The distance between fixed points of some pairs of maps in Banach spaces and applications to differential systems

Mortici, Cristinel

Czechoslovak Mathematical Journal, Tome 56 (2006) no. 2, pp. 689-695.

Résumé

Let $T$ be a $\gamma$-contraction on a Banach space $Y$ and let $S$ be an almost $\gamma$-contraction, i.e. sum of an $(\epsilon ,\gamma )$-contraction with a continuous, bounded function which is less than $\epsilon$ in norm. According to the contraction principle, there is a unique element $u$ in $Y$ for which $u=Tu.$ If moreover there exists $v$ in $Y$ with $v=Sv$, then we will give estimates for $\Vert u-v\Vert .$ Finally, we establish some inequalities related to the Cauchy problem.

Détail

Classification : 34A12, 34C11, 34L30, 47H10, 47N20
Mots-clés : contraction principle; Cauchy problem