Mervan Pašić

Minkowski–Bouligand dimension of solutions of the one-dimensional p-Laplacian

The upper Minkowski–Bouligand dimension of the graph of continuous solutions, denoted by dimMy, is studied for a class of nonlinear one-dimensional p-Laplacian. Some sufficient conditions on the nonlinearities are given such that dimMy takes the prescribed fractional values. Next, a relation between dimMy and the order of growth for singular behaviour of Lp norm of derivatives of solutions is given. Finally, the change and stability of dimMy are considered by means of a double-parametric problem.

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.

Rectifiable and Unrectifiable Oscillations for a Generalization of the Riemann–Weber Version of Euler Differential Equation

Abstract The following generalization (for α = 2 we suppose that β = 2 and λ > 1/4), of the Riemann–Weber version of Euler differential equation is introduced and it is considered together with a suitable boundary layer condition depending on α near 𝑥 = 0. It is shown that this problem is rectifiable (resp., unrectifiable) oscillatory on (0, 𝑏) provided α ∈ [2, 4) (resp., α ≥ 4). It is a kind of geometrical oscillations on (0, 𝑏) for which the finite (resp., infinite) length of the graphs of all its solutions is proposed.

Sign-changing first derivative of positive solutions of forced second-order nonlinear differential equations

Abstract Some oscillatory phenomena in physics, population, biomedicine and biochemistry are described by positive functions having sign-changing first derivative. Here, it is studied for all positive not necessarily periodic solutions of a large class of second-order nonlinear differential equations. It is based on a new reciprocal principle by which the classic oscillations of corresponding reciprocal linear equation causes the sign-changing first derivative of every positive solution of the main equation.

New interval oscillation criteria for forced second- order differential equations with nonlinear damping

Our aim in this paper is to prove, under some growth conditions on the datas, the solvability in a Gevrey class of a polynomially nonlinear functional differential equation.

Control of essential infimum and supremum of solutions of quasilinear elliptic equations

Abstract We consider a class of quasilinear elliptic equations of Leray-Lions type that allow us to control the ess inf and ess sup of solutions. Precisely, for arbitrary given four constants m0

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.

Symmetrization and existence results for some quasilinear elliptic equations

Abstract We prove the symmetrization and existence results and the precise a priori estimates in w01, p (Ω) ∩ L∞ (Ωt) for the solution of some quasilinear elliptic equations associated with Leray-Lions operators.

Nonautonomous Differential Equations in Banach Space and Nonrectifiable Attractivity in Two-Dimensional Linear Differential Systems

We study the asymptotic behaviour on a finite interval of a class of linear nonautonomous singular differential equations in Banach space by the nonintegrability of the first derivative of its solutions. According to these results, the nonrectifiable attractivity on a finite interval of the zero solution of the two-dimensional linear integrable differential systems with singular matrix-elements is characterized.

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.