1,720,997 research outputs found
Prescribing the nodal behaviour of periodic solutions of a superlinear equation with indefinite weight
No full textThis paper deals with the existence of periodic solutions to the differential equation x'' + q(t)g(x) = 0. Here g is Lipschitz, xg(x) > 0 for all non vanishing x, g has superlinear growth at infinity and q is continuous and is allowed to change sign finitely many times. We prove that there are two periodic solutions with a precise number of zeros in each interval of positivity of q and that, moreover, for each interval of negativity, one can fix a priori whether the solution will have exactly one zero and be strictly monotone or will have no zeros and exactly one zero of the derivative. The techniques are based on the study of the Poincaré map and a careful phase plane analysis. Generalizations are discussed in order to treat more gereal Floquet-type boundary conditions
Boundary Value Problems for Second Order Differential Equations with Superlinear Terms: a Topological Approach
No full textPeriodic points and chaotic-like dynamics of planar maps associated to the nonlinear Hill's equations with indefinite weight
No full textInfinitely many solutions for a Floquet-type BVP with superlinearity indefinite in sign
No full textBy the application of a technique developed by G. J. Butler we find infinitely many solutions of a Floquet-type BVP for the equation x''+q(t)g(x)=0, where q is a weight function that is allowed to change sign, g is is superlinear and such that g(x)x>0 for all non-zero x. The boundary condition is (x(b),x'(b))=L(x(a),x'(a)), where L is a continuous, positively homogeneous, and nondegenerate map. At first we apply the main result to obtain solutions with a prescribed large number of zeros when L is the rotation of a fixed angle l; second, we find infinitely many subharmonic solutions of any order and, again, solutions with a prescribed large number of zeros for the periodic problem associated to the equation x''+cx'+q(t)g(x)=0, with q and g as above and a constant c
Complex Dynamics in a ODE Model Related to Phase Transition
Full text linkMotivated by some recent studies on the Allen–Cahn phase transition model with a periodic nonautonomous term, we prove the existence of complex dynamics for the second order equation −x ̈+(1+ε−1A(t))G′(x)=0, where A(t) is a nonnegative T-periodic function and ε>0 is sufficiently small. More precisely, we find a full symbolic dynamics made by solutions which oscillate between any two different strict local minima x0 and x1 of G(x). Such solutions stay close to x0 or x1 in some fixed intervals, according to any prescribed coin tossing sequence. For convenience in the exposition we consider (without loss of generality) the case x0=0 and x1=1
Analysis of a lagoon ecological model with anoxic crises and impulsive harvesting
Full text linkWe analyze mathematically a system of impulsive nonlinear parabolic equations that model a shallow lagoon subject to anoxic crises and two types of impulsive harvesting. The main focus is on the existence and properties of periodic solutions. In particular we give conditions that ensure the existence of such solutions and examine the effect of harvesting on the occurrence of anoxic crises. Our approach is based on estimates of the principal eigenvalue of associated linear problems, and on results from Nonlinear Functional Analysis. In particular, we obtain explicit criteria that involve the integrals of coefficients rather than maxima and minima. This is significant due to the large seasonal variations in the coefficient values
Periodic solutions of asymptotically linear second order equations with changing sign weight
No full textIn this paper we study the ordinary differential equation x'' + q(t)g(x) = 0, where g is a locally Lipschitz continuous function that satisfies g(x)x > 0 for all non zero x and is asymptotically linear, while q is a continuous, π-periodic and changing sign weight. By the application of a recent result on the existence and multiplicity of fixed points of planar maps, we give conditions on q and on the behavior of the ratio g(x)/x near zero and near infinity in order to obtain multiple periodic solutions with the prescribed number of zeros in the intervals of positivity and negativity of q, as well as multiple subharmonics of any order and uncountably many bounded solutions
A topological approach to superlinear indefinite boundary value problems
No full textWe obtain the existence of infinitely many solutions with prescribed nodal properties for some boundary value problems associated to the second order scalar equation x''+q(t)g(x)=0, where g(x) has superlinear growth at infinity and q(t) changes sign
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