# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $1-\alpha\eta>0$.

##### MSC:
 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 34B18 Positive solutions of nonlinear boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations
##### Keywords:
positive solution; fixed point theorem; cone
Full Text:
##### References:
 [1] Agarwal, R. P.; O’regan, D.: Periodic solutions to nonlinear integral equations on the infinite interval modelling infectious disease. Nonlinear anal. 40, 21-35 (2000) · Zbl 0958.45011 [2] Agarwal, R. P.; O’regan, D.; Wong, P. J. Y.: Positive solutions of differential, difference and integral equations. (1999) [3] Anderson, D. R.; Avery, R. I.: Fixed point theorem of cone expansion and compression of functional type. J. differ. Equations appl. 8, 1073-1083 (2002) · Zbl 1013.47019 [4] Anderson, D. R.; Davis, J. M.: Multiple solutions and eigenvalues for third-order right focal boundary value problems. J. math. Anal. appl. 267, 135-157 (2002) · Zbl 1003.34021 [5] Erbe, L. H.; Wang, H.: On the existence of positive solutions of ordinary differential equations. Proc. amer. Math. soc. 120, 743-748 (1994) · Zbl 0802.34018 [6] Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cones. (1988) · Zbl 0661.47045 [7] Gupta, C. P.; Trofimchuk, S. I.: Existence of a solution of a three-point boundary value problem and the spectral radius of a related linear operator. Nonlinear anal. 34, 489-507 (1998) · Zbl 0944.34009 [8] Infante, G.; Webb, J. R. L.: Nonzero solutions of Hammerstein integral equations with discontinuous kernels. J. math. Anal. appl. 272, 30-42 (2002) · Zbl 1008.45004 [9] Leggett, R. W.; Williams, L. R.: A fixed point theorem with application to an infectious disease model. J. math. Anal. appl. 76, 91-97 (1980) · Zbl 0448.47044 [10] Lian, W. C.; Wong, F. H.; Yeh, C. C.: On the existence of positive solutions of nonlinear second order differential equations. Proc. amer. Math. soc. 124, 1117-1126 (1996) · Zbl 0857.34036 [11] Ma, R.: Existence theorems for a second order three-point boundary value problem. J. math. Anal. appl. 212, 430-442 (1997) · Zbl 0879.34025 [12] Meehan, M.; O’regan, D.: Positive lp solutions of Hammerstein integral equations. Arch. math. 76, 366-376 (2001) · Zbl 0981.45001 [13] Zima, M.: On positive solutions of boundary value problems on the half-line. J. math. Anal. appl. 259, 127-136 (2001) · Zbl 1003.34024