Journal of Fixed Point Theory and Applications

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Applications of theorems of Jean Leray to the Einstein-scalar field equations

Yvonne Choquet-Bruhat, James Isenberg, Daniel Pollack

Journal of Fixed Point Theory and Applications > 2007 > 1 > 1 > 31-46

. The Einstein-scalar field theory can be used to model gravitational physics with scalar field sources. We discuss the initial value formulation of this field theory, and show that the ideas of Leray can be used to show that the Einstein-scalar field system of partial differential equations is well-posed as an evolutionary system. We also show that one can generate solutions of the Einstein-scalar...

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Solitary waves in the nolinear wave equation and in gauge theories

Vieri Benci, Donato Fortunato

Journal of Fixed Point Theory and Applications > 2007 > 1 > 1 > 61-86

. Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time. This paper is an introduction to the study of solitary waves relative to the nonlinear wave equation and to the Abelian gauge theories. Abelian gauge theories consist of a class of field equations obtained by coupling in a suitable way the nonlinear...

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The global geometry of Riemannian manifolds with commuting curvature operators

M. Brozos-Vázquez, P. Gilkey

Journal of Fixed Point Theory and Applications > 2007 > 1 > 1 > 87-96

. We give manifolds whose Riemann curvature operators commute, i.e. which satisfy $${\mathcal{R}}(x_{1} ,x_{2} ){\mathcal{R}}(x_{3} ,x_{4} ) = {\mathcal{R}}(x_{3} ,x_{4} ){\mathcal{R}}(x_{1} ,x_{2} )$$ for all tangent vectors x _i in both the Riemannian and the higher signature settings. These manifolds have global geometric phenomena which are quite different for higher signature manifolds than...

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Conley indices on convex sets, attractor-repeller pairs and applications to multiple solutions of nonlinear elliptic equations

E. N. Dancer

Journal of Fixed Point Theory and Applications > 2007 > 1 > 1 > 47-60

. We discuss the calculation of homotopy indices on closed convex sets in a Banach space and then use these ideas to prove the existence of multiple solutions of nonlinear elliptic equations. We obtain more information on the solutions than in earlier work. A key idea is to calculate the homotopy index of the stable and mountain pass solutions and the connections joining them.

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Editorial

Journal of Fixed Point Theory and Applications > 2007 > 1 > 1 > 1-2

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Le théorème de Lefschetz–Hopf pour les applications compactes des espaces ULC

Robert Cauty

Journal of Fixed Point Theory and Applications > 2007 > 1 > 1 > 123-134

. We prove the Lefschetz–Hopf fixed point theorem for compact maps of locally equiconnected spaces.

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The degree of maps of free G-manifolds and the average

Jan Jaworowski

Journal of Fixed Point Theory and Applications > 2007 > 1 > 1 > 111-121

. Suppose that G is a compact Lie group, M and N are orientable, free G-manifolds and f : M → N is an equivariant map. We show that the degree of f satisfies a formula involving data given by the classifying maps of the orbit spaces M/G and N/G. In particular, if the generator of the top dimensional cohomology of M/G with integer coefficients is in the image of the cohomology map induced by the classifying...

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The fibrewise Leray–Schauder index

M. C. Crabb

Journal of Fixed Point Theory and Applications > 2007 > 1 > 1 > 3-30

. The index constructed by Leray and Schauder in 1934 admits generalizations in two directions to infinite-dimensional fixed-point and vector field indices. We present the constructions of fibrewise equivariant indices of both types and illustrate the definitions by applications to the stable homo-topy Fuller index and Seiberg–Witten invariant.

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Viscosity approximation methods for resolvents of accretive operators in Banach spaces

Wataru Takahashi

Journal of Fixed Point Theory and Applications > 2007 > 1 > 1 > 135-147

. In this paper, we first prove a strong convergence theorem for resolvents of accretive operators in a Banach space by the viscosity approximation method, which is a generalization of the results of Reich [J. Math. Anal. Appl. 75 (1980), 287–292], and Takahashi and Ueda [J. Math. Anal. Appl. 104 (1984), 546–553]. Further using this result, we consider the proximal point algorithm in a Banach space...

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Two results in metric fixed point theory

Daniel Reem, Simeon Reich, Alexander J. Zaslavski

Journal of Fixed Point Theory and Applications > 2007 > 1 > 1 > 149-157

. We establish two fixed point theorems for certain mappings of contractive type. The first result is concerned with the case where such mappings take a nonempty, closed subset of a complete metric space X into X, and the second with an application of the continuation method to the case where they satisfy the Leray–Schauder boundary condition in Banach spaces.

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Integer-valued fixed point index for compositions of acyclic maps

E. G. Sklyarenko, G. S. Skordev

Journal of Fixed Point Theory and Applications > 2007 > 1 > 1 > 97-109

. An integer-valued fixed point index for compositions of acyclic multivalued maps is constructed. This index has the additivity, homotopy invariance, normalization, commutativity, and multiplicativity properties. The acyclicity is with respect to the Čech cohomology with integer coefficients. The technique of chain approximation is used.

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Boundary interface for the Allen–Cahn equation

A. Malchiodi, Juncheng Wei

Journal of Fixed Point Theory and Applications > 2007 > 1 > 2 > 305-336

. We consider the Allen–Cahn equation $$ \varepsilon^{2}\Delta u + u - u^3 = 0 \quad {\rm in}\,\Omega, \quad \frac{\partial u}{\partial v} = 0 \quad {\rm on}\,\partial\Omega, $$ where Ω is a smooth and bounded domain in $${\mathbb{R}}^n$$ such that the mean curvature is positive at each boundary point. We show that there exists a sequence ε _j → 0 such that the Allen–Cahn equation has...

