Periodic solutions of the nonlinear telegraph equations with bounded nonlinearities

In this article, using the Leray-Schauder degree theory, we discuss existence, nonexistence and multiplicity for the periodic solutions of the nonlinear telegraph equation

utt−uxx+cut+Φ(u)=f(t,x)+s,     

On the approximation of periodic solutions of equations with bounded nonlinearities

We present a method for obtaining a closed-form approximation to harmonic periodic solutions of a class of ordinary differential equations containing a bounded, Lipschitzian nonlinearity and a sinusoidal forcing term. Our technique replaces the continuous nonlinearity with a suitable step-function nonlinearity and uses the Phase-Shift-Averaging Method to write the solution of the piecewise-linear problem in closed form.     

Homogenization with jumping nonlinearities

Summary Let a bounded regular open set of R ^N (N1),and {A ,0} be a sequence of second order, uniformly elliptic operators, which G-converges to A ₀.Let gC(R, R)be a nonlinear function with «jumping nonlinearities» (that is ).For h L ² () given, we obtain some results of convergence of the (eventual) solutions of the equation A u =g(u) +h.For instance, we study the case so-called «Ambrosetti-Prodi equation», that is when –< ^– < ₁ < ⁺ < ₂ where ₁ and ₂ are the firts and the second) eigenvalues of A ₀.

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Résumé Soient un ouvert borné régulier de R ^N (N1), et {A ,0} une suite d'opérateurs sur , du 2éme ordre, uniformément elliptiques et qui G-converge vers A ₀.Soit gC (R, R)une fonction demi-linéaire à l'infini (c'est à dire telle que ).Pour hL ² () donné, on obtient des résultats de convergence pour les solutions (éventuelles) de l'équation A u =g(u) +h.Par exemple, on étudie le cas de «l'équation d' Ambrosetti-Prodi», c'est à dire le cas – _– < ₁) ⁺ < ₂,où ₁ et ₂ sont les lère et 2éme valeurs propres de A ₀.

     

Blow-up with logarithmic nonlinearities

We study the asymptotic behaviour of nonnegative solutions of the nonlinear diffusion equation in the half-line with a nonlinear boundary condition,

     

Variational inequalities with strong nonlinearities

We study a connection between critical values and topological characteristics of non-smooth functionals. We establish analogs of theorems about regular interval and “nek”. We also find lower estimates of solutions to variational inequalities with odd potential operators.     

Dirichlet problem with indefinite nonlinearities

We consider the following nonlinear elliptic equation in a bounded domain with the Dirichlet boundary condition, and , g₁(u)u and g₂(u)u are positive for |u| > > 1. Some existence results are given for superlinear g₁ and g₂ via the Morse theory.Received: 16 Januray 2003, Accepted: 26 August 2003, Published online: 24 November 2003Mathematics Subject Classification (2000): 35J20, 35J25, 58E05Parts of the work were completed while the authors were visiting the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. The authors thank the hospitality of ICTP. Both authors are supported by NSFC, RFDP, MCME, the second author is also supported by the Foundation for University Key Teacher of the Ministry of Education of China and the 973 project of the Ministry of Science and Technology of China.     

Nonlinear equations with expansive nonlinearities

Summary We prove existence and multiplicity theorems for nonlinear equations at resonance with expansive nonlinearities.

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Riassunto Si provano teoremi di esistenza e molteplicità per equazioni nonlineari in risonanza con nonlinearità espansiva.

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These results were obtained while the second author was visiting the University of Ferrara through a grant of C.N.R.

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Supported by C.N.R., G.N.A.F.A.     

Semilinear elliptic equations with singular nonlinearities

We prove existence, regularity and nonexistence results for problems whose model is $$-\Delta u = \frac{f(x)}{u^{\gamma}}\quad {{\rm in}\,\Omega},$$ with zero Dirichlet conditions on the boundary of an open, bounded subset Ω of ${\mathbb{R}^{N}}$ . Here γ > 0 and f is a nonnegative function on Ω. Our results will depend on the summability of f in some Lebesgue spaces, and on the values of γ (which can be equal, larger or smaller than 1).     

On elliptic systems with discontinuous nonlinearities

In this paper we deal with elliptic systems with discontinuous nonlinearities. The discontinuous nonlinearities are assumed to satisfy quasimonotone conditions. We shall use the method of upper and lower solutions with fixed point theorems on increasing operators in ordered Banach spaces to show some existence theorems.     

Evolution differential equations with non-Lipschitz nonlinearities

We suggest a generalization of the Peano theorem (on the existence of a solution of the Cauchy problem for a differential system) to the case of an infinite-dimensional phase space.     

On wave equations with supercritical nonlinearities

We prove that the solution operators e_t (f, y){\cal e}_t (\phi , \psi ) for the nonlinear wave equations with supercritical nonlinearities are not Lipschitz mappings from a subset of the finite-energy space ([(H)\dot]¹ ?L_r+1) ×L₂(\dot {H}^1 \cap L_{\rho +1}) \times L_2 to [(H)\dot]^s_q￠\dot {H}^s_{q'} for t 1 0t\neq 0, and 0 ￡ s ￡ 1,0\leq s\leq 1, (n+1)/(1/2-1/q￠) = 1(n+1)/(1/2-1/q')= 1. This is in contrast to the subcritical case, where the corresponding operators are Lipschitz mappings ([3], [6]). Here e_t(f, y)=u(·, t){\cal e}_t(\phi , \psi )=u(\cdot , t), where u is a solution of {

Estimation and control with cubic nonlinearities

The nonlinearities in a dynamic system and its measurement equations are assumed to be cubic and small, i.e., all proportional to a single scalar small parameter . The optimal digital nonlinear feedback control law is carried through the first power of , taking into account the non-Gaussian character of the state conditional distribution. The optimal law involves cubic and linear terms in the state estimate, as well as higher moments of the state conditional distribution.