The nonlinear behavior of thin-walled, sinusoidal, slightly curved beams with pinned ends under lateral sinusoidal loading is considered numerically, based on Eringen's nonlocal elasticity theory and the Euler-Bernoulli beam theory. Geometrically nonlinear governing differential equations are solved by using the finite difference method and the displacement-controlled Newton-Raphson method. The influences of the nonlocal parameter, the initial curvature, and the span of the slightly curved beam on the buckling values are examined. Unlike the relevant previous studies, it is stated in this study that the values of these parameters must remain within certain limits in order to have purely snap-through buckling behavior. Moreover; the graphs of the deformed shapes of the slightly curved beams, the internal forces and the horizontal support reactions corresponding to the various phases of the equilibrium paths (including the prebuckling, buckling and postbuckling stages) are drawn for various values of the nonlocal parameter and the effect of the nonlocal parameter on these graphs are investigated in detail. The study contains insightful results and concluding remarks improving the comprehension of the snap-through phenomenon of the slightly curved beams under lateral loading.