# Generalized Nonsmooth Exponential-Type Vector Variational-Like Inequalities and Nonsmooth Vector Optimization Problems in Asplund Spaces

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2**

**.**Suppose that Ω is a nonempty subset of a normed vector space X. Then, for any $x\in X$ and $\epsilon \ge 0$, the set of ε-normals to Ω at x is defined as

**Remark**

**1**

**.**It is noted that the Clarke subdifferential is larger class than the Fréchet subdifferential and the limiting subdifferential with the relation ${\partial}_{F}f\left(x\right)\subseteq {\partial}_{L}f\left(x\right)\subseteq {\partial}_{C}f\left(x\right)$.

**Definition**

**3.**

**Remark**

**2.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6**

**.**Suppose that K is a subset of a topological vector space Y. A set-valued mapping $T:K\u27f6{2}^{Y}$ is called a KKM-mapping if, for each nonempty finite subset $\{{y}_{1},{y}_{2},\cdots ,{y}_{n}\}\subset K$, we have

**Theorem**

**1**

**.**Suppose that K is a subset of a topological vector space Y and $T:K\u27f6{2}^{Y}$ is a KKM-mapping. If, for each $y\in K,T\left(y\right)$ is closed and for at least one $y\in K,T\left(y\right)$ is compact, then

**Definition**

**7.**

**Definition**

**8.**

**Remark**

**3.**

**Problem**

**1.**

**Definition**

**9.**

- (i)
- an efficient solution $\left(Pareto\phantom{\rule{3.33333pt}{0ex}}solution\right)$ of (P
_{1}) if and only if$$f\left(y\right)-f\left(\overline{x}\right)\notin -C\backslash \left\{0\right\},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall y\in K;$$ - (ii)
- a weak efficient solution $\left(weak\phantom{\rule{3.33333pt}{0ex}}Pareto\phantom{\rule{3.33333pt}{0ex}}solution\right)$ of (P
_{1}) if and only if$$f\left(y\right)-f\left(\overline{x}\right)\notin -intC,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall y\in K.$$

**Problem**

**2.**

**Problem**

**3.**

**Special Cases:**

- (i)
- If $\theta \equiv 0$ and ${\partial}_{L}f(\xb7)=\partial f(\xb7)$, i.e., the Clarke subdifferential operator, then (P
_{2}) and (P_{3}) reduces to nonsmooth exponential-type vector variational like inequality problem and nonsmooth exponential-type weak vector variational like inequality problem considered and studied by Jayswal and Choudhury [17]. - (ii)
- For $p=0$, a similar analogue of problems (P
_{2}) and (P_{3}) was introduced and studied by Oveisiha and Zafarani [13].

_{2}) is also a solution of (P

_{3}). We construct the following example in support of (P

_{2}).

**Example**

**1.**

_{2}) is to find a point $\overline{x}\in K$ such that

_{2}).

## 3. Main Results

_{2}) is an efficient solution of (P

_{1}).

**Theorem**

**2.**

_{2}), then $\overline{x}$ is an efficient solution of (P

_{1}).

**Proof.**

_{2}). We will prove that $\overline{x}\in K$ is an efficient solution of (P

_{1}). Indeed, let us assume that $\overline{x}\in K$ is not an efficient solution of (P

_{1}). Then, $\exists y\in K$ such that

_{2}). Hence, $\overline{x}$ is an efficient solution of (P

_{1}). This completes the proof. □

**Theorem**

**3.**

_{1}), then $\overline{x}$ is a solution of (P

_{2}).

**Proof.**

_{1}). On contrary suppose that $\overline{x}$ is not a solution of (P

_{2}). Then, each $\beta $ ensures the existence of ${x}_{\beta}$ satisfying

_{1}). Therefore, $\overline{x}$ is a solution of (P

_{2}). This completes the proof. □

_{1}) and (P

_{3}).

**Theorem**

**4.**

_{1}), then $\overline{x}\in K$ is also a solution of (P

_{3}). Conversely, if each ${f}_{i}\phantom{\rule{3.33333pt}{0ex}}(1\le i\le n)$ is limiting $(p,r)$-${\alpha}_{i}$-$(\eta ,\theta )$-invex mapping with respect to η and θ and $\overline{x}\in K$ is the solution of (P

_{3}), then $\overline{x}\in K$ is also a weak efficient solution of (P

_{1}).

**Example**

**2.**

_{3}) is to find $\overline{x}\in [0,1]$ such that

_{3}). One can easily show that $\overline{x}=0$ is a weakly efficient solution of vector optimization problem (7) by using Theorem 4.

_{3}) by employing the Fan-KKM Theorem.

**Theorem**

**5.**

_{3}) admits a solution.

**Proof.**

_{3}) has a solution. This completes the proof. □

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Irfan, S.S.; Rahaman, M.; Ahmad, I.; Ahmad, R.; Husain, S. Generalized Nonsmooth Exponential-Type Vector Variational-Like Inequalities and Nonsmooth Vector Optimization Problems in Asplund Spaces. *Mathematics* **2019**, *7*, 345.
https://doi.org/10.3390/math7040345

**AMA Style**

Irfan SS, Rahaman M, Ahmad I, Ahmad R, Husain S. Generalized Nonsmooth Exponential-Type Vector Variational-Like Inequalities and Nonsmooth Vector Optimization Problems in Asplund Spaces. *Mathematics*. 2019; 7(4):345.
https://doi.org/10.3390/math7040345

**Chicago/Turabian Style**

Irfan, Syed Shakaib, Mijanur Rahaman, Iqbal Ahmad, Rais Ahmad, and Saddam Husain. 2019. "Generalized Nonsmooth Exponential-Type Vector Variational-Like Inequalities and Nonsmooth Vector Optimization Problems in Asplund Spaces" *Mathematics* 7, no. 4: 345.
https://doi.org/10.3390/math7040345