Tesis por compendio / [EN] Fuzzy mathematics has constituted a wide field of research, since L. A. Zadeh introduced in 1965 the concept of fuzzy set. In particular, the problem of constructing a satisfactory theory of fuzzy metric spaces has been investigated by several authors. In 1994, George and Veeramani introduced and studied a notion of fuzzy metric space that constituted a modification of the one given by Kramosil and Michalek. Several authors have contributed to the study of this kind of fuzzy metrics, from the mathematical point of view and for their applications. In this thesis we have contributed to develop the study of these fuzzy metrics, from the mathematical point of view, and we approached the problem of measuring perceptual colour-difference between samples of colour using one of these fuzzy metrics. The contributions of the study carried out in this thesis is summarized as follows: \begin{enumerate} \item[(i)] We have made a detailed study of the fuzzy metric space $(X,M,\cdot)$ where $M$ is given on $X=[0,\infty[$ by $M(x,y,t)=\frac{\min\{x,y\}+t}{\max\{x,y\}+t}$ and others related to it. As a consequence we have introduced five questions in fuzzy metrics related to continuity, extension, contractivity and completion. \item[(ii)] We have answered an open question constructing a fuzzy metric space $(X,M,\ast)$ in which the assignment $f(t)=\lim_n M(a_n,b_n,t)$, where $\{a_n\}$ and $\{b_n\}$ are $M$-Cauchy sequences in $X$, is not a continuous function on $t$. The response to this question has allowed us to characterize the class of completable strong fuzzy metric spaces. \item[(iii)] We have introduced and studied a stronger concept than convergence of sequences in fuzzy metric spaces, which we call $s$-convergence. In our study, we have gotten a characterization of those spaces in which every convergent sequence is $s$-convergent and we have given a classification of fuzzy metrics attending to the behaviour of the fuzzy metric with respect to the different types of convergence. \item[(iv)] We have studied, in the context of fuzzy metric spaces, when certain families of open balls centered at a point are local bases for this point. \item[(v)] We have answered two open questions related to standard convergence, a stronger concept than convergence of sequences in fuzzy metric spaces, introduced in a natural way attending to the concept of standard Cauchy sequence (introduced in \cite{adomain}). These responses have led us to establish conditions under which Cauchyness and convergence should be considered \textit{compatible}. \item[(vi)] As a practical application, we have shown that a certain fuzzy metric is useful for measuring perceptual colour-differences between colour samples. \end{enumerate} / [ES] La matemática fuzzy ha constituido un amplio campo en la investigación, desde que en 1965 L. A. Zadeh introdujo el concepto de conjunto fuzzy. En particular, la construcción de una teoría satisfactoria de espacios métricos fuzzy ha sido un problema investigado por muchos autores. En 1994, George y Veeramani introdujeron y estudiaron una noción de espacio métrico fuzzy que constituía una modificación de la anteriormente dada por Kramosil y Michalek. Muchos autores han contribuido al estudio de este tipo de métricas fuzzy, desde el punto de vista matemático y de sus aplicaciones. En esta tesis hemos contribuido al desarrollo del estudio de estas métricas fuzzy, desde el punto de vista matemático, y hemos abordado el problema de la medida de la diferencia perceptual de color utilizando una de estas métricas. Las contribuciones que aportamos en esta tesis a dicho estudio, se resumen a continuación: \begin{enumerate} \item[(i)] Hemos hecho un estudio detallado del espacio métrico fuzzy $(X,M,\cdot)$ donde $M$ está dada sobre $[0,\infty[$ por la expresión $M(x,y,t)=\frac{\min\{x,y\}+t}{\max\{x,y\}+t}$ y de otros espacios métricos fuzzy relacionados con el. Como consecuencia de este estudio hemos introducido cinco cuestiones en la teoría de las métricas fuzzy relacionadas con continuidad, extensión, contractividad y completación. \item[(ii)] Hemos respondido a una cuestión abierta construyendo un espacio métrico fuzzy $(X,M,\ast)$ en el cual la asignación $f(t)=\lim_n M(a_n,b_n,t)$, donde $\{a_n\}$ y $\{b_n\}$ son sucesiones $M$-Cauchy, no es una función continua sobre $t$. La respuesta a esta cuestión nos ha permitido caracterizar la clase de los espacios métricos fuzzy strong completables. \item[(iii)] Hemos introducido y estudiado un concepto más fuerte que el de convergencia de sucesiones en espacios métricos fuzzy, al que hemos llamado $s$-convergencia. En nuestro estudio hemos conseguido una