Topological Methods in Nonlinear Analysis Articles (Project Euclid)
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The latest articles from Topological Methods in Nonlinear Analysis on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2016 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Mon, 21 Mar 2016 15:31 EDTMon, 21 Mar 2016 15:31 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Coincidence of maps on torus fiber bundles over the circle
http://projecteuclid.org/euclid.tmna/1458588650
<strong>João Peres Vieira</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Volume 46, Number 2, 507--548.</p><p><strong>Abstract:</strong><br/> The main purpose of this work is to study coincidences of
fibre-preserving self-maps over the circle $S^1$ for spaces which
are fibre bundles over $S^1$ and the fibre is the torus $T$. We
classify all pairs of self-maps over $S^1$ which can be deformed
fibrewise to a pair of coincidence free maps. </p>projecteuclid.org/euclid.tmna/1458588650_20160321153103Mon, 21 Mar 2016 15:31 EDTOn a singular semilinear elliptic problem: multiple solutions via critical point theoryhttps://projecteuclid.org/euclid.tmna/1527213956<strong>Francesca Faraci</strong>, <strong>George Smyrlis</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 33 pp..</p><p><strong>Abstract:</strong><br/>
We study existence and multiplicity of solutions of a semilinear elliptic problem involving a singular term. Combining various techniques from critical point theory, under different sets of assumptions, we prove the existence of $k$ solutions ($k\in\mathbb N$) or infinitely many weak solutions.
</p>projecteuclid.org/euclid.tmna/1527213956_20180524220607Thu, 24 May 2018 22:06 EDTA gradient flow generated by a nonlocal model of a neural field in an unbounded domainhttps://projecteuclid.org/euclid.tmna/1527213957<strong>Severino Horacio da Silva</strong>, <strong>Antônio Luiz Pereira</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 16 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we consider the nonlocal evolution equation \[ \frac{\partial u(x,t)}{\partial t} + u(x,t)= \int_{\mathbb{R}^{N}}J(x-y)f(u(y,t))\rho(y)\,dy+ h(x). \] We show that this equation defines a continuous flow in both the space $C_{b}(\mathbb{R}^{N})$ of bounded continuous functions and the space $C_{\rho}(\mathbb{R}^{N})$ of continuous functions $u$ such that $u \cdot \rho$ is bounded, where $\rho $ is a convenient ``weight function''. We show the existence of an absorbing ball for the flow in $C_{b}(\mathbb{R}^{N})$ and the existence of a global compact attractor for the flow in $C_{\rho}(\mathbb{R}^{N})$, under additional conditions on the nonlinearity. We then exhibit a continuous Lyapunov function which is well defined in the whole phase space and continuous in the $C_{\rho}(\mathbb{R}^{N})$ topology, allowing the characterization of the attractor as the unstable set of the equilibrium point set. We also illustrate our result with a concrete example.
</p>projecteuclid.org/euclid.tmna/1527213957_20180524220607Thu, 24 May 2018 22:06 EDTHeteroclinic solutions of Allen-Cahn type equations with a general elliptic operatorhttps://projecteuclid.org/euclid.tmna/1527213958<strong>Karol Wroński</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 10 pp..</p><p><strong>Abstract:</strong><br/>
We consider a generalization of the Allen--Cahn type equation in divergence form $-{\rm div}(\nabla G(\nabla u(x,y)))+F_u(x,y,u(x,y))=0$. This is more general than the usual Laplace operator. We prove the existence and regularity of heteroclinic solutions under standard ellipticity and $m$-growth conditions.
</p>projecteuclid.org/euclid.tmna/1527213958_20180524220607Thu, 24 May 2018 22:06 EDTNonlinear Unilateral Parabolic Problems in Musielak--Orlicz spaces with $L^1$ datahttps://projecteuclid.org/euclid.tmna/1527645618<strong>Mustafa Ait Khellou</strong>, <strong>Sidi Mohamed Douiri</strong>, <strong>Youssef Elhadfi</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 29 pp..</p><p><strong>Abstract:</strong><br/>
We study, in Musielak-Orlicz spaces, the existence of solutions for some strongly nonlinear parabolic unilateral problem with $L^1$ data and without sign condition on nonlinearity.
</p>projecteuclid.org/euclid.tmna/1527645618_20180529220030Tue, 29 May 2018 22:00 EDTExistence and uniquenes results for systems of impulsive functional stochastic differential equations driven by fractional Brownian motion with multiple delayhttps://projecteuclid.org/euclid.tmna/1527645619<strong>Mohamed Ferhat</strong>, <strong>Tayeb Blouhi</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 29 pp..</p><p><strong>Abstract:</strong><br/>
We present some existence and uniqueness results on impulsive functional differential equations with multiple delay with fractional Brownian motion. Our approach is based on the Perov fixed point theorem and a new version of Schaefer's fixed point in generalized metric and Banach spaces.
</p>projecteuclid.org/euclid.tmna/1527645619_20180529220030Tue, 29 May 2018 22:00 EDTMarek Burnat --- Life and Researchhttps://projecteuclid.org/euclid.tmna/1532484285<strong>Andrzej Palczewski</strong>, <strong>Zbigniew Peradzyński</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 5 pp..</p><p><strong>Abstract:</strong><br/>
</p>projecteuclid.org/euclid.tmna/1532484285_20180724220506Tue, 24 Jul 2018 22:05 EDTRayleigh-Bénard problem for thermomicropolar fluidshttps://projecteuclid.org/euclid.tmna/1532484286<strong>Piotr Kalita</strong>, <strong>Grzegorz Łukaszewicz</strong>, <strong>Jakub Siemianowski</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 38 pp..</p><p><strong>Abstract:</strong><br/>
The two-dimensional Rayleigh-Bénard problem for a thermomicropolar fluids model is considered. The existence of suitable weak solutions which may not be unique, and the existence of the unique strong solution are proved. The global attractor for the m-semiflow associated with weak solutions and the global attractor for semiflow associated with strong solutions are shown to be equal.
</p>projecteuclid.org/euclid.tmna/1532484286_20180724220506Tue, 24 Jul 2018 22:05 EDTBlowup versus global in time existence of solutions for nonlinear heat equationshttps://projecteuclid.org/euclid.tmna/1532484287<strong>Piotr Cezary Biler</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 14 pp..</p><p><strong>Abstract:</strong><br/>
This note is devoted to a simple proof of blowup of solutions for a nonlinear heat equation. The criterion for a blowup is expressed in terms of a Morrey space norm and is in a sense complementary to conditions guaranteeing the global in time existence of solutions. The method goes back to H. Fujita and extends to other nonlinear parabolic equations.
