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Vector equilibrium problems with new types of generalized monotonicity

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In this paper, we introduce the concept of generalized relaxed α-pseudomonotonicity for vector valued bi-functions. By using the KKM technique, we obtain some substantial results of the vector equilibrium problems with generalized relaxed α-pseudomonotonicity assumptions in reflexive Banach spaces. Several examples are provided to illustrate our investigations. | Yugoslav Journal on Operations Research 23(2013) Number 2, 213–220 DOI: 10.2298/YJOR130107024M VECTOR EQUILIBRIUM PROBLEMS WITH NEW TYPES OF GENERALIZED MONOTONICITY Nihar Kumar MAHATO Department of Mathematics, Indian Institute of Technology Kharagpur Kharagpur-721302, India. nihariitkgp@gmail.com Chandal NAHAK1 Department of Mathematics, Indian Institute of Technology Kharagpur Kharagpur-721302, India. cnahak@maths.iitkgp.ernet.in Received: Junuary 2013 / Accepted: May 2013 Abstract: In this paper, we introduce the concept of generalized relaxed αpseudomonotonicity for vector valued bi-functions. By using the KKM technique, we obtain some substantial results of the vector equilibrium problems with generalized relaxed α-pseudomonotonicity assumptions in reflexive Banach spaces. Several examples are provided to illustrate our investigations. Keywords: Vector equilibrium problem, generalized relaxed α-pseudomonotonicity, KKM mapping. MSC: 49J40; 47H10; 91B52; 90C30 1 Corresponding author. 213 214 N.K. Mahato and C. Nahak / Vector Equilibrium Problems 1. INTRODUCTION Equilibrium problems in the sense of Blum and Oettli [1] has vast applications in the several branches of pure and applied sciences. The equilibrium problem includes many mathematical problems as its particular cases, e.g., mathematical programming problems, complementary problems, variational inequality problems, and fixed point problems. Inspired by the notion of vector variational inequality problem introduced and studied by Giannessi [2], Chen and Yang [3], the equilibrium problem has been extended to vector equilibrium problem. The vector equilibrium problem contains vector optimization problems, vector variational inequality problems, and vector complementarity problems as a special case. Let Y be a real Banach space and C be a nonempty subset of Y . C is called a cone if λC ⊂ C, for any λ ≥ 0. Further, the cone C is called convex cone if C + C ⊂ C. C is pointed cone if C is cone and C ∩ .