caracterización de aquellos espacios métricos fuzzy en los cuales toda sucesión convergente es $s$-convergente y hemos dado una clasificación de los espacios métricos fuzzy atendiendo a su comportamiento con respecto a los diferentes tipos de convergencia que se da en él. \item[(iv)] Hemos estudiado, en el contexto de los espacios métricos fuzzy, cuando ciertas familias de bolas abiertas centradas en un punto son base local de este punto. \item[(v)] Hemos respondido a dos cuestiones abiertas relacionadas con la convergencia standard, un concepto más fuerte que el de convergencia de sucesiones en espacios métricos fuzzy, introducido de forma natural a partir del concepto de sucesión de Cauchy standard (introducido en \cite{adomain}). Estas respuestas nos han llevado a establecer unas condiciones bajo las cuales un concepto relacionado con el concepto de sucesión de Cauchy y un concepto relacionado con el de convergencia deberían satisfacer para ser consideradas \textsl{compatibles}. \item[(vi)] Como aplicación práctica, hemos mostrado que una cierta métrica fuzzy es útil para medir diferencia perceptual de color entre muestras de color. \end{enumerate} / [CA] La matemàtica fuzzy ha constituït un ampli camp en la investigació, des que el 1965 L. A. Zadeh va introduir el concepte de conjunt fuzzy. En particular, la construcció d'una teoria satisfactòria d'espais mètrics fuzzy ha estat un problema investigat per molts autors. El 1994, George i Veeramani introduiren i estudiaren una noció d'espai mètric fuzzy que constituïa una modificació de la donada per Kramosil i Michalek anteriorment. Molts autors han contribuït a l'estudi d'aquest tipus de mètriques fuzzy, des del punt de vista matemàtic i de les seves aplicacions. En aquesta tesi hem contribuït al desenvolupament de l'estudi d'aquestes mètriques fuzzy, des del punt de vista matemàtic, i hem abordat el problema de la mesura de la diferència perceptiva de color utilitzant aquestes mètriques. Les contribucions que aportem en aquesta tesi a tal estudi es resumeixen a continuació: \begin{enumerate} \item[(i)] Hem fet un estudi detallat de l'espai mètric fuzzy $(X,M,\cdot)$ on $M$ està donada sobre $[0,\infty[$ per l'expressió $M(x,y,t)=\frac{\min\{x,y\}+t}{\max\{x,y\}+t}$ i d'altres espais mètrics fuzzy relacionats amb ell. Com a conseqüència d'aquest estudi hem introduït cinc qüestions en la teoria de les mètriques fuzzy relacionades amb continuïtat, extensió, contractividad i completació. \item[(ii)] Hem respost a una qüestió oberta construint un espai mètric fuzzy $ (X, M, \ast) $ en el qual l'assignació $ f (t) = \lim_n M (a_n, b_n, t) $, on $ \{a_n\} $ i $ \{b_n \} $ són successions $ M $-Cauchy, no és una funció contínua sobre $ t $. La resposta a aquesta qüestió ens ha permès caracteritzar la classe dels espais mètrics fuzzy strong completables. \item[(iii)] Hem introduït i estudiat un concepte més fort que el de convergència de successions en espais mètrics fuzzy, al qual hem anomenat $ s $-Convergència. En el nostre estudi hem aconseguit una caracterització d'aquells espais mètrics fuzzy en els quals tota successió convergent és $ s $-convergente i hem donat una classificació dels espais mètrics fuzzy atenent al seu comportament respecte als diferents tipus de convergència que es dóna en ell. \item[(iv)] Hem estudiat, en el context dels espais mètrics fuzzy, quan certes famílies de boles obertes centrades en un punt són base local d'aquest punt. \item[(v)] Hem respost a dues qüestions obertes relacionades amb la convergència estàndard, un concepte més fort que el de convergència de successions en espais mètrics fuzzy, introduït de forma natural a partir del concepte de successió de Cauchy estàndard (introduït en \cite{adomain}). Aquestes respostes ens han portat a establir unes condicions sota les quals un concepte relacionat amb el concepte de successió de Cauchy i un concepte relacionat amb el de convergència haurien de satisfer per a ser considerats \textsl{compatibles}. \item[(vi)] Com a aplicació pràctica, hem mostrat que una certa mètrica fuzzy és útil per mesurar la diferència perceptiva de color entre mostres de color. \end{enumerate} / Miñana Prats, JJ. (2015). Fuzzy metric spaces and applications to perceptual colour-differences [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/50612 / Compendio

