There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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

Let (X; d) be a metric space and T : X! X be a mapping. In this work, we introduce the mapping : (0;1) (0;1)! R, called the simulation function and the notion ofZ-contraction with respect to which generalize the Banach contraction principle and unify several known types of contractions involving the combination of d(Tx; Ty) and d(x; y): The related fixed point theorems are also proved.

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A New Approach to the Study of Fixed Point Theory for Simulation Functions

Preface On the Semilocal Convergence of Newton's Method Under Unifying Conditions Inequalities and Fixed Points in Menger Convex Metric Spaces Applications of the Perov's Fixed Point Theorem to Delay Integro-Differential Equations On Vector Equilibrium Problems with Multifunctions Fixed Points in Generalised Metric Spaces and the Stability of a Cubic Functional Equation Continuous Selection and Coincidence Theorems on Product G-Convex Space Sensitivity Analysis of Solution Set for a New Class of Generalised Implicit Quasi-Variational Inclusions Fixed Points in Probabilistic-Quasi-Metric Spaces Common Fixed Point Theorems for Condensed Mappings The Strongly Convergence Theorems of Fixed Points for Local Strictly F-Pseudocontractive On Generalised Non-linear Variational Inequalities A Note on a Paper of Hadic and Pap The Ishiwaka and Mann Iteration Methods On Certain Applications of Leray-Schauder Alternate Remarks on Concepts of Generalised Convex Spaces Generic Existence and Non-Existence of Approximate Fixed Points General System of Relaxed g-?-r-Pseudococoercive Non-linear Variational Inequalities and Projection Methods Three-Step Iteration Methods with Errors for Non-expansive Mappings in Uniformly Convex Banach Spaces On Nonnegative Linear Composite Games of NTU-Game Probabilistic Contractor and Non-linear Operator Equations with Set-Valued Operator in Probabilistic Normed Spaces Index.

Fixed Point Theory and Applications

The aim of this work is to modify the notions of α-admissible and α-ψ-contractive mappings and establish new fixed point theorems for such mappings in complete metric spaces. Presented theorems provide main results of Karapinar and Samet (Abstr. Appl. Anal. 2012:793486, 2012) and Samet et al. (Nonlinear Anal. 75:2154-2165, 2012) as direct corollaries. Moreover, some examples and applications to integral equations are given here to illustrate the usability of the obtained results.

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Modified α-ψ-contractive mappings with applications

In this paper, we introduce the notion of α-ψ-Meir-Keeler contractive mappings via a triangular α-admissible mapping. We discuss the existence and uniqueness of a fixed point of such a mapping in the setting of complete metric spaces. We state a number of examples to illustrate our results.

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On α-ψ-Meir-Keeler contractive mappings

We first introduce the concept of b-metric-like space which generalizes the notions of partial metric space, metric-like space and b-metric space. Then we establish the existence and uniqueness of fixed points in a b-metric-like space as well as in a partially ordered b-metric-like space. As an application, we derive some new fixed point and coupled fixed point results in partial metric spaces, metric-like spaces and b-metric spaces. Moreover, some examples and an application to integral equations are provided to illustrate the usability of the obtained results. MSC: Primary 47H10; 54H25; 55M20

/pdf/fixed-point-and-coupled-fixed-point-theorems-on-b-metric-2w226phhs4.pdf

Fixed point and coupled fixed point theorems on b-metric-like spaces

The aim of this paper is to introduce new concepts of --complete metric space and --continuous function and establish fixed point results for modified ---rational contraction mappings in --complete metric spaces. As an application, we derive some Suzuki type fixed point theorems and new fixed point theorems for -graphic-rational contractions. Moreover, some examples and an application to integral equations are given here to illustrate the usability of the obtained results.

/pdf/fixed-point-theory-in-complete-metric-spaces-with-1gll2rf8s2.pdf

Fixed Point Theory in -Complete Metric Spaces with Applications

In this paper, we prove the existence and uniqueness of a fixed point for certain α-admissible contraction mappings Our results generalize and extend some well-known results on the topic in the literature We consider some examples to illustrate the usability of our results MSC: 46N40; 47H10; 54H25; 46T99

/pdf/a-admissible-mappings-and-related-fixed-point-theorems-20pij63rdy.pdf

Peyman Salimi

Papers

On α-ψ-Meir-Keeler contractive mappings

Modified α-ψ-contractive mappings with applications

Fixed point and coupled fixed point theorems on b-metric-like spaces

Fixed Point Theory in -Complete Metric Spaces with Applications

α-admissible mappings and related fixed point theorems