A note on G-preinvex functions

Abstract

With the equivalent relationships between the G-generalized invexities and general invexities on the hand, we present two characterizations for G-preinvexity; we also discuss the relationships between different G-generalized invexities such as G-preinvexity, strict G-preinvexity and semistrict G-preinvexity. Note that our results are proved by applying the results from general invexities introduced in the literatures.

1 Introduction

Recently, Antczak [1, 2] introduced the concept of the G-preinvexity, which included the preinvexity [3] and the r-preinvexity [4] as special cases. Relation of this G-preinvexity to preinvexity and some properties of this class of functions were studied in [2]. In another recent paper, Luo and Wu [5] introduced a new class of functions, named semistrictly G-preinvex functions. The relationships between semistrictly G-preinvex functions and G-preinvex functions were investigated under mild assumptions. Their results improved and extended the existing ones in the literature. Also, the properties of semistrictly G-preinvex functions were further considered by Peng in [6].

In this note, we are interested in the relationships between three kinds of G-generalized invexities. For this purpose, we firstly investigate the relation between the G-generalized invexities and the corresponding general generalized invexities. Then we characterize these G-generalized invexities by applying the well-known results from the preinvexity, the strict preinvexity and the semistrict preinvexity. Moreover, we point out that our method is different from the one used by Luo and Wu in [5]. The rest of this note is organized as follows. In Section 2, we give some definitions and some preliminaries; moreover, we establish the useful Lemma 1. Section 3 presents two characterizations for G-preinvex functions and proves that, under certain conditions, the G-preinvexity is equivalent with prequasi-invexity when an intermediate-point G-preinvexity is required. In Section 4, we obtain relationships between different G-generalized invexities. Section 5 gives some conclusions.

2 Definitions and preliminaries

In this section, we provide some definitions and some notations. Moreover, we establish an important lemma.

Definition 1 [3]

Let $X\subset {\mathbb{R}}^n$, $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$. The set X is said to be invex at $u\in X$ with respect to η if for all $x\in X$ such that

$$u+\lambda \eta (x,u)\in X,\mathrm{\forall }\lambda \in [0,1].$$

X is said to be invex set with respect to η if X is invex at each $u\in X$.

Definition 2 [7]

Let X be a nonempty invex subset of ${\mathbb{R}}^n$ with respect to η. A function $f:X\to \mathbb{R}$ is said to be preinvex at $u\in X$ with respect to η if

$$f(u+\lambda \eta (x,u))\le \lambda f(x)+(1-\lambda )f(u),\mathrm{\forall }\lambda \in [0,1],\mathrm{\forall }x\in X.$$

(1)

The function f is said to be preinvex on X with respect to η if f is preinvex at each $u\in X$ with respect to η; f is said to be strictly preinvex on X with respect to η if the inequality (1) strictly holds for all $x,u\in X$ such that $x\ne u$; f is said to be semistrictly preinvex on X with respect to η if the inequality (1) strictly holds for all $x,u\in X$ such that $f(x)\ne f(u)$.

Definition 3 [1, 2, 5]

Let X be a nonempty invex subset of ${\mathbb{R}}^n$ with respect to η. A function $f:X\to \mathbb{R}$ is said to be G-preinvex at u on X with respect to η if there exists a continuous function $G:\mathbb{R}\to \mathbb{R}$ such that $G:I_f(X)\to \mathbb{R}$ is a strictly increasing function on its domain, and

$$f(u+\lambda \eta (x,u))\le G^{-1}(\lambda G(f(x))+(1-\lambda )G(f(u))),\mathrm{\forall }\lambda \in [0,1],\mathrm{\forall }x\in X.$$

(2)

The function f is said to be G-preinvex on X with respect to η if f is G-preinvex at each $u\in X$ with respect to η; f is said to be strictly G-preinvex on X with respect to η if the inequality (2) strictly holds for all $x,u\in X$ such that $x\ne u$; f is said to be semistrictly G-preinvex on X with respect to η if the inequality (2) strictly holds for all $x,u\in X$ such that $f(x)\ne f(u)$.

From Definition 3, G is a strictly increasing function because $G^{-1}$ must exist. Hence, let G be a strictly increasing function throughout this note. Now we present a useful lemma.