Recall that a Banach space X has the Dunford-Pettis property if every weakly compact operator defined on X takes weakly compact sets into norm compact sets. Some valuable characterisations of Banach spaces with the Dunford-Pettis property are: X has the DPP if and only if for all Banach spaces Y, every weakly compact operator from X to Y sends weakly convergent sequences onto norm convergent sequences (i.e. it requires that weakly compact operators on X are completely continuous) and this is equivalent to “if (xn) and (x*n) are sequences in X and X* respectively and limn xn = 0 weakly and limn x*n = 0 weakly then limn x*n xn = 0". A striking application of the Dunford-Pettis property (as was observed by Grothendieck) is to prove that if X is a linear subspace of L() for some finite measure  and X is closed in some Lp() for 1 ≤ p < , then X is finite dimensional. The fact that the well known spaces L1() and C() have this property (as was proved by Dunford and Pettis) was a remarkable achievement in the early history of Banach spaces and was motivated by the study of integral equations and the hope to develop an understanding of linear operators on Lp() for p ≥ 1. In fact, it played an important role in proving that for each weakly compact operator T : L1()  L1() or T : C()  C(), the operator T2 is compact, a fact which is important from the point of view that there is a nice spectral theory for compact operators and operators whose squares are compact. There is an extensive literature involving the Dunford-Pettis property. Almost all the articles and books in our list of references contain some information about this property, but there are plenty more that could have been listed. The reader is for instance referred to [4], [5], [7], [8], [10], [17] and [24] for information on the role of the DPP in different areas of Banach space theory. In this dissertation, however, we are motivated by the two papers [7] and [8] to study alternative Dunford-Pettis properties, to introduce a scale of (new) alternative Dunford-Pettis properties, which we call DP*-properties of order p (briefly denoted by DP*P), and to consider characterisations of Banach spaces with these properties as well as applications thereof to polynomials and holomorphic functions on Banach spaces. In the paper [8] the class Cp(X, Y) of p-convergent operators from a Banach space X to a Banach space Y is introduced. Replacing the requirement that weakly compact operators on X should be completely continuous in the case of the DPP for X (as is mentioned above) by “weakly compact operators on X should be p-convergent", an alternative Dunford-Pettis property (called the Dunford-Pettis property of order p) is introduced. More precisely, if 1 ≤ p ≤ , a Banach space X is said to have DPPp if the inclusion W(X, Y)  Cp(X, Y) holds for all Banach spaces Y . Here W(X, Y) denotes the family of all weakly compact operators from X to Y. We now have a scale of “Dunford-Pettis like properties" in the sense that all Banach spaces have the DPP1, if p < q, then each Banach space with the DPPq also has the DPPp and the strongest property, namely the DPP1 coincides with the DPP. In the paper [7] the authors study a property on Banach spaces (called the DP*-property, or briey the DP*P) which is stronger than the DPP, in the sense that if a Banach space has this property then it also has DPP. We say X has the DP*P, when all weakly compact sets in X are limited, i.e. each sequence (x*n)  X * in the dual space of X which converges weak* to 0, also converges uniformly (to 0) on all weakly compact sets in X. It turns out that this property is equivalent to another property on Banach spaces which is introduced in [17] (and which is called the *-Dunford-Pettis property) as follows: We say a Banach space X has the *-Dunford-Pettis property if for all weakly null sequences (xn) in X and all weak* null sequences (x*n) in X*, we have x*n(xn) n 0. After a thorough study of the DP*P, including characterisations and examples of Banach spaces with the DP*P, the authors in [7] consider some applications to polynomials and analytic functions on Banach spaces. Following an extensive literature study and in depth research into the techniques of proof relevant to this research field, we are able to present a thorough discussion of the results in [7] and [8] as well as some selected (relevant) results from other papers (for instance, [2] and [17]). This we do in Chapter 2 of the dissertation. The starting point (in Section 2.1 of Chapter 2) is the introduction of the so called p-convergent operators, being those bounded linear operators T : X  Y which transform weakly p-summable sequences into norm-null sequences, as well as the so called weakly p-convergent sequences in Banach spaces, being those sequences (xn) in a Banach space X for which there exists an x  X such that the sequence (xn - x) is weakly p-summable. Using these concepts, we state and prove an important characterisation (from the paper [8]) of Banach spaces with DPPp. In Section 2.2 (of Chapter 2) we continue to report on the results of the paper [7], where the DP*P on Banach spaces is introduced. We focus on the characterisation of Banach spaces with DP*P, obtaining among others that a Banach space X has DP*P if and only if for all weakly null sequences (xn) in X and all weak* null sequences (x*n) in X*, we have x*n(xn) n 0. An important characterisation of the DP*P considered in this section is the fact that X has DP*P if and only if every T  L(X, c0) is completely continuous. This result proves to be of fundamental importance in the study of the DP*P and its application to results on polynomials and holomorphic functions on Banach spaces. To be able to report on the applications of the DP*P in the context of homogeneous polynomials and analytic functions on Banach spaces, we embark on a study of “Complex Analysis in Banach spaces" (mostly with the focus on homogeneous polynomials and analytic functions on Banach spaces). This we do in Chapter 3; the content of the chapter is mostly based on work in the books [23] and [14], but also on the work in some articles such as [15]. After we have discussed the relevant theory of complex analysis in Banach spaces in Chapter 3, we devote Chapter 4 to considering properties of polynomials and analytic functions on Banach spaces with DP*P. The discussion in Chapter 4 is based on the applications of DP*P in the paper [7]. Finally, in Chapter 5 of the dissertation, we contribute to the study of “Dunford-Pettis like properties" by introducing the Banach space property “DP*P of order p", or briefly the DP*Pp for Banach spaces. Using the concept “weakly p-convergent sequence in Banach spaces" as is defined in [8], we define weakly-p-compact sets in Banach spaces. Then a Banach space X is said to have the DP*-property of order p (for 1 ≤ p ≤ ) if all weakly-p-compact sets in X are limited. In short, we say X has DP*Pp. As in [8] (where the DPPp is introduced), we now have a scale of DP*P-like properties, in the sense that all Banach spaces have DP*P1 and if p < q and X has DP*Pq then it has DP*Pp. The strongest property DP*P coincides with DP*P. We prove characterisations of Banach spaces with DP*Pp, discuss some examples and then consider applications to polynomials and analytic functions on Banach spaces. Our results and techniques in this chapter depend very much on the results obtained in the previous three chapters, but now we have to find our own correct definitions and formulations of results within this new context. We do this with some success in Sections 5.1 and 5.2 of Chapter 5. Chapter 1 of this dissertation provides a wide range of concepts and results in Banach spaces and the theory of vector sequence spaces (some of them very deep results from books listed in the bibliography). These results are mostly well known, but they are scattered in the literature - they are discussed in Chapter 1 (some with proof, others without proof, depending on the importance of the arguments in the proofs for later use and depending on the detail with which the results are discussed elsewhere in the literature) with the intention to provide an exposition which is mostly self contained and which will be comfortably accessible for graduate students. The dissertation reflects the outcome of our investigation in which we set ourselves the following goals: 1. Obtain a thorough understanding of the Dunford-Pettis property and some related (both weaker and stronger) properties that have been studied in the literature. 2. Focusing on the work in the paper [8], understand the role played in the study of difierent classes of operators by a scale of properties on Banach spaces, called the DPPp, which are weaker than the DP-property and which are introduced in [8] by using the weakly p-summable sequences in X and weakly null sequences in X*. 3. Focusing on the work in the paper [7], investigate the DP*P for Banach spaces, which is the exact property to answer a question of Pelczynsky's regarding when every symmetric bilinear separately compact map X x X  c0 is completely continuous. 4. Based on the ideas intertwined in the work of the paper [8] in the study of a scale of DP-properties and the work in the paper [7], introduce the DP*Pp on Banach spaces and investigate their applications to spaces of operators and in the theory of polynomials and analytic mappings on Banach spaces. Thereby, not only extending the results in [7] to a larger family of Banach spaces, but also to find an answer to the question: “When will every symmetric bilinear separately compact map X x X  c0 be p-convergent?" / Thesis (M.Sc. (Mathematics))--North-West University, Potchefstroom Campus, 2012.