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A note on weak convergence in $$L^{1}_{\rm loc}({\mathbb{R}})$$

Gioconda Moscariello, Carlo Sbordone

Journal of Fixed Point Theory and Applications > 2007 > 1 > 2 > 337-350

. Let $$a_j : {\mathbb{R}} \rightarrow {\mathbb{R}}$$ be a sequence of Borel measurable functions satisfying, for a function $$K \in L_{\rm loc}^{1}, K : {\mathbb{R}} \rightarrow [1,\infty),$$ the inequalities $$ 1/K(x) \leq a_j (x) \leq K(x)\quad {\rm a.e.}\, x \in {\mathbb{R}} $$ and suppose $$ a_j \rightharpoonup a\quad {\rm in}\,\sigma(L^1, L^\infty). $$ Then there exists...

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Analytic symmetrizations of 3 × 3 systems of first order partial differential equations with reduced dimension 4

Jean Vaillant

Journal of Fixed Point Theory and Applications > 2007 > 1 > 2 > 221-238

. Let L(y, D) be a first order 3 × 3 system with reduced dimension 4 depending on a real vector parameter y. Assume that, at every frozen y, L(y, D) is symmetrizable. We state geometrical conditions in order that L(y, D) is smoothly (or analytically) symmetrizable.

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Global existence of analytic solutions to the Cauchy problem in a complex domain

M. Frigon

Journal of Fixed Point Theory and Applications > 2007 > 1 > 2 > 189-194

. We present results on the global existence of analytic solutions to the Cauchy problem in starshaped or convex complex domains. No growth conditions are imposed. Our results rely on a notion of solution-tube that we introduce.

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Asymptotic structure for solutions of the Cauchy problem for Burgers type equations

G. M. Henkin

Journal of Fixed Point Theory and Applications > 2007 > 1 > 2 > 239-291

. Large time asymptotic structure for solutions of the Cauchy problem for a generalized Burgers equation is determined. In particular, Gelfand’s question about location of viscous shock waves for such equations is answered.

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Notes on the bifurcation theorem

Kung-Ching Chang, Zhi-Qiang Wang

Journal of Fixed Point Theory and Applications > 2007 > 1 > 2 > 195-208

. We reexamine bifurcation questions for potential operators. Based on the Conley index theory we present a unified approach to bifurcations both at the origin and at infinity. Our results provide improvements of the existing theory.

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Le prolongement analytique de la solution du problème de Cauchy pour un certain opérateur différentiel à coefficients polynomiaux

Yûsaku Hamada

Journal of Fixed Point Theory and Applications > 2007 > 1 > 2 > 209-220

. In this article, we study analytic continuations of the solution of the Cauchy problem for a certain differential operator in the complex domain.

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Jean Leray and nonlinear integral equations: A partially forgotten legacy

Jean Mawhin

Journal of Fixed Point Theory and Applications > 2007 > 1 > 2 > 159-187

. Some applications given in Jean Leray’s thesis of 1933 seem to have been forgotten, before being rediscovered and then widely developed. The reason may be found in the fact that the Arzelà–Schmidt’s method introduced in the thesis was superseded one year later by the more general and fruitful Leray-Schauder’s method. We also show that, despite its fame and a steadily increasing number of applications,...

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The Leray separating operator in the theory of the Cauchy problem

L. R. Volevich

Journal of Fixed Point Theory and Applications > 2007 > 1 > 2 > 293-304

. The paper is devoted to the presentation of Leray’s approach to the Cauchy problem for strictly hyperbolic operators. In the first section we give the main definitions of strictly hyperbolic operators and separating operators corresponding to them. We present the plan of derivation of the a priori estimates necessary for the proof of solvability of the Cauchy problem. In the second section we generalize...

INFONA - science communication portal

Journal of Fixed Point Theory and Applications

Applications of theorems of Jean Leray to the Einstein-scalar field equations

Solitary waves in the nolinear wave equation and in gauge theories

The global geometry of Riemannian manifolds with commuting curvature operators

Conley indices on convex sets, attractor-repeller pairs and applications to multiple solutions of nonlinear elliptic equations

Editorial

Le théorème de Lefschetz–Hopf pour les applications compactes des espaces ULC

The degree of maps of free G-manifolds and the average

The fibrewise Leray–Schauder index

Viscosity approximation methods for resolvents of accretive operators in Banach spaces

Two results in metric fixed point theory

Integer-valued fixed point index for compositions of acyclic maps

Boundary interface for the Allen–Cahn equation

A note on weak convergence in $$L^{1}_{\rm loc}({\mathbb{R}})$$

Analytic symmetrizations of 3 × 3 systems of first order partial differential equations with reduced dimension 4

Global existence of analytic solutions to the Cauchy problem in a complex domain

Asymptotic structure for solutions of the Cauchy problem for Burgers type equations

Notes on the bifurcation theorem

Le prolongement analytique de la solution du problème de Cauchy pour un certain opérateur différentiel à coefficients polynomiaux

Jean Leray and nonlinear integral equations: A partially forgotten legacy

The Leray separating operator in the theory of the Cauchy problem

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