A class of semilocal E-preinvex maps in Banach spaces with applications to nondifferentiable vector optimization

In this paper, a new class of semilocal E-preinvex and related maps in Banach spaces is introduced for a nondifferentiable vector optimization problem with restrictions of inequalities and some of its basic properties are studied. Furthermore, as its applications, some optimality conditions and duality results are established for a nondifferentiable vector optimization under the aforesaid maps assumptions.


Introduction
In recent years, generalizations of convexity in connection with optimality conditions and duality theory have been of much interest and many contributions have been made to this development.See, e.g., [1][2][3][4][5][6] and the references therein.
Preda and Stancu-Minasian [7] derived optimality conditions for weak vector minima under semilocal preinvexity and η-semidifferentiability conditions and extended the Wolfe and Mond-Weir duals.Later, Preda [8] further established optimality conditions and duality results for a nonlinear fractional multiple objective programming problem with semilocal preinvex functions involving η-semidifferentiability.Additionally, Batista et al. [9] introduced the notions of preinvex maps in Banach spaces and obtained optimality conditions for vector problem.Subsequently, Yu and Liu [10] studied optimality and duality for the same vector problem involving the generalized type-I maps in Banach spaces.
On the other hand, Chen [11] proposed a class of semi-E-convex functions and discussed its basic properties.On the basis of this notion, Hu et al. [12] also brought forward the concept of semilocal E-convexity, and studied its some characterizations, and established some optimality conditions and duality results for semilocal Econvex programming.Recently, Fulga and Preda [13] extended the E-convexity to E-preinvexity and local E-preinvexity, and studied some of their properties and an application.More recently, Luo and Jian [14] presented semi E-preinvex maps in Banach spaces and discussed their properties.
Motivated by research works of [10,12,14] and references therein, in present paper, I introduce the concepts of semilocal E-(pre)invexity, E-ηsemidifferentiablity and E-type-I maps in Banach spaces and study some of their important properties.
Additionally, I establish some optimality conditions for a nondifferentialbe vector optimization problem with restrictions of inequalities under semilocal E-preinvexity, semilocal E-invexity and E-type-I assumptions, respectively.Furthermore, I formulate a dual type for this optimization problem and obtain weak and converse duality results using E-type-I maps.This work partially extends earlier works of [10,14] to a wider class of maps.

Preliminaries and Definitions
Throughout this paper, let X, Y and Z j , j ∈ M = {1, 2, . . ., m} be real Banach spaces with topological duals X * , Y * and Z * j , respectively, E : X → X and η : X × X → X be two fixed mappings.
Consider the following optimization problem: where f : X → Y and g j : X → Z j are maps, K and D j are subsets of X and Z j .Denote the feasible set of (P ) by We assume that the spaces Y and Z j are ordered by cones C ⊂ Y , D j ⊂ Z j and that these cones are pointed, closed, convex, and with nonempty interior.The dual cone of C is denoted by Analogously, D j induces a partial order on Z j .Recall some definitions and results that will be used in the sequel.
We below introduce the concepts of local starshaped E-convex set, local E-invex set, semilocal E-convex map and local E-preinvex map in Banach spaces.Especially, if X = R n , Y = R, these concepts were given by earlier research (see [12,13]).Definition 4. A set K ⊂ X is said to be local starshaped E-convex, if there is a map E such that corresponding to each pair of points x, y ∈ K, there is a maximal positive number a(x, y) ≤ 1 satisfying for each pair of x, y ∈ K(with a maximal positive number a(x, y) ≤ 1 satisfying (2)), there exists a positive number b(x, y) ≤ a(x, y) satisfying

