Perturbed Mixed Variational-like Inequalities and Auxiliary Principle Pertaining to a Fuzzy Environment

: Convex and non-convex fuzzy mappings are well known to be important in the research of fuzzy optimization. Symmetry and the idea of convexity are closely related. Therefore, the concept of symmetry and convexity is important in the discussion of inequalities because of how its deﬁnition behaves. This study aims to consider new class of generalized fuzzy variational-like inequality for fuzzy mapping which is known as perturbed fuzzy mixed variational-like inequality. We also introduce strongly fuzzy mixed variational inequality, as a particular case of perturbed fuzzy mixed variational-like inequality which is also a new one. Furthermore, by using the generalized auxiliary principle technique and some new analytic techniques, some existence results and efﬁcient numerical techniques of perturbed fuzzy mixed variational-like inequality are established. As exceptional cases, some known and new results are obtained. Results obtained in this paper can be viewed as reﬁnement and improvement of previously known results.


Introduction
In [1], Zadeh introduced fuzzy set theory, which has been extensively used to simulate the uncertainty found in practical applications.The expansion of fuzzy set theory and its applications have drawn the interest of numerous academics, see [2].With the passage of time, these have been expanded and generalized to analyze a vast class of problems like in optimization and control system, mechanics, economics and transportation, physics and so forth.The idea of variational inequality was initiated by Hartman and Stampacchia [3] in 1964.A useful generalization of variational inequality is generalized mixed variational-like inequality (in short, GMVLI).The "GMVLI" have capability and noteworthy applications in different fields like structural analysis [4], optimization theory [5][6][7][8][9][10][11], and economics [12,13].Motivated and inspired by going on research work, many authors discussed fuzzy variational-like inequalities and its generalizations, and its applications in different fields.For more informations related to fuzzy theory and inequalities, see [14][15][16][17].In fuzzy optimization, Noor [18,19] studied the characterization of minimum of convex fuzzy mapping through fuzzy variational inequality and fuzzy mixed variational inequality, and obtained some iterative algorithms.It is worthy to mention one of the most considered generalization of convex fuzzy mapping is preinvex fuzzy mapping.The idea of fuzzy preinvex mapping on the invex set was introduced and studied by Noor [20], Moreover, any local minimum of a preinvex fuzzy mapping is a global minimum on invex set and necessary and sufficient condition for fuzzy mapping is to be preinvex if its epigraph is an invex set.Furthermore, it has been verified that a fuzzy optimality conditions of differentiable fuzzy preinvex mappings can be distinguished by variational-like inequalities.Then many another's extended this concept and discussed the applications of fuzzy variational-like inequalities.Chang [21], Chang and Zhu [22], and Chang et al. [23] and Kumam and Petrot [24], studied the idea of "GMVLI" and complementarity problems for fuzzy mappings in different context.By using the Berge maximum theorem in finite and infinite dimensional space, some particular cases of "GMVLI" are studied by Tian [25], Parida and Sen [26], and Yao [27].It is convenient to mention that their methods are not productive.Therefore, the enlargement of a well-organized and applicable technique for solving variational-like inequality is one of the impotent and engaging problem.Although, there exist substantial numbers of numerical methods as well as projection method and its alternative forms, Newton's methods, linear approximation and descent.But the projection type method can be used to recommend iterative method for variational-like inequalities.To overcome this drawback of above mentioned method, Glowinski, Lions and Tremolieres [28], was suggested auxiliary principle technique.Then many authors expended the auxiliary principle technique to study the existence and uniqueness of a solution of "GMVLI" for set valued mappings with compact and non-compact values and single valued mappings, see [29][30][31][32][33][34][35][36][37][38].By using auxiliary principle technique, "GMVLI" can easily be handled as well as its particular cases.
Motivated and inspired by going on research work in this interesting and fascinating field, the objective of this article to introduce new class of "GMVLI", which is known as perturbed fuzzy mixed variational-like inequality (PFMVLI) and to study some existence theorems.We also prove the existence of auxiliary problem for "PFMVLI".By utilizing the theorems, we construct an iterative algorithms auxiliary problem for "PFMVLI".The results given in this paper are up to date and they generalize, refine and consolidate a number of recent results in [3,21,28,31].

