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The bounce problem, on n-dimensional Riemannian manifolds

Giuseppe Buttazzo, Danilo Percivale (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In questo lavoro vengono generalizzati i risultati relativi al problema del rimbalzo unidimensionale studiato in [5]. Precisamente si considera un punto mobile su una varietà Riemanniana V n -dimensionale, soggetto all’azione di un potenziale variabile nel tempo e vincolato a restare in una parte W di V avente un bordo di classe C 3 contro cui il punto «rimbalza»....

The Conley index in Hilbert spaces and its applications

K. Gęba, M. Izydorek, A. Pruszko (1999)

Studia Mathematica

We present a generalization of the classical Conley index defined for flows on locally compact spaces to flows on an infinite-dimensional real Hilbert space H generated by vector fields of the form f: H → H, f(x) = Lx + K(x), where L: H → H is a bounded linear operator satisfying some technical assumptions and K is a completely continuous perturbation. Simple examples are presented to show how this new invariant can be applied in searching critical points of strongly indefinite functionals having...

The distance between fixed points of some pairs of maps in Banach spaces and applications to differential systems

Cristinel Mortici (2006)

Czechoslovak Mathematical Journal

Let T be a γ -contraction on a Banach space Y and let S be an almost γ -contraction, i.e. sum of an ε , γ -contraction with a continuous, bounded function which is less than ε in norm. According to the contraction principle, there is a unique element u in Y for which u = T u . If moreover there exists v in Y with v = S v , then we will give estimates for u - v . Finally, we establish some inequalities related to the Cauchy problem.

The existence of limit cycle for perturbed bilinear systems

Hanen Damak, Mohamed Ali Hammami, Yeong-Jeu Sun (2012)

Kybernetika

In this paper, the feedback control for a class of bilinear control systems with a small parameter is proposed to guarantee the existence of limit cycle. We use the perturbation method of seeking in approximate solution as a finite Taylor expansion of the exact solution. This perturbation method is to exploit the “smallness” of the perturbation parameter ε to construct an approximate periodic solution. Furthermore, some simulation results are given to illustrate the existence of a limit cycle for...

The first derivative of period function of a plane vector field.

Jean-Pierre Françoise (1997)

Publicacions Matemàtiques

The algorithm of the successive derivatives introduced in [5] was implemented in [7], [8]. This algorithm is based on the existence of a decomposition of 1-forms associated to the relative cohomology of the Hamiltonian function which is perturbed. We explain here how the first step of this algorithm gives also the first derivative of the period function. This includes, for instance, new presentations of formulas obtained by Carmen Chicone and Marc Jacobs in [3].

The fixed point theorem and the boundedness of solutions of differential equations in the Banach space

František Tumajer (1993)

Mathematica Bohemica

The properties of solutions of the nonlinear differential equation x ' = A ( s ) x + f ( s , x ) in a Banach space and of the special case of the homogeneous linear differential equation x ' = A ( s ) x are studied. Theorems and conditions guaranteeing boundedness of the solution of the nonlinear equation are given on the assumption that the solutions of the linear homogeneous equation have certain properties.

The General Differential Operators Generated by a Quasi-Differential Expressions with their Interior Singular Points

El-sayed Ibrahim, Sobhy (1999)

Serdica Mathematical Journal

The general ordinary quasi-differential expression M of n-th order with complex coefficients and its formal adjoint M + are considered over a regoin (a, b) on the real line, −∞ ≤ a < b ≤ ∞, on which the operator may have a finite number of singular points. By considering M over various subintervals on which singularities occur only at the ends, restrictions of the maximal operator generated by M in L2|w (a, b) which are regularly solvable with respect to the minimal operators T0 (M ) and T0...

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