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1

Existence and uniqueness theorems for fourth-order boundary value problems

100%

Przybycin J.

Annales Polonici Mathematici

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1997

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tom 67

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nr 1

59-64

EN

We establish the existence and uniqueness theorems for a linear and a nonlinear fourth-order boundary value problem. The results obtained generalize the results of Usmani [4] and Yang [5]. The methods used are based, in principle, on [3], [5].

2

Nonlinear eigenvalue problems for fourth order ordinary differential equations

100%

Przybycin J.

Annales Polonici Mathematici

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1994-1995

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tom 60

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nr 3

249-253

EN

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.

3

Bifurcation theorems of Rabinowitz type for certain differential operators of the fourth order

100%

Przybycin J.

Annales Polonici Mathematici

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1992

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tom 57

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nr 1

21-28

EN

This paper was inspired by the works of P. H. Rabinowitz. We study nonlinear eigenvalue problems for some fourth order elliptic partial differential equations with nonlinear perturbation of Rabinowitz type. We show the existence of an unbounded continuum of nontrivial positive solutions bifurcating from (μ₁,0), where μ₁ is the first eigenvalue of the linearization about 0 of the considered problem. We also prove the related theorem for bifurcation from infinity. The results obtained are similar to those proved by Rabinowitz for second order elliptic partial differential equations ([5]-[7]). The methods used are based, in principle, on the results of [1], [5], [6].

4

On bifurcation intervals for nonlinear eigenvalue problems

100%

Przybycin J.

Annales Polonici Mathematici

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1999

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tom 71

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nr 1

39-46

EN

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.

5

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The connection between number and form of bifurcation points and properties of the nonlinear perturbation of Berestycki type

88%

Przybycin J.

Annales Polonici Mathematici

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1989-1990

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tom 50

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nr 2

129-136

6

Some applications of bifurcation theory to ordinary differential equations of the fourth order

63%

Przybycin J.

Annales Polonici Mathematici

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1991

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tom 53

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nr 2

153-160

Ograniczanie wyników

6 Annales Polonici Mathematici

6 Przybycin J.

1 1999

1 1997

1 1995

1 1992

1 1991

1 1989

Existence and uniqueness theorems for fourth-order boundary value problems

Nonlinear eigenvalue problems for fourth order ordinary differential equations

Bifurcation theorems of Rabinowitz type for certain differential operators of the fourth order

On bifurcation intervals for nonlinear eigenvalue problems

The connection between number and form of bifurcation points and properties of the nonlinear perturbation of Berestycki type

Some applications of bifurcation theory to ordinary differential equations of the fourth order