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1.
Let \(\mathcal {U}=\{U(t,s)\}_{t\ge s\ge 0}\) be a strongly continuous and exponentially bounded evolution family acting on a complex Banach space X and let \(\mathcal {X}\) be a certain Banach function space of X-valued functions. We prove that the growth bound of the family \(\mathcal {U}\) is less than or equal to \(-\frac{1}{c(\mathcal {U}, \mathcal {X})}\) provided that the convolution operator \(f\mapsto \mathcal {U}*f\) acts on \(\mathcal {X}.\) It is well known that under the latter assumption, the convolution operator is bounded and then \(c(\mathcal {U}, \mathcal {X})\) denotes (ad-hoc) its norm in \(\mathcal {L}(\mathcal {X}).\) As a consequence, we prove that if \(\sup \nolimits _{s\ge 0}\int \nolimits _{s}^\infty \Vert U(t,s)\Vert dt=u_1(\mathcal {U})<\infty ,\) then \(\omega _0(\mathcal {U})u_1(\mathcal {U})\le -1.\) Finally, we give an example showing that the accuracy of the estimates may be quite accurate. 相似文献
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Manas Das Dr. Atul Manvar Dr. Maïwenn Jacolot Dr. Marco Blangetti Dr. Roderick C. Jones Prof. Dr. Donal F. O'Shea 《Chemistry (Weinheim an der Bergstrasse, Germany)》2015,21(24):8737-8740
The synthesis of bench‐stable α,α‐bis(trimethylsilyl)toluenes and tris(trimethylsilyl)methane is described and their use in stereoselective Peterson olefinations has been achieved with a wide substrate scope. Product stereoselectivity was poor with carbonyl electrophiles (E/Z ~1:1 to 4:1) though this was significantly improved by employing the corresponding substituted N‐benzylideneaniline (up to 99:1) as an alternative electrophile. The olefination byproduct was identified as N,N‐bis(trimethylsilyl)aniline and could be easily separated from product by aqueous acid extraction. Evidence for an autocatalytic cycle has been obtained. 相似文献
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Two theorems concerning the existence of positive solutions for the singular equation
are presented. The results are obtained by using the nonlinear Leray-Schauder alternative and the lower-upper solution method. 相似文献
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In this paper we establish some new dynamic inequalities on time scales which contain in particular generalizations of integral and discrete inequalities due to Hardy, Littlewood, P′olya, D'Apuzzo, Sbordone and Popoli. We also apply these inequalities to prove a higher integrability theorem for decreasing functions on time scales. 相似文献
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Donal O'Regan 《Applicable analysis》2013,92(3):565-574
New fixed point results and Leray-Schauder alternatives are presented for countably P -concentrative multimaps. The notion of an essential map is also introduced for a wide class of maps. 相似文献
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In this paper, by using the modification function technique and the Leray-Schauder degree method some existence and multiplicity results for sign- changing solutions of certain three-point boundary value problems are obtained. The main result of this paper is an improvement on a former existence result for sign-changing solutions of certain three-point boundary value problems. 相似文献