Lemma 1 Let $f:X\to \mathbb{R}$. Suppose $G:I_f(X)\to \mathbb{R}$ is a strictly increasing function on its domain. Then

• (i) f is G-preinvex on X with respect to η if and only if $G(f)$ is preinvex on X with respect to η;

• (ii) f is strictly G-preinvex on X with respect to η if and only if $G(f)$ is strictly preinvex on X with respect to η;

• (iii) f is semistrictly G-preinvex on X with respect to η if and only if $G(f)$ is semistrictly preinvex on X with respect to η.

Proof (i) By the monotonicity of G, we know that the inequality (2) is equivalent to

$$G\left(f(u+\lambda \eta (x,u))\right)\le \lambda G(f(x))+(1-\lambda )G(f(u)),\mathrm{\forall }\lambda \in [0,1],\mathrm{\forall }x\in X.$$

Therefore, by Definitions 2 and 3, f is G-preinvex on X with respect to η if and only if $G(f)$ is preinvex on X with respect to η.

Similar to part (i), we can prove part (ii) and (iii). □

3 Semicontinuity and G-preinvexity

In this section, two conditions that determine the G-preinvexity of a function via an intermediate-point G-preinvexity check under conditions of upper and lower semicontinuity, respectively, are presented; moreover, equivalent relationship between G-preinvexity and prequasi-invexity is proved under the intermediate-point G-preinvexity assumption. Here, we need the following Condition C, which was introduced by Mohan and Neogy in [8]. The function $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ satisfies Condition C if

$$\eta (y,y+\lambda \eta (x,y))=-\lambda \eta (x,y),$$

(3)

$$\eta (x,y+\lambda \eta (x,y))=(1-\lambda )\eta (x,y)$$

(4)

hold for any $x,y\in X$ and for any $\lambda \in [0,1]$.

The upper and lower semicontinuity of a real function f is defined as follows.

Definition 4 [9]

Let X be a nonempty subset of ${\mathbb{R}}^n$. A function $f:X\to \mathbb{R}$ is said to be upper semicontinuous at $\overline{x}\in X$ if, for every $ϵ>0$, there exists a $\delta >0$ such that for all $x\in X$, if $\parallel x-\overline{x}\parallel <\delta$, then

$$f(x)<f(\overline{x})+ϵ.$$

If −f is upper semicontinuous at $\overline{x}\in X$, then f is said to be lower semicontinuous at $\overline{x}\in X$.

We also need the following Lemma 2, which is Lemma 3.2 in [9].

Lemma 2 Let X be a nonempty, open and invex set in ${\mathbb{R}}^n$ with respect to $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, where η satisfies Condition C. Assume that $f:X\to \mathbb{R}$ satisfies

$$f(y+\eta (x,y))\le f(x),\mathrm{\forall }x,y\in X.$$

Moreover, there exists an $\alpha \in (0,1)$ such that for every $x,y\in X$ the inequality

$$f(y+\alpha \eta (x,y))\le \alpha f(x)+(1-\alpha )f(y)$$

(5)

holds. Then the set $A:=\{\lambda \in [0,1]|f(y+\lambda \eta (x,y))\le \lambda f(x)+(1-\lambda )f(y),\mathrm{\forall }x,y\in X\}$ is dense in $[0,1]$.

Under semicontinuity conditions, Yang proved from Lemma 2 that judging a function to be preinvex or not can be reduced to checking intermediate-point preinvexity for the function; see the following Lemmas 3 and 4, which are taken from Theorems 3.1 and 3.2 in [9], respectively.

Lemma 3 Let X be a nonempty open invex set in ${\mathbb{R}}^n$ with respect to $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, where η satisfies Condition C. Assume that the function $f:X\to \mathbb{R}$ is upper semicontinuous on X and satisfies

$$f(y+\eta (x,y))\le f(x),\mathrm{\forall }x,y\in X.$$

Then f is a preinvex function on X if and only if there exists an $\alpha \in (0,1)$ such that

$$f(y+\alpha \eta (x,y))\le \alpha f(x)+(1-\alpha )f(y),\mathrm{\forall }x,y\in X.$$

Lemma 4 Let X be a nonempty invex set in ${\mathbb{R}}^n$ with respect to $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, where η satisfies Condition C. Assume that the function $f:X\to \mathbb{R}$ is lower semicontinuous on X and satisfies

$$f(y+\eta (x,y))\le f(x),\mathrm{\forall }x,y\in X.$$

Then f is a preinvex function on X if and only if for any $x,y\in X$, there exists an $\alpha \in (0,1)$ such that

$$f(y+\alpha \eta (x,y))\le \alpha f(x)+(1-\alpha )f(y).$$

With Lemmas 2-4 on hand, we can prove the following Theorems 1-3, respectively.