Recall that a Banach space X has the Dunford-Pettis property if every weakly compact operator defined on X takes weakly compact sets into norm compact sets. Some valuable characterisations of Banach spaces with the Dunford-Pettis property are: X has the DPP if and only if for all Banach spaces Y, every weakly compact operator from X to Y sends weakly convergent sequences onto norm convergent sequences (i.e. it requires that weakly compact operators on X are completely continuous) and this is equivalent to “if (xn) and (x*n) are sequences in X and X* respectively and limn xn = 0 weakly and limn x*n = 0 weakly then limn x*n xn = 0". A striking application of the Dunford-Pettis property (as was observed by Grothendieck) is to prove that if X is a linear subspace of L() for some finite measure  and X is closed in some Lp() for 1 ≤ p < , then X is finite dimensional. The fact that the well known spaces L1() and C() have this property (as was proved by Dunford and Pettis) was a remarkable achievement in the early history of Banach spaces and was motivated by the study of integral equations and the hope to develop an understanding of linear operators on Lp() for p ≥ 1. In fact, it played an important role in proving that for each weakly compact operator T : L1()  L1() or T : C()  C(), the operator T2 is compact, a fact which is important from the point of view that there is a nice spectral theory for compact operators and operators whose squares are compact. There is an extensive literature involving the Dunford-Pettis property. Almost all the articles and books in our list of references contain some information about this property, but there are plenty more that could have been listed. The reader is for instance referred to [4], [5], [7], [8], [10], [17] and [24] for information on the role of the DPP in different areas of Banach space theory. In this dissertation, however, we are motivated by the two papers [7] and [8] to study alternative Dunford-Pettis properties, to introduce a scale of (new) alternative Dunford-Pettis properties, which we call DP*-properties of order p (briefly denoted by DP*P), and to consider characterisations of Banach spaces with these properties as well as applications thereof to polynomials and holomorphic functions on Banach spaces. In the paper [8] the class Cp(X, Y) of p-convergent operators from a Banach space X to a Banach space Y is introduced. Replacing the requirement that weakly compact operators on X should be completely continuous in the case of the DPP for X (as is mentioned above) by “weakly compact operators on X should be p-convergent", an alternative Dunford-Pettis property (called the Dunford-Pettis property of order p) is introduced. More precisely, if 1 ≤ p ≤ , a Banach space X is said to have DPPp if the inclusion W(X, Y)  Cp(X, Y) holds for all Banach spaces Y . Here W(X, Y) denotes the family of all weakly compact operators from X to Y. We now have a scale of “Dunford-Pettis like properties" in the sense that all Banach spaces have the DPP1, if p < q, then each Banach space with the DPPq also has the DPPp and the strongest property, namely the DPP1 coincides with the DPP. In the paper [7] the authors study a property on Banach spaces (called the DP*-property, or briey the DP*P) which is stronger than the DPP, in the sense that if a Banach space has this property then it also has DPP. We say X has the DP*P, when all weakly compact sets in X are limited, i.e. each sequence (x*n)  X * in the dual space of X which converges weak* to 0, also converges uniformly (to 0) on all weakly compact sets in X. It turns out that this property is equivalent to another property on Banach spaces which is introduced in [17] (and which is called the *-Dunford-Pettis property) as follows: We say a Banach space X has the *-Dunford-Pettis property if for all weakly null sequences (xn) in X and all weak* null sequences (x*n) in X*, we have x*n(xn) n 0. After a thorough study of the DP*P, including characterisations and examples of Banach spaces with the DP*P, the authors in [7] consider some applications to polynomials and analytic functions on Banach spaces. Following an extensive literature study and in depth research into the techniques of proof relevant to this research field, we are able to present a thorough discussion of the results in [7] and [8] as well as some selected (relevant) results from other papers (for instance, [2] and [17]). This we do in Chapter 2 of the dissertation. The starting point (in Section 2.1 of Chapter 2) is the introduction of the so called p-convergent operators, being those bounded linear operators T : X  Y which transform weakly p-summable sequences into norm-null sequences, as well as the so called weakly p-convergent sequences in Banach spaces, being those sequences (xn) in a Banach space X for which there exists an x  X such that the sequence (xn - x) is weakly p-summable. Using these concepts, we state and prove an important characterisation (from the paper [8]) of Banach spaces with DPPp. In Section 2.2 (of Chapter 2) we continue to report on the results of the paper [7], where the DP*P on Banach spaces is introduced. We focus on the characterisation of Banach spaces with DP*P, obtaining among others that a Banach space X has DP*P if and only if for all weakly null sequences (xn) in X and all weak* null sequences (x*n) in X*, we have x*n(xn) n 0. An important characterisation of the DP*P considered in this section is the fact that X has DP*P if and only if every T  L(X, c0) is completely continuous. This result proves to be of fundamental importance in the study of