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A Fixed Point Theorem of Krasnoselskii—Schaefer Type
Author(s) -
Burton T. A.,
Kirk Colleen
Publication year - 1998
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19981890103
Subject(s) - mathematics , fixed point theorem , contraction mapping , contraction (grammar) , schauder fixed point theorem , brouwer fixed point theorem , sign (mathematics) , fixed point , type (biology) , discrete mathematics , combinatorics , pure mathematics , mathematical analysis , medicine , ecology , biology
In this paper we focus on three fixed point theorems and an integral equation. Schaefer's fixed point theorem will yield a T‐periodic solution of (0.1) x ( t ) = a (t) + t t‐h D(t,s)g(s,x(s))ds if D and g satisfy certain sign conditions independent of their magnitude. A combination of the contraction mapping theorem and Schauder's theorem (known as Krasnoselskii's theorem) will yield a T‐periodic solution of (0.2) x ( t ) = f(t,x(t)) + t t‐h D(t,s)g(s,x(s))ds if f defines a contraction and if D and g are small enough. We prove a fixed point theorem which is a combination of the contraction mapping theorem and Schaefer's theorem which yields a T‐periodic solution of (0.2) when / defines a contraction mapping, while D and g satisfy the aforementioned sign conditions.