Theorem 1 Let X be a nonempty invex set in ${\mathbb{R}}^n$ with respect to $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, where η satisfies Condition C. Assume that $f:X\to \mathbb{R}$ satisfies

$$f(y+\eta (x,y))\le f(x),\mathrm{\forall }x,y\in X.$$

Suppose the function G is increasing on $I_f(X)$. Moreover, there exists an $\alpha \in (0,1)$ such that for every $x,y\in X$ the inequality

$$G\left(f(y+\alpha \eta (x,y))\right)\le \alpha G(f(x))+(1-\alpha )G(f(y))$$

(6)

holds. Then the set $A:=\{\lambda \in [0,1]|G(f(y+\lambda \eta (x,y)))\le \lambda G(f(x))+(1-\lambda )G(f(y)),\mathrm{\forall }x,y\in X\}$ is dense in $[0,1]$.

Proof

From the assumption of this theorem, we have

$$G\left(f(y+\eta (x,y))\right)\le G(f(x)),\mathrm{\forall }x,y\in X.$$

Hence, we can deduce the result from Lemma 2. □

Theorem 2 Let X be a nonempty open invex set in ${\mathbb{R}}^n$ with respect to $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, where η satisfies Condition C. Assume that a function $f:X\to \mathbb{R}$ is upper semicontinuous on X and satisfies

$$f(y+\eta (x,y))\le f(x),\mathrm{\forall }x,y\in X.$$

Moreover, the function G is both continuous and increasing on $I_f(X)$. Then f is G-preinvex on X if and only if there exists an $\alpha \in (0,1)$ such that

$$G\left(f(y+\alpha \eta (x,y))\right)\le \alpha G(f(x))+(1-\alpha )G(f(y)),\mathrm{\forall }x,y\in X.$$

Proof By assumption, we know that the function $G(f)$ is upper semicontinuous on X and it satisfies

$$G\left(f(y+\eta (x,y))\right)\le G(f(x)),\mathrm{\forall }x,y\in X.$$

Replacing f by $G(f)$ in Lemma 3 and combining Lemma 1(i), we obtain the desired result. □

If f is continuous on X, then the above Theorem 2 is Theorem 10 in [1]. However, our proof is simpler than the proof of Theorem 10 in [1], since we apply the result pertaining to the preinvexity as defined in Definition 2.

Theorem 3 Let X be a nonempty invex set in ${\mathbb{R}}^n$ with respect to $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, where η satisfies Condition C. Assume that the function $f:X\to \mathbb{R}$ is lower semicontinuous on X and satisfies

$$f(y+\eta (x,y))\le f(x),\mathrm{\forall }x,y\in X.$$

Moreover, the function G is both continuous and increasing on $I_f(X)$. Then f is G-preinvex on X if and only if for any $x,y\in X$, there exists an $\alpha \in (0,1)$ such that

$$G\left(f(y+\alpha \eta (x,y))\right)\le \alpha G(f(x))+(1-\alpha )G(f(y)).$$

Proof

By the assumption of the theorem, it is easy to check that

$$G\left(f(y+\eta (x,y))\right)\le G(f(x)),\mathrm{\forall }x,y\in X.$$

Moreover, $G(f)$ is lower semicontinuous on X. Now, with Lemma 1(i) and Lemma 4, we derive the desired result. □

The above Theorems 2 and 3 illustrate that, to justify G-preinvexity of a function, it is sufficient to check intermediate-point G-preinvexity for the function. Our development extends the results of general preinvexity to the G-preinvexity. Note that Theorems 1-3 generalize Lemmas 2-4 from the preinvex case to the G-preinvex situation, respectively.

On the relationship between the preinvexity and prequasi-invexity, where the prequasi-invexity concept is presented in Definition 5, Yang et al. obtained an interesting result (see Lemma 5).

Definition 5 Let X be a nonempty invex subset of ${\mathbb{R}}^n$ with respect to η. A function $f:X\to \mathbb{R}$ is said to be prequasi-invex on X if

$$f(y+\lambda \eta (x,y))\le max\{f(x),f(y)\},\mathrm{\forall }\lambda \in [0,1],\mathrm{\forall }x,y\in X.$$

Remark 1 If the function G is strictly increasing on $I_f(X)$, then f is prequasi-invex on X if and only if $G_f(f)$ is prequasi-invex on X.

