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We consider a constrained semilinear evolution inclusion of parabolic type involving an \begin{document}$m$\end{document} -dissipative linear operator and a source term of multivalued type in a Banach space and topological properties of... more
We consider a constrained semilinear evolution inclusion of parabolic type involving an \begin{document}$m$\end{document} -dissipative linear operator and a source term of multivalued type in a Banach space and topological properties of the solution map. We establish the \begin{document}$R_δ$\end{document} -description of the set of solutions surviving in the constraining area and show a relation between the fixed point index of the Krasnosel'skii-Poincare operator of translation along trajectories associated with the problem and the appropriately defined constrained degree of the right-hand side in the equation. This provides topological tools appropriate to obtain results on the existence of periodic solutions to studied differential problems.
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The Darboux theorem, one of the fundamental results in analysis, states that the derivative of a real (not necessarily continuously) differentiable function defined on a compact interval has the intermediate value property, i.e. attains... more
The Darboux theorem, one of the fundamental results in analysis, states that the derivative of a real (not necessarily continuously) differentiable function defined on a compact interval has the intermediate value property, i.e. attains each value between the derivatives at the endpoints. The Bolzano intermediate value theorem, which implies Darboux’s theorem when the derivative is continuous, states that a continuous real-valued function $f$ defined on $[-1,1]$ satisfying $f(-1)<0$ and $f(1)>0$ 0, has a zero, i.e. $f(x) = 0$ for at least one number $-1
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We study several aspects of a generalized Perron-Frobenius and Krein-Rutman theorems concerning spectral properties of a (possibly unbounded) linear operator on a cone in a Banach space. The operator is subject to the so-called tangency... more
We study several aspects of a generalized Perron-Frobenius and Krein-Rutman theorems concerning spectral properties of a (possibly unbounded) linear operator on a cone in a Banach space. The operator is subject to the so-called tangency or weak range assumptions implying the resolvent invariance of the cone. The further assumptions rely on relations between the spectral and essential spectral bounds of the operator. In general we do not assume that the cone induces the Banach lattice structure into the underlying space.
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The reaction of aryl iodides with 1,1-diphenyl-silacyclobutane in the presence of a catalytic amount of Pd(PPh3)4 affords unexpected ring-opening adducts, 1- and 2-propenyl(triaryl)-silanes, in good yields. On the other hand, the... more
The reaction of aryl iodides with 1,1-diphenyl-silacyclobutane in the presence of a catalytic amount of Pd(PPh3)4 affords unexpected ring-opening adducts, 1- and 2-propenyl(triaryl)-silanes, in good yields. On the other hand, the PdCl2(PhCN)2-catalyzed reaction of 1,1-diphenylsilacyclobutanes with aryl halides gives unexpected products, triarylsilanols, after hydrolysis in moderate yields. The catalysis involves the reaction of aryl–palladium intermediates with silacyclobutanes along with regioselective aryl–silicon bond formation. Copyright © 2001 John Wiley & Sons, Ltd.
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Summary. - In this paper we study the existence of solutions for a system of upper semicontinu-ous, non necessarily convex, multivalued maps defined in abstract spaces. To this aim, we investigate the properties of the solution set of a... more
Summary. - In this paper we study the existence of solutions for a system of upper semicontinu-ous, non necessarily convex, multivalued maps defined in abstract spaces. To this aim, we investigate the properties of the solution set of a multivalued equation. In ...
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We present some abstract theorems on the existence of selections and graph-approximations of set-valued mappings with convex values in the equivariant setting, i.e maps commuting with the action of a compact group. Some known results of... more
We present some abstract theorems on the existence of selections and graph-approximations of set-valued mappings with convex values in the equivariant setting, i.e maps commuting with the action of a compact group. Some known results of the Michael, Browder and Cellina type are generalized to this context. The equivariant measurable as well as Caratheodory selection/approximation problems are also studied.
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We prove the existence of solutions $u$ in $H^1(\mathbb{R}^N,\mathbb{R}^M)$ of the following strongly coupled semilinear system of second order elliptic PDEs on $\mathbb{R}^N$ \[ \mathcal{P}[u] = f(x,u,\nabla u), \quad x\in \mathbb{R}^N,... more
We prove the existence of solutions $u$ in $H^1(\mathbb{R}^N,\mathbb{R}^M)$ of the following strongly coupled semilinear system of second order elliptic PDEs on $\mathbb{R}^N$ \[ \mathcal{P}[u] = f(x,u,\nabla u), \quad x\in \mathbb{R}^N, \] whith pointwise constraints. We present the construction of the suitable topoligical degree which allows us to solve the above system on bounded domains. The key step in the proof consists of showing that the sequence of solutions of the truncated system is compact in $H^1$ by the use of the so-called tail estimates.
