Global structure of positive and sign-changing periodic solutions for the equations with Minkowski-curvature operator

We show the existence of unbounded connected components of 2 pi-periodic positive solutions for the equations with one-dimensional Minkowski-curvature operator-(u'/root 1-u'(2))' = lambda a(x)f (u,u'), x is an element of R, where lambda > 0 is a parameter, a is an element of C (R, R) is a 2 pi-periodic sign-changing function with integral(2 pi)(0) (0) a(x)dx < 0, f is an element of C(RxR, R) satisfies a generalized regular-oscillation condition. Moreover, for the special case that f does not contain derivative term, we also investigate the global structure of 2 pi-periodic odd/even sign-changing solutions set undersome parity conditions. The proof of our main results are based upon bifurcation techniques.