Existence results for vector variational inequality problems on Hadamard manifolds
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Existence results for vector variational inequality problems on Hadamard manifolds Sheng-lan Chen1 Received: 1 July 2019 / Accepted: 24 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we introduce and study vector variational inequality problems (VVIP) on Hadamard manifolds. The concepts of C-pseudomonotone, v-hemicontinuous and v-coercive operators are given. Some existence results for VVIP are obtained with the assumptions of C-pseudomonotonicity and v-hemicontinuity. These new results extend some corresponding known results given in literatures. Keywords Vector variational inequality · C-pseudomonotone · v-hemicontinuous · v-coercive · Geodesic convex · Hadamard manifold
1 Introduction The vector variational inequalities was first introduced and studied by Giannessi [1] in a finite dimensional space. Since then, vector variational inequalities have been investigated extensively by many authors due to its potential applications in vector optimization problems and vector traffic equilibrium problems, as well as management science. One basic problem for vector variational inequality problem is the existence issue of solutions. A large number of important results concerned with the existence of solutions for (vector) variational inequalities have been appeared in the literature (see, for example [2–10] and the references therein). In the last few years, many important concepts and methods of optimization problems have been extended from linear spaces to Riemannian manifolds (in particular Hadamard manifolds), see [11–32] and the references therein. As we know, in general, a manifold does not have a linear structure. In this setting, the linear space is replaced by a Riemannian manifold, and the line segment is replaced by a geodesic [27,31]. It is noteworthy that the generalization of optimization problems from Euclidean spaces
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Sheng-lan Chen [email protected] Key Lab of Intelligent Analysis and Decision on Complex Systems, School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, People’s Republic of China
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to Riemannian manifolds has some important advantages, for example, non-convex minimization problems can be reduced to convex problems on Riemannian manifolds, and non-monotone vector fields can be transformed into monotone by choosing an appropriate Riemannian metric; see [18,22,28,31]. On the other hand, as mentioned by Nemeth ´ [26], there are numerous problems in applied fields that can be formulated as variational inequalities or boundary value problems on manifolds. Therefore, extension of some concepts, techniques as well as methods of the theory of variational inequalities and related topics from Euclidean spaces to Riemannian manifolds is natural. Recent interests of a number of researchers are focused on this topic (see [15,16,23,24,26] and the references therein). In particular, Nemeth ´ [26] first introduced and studied variational inequality problems (VIP) on Hadamard manifolds as follows: fi
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