On a general symptotic problem associated with Leray-Lions operators

Firstly, we prove a pointwise comparison result for the suitable symmetrized problem that depends ona small positive parameter λ. Then, by these results and by thze Schwarz symmetrization, we obtain some asymtotic relationship between the solutions uε of a general ε and a sequence of real numbers λε. Finally, it is shown an application of the preceding results to getting a priori estimates in the homogenization theory

Fite-Wintner-Leighton-Type Oscillation Criteria for Second-Order Differential Equations with Nonlinear Damping

Some new oscillation criteria for a general class of second-order differential equations with nonlinear damping are shown. Except some general structural assumptions on the coefficients and nonlinear terms, we additionally assume only one sufficient condition (of Fite-Wintner-Leighton type). It is different compared to many early published papers which use rather complex sufficient conditions. Our method contains three items: classic Riccati transformations, a pointwise comparison principle, and a blow-up principle for sub- and supersolutions of a class of the generalized Riccati differential equations associated to any nonoscillatory solution of the main equation

Fractal Oscillations of Chirp Functions and Applications to Second-Order Linear Differential Equations

We derive some simple sufficient conditions on the amplitude , the phase and the instantaneous frequency such that the so-called chirp function is fractal oscillatory near a point , where and is a periodic function on . It means that oscillates near , and its graph is a fractal curve in such that its box-counting dimension equals a prescribed real number and the -dimensional upper and lower Minkowski contents of are strictly positive and finite. It numerically determines the order of concentration of oscillations of near . Next, we give some applications of the main results to the fractal oscillations of solutions of linear differential equations which are generated by the chirp functions taken as the fundamental system of all solutions

Fractal oscillations for a class of second order linear differential equations of Euler type

AbstractThe s-dimensional fractal oscillations for continuous and smooth functions defined on an open bounded interval are introduced and studied. The main purpose of the paper is to establish this kind of oscillations for solutions of a class of second order linear differential equations of Euler type. Next, it will be shown that the dimensional number s only depends on a positive real parameter α appearing in a singular term of the main equation. It continues some recent results on the rectifiable and unrectifiable oscillations given in Pašić [M. Pašić, Rectifiable and unrectifiable oscillations for a class of second-order linear differential equations of Euler type, J. Math. Anal. Appl. 335 (2007) 724–738] and Wong [J.S.W. Wong, On rectifiable oscillation of Euler type second order linear differential equations, Electron. J. Qual. Theory Differ. Equ. 20 (2007) 1–12]

Parametrically Excited Oscillations of Second-Order Functional Differential Equations and Application to Duffing Equations with Time Delay Feedback

We study oscillatory behaviour of a large class of second-order functional differential equations with three freedom real nonnegative parameters. According to a new oscillation criterion, we show that if at least one of these three parameters is large enough, then the main equation must be oscillatory. As an application, we study a class of Duffing type quasilinear equations with nonlinear time delayed feedback and their oscillations excited by the control gain parameter or amplitude of forcing term. Finally, some open questions and comments are given for the purpose of further study on this topic

New Oscillation Criteria for Second-Order Forced Quasilinear Functional Differential Equations

We establish some new interval oscillation criteria for a general class of second-order forced quasilinear functional differential equations with ϕ-Laplacian operator and mixed nonlinearities. It especially includes the linear, the one-dimensional p-Laplacian, and the prescribed mean curvature quasilinear differential operators. It continues some recently published results on the oscillations of the second-order functional differential equations including functional arguments of delay, advanced, or delay-advanced types. The nonlinear terms are of superlinear or supersublinear (mixed) types. Consequences and examples are shown to illustrate the novelty and simplicity of our oscillation criteria

Characterization for Rectifiable and Nonrectifiable Attractivity of Nonautonomous Systems of Linear Differential Equations

We study a new kind of asymptotic behaviour near for the nonautonomous system of two linear differential equations: , , where the matrix-valued function has a kind of singularity at . It is called rectifiable (resp., nonrectifiable) attractivity of the zero solution, which means that as and the length of the solution curve of is finite (resp., infinite) for every . It is characterized in terms of certain asymptotic behaviour of the eigenvalues of near . Consequently, the main results are applied to a system of two linear differential equations with polynomial coefficients which are singular at

Oscillations of a Class of Forced Second-Order Differential Equations with Possible Discontinuous Coefficients

We study the oscillation of all solutions of a general class of forced second-order differential equations, where their second derivative is not necessarily a continuous function and the coefficients of the main equation may be discontinuous. Our main results are not included in the previously published known oscillation criteria of interval type. Many examples and consequences are presented illustrating the main results