1,721,030 research outputs found
The invariance of domain theorem for compact perturbations of nonlinear Fredholm maps of index zero.
No full textOn existence and uniqueness of solutions for ordinary differential equations with nonlinear boundary conditions
No full textHeteroclinic solutions of boundary value problems on the real line involving singular Φ-Laplacian operators
No full textAbstractWe discuss the solvability of the following strongly nonlinear BVP:{(a(x(t))Φ(x′(t)))′=f(t,x(t),x′(t)),t∈R,x(−∞)=α,x(+∞)=β where α<β, Φ:(−r,r)→R is a general increasing homeomorphism with bounded domain (singular Φ-Laplacian), a is a positive, continuous function and f is a Carathéodory nonlinear function. We give conditions for the existence and non-existence of heteroclinic solutions in terms of the behavior of y↦f(t,x,y) and y↦Φ(y) as y→0, and of t↦f(t,x,y) as |t|→+∞. Our approach is based on fixed point techniques suitably combined to the method of upper and lower solutions
On fourth order retarded equations with functional boundary conditions: A unified approach
No full textBy means of a recent Birkhoff-Kellogg type theorem, we discuss the solvability of a fairly general class of parameter-dependent fourth order retarded differential equations subject to functional boundary conditions. We seek solutions within a translate cone of nonnegative functions. We provide an example to illustrate our theoretical results
On the solvability of parameter-dependent elliptic functional BVPs on annular-like domains
Full text linkWe investigate the existence of nontrivial solutions of parameter-dependent elliptic equations with deviated argument in annular-like domains in Rn, with n ≥ 2, subject to functional boundary conditions. In particular we consider a boundary value problem that may be used to model heat-flow problems. We obtain an existence result by means of topological methods; in particular, we make use of a recent variant in affine cones of the celebrated Birkhoff–Kellogg theorem. Using an ODE argument, we illustrate in an example the applicability of our theoretical result
Mel’nikov methods and homoclinic orbits in discontinuous systems
No full textWe consider a discontinuous system exhibiting a, possibly non-smooth,
homoclinic trajectory. We assume that the critical point lies on
the discontinuity level. We study the persistence of such a
trajectory when the system is subject to a smooth
non-autonomous perturbation.
We use a Mel'nikov type approach and we introduce conditions
which enable us to reformulate the problem in the setting of smooth systems so
that we can follow the outline of the classical theory
Carathéodory periodic perturbations of degenerate systems
Full text linkWe study the structure of the set of harmonic solutions to T periodically perturbed coupled differential equations on differentiable manifolds, where the perturbation is allowed to be of Carathéodory-type regularity. Employing degree-theoretic methods, we prove the existence of a noncompact connected set of nontrivial T-periodic solutions that, in a sense, emanates from the set of zeros of the unperturbed vector field. The latter is assumed to be “degenerate”: Meaning that, contrary to the usual assumptions on the leading vector field, it is not required to be either trivial nor to have a compact set of zeros. In fact, known results in the “nondegenerate” case can be recovered from our ones. We also provide some illustrating examples of Liénard-and φ-Laplacian-type perturbed equation
An infinite dimensional version of the Kronecker index and its relation with the Leray–Schauder degree
Full text linkLet f be a compact vector field of class C1 on a real Hilbert space H. Denote by B the open unit ball of H and by S = ∂B the unit sphere. Given a point q ∉ f(S), consider the self-map of S defined by (Equation Presented) If H is finite dimensional, then S is an orientable, connected, compact differentiable manifold. Therefore, the Brouwer degree, degBr(fq∂) is well defined, no matter what orientation of S is chosen, assuming it is the same for S as domain and codomain of fq∂. This degree may be considered as a modern reformulation of the Kronecker index of the map fq∂. Let degBr(f,B,q) denote the Brouwer degree of f on B with target q. It is known that one has the equality (Equation Presented) Our purpose is an extension of this formula to the infinite dimensional context. Namely, we will prove that (Equation Presented) where degLS(∙) denotes the Leray–Schauder degree and degbf(∙) is the degree earlier introduced by M. Furi and the first author, which extends, to the infinite dimensional case, the Brouwer degree and the Kronecker index. In other words, here, we extend to the Leray–Schauder degree the boundary dependence property which holds for the Brouwer degree in the finite dimensional context
Nontrivial Solutions of a Parameter-Dependent Heat-Flow Problem with Deviated Arguments
Full text linkBy means of a recent Birkhoff–Kellogg-type theorem set in affine cones, we discuss the solvability of a parameter-dependent thermostat problem subject to deviated arguments. We illustrate in a specific example the constants that occur in our theory
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