Continuous selections of solution sets to Volterra integral inclusions in Banach spaces

We consider a nonlinear Volterra integral equation governed by an m-accretive operator and a multivalued perturbation in a separable Banach. The existence of a continuous selection for the corresponding solution map is proved. The case when the m-accretive operator in the integral inclusion depends on time is also discussed.Universidade de AveiroFCTFEDER POCTI/MAT/55524/200

Second order ancillary: A differential view from continuity

Second order approximate ancillaries have evolved as the primary ingredient for recent likelihood development in statistical inference. This uses quantile functions rather than the equivalent distribution functions, and the intrinsic ancillary contour is given explicitly as the plug-in estimate of the vector quantile function. The derivation uses a Taylor expansion of the full quantile function, and the linear term gives a tangent to the observed ancillary contour. For the scalar parameter case, there is a vector field that integrates to give the ancillary contours, but for the vector case, there are multiple vector fields and the Frobenius conditions for mutual consistency may not hold. We demonstrate, however, that the conditions hold in a restricted way and that this verifies the second order ancillary contours in moderate deviations. The methodology can generate an appropriate exact ancillary when such exists or an approximate ancillary for the numerical or Monte Carlo calculation of $p$-values and confidence quantiles. Examples are given, including nonlinear regression and several enigmatic examples from the literature.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ248 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Constant sign and nodal solutions for nonlinear elliptic equations with combined nonlinearities

We study a parametric nonlinear Dirichlet problem driven by a nonhomogeneous differential operator and with a reaction which is ”concave” (i.e., (p − 1)− sublinear) near zero and ”convex” (i.e., (p − 1)− superlinear) near ±1. Using variational methods combined with truncation and comparison techniques, we show that for all small values of the parameter > 0, the problem has at least five nontrivial smooth solutions (four of constant sign and the fifth nodal). In the Hilbert space case (p = 2), using Morse theory, we produce a sixth nontrivial smooth solution but we do not determine its sign

Periodic problems with a reaction of arbitrary growth

We consider nonlinear periodic equations driven by the scalar p-Laplacian and with a Carath eodory reaction which does not satisfy a global growth condition. Using truncation-perurbation techniques, variational methods and Morse theory, we prove a "three solutions theorem", providing sign information for all the solutions. In the semilinear case (p = 2), we produce a second nodal solution, for a total of four nontrivial solutions. We also cover problems which are resonant at zero

Positive solutions for parametric nonlinear periodic problems with competing nonlinearities

We consider a nonlinear periodic problem driven by a nonhomogeneous differential operator plus an indefinite potential and a reaction having the competing effects of concave and convex terms. For the superlinear (concave) term we do not employ the usual in such cases Ambrosetti-Rabinowitz condition. Using variational methods together with truncation, perturbation and comparison techniques, we prove a bifurcation-type theorem describing the set of positive solutions as the parameter varies

Three nontrivial solutions for nonlocal anisotropic inclusions under nonresonance

In this article, we study a pseudo-differential inclusion driven by a nonlocal anisotropic operator and a Clarke generalized subdifferential of a nonsmooth potential, which satisfies nonresonance conditions both at the origin and at infinity. We prove the existence of three nontrivial solutions: one positive, one negative and one of unknown sign, using variational methods based on nosmooth critical point theory, more precisely applying the second deformation theorem and spectral theory. Here, a nosmooth anisotropic version of the Hölder versus Sobolev minimizers relation play an important role

Semilinear neumann equations with indefinite and unbounded potential

We consider a semilinear Neumann problem with an indefinite and unbounded potential, and a Carathéodory reaction term. Under asymptotic conditions on the reaction which make the energy functional coercive, we prove multiplicity theorems producing three or four solutions with sign information on them. Our approach combines variational methods based on the critical point theory with suitable perturbation and truncation techniques, and with Morse theory

Nonlinear Robin problems with locally defined reaction

We consider a nonlinear Robin problem driven by a p− Laplacian. The reaction consistes of two terms. The first one is parametric and only locally defined, while the second one is (p − 1)- superlinear. Using cutt-off techniques together with critical point theory and critical groups, we show that for big values of the parameter λ > 0, the problem has at least three nontrivial solutions, all with sign information (positive, negative and nodal). In the semilinear case (p = 2), we produce a second nodal solution, for a total of four nontrivial solutions, all with sign information.publishe

Multiple solutions with sign information for (p, 2)−equations with asymmetric resonant reaction

We consider a nonlinear nonhomogeneous Dirichlet problem driven by the sum of a p−Laplacian and a Laplacian (a (p, 2)− equation). The reaction is the sum of two competing terms, a parametric (p − 1)−sublinear term and an asymmetric (p − 1)−linear perturbation which is resonant at −∞. Using variational methods together with truncations and comparison techniques and Morse theory (critical groups), we prove two multiplicity theorems which provide sign information for all the solutions.publishe