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

This chapter relates the notions of mutations with the concept of graphical derivatives of set-valued maps and more generally links the above results of morphological analysis with some basic facts of set-valued analysis that we shall recall.

Set-valued analysis

Elementary Fixed Point Theorems * Theorem of Borsuk and Topological Transversality * Homology and Fixed Points * Leray-Schauder Degree and Fixed Point Index * The Lefschetz-Hopf Theory * Selected Topics * Index

Some Results on Fixed Points

We extend the celebrated result of W. A. Kirk that a metric space is complete if and only if every Caristi self-mapping for has a fixed point, to partial metric spaces.

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A Kirk Type Characterization of Completeness for Partial Metric Spaces

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

Samet et al. (Nonlinear Anal. 75:2154-2165, 2012) introduced α-ψ-contractive mappings and proved some fixed point results for these mappings. More recently Salimi et al. (Fixed Point Theory Appl. 2013:151, 2013) modified the notion of α-ψ-contractive mappings and established certain fixed point theorems. Here, we continue to utilize these modified notions for single-valued Geraghty and Meir-Keeler-type contractions, as well as multi-valued contractive mappings. Presented theorems provide main results of Hussain et al. (J. Inequal. Appl. 2013:114, 2013), Karapinar et al. (Fixed Point Theory Appl. 2013:34, 2013) and Asl et al. (Fixed Point Theory Appl. 2012:212, 2012) as corollaries. Moreover, some examples are given here to illustrate the usability of the obtained results. MSC:46N40, 47H10, 54H25, 46T99.

/pdf/fixed-point-results-for-single-and-set-valued-a-e-ps-375y48heam.pdf

Fixed point results for single and set-valued α-η-ψ-contractive mappings

In this paper, a regularization projection algorithm is investigated for solving common elements of an equilibrium problem, a variational inequality problem and a fixed point problem of a strictly pseudocontractive mapping. Strong convergence theorems are established in the framework of real Hilbert spaces.

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A regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces

In this paper, we introduce a modified Suzuki α-ψ-proximal contraction. Then we establish certain best proximity point theorems for such proximal contractions in metric spaces. As an application, we deduce best proximity and fixed point results in partially ordered metric spaces. The results presented generalize and improve various known results from best proximity and fixed point theory. Moreover, some examples are given to illustrate the usability of the obtained results.

/pdf/best-proximity-point-results-for-modified-suzuki-a-ps-26gn5xnvmn.pdf

Best proximity point results for modified Suzuki α-ψ-proximal contractions

Some common fixed point theorems due to Bhatt et al. [A. Bhatt, et al., Common fixed point theorems for occasionally weakly compatible mappings under relaxed conditions, Nonlinear Anal. 73 (2010) 176–182], Jungck and Rhoades [G. Jungck and B. E. Rhoades, Fixed point theorems for occasionally weakly compatible mappings, Fixed Point Theory 7 (2) (2006) 287–296. Fixed Point Theory 9 (2008) 383–384 (erratum)] and Imdad and Soliman [M. Imdad, A.H. Soliman, Some common fixed point theorems for a pair of tangential mappings in symmetric spaces, Appl. Math. Lett. 23 (2010) 351–355] are extended to two new classes of non-commuting selfmaps which contain the occasionally weakly compatible and weakly biased selfmaps as proper subclasses. Some illustrative examples are also provided to highlight the realized improvements.

/pdf/common-fixed-points-for-j-operators-and-occasionally-weakly-4zmxpx2qzw.pdf

Abdul Latif

Papers

Modified α-ψ-contractive mappings with applications

Fixed point results for single and set-valued α-η-ψ-contractive mappings

A regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces

Best proximity point results for modified Suzuki α-ψ-proximal contractions

Common fixed points for Jℋ-operators and occasionally weakly biased pairs under relaxed conditions