36].) Let C : X → 2Y be a multi-valued mapping. A mapping h : X × X → Y is said to be 0-C(x)-diagonally convex with respect to the second argument if, for any finite subset {y1, y2 , . . . , yn } in X and any x ∈ X with x = ni=1 αi yi , αi ≥ 0, ni=1 αi = 1, we have n
αi h(x, yi ) ∈ C(x).
i=1
A class of physical and economic situations are most naturally modeled by saying that certain pairs of inequality constraints must be complementary, in the sense that at least one must hold with equality. Complementarity problems can be seen as extensions of square systems of nonlinear equations that incorporate a mixture of equations and inequalities. Due to important applications of complementarity problems and fuzzy set theory, we introduce the following problem. Let T : K → F(L(X, Y )) be a closed fuzzy mapping and α : K → [0, 1] a mapping. Let T˜ : K → 2L(X,Y ) be the multi-valued mapping induced by the fuzzy mapping T such that T˜ (x) = (Tx )α(x) ,
∀x ∈ K.
We introduce the following problem: Find x ∈ K, t ∈ (Tx )α(x) such that t, f (x, x) + H (x) = 0 and t, f (y, y) + H (y) ∈ C(x),
∀y ∈ K,
(2.1)
where f : K × K → K and H : K → Y are the mappings. We call (2.1) as generalized vector complementarity problem with fuzzy mappings. Special cases: (i) If T˜ : K → 2L(X,Y ) is the classical multi-valued mapping, then we can define the fuzzy mapping T : K → F(L(X, Y )) by x → X (T (x)), where X (T (x)) is the characteristic function of T (x). If α(x) = 1, T is singlevalued and f (x, x) = x, then the problem (2.1) reduces to the following vector complementarity problem: Find x ∈ K such that T (x), x + H (x) = 0 and T (x), y + H (y) ∈ C(x), ∀y ∈ K. (2.2) (ii) If Y = R, C = R+ , then the problem (2.2) reduces to the problem of finding x ∈ K such that T (x), x + H (x) = 0 and T (x), y + H (y) ≥ 0, ∀y ∈ K,
(2.3)
which was introduced by Yin et al. [34]. (iii) If H = 0, then the problem (2.2) reduces to the following problem: Find x ∈ K such that T (x), x = 0 and T (x), y ∈ C(x), ∀y ∈ K.
(2.4)
The problem (2.4) was introduced and studied by Chen and Yang [8] for a constant cone i.e., C(x) = C. Clearly, generalized vector complementarity problem with fuzzy mappings includes many complementarity problems studied in the recent past. In support of our problem (2.1), we provide the following example. Example 2.1. Let X = Y = K = C = [0, 1], and we define the closed fuzzy mapping T : K −→ F(L(X, Y )), for t ∈ [0, 1] as
0, if x ∈ [0, 12 ); Tx (t) = x+t 1 2 , if x ∈ [ 2 , 1],
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and the mapping α : K −→ [0, 1] as
0, if x ∈ [0, 12 ); α(x) = x 1 2 , if x ∈ [ 2 , 1]. Clearly, Tx (t) ≥ α(x), for all x ∈ K, i.e., t ∈ (Tx )α(x) . Now, we define the mappings f : K × K −→ Y and H : K −→ Y as follows:
0, if x ∈ [0, 12 ); f (x, y) = x − y and H (x) = 1−x 1 x , if x ∈ [ 2 , 1]. Then, any point of the interval [0, 12 ) satisfies the problem t, f (x, x) + H (x) = 0, whereas for x ∈ [ 12 , 1], we have t, f (y, y) + H (y) ∈ C(x), for all y ∈ K. Thus, generalized vector complementarity problem with fuzzy mappings (2.1) is satisfied. We now introduce the following generalized vector variational inequality problem with fuzzy mappings: Find x ∈ K, t ∈ (Tx )α(x) such that t, f (y, x) + H (y) − H (x) ∈ C(x), ∀y ∈ K.
