We prove the simplicity and isolation of the first eigenvalue for the problem <mml:math alttext="$\Delta_p u = |u|^{p-2}u$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Δ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>u</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi></mml:mrow></mml:math> in a bounded smooth domain <mml:math alttext="$\Omega\subset\mathbb{R}^N$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Ω</mml:mi><mml:mo>⊂</mml:mo><mml:msup><mml:mi>ℝ</mml:mi><mml:mi>N</mml:mi></mml:msup></mml:mrow></mml:math>, with a nonlinear boundary condition given by <mml:math alttext="$|\nabla u|^{p-2}\partial u/\partial\nu =\lambda|u|^{p-2} u$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mo>∇</mml:mo><mml:mi>u</mml:mi></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>∂</mml:mo><mml:mi>u</mml:mi><mml:mo>/</mml:mo><mml:mo>∂</mml:mo><mml:mi>v</mml:mi><mml:mo>=</mml:mo><mml:mi>λ</mml:mi><mml:msup><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>u</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi></mml:mrow></mml:math> on the boundary of the domain.

Sandra Martínez

Julio D. Rossi

Isolation and simplicity for the first eigenvalue of the <svg style="vertical-align:-3.50804pt" height="13.3" version="1.1" viewBox="0 0 10.2 13.3" width="10.2" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,8.863)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36&#xA;t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" /></g></svg>-Laplacian with a nonlinear boundary condition


Guang Zhang

Sui Sun Cheng

Positive periodic solutions of nonautonomous functional differential equations depending on a parameter

We study quasilinear hemivariational inequalities involving the <mml:math alttext="$p$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-Laplacian. We prove two existence theorems. In the first, we allow “crossing” of the principal eigenvalue by the generalized potential, while in the second, we incorporate problems at
resonance. Our approach is based on the nonsmooth critical point
theory for locally Lipschitz energy functionals.

Nikolaos C. Kourogenis

Nikolaos S. Papageorgiou

Existence theorems for elliptic hemivariational inequalities involving the <svg style="vertical-align:-3.50804pt" height="13.3" version="1.1" viewBox="0 0 10.2 13.3" width="10.2" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,8.863)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36&#xA;t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" /></g></svg>-Laplacian


The basic higher order commutator theorem is formulated for the real interpolation methods associated with the quasi-power parameters, that is, the function spaces on which Hardy inequalities are valid. This theorem unifies and extends various results given by Cwikel, Jawerth, Milman, Rochberg, and others, and incorporates some results of Kalton to the context of commutator estimates for the real interpolation methods.

Ming  Fan

Commutators in real interpolation with quasi-power parameters

F. Bucci

I. Lasiecka

R. Triggiani

Singular estimates and uniform stability of coupled systems of
hyperbolic/parabolic PDEs

This work is devoted to the study of a quasilinear elliptic
system of resonant type. We prove the existence of infinitely
many solutions of a related nonlinear eigenvalue problem.
Applying an abstract minimax theorem, we obtain a solution of the
quasilinear system <mml:math alttext="$-\Delta _{p}u = F_{u}(x,u,v),-\Delta_{q}v = F_{v}(x,u,v)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>&#8722;</mml:mo><mml:msub><mml:mi>&#916;</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mtext>&#8201;</mml:mtext><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mtext>&#8201;</mml:mtext><mml:mi>v</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mtext>&#8201;</mml:mtext><mml:mo>&#8722;</mml:mo><mml:msub><mml:mi>&#916;</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mi>v</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mtext>&#8201;</mml:mtext><mml:mi>v</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mtext>&#8201;</mml:mtext><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mtext>&#8201;</mml:mtext><mml:mi>v</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, under conditions involving the first and the second eigenvalues.

Pablo L. de N&#224;poli

M. Cristina Mariani

Quasilinear elliptic systems of resonant type and nonlinear eigenvalue problems

We study minimal solutions for one-dimensional variational
problems on a torus. We show that, for a generic
integrand and any rational number <mml:math alttext="$\alpha$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#945;</mml:mi></mml:math>, there exists a unique
(up to translations) periodic minimal solution with rotation number <mml:math alttext="$\alpha$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#945;</mml:mi></mml:math>.

Alexander J. Zaslavski

Generic uniqueness of a minimal solution for variational problems on a torus

The integral wavelet transform is defined in weighted Sobolev
spaces, in which some properties of the transform as well as its
asymptotical behaviour for small dilation parameter are studied.

Nguyen Minh Chuong

Ta Ngoc Tri

The integral wavelet transform in weighted Sobolev spaces

The problem <mml:math alttext="$\dot{x} = f(t,x),x(-\infty)= x(+\infty)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo>&#729;</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mtext>&#8201;</mml:mtext><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>&#8722;</mml:mo><mml:mi>&#8734;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>+</mml:mo><mml:mi>&#8734;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, where <mml:math alttext="$x(\pm\infty):=\lim_{t\rightarrow\pm\infty}x(t) \in\mathbb{R}^{n}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>&#177;</mml:mo><mml:mi>&#8734;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mo>lim</mml:mo></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#8594;</mml:mo><mml:mo>&#177;</mml:mo><mml:mi>&#8734;</mml:mi></mml:mrow></mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#8712;</mml:mo><mml:msup><mml:mi>&#8477;</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math>, is considered. Some existence results for this
problem are established using the fixed point method and
topological degree theory.

Cezar  Avramescu

Existence problems for homoclinic solutions

About this journal

<p><em>Abstract and Applied Analysis</em> publishes research with an emphasis on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimisation theory, and control theory.</p>


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<p>Chief Editor, Dr Wong, is an associate professor at Nanyang Technological University, Singapore. Her research interests include differential equations, difference equations, integral equations, and numerical mathematics.</p>


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Abstract and Applied Analysis - Table of contents 2002

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Abstract and Applied Analysis publishes research with an emphasis on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimisation theory, and control theory.

Abstract and Applied Analysis