</p>projecteuclid.org/euclid.tmna/1532484287_20180724220506Tue, 24 Jul 2018 22:05 EDTExistence, uniqueness and properties of global weak solutions to interdiffusion with Vegard rulehttps://projecteuclid.org/euclid.tmna/1532484289<strong>Lucjan Sapa</strong>, <strong>Bogusław Bożek</strong>, <strong>Marek Danielewski</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 26 pp..</p><p><strong>Abstract:</strong><br/>
We consider the diffusional transport in an $r$-component solid solution. The model is expressed by the nonlinear system of strongly coupled parabolic differential equations with initial and nonlinear boundary conditions. The techniques involved are the local mass conservation law for fluxes, which are a sum of the diffusional and Darken drift terms, and the Vegard rule. The considered boundary conditions allow the physical system to be not only closed but also open. The theorems on existence, uniqueness and properties of global weak solutions are proved. The main tool used in the proof of the existence result is the Galerkin approximation method. The agreement between the theoretical results, numerical simulations and experimental data is shown.
</p>projecteuclid.org/euclid.tmna/1532484289_20180724220506Tue, 24 Jul 2018 22:05 EDTOn the Faedo-Galerkin method for a free boundary problem for incompressible viscous magnetohydrodynamicshttps://projecteuclid.org/euclid.tmna/1532484290<strong>Piotr Kacprzyk</strong>, <strong>Wojciech M. Zajączkowski</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 30 pp..</p><p><strong>Abstract:</strong><br/>
The motion of incompressible magnetohydrodynamics (mhd) in a domain bounded by a free surface and coupled through it with external electromagnetic field is considered. Transmission conditions for electric currents and magnetic fields are prescribed on the free surface. Although we show the idea of a proof of local existence by the method of successive approximations, we are not going to prove neither local nor global existence of solutions. The existence of solutions of the linearized problems (the Stokes system for velocity and pressure and the linear transmission problem for the electromagnetic fields) is the main step in the proof of existence to the considered problem. This can be done either by the Faedo-Galerkin method or by the technique of regularizer. We concentrate our considerations to the Faedo-Galerkin method. For this we need an existence of a fundamental basis. We have to find the basis for the Stokes system and mhd system. We concentrate our considerations on the mhd system because this for the Stokes system is well known. We have to emphasize that the considered mhd system is obtained after linearization and transformation to the initial domains by applying the Lagrangian coordinates. This is the main aim of this paper.
</p>projecteuclid.org/euclid.tmna/1532484290_20180724220506Tue, 24 Jul 2018 22:05 EDTConcentration-compactness for singular nonlocal Schrödinger equations with oscillatory nonlinearitieshttps://projecteuclid.org/euclid.tmna/1532484291<strong>João Marcos do Ó</strong>, <strong>Diego Ferraz</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 49 pp..</p><p><strong>Abstract:</strong><br/>
The paper is dedicated to the theory of concentration-compactness principles for inhomogeneous fractional Sobolev spaces. This subject for the local case has been studied since the publication of the celebrated works due to P.-L. Lions, which laid the broad foundations of the method and outlined a wide scope of its applications. Our study is based on the analysis of the profile decomposition for the weak convergence following the approach of dislocation spaces, introduced by K. Tintarev and K.-H. Fieseler. As an application, we obtain existence of nontrivial and nonnegative solutions and ground states for fractional Schrödinger equations for a wide class of possible singular potentials, not necessarily bounded away from zero. We consider possible oscillatory nonlinearities for both cases, subcritical and critical which are superlinear at the origin, without the classical Ambrosetti and Rabinowitz growth condition. In some of our results we prove existence of solutions by means of compactness of Palais-Smale sequences of the associated functional at the mountain pass level. To this end we study and provide the behavior of the weak profile decomposition convergence under the related functionals. Moreover, we use a Pohozaev type identity in our argument to compare the minimax levels of the energy functional with the ones of the associated limit problem. Motivated by this fact, in our work we also prove that this kind of identities hold for a larger class of potentials and nonlinearities for the fractional framework.
</p>projecteuclid.org/euclid.tmna/1532484291_20180724220506Tue, 24 Jul 2018 22:05 EDTRegularity problem for $2m$-order quasilinear parabolic systems with non smooth in time principal matrix. $(A(t),m)$-caloric approximation methodhttps://projecteuclid.org/euclid.tmna/1532484292<strong>Arina A. Arkhipova Arkhipova</strong>, <strong>Jana Stará</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 36 pp..</p><p><strong>Abstract:</strong><br/>
Partial regularity of solutions to a class of $2m$-order quasilinear parabolic systems and full interior regularity for $2m$-order linear parabolic systems with non smooth in time principal matrices is proved in the paper. The coefficients are assumed to be bounded and measurable in the time variable and VMO-smooth in the space variables uniformly with respect to time. To prove the result, we apply the $(A(t),m)$-caloric approximation method, $m\geq 1$. It is both an extension of the $A(t)$-caloric approximation applied by the authors earlier to study regularity problem for systems of the second order with non-smooth coefficients and an extension of the $A$-polycaloric lemma proved by V. Bögelein in [6] to systems of $2m$-order.
</p>projecteuclid.org/euclid.tmna/1532484292_20180724220506Tue, 24 Jul 2018 22:05 EDTStrict $C^1$-triangulations in o-minimal structureshttps://projecteuclid.org/euclid.tmna/1532484293<strong>Małgorzata Czapla</strong>, <strong>Wiesław Pawłucki</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 9 pp..</p><p><strong>Abstract:</strong><br/>
Inspired by the recent articles of T. Ohmoto and M. Shiota [9], [10] on $\mathcal C^1$-triangulations of semialgebraic sets, we prove here by using different methods the following theorem: Let $R$ be a real closed field and let an expansion of $R$ to an o-minimal structure be given. Then for any closed bounded definable subset $A$ of $ R^n$ and a finite family $B_1,\dots,B_r$ of definable subsets of $A$ there exists a definable triangulation $h\colon |\mathcal K|\rightarrow A$ of $A$ compatible with $B_1,\dots,B_r$ such that $\mathcal K$ is a simplicial complex in $R^n$, $h$ is a $C^1$-embedding of each {\rm(}open{\rm)} simplex $\Delta\in \mathcal K$ and $h$ extends to a definable $C^1$-mapping defined on a definable open neighborhood of $|\mathcal K|$ in $R^n$. This improves Ohmoto-Shiota's theorem in three ways; firstly, $h$ is a $\mathcal C^1$-embedding on each simplex; secondly, the simplicial complex $\mathcal K$ is in the same space as $A$ and thirdly, our proof is performed for any o-minimal structure. The possibility to have $h$ with the first of these properties was stated by Ohmoto and Shiota as an open problem (see [9]).