(2.5)
For the mapping f , the following condition is required to prove the main results of this paper, which states that f does not vanishes at diagonal. Condition. (): f (y, x) = f (y, y) − f (x, x), ∀x, y ∈ K. 3. Existence results We begin this section by showing the equivalence of generalized vector complementarity problem with fuzzy mappings (2.1) with generalized vector variational inequality problem with fuzzy mappings (2.5) by using condition (). Theorem 3.1. If H is positive homogeneous and f is positive homogeneous of order 1 and condition () is satisfied, then problems (2.1) and (2.5) are equivalent. Proof. (I) Let x ∈ K, t ∈ (Tx )α(x) be a solution of (2.1), i.e., t, f (x, x) + H (x) = 0 and t, f (y, y) + H (y) ∈ C(x),
∀y ∈ K.
Using condition (), it follows that t, f (y, x) + H (y) − H (x) = t, f (y, y) − f (x, x) + H (y) − H (x) = t, f (y, y) + H (y) − t, f (x, x) + H (x) = t, f (y, y) + H (y) ∈ C(x), i.e., t, f (y, x) + H (y) − H (x) ∈ C(x). Thus (2.1) implies (2.5). (II) Let x ∈ K, t ∈ (Tx )α(x) be a solution of (2.5) i.e., t, f (y, x) + H (y) − H (x) ∈ C(x). Putting y = 12 x and y = 2x in (2.5), respectively and using condition (), we have t, f (x, x) + H (x) ∈ −C(x), t, f (x, x) + H (x) ∈ C(x). Combining (3.1) and (3.2), we have t, f (x, x) + H (x) ∈ C(x) ∩ −C(x) .
(3.1) (3.2)
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Since C is a pointed cone, we have t, f (x, x) + H (x) = 0. Also
t, f (y, y) + H (y) = t, f (y, y) + H (y) − t, f (x, x) + H (x) = t, f (y, y) − f (x, x) + H (y) − H (x) = t, f (y, x) + H (y) − H (x) ∈ C(x).
That is, we have t, f (x, x) + H (x) = 0 and t, f (y, y) + H (y) ∈ C(x), Thus, (2.5) implies (2.1).
(3.3) (3.4) (3.5)
∀y ∈ K.
2
It follows from Theorem 3.1 that generalized vector complementarity problem with fuzzy mappings (2.1) and generalized vector variational inequality problem with fuzzy mappings (2.5) are equivalent. We prove the following existence result for generalized vector variational inequality problem with fuzzy mappings (2.5). Theorem 3.2. Assume that (a) for all A ∈ P(K), the multi-valued mapping on GA : CoA → 2K defined by GA (y) = x ∈ K : t, f (y, x) + H (y) − H (x) ∈ C(x) , for all y ∈ K and t ∈ (Tx )α(x) , is transfer closed-valued mapping; (b) (i) there exists a mapping h : K × K → Y such that h is 0-C(x)-diagonally convex in the second argument; (ii) t, f (y, x) + H (y) − H (x) − h(x, y) ∈ C(x), for all x, y ∈ K and t ∈ (Tx )α(x) ; (c) let the mapping T˜ : K → 2L(X,Y ) is upper semi-continuous, compact valued and f, H are hemicontinuous; (d) there exist a nonempty compact subset B and a non-empty compact convex subset D of K such that for each x ∈ K \ B, there exists y ∈ D such that t, f (y, x) + H (y) − H (x) ∈ / C(x), for all x, y ∈ K and t ∈ (Tx )α(x) . Then, the generalized vector variational inequality problem with fuzzy mappings (2.5) is solvable. Moreover, the solution set is compact. ˆ : K → 2K as Proof. We define multi-valued mappings G, G G(y) = x ∈ K : t, f (y, x) + H (y) − H (x) ∈ C(x) , ˆ G(y) = x ∈ K : h(x, y) ∈ C(x) , ∀y ∈ K.
