Fixed Point Theorems for Different Mappings in Various Spaces

Abstract

Fixed-point theory is an important branch of nonlinear analysis. It is used to investigate the conditions under which single-valued or multivalued mappings have solutions. Numerous problems occuring in different branches of mathematics, such as differential equations, optimization theory and variational analysis can be modeled by the equation u=Du where $D$ is a nonlinear operator defined on a metric space. The solutions of this equation are called fixed points of $D$. The first fixed point theorem in metric spaces for contraction mappings was proved by a Polish mathematician Banach in 1922. This theorem is known as Banach's fixed point theorem or the Banach contraction principle. Due to the simplicity and usefulness of this basic theorem, it has become a very popular tool for proving the existence and uniqueness theorems in various branches of mathematical analysis.

In 1968, Kannan proved that there are mappings that have a discontinuity in their domain but still have fixed point, although such mappings are continuous at their fixed point. This paper of Kannan lead to lot of improvements and extensions of Banach contraction principle.

In the present thesis, several fixed point results in various abstract spaces such as Quasi partial metric spaces, $b$-metric-like spaces, partially ordered metric spaces and Partial Hausdorff metric spaces and for various types of contraction and expansion mappings have been discussed and thereby many existing results have been extended and generalized. The present thesis consists of seven chapters.

Chapter 1 is introductory. In this chapter, apart from setting up the notations and terminologies to be used in the sequel, a brief review of the work done in the area of fixed point theory is presented. Further, a systematic plan of the results presented in the subsequent chapters is given towards the end of this chapter.

In Chapter 2, Some fixed point results in Hausdorff metric spaces using $\alpha_H$-$\psi_H$-multivalued contractive type mappings are investigated. Various theorems regarding this class of contractive pair of mappings have been studied in this chapter. The results presented in this chapter extend some well-known relevant results (such as Kikkawa, Nadler , Samet \emph{et al.}) existing in literature. Some illustrative examples are provided to demonstrate the main results. A result in homotopy theory is presented as an application of these results. In attempt to give extension to the results of Shahi \textit{et al.}, some fixed point results in partially ordered metric spaces for $(\xi,\alpha)$-expansive mappings are studied. Along with these results, an application to periodic boundary value problem is presented to show the usefulness of our main results.

Chapter 3 deals with some interesting fixed results in Quasi partial metric spaces. In this chapter, a new approach in the field of aggregation theory and metric aggregation is introduced. Firstly, the notion of expansion between quasi partial metric spaces through distance aggregation perspective along with suitable aggregation properties is defined. After that, with the help of aggregation functions, the concept of projective $\Psi$-expansion has been introduced and several fixed point results in Quasi partial metric spaces are obtained through this notion. Furthermore, sufficient conditions are also provided to characterize aggregation operator to ensure the existence and uniqueness of fixed point. The results derived in this chapter generalize the results presented by Borsik and Dobo$\check{s}$, Massanet and Valero, Mayor and Valero. Some comparative examples are also given to check the efficacy of obtained results. Moreover, an application to asymptotic complexity analysis has also been presented in the end of this chapter.

In Chapter 4, the concept of contraction has been extended by introducing $\mathfrak{D}$-Contraction defined on a family $\mathfrak{F}$ of bounded functions. Also, a new notion of fixed function for a metric space is introduced. Some fixed function theorems have been obtained by using different forms of $\mathfrak{D}$-Contraction along with demonstrative examples. In order to support the applicability of these results, an application to intensity modulated radiation therapy (IMRT) has also been presented. This application is based on determining the best suitable treatment plan for tumor patients getting intensity modulated radiation therapy(IMRT). The fixed function obtained in this way represents the suitable doses for a number of tumor patients at the same time.

Chapter 5 deals with the generalization of the results of Wang \textit{et al.} and Jungck by introducing a new notion of $\mathfrak{P}$-expansion defined on a family $\mathfrak{F}$ of bounded functions. Some fixed function theorems in complete metric spaces have been investigated by using various kinds of generalized expansive conditions. In addition to these results, some common fixed function theorems for a pair of weakly compatible mappings are also derived. Moreover, an application based on deciding the suitable doses of intensity values for the patients under Tomotherapy is also given in the end of the chapter.

In Chapter 6, the notion of $\mathcal{F}$-generalized multivalued contractive type mappings is introduced by using $\mathcal{C}$-class functions. Some common fixed point results for weakly isotone increasing set-valued mappings in the setting of ordered partial metric spaces are investigated for this new class. The results presented in this chapter generalize various relevant results from the current literature \emph{e.g.} Ansari, Nashine, Nazari \emph{et al.}, Wang \emph{et al.} and references therein.

Chapter 7 is devoted to some common and coincidence fixed point results for a sequence of functions in complete metric spaces. There exists vast literature showing the existence of fixed points using expansive mappings. But the existence of common and coincidence fixed points for a sequence of functions using expansive mappings is still not explored much. In this chapter, some results for a sequence of mappings satisfying generalized weakly expansive conditions in the setting of quasi partial metric spaces have been investigated. To demonstrate the usability of presented results, some useful deductions along with some examples are also provided. In the end of this chapter, some common coupled fixed point theorems are investigated in the framework of $b$-metric-like spaces in order to generalize the results of Bhaskar and Lakshmikantham, Alghamdi \emph{et al.}. Towards the end of this chapter, some relevant topics for further research have been suggested based on the present study. The thesis is concluded by listing the Bibliography with various publications which are cited in this research work.