Lemma 5 [9]

Let X be a nonempty invex set in ${\mathbb{R}}^n$ with respect to $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, where η satisfies Condition C. Then a function $f:X\to \mathbb{R}$ is preinvex on X if and only if it is a prequasi-invex function on X and there exists an $\alpha \in (0,1)$ such that

$$f(y+\alpha \eta (x,y))\le \alpha f(x)+(1-\alpha )f(y),\mathrm{\forall }x,y\in X.$$

Next, we extend the result obtained by Yang et al. to the G-preinvex situation in the following theorem, which reveals that, under an intermediate-point G-preinvexity assumption, the G-preinvexity is equivalent with prequasi-invexity.

Theorem 4 Let X be a nonempty invex set in ${\mathbb{R}}^n$ with respect to $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ and $f:X\to \mathbb{R}$, where η satisfies Condition C. Suppose that G is a strictly increasing function on $I_f(X)$. Then f is a G-preinvex function on X if and only if it is a prequasi-invex function on X and there exists an $\alpha \in (0,1)$ such that

$$G\left(f(y+\alpha \eta (x,y))\right)\le \alpha G(f(x))+(1-\alpha )G(f(y)),\mathrm{\forall }x,y\in X.$$

Proof By Remark 1, one obtains that f is a prequasi-invex function on X if and only if $G(f)$ is a prequasi-invex function on X. Thus, we have the desired result from Lemma 1(i) and Lemma 5. □

4 Relationships among G-generalized preinvexities

In this section, we discuss the relationships between G-invexities under Condition C. To this end, we will use the following results proved in the literatures.

Theorem 5 [[10], Theorem 2.3]

Let X be nonempty invex set in ${\mathbb{R}}^n$ with respect to $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, where η satisfies Condition C. Suppose function $f:X\to \mathbb{R}$ is semistrictly preinvex on X with respect to η. If there exists a $\lambda \in (0,1)$ such that

$$f(y+\lambda \eta (x,y))\le \lambda f(x)+(1-\lambda )f(y),\mathrm{\forall }x,y\in X,$$

then f is a preinvex function on X with respect to the same η.

Theorem 6 [[10], Theorem 2.5]

Let X be nonempty invex set in ${\mathbb{R}}^n$ with respect to $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, where η satisfies Condition C. Suppose that $f:X\to \mathbb{R}$ is a preinvex function on X with respect to η. For each pair $x,y\in X$, $x\ne y$, if there exists a $\lambda \in (0,1)$ such that

$$f(y+\lambda \eta (x,y))<\lambda f(x)+(1-\lambda )f(y),\mathrm{\forall }x,y\in X,$$

then f is a strictly preinvex function on X with respect to the same η.

Theorem 7 [[7], Theorem 4.2]

Let X be a nonempty invex set in ${\mathbb{R}}^n$ with respect to $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, where η satisfies Condition C. Assume that $f:X\to \mathbb{R}$ is a preinvex function on X with respect to η. If there exists a $\lambda \in (0,1)$ such that for every $x,y\in X$, $f(x)\ne f(y)$, the inequalities

$$\begin{array}{r}f(y+\lambda \eta (x,y))<\lambda f(x)+(1-\lambda )f(y),\\ f(y+(1-\alpha )\eta (x,y))<\alpha f(y)+(1-\alpha )f(x)\end{array}$$

hold, then f is a semistrictly preinvex function on X with respect to the same η.

Theorem 8 Let X be a nonempty invex set in ${\mathbb{R}}^n$ with respect to $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, where η satisfies Condition C. Suppose that $f:X\to \mathbb{R}$ is a semistrictly G-preinvex function on X with respect to η. If there exists a $\lambda \in (0,1)$ such that

$$G\left(f(y+\lambda \eta (x,y))\right)\le \lambda G(f(x))+(1-\lambda )G(f(y)),\mathrm{\forall }x,y\in X,$$

(7)

then f is a G-preinvex function on X with respect to the same η.

Proof Since f is a semistrictly G-preinvex function on X with respect to η. Then, by Lemma 1(iii), $G(f)$ is a semistrictly preinvex function on X with respect to η. Replacing f by $G(f)$ in Theorem 5, we deduce that $G(f)$ is a preinvex function on X with respect to η. Again, from Lemma 1(i), f is a G-preinvex function on X with respect to the same η. □

Recall that Theorem 8 was also presented in [5]. But our method of proof is different from [5]. Note that we establish the result by applying the above Theorem 8, which is an existed result for semistrictly preinvex function.

