Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

Fixed Point Theory

Wardowski (Fixed Point Theory Appl., 2012:94) introduced a new concept of contraction and proved a fixed point theorem which generalizes Banach contraction principle. Following this direction of research, we will present some fixed point results for closed multi-valued F-contractions or multi-valued mappings which satisfy an F-contractive condition of Hardy-Rogers-type, in the setting of complete metric spaces or complete ordered metric spaces. An example and two applications, for the solution of certain functional and integral equations, are given to illustrate the usability of the obtained results.

/pdf/multi-valued-f-contractions-and-the-solutions-of-certain-zld6wbfa9d.pdf

Multi-valued F-contractions and the solutions of certain functional and integral equations

Abstract In this paper, we consider generalized α - ψ {\alpha-\psi} -Geraghty contractive type mappings and investigate the existence and uniqueness of a fixed point for mappings involving such contractions. In particular, we extend, improve and generalize some earlier results in the literature on this topic. An application concerning the existence of an integral equation is also considered to illustrate the novelty of the main result.

On generalized α-ψ-Geraghty contractions on b-metric spaces

In this paper, we consider the setting of b-metric spaces to establish results regarding the common fixed points of two mappings, using a contraction condition defined by means of a comparison function. An example is presented to support our results comparing with existing ones.

/pdf/contraction-conditions-using-comparison-functions-on-b-1ycf423nlf.pdf

Contraction conditions using comparison functions on b-metric spaces

In this paper, owing to the concept of F-contraction, we define two new classes of functions M(S, T )a ndN(S, T), and we prove some new fixed point results for single-valued and multivalued mappings in complete metric spaces. Our results extend, generalize and unify several known results in the literature. We include an example to show that the generalization is proper. MSC: Primary 30C45; 30C10; secondary 47B38

/pdf/new-fixed-point-theorems-for-generalized-f-contractions-in-1rfvsb8m1a.pdf

New fixed point theorems for generalized F-contractions in complete metric spaces

Recently, Wardowski introduced a new concept of contraction and proved a fixed
 point theorem which generalizes Banach contraction principle. Following this
 direction of research, in this paper, we will present some fixed point
 results of Hardy-Rogers-type for self-mappings on complete metric spaces or
 complete ordered metric spaces. Moreover, an example is given to illustrate
 the usability of the obtained results.

/pdf/fixed-point-results-for-f-contractive-mappings-of-hardy-216l5dg6cb.pdf

Fixed Point Results for F-Contractive Mappings of Hardy-Rogers-Type

The purpose of this paper is to study the existence of solutions set of integrodifferential problems in Banach spaces. We obtain our results by using fixed point theorems for multivalued mappings, under new contractive conditions, in the setting of complete b-metric spaces. Also, we present a data dependence theorem for the solutions set of fixed point problems.

/pdf/solvability-of-integrodifferential-problems-via-fixed-point-4r95bmwlq0.pdf

Monica Cosentino

Papers

Fixed Point Results for F-Contractive Mappings of Hardy-Rogers-Type

Solvability of integrodifferential problems via fixed point theory in b -metric spaces

Fixed point results on metric-type spaces