∀y ∈ K, t ∈ (Tx )α(x) ;
ˆ satisfy all conditions of Lemma 2.1. From assumption (b)(ii), we have G(y) ˆ We show that G, G ⊆ G(y), for all y ∈ K. ˆ ˆ Next, we show that G is a KKM-mapping. Suppose to contrary that G is not a KKM-mapping. Then, there exists a finite subset A = {y1 , y2 , · · · , yn } of K and λi ≥ 0, (i = 1, 2, · · · , n) with ni=1 λi = 1 and z = ni=1 λi yi such that z∈ /
n
ˆ i ). G(y
i=1
It follows that h(z, yi ) ∈ / C(z),
i = 1, 2, · · · , n,
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and n
h(z, yi ) ∈ / C(z),
i=1
ˆ is a KKM-mapping. which contradicts the 0-C(x)-diagonally convexity of h in the second argument. Thus, G From assumption (a), it follows that G is a transfer closed-valued on CoA i.e., G(x) ∩ CoA = cl G(x) ∩ CoA . x∈CoA
x∈CoA
In order to show that condition (iv) of Lemma 2.1 is satisfied, let A ∈ P(K), x, y ∈ CoA. Since T˜ is upper semi-continuous and compact valued, let {xα } be a net in K such that xα → x0 and t ∈ T˜ (xα ), for all α i.e., for λ ∈ [0, 1], we have t , f λx + (1 − λ)y, xα + H λx + (1 − λ)y − H (xα ) ∈ C(xα ) By Lemma 2.2, there exists t ∈ T˜ (x0 ) and a subset {t } of {t } such that t → t and a subset {xα } of {xα } such that xα → x0 . By the hemicontinuity of f and H , it follows that t , f λx + (1 − λ)y, xα + H λx + (1 − λ)y − H xα −→ t, f (y, x0 ) + H (y) − H (x0 ) ∈ C(x0 ). That is,
G(x) ∩ CoA = clK
x∈CoA
G(x) ∩ CoA;
A ∈ P(K).
x∈CoA
ˆ satisfy all the conFrom (d), we deduce that clK ( x∈A G(x)) ⊆ B i.e., clK ( x∈B G(x)) is compact. Thus G, G ditions of Lemma 2.1 and consequently, we have G(x) = ∅, x∈K
which shows that generalized vector variational inequality problem with fuzzy mappings (2.5) is solvable. Also, it is clear from the proof above that the solution set of generalized vector variational inequality problem with fuzzy mappings is compact. This completes the proof. 2 Remark 3.1. Most existing complementarity problems involve a single-valued mapping and are solved by using some types of monotonicity, convexity and classical KKM-Theorem. We obtain the solution of generalized vector complementarity problem for fuzzy mappings (2.1), which involves a multi-valued mapping induced by a fuzzy mapping and a bi-mapping, by applying a generalized version of KKM-Theorem [14], weaker convexity assumption i.e., 0-C(x)-diagonally convexity. Moreover, we do not use monotonicity and continuity assumptions in our results. Thus, our results can be viewed as refinement and extension of several existing results, see e.g. [12,13,27]. Combining Theorem 3.1 and Theorem 3.2, we have the following theorem. Theorem 3.3. If all the assumptions of Theorem 3.1 and Theorem 3.2 are satisfied, then generalized vector complementarity problem with fuzzy mappings (2.1) is solvable. Moreover, the solution set is compact. 4. Conclusion and discussion The concept of complementarity theory has received great attention during last fifty years and it is well known that both the linear and nonlinear programs can be characterized by a class of complementarity problems. Fuzzy sets and fuzzy logic are powerful mathematical tools for modeling and controlling uncertain systems in industry, humanity and nature: They are expeditious for approximating reasoning in decision making in the absence of complete and precise information.
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Due to interesting applications of both the above discussed concepts, in this paper, we introduce and solve a generalized vector complementarity problem with fuzzy mapping and prove some existence results which may be seen as generalization of many corresponding existing results. We refer that problem (2.1) can be further generalized and studied in higher dimensional spaces by using the concepts of functional analysis, and one might develop new mathematical tools to solve the problems occurring in mathematical sciences. Acknowledgements The authors are grateful to the referees and editors for their valuable suggestions which improve this paper a lot. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]
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