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Fixed point theorem of Leggett-Williams type and its application. (English) Zbl 1066.47059
One of the generalizations of Krasnoselskii’s theorem on cone expansion and compression was obtained in [{\it R. W. Leggett, L. R. Williams}, J. Math. Anal. Appl., Vol. 76, 91--97 (1980; Zbl 0448.47044)]. In the present paper, the author proves the following Leggett-Williams type theorem: Theorem. Let $P$ be a normal cone in a Banach space $E$ and let $\gamma$ be the normal constant of $P$. Assume that $\Omega_1$ and $\Omega_2$ are bounded open sets in $E$ such that $0\in \Omega_1$ and $\overline{\Omega}_1\subset \Omega_2$. Let $F:P\cap(\overline{\Omega}_2\backslash\Omega_1)\to P$ be a completely continuous operator and $u_0\in P\backslash {0}$. If either $ \gamma\Vert x\Vert \leq \Vert Fx\Vert $ for $x\in P(u_0)\cap\partial\Omega_1$ and $\Vert Fx\Vert \leq\Vert x\Vert $ for $x\in P\cap\partial\Omega_2$, or $\Vert Fx\Vert \leq\Vert x\Vert $ for $x\in P\cap\partial\Omega_1$ and $\gamma\Vert x\Vert \leq \Vert Fx\Vert $ for $x\in P(u_0)\cap\partial\Omega_2$ is satisfied, then $F$ has a fixed point in the set $P\cap (\overline{\Omega}_2 \backslash \Omega_1)$. The author applies this theorem to prove the existence of positive solutions to the following second order three-point boundary value problem: $$ x^{\prime\prime}(t)+f(t, x(t))=0, $$ $$ x(0)=0, \ \ \alpha x(\eta)=x(1), $$ where $t\in [0,1]$, $f$ is a continuous function, $\eta\in (0,1)$, $\alpha\geq 0$, and $1-\alpha\eta>0$.

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
34B18Positive solutions of nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
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