Preliminaries
Let H be a real Hilbert space and ∅ = K ⊂ H be a convex set.we denote the collection CB(H) of all nonempty bounded and closed subsets of H and D(., .) is the Hausdorff metric on CB(H) defined by A fuzzy set on H is a mapping ψ : H → [0, 1] , for each fuzzy set and α ∈ (0, 1], then α-level sets of ψ is denoted and defined as follows In what follows, F (H) = {A : H → I = [0, 1]} denote the family of all fuzzy sets on H.A mapping T from H to F (H) is called a fuzzy mapping.If T : H → F (H) is a fuzzy mapping, then the set T (u), for u ∈ H is a fuzzy set in F (H) (in the sequel we denote T (u) by T u ) and T u (ϑ) , ϑ ∈ H is the degree of membership of ϑ in T u .Definition 1. (i) If for each u ∈ H, the function ϑ → T u (ϑ) is upper semicontinuous, then fuzzy mapping T is called closed.If {ϑ α } ⊂ H is a net and satisfying ϑ α → ϑ 0 ∈ H , then T u have following property, we have limsup α T u (ϑ α ) ≤ T u (ϑ 0 ).
(ii) A closed fuzzy mapping T : H → F (H) is said to satisfy the condition ( * ), if there exists a function α : H → [0, 1] such that for each u ∈ H the set is a bounded subset of H.
This implies that ϑ 0 ∈ [T u ] α(u) and so where J : H → R ∪ {+∞} is a nondifferentiable function and T : H → F (H) is a fuzzy map- ping satisfying condition ( * ) with function α : H → [0, 1] , such that The inequality (3) is known as perturbed fuzzy mixed variational-like inequality (in short, PFMVLI).It can easily be seen that; this inequality is more general than fuzzy mixed variationallike inequality (in short, FMVLI) and include classical mixed variational-like inequality (in short, MVLI) and associated fuzzy optimizations problems are particular cases.For applications, see [3,17,21,22,26] and the references therein.Now we study some certain cases of Problem (1).
(1) Let T : H → F (H) be an ordinary multivalued mapping and J , ξ be the mapping in problem (1).Now we define a fuzzy mapping T (.) : H → F (H) as follows where X T u is the characteristic functions of the T (u).From (4), it can straightforwardly be noticed that the T is a closed fuzzy mapping fulfilling condition ( * ) with constant function This kind of problem is called the set valued "PMVLI".This inequality is also new one.
If M = I (identity mapping), p ∈ H and T : H → H is single valued, then inequality ( 5) is parallel to finding a u ∈ H such that This is known as "PMVLI" and studied by Noor et al. [31].1) is called strongly fuzzy mixed variational inequality and is parallel to finding This class of "FMVI" is also new one.This inequality is more general as well as classical "FMVI" and related fuzzy optimization problems as particular cases.In case of ordinary set-valued mapping as flourished by Noor [9].
• When M = I and T : H → F (H) is a fuzzy mapping, then inequality ( 9) is parallel to finding This is known as "FMVI", see [18].
• If J (.) is an indicator mapping of a closed invex set K ξ in H, that is which is known as "FVLI".
which is known as "FVI".
From above discussion, it can easily be seen that inequalities ( 7)-( 13) are particular cases of "PFMVLI" (3).In fact, "PFMVLI" is more generalize and unifying one, which is main motivation of our work.For a proper and suitable choice of T , ξ and J , we can choose a number of known and unknown "FVLI" and complementary problems.
Next, we will use mathematical terminologies S-monotone and L-continuous for strongly monotone and Lipschitz continuous, respectively.Definition 2. If fuzzy mapping T : H → F (H) is closed and fulfil the condition ( * ) with function α : H → [0, 1], then nonlinear mapping M : H → H is said to be: where D(., .) is the Housdroff metric on F (H).
In particular, from (a) and (b), we have which implies that β γλ.
From (d) and (e), we can observe that µ δ.
Lemma 1. [8] Let C be an arbitrary nonempty in a topological vector space G, and let T : C → 2 H is a KKM mapping.If T(ϑ) is closed for all ϑ ∈ C and is compact for at least one ϑ ∈ C , then Theorem 1. [21] Let G be a locally convex Hausdorff topologically vector space and g : K → R ∪ {+∞} be a properly convex functional.Then g is lower semicontinuous on G if and only if, g is weakly lower semicontinuous on G.
Lemma 2. [11] Let (X, d) be a complete metric space and C 1 , C 2 ∈ CB(X ) and r ≥ 1 be any real number.Then, for every c 1 ∈ C 1 then there exist c 2 ∈ C 2 such that d(c 1 , c 2 ) ≤ rD(C 1 , C 2 ).
In next sections, we will use the above results.