</p>projecteuclid.org/euclid.tmna/1532484293_20180724220506Tue, 24 Jul 2018 22:05 EDTPointwise estimates in the Filippov lemma and Filippov-Ważewski theorem for fourth order differential inclusionshttps://projecteuclid.org/euclid.tmna/1533780028<strong>Grzegorz Bartuzel</strong>, <strong>Andrzej Fryszkowski</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 26 pp..</p><p><strong>Abstract:</strong><br/>
In this work we give a generalization of the Filippov-Ważewski Theorem to the fourth order differential inclusions in a separable complex Banach space $\mathbb{X}$ \begin{equation*} \mathcal{D}y=y^{\prime \prime \prime \prime }-( A^{2}+B^{2}) y^{\prime \prime }+A^{2}B^{2}y\in F( t,y) , \end{equation*} with the initial conditions in $c\in \[ 0,T] \begin{equation} y( c) =\alpha ,\qquad y^{\prime }( c) =\beta ,\qquad y^{\prime \prime }( c) =\gamma ,\qquad y^{\prime \prime \prime }( c) =\delta , \label{*} \end{equation} We assume that the multifunction $F\colon\[ 0,T\] \times \mathbb{X} \leadsto c( \mathbb{X}) $ is Lipschitz continuous in $y$ with the integrable Lipschitz constant $l( .) $, while $A^{2},B^{2}\in B( \mathbb{X}) $ are the infinitesimal generators of two cosine families of operators. The main result is the following version of Filippov Lemma: Theorem: Let $y_{0}\in W^{4,1}=W^{4,1}([ 0,T] ,\mathbb{X}) $ be such function with \eqref{*} that \begin{equation*} \mathrm{dist}( \mathcal{D}y_{0}( t) ,F( t,y_{0}( t) ) ) \leq p_{0}( t) \quad \text{a.e.\ in } [ c,d] \subset [ 0,T] , \end{equation*} where $p_{0}\in L^{1}[ 0,T] $. Then there are $\mathcal{\sigma }_{0}$ {\rm(}depending on $p_{0})$ and $\varphi $ such that for each $\varepsilon >0$ there exists a solution $y\in W^{4,1}$ of the above problem such that almost everywhere in $t\in [c,d]$ we have $\vert \mathcal{D}y( t) -\mathcal{D}y_{0}( t) \vert \leq \mathcal{\sigma }_{0}( t) $, \begin{alignat*}2 \vert y( t) -y_{0}( t) \vert &\leq(\varphi \ast _{c}\sigma _{0})( t) , &\qquad \vert y^{\prime }( t) -y_{0}^{\prime }( t) \vert \leq ( \varphi ^{\prime }\ast _{c}\sigma _{0}t) ( t ) , \\ \vert y^{\prime \prime }( t) -y_{0}^{\prime \prime }( t) \vert &\leq ( \varphi ^{\prime \prime }\ast _{c}\sigma _{0}) ( t) &\qquad \vert y^{\prime \prime \prime }( t) -y_{0}^{\prime \prime \prime }( t) \vert \leq( \varphi ^{\prime \prime \prime }\ast _{c}\sigma _{0})( t) , \end{alignat*} where $\ast _{c}$ stands for the convolution started at $c$. Our estimates are constructive and more precise then those in the known versions of Filippov Lemma.
</p>projecteuclid.org/euclid.tmna/1533780028_20180808220059Wed, 08 Aug 2018 22:00 EDTRealization of a graph as the Reeb graph of a Morse function on a manifoldhttps://projecteuclid.org/euclid.tmna/1533780029<strong>Łukasz Patryk Michalak</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 14 pp..</p><p><strong>Abstract:</strong><br/>
We investigate the problem of the realization of a given graph as the Reeb graph $\mathcal{R}(f)$ of a smooth function $f\colon M\rightarrow \mathbb{R}$ with finitely many critical points, where $M$ is a closed manifold. We show that for any $n\geq2$ and any graph $\Gamma$ admitting the so-called good orientation there exist an $n$-manifold $M$ and a Morse function $f\colon M\rightarrow \mathbb{R} $ such that its Reeb graph $\mathcal{R}(f)$ is isomorphic to $\Gamma$, extending previous results of Sharko and Masumoto-Saeki. We prove that Reeb graphs of simple Morse functions maximize the number of cycles. Furthermore, we provide a complete characterization of graphs which can arise as Reeb graphs of surfaces.
</p>projecteuclid.org/euclid.tmna/1533780029_20180808220059Wed, 08 Aug 2018 22:00 EDTStrong solutions in $L^2$ framework for fluid-rigid body interaction problem. Mixed casehttps://projecteuclid.org/euclid.tmna/1533780030<strong>Hind Al Baba</strong>, <strong>Nikolai V. Chemetov</strong>, <strong>Šárka Nečasová</strong>, <strong>Boris Muha</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 14 pp..</p><p><strong>Abstract:</strong><br/>
The paper deals with the problem describing the motion of a rigid body inside a viscous incompressible fluid when the mixed boundary conditions are considered. At the fluid-rigid body interface the slip Navier boundary condition is prescribed, having the continuity of velocity just in the normal component, and the Dirichlet condition is given on the boundary of the fluid domain. The existence and uniqueness of the local strong solution is proven by the local transformation and the fixed point argument.
</p>projecteuclid.org/euclid.tmna/1533780030_20180808220059Wed, 08 Aug 2018 22:00 EDTGlobal existence of a diffusion limit with damping for the compressible radiative Euler system coupled to an electromagnetic fieldhttps://projecteuclid.org/euclid.tmna/1533780032<strong>Xavier Blanc</strong>, <strong>Bernard Ducomet</strong>, <strong>Šárka Nečasová</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 25 pp..</p><p><strong>Abstract:</strong><br/>
We study the Cauchy problem for a system of equations corresponding to a singular limit of radiative hydrodynamics, namely the 3D radiative compressible Euler system coupled to an electromagnetic field through the MHD approximation. Assuming the presence of damping together with suitable smallness hypotheses for the data, we prove that this problem admits a unique global smooth solution.