Theorem 9 Let X be a nonempty invex set in ${\mathbb{R}}^n$ with respect to $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, where η satisfies Condition C. Suppose that $f:X\to \mathbb{R}$ is a G-preinvex function on X with respect to η. For each pair $x,y\in X$, $x\ne y$, if there exists a $\lambda \in (0,1)$ such that

$$G\left(f(y+\lambda \eta (x,y))\right)<\lambda G(f(x))+(1-\lambda )G(f(y)),$$

then f is a strictly G-preinvex function on X with respect to the same η.

Proof Note that f is G-preinvex on X with respect to η. By Lemma 1(i), $G(f)$ is preinvex on X with respect to η. Now, we deduce from Theorem 6 that $G(f)$ is strictly preinvex on X with respect to η. Therefore, one obtains from Lemma 1(ii) that f is strictly G-preinvex on X with respect to the same η. □

Lemma 6 Let X be a nonempty invex set in ${\mathbb{R}}^n$ with respect to $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, where η satisfies Condition C. Suppose function $f:X\to \mathbb{R}$ is G-preinvex on X with respect to η. If there exists an $\alpha \in (0,1)$ such that for every $x,y\in X$, $f(x)\ne f(y)$,

$$G\left(f(y+\alpha \eta (x,y))\right)<\alpha G(f(x))+(1-\alpha )G(f(y)).$$

(8)

Then for every $x,y\in X$, $f(x)\ne f(y)$

$$G\left(f(y+(1-\alpha )\eta (x,y))\right)<\alpha G(f(y))+(1-\alpha )G(f(x)).$$

(9)

Proof (i) If $\alpha =\frac{1}{2}$, the inequality (8) is the inequality (9).

• (ii) If $\alpha <\frac{1}{2}$, then $\alpha <1-\alpha <1$. Denote by $u_1=1$, $u_2=\alpha$ and $\beta =\frac{1-2\alpha }{1-\alpha }$, then $1-\alpha =\beta u_1+(1-\beta )u_2$.

From Condition C, we have

$$\begin{array}{r}y+u_2\eta (x,y)+\beta \eta (y+u_1\eta (x,y),y+u_2\eta (x,y))\\ =y+u_2\eta (x,y)+\beta \eta (y+u_1\eta (x,y),y+u_1\eta (x,y)-(u_1-u_2)\eta (x,y))\\ =y+u_2\eta (x,y)+\beta \eta (y+u_1\eta (x,y),y+u_1\eta (x,y)+\frac{u_1-u_2}{u_1}\eta (y,y+u_1\eta (x,y)))\\ =y+u_2\eta (x,y)-\beta \frac{u_1-u_2}{u_1}\eta (y,y+u_1\eta (x,y))\\ =y+(u_2+\beta (u_1-u_2))\eta (x,y)=y+(1-\alpha )\eta (x,y).\end{array}$$

Note that the identity (3) in Condition C is used in the second, third and fourth equalities. Hence, from (8) and the G-preinvexity of f, we obtain

$$\begin{array}{r}G\left(f(y+(1-\alpha )\eta (x,y))\right)\\ =G\left(f(y+u_2\eta (x,y)+\beta \eta (y+u_1\eta (x,y),y+u_2\eta (x,y)))\right)\\ \le \beta G\left(f(y+u_1\eta (x,y))\right)+(1-\beta )G\left(f(y+u_2\eta (x,y))\right)\\ <\beta (u_{1}G(f(x))+(1-u_1)G(f(y)))+(1-\beta )(u_{2}G(f(x))+(1-u_2)G(f(y)))\\ =(1-\alpha )G(f(x))+\alpha G(f(y)).\end{array}$$

• (iii) If $\alpha >\frac{1}{2}$, then $0<1-\alpha <\alpha$. Denote by $u_1=\alpha$, $u_2=0$ and $\beta =\frac{1-\alpha }{\alpha }$, then $1-\alpha =\beta u_1+(1-\beta )u_2$. Similar to (ii), we can prove that the inequality (9) still holds. □

Theorem 10 Let X be a nonempty invex set in ${\mathbb{R}}^n$ with respect to $\eta :{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, where η satisfies Condition C. Assume that $f:X\to \mathbb{R}$ is a G-preinvex function on X with respect to η. If there exists a $\lambda \in (0,1)$ such that for every $x,y\in X$, $f(x)\ne f(y)$, the inequality

$$G\left(f(y+\lambda \eta (x,y))\right)<\lambda G(f(x))+(1-\lambda )G(f(y))$$

(10)

holds, then f is a semistrictly G-preinvex function on X with respect to the same η.

