Singular periodic problem for nonlinear ordinary differential equations with ϕ-Laplacian
Texas State University-San Marcos, Department of Mathematics
We investigate the singular periodic boundary-value problem with ϕ-Laplacian, (ϕ(u′))′ = ƒ(t, u, u′), u(0) = u(T), u′(0) = u′(T), where ϕ is an increasing homeomorphism, ϕ(ℝ) = ℝ, ϕ(0) = 0. We assume that ƒ satisfies the Carathéodory conditions on each set [α, b] ⊂ (0, T) and ƒ does not satisfy the Carathéodory conditions on [0, T] x ℝ2, which means that ƒ has time singularities at t = 0, t = T. We provide sufficient conditions for the existence of solutions to the above problem belonging to C1 [0, T]. We also find conditions which guarantee the existence of a sign-changing solution to the problem.
Singular periodic problem, ϕ-Laplacian, Smooth sign-changing solutions, Lower and upper functions
Polásek, V., & Rachunková, I. (2006). Singular periodic problem for nonlinear ordinary differential equations with ϕ-Laplacian. <i>Electronic Journal of Differential Equations, 2006</i>(27), pp. 1-12.