</p>projecteuclid.org/euclid.tmna/1533780032_20180808220059Wed, 08 Aug 2018 22:00 EDTSteady solutions to the Navier-Stokes-Fourier system for dense compressible fluidhttps://projecteuclid.org/euclid.tmna/1533780033<strong>Šimon Axmann</strong>, <strong>Piotr B. Mucha</strong>, <strong>Milan Pokorný Nečasová</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 25 pp..</p><p><strong>Abstract:</strong><br/>
We establish existence of strong solutions to the stationary Navier-Stokes-Fourier system for compressible flows with density dependent viscosities in regime of heat conducting fluids with very high densities. In comparison to the known results considering the low Mach number case, we work in the $L^p$-setting combining the methods for the weak solutions with the method of decomposition. Moreover, the magnitude of gradient of the density as well as other data are not limited, our only assumption is the given total mass must be sufficiently large.
</p>projecteuclid.org/euclid.tmna/1533780033_20180808220059Wed, 08 Aug 2018 22:00 EDTIntegrability of the derivative of solutions to a singular one-dimensional parabolic problemhttps://projecteuclid.org/euclid.tmna/1533780035<strong>Atsushi Nakayasu</strong>, <strong>Piotr Rybka</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 19 pp..</p><p><strong>Abstract:</strong><br/>
We study integrability of the derivative of a solution to a singular one-dimensional parabolic equation with initial data in $W^{1,1}$. In order to avoid additional difficulties we consider only the periodic boundary conditions. The problem we study is a gradient flow of a convex, linear growth variational functional. We also prove a similar result for the elliptic companion problem, i.e. the time semidiscretization.
</p>projecteuclid.org/euclid.tmna/1533780035_20180808220059Wed, 08 Aug 2018 22:00 EDTNonzero positive solutions of a multi-parameter elliptic system with functional BCshttps://projecteuclid.org/euclid.tmna/1511751650<strong>Gennaro Infante</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 11 pp..</p><p><strong>Abstract:</strong><br/>
We prove, by topological methods, new results on the existence of nonzero positive weak solutions for a class of multi-parameter second order elliptic systems subject to functional boundary conditions. The setting is fairly general and covers the case of multi-point, integral and nonlinear boundary conditions. We also present a non-existence result. We provide some examples to illustrate the applicability of our theoretical results.
</p>projecteuclid.org/euclid.tmna/1511751650_20180809220227Thu, 09 Aug 2018 22:02 EDTThree-dimensional thermo-visco-elasticity with the Einstein-Debye $(\theta^3+\theta)$-law for the specific heat. Global regular solvabilityhttps://projecteuclid.org/euclid.tmna/1534557630<strong>Irena Pawłow</strong>, <strong>Wojciech M. Zajączkowski</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 33 pp..</p><p><strong>Abstract:</strong><br/>
A three-dimensional thermo-visco-elastic system for the Kelvin-Voigt type material at small strain is considered. The system involves the constant heat conductivity and the specific heat satisfying the Einstein-Debye $(\theta^3+\theta)$-law. Such a nonlinear law, relevant at relatively low temperatures, represents the main novelty of the paper. The existence of global regular solutions is proved without the small data assumption. The crucial part of the proof is the strictly positive lower bound on the absolute temperature $\theta$. The problem remains open in the case of the Debye $\theta^3$-law. The existence of local in time solutions is proved by the Banach successive approximations method. The global a priori estimates are derived with the help of the theory of anisotropic Sobolev spaces with a mixed norm. Such estimates allow to extend the local solution step by step in time.
</p>projecteuclid.org/euclid.tmna/1534557630_20180817220054Fri, 17 Aug 2018 22:00 EDT$L_2$-theory for two incompressible fluids separated by a free interfacehttps://projecteuclid.org/euclid.tmna/1534557631<strong>Irina V. Denisova</strong>, <strong>Vsevolod A. Solonnikov</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 26 pp..</p><p><strong>Abstract:</strong><br/>
The paper is devoted to the problem of non-stationary motion of two viscous incompressible fluids separated by a free surface and contained in a bounded vessel. It is assumed that the fluids are subject to mass forces and capillary forces at the interface. We prove the stability of a rest state under the assumption that initial velocities are small, a free interface is close to a sphere at an initial instant of time, and mass forces decay as $t\to\infty$.
</p>projecteuclid.org/euclid.tmna/1534557631_20180817220054Fri, 17 Aug 2018 22:00 EDTRelative entropy method for measure-valued solutions in natural scienceshttps://projecteuclid.org/euclid.tmna/1534557632<strong>Tomasz Dębiec</strong>, <strong>Piotr Gwiazda</strong>, <strong>Kamila Łyczek</strong>, <strong>Agnieszka Świerczewska-Gwiazda</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 25 pp..</p><p><strong>Abstract:</strong><br/>
We describe the applications of the relative entropy framework introduced in [10]. In particular the uniqueness of an entropy solution is proven for a scalar conservation law, using the notion of measure-valued entropy solutions. Further we survey recent results concerning measure-valued-strong uniqueness for a number of physical systems - incompressible and compressible Euler equations, compressible Navier-Stokes, polyconvex elastodynamics and general hyperbolic conservation laws, as well as long-time asymptotics of the McKendrick-Von Foerster equation.
</p>projecteuclid.org/euclid.tmna/1534557632_20180817220054Fri, 17 Aug 2018 22:00 EDTThis volume is dedicated to the memory of Marek Burnathttps://projecteuclid.org/euclid.tmna/1538791339<strong>Wojciech Kryszewski</strong>, <strong>Wojciech Zajączkowski</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Volume 52, Number 1, 1--3.</p><p><strong>Abstract:</strong><br/>
The present issue of Topological Methods in Nonlinear Analysis is devoted to the memory of the late Professor Marek Burnat. The untimely death of Professor Burnat, the mathematician, the teacher and a good friend of many of us was a great loss to the mathematical community. It was acknowledged with much sorrow and regret.
Professor Marek Burnat passed away at age 81 on December 19, 2015, in Warsaw. He was Professor Emeritus at the Faculty of Mathematics, Computer Sciences and Mechanics of the University of Warsaw and a distinguished specialist in the area of partial differential equations and their applications. Among his numerous interests there were: geometrical theory of continuum mechanics and their Riemann invariants, numerical methods of solving equations of fluid dynamics and equations of elasticity theory, models of quantum dynamics on Hilbert spaces and application of these methods in the crystal theory.