Proof By Lemma 1(iii), it is sufficient to prove that $G(f)$ is semistrictly preinvex on X with respect to η. From Lemma 6 and the assumption of Theorem 10, we know that the assumption of Theorem 7 holds. Using Theorem 7 and Lemma 1(iii), we can deduce the result. □

Theorem 11 Let X be a nonempty invex set in ${\mathbb{R}}^n$ with respect to $\eta :X\times X\to {\mathbb{R}}^n$, where η satisfies Condition C; suppose the function $f:X\to \mathbb{R}$ is lower semicontinuous and satisfies

$$f(y+\eta (x,y))\le f(x)$$

for any $x,y\in X$. Moreover, the function G is both continuous and increasing on $I_f(X)$. If there exists an $\alpha \in (0,1)$ such that for every $x,y\in X$, $f(x)\ne f(y)$, the inequality

$$G\left(f(y+\alpha \eta (x,y))\right)<\alpha G(f(x))+(1-\alpha )G(f(y))$$

(11)

holds, then f is both G-preinvex and semistrictly G-preinvex on X.

Proof Firstly, we shall prove that f is a G-preinvex function on X. Recalling Theorem 4, we need to show that there exists a $\lambda \in (0,1)$ such that for every $x,y\in X$ the inequality (7) holds. Assume, by a contradiction, there exist $x,y\in X$ such that

$$G\left(f(y+\lambda \eta (x,y))\right)>\lambda G(f(x))+(1-\lambda )G(f(y)),\mathrm{\forall }\lambda \in (0,1).$$

(12)

Since G is both continuous and increasing, then $G(f)$ is lower semicontinuous and satisfies

$$G\left(f(y+\eta (x,y))\right)\le G(f(x)).$$

We need to consider the following cases.

Case (i) $f(x)\ne f(y)$. According to (11), we must have

$$G\left(f(y+\alpha \eta (x,y))\right)<\alpha G(f(x))+(1-\alpha )G(f(y)),$$

which contradicts to (12).

Case (ii) $f(x)=f(y)$. Since $\alpha \in (0,1)$, then $2\alpha -{\alpha }^2=1-{(\alpha -1)}^2\in (0,1)$. Let $\lambda =2\alpha -{\alpha }^2$ and $\lambda =\alpha$ in (12), respectively. Then we have

$$G\left(f(y+(2\alpha -{\alpha }^2)\eta (x,y))\right)>G(f(x)),$$

(13)

$$G\left(f(y+\alpha \eta (x,y))\right)>G(f(x)).$$

(14)

From (14), we obtain $G(f(y+\alpha \eta (x,y)))\ne G(f(x))$. Therefore, according to (11), we obtain from Condition C the following inequality:

$$\begin{array}{r}G\left(f(y+(2\alpha -{\alpha }^2)\eta (x,y))\right)\\ =G\left(f(y+\alpha \eta (x,y)+\alpha \eta (x,y+\alpha \eta (x,y)))\right)\\ <(1-\alpha )G\left(f(y+\alpha \eta (x,y))\right)+\alpha G(f(x))\\ <(1-\alpha )((1-\alpha )G(f(y))+\alpha G(f(x)))+\alpha G(f(x))\\ =[(1-\alpha )(1-\alpha )+(1-\alpha )\alpha +\alpha ]G(f(x))=G(f(x)),\end{array}$$

which contradicts to (13). Therefore, from Theorem 4, f is a G-preinvex function on X with respect to η.

Further, from the above Theorem 10, f is also a semistrictly G-preinvex function on X with respect to η. □

5 Conclusions

In this note, our purpose is to investigate the G-generalized invexities introduced by researchers in the past few years. To apply the existed results from the general invexities to deal with the G-generalized ones, we have established the useful Lemma 1, which discloses the relationships between G-generalized invexities and the general invexities. With this important lemma on hand, we have extended the acknowledged results pertaining to the general invexities to the corresponding G-generalized invexities. More exactly, some characterizations for G-preinvex functions have been deduced; when an intermediate-point G-preinvexity is satisfied, two equivalent relationships between G-preinvexity and prequasi-invexity have been established (see Theorems 2 and 3). Using the existed results (Theorems 5, 6 and 7) relating to the general invexities, we deduce the similar results for G-generalized invexities; see Theorems 8, 10 and 11. Note that Theorems 8, 10 and 11 are also presented in [5]. However, our method is different from the one used by Luo and Wu in [5]. Here, we prove the results by applying the well-known results of the general invexities presented in the literatures.