the DP*P and its application to results on polynomials and holomorphic functions on Banach spaces. To be able to report on the applications of the DP*P in the context of homogeneous polynomials and analytic functions on Banach spaces, we embark on a study of “Complex Analysis in Banach spaces" (mostly with the focus on homogeneous polynomials and analytic functions on Banach spaces). This we do in Chapter 3; the content of the chapter is mostly based on work in the books [23] and [14], but also on the work in some articles such as [15]. After we have discussed the relevant theory of complex analysis in Banach spaces in Chapter 3, we devote Chapter 4 to considering properties of polynomials and analytic functions on Banach spaces with DP*P. The discussion in Chapter 4 is based on the applications of DP*P in the paper [7]. Finally, in Chapter 5 of the dissertation, we contribute to the study of “Dunford-Pettis like properties" by introducing the Banach space property “DP*P of order p", or briefly the DP*Pp for Banach spaces. Using the concept “weakly p-convergent sequence in Banach spaces" as is defined in [8], we define weakly-p-compact sets in Banach spaces. Then a Banach space X is said to have the DP*-property of order p (for 1 ≤ p ≤ ) if all weakly-p-compact sets in X are limited. In short, we say X has DP*Pp. As in [8] (where the DPPp is introduced), we now have a scale of DP*P-like properties, in the sense that all Banach spaces have DP*P1 and if p < q and X has DP*Pq then it has DP*Pp. The strongest property DP*P coincides with DP*P. We prove characterisations of Banach spaces with DP*Pp, discuss some examples and then consider applications to polynomials and analytic functions on Banach spaces. Our results and techniques in this chapter depend very much on the results obtained in the previous three chapters, but now we have to find our own correct definitions and formulations of results within this new context. We do this with some success in Sections 5.1 and 5.2 of Chapter 5. Chapter 1 of this dissertation provides a wide range of concepts and results in Banach spaces and the theory of vector sequence spaces (some of them very deep results from books listed in the bibliography). These results are mostly well known, but they are scattered in the literature - they are discussed in Chapter 1 (some with proof, others without proof, depending on the importance of the arguments in the proofs for later use and depending on the detail with which the results are discussed elsewhere in the literature) with the intention to provide an exposition which is mostly self contained and which will be comfortably accessible for graduate students. The dissertation reflects the outcome of our investigation in which we set ourselves the following goals: 1. Obtain a thorough understanding of the Dunford-Pettis property and some related (both weaker and stronger) properties that have been studied in the literature. 2. Focusing on the work in the paper [8], understand the role played in the study of difierent classes of operators by a scale of properties on Banach spaces, called the DPPp, which are weaker than the DP-property and which are introduced in [8] by using the weakly p-summable sequences in X and weakly null sequences in X*. 3. Focusing on the work in the paper [7], investigate the DP*P for Banach spaces, which is the exact property to answer a question of Pelczynsky's regarding when every symmetric bilinear separately compact map X x X  c0 is completely continuous. 4. Based on the ideas intertwined in the work of the paper [8] in the study of a scale of DP-properties and the work in the paper [7], introduce the DP*Pp on Banach spaces and investigate their applications to spaces of operators and in the theory of polynomials and analytic mappings on Banach spaces. Thereby, not only extending the results in [7] to a larger family of Banach spaces, but also to find an answer to the question: “When will every symmetric bilinear separately compact map X x X  c0 be p-convergent?" / Thesis (M.Sc. (Mathematics))--North-West University, Potchefstroom Campus, 2012.

The purpose of the thesis is to study completeness of abstract spaces. In particular, we study completeness in partial metric spaces, partial metric type spaces, dislocated metric spaces, dislocated metric type spaces and symmetric spaces that are generalizations of metric spaces. It is well known that complete metric spaces have a wide range of applications. For instance, the classical Banach contraction principle is phrased in the context of complete metric spaces. Analogously, the Banach's xed point theorem and xed point results for Lipschitzian maps are discussed in this context, namely in, partial metric spaces and metric type spaces. Finally, xed point results are presented for symmetric spaces / Mathematical Sciences / Ph. D. (Mathematics)

The purpose of the thesis is to study completeness of abstract spaces. In particular, we study completeness in partial metric spaces, partial metric type spaces, dislocated metric spaces, dislocated metric type spaces and symmetric spaces that are generalizations of metric spaces. It is well known that complete metric spaces have a wide range of applications. For instance, the classical Banach contraction principle is phrased in the context of complete metric spaces. Analogously, the Banach's xed point theorem and xed point results for Lipschitzian maps are discussed in this context, namely in, partial metric spaces and metric type spaces. Finally, xed point results are presented for symmetric spaces. / Geography / Ph. D. (Mathematics)