<p>This paper considers the Tikhonov regularized regularization for infeasible absolute value equations (AVE), whose objective function is the sum of a non-differentiable fractional term and a quadratic term. We first reformulate this problem into a well-conditioned single-ratio fractional programming problem and apply the proximal gradient method (PGM) to solve it. Notably, we demonstrate that the objective function is naturally a Kurdyka-Łojasiewicz (KŁ) function, allowing us to establish global convergence of PGM without additional assumptions, such as the function being Lipschitz continuous or the gradient being locally Lipschitz continuous, that are indispensable in the existing literature. In addition, when the proximity parameter is adaptively chosen based on the iteration, PGM with extrapolation can also be used to solve this problem. To address the computational challenges posed by the absolute value in the proximal operator, we employ the smoothing technique to derive approximate local solutions. We conclude with numerical experiments that illustrate the effectiveness of the proposed algorithms.</p>