| Fuzzy multiple attribute decision making theory is an important branch of modern decision science. Due to the complexity and uncertainty of the objective things, the evaluation provided by decision makers often is in the form of fuzzy numbers during the multi-attribute decision making. As a kind of fuzzy numbers, hesitant fuzzy sets set a proper interpretation of the variability of realistic environment and the limitations of human cognition. As a generalization of Banach contraction mapping principle, the fixed point theory of fuzzy mappings is an important tool for combining the fuzzy nonlinear analysis with other subjects to serve the real world. With the development of modern society and economy, the decision making problems are becoming more and more complex and unpredictable, carrying out the study on hesitant fuzzy multi-attribute decision making problems and fuzzy nonlinear analysis methods has great significance in theory and application. The purpose of this work is to investigate fuzzy fixed point theory and hesitant fuzzy multi-attribute decision making methods, the main contents are as follows:(1)The fixed point problems of fuzzy mappings are studied. In view of the fact that nonlinear problems often need to consider a variety of hybrid factors, we propose the fixed point problems of the hybrid pair of single-valued mapping and fuzzy mapping. Based on the weak commutativity and compatibility, the existence of fuzzy coupled coincidence point and fuzzy coupled common fixed point of the hybrid pair are discussed. We also introduce the concept and topological properties of a new fuzzy-metric space and the existence of common fixed point of a family of fuzzy self-mappings in this space.(2)The information aggregation methods of linguistic type hesitant fuzzy decision making are studied. First, we discuss the linguistic term hesitant fuzzy multi-attribute decision making problems by combining probability theory with distance measures. The concept and operational laws of linguistic term hesitant fuzzy sets are introduced on the basis of the hesitant fuzzy linguistic term sets. By extending the probabilistic ordered weighted averaging distance operator to linguistic term hesitant fuzzy environment, we study the aggregation of linguistic term hesitant fuzzy information and the sequencing of alternatives. Second, based on the cloud model theory, we define the comprehensive cloud and the distance measure of linguistic hesitant fuzzy sets. In such a way, we discuss the generalization of the POWER operators in linguistic hesitant fuzzy environment and propose the related properties of linguistic hesitant fuzzy POWER aggregation operators and their applications to multi-attribute decision making.(3)The attribute reductions of hesitant fuzzy information systems are studied. By using the hesitant fuzzy similarity relation, we present the attribute reductions methods of complete hesitant fuzzy information systems. We also introduce an approach of comparing the hesitant fuzzy elements and linguistic term hesitant fuzzy elements. The discernibility matrix is established according to the limited dominance relation and the attribute reductions methods of incomplete hesitant fuzzy hybrid information systems are discussed.(4)The hesitant fuzzy risky multi-attribute decision making problems are studied. On the premise of giving reference points of attributes, the method for hesitant fuzzy multi-attribute decision making based on prospect theory is proposed by introducing the idea of hesitant fuzzy into the risky decision making problems with various natural state. Taking into account the risk preference of the decision maker, we introduce the applications of the improved prospect theory, i.e. TODIM method in hesitant fuzzy uncertain linguistic multi-attribute decision making problems when there are no given reference points.(5)The hesitant fuzzy multi-attribute decision making problems is studied in which the information about attribute weights is completely unknown. Using the approximate method and attribute reductions method of rough set theory, we propose the approach of ascertaining attribute weight for the unsure attribute weight of hesitant fuzzy multiple attribute decision making problems. Furthermore, based on the prioritized weighted average operator and prioritized ordered weighted average operator, we discuss the hesitant fuzzy multi-attribute decision making problems in which the attribute weights is unknown but there are priority relationships between attributes.(6)The hesitant fuzzy multi-attribute decision making method is applied to the evaluating of the college students’ comprehensive quality. According to the different purposes of the comprehensive quality evaluation, the index attribute values are aggregated and the comprehensive evaluation value and the rank of students are reached by utilizing the linguistic hesitant fuzzy POWER ordered weighted average operator and prioritized weighted average operator respectively.
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