Fractional absolute value equations

We introduce the fractional absolute value equation (FAVE), which is an absolute value equation containing fractional terms of the unknown variables. For simplicity, we first consider a simple 1-dimensional FAVE and then generalize the results to problems of arbitrary dimension n. In this context, we investigate the relationship between the FAVE and the weighted generalization of the horizontal linear complementarity problem. Moreover, we provide conditions that ensure the solvability of the FAVE by characterizing the number and the sign pattern of the solutions in several situations.