Auxiliary Principle and Algorithm
In this portion, we explore the auxiliary principle technique which is mainly due to Glowinski, Lions and Tremolieres [28], as flourished and improved by Noor [32], and Huang and Deng [17], to examine "PFMVLI".We give an existence result of the solution of auxiliary problem for the "PFMVLI" (3).Furthermore, based on this existence result, we suggest an iterative algorithm for the "PFMVLI" (3).
For a given u ∈ K, p ∈ [T u ] α(u) satisfying the problem (1), we consider the problem of finding z ∈ H, such that ∀ ϑ ∈ H, where ρ > 0 is a constant.The inequality ( 14) is also called auxiliary "PFMVLI".
Proof.Let given u ∈ K, p ∈ [T u ] α(u) , we consider the mapping T : H → 2 H define by for all ϑ ∈ H.
Replacing ϑ by z 1 in ( 19) and ϑ by z in ( 20), we have and (22)   using the Assumption 1, that ξ(u 1 , u 2 ) = −ξ(u 2 , u 1 ) and then adding the ( 21) and ( 22), we have which implies that z 1 = z is the uniqueness of the solution of auxiliary problem ( 14).This complete the proof of Theorem 2.
Algorithm 1.At n = 0, start with initial value u 0 ∈ H,p 0 ∈ [T u 0 ] α(u 0 ) , from Theorem 2, the auxiliary problem ( 14) has a unique solution u 1 ∈ H , such that , then by Nadler's Lemma 2, there exist p 1 ∈ [T u 1 ] α(u 1 ) , such that ,again from Theorem 2, the auxiliary problem ( 14) has a unique solution u 2 ∈ H, such that . and Similarly, by the S-monotonicity and L-continuity of bifunction ξ, we have Combining ( 29), ( 30) and ( 31) and using the γ-L-continuity of M and T -L-continuity of T and ξ, respectively, we have which implies that It follows that, where Clearly where From (24), it follows that ϕ < 1.Hence, it follows from ( 32) and ( 33) that {u n } is Cauchy sequence in K. Hence it converges to some point.Since K is closed convex set in K, then there exist a u in K such that u n → u, which satisfy the "PFMVLI" (3).On the other hand, from algorithm 1, we have This implies that {p n } is Cauchy sequence in H, since {u n } is converging sequence.Then, we consider that p n → p, when n → ∞ .Since p n ∈ [T u n ] α(u n ) , so we have When n → ∞ , we have . Now finally we show that We again study (23), as follows Now from Lipchitz continuity and convexity of J and by Assumption 1, we get This completes the proof.

Special Cases:
Theorem 4. Let the operator T : H → CB(H) be T -S-monotone with constant β > 0 and T -L-continuous with constant λ > 0 respectively.Let nonlinear continuous mapping M : H → H be β -S-monotone and γ -L-continuous with constants β > 0 and γ > 0, respectively.Let J : K → R ∪ {+∞} be a properly convex functional and ∂ -L-continuous with constant ∂ and J (.) is nondifferentiable.If the bi-function ξ(., .) is S-monotone with constant µ > 0 and Lcontinuous with constant δ > 0 , respectively.If Assumption 1 hold, then for constant ρ > 0 , such that Then there exist u ∈ K,p ∈ [T u ] α(u) satisfying the set-valued "PMVLI" (5) and the sequences {u n } and {p n } , generated by ( 23) converge strongly to u and p, respectively.Proof .By using the set valued mapping F : H → CB(H), we define the fuzzy mapping T : H → F (H) , as follows T u = X T (u), where X T (u) is a characteristic function of the sets T (u).It is easy to see that T is a closed fuzzy mapping satisfying condition ( * ) with constant functions α(u) = 1, for all u ∈ H.
Therefore, the conclusion of Theorem 4 can be obtained from Theorem 2 and Theorem 3 immediately.
Remark 2. At the end, we would like to mention that many earlier defined familiar methods as well as decent, projection techniques and its mixed forms, relaxation, and Newton's methods that can be obtained form auxiliary "PFMVLI".Similarly, for suitable choice of the fuzzy mapping T : H → F (H) and bi-function ξ(., .): H × H → H, many old of (fuzzy) "VI" and (fuzzy) variational inclusion and corresponding (fuzzy) optimization problems from the "PFMVLI" (3) can be obtain.In future, we will try to find "PFMVLI" for higher order strongly-preinvex fuzzy mappings.

Conclusions
In this paper, we have proposed the idea of "PFMVLI".As a particular case of "PFMVLI", strongly fuzzy mixed variational inequality are also introduced.With the help of generalized auxiliary principle technique and some new analytic techniques, some existence theorems of auxiliary "PFMVLI" are studied for "PFMVLI" and some iterative methods are obtained for the solution of "PFMVLI".Then we have obtained some known and new results.There is much room for further study to explore this concept because for suitable choice of the fuzzy mapping T (.), ξ(., .),J (.) and K many new classes of (fuzzy) "VI" and (fuzzy) variational inclusion and corresponding (fuzzy) optimization problems can be obtain from the "PFMVLI" (3).