{"id":436,"date":"2017-09-01T06:12:55","date_gmt":"2017-09-01T06:12:55","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=436"},"modified":"2017-09-01T06:13:27","modified_gmt":"2017-09-01T06:13:27","slug":"krasnoselskiiguanyufuhehanshulianxuxingdeyigedingli","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=436","title":{"rendered":"Krasnoselskii\u5173\u4e8e\u590d\u5408\u51fd\u6570\u8fde\u7eed\u6027\u7684\u4e00\u4e2a\u5b9a\u7406"},"content":{"rendered":"

\u6458\u8981<\/span>.<\/span> Krasnoselskii\u5728[1<\/a>,Thm.~I.2.1]\u4e2d\u7ed9\u51fa\u4e86\u6709\u754cNemytskii\u7b97\u5b50\u8fde\u7eed\u6027\u7684\u4e00\u4e2a\u5224\u65ad, Nemytskii\u7b97\u5b50\u4e00\u822c\u800c\u8a00\u4e0d\u662f\u7ebf\u6027\u7684, \u6545\u8fd9\u4e00\u7ed3\u679c\u662f\u975e\u5e38\u91cd\u8981\u7684\u3002
\n<\/div>
\n\n\u6211\u4eec\u76f4\u63a5\u9648\u8ff0\u5b9a\u7406\u5982\u4e0b\u3002
\n
\u5b9a\u7406 1<\/span>.<\/span> \u5047\u8bbe$g:\\Omega\\times \\mathbb{R}^n\\to \\mathbb{R}$\u662f\u4e00\u4e2aCarath\\’eodory\u51fd\u6570, \u5373$g(x,u)$\u5173\u4e8e$x\\in\\Omega$\u662f\u53ef\u6d4b\u7684, \u5173\u4e8e$u\\in \\mathbb{R}^n$\u662f\u8fde\u7eed\u7684\u3002\u82e5$g$\u6ee1\u8db3\u5982\u4e0b\u589e\u957f\u6761\u4ef6: \u5bf9\u67d0\u4e2a$s\\geq1$,
\n \\begin{equation}\\label{eq:growth-condi}
\n \\lvert g(x,u) \\rvert\\leq C(1+\\lvert u \\rvert^s).
\n \\end{equation}
\n \u5219\u5bf9\u4efb\u4f55\u7684$p\\in[1,+\\infty)$, Nemytskii\u7b97\u5b50
\n \\begin{align*}
\n \\mathcal{N}\\mathpunct{:}L^{sp}(\\Omega)&\\to L^p(\\Omega)\\\\
\n u&\\mapsto g(\\cdot, u(\\cdot))
\n \\end{align*}
\n \u662f\u8fde\u7eed\u7684\u3002
\n<\/div>
\n
\n
\u8bc1\u660e<\/span>.<\/span> \u5f53$\\lvert\\Omega\\rvert<+\\infty$\u65f6, \u6ce8\u610f\u5230,\n \\[\n \\lvert g(x,u) \\rvert^p\\leq C^p(1+\\lvert u \\rvert^s)^p\\leq(2C)^p\\left( 1+\\lvert u \\rvert^{ps} \\right),\n \\]\n \u53ef\u89c1, $g\\in L^p(\\Omega)$. \n\n \u4e3a\u4e86\u8bc1\u660e\u7b97\u5b50$\\mathcal{N}$\u7684\u8fde\u7eed\u6027, \u5047\u8bbe$u_n\\to u\\in L^{ps}(\\Omega)$, \u6211\u4eec\u9700\u8981\u8bc1\u660e$\\lVert g(x,u_n(x))-g(x,u(x))\\rVert_{L^p(\\Omega)}\\to0$. \u53cd\u8bbe\u7ed3\u8bba\u4e0d\u6210\u7acb, \u5219\u5b58\u5728$\\epsilon>0$, \u4ee5\u53ca\u5b50\u5e8f\u5217$\\left\\{ v_n \\right\\}\\subset\\left\\{ u_n \\right\\}$\u4f7f\u5f97
\n \\begin{equation}\\label{eq:contra}
\n \\lVert g(x,v_n(x))-g(x,u(x))\\rVert_{L^p(\\Omega)}\\geq\\epsilon.
