‎In this paper‎, ‎we extend the notion of approximate convexity to set-valued maps and obtain some relations between approximate convexity and approximate monotonicity of their normal subdifferential‎.

This paper is devoted to consider the notions of complementary problem (CP) and set-valued complementary problem (SVCP). The set-valued complementary problem is compared with the classical single-valued complementary problem. Also, the solution set of the set-valued complementary problem is characterized. Our results illustrated by some examples.

In this work, we introduce the idea of generalized interval-valued neutrosophic sets. After providing the fundamental definitions of operations related to these sets, we establish several key properties and explore the relationships between generalized interval-valued neutrosophic sets and other related concepts. Finally, we extend the notion of generalized neutrosophic topological spaces to incorporate generalized interval-valued neutrosophic topological spaces. We define the concept of generalized interval-valued neutrosophic g-continuous function between two generalized interval-valued neutrosophic topological spaces. Lastly, an application has been shown in decision making problem.

Interval-valued Pythagorean fuzzy set (IVPFS), originally proposed by Peng and Yang, is a novel tool to deal with vagueness and incertitude. As a generalized set, IVPFS has close relationship with interval-valued intuitionistic fuzzy set (IVIFS). IVPFS can be reduced to IVIFS satisfying the condition $\mu^++\nu^+ \leq 1$. However, the related operations of IVPFS do not take different conditions into consideration. In this paper, we initiate some new interval-valued Pythagorean fuzzy operators ($\diamondsuit, \Box, \spadesuit, \clubsuit, \maltese, \rightarrow, \$ $) and discuss their properties in relation with some existing operators $(\cup, \cap, \oplus, \otimes)$ in detail. It will promote the development of interval-valued Pythagorean fuzzy operators. Later, we propose an algorithm to deal with multi-attribute decision making (MADM) problem based on proposed $\spadesuit$ operator. Finally, the effectiveness and feasibility of approach is demonstrated by mine emergency decision making example.