When on October 29, 1991 the Schauder Center was established at the Nicolaus Copernicus University Professor Burnat was designated to the first Scientific Council of the Center. For many years he was also the member of the Editorial Committee of the journal Topological Methods in Nonlinear Analysis published by the Center.
</p>projecteuclid.org/euclid.tmna/1538791339_20181005220239Fri, 05 Oct 2018 22:02 EDTExistence of positive ground solutions for biharmonic equations via Pohožaev-Nehari manifoldhttps://projecteuclid.org/euclid.tmna/1541473235<strong>Liping Xu</strong>, <strong>Haibo Chen</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 20 pp..</p><p><strong>Abstract:</strong><br/>
We investigate the following nonlinear biharmonic equations with pure power nonlinearities: \begin{equation*} \begin{cases} \triangle^2u-\triangle u+V(x)u= u^{p-1}u & \text{in } \mathbb{R}^N,\\ u>0 &\text{for } u\in H^2(\mathbb{R}^N), \end{cases} \end{equation*} where $2 < p< 2^*={2N}/({N-4})$. Under some suitable assumptions on $V(x)$, we obtain the existence of ground state solutions. The proof relies on the Pohožaev-Nehari manifold, the monotonic trick and the global compactness lemma, which is possibly different to other papers on this problem. Some recent results are extended.
</p>projecteuclid.org/euclid.tmna/1541473235_20181105220057Mon, 05 Nov 2018 22:00 ESTPositive solutions for singular impulsive Dirichlet boundary value problemshttps://projecteuclid.org/euclid.tmna/1541473236<strong>Liang Bai</strong>, <strong>Juan J. Nieto</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 24 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, a class of singular impulsive Dirichlet boundary value problems is considered. By using variational method and critical point theory, different parameter ranges are obtained to guarantee existence and multiplicity of positive classical solutions of the problem when nonlinearity exhibits different growths.
</p>projecteuclid.org/euclid.tmna/1541473236_20181105220057Mon, 05 Nov 2018 22:00 ESTExistence of solutions for the semilinear corner degenerate elliptic equationshttps://projecteuclid.org/euclid.tmna/1541473237<strong>Jae-Myoung Kim</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 13 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we are concerned with the following elliptic equations: \begin{equation*}\label{e:JG} \begin{cases} -\Delta_{\mathbb{M}}u = \lambda f &\textmd{in } z:= (r,x,t) \in \mathbb{M}_0,\\ u= 0 &\text{on } \partial\mathbb{M}. \end{cases} \end{equation*} Here, $\lambda >0$ and $M=[0,1)\times X\times[0,1)$ as a local model of stretched corner-manifolds, that is, the manifolds with corner singularities with dimension $N=n+2\geq 3$. Here $X$ is a closed compact submanifold of dimension $n$ embedded in the unit sphere of $\mathbb{R}^{n+1}$. We study the existence of nontrivial weak solutions for the semilinear corner degenerate elliptic equations without the Ambrosetti and Rabinowitz condition via the mountain pass theorem and fountain theorem.
</p>projecteuclid.org/euclid.tmna/1541473237_20181105220057Mon, 05 Nov 2018 22:00 ESTContractibility of manifolds by means of stochastic flowshttps://projecteuclid.org/euclid.tmna/1541473238<strong>Alexandra Antoniouk</strong>, <strong>Sergiy Maksymenko</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 13 pp..</p><p><strong>Abstract:</strong><br/>
In the paper [Probab. Theory Relat. Fields, 100 (1994), 417-428] Xue-Mei Li has shown that the moment stability of an SDE is closely connected with the topology of the underlying manifold. In particular, she gave sufficient condition on SDE on a manifold $M$ under which the fundamental group $\pi_1 M=0$. We prove that under similar analytical conditions the manifold $M$ is contractible, that is all homotopy groups $\pi_n M$, $n\geq1$, vanish.
</p>projecteuclid.org/euclid.tmna/1541473238_20181105220057Mon, 05 Nov 2018 22:00 ESTContractibility of manifolds by means of stochastic flowshttps://projecteuclid.org/euclid.tmna/1541473239<strong>Marcio Colombo Fenille</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 17 pp..</p><p><strong>Abstract:</strong><br/>
Let $f$ be a map from a one-relator model two-complex $K_{\mathcal{P}}$ into the real projective plane. The composition $\varrho\circ f_{\#}$ of the homomorphism $f_{\#}$ induced by $f$ on fundamental groups with the action $\varrho$ of $\pi_1(\mathbb{R}\mathrm{P}^2)$ over $\pi_2(\mathbb{R}\mathrm{P}^2)$ provides a local integer coefficient system $f_{\#}^{\varrho}$ over $K_{\mathcal{P}}$. We prove that if the twisted integer cohomology group $H^2(K_{\mathcal{P}};_{f_{\#}^{\varrho}}\mathbb Z)=0$, then $f$ is homotopic to a non-surjective map. As an intermediary step for the proof, we show that if $H^2(K_{\mathcal{P}};_{\beta}\mathbb Z)=0$ for some local integer coefficient system $\beta$ over $K_{\mathcal{P}}$, then $K_{\mathcal{P}}$ is aspherical.
</p>projecteuclid.org/euclid.tmna/1541473239_20181105220057Mon, 05 Nov 2018 22:00 ESTOn spectral convergence for some parabolic problems with locally large diffusionhttps://projecteuclid.org/euclid.tmna/1541473240<strong>Maria C. Carbinatto</strong>, <strong>Krzysztof P. Rybakowski</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 34 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, which is a sequel to [1], we extend the spectral convergence result from [5] to a larger class of singularly perturbed families of scalar linear differential operators. This also extends the Conley index continuation principles from [1].
</p>projecteuclid.org/euclid.tmna/1541473240_20181105220057Mon, 05 Nov 2018 22:00 ESTApproximate controllability for abstract semilinear impulsive functional differential inclusions based on Hausdorff product measureshttps://projecteuclid.org/euclid.tmna/1543114845<strong>Jian-Zhong Xiao</strong>, <strong>Xing-Hua Zhu</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 20 pp..</p><p><strong>Abstract:</strong><br/>
A second order semilinear impulsive functional differential inclusion in a separable Hilbert space is considered. Without imposing hypotheses of the compactness on the cosine families of operators, some sufficient conditions of approximate controllability are formulated in the case where the multivalued nonlinearity of the inclusion is a completely continuous map dominated by a function. By the use of resolvents of controllability Gramian operators and developing appropriate computing techniques for the Hausdorff product measures of noncompactness, the results of approximate controllability for position and velocity are derived. An example is also given to illustrate the application of the obtained results.
