Electronic Journal of Differential Equations https://digital.library.txstate.edu/handle/10877/86 2020-05-30T00:21:40Z Nontrivial Periodic Solutions of Asymptotically Linear Hamiltonian Systems https://digital.library.txstate.edu/handle/10877/9330 Nontrivial Periodic Solutions of Asymptotically Linear Hamiltonian Systems Fei, Guihua We study the existence of periodic solutions for some asymptotically linear Hamiltonian systems. By using the Saddle Point Theorem and Conley index theory, we obtain new results under asymptotically linear conditions. 2001-11-19T00:00:00Z Stability Properties of Positive Solutions to Partial Differential Equations with Delay https://digital.library.txstate.edu/handle/10877/9329 Stability Properties of Positive Solutions to Partial Differential Equations with Delay Farkas, Gyula; Simon, Peter L We investigate the stability of positive stationary solutions of semilinear initial-boundary value problems with delay and convex or concave nonlinearity. If the nonlinearity is monotone, then in the convex case <i>f</i>(0) ≤ 0 implies instability and in the concave case <i>f</i>(0) ≥ 0 implies stability. Special cases are shown where the monotonicity assumption can be weakened or omitted. 2001-10-08T00:00:00Z Global Bifurcation Result for the p-Biharmonic Operator https://digital.library.txstate.edu/handle/10877/9328 Global Bifurcation Result for the p-Biharmonic Operator Drabek, Pavel; Otani, Mitsuharu We prove that the nonlinear eigenvalue problem for the p-biharmonic operator with p > 1, and Ω a bounded domain in ℝN with smooth boundary, has principal positive eigenvalue λ1 which is simple and isolated. The corresponding eigenfunction is positive in Ω and satisfies ∂u/∂n < 0 on ∂Ω, ∆u1 < 0 in Ω. We also prove that (λ1, 0) is the point of global bifurcation for associated nonhomogeneous problem. In the case <i>N</i> = 1 we give a description of all eigenvalues and associated eigenfunctions. Every such an eigenvalue is then the point of global bifurcation. 2001-07-03T00:00:00Z Asymptotic Behavior of Solutions to Wave Equations with a Memory Condition at the Boundary https://digital.library.txstate.edu/handle/10877/9327 Asymptotic Behavior of Solutions to Wave Equations with a Memory Condition at the Boundary Santos, Mauro de Lima In this paper, we study the stability of solutions for wave equations whose boundary condition includes a integral that represents the memory effect. We show that the dissipation is strong enough to produce exponential decay of the solution, provided the relaxation function also decays exponentially. When the relaxation function decays polynomially, we show that the solution decays polynomially and with the same rate. 2001-11-26T00:00:00Z