\n \\end{equation}
\n \u73b0\u5728\u7531\u4e8e$u_n\\to u\\in L^{ps}(\\Omega)$, \u6211\u4eec\u77e5\u9053$v_n\\to u\\in L^{ps}(\\Omega)$. \u9009\u53d6$\\left\\{ w_n \\right\\}\\subset\\left\\{ v_n \\right\\}$\u4f7f\u5f97$\\lVert w_{n+1}-w_n\\rVert_{L^{ps}(\\Omega)}\\leq 2^{-n}$, \u5e76\u4ee4
\n \\[
\n W(x)=\\lvert w_1(x) \\rvert+\\sum_{n=1}^\\infty\\lvert w_{n+1}(x)-w_n(x)\\rvert,
\n \\]
\n\u5219\u6613\u89c1
\n\\[
\n \\lvert w_n \\rvert\\leq W(x),\\quad \\lvert u \\rvert\\leq W(x),\\quad W(x)\\in L^{ps}(\\Omega).
\n\\]
\n\u73b0\u5728, \u6211\u4eec\u6ce8\u610f\u5230
\n\\begin{align*}
\n \\lvert g(x,w_n(x))-g(x,u(x)) \\rvert^p&\\leq 2^p\\left( \\lvert g(x,w_n(x))\\rvert^p+\\lvert g(x,u(x)) \\rvert^p \\right)\\\\
\n &\\leq (4C)^p\\left( 1+ \\lvert w_n \\rvert^{ps}+1+\\lvert u \\rvert^{ps} \\right)\\\\
\n &\\leq (4C)^p\\left( 2+ \\lvert W \\rvert^{ps} \\right)
\n\\end{align*}
\n\u7531\u4e8e\u53f3\u7aef\u662f\u53ef\u79ef\u7684, \u6545\u7531Lebesgue\u63a7\u5236\u6536\u655b\u5b9a\u7406\u77e5$g(x,w_n(x))\\to g(x,u(x))$\u4f9d$L^p(\\Omega)$\u6536\u655b\u3002\u8fd9\u4e0e\\eqref{eq:contra}\u77db\u76fe\u3002
\n<\/div>
\n
\u6ce8\u8bb0 1<\/span>.<\/span> \u4e0a\u9762\u8bc1\u660e\u4e2d\u5047\u8bbe\u533a\u57df\u662f\u6709\u754c\u7684\u4e3b\u8981\u662f\u56e0\u4e3a$g$\u7684\u589e\u957f\u6027\u6761\u4ef6\u4e2d\u6709\u5e38\u6570\u9879\u3002\u82e5\u6ca1\u6709\u5e38\u6570\u9879, \u5219\u8be5\u6761\u4ef6\u53ef\u4ee5\u53bb\u6389\u3002
\n<\/div>
\n
\u6ce8\u8bb0 2<\/span>.<\/span> \u601d\u8003\uff1a\u5982\u4f55\u6539\u9020\u4e0a\u8ff0\u8bc1\u660e, \u4f7f\u5f97\u7ed3\u8bba\u5bf9$\\lvert \\Omega \\rvert=+\\infty$\u4e5f\u5bf9\uff1f
\n<\/div><\/p>\n
\n\t
\u53c2\u8003\u6587\u732e<\/div>\n\t
  1. M. Krasnosel skii<\/span>, Topological methods in the theory of nonlinear integral equations<\/a>, Translated by A. H. Armstrong; translation edited by J. Burlak. A Pergamon Press Book<\/span>, The Macmillan Co., New York<\/span>, 1964<\/span>. xi + 395<\/span>. MR0159197<\/a><\/li><\/ol><\/div>","protected":false},"excerpt":{"rendered":"

    \u6458\u8981. Krasnoselskii\u5728[1,Thm.~I.2.1]\u4e2d\u7ed9\u51fa\u4e86\u6709\u754cNemytskii\u7b97\u5b50\u8fde\u7eed\u6027\u7684\u4e00\u4e2a… \u7ee7\u7eed\u9605\u8bfbKrasnoselskii\u5173\u4e8e\u590d\u5408\u51fd\u6570\u8fde\u7eed\u6027\u7684\u4e00\u4e2a\u5b9a\u7406<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[91,93,92],"class_list":["post-436","post","type-post","status-publish","format-standard","hentry","category-math","tag-krasnoselskii","tag-lianxuxing","tag-feixianxingsuanzi","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/436","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=436"}],"version-history":[{"count":2,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/436\/revisions"}],"predecessor-version":[{"id":438,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/436\/revisions\/438"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=436"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=436"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=436"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}