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An approximative approach to a generalized theory of the topological mapping degree is presented. Some new wide classes of operators acting in Banach spaces, which include ^-proper mappings of Petryshyn, are introduced and studied from... more
An approximative approach to a generalized theory of the topological mapping degree is presented. Some new wide classes of operators acting in Banach spaces, which include ^-proper mappings of Petryshyn, are introduced and studied from the viewpoint of the homotopic properties of the topological degree. The results are applied in some existence aspects of abstract nonlinear equations. Introduction One of the main tools for the study of existence problems for a nonlinear equation f(y) = x is the so-called topological degree of a mapping /. Generally speaking, the degree of / is an element of a suitably chosen abelian group G which is invariant under homotopies and, in the case when it does not vanish, the equation f(y) = x has at least an approximative solution. If the map acts in a finite-dimensional space, then assuming the continuity of / we can define the Brouwer degree [1, 14], which is integer valued. The direct generalization of the Brouwer degree theory to the infinite-dimens...
We consider a constrained evolution inclusions of parabolic type \eqref{inkluzja-rozn} involving an $m$-dissipative linear operator and the source term of multivalued type in a Banach space and topological properties of the solution map.... more
We consider a constrained evolution inclusions of parabolic type \eqref{inkluzja-rozn} involving an $m$-dissipative linear operator and the source term of multivalued type in a Banach space and topological properties of the solution map. We show a relation between the constrained fixed point index of the Krasnosel'skii--Poincar\'{e} operator of translation along trajectories associated with \eqref{inkluzja-rozn} and the appropriately defined constrained degree of $A + F\le 0 , \cdot \pr $ of the right-hand side in \eqref{inkluzja-rozn}. Our results extend those of \cite{cw} and \cite{gab-krysz}.
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Consider the semilinear Schrodinger equation (*) $-\Delta u + V(x)u = f(x,u)$, $u\in H^1(\mathbf {R} ^N)$. It is shown that if $f$, $V$ are periodic in the $x$-variables, $f$ is superlinear at $u=0$ and $\pm\infty$ and 0 lies in a... more
Consider the semilinear Schrodinger equation (*) $-\Delta u + V(x)u = f(x,u)$, $u\in H^1(\mathbf {R} ^N)$. It is shown that if $f$, $V$ are periodic in the $x$-variables, $f$ is superlinear at $u=0$ and $\pm\infty$ and 0 lies in a spectral gap of $-\Delta+V$, then (*) has at least one nontrivial solution. If in addition $f$ is odd in $u$, then (*) has infinitely many (geometrically distinct) solutions. The proofs rely on a degree-theory and a linking-type argument developed in this paper.
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In the paper, we provide the construction of a coincidence degree being a homotopy invariant detecting the existence of solutions of equations or inclusions of the form Ax ∈ F ( x ), x ∈ U , where A : D ( A ) ⊸ E is an m -accretive... more
In the paper, we provide the construction of a coincidence degree being a homotopy invariant detecting the existence of solutions of equations or inclusions of the form Ax ∈ F ( x ), x ∈ U , where A : D ( A ) ⊸ E is an m -accretive operator in a Banach space E , F : K ⊸ E is a weakly upper semicontinuous set-valued map constrained to an open subset U of a closed set K ⊂ E . Two different approaches are presented. The theory is applied to show the existence of non-trivial positive solutions of some nonlinear second-order partial differential equations with discontinuities. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.
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(~) 1994 Kluwer Academic Publishers. Printed in the Netherlands. ... Fixed-Point Index for Compositions of Set-Valued ... Maps with Proximally oc-Connected Values on ... RALF BADER and WOJCIECH KRYSZEWSKI 2 I Mathematisches Institut,... more
(~) 1994 Kluwer Academic Publishers. Printed in the Netherlands. ... Fixed-Point Index for Compositions of Set-Valued ... Maps with Proximally oc-Connected Values on ... RALF BADER and WOJCIECH KRYSZEWSKI 2 I Mathematisches Institut, Universittit Mlinchen, Theresienstrasse 39, D-...
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ABSTRACT A subset K of a Banach space E is called an ℒ-retract if there exists a neighborhood U of K, a retraction r:U→K, a neighborhood V⊆U of K and a constant L&amp;gt;0 such that x-r(x)≤Ldx,K for all x∈V· In this paper the... more
ABSTRACT A subset K of a Banach space E is called an ℒ-retract if there exists a neighborhood U of K, a retraction r:U→K, a neighborhood V⊆U of K and a constant L&amp;gt;0 such that x-r(x)≤Ldx,K for all x∈V· In this paper the following theorem is shown: Let K⊆E be a compact ℒ-retract with a nontrivial Euler characteristic χ(K)≠0. Any norm-to-weak * upper semicontinuous set-valued map Φ:K→2 X * with nonempty, weak * -compact convex values admits a generalized equilibrium, i.e., a point x 0 ∉K such that Φx 0 ∩N K x 0 ≠∅, where N K x 0 is the Clarke normal cone at x 0 . The class of ℒ-retracts is large, containing closed convex sets, epi-Lipschitzian sets etc., thus the above result generalizes many previously known results. Also, new classes of “regular” and “strictly regular” sets are introduced and studied. Existence of equilibria is shown for sets in the class, without assuming that the these sets are compact.
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(Received: 9 November 1999; in final form: 5 December 2000) Abstract. We study the existence and the structure of solutions to differential inclusions with constraints. We show that the set of all viable solutions to the Cauchy problem... more
(Received: 9 November 1999; in final form: 5 December 2000) Abstract. We study the existence and the structure of solutions to differential inclusions with constraints. We show that the set of all viable solutions to the Cauchy problem for a Carathéodory-type differential inclusion ...