</p>projecteuclid.org/euclid.tmna/1543114845_20181124220058Sat, 24 Nov 2018 22:00 ESTLipschitz retractions onto sphere vs spherical cup in a Hilbert spacehttps://projecteuclid.org/euclid.tmna/1543114846<strong>Jumpot Intrakul</strong>, <strong>Phichet Chaoha</strong>, <strong>Wacharin Wichiramala</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 15 pp..</p><p><strong>Abstract:</strong><br/>
We prove that, in every infinite dimensional Hilbert space, there exists $t_0>-1$ such that the smallest Lipscthiz constant of retractions from the unit ball onto its boundary is the same as the smallest Lipschitz constant of retractions from the unit ball onto its $t$-spherical cup for all $t\in[-1,t_0]$.
</p>projecteuclid.org/euclid.tmna/1543114846_20181124220058Sat, 24 Nov 2018 22:00 ESTA continuation lemma and the existence of periodic solutions of perturbed planar Hamiltonian systems with sub-quadratic potentialshttps://projecteuclid.org/euclid.tmna/1543114847<strong>Zaihong Wang</strong>, <strong>Tiantian Ma</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 14 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we study the existence of periodic solutions of perturbed planar Hamiltonian systems of the form $$ \begin{cases} x'=f(y)+p_1(t,x,y), \\ y'=-g(x)+p_2(t,x,y). \end{cases} $$ We prove a continuation lemma for a given planar system and further use it to prove that this system has at least one $T$-periodic solution provided that $g$ has some sub-quadratic potentials.
</p>projecteuclid.org/euclid.tmna/1543114847_20181124220058Sat, 24 Nov 2018 22:00 ESTUniform stability for fractional Cauchy problems and applicationshttps://projecteuclid.org/euclid.tmna/1543114848<strong>Luciano Abadias</strong>, <strong>Edgardo Álvarez</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 22 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we give uniform stable spatial bounds for the resolvent operator families of the abstract fractional Cauchy problem on $\mathbb{R}_+$. Such bounds allow to prove existence and uniqueness of $\mu$-pseudo almost automorphic $\epsilon$-mild regular solutions to the nonlinear fractional Cauchy problem in the whole real line. Finally, we apply our main results to the fractional heat equation with critical nonlinearities.
</p>projecteuclid.org/euclid.tmna/1543114848_20181124220058Sat, 24 Nov 2018 22:00 ESTEquivalence between uniform $L^{2^\star}(\Omega)$ a-priori bounds and uniform $L^{\infty}(\Omega)$ a-priori bounds for subcritical elliptic equationshttps://projecteuclid.org/euclid.tmna/1547434817<strong>Alfonso Castro</strong>, <strong>Nsoki Mavinga</strong>, <strong>Rosa Pardo</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 14 pp..</p><p><strong>Abstract:</strong><br/>
We provide sufficient conditions for a uniform $L^{2^\star}(\Omega)$ bound to imply a uniform $L^\infty (\Omega)$ bound for positive classical solutions to a class of subcritical elliptic problems in bounded $C^2$ domains in ${\mathbb R}^N$. We also establish an equivalent result for sequences of boundary value problems.
</p>projecteuclid.org/euclid.tmna/1547434817_20190113220034Sun, 13 Jan 2019 22:00 ESTA diffusive logistic equation with U-shaped density dependent dispersal on the boundaryhttps://projecteuclid.org/euclid.tmna/1547434818<strong>Jerome Goddard II</strong>, <strong>Quinn Morris</strong>, <strong>Catherine Payne</strong>, <strong>Ratnasingham Shivaji</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 15 pp..</p><p><strong>Abstract:</strong><br/>
We study positive solutions to the steady state reaction diffusion equation: \begin{equation*} \begin{cases} - \Delta v = \lambda v(1-v), & x \in \Omega_0, \\ \frac{\partial v}{\partial \eta} + \gamma \sqrt{\lambda} ( v-A)^2 v =0 , & x \in \partial \Omega_0, \end{cases} \end{equation*} where $\Omega_0$ is a bounded domain in $\mathbb{R}^n$; $n \ge 1$ with smooth boundary $\partial \Omega_0$, ${\partial }/{\partial \eta}$ is the outward normal derivative, $A \in (0,1)$ is a constant, and $\lambda$, $\gamma$ are positive parameters. Such models arise in the study of population dynamics when the population exhibits a U-shaped density dependent dispersal on the boundary of the habitat. We establish existence, multiplicity, and uniqueness results for certain ranges of the parameters $\lambda$ and $\gamma$. We obtain our existence and mulitplicity results via the method of sub-super solutions.
</p>projecteuclid.org/euclid.tmna/1547434818_20190113220034Sun, 13 Jan 2019 22:00 ESTZero-dimensional compact metrizable spaces as attractors of generalized iterated function systemshttps://projecteuclid.org/euclid.tmna/1547434819<strong>Filip Strobin</strong>, <strong>Łukasz Maślanka</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 41 pp..</p><p><strong>Abstract:</strong><br/>
R. Miculescu and A. Mihail in 2008 introduced the concept of a generalized iterated function system (GIFS in short), a particular extension of the classical IFS. The idea is that, instead of families of selfmaps of a metric space $X$, GIFSs consist of maps defined on a finite Cartesian $m$-th power $X^m$ with values in $X$ (in such a case we say that a GIFS is of order $m$). It turned out that a great part of the classical Hutchinson theory has natural counterpart in this GIFSs' framework. On the other hand, there are known only few examples of fractal sets which are generated by GIFSs, but which are not IFSs' attractors. In the paper we study $0$-dimensional compact metrizable spaces from the perspective of GIFSs' theory. Such investigations for classical IFSs have been undertaken in the last several years, for example by T. Banakh, E. D'Aniello, M. Nowak, T.H. Steele and F. Strobin.
We prove that each such space $X$ is homeomorphic to the attractor of some GIFS on the real line. Moreover, we prove that $X$ can be embedded into the real line $\R$ as the attractor of some GIFS of order $m$ and (in the same time) a nonattractor of any GIFS of order $m-1$, as well as it can be embedded as a nonattractor of any GIFS. Then we show that a relatively simple modifications of $X$ deliver spaces whose each connected component is ``big'' and which are GIFS's attractors not homeomorphic with IFS's attractors. Finally, we use obtained results to show that a generic compact subset of a Hilbert space is not the attractor of any Banach GIFS.
