IJPAM: Volume 56, No. 1 (2009)

FIXED POINTS ON THE CLOSURE
OF OPEN, CONVEX AND BOUNDED SETS

Domenico Delbosco$^1$, Gabriella Viola$^2$
$^{1,2}$Department of Mathematics
University of Turin
10, Via C. Alberto, Turin, 10123, ITALY
$^1$e-mail: [email protected]
$^2$e-mail: [email protected]


Abstract.Let $X$ be a reflexive and separable Banach space, $U$ a nonempty, open, convex and bounded subset of $X$ containing the origin $O$ and and $f:\overline {U} \rightarrow X$ a continuous mapping in the weak topology of $X$. The boundary condition, here introduced, has the following statement: There exists $p > 1$ such that for all $ x \in \partial U$ one has

\begin{displaymath} \Vert x - fx \Vert^p \geq \Vert fx \Vert^p - \Vert x \Vert^p \end{displaymath}

or

\begin{displaymath} \Vert x - fx \Vert > \Vert fx \Vert- \Vert x \Vert. \end{displaymath}

This paper contains some fixed points theorems under the boundary conditions, giving an extension of previous results concerning a ball $B_r$ to a general closed and bounded sets in a reflexive and separable Banach space $X$.

Received: August 15, 2009

AMS Subject Classification: 47H10, 54H25

Key Words and Phrases: reflexive and separable Banach space, fixed points theorems

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 56
Issue: 1