Semilocal E-preinvex and Related Maps
In this section, we introduce the concepts of semilocal E-preinvex and related maps in Banach spaces and study some of their basic properties.
Definition 8.A map f : X → Y is said to be semilocal E-preinvex on k ⊂ X with respect to η if for any x, y ∈ K(with a maximal positive number a(x, y) ≤ 1 satisfying (3)), there exists 0 < b(x, y) ≤ a(x, y) such that K is a local E-invex set and If the inequality sign above is strict for any x, y ∈ K and x = y, then f is called as a strict semilocal E-preinvex map.
Remark 2. Every semilocal E-convex map is a semilocal E-preinvex map, where η(x, y) = x − y, ∀x, y ∈ X.Every semi E-preinvex map with respect to η is a semilocal E-preinvex map, where a(x, y) = b(x, y) = 1, ∀x, y ∈ X.But their converses are not necessarily true.
See the following example.
Example 1.Let the map E : R → R be defined as and the map η : R × R → R be defined as Obviously, R is a local starshaped E-convex set and a local E-invex set with respect to η.Let f : R → R be defined as We can prove that f is semilocal E-preinvex on R with respect to η.However, when x 0 = 2, y 0 = 3, and for any b ∈ (0, 1], there exists a sufficiently small λ 0 ∈ (0, b] satisfying Similarly, taking x 1 = 1, y 1 = 4, we have Definition 10.A map f defined on a local Einvex set K ⊂ X is said to be pseudo-semilocal E-preinvex (with respect to η)if for all x, y ∈ K(with a maximal positive number a(x, y) ≤ 1 satisfying (3)) satisfying f (x) < C f (y), there are a positive number b(x, y) ≤ a(x, y) and a positive number c(x, y) such that Remark 3. Every semilocal E-preinvex map on a local E-invex set K with respect to η is both a quasi-semilocal E-preinvex map and a pseudosemilocal E-preinvex map.
Proof.Suppose that f is a semilocal E-preinvex map on set K with respect to η, then for each pair of points x, y ∈ K (with a maximal positive number a(x, y) ≤ 1 satisfying (3)), there exists a positive number b(x, y) ≤ a(x, y) satisfying Conversely, assume that f is a local E-preinvex map on a local E-invex set K, then for any x, y ∈ K, there exist a(x, y) ∈ (0, 1] satisfying (3) and b(x, y) ∈ (0, a(x, y)] such that The proof is completed.
and f : R → R be defined by said to be a local Einvex set with respect to η corresponding to X if there are two maps η, E and a maximal positive number a((x, α 1 ), (y, α 2 )) ≤ 1, for each (x, α 1 ), (y, α 2 ) ∈ G such that In addition, in view of f being a semilocal Epreinvex map on K with respect to η, there is a positive number b(x, y) ≤ a(x, y) such that Conversely, if G f is a local E-invex set with respect to η corresponding to X, then for any points (x, f (x)), (y, Thus, K is a local E-invex set and f is a semilocal E-preinvex map on K. Proof.For any α ∈ Y and x, y ∈ S α , then x, y ∈ K and f (x) ≦ C α, f (y) ≦ C α. Since K is a local E-invex set, there is a maximal positive number a(x, y) ≤ 1 such that In addition, due to the semilocal E-preinvexity of f , there is a positive number b(x, y) ≤ a(x, y) Then f is a semilocal E-preinvex map with respect to η if and only if for each pair of points x, y ∈ K(with a maximal positive number a(x, y) ≤ 1 satisfying (3)), there exists a positive number b(x, y) ≤ a(x, y) Proof.Let x, y ∈ K and α, β ∈ Y such that f (x) < C α, f (y) < C β. Due to the local Einvexity of K, there is a maximal positive number a(x, y) ≤ 1 such that In addition, owing to the semilocal E-preinvexity of f , there is a positive number b(x, y) ≤ a(x, y) such that f (E(y) + λη(E(x), E(y))) and f (y) < C β +ǫ hold for any ǫ > C 0. According to the hypothesis, for x, y ∈ K(with a positive number a(x, y) ≤ 1satisfying (3)), there exists a positive number b(x, y) ≤ a(x, y) such that Therefore, G f is a local E-invex set corresponding to X. From Theorem 2, it follows that f is semilocal E-preinvex on K with respect to η .Theorem 5. Assume that a map f : X → Y is semilocal E-preinvex on a local E-invex set F ⊂ X with respect to η.If E(x) = x ∈ F is a local weakly efficient solution for problem (P ), then x is a global weakly efficient solution for (P ) on F .
Proof.By contradiction, suppose that E(x) = x ∈ F is not a global weakly efficient solution for (P ) on F , then there exists y ∈ F such that Since map f is semilocal E-preinvex on local Einvex set F , there exist positive number b(x, y) ≤ a(x, y) ≤ 1 such that E(x) + λη(E(y), E(x)) ∈ F for any λ ∈ (0, a(x, y)] and or equivalently, Since C is a pointed cone, from ( 9) and (10), we obtain η(E(y), x) = 0. We observe which contradicts the local weakly efficiency of x for problem (P ).Thus, x is a global weakly efficient solution for problem (P ) on F .Remark 5. a(x, y) = b(x, y) = 1, ∀x, y ∈ X, the results presented in this section reduce to the results given in [14].

