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Decaying Regularly Varying Solutions of Third-order Differential Equations with a Singular Nonlinearity

Ivana Kučerová (2014)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

This paper is concerned with asymptotic analysis of strongly decaying solutions of the third-order singular differential equation x ' ' ' + q ( t ) x - γ = 0 , by means of regularly varying functions, where γ is a positive constant and q is a positive continuous function on [ a , ∞ ) . It is shown that if q is a regularly varying function, then it is possible to establish necessary and sufficient conditions for the existence of slowly varying solutions and regularly varying solutions of (A) which decrease to 0 as t → ∞ and to acquire...

Derivative of the norm of a linear mapping and its application to differential equations

František Tumajer (1992)

Applications of Mathematics

In this paper the notion of the derivative of the norm of a linear mapping in a normed vector space is introduced. The fundamental properties of the derivative of the norm are established. Using these properties, linear differential equations in a Banach space are studied and lower and upper estimates of the norms of their solutions are derived.

Differential inclusions and multivalued integrals

Kinga Cichoń, Mieczysław Cichoń, Bianca Satco (2013)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we consider the nonlocal (nonstandard) Cauchy problem for differential inclusions in Banach spaces x'(t) ∈ F(t,x(t)), x(0)=g(x), t ∈ [0,T] = I. Investigation over some multivalued integrals allow us to prove the existence of solutions for considered problem. We concentrate on the problems for which the assumptions are expressed in terms of the weak topology in a Banach space. We recall and improve earlier papers of this type. The paper is complemented...

Dynamic analysis of an impulsive differential equation with time-varying delays

Ying Li, Yuanfu Shao (2014)

Applications of Mathematics

An impulsive differential equation with time varying delay is proposed in this paper. By using some analysis techniques with combination of coincidence degree theory, sufficient conditions for the permanence, the existence and global attractivity of positive periodic solution are established. The results of this paper improve and generalize some previously known results.

Dynamical systems method for solving linear finite-rank operator equations

N. S. Hoang, A. G. Ramm (2009)

Annales Polonici Mathematici

A version of the dynamical systems method (DSM) for solving ill-conditioned linear algebraic systems is studied. An a priori and an a posteriori stopping rules are justified. An iterative scheme is constructed for solving ill-conditioned linear algebraic systems.

Dynamics and density evolution in piecewise deterministic growth processes

Michael C. Mackey, Marta Tyran-Kamińska (2008)

Annales Polonici Mathematici

A new sufficient condition is proved for the existence of stochastic semigroups generated by the sum of two unbounded operators. It is applied to one-dimensional piecewise deterministic Markov processes, where we also discuss the existence of a unique stationary density and give sufficient conditions for asymptotic stability.

Embedded eigenvalues and resonances of Schrödinger operators with two channels

Xue Ping Wang (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

In this article, we give a necessary and sufficient condition in the perturbation regime on the existence of eigenvalues embedded between two thresholds. For an eigenvalue of the unperturbed operator embedded at a threshold, we prove that it can produce both discrete eigenvalues and resonances. The locations of the eigenvalues and resonances are given.

Equations with discontinuous nonlinear semimonotone operators

Nguyen Buong (1999)

Commentationes Mathematicae Universitatis Carolinae

The aim of this paper is to present an existence theorem for the operator equation of Hammerstein type x + K F ( x ) = 0 with the discontinuous semimonotone operator F . Then the result is used to prove the existence of solution of the equations of Urysohn type. Some examples in the theory of nonlinear equations in L p ( Ω ) are given for illustration.

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