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Nonlinear eigenvalue problems for fourth order ordinary differential equations

100%

Przybycin J.

Annales Polonici Mathematici

1994-1995

tom 60

nr 3

249-253

This paper was inspired by the works of Chiappinelli ([3]) and Schmitt and Smith ([7]). We study the problem ℒu = λau + f(·,u,u',u'',u''') with separated boundary conditions on [0,π], where ℒ is a composition of two operators of Sturm-Liouville type. We assume that the nonlinear perturbation f satisfies the inequality |f(x,u,u',u'',u''')| ≤ M|u|. Because of the presence of f the considered equation does not in general have a linearization about 0. For this reason the global bifurcation theorem of Rabinowitz ([5], [6]) is not applicable here. We use the properties of Leray-Schauder degree to establish the existence of nontrivial solutions and describe their location. The results obtained are similar to those proved by Chiappinelli for Sturm-Liouville operators.

On bifurcation intervals for nonlinear eigenvalue problems

75%

Przybycin J.

Annales Polonici Mathematici

1999

tom 71

nr 1

39-46

We give a sufficient condition for [μ-M, μ+M] × {0} to be a bifurcation interval of the equation u = L(λu + F(u)), where L is a linear symmetric operator in a Hilbert space, μ ∈ r(L) is of odd multiplicity, and F is a nonlinear operator. This abstract result provides an elementary proof of the existence of bifurcation intervals for some eigenvalue problems with nondifferentiable nonlinearities. All the results obtained may be easily transferred to the case of bifurcation from infinity.

Ograniczanie wyników

2 Annales Polonici Mathematici

2 Przybycin J.

1 1999

1 1995

Wyniki wyszukiwania

Nonlinear eigenvalue problems for fourth order ordinary differential equations

On bifurcation intervals for nonlinear eigenvalue problems