Optimality Criteria
In this section, we establish some optimality conditions for the vector optimization problem (P ) involving semilocal E-preinvex, semilocal Einvex and E-type-I maps, respectively.
First, we give an optimality condition for (P ) involving semilocal E-preinvex maps.Theorem 6. Assume that a map f : X → Y is E-preinvex on local E-invex set F ⊂ X with respect to η and x is a weakly efficient solution for the following optimization problem: Then, E(x) is a weakly efficient solution for problem (P ).
Proof.Since x is a weakly efficient solution for problem (P E ), then there exists no x ∈ F such that Suppose to the contrary that E(x) is not a weakly efficient solution for (P ), then there exists a point x ∈ F such that From necessity of Theorem 1, it follows that Thus, we have which is in contradiction with the weakly efficiency of x for problem (P E ).Hence, the theorem is proved.
Next, we introduce some concepts that will be used in the sequel.Definition 12. Let f : K → Y be a map, where K ⊂ X is a local E-invex set with respect to η.We say that f is E-η-semidifferentiable at ∈ K if f ′ (E(x); η(E(x), E(x))) exists for each x ∈ K, where (the right derivative at E(x) along the direction η(E(x), E(x))).
Remark 6.If E is an identity map, the E-ηsemidifferentiability is the η-semidifferentiability notion [16].If is an identity map and η(x, x) = x − x, the E-η-semidifferentiability is the semidnotion.If a function is directionally differentiable, then it is semidifferentiable , but the converse is not true.
x and Below, we give a necessary and sufficient optimality condition for problem (P ) involving semilocal E-invex maps.
Then u is a weakly efficient solution of map f on K if and only if u satisfies the inequality Proof.Since the map f is semilocal E-invex at u ∈ K on K with respect to η, If u ∈ K satisfies the inequality (11), then which means u ∈ K is the weakly efficient solution of map f on K. Conversely, assume that u is a weakly efficient solution of map f on K, then, Since K is a local E-invex set, for any v ∈ K, there exists a(u, v) ∈ (0, 1] such that That is, Dividing the above inequality by λ and taking λ → 0 + , we get which is the desirable result (11).Therefore, the proof is completed.
By [9, Lemma 2.3], Definition 13 is also equivalent to the next definition.
x and for any x ∈ K, µ * ∈ C * such that Throughout the remainder of this paper, we always assume that f : X → Y and g j : X → Z j , j ∈ M are E-η-semidifferentiable.Now, we extend the generalized type-I maps in [10] as follows.
Definition 15. (f, g) is said to be E-type-I at x ∈ K with respect to η, if E(x) = x and for each x ∈ K, there exist two maps E and η, such that for all µ * ∈ C * and v Definition 16. (f, g) is said to be quasi E-type-I at x ∈ K with respect to η, if E(x) = x and for each x ∈ K, there exist two maps E and η, such that for all µ * ∈ C * and v Definition 17. (f, g) is said to be pseudo E-type-I at x ∈ K with respect to η, if E(x) = x and for each x ∈ K, there exist two maps E and η, such that for all µ * ∈ C * and v Definition 18. (f, g) is said to be quasipseudo E-type-I at x ∈ K with respect to η, if E(x) = x and for each x ∈ K, there exist two maps E and η, such that for all µ * ∈ C * and v in the above relation, we have Then, we say that (f, g) is quasistrictlypseudo Etype-I at x ∈ K.
Definition 19.(f, g) is said to be pseudoquasi E-type-I at x ∈ K with respect to η, if E(x) = x and for each x ∈ K, there exist two maps E and η, such that for all µ * ∈ C * and v is an identity map and m = 1, the above definitions reduce to the definitions of generalized type-I maps in [10].
Now, we establish the sufficient optimality conditions for (P ) involving E-type-I maps.
Theorem 8. Assume that there exist x ∈ F and µ * ∈ C * \ {0 Y * } [or, µ * ∈ intC * ], v * j ∈ D * j , j ∈ M such that the following two relations are satisfied, Furthermore, if any one of the following conditions holds: (a) (f, g) is E-type-I at x ∈ F with respect to the same η; (b) (f, g) is pseudoquasi E-type-I at x ∈ F with respect to the same η; (c) (f, g) is quasistrictlypseudo E-type-I at x ∈ F with respect to the same η.Then x is a weakly efficient solution [or, an efficient solution] of (P ).
Proof.By contradiction, we assume that x is not a weakly efficient solution [or, an efficient solution] of (P ).Then there is a feasible solution x of problem (P ) such that If condition (a) holds, then from relation (12), it follows that According to relations ( 13) and (24), we obtain Adding ( 26) and ( 27), we have which is in contradiction with (23).
If condition (c) holds, then from relations (18) and ( 25), it follows that Combining the above inequality with (23), we get From (20), it leads to . Therefore, the theorem is proved.
Remark 9.If E is an identity map and m = 1, the results obtained in the above theorem become the results of Yu and Liu [10].

