The paper defines, with the help of generalized topologies and generalized neighbourhood systems, two kinds of generalized continuity, by giving in this way a general form to various concepts discussed in the literature. In particular, generalized continuity admits a characterization furnishing a known charcterization of θ-continuous maps.

Generalized topology, generized continuity

We present some fixed point results for nondecreasing and weakly increasing operators in a partially ordered metric space using implicit relations. Also we give an existence theorem for common solution of two integral equations.

/pdf/some-fixed-point-theorems-on-ordered-metric-spaces-and-45ab5cujjr.pdf

Some Fixed Point Theorems on Ordered Metric Spaces and Application

The concept of weak contraction from the case of single-valued mappings is extended to multi-valued mappings and then corresponding convergence theorems for the Picard iteration associated to a multi-valued weak contraction are obtained. The main results in this paper extend, improve and unify a multitude of classical results in the fixed point theory of single and multi-valued contractive mappings and also improve recent results from the paper [P.Z. Daffer, H. Kaneko, Fixed points of generalized contractive multi-valued

On a general class of multi-valued weakly Picard

General topology I , edited by A. V. Arkhangel’skii and L. S. Pontryagin. Pp 230. DM128. 1990. ISBN 3-540-18178-4 (Springer)

In Hatir and Noiri [Hatir E, Noiri T. Decompositions of continuity and complete continuity. Acta Math Hungary 113(4);2006:281–287], δ–β-continuity has given to obtain a decomposition of continuity. In this paper, we investigate the properties of δ–β-continuous functions and discuss characterizations and the relationships with related functions.

On δ–β-continuous functions

Quite recently, by using semi-open (resp.α-open, preopen,β-open) sets in a topological space, the notions ofsg*-closed (resp.αg*-closed,pg*-closedβg*-closed) sets are indroduced and investigated in [8]. These subsets place between closed sets andg-closed sets due to Levine [5]. In this paper, we introduce the notion ofmg*-closed sets and obtain the unified theory for collections of subsets between closed sets andg-closed sets.

Between closed sets and g -closed sets

It is well known that ( C B ( X ) , H) is a metric space and the completeness of (X,d) implies the completeness of ( C B ( X ) , H ) . Let A, B € CB(X) and k > 1. In the sequel the following well known fact will be used [4]: for each a € A, there is 6 € B such that d(a, b) < kH(A, B). Let 5 and T be two self mappings of a metric space (X,d). Sessa [5] defines 5 and T to be weakly commuting if d(STx,TSx) < d(Tx,Sx) for all x in X. Jungck [2] defines S and T to be compatible if

A general coincidence theorem for compatible multivalued mappings satisfying an implicit relation

We prove a common fixed point theorem for two pairs of occasionally weakly compatible mappings satisfying an implicit relation of a new type that involves an altering distance, so generalizing a theorem of Aliouche and Djoudi (Common fixed point theorems for mappings satisfying an implicit relation without decreasing assumption, Hacettepe J. Math. Stat. 36(1), 11–18, 2007). As a consequence, we obtain a fixed point theorem for two pairs of mappings satisfying an implicit relation of integral type, providing a strong generalization to a known result from Kumar, Chugh and Kumar (Fixed point theorems for compatible mappings satisfying a contractive condition of integral type, Soochow J. Math. 33(2), 181–185, 2007).

/pdf/altering-distance-and-common-fixed-points-under-implicit-jtqgtkfuwk.pdf

Altering Distance and Common Fixed Points under Implicit Relations

In this paper we introduce a new notion of weaklyM-continuous functions as functions from a set satisfying some minimal conditions into a set satisfying some minimal conditions. We obtain some characterizations and several properties of such functions. This function leads to the formulation of a unified theory of weak continuity [27], almosts-continuity [43],p(θ)-continuity [10] andp-continuity [59].

A unified theory of weak continuity for functions

In this paper the authors define a multifunction \(F:X \mapsto Y\) to be upper (respectively, lower) \(\beta\)-continuous if \(F^+(V)\) (resp. \(F^-(V))\) is \(\beta\)-open in \(X\) for every open set \(V\) of \(Y\). They obtain some characterizations and several properties concerning upper (resp. lower) \(\beta\)-continuous multifunctions. The relationships between these multifunctions and quasi continuous multifunctions are investigated.

/pdf/on-upper-and-lower-b-continuous-multifunctions-m5otj1wrav.pdf

On upper and lower β-continuous multifunctions

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