Eigenvalue problems for nonlinear differential equations on a measure chain

Values of lambda are determined for which there exist positive solutions of the second order differential equation on a measure chain, u(Delta Delta)( t) + lambda a(t)f(u(sigma(t))) = 0, t is an element of [0, 1], satisfying e ither the conjugate boundary conditions u(0)= u(sigma(1)) = 0 or the right focal boundary conditions u(0)= u(Delta)(sigma(1)) = 0, where a and f are p ositive valued, and both lim(x --> 0)(+) f(x)/x and lim(x --> infinity) f(x )/x exist. (C) 2000 Academic Press.