Functional inequalities motivated by the Lax-Milgram theorem

By adopting some ideas of the theory of functional inequalities we obtain a modification of the Lax-Milgram theorem. We drop linearity assumptions by assuming that respective inequalities hold true and we impose only a mild regularity assumption. Moreover, we do not assume the coercivity, however, as a consequence, our assertions are adequately weaker. One of our tools is a selection theorem for multifunctions.