Pierpaolo Omari

Periodic solutions of the Liénard equation with one-sided growth restrictions

In this article we are concerned with the problem of the existence of periodic solutions to the periodically forced scalar Lienard equation x” + f(x) x’ + g(x) = e( t ) (’ = d/dt), (1) where ,f, g, e: R + R are continuous functions and e(.) is periodic. We are looking for solutions of (1) having the same period of the forcing term e(.). Throughout the paper, we assume, without loss of generality (see e.g. [22; 36, p. 693]), that

Periodic solutions of a second order ordinary differential equation: A necessary and sufficient condition for nonresonance

The nonlinearity g in (1.1) is a continuous function from R to R and the forcing term h is taken in L”(O,27r). Nonresonance means that (1.1) admits at least one solution x for any given h. Integrating Eq. (1.1) over a period, one immediately sees that a necessary condition for nonresonance is that the function g be unbounded from above and from below on R. It will appear later (cf. (1.6)) that this unboundedness can be looked at as a condition relating the behaviour at infinity of the nonlinearity g with respect to the first eigenvalue 1, = 0 of the associated linear problem: -xf’= Ax in [0, 27r], x(0) = x(27r), x’(0) = x1(271). (1.2)

Multiple positive solutions of a one-dimensional prescribed mean curvature problem

We discuss existence, non-existence and multiplicity of positive solutions of the Dirichlet problem for the one-dimensional prescribed curvature equation [GRAPHICS] in connection with the changes of concavity of the function f. The proofs are based on an upper and lower solution method, we specifically develop for this problem, combined with a careful analysis of the time-map associated with some related autonomous equations.

Nonlinear perturbations of differential operators with nontrivial kernel and applications to third-order periodic boundary value problems

This paper deals with the solvability of the nonlinear operator equations in normed spaces Yx = EGx + J where dp is a linear map with possible nontrivial kernel. Applications are given to the existence of periodic solutions for the thirdorder scalar differential equation x”’ + ax” + bx’ + cx + g(t, x) = p(t) under various conditions on the interaction of g(t, x)/x with spectral contigurations of a, b, and c.

Positive solutions of an indefinite prescribed mean curvature problem on a general domain

Abstract The existence of positive solutions is proved for the prescribed mean curvature problem where Ω ⊂ℝN is a bounded smooth domain, not necessarily radially symmetric. We assume that ∫0u f(x, s) ds is locally subquadratic at 0, ∫0u g(x, s) ds is superquadratic at 0 and λ > 0 is sufficiently small. A multiplicity result is also obtained, when ∫0u f(x, s) ds has an oscillatory behaviour near 0. We allow f and g to change sign in any neighbourhood of 0.

Positive Solutions of the Dirichlet Problem for the One-dimensional Minkowski-Curvature Equation

Abstract We discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential equation . Depending on the behaviour of f = f (t, s) near s = 0, we prove the existence of either one, or two, or three, or infinitely many positive solutions. In general, the positivity of f is not required. All results are obtained by reduction to an equivalent non-singular problem to which variational or topological methods apply in a classical fashion.

Existence and localization of solutions of second order elliptic problems using lower and upper solutions in the reversed order

where Ω is a bounded domain in R ,L is a linear second order elliptic operator for which the maximum principle holds, B is a linear first order boundary operator and f is a nonlinear Caratheodory function. We are concerned with the solvability of (1.1) in presence of lower and upper solutions. A classical basic result in this context says that if α is a lower solution and β is an upper solution satisfying

A necessary and sufficient condition of nonresonance for a semilinear Neumann problem

We consider the Neumann problem −Δu=g(u)+h(x in Ω, ∂u/∂=0 on bdry Ω. Assuming some growth restriction on the nonlinearity g, we prove that a necessary and sufficient condition for the existence of a solution for every given h∈L ∞(Ω) is that g be unbounded from above and from below on R

A note on nonlinear oscillations at resonance

AbstractExistence of 2π-periodic solutions to the equation