Liaqat Ali Khan

Caristi’s Fixed Point Theorem and Subrahmanyam’s Fixed Point Theorem in -Generalized Metric Spaces

We discuss the completeness of -generalized metric spaces in the sense of Branciari. We also prove generalizations of Subrahmanyam’s and Caristi’s fixed point theorem.

Fixed point results for modified weak and rational α-ψ-contractions in ordered 2-metric spaces

AbstractWe first introduce the concept of a triangular 2-α-η-admissible mapping which extends the notion of α-admissible mapping with respect to η to 2-metric spaces. Next, we introduce the concepts of modified weak and modified rational α-ψ-contractions and establish the existence and uniqueness of fixed points for such mappings in complete 2-metric spaces. As an application of the obtained results, we prove some fixed point results in partially ordered 2-metric spaces. The presented theorems generalize and improve certain existing results in the literature and provide main results in Dung and Hang (Fixed Point Theory Appl. 2013:161, 2013) as corollaries. Moreover, some examples and an application to integral equations are provided to illustrate the usability of the obtained results. MSC:46N40, 47H10, 54H25, 46T99.

Separability in Function Spaces

Abstract Let X be a completely regular Hausdorff space and E a real Hausdorff topological vector space, and let β0 and κ be the substrict and compact-open topologies on Cb(X, E) and C(X, E), respectively. If E is separable with nontrivial dual E′, then the following are shown to be equivalent: (1) X is separably submetrizable; (2) (Cb(X) ⊗ E, β0) is separable; (3) (C(X)⊗ E, κ) is separable. For E a locally bounded space, (Cb(X, E), β0) is separable iff (C(X, E), κ) is so.

Quasimultipliers on -Algebras

We investigate the extent to which the study of quasimultipliers can be made beyond Banach algebras. We will focus mainly on the class of 𝐹 -algebras, in particular on complete 𝑘 -normed algebras, 0 𝑘 ≤ 1 , not necessarily locally convex. We include a few counterexamples to demonstrate that some of our results do not carry over to general 𝐹 -algebras. The bilinearity and joint continuity of quasimultipliers on an 𝐹 -algebra 𝐴 are obtained under the assumption of strong factorability. Further, we establish several properties of the strict and quasistrict topologies on the algebra 𝑄 𝑀 ( 𝐴 ) of quasimultipliers of a complete 𝑘 -normed algebra 𝐴 having a minimal ultra-approximate identity.

Random fixed point theorems for multivalued acyclic random maps

Some random fixed point theorems for acyclic valued random operators are proved. The class of acyclic – valued random operators includes convex – valued and star – shaped – valued random operators. This leads to the discovery of some new results

Multipliers of Modules of Continuous Vector-Valued Functions

In 1961, Wang showed that if is the commutative -algebra with a locally compact Hausdorff space, then . Later, this type of characterization of multipliers of spaces of continuous scalar-valued functions has also been generalized to algebras and modules of continuous vector-valued functions by several authors. In this paper, we obtain further extension of these results by showing that where and are -normed spaces which are also essential isometric left -modules with being a certain commutative -algebra, not necessarily locally convex. Our results unify and extend several known results in the literature.

Random fixed point theorems for composites of acyclic multifunctions

In this paper we obtain random versions of Kakutani–Fan type fixed point theorems for a class V c + of multifunctions which contains Kakutani factorizable maps and composites of acyclic maps. As applications, we derive some random approximation theorems.