On Semi- B,G -Preinvex Functions

and Applied Analysis 3 Example 2.3. Let X be a subset in Rn defined as follows: X : { x1, x2 | 0 < x2 < x2 1, 0 < x1 < 2 } ∪ { 0, 0 }. 2.2 Consider the point u 0, 0 . Since the tangent line of the curve x2 x2 1 at point u is the line x2 0. Then, for any x ∈ X \ {u}, there exists 0 < λ0 < 1 such that u λη x, u / ∈ X, ∀λ ∈ 0, λ0 . 2.3 Therefore, there exists no vector-valued function η x, u / 0 such that u λη x, u ∈ X, ∀λ ∈ 0, 1 . 2.4 However, define η x, u;λ : x1, 1/2 λx2 for x x1, x2 , then u λη x, u, λ ∈ X, ∀λ ∈ 0, 1 . 2.5 Hence, X is semi-invex at uwith respect to η. Definition 2.4 see 33 . LetX be a nonempty semi-invex subset ofR. A real-valued function f : X → R is said to be semi-B-preinvex at u ∈ X with respect to η if there exist vector-valued function η : X ×X × 0, 1 → R and real functions b1, b2 : X ×X × 0, 1 → R such that for all x ∈ X f ( u λη x, u, λ ) ≤ b1 x, u;λ f x b2 x, u;λ f u , b1 x, u; 1 b2 x, u; 0 1, b1 x, u;λ b2 x, u;λ 1, λ ∈ 0, 1 , 2.6 where limλ→ 0λη x, u, λ 0. The real-valued function f is said to be semi-B-preinvex on X with respect to η if f is semi-B-preinvex at each u ∈ X with respect to η; f is said to be strictly semi-B-preinvex on X with respect to η if strict inequality 2.6 holds for all x, u ∈ X such that x / u; f is said to be explicitly semi-B-preinvex on X with respect to η if strict inequality 2.6 holds for all x, u ∈ X such that f x / f u . Remark 2.5. Note that semi-B-preinvexity is a special kind of φ1, φ2 convexity defined in 11, 12 . Furthermore, assume thatX is an invex subset. Then semi-B-preinvexity is B-preinvexity 14 ; explicitly semi-B-preinvexity is explicitly B-preinvexity 30 ; strictly semi-B-preinvexity is strictly B-preinvexity 34 . Moreover, if X be a convex set, then semi-B-preinvexity is Bvexity defined in 8, 9 . Definition 2.6. Let X be a nonempty semi-invex subset of R. A real-valued function f : X → R is said to be semiB,G -preinvex at u on X with respect to η if there exists a continuous real-valued functionG : If X → R such thatG is a strictly increasing function on its domain, 4 Abstract and Applied Analysis a vector-valued function η : X×X× 0, 1 → R, and real functions b1, b2 : X×X× 0, 1 → R such that for all x ∈ X f ( u λη x, u, λ ) ≤ G−1 ( b1 x, u;λ G ( f x ) b2 x, u;λ G ( f u )) , b1 x, u; 1 b2 x, u; 0 1, b1 x, u;λ b2 x, u;λ 1, λ ∈ 0, 1 . 2.7 If inequality 2.7 holds for any u ∈ X, then f is semiB,G -preinvex on X with respect to η; f is said to be strictly semiB,G -preinvex on X with respect to η if strict inequality 2.7 holds for all x, u ∈ X such that x / u; f is said to be explicitly semiB,G -preinvex on X with respect to η if strict inequality 2.7 holds for all x, u ∈ X such that f x / f u . Remark 2.7. Let X be an invex subset. Then semiB,G -preinvexity, strictly semiB,G -preinvexity, and explicitly semiB,G -preinvexity are called B,G -preinvexity, strictly B,G preinvexity, and explicitly B,G -preinvexity, respectively. Remark 2.8. EveryG-preinvex function with respect to η introduced in 19, 22 is semiB,G preinvex function with respect to η, where b1 x, u;λ λ, b2 x, u;λ 1 − λ, λ ∈ 0, 1 ; every semi-B-preinvex function with respect to η introduced in 14 is semiB,G -preinvex function with respect to η, where G a a, a ∈ R. The converse results are, in general, not true, see Example 2.10. Remark 2.9. Every semistrictly G-preinvex function with respect to η introduced in 24 is explicitly B,G -preinvex function with respect to η, where b1 x, u;λ λ, b2 x, u;λ 1 − λ, λ ∈ 0, 1 ; every explicitly semi-B-preinvex function with respect to η introduced in 30 is explicitly semiB,G -preinvex function with respect to η, where G a a, a ∈ R. The converse results are, in general, not true. See Example 2.10 too. Example 