Duality
In this section, we provide weak and converse duality results utilizing E-type-I maps.Consider the following dual type for problem (P ): Denote the feasible set of problem (D) by G, i.e., Furthermore, if any one of the following conditions holds: (a) (f, g) is E-type-I at y ∈ F with respect to the same η; (b) (f, g) is pseudoquasi E-type-I at y ∈ F with respect to the same η; (c) (f, g) is quasistrictlypseudo E-type-I at y ∈ F with respect to the same η.
Proof.Assume to the contrary that there exist According to the first inequality in (28) and x ∈ F , we get Utilizing relations (29), (30) and condition (a), we obtain Summing the above two inequalities, we have Taking (31) into account, we obtain By condition (b) again, the above relation means that µ * , f (x) − f (y) ≥ 0, which contradicts (29).
Remark 10.If E is an identity map and m = 1, results presented in this section reduce to the results given by Yu and Liu [10] .

Conclusions
In this paper, we have introduced a new concept of semilocal E-preinvex maps in Banach spaces, which extend the semi E-preinvex maps presented by Luo and Jian (2011) [14].Simultaneously, we have derived some of its basic properties.Next, we have defined an E-ηsemidifferentiable map, which generalizes the ηsemidifferentiable function introduced by Preda and Stancu-Minasian (1997) [7].Based on this, we have proposed a new notion of nondifferentiable maps called semilocal E-invex maps and the concepts of E-type-I maps, which extend the generalized type-I maps brought forward by Yu and Liu (2007) [10].In the framework of the new concepts, we have established some optimality conditions for the nondifferentiable vector optimization problem with inequalities constraints using semilocal E-preinvex, semilocal Einvex and E-type-I maps, respectively.Moreover, we have proved weak and converse duality results under various types of E-type-I maps requirements.The results presented in this paper extend and improve many results of [10,14] and generalize results obtained in the literatures on this topic.