Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica
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Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2016, Issue 5(36), Pages 24–28
DOI: https://doi.org/10.15688/jvolsu1.2016.5.3
(Mi vvgum128)
 

Mathematics

A fixed point theorem for $L$-contractions

A. G. Korolåv

Volgograd State University
References:
Abstract: Our goal is to introduce a new fixed point theorem for operators acting on the space $C([0,T];X)$. This result can be considered as a generalization of the celebrated Banach Contraction Principle.
Let $X$ be a Banach space, $T > 0$ and consider the space $C([0,T];X)$ of continuous $X$-valued functions from the segment $I=[0,T]$ to $X$ equipped with the uniform norm:

\begin{equation*} ||{u}||=\max_{t\in [0,T]} ||u(t)||_{X}. \end{equation*}

Let $F$ be a closed subset of $C([0,T];X)$. Consider a continuous non-linear operator $N\colon F\to F$ that maps $F$ to itself.
We say that the operator $N$ is $L$-contraction on $F$ if for any $u,v\in F$ it satisfies the so called $L$-condition:

\begin{equation*} ||N(u)(t)-N(v)(t)||_{X} \leq L (||u(t)-v(t)||_{X}), \end{equation*}
where $L\colon C[0,T]\to C[0,T]$ is a linear positive monotone operator acting on the space $C([0,T]; \mathbb R)$ of the real-valued continuous functions and having the spectral radius $\rho(L) < 1$.
Our main result is the following theorem.
Theorem. Suppose that an operator $N$ is $L$-contraction on $F$. Then $N$ has a fixed point in $F$.
Keywords: nonlinear equations, fixed point theorems, Banach contraction principle, generalized contractions, method of successive approximations.
Document Type: Article
UDC: 517.98
BBC: 22.162
Language: Russian
Citation: A. G. Korolåv, “A fixed point theorem for $L$-contractions”, Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2016, no. 5(36), 24–28
Citation in format AMSBIB
\Bibitem{Kor16}
\by A.~G.~Korolåv
\paper A fixed point theorem for $L$-contractions
\jour Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica
\yr 2016
\issue 5(36)
\pages 24--28
\mathnet{http://mi.mathnet.ru/vvgum128}
\crossref{https://doi.org/10.15688/jvolsu1.2016.5.3}
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