2.10. LetX be the subset defined in Example 2.3, x x1, x2 , u u1, u2 ∈ X. Define η x, u, λ ⎧ ⎪⎨ ⎪⎩ ( x1, 1 2 λx2 ) , u 0, 0 , x0 − u, u/ 0, 0 , 2.8 where x0 ∈ X is a point on the line between u and x, which is different from u, such that ⋃ u, ‖u − x0‖ ⊂ X. Define f x ln x1 x2 2 , x x1, x2 ∈ X, b1 x, u;λ λ, b2 x, u;λ 1 − λ, λ ∈ 0, 1 , G a e, a ∈ R. 2.9 Then, it is easy to check that f is both an explicitly semiB,G -preinvex function and a semiB,G -preinvex function on X with respect to η. However, f is not a G-preinvex function on Abstract and Applied Analysis 5and Applied Analysis 5 X with respect to η and f is also not a semistrictly G-preinvex function on X with respect to η, because X is not an invex set. Moreover, by letting u 0, 0 , x 1, 1/2 , λ 1/2, we have f ( u λη x, u, λ ) f ( 1 2 , 1 16 ) ln ( 41 16 ) > 1 2 ln 2 1 2 ln ( 7 2 ) λf x 1 − λ f u . 2.10 Hence, f is not an explicitly semi-B-preinvex function and f is also not a semi-B-preinvex function on X with respect to η. From Definition 2.6, the inverse of function G must exist. Hence function G must be a strictly increasing one. Thus, we can assume that function G is a strictly increasing function on its domain. Now we give the following useful lemma. Lemma 2.11. Let f : X → R. Then: i f is semiB,G -preinvex on X with respect to η if and only if G f is semi-B-preinvex on X with respect to η; ii f is strictly semiB,G -preinvex on X with respect to η if and only if G f is strictly semi-B-preinvex on X with respect to η; iii f is explicitly semiB,G -preinvex on X with respect to η if and only if G f is explicitly semi-B-preinvex on X with respect to η. Proof. i By the monotonicity of G, we know that the inequality 2.7 is equivalent with G ( f ( u λη x, u, λ )) ≤ b1 x, u;λ G(f x ) b2 x, u;λ G(f u ), b1 x, u; 1 b2 x, u; 0 1, b1 x, u;λ b2 x, u;λ 1, λ ∈ 0, 1 . 2.11 Therefore, by Definitions 2.6 and 2.4, f is semiB,G -preinvex on X with respect to η if and only if G f is semi-B-preinvex on X with respect to η. Similar to part i , we can prove ii and iii . This completes the proof. Theorems 2.12 and 2.13, present the optimality properties for semiB,G -preinvex functions and explicitly semiB,G -preinvex functions, respectively. Theorem 2.12. Let X be a nonempty semi-invex set in R with respect to η : X ×X × 0, 1 → R, and f : X → R be a semiB,G -preinvex function on X with respect to η. If x ∈ X is a local minimum to the problem of minimizing f x subject to x ∈ X, then x is a global one. Proof. Let f be a semiB,G -preinvex function on X with respect to η. Then, by Lemma 2.11 i , G f is a semi-B-preinvex function on X with respect to η. Since G is increasing on its domain If x , then x ∈ X is a local minimum to the problem of minimizing f x subject to x ∈ X if and only if x ∈ X is a local minimum to the problem of minimizing G f x subject to x ∈ X. Therefore, by Theorem 3.1 in 33 , x ∈ X is a global one to the problem of minimizing G f x subject to x ∈ X. Hence x ∈ X is a global one for the problem of minimizing f x subject to x ∈ X. This completes the proof. 6 Abstract and Applied Analysis Theorem 2.13. Let X be a nonempty semi-invex set in R with respect to η : X ×X × 0, 1 → R, and f : X → R be an explicitly semiB,G -preinvex function on X with respect to η. If x ∈ X is a local minimum to the problem of minimizing f x subject to x ∈ X, then x is a global one. Proof. Similar to the proof of Theorem 2.12, from Theorem 3.1 in 17 , we can establish the result. From Example 2.10, Theorems 2.12 and 2.13, we can conclude that these new generalized convex functions constitutes an important class of generalized convex functions in mathematical programming. 