</p>projecteuclid.org/euclid.tmna/1547434819_20190113220034Sun, 13 Jan 2019 22:00 ESTParabolic equations with localized large diffusion: Rate of convergence of attractorshttps://projecteuclid.org/euclid.tmna/1550631829<strong>Alexandre N. Carvalho</strong>, <strong>Leonardo Pires</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 23 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we study the asymptotic nonlinear dynamics of scalar semilinear parabolic problems of reaction-diffusion type when the diffusion coefficient becomes large in a subregion in the interior to the domain. We obtain, under suitable assumptions, that the family of attractors behaves continuously and we exhibit the rate of convergence. An accurate description of the localized large diffusion is necessary.
</p>projecteuclid.org/euclid.tmna/1550631829_20190219220420Tue, 19 Feb 2019 22:04 ESTFinite-time blow-up in a quasilinear chemotaxis system with an external signal consumptionhttps://projecteuclid.org/euclid.tmna/1550631830<strong>Pan Zheng</strong>, <strong>Chunlai Mu</strong>, <strong>Xuegang Hu</strong>, <strong>Liangchen Wang</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 17 pp..</p><p><strong>Abstract:</strong><br/>
This paper deals with a quasilinear chemotaxis system with an external signal consumption \begin{equation*} \begin{cases} u_t=\nabla\cdot(\varphi(u)\nabla u)-\nabla\cdot(u\nabla v), & (x,t)\in \Omega\times (0,\infty), \\ 0=\Delta v+u-g(x), & (x,t)\in \Omega\times (0,\infty), \end{cases} \end{equation*} under homogeneous Neumann boundary conditions in a ball $\Omega\subset \mathbb{R}^{n}$, where $\varphi(u)$ is a nonlinear diffusion function and $g(x)$ is an external signal consumption. Under suitable assumptions on the functions $\varphi$ and $g$, it is proved that there exists initial data such that the solution of the above system blows up in finite time.
</p>projecteuclid.org/euclid.tmna/1550631830_20190219220420Tue, 19 Feb 2019 22:04 ESTOptimal retraction problem for proper $k$-ball-contractive mappings in $C^{m} [0,1]$https://projecteuclid.org/euclid.tmna/1550631831<strong>Diana Caponetti</strong>, <strong>Alessandro Trombetta</strong>, <strong>Giulio Trombetta</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 15 pp..</p><p><strong>Abstract:</strong><br/>
In this paper for any $\epsilon >0$ we construct a new proper $k$-ball-contractive retraction of the closed unit ball of the Banach space $C^m [0,1]$ onto its boundary with $k < 1+ epsilon$, so that the Wośko constant $W_\gamma (C^m [0,1])$ is equal to $1$.
</p>projecteuclid.org/euclid.tmna/1550631831_20190219220420Tue, 19 Feb 2019 22:04 ESTOn the Lyapunov stability theory for impulsive dynamical systemshttps://projecteuclid.org/euclid.tmna/1550631832<strong>Everaldo Mello Bonotto</strong>, <strong>Ginnara M. Souto</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 24 pp..</p><p><strong>Abstract:</strong><br/>
In this work, we establish necessary and sufficient conditions for the uniform and orbital stability of a special class of sets on impulsive dynamical systems. The results are achieved by means of Lyapunov functions.
</p>projecteuclid.org/euclid.tmna/1550631832_20190219220420Tue, 19 Feb 2019 22:04 ESTMultiplicity of positive solutions for a fractional Laplacian equations involving critical nonlinearityhttps://projecteuclid.org/euclid.tmna/1550631833<strong>Jinguo Zhang</strong>, <strong>Xiaochun Liu</strong>, <strong>Hongying Jiao</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 32 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the following problem involving fractional Laplacian operator \begin{equation*} (-\Delta)^{s} u=\lambda f(x)|u|^{q-2}u+|u|^{2^{*}_{s}-2}u\quad \text{in } \Omega,\qquad u=0\quad \text{on } \partial\Omega, \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$, $0< s< 1$, $2^*_{s}={2N}/({N-2s})$, and $(-\Delta)^{s}$ is the fractional Laplacian. We will prove that there exists $\lambda_{*}>0$ such that the problem has at least two positive solutions for each $\lambda\in (0,\lambda_{*})$. In addition, the concentration behavior of the solutions are investigated.
</p>projecteuclid.org/euclid.tmna/1550631833_20190219220420Tue, 19 Feb 2019 22:04 ESTGlobal existence for reaction-diffusion systems modeling ions electroe-migration through biological membranes with mass control and critical growth with respect to the gradienthttps://projecteuclid.org/euclid.tmna/1550631834<strong>Bassam Al-Hamzah</strong>, <strong>Naji Yebari</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 32 pp..</p><p><strong>Abstract:</strong><br/>
This paper studies the existence of global weak solutions for reaction-diffusion systems depending on two main assumptions: the non-negative of solutions and the total mass of components are preserved with time, the non-linearities have critical growth with respect to the gradient. This work is a generalization of the work developed by Alaa and Lefraich [2] without the presence of the gradient in the kinetic reaction terms.
</p>projecteuclid.org/euclid.tmna/1550631834_20190219220420Tue, 19 Feb 2019 22:04 ESTGlobal existence for reaction-diffusion systems modeling ions electroe-migration through biological membranes with mass control and critical growth with respect to the gradienthttps://projecteuclid.org/euclid.tmna/1550631835<strong>Ethan Akin</strong>, <strong>Sławomir Plaskacz</strong>, <strong>Joanna Zwierzchowska</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 11 pp..</p><p><strong>Abstract:</strong><br/>
Adapting methods introduced by Steven Smale, we describe good strategies for a symmetric version of the Iterated Prisoner's Dilemma with $n$ players.
</p>projecteuclid.org/euclid.tmna/1550631835_20190219220420Tue, 19 Feb 2019 22:04 ESTStability of multivalued attractorshttps://projecteuclid.org/euclid.tmna/1551063639<strong>Miroslav Rypka</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 13 pp..</p><p><strong>Abstract:</strong><br/>
Stimulated by recent problems in the theory of iterated function systems, we provide a variant of the Banach converse theorem for multivalued maps. In particular, we show that attractors of continuous multivalued maps on metric spaces are stable. Moreover, such attractors in locally compact, complete metric spaces may be obtained by means of the Banach theorem in the hyperspace.
