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Plus-operators in Krein spaces and dichotomous behavior of irreversible dynamical systems with discrete time

V. Khatskevich, L. Zelenko (2006)

Studia Mathematica

We study dichotomous behavior of solutions to a non-autonomous linear difference equation in a Hilbert space. The evolution operator of this equation is not continuously invertible and the corresponding unstable subspace is of infinite dimension in general. We formulate a condition ensuring the dichotomy in terms of a sequence of indefinite metrics in the Hilbert space. We also construct an example of a difference equation in which dichotomous behavior of solutions is not compatible with the signature...

Point fixe d'une application non contractante.

Pierre Gilles Lemarié-Rieusset (2006)

Revista Matemática Iberoamericana

We study a multilinear fixed-point equation in a closed ball of a Banach space where the application is 1-Lipschitzian: existence, uniqueness, approximations, regularity.

Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order boundary value problem

Qingliu Yao (2013)

Applications of Mathematics

We consider the classical nonlinear fourth-order two-point boundary value problem u ( 4 ) ( t ) = λ h ( t ) f ( t , u ( t ) , u ' ( t ) , u ' ' ( t ) ) , 0 < t < 1 , u ( 0 ) = u ' ( 1 ) = u ' ' ( 0 ) = u ' ' ' ( 1 ) = 0 . In this problem, the nonlinear term h ( t ) f ( t , u ( t ) , u ' ( t ) , u ' ' ( t ) ) contains the first and second derivatives of the unknown function, and the function h ( t ) f ( t , x , y , z ) may be singular at t = 0 , t = 1 and at x = 0 , y = 0 , z = 0 . By introducing suitable height functions and applying the fixed point theorem on the cone, we establish several local existence theorems on positive solutions and obtain the corresponding eigenvalue intervals.

Positive solutions and eigenvalue intervals of a singular third-order boundary value problem

Qingliu Yao (2011)

Annales Polonici Mathematici

This paper studies positive solutions and eigenvalue intervals of a nonlinear third-order two-point boundary value problem. The nonlinear term is allowed to be singular with respect to both the time and space variables. By constructing a proper cone and applying the Guo-Krasnosel'skii fixed point theorem, the eigenvalue intervals for which there exist one, two, three or infinitely many positive solutions are obtained.

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