Existence theorem of positive solution to a nonlinear Sturm-Liouville problem (Chinese)
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摘要: 研究了非线性~Sturm-Liouville~边值问题的正解存在性,~%其中非线性项~$f(t,u)$~可以在~$t = 0,\,t = 1$~处奇异.~%通过引入非线性项在有界集合上的高度函数的积分来描述非线性项的增长变化.~%在极限函数~$\mathop {\lim }\limits_{u \to + 0} f(t,u) / u$,$\mathop{\lim }\limits_{u \to + \infty } f(t,u) /u$~存在的情况下利用度理论中的~Krasnosel'skii~不动点定理和实变函数论中的控制收敛定理证明了一个正解存在定理.Abstract: The existence of positive solution was studied for thenonlinear Sturm-Liouville boundary value problem, where thenonlinear term $f(t,u)$ may be singular at $t = 0,\,t = 1$. Byintroducing the integrations of height functions of nonlinear termon bounded set the growths of nonlinear term were described. Byapplying the Krasnoselskii fixed point theorem in degree theory andthe dominated convergence theorems in real variable, an existencetheorem of positive solution was proved when there are limitfunctions $\mathop {\lim }\limits_{u \to + 0} f(t,u) / u$ and$\mathop {\lim }\limits_{u \to + \infty } f(t,u) / u$.
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