Fixed Points of Almost Generalized (a, y)-Contractions with Rational Expressions

In this paper, we introduce almost generalized (a, y)-contractions with rational expression type mappings and establish the existence of fixed points for such mappings in complete partially ordered metric spaces. Further, we define `Condition (H)’ and prove the existence of unique fixed point under the additional assumption `Condition (H)’. Our results generalize the results of Arshad, Karapinar and Ahmad [1] and Harjani, Lopez and Sadarangani [2]

Comparison of fastness of the convergence among Krasnoselskij, Mann, and Ishikawa iterations in arbitrary real Banach spaces

<p/> <p>Let <inline-formula><graphic file="1687-1812-2006-35704-i1.gif"/></inline-formula> be an arbitrary real Banach space and <inline-formula><graphic file="1687-1812-2006-35704-i2.gif"/></inline-formula> a nonempty, closed, convex (not necessarily bounded) subset of <inline-formula><graphic file="1687-1812-2006-35704-i3.gif"/></inline-formula>. If <inline-formula><graphic file="1687-1812-2006-35704-i4.gif"/></inline-formula> is a member of the class of Lipschitz, strongly pseudocontractive maps with Lipschitz constant <inline-formula><graphic file="1687-1812-2006-35704-i5.gif"/></inline-formula>, then it is shown that to each Mann iteration there is a Krasnosleskij iteration which converges faster than the Mann iteration. It is also shown that the Mann iteration converges faster than the Ishikawa iteration to the fixed point of <inline-formula><graphic file="1687-1812-2006-35704-i6.gif"/></inline-formula>.</p