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On Existence and Asymptotic Properties of Kneser Solutions to Singular Second Order ODE.

Jana Vampolová (2013)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

We investigate an asymptotic behaviour of damped non-oscillatory solutions of the initial value problem with a time singularity p ( t ) u ' ( t ) ' + p ( t ) f ( u ( t ) ) = 0 , u ( 0 ) = u 0 , u ' ( 0 ) = 0 on the unbounded domain [ 0 , ∞ ) . Function f is locally Lipschitz continuous on ℝ and has at least three zeros L 0 < 0 , 0 and L > 0 . The initial value u 0 ∈ ( L 0 , L ) ∖ { 0 } . Function p is continuous on [ 0 , ∞ ) , has a positive continuous derivative on ( 0 , ∞ ) and p ( 0 ) = 0 . Asymptotic formulas for damped non-oscillatory solutions and their first derivatives are derived under some additional assumptions. Further, we provide...

On existence and uniqueness of solutions for ordinary differential equations with nonlinear boundary conditions

Alessandro Calamai (2004)

Bollettino dell'Unione Matematica Italiana

We prove an existence and uniqueness theorem for a nonlinear functional boundary value problem, that is, an ordinary differential equation with a nonlinear boundary condition. The proof is based on a Global Inversion Theorem of Ambrosetti and Prodi, which is applied to the boundary operator restricted to the manifold of the global solutions to the equation. Our result is a generalization of an analogous existence and uniqueness theorem of G. Vidossich, as it is shown with some examples.

On existence theorems for semilinear equations and applications

Fang Zhang, Feng Wang (2013)

Annales Polonici Mathematici

Existence results for semilinear operator equations without the assumption of normal cones are obtained by the properties of a fixed point index for A-proper semilinear operators established by Cremins. As an application, the existence of positive solutions for a second order m-point boundary value problem at resonance is considered.

On exponential stability of second order delay differential equations

Ravi P. Agarwal, Alexander Domoshnitsky, Abraham Maghakyan (2015)

Czechoslovak Mathematical Journal

We propose a new method for studying stability of second order delay differential equations. Results we obtained are of the form: the exponential stability of ordinary differential equation implies the exponential stability of the corresponding delay differential equation if the delays are small enough. We estimate this smallness through the coefficients of this delay equation. Examples demonstrate that our tests of the exponential stability are essentially better than the known ones. This method...

On families of trajectories of an analytic gradient vector field

Adam Dzedzej, Zbigniew Szafraniec (2005)

Annales Polonici Mathematici

For an analytic function f:ℝⁿ,0 → ℝ,0 having a critical point at the origin, we describe the topological properties of the partition of the family of trajectories of the gradient equation ẋ = ∇f(x) attracted by the origin, given by characteristic exponents and asymptotic critical values.

Currently displaying 341 – 360 of 1662