On Diviccaro, Fisher and Sessa open questions

summary:Let $K$ be a closed convex subset of a complete convex metric space $X$ and $T, I: K \rightarrow K$ two compatible mappings satisfying following contraction definition: $Tx, Ty)\le (Ix, Iy)+(1-a)\max \ \lbrace Ix.Tx),\ Iy, Ty)\rbrace$ for all $x,y$ in $K$, where $0<a<1/2^{p-1}$ and $p\ge 1$. If $I$ is continuous and $I(K)$ contains $[T(K)]$ , then $T$ and $I$ have a unique common fixed point in $K$ and at this point $T$ is continuous. This result gives affirmative answers to open questions set forth by Diviccaro, Fisher and Sessa in connection with necessarity of hypotheses of linearity and non-expansivity of $I$ in their Theorem [3] and is a generalisation of that Theorem. Also this result generalizes theorems of Delbosco, Ferrero and Rossati [2], Fisher and Sessa [4], Gregus [5], G. Jungck [7] and Mukherjee and Verma [8]. Two examples are presented, one of which shows the generality of this result

On a generalization of a Greguš fixed point theorem

summary:Let $C$ be a closed convex subset of a complete convex metric space $X$. In this paper a class of selfmappings on $C$, which satisfy the nonexpansive type condition $(2)$ below, is introduced and investigated. The main result is that such mappings have a unique fixed point

Faster Multistep Iterations for the Approximation of Fixed Points Applied to Zamfirescu Operators

By taking a counterexample, we prove that the multistep iteration process is faster than the Mann and Ishikawa iteration processes for Zamfirescu operators

Contingency Analysis of the Agonistic Behavioral Sequence in Sceloporus undulatus hyacinthinus

A contingency analysis of the sequence of actions associated with agonistic behavior of resident and introduced male Sceloporus undulatus nya irthirus was carried out to determine the degree of stereotypy. During the time period April to June 1984 and 1985, caged pairs of adult male and juvenile female lizards were filmed on three occasions in order to gather data on resident males not involved in aggressive interactions. During the same threemonth time period, a different nonresident male was introduced singly into a given resident\u27s cage on three occasions and the interactions filmed. The films were analyzed to determine which actions were agonistic, the sequence of actions and the transition frequencies from one action to another. The sequence of actions was determined to be probabilistic, that is, the occurrence of one action influenced the probability that a specific action would follow. The most probable behavioral sequence for resident males during male-male interactions was head turn-crawl-jaw snap-lateral compression-bobbing-attack. The latter three actions were unique to agonistic encounters. The introduced males expressed a similar behavioral sequence except that they very rarely attacked. The behavioral sequence of resident females during male-male interactions consisted of head turn-crawl-jaw snap-bobbing. Bobbing generally preceded jaw snapping in females when such male-male interactions were not taking place. The probabilistic nature of the display action pattern during agonistic behavior is similar to that observed in other iguanid lizards studied. Three levels of arousal were evident in resident males during aggressive interactions. These were 1) heightened awareness, 2) display and 3) attack. Again, this observation is consistent with those in studies carried out in other iguanid lizards