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We introduce two new concepts of weakly relaxed η-α monotone mappings and weakly relaxed η-α semimonotone mappings. Using the KKM technique, the existence of solutions for variational-like problems with weakly relaxed η-α monotone mapping in reflexive Banach spaces is established. Also, we obtain the existence of solution for variational-like problems with weakly relaxed η-α semimonotone mappings in arbitrary Banach spaces by using the Kakutani-Fan-Glicksberg fixed-point theorem

abstract and applied analysis

On the Solution Existence of Variational-Like Inequalities Problems for Weakly Relaxed<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>η</mml:mi><mml:mtext>−</mml:mtext><mml:mi>α</mml:mi></mml:mrow></mml:math>Monotone Mapping

On the Solution Existence of Variational-Like Inequalities Problems for Weakly Relaxedη−αMonotone Mapping