In this paper, we provide the proof of the stability of an extremum-seeking control scheme enforced by a sliding mode control strategy with periodic search signals. The stability conditions are given for a general plant modeled as a static nonlinear map possibly cascaded with a linear dynamic block. Under the conditions given, either the input cost map or output cost map converges to its maximum or minimum point stably within a finite time interval. Extremum-seeking structures are studied and simulations are done under both sinusoidal and sawtooth periodic search signals. To compensate the inherit limitation of the steady state oscillation, a possible online estimation method is studied which provides a good result for a large set of cost functions.