In this paper we study the periodic boundary value problem associated with a first order ODE of the form x' + g(t, x) = s where s is a real parameter and g is a continuous function, T-periodic in the variable t. We prove an Ambrosetti-Prodi type result in which the classical uniformity condition on g(t, x) at infinity is considerably relaxed. The Carathéodory case is also discussed.
We are concerned with multiplicity and bifurcation results for solutions of nonlinear second order diff erential equations with general linear part and periodic boundary conditions. We impose asymptotic conditions on the nonlinearity and let the parameter vary. We then proceed to establish a priori estimates and prove multiplicity results (for large-norm solutions) when the parameter belongs to a (nontrivial) continuum of real numbers. Our results extend and complement those in the literature. The proofs are based on degree theory, continuation methods, and bifurcation from infinity techniques.
This paper is devoted, with my great esteem, to Jean Mawhin. Jean Mawhin, who is for me a great teacher and a very good friend, is a fundamental reference for the research in nonlinear differential problems dealt both with topological and variational methods. Here, owing to this occasion in honor of Jean Mawhin, Dirichlet problems depending on a parameter are investigated, ensuring the existence of non-zero solutions without requiring asymptotic conditions neither at zero nor at infinity on the nonlinear term which, in addition, is not forced by subcritical or critical growth. The approach is based on a combination of variational and topological tools that in turn are developed by starting from a fundamental estimate.
The purpose of this paper is to consider boundary value problems for second order ordinary diff erential equations where the solutions sought are subject to a host of linear constraints (such as multipoint constraints) and to present a unifying framework for studying such. We show how Leray-Schauder continuation techniques may be used to obtain existence results for nontrivial solutions of a variety of nonlinear second order diff erential equations. A typical example may be found in studies of the four-point boundary value problem for the diff erential equation y’’(t)+a(t)f(y(t)) = 0 on [0, 1], where the values of y at 0 and 1 are each some multiple of y(t) at two interior points of (0, 1). The techniques most often used in such studies have their origins in fixed point theory. By embedding such problems into parameter dependent ones, we show that detailed information may be obtained via global bifurcation theory. Of course, such techniques, as they are consequences of properties of the topological degree, are similar in nature.
We are concerned with a Dirichlet system, involving the mean curvature operator in Minkowski space
M(w) = div (∇w / 1−|∇w|2)
in a ball in RN. Using topological degree arguments, critical point theory and lower and upper solutions method, we obtain non existence, existence and multiplicity of radial, positive solutions. The examples we provide involve Lane-Emden type nonlinearities in both sublinear and superlinear cases.
We review recent results on the existence of weak 2π- periodic solutions in time and space for a class of semilinear wave equations with non-monotone nonlinearity. Similar results exist for Dirichlet-periodic boundary conditions but, for the sake of clarity, we exclude them in this presentation.