New existence results for some periodic and Neumann-Steklov boundary value problems with ϕ-Laplacian (original) (raw)

Periodic solutions for some nonautonomous semilinear boundary evolution equations

Nonlinear Analysis: Theory, Methods & Applications, 2009

In this work we study the Massera problem for the existence of a periodic mild solution of a class of nonautonomous semilinear boundary evolution equations    x (t) = A m (t)x(t) + f (t, x(t)) , t ≥ 0, L(t)x(t) = Φ(t)x(t) + g (t, x(t)) , t ≥ 0, x(0) = x 0. (0.1) First, we prove the existence of a periodic solution for nonhomogeneous boundary evolution equations under the existence of a bounded solution on the right half real line. Next, by using a fixed point theorem, we investigate the existence of periodic solutions in the semilinear case. We end with an application to a periodic heat equation with semilinear boundary conditions.

On the Solutions of a Singular Nonlinear Periodic Boundary Value Problem

2001

In this paper, we consider the following singular nonlinear problem 1 A (Au) − qu = −f (• , u) a.e. on (0,1), u(0) = u(1), Au (0) = Au (1), (P) where A is a positive continuous function on (0,1), q is a nonnegative measurable function on [0,1] and f is a nonnegative regular function on (0, 1) × (0, ∞). We suppose that 1 0 dt/A(t) < ∞ and 0 < 1 0 A(t)q(t) dt < ∞. Then we prove the existence and the uniqueness of a positive solution of this problem (P). Our approach is based on the use of the Green's function and the Schauder's fixed point theorem.

Periodic solutions of quasi-differential equations

1996

Existence principles and theorems are established for the nonlinear problem Luf(t, u) where Lu--(pu')'+ hu is a quasi-differential operator and f is a Carathodory function. We prove a maximum principle for the operator L and then we show the validity of the upper and lower solution method as well as the monotone iterative technique.

Existence of positive periodic solutions to nonlinear second order differential equations

Applied Mathematics Letters, 2005

In this paper, we discuss the existence of positive periodic solutions to the nonlinear differential equation u (t) + a(t)u(t) = f (t, u(t)), t ∈ R, where a : R → [0, +∞) is an ω-periodic continuous function with a(t) ≡ 0, f : R × [0, +∞) → [0, +∞) is continuous and f (•, u) : R → [0, +∞) is also an ω-periodic function for each u ∈ [0, +∞). Using the fixed point index theory in a cone, we get an essential existence result because of its involving the first positive eigenvalue of the linear equation with regard to the above equation.

Existence of Solutions for a Class of Nonvariational Quasilinear Periodic Problems

Set-Valued Analysis, 2008

We consider a nonlinear nonvariational periodic problem with a nonsmooth potential. Using the spectrum of the asymptotic (as |x| → ∞) differential operator and degree theoretic methods based on the degree map for multivalued perturbations of (S) + operators, we establish the existence of a nontrivial smooth solution.