Abstract
In this paper, some new mappings called relaxed η-α quasimonotone and a relaxed η-α properly quasimonotone operator are first introduced. The relationships between them are obtained. After this, the variational-like inequality problem and the relaxed Minty variational-like inequality problem are discussed by use of the proposed generalized monotone operators. Furthermore, we give the gap function of the two variational-like inequalities and two kinds of optimization problems. Finally, we point out that the two optimization problems are equivalent under some conditions.
Highlights
1 Introduction As we know, variational inequality theory plays an important role in many fields, such as optimal control, mechanics, economics, transportation equilibrium, engineering sciences
It is well known that the role of generalized monotonicity of the operator in the variational inequality problem corresponds to the role of generalized convexity of the objective function in the mathematical programming problem
In [ ], Fang and Huang introduced a new concept of relaxed η-α monotonicity and obtained the existence for variational-like inequalities with relaxed η-α monotone mappings in a reflexive Banach space
Summary
Variational inequality theory plays an important role in many fields, such as optimal control, mechanics, economics, transportation equilibrium, engineering sciences. With a more weakly monotone assumption, the existence for variational-like inequalities with a relaxed η-α quasimonotone mapping in a reflexive Banach space is discussed. In Section , the existence for variational-like inequalities with relaxed η-α quasimonotone mappings in a reflexive Banach space is established.
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