</p>projecteuclid.org/euclid.tmna/1551063639_20190224220051Sun, 24 Feb 2019 22:00 ESTMultiplicity and concentration for Kirchhoff type equations around topologically critical points in potentialhttps://projecteuclid.org/euclid.tmna/1551063640<strong>Yu Chen</strong>, <strong>Yanheng Ding</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 41 pp..</p><p><strong>Abstract:</strong><br/>
We consider the multiplicity and concentration of solutions for the Kirchhoff Type Equation \[ -\varepsilon^2 M\bigg( \varepsilon^{2-N}\int_{\mathbb{R}^N} |\nabla v|^2\,dx \bigg) \Delta v+V(x)v=f(v), \quad \mathrm{in }\ \mathbb{R}^N. \] Under suitable conditions on functions $M$, $V$ and $f$, we obtain the existence of positive solutions concentrating around the local maximum points of $V$, which gives an affirmative answer to the problem raised in [21]. Moreover, we also obtain multiplicity of solutions which are affected by the topology of critical points set of potential $V$.
</p>projecteuclid.org/euclid.tmna/1551063640_20190224220051Sun, 24 Feb 2019 22:00 ESTPositive ground states for a subcritical and critical coupled system involving Kirchhoff-Schrödinger equationshttps://projecteuclid.org/euclid.tmna/1551063641<strong>José Carlos de Albuquerque</strong>, <strong>João Marcos do Ó</strong>, <strong>Giovany M. Figueiredo</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 17 pp..</p><p><strong>Abstract:</strong><br/>
In this paper we prove the existence of positive ground state solution for a class of linearly coupled systems involving Kirchhoff-Schrödinger equations. We study the subcritical and critical case. Our approach is variational and based on minimization technique over the Nehari manifold. We also obtain a nonexistence result using a Pohozaev identity type.
</p>projecteuclid.org/euclid.tmna/1551063641_20190224220051Sun, 24 Feb 2019 22:00 ESTMarkov perfect equilibria in OLG models with risk sensitive agentshttps://projecteuclid.org/euclid.tmna/1551063642<strong>Łukasz de Balbus</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 25 pp..</p><p><strong>Abstract:</strong><br/>
In this paper, we present an overlapping generation model (OLG for short) of resource extraction with a random production function and an altruism having both paternalistic and non-paternalistic features. All generations are risk-sensitive with a constant coefficient of absolute risk aversion. The preferences are represented by a possibly dynamic inconsistent dynamic recursive utility function with non-cooperating generations. Under general conditions on the aggregator and transition probability, we examine the existence and the uniqueness of a recursive utility function and the existence of a stationary mixed Markov Perfect Nash Equilibria.
</p>projecteuclid.org/euclid.tmna/1551063642_20190224220051Sun, 24 Feb 2019 22:00 ESTNonautonomous Conley index theory. The homology index and attractor-repeller decompositionshttps://projecteuclid.org/euclid.tmna/1552356033<strong>Axel Janig</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 21 pp..</p><p><strong>Abstract:</strong><br/>
In a previous work, the author established a nonautonomous Conley index based on the interplay between a nonautonomous evolution operator and its skew-product formulation. This index is refined to obtain a~Conley index for families of nonautonomous evolution operators. Different variants such as a categorial index, a homotopy index and a homology index are obtained. Furthermore, attractor-repeller decompositions and conecting homomorphisms are introduced for the nonautonomous setting.
</p>projecteuclid.org/euclid.tmna/1552356033_20190311220052Mon, 11 Mar 2019 22:00 EDTNonautonomous Conley index theory. Continuation of Morse-decompositionshttps://projecteuclid.org/euclid.tmna/1552356034<strong>Axel Janig</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 18 pp..</p><p><strong>Abstract:</strong><br/>
In previous works the author established a nonautonomous Conley index based on the interplay between a nonautonomous evolution operator and its skew-product formulation. In this paper, the treatment of attractor-repeller decomposition is refined. The more general concept of partially ordered Morse-decompositions is used. It is shown that, in the nonautonomous setting, these Morse-decompositions persist under small perturbations. Furthermore, a continuation property for these Morse decompositions is established. Roughly speaking, the index of every Morse-set and every connecting homomorphism continue as the nonautonomous problem, depending continuously on a parameter, changes.
</p>projecteuclid.org/euclid.tmna/1552356034_20190311220052Mon, 11 Mar 2019 22:00 EDTA generic result on Weyl tensorhttps://projecteuclid.org/euclid.tmna/1552356035<strong>Anna Maria Micheletti</strong>, <strong>Angela Pistoia</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 13 pp..</p><p><strong>Abstract:</strong><br/>
Let $M$ be a connected compact $C^\infty$ manifold of dimension $n\ge 4$ without boundary. Let $ \mathcal{M}^k$ be the set of all $C^k$ Riemannian metrics on $M$. Any $g\in\mathcal{M}^k$ determines the Weyl tensor $$ \mathcal W^g\colon M\to \mathbb R^{4n},\qquad \mathcal W^g(\xi):=(W^g_{ijkl}(\xi))_{i,j,k,l=1,\dots,n}.$$ We prove that the set $$\mathcal{A}:=\big\{g\in \mathcal{M}^k : |\mathcal W^g(\xi)|+|D \mathcal W^g(\xi)|+|D^2 \mathcal W^g(\xi)|>0\ \hbox{for any}\ \xi\in M\big\}$$ is an open dense subset of $\mathcal{M}^k$.
</p>projecteuclid.org/euclid.tmna/1552356035_20190311220052Mon, 11 Mar 2019 22:00 EDTNew results of mixed monotone operator equationshttps://projecteuclid.org/euclid.tmna/1552356036<strong>Tian Wang</strong>, <strong>Zhaocai Hao</strong>. <p><strong>Source: </strong>Topological Methods in Nonlinear Analysis, Advance publication, 19 pp..</p><p><strong>Abstract:</strong><br/>
In this article, we study the existence and uniqueness of fixed points for some mixed monotone operators and monotone operators with perturbation. These mixed monotone operators and monotone operators are $e$-concave-convex operators and $e$-concave operators respectively. Without using compactness or continuity, we obtain the existence and uniqueness of fixed points by monotone iterative techniques and properties of cones. Our main results extended and improved some existing results. Also, we applied the results to some differential equations.
</p>projecteuclid.org/euclid.tmna/1552356036_20190311220052Mon, 11 Mar 2019 22:00 EDT