60,070 research outputs found
Pairs of positive solutions for the periodic scalar p-Laplacian
No full textAbstract. We study a nonlinear periodic problem driven by the scalar p-
Laplacian and having a nonsmooth potential (hemivariational inequality).
Using a combination of variational techniques and degree-theoretic methods
based on a degree map for certain multivalued perturbations of (S)+-
operators, we establish the existence of two positive solutions
EXISTENCE, NONEXISTENCE AND MULTIPLICITY OF POSITIVE SOLUTIONS FOR PARAMETRIC NONLINEAR ELLIPTIC EQUATIONS
No full textWe consider a parametric nonlinear elliptic equation driven by the Dirichlet p-Laplacian. We study the existence, nonexistence and multiplicity of positive solutions as the parameter λ varies in R_0^+ and the potential exhibits a p-superlinear growth, without satisfying the usual in such cases Ambrosetti–Rabinowitz condition. We prove a bifurcation-type result when the reaction has (p - 1)-sublinear terms near zero (problem with concave and convex nonlinearities). We show that a similar bifurcation-type result is also true, if near zero the right hand side is (p - 1)-linear
Positive solutions for generalized nonlinear logistic equations of superdiffusive type
No full textWe consider a generalized version of the p-logistic equation. Using variational methods based on the critical point theory and truncation techniques, we prove a bifurcation-type theorem for the equation. So, we show that there is a critical value lambda*> 0 of the parameter lambda> 0 such that the following holds: if lambda> lambda*, then the problem has two positive solutions; if lambda= lambda*, then there is a positive solution; and finally, if 0 < lambda< lambda*, then there are no positive solutions
Existence of two solutions for quasilinear periodic differential equations with discontinuities
No full textsummary:In this paper we examine a quasilinear periodic problem driven by the one- dimensional -Laplacian and with discontinuous forcing term . By filling in the gaps at the discontinuity points of we pass to a multivalued periodic problem. For this second order nonlinear periodic differential inclusion, using variational arguments, techniques from the theory of nonlinear operators of monotone type and the method of upper and lower solutions, we prove the existence of at least two non trivial solutions, one positive, the other negative
On a nonstationary discrete time infinite horizon growth model with uncertainty
No full textsummary:In this paper we examine a nonstationary discrete time, infinite horizon growth model with uncertainty. Under very general hypotheses on the data of the model, we establish the existence of an optimal program and we show that the values of the finite horizon problems tend to that of the infinite horizon as the end of the planning period approaches infinity. Finally we derive a transversality condition for optimality which does not involve dual variables (prices)
Existence and Relaxation for Finite-Dimensional Optimal Control Problems Driven by Maximal Monotone Operators
No full textIn this paper we study the optimal control of a class of nonlinear finite-dimensional optimal control problems driven by a maximal monotone operator which is not necessarily everywhere defined. So our model problem incorporates systems monitored by variational inequalities. First we prove an existence theorem using the reduction method of Berkovitz and Cesari. This requires a convexity hypothesis. When this convexity condition is not satisfied, we have to pass to an augmented, convexified problem known as the "relaxed problem". We present four relaxation methods. The first uses Young measures, the second uses multi-valued dynamics, the third is based on Caratheodory's theorem for convex sets in R-N and the fourth uses lower semicontinuity arguments and Gamma-limits. We show that they are equivalent and admissible, which roughly speaking means that the corresponding relaxed problem is in a sense the "closure" of the original one
On the solution set of nonconvex subdifferential evolution inclusions
No full textsummary:We consider nonlinear systems with a priori feedback. We establish the existence of admissible pairs and then we show that the Lagrange optimal control problem admits an optimal pair. As application we work out in detail two examples of optimal control problems for nonlinear parabolic partial differential equations
Periodic problems and problems with discontinuities for nonlinear parabolic equations
No full textsummary:In this paper we study nonlinear parabolic equations using the method of upper and lower solutions. Using truncation and penalization techniques and results from the theory of operators of monotone type, we prove the existence of a periodic solution between an upper and a lower solution. Then with some monotonicity conditions we prove the existence of extremal solutions in the order interval defined by an upper and a lower solution. Finally we consider problems with discontinuities and we show that their solution set is a compact -set in
Existence, nonexistence and multiplicity of positive solutions for nonlinear, nonhomogeneous Neumann problems
No full textWe consider a nonlinear parametric Neumann problem driven by a nonhomogeneous differential operator and a strictly sublinear reaction term. We prove a bifurcation -type result establishing the existence of a critical parameter value such that for all the problem has at least two positive solutions, for it has at least one positive solution and for there are no positive solutions. Also, for we show that the problem has a smallest positive solution and we investigate the continuity and monotonicity properties of the map
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