3. Properties of SemiB,G -Preinvex Functions In this section, we first discuss the relations between our new kinds of generalized convex functions. By definitions of strictly semiB,G -preinvexity, explicitly semiB,G preinvexity, and semiB,G -preinvexity, the following result is obviously true. Theorem 3.1. If f is strictly semiB,G -preinvex function on X with respect to η, thenf is both an explicitly semiB,G -preinvex function and a semiB,G -preinvex function on X with respect to η. The following example illustrates that semiB,G -preinvexity does not imply strictly semiB,G -preinvexity; also explicitly semiB,G -preinvexity does not imply strictly semiB,G -preinvexity. Example 3.2. Let X be the set defined in Example 2.3; let η x, u, λ , b1 x, u;λ , and b2 x, u;λ be functions defined in Example 2.10. define f x ⎧ ⎨ ⎩ 1, x 0, 0 , 0, x / 0, 0 . 3.1 Then f is both an explicitly semiB,G -preinvex function and a semiB,G -preinvex function on X with respect to η, but f is not a strictly semiB,G -preinvex function on X with respect to η, where G a a, a ∈ R. Note that B-preinvex function is semiB,G -preinvex, and explicitly B-preinvex function is explicitly semiB,G -preinvex, where G a a, a ∈ R. Examples 2.1 and 2.2 in 30 can illustrate that semiB,G -preinvexity does not imply explicitly semiB,G -preinvexity, and also explicitly semiB,G -preinvexity does not imply semiB,G -preinvexity. Next, we present properties of semiB,G -preinvex functions and explicitly semiB,G -preinvex functions. Theorem 3.3. Let X be a nonempty semi-invex set in R with respect to η : X × X × 0, 1 → R , f : X → R be an explicitly semiB,G1 -preinvex function on X with respect to η, and G2 : IG1 f X → R be both a convex function and an increasing function. Then f is an explicitly semiB,G2 G1 -preinvex function on X with respect to the same η. Proof. If f is an explicitly semiB,G1 -preinvex function on X with respect to η. Then, by Lemma 2.11 i , G1 f is an explicitly semi-B-preinvex function on X with respect to η. Abstract and Applied Analysis 7 Therefore, there exist b1, b2 : X ×X × 0, 1 → R such that, for any x, u ∈ X, f x / f u , the inequalityand Applied Analysis 7 Therefore, there exist b1, b2 : X ×X × 0, 1 → R such that, for any x, u ∈ X, f x / f u , the inequality G1 ( f ( u λη x, u, λ )) < b1 x, u;λ G1 ( f x ) b2 x, u;λ G1 ( f u ) , b1 x, u; 1 b2 x, u; 0 1, b1 x, u;λ b2 x, u;λ 1, λ ∈ 0, 1 3.2 holds. Note the convexity and monotonicity of G2, we have G2 ( G1 ( f ( u λη x, u, λ ))) < G2 ( b1 x, u;λ G1 ( f x ) b2 x, u;λ G1 ( f u )) ≤ b1 x, u;λ G2 ( G1 ( f x )) b2 x, u;λ G2 ( G1 ( f u )) . 3.3 Hence, G2 G1 f is an explicitly semi-B-preinvex function on X with respect to η. Again, by Lemma 2.11 i , f is an explicitly semiB,G2 G1 -preinvex function on X with respect to η. This completes the proof. Theorem 3.4. Let X be a nonempty semi-invex set in R with respect to η : X × X × 0, 1 → R, fi : X → R i ∈ K {1, . . . , k} be semiB,G -preinvex function on X with respect to the same η, G, b1, and b2. Moreover, G is both a convex function and a concave function on R. Then, for any λi > 0, ∑k i 1 λi 1, the function h x : ∑k i 1 λifi x is semiB,G -preinvex on X with respect to the same η,G, b1, and b2. Further, if there exists i0 ∈ K such that fi0 is explicitly semiB,G -preinvex on X with respect to the same η, G, b1, and b2, then h is explicitly semiB,G -preinvex on X with respect to the same η, G, b1, and b2. Proof. If fi is semiB,G -preinvex on X with respect to the same η, G, b1, and b2, i ∈ K. Then, by Lemma 2.11 i , G fi is a semi-B-preinvex function on X with respect to the same η, b1, and b2, i ∈ K. Therefore, for any x, u ∈ X, the inequality G ( fi ( u λη x, u, λ )) ≤ b1 x, u;λ Gfi x ) b2 x, u;λ G ( fi u ) , b1 x, u; 1 b2 x, u; 0 1, b1 x, u;λ b2 x, u;λ 1, λ ∈ 0, 1 3.4 holds for i ∈ K. Since G is both a convex function and a concave function on R, then