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We study some common results in C*-algebra-valued Sb-metric spaces. We also present an interesting application of an existing and unique result for one type of integral equation.

Many authors used the concept of F-contraction introduced by Wardowski in 2012 in order to define and prove new results on fixed points in complete metric spaces. In some later papers (for example Proinov P.D., J. Fixed Point Theory Appl. (2020)22:21, doi:10.1007/s11784-020-0756-1) it is shown that conditions (F2) and (F3) are not necessary to prove Wardowski's results. In this article we use a new approach in proving that the Picard-Jungck sequence is a Cauchy one. It helps us obtain new Jungck-Fisher-Wardowski type results using Wardowski's condition (F1) only, but in a way that differs from the previous approaches. Along with that, we came to several new contractive conditions not known in the fixed point theory so far. With the new results presented in the article, we generalize, extend, unify and enrich methods presented in the literature that we cite.

In this paper, we have proved the existence of unique common fixed point of four contractive maps on cone Banach space through an upper semi continuous contractive modulus and weakly compatible maps.

In this paper, we have proved the existence of unique common fixed point of four contractive maps on 2-cone Banach space through a contractive modulus and weakly compatible maps.

In this paper, we have proved the existence of unique common fixed point of four contractive maps on cone Banach space through an upper semi continuous contractive modulus and weakly compatible maps.

In this paper, we have proved the existence of unique common fixed point of four contractive maps on 2-cone Banach space through a contractive modulus and weakly compatible maps.

In this paper, we have proved the existence of unique common fixed point of four contractive maps on 2-cone Banach space through a contractive modulus and weakly compatible maps.

In this paper, we have proved the existence of unique common fixed point of four contractive maps on cone Banach space through an upper semi continuous contractive modulus and weakly compatible maps.

Ова дисертација је прилог Метричкој теорији фиксне тачке, области која се у последње време интензивно развија. Састоји се из пет поглавља. У првом поглављу доказана је једна позната лема, која се примењује у доказу Banach-овог принципа контракције, исказаног за орбитално комплетне метричке просторе. У другом поглављу доказано је осам теорема које генералишу неке познате резултате о фиксним тачкама у метричким просторима (Boyd-Wоng-ов, Ćirićev, Pant-ов и друге). Неке од тих теорема су модификације познатих, а три су потпуно нове. У трећем поглављу доказане су три теореме које генералишу резултат Немыцкий-а о фиксним тачкама пресликавања у компактним метричким просторима, као и једно уопштење теореме Edelstein-а. Код неких теорема су оригинални доказ и исказане последице. У четвртом поглављу разматрају се b-метрички простори, као уопштења метричких. Изведена је генерализација теореме Zamfirescu-а у b-метричком простору и неке њене примене. Дат је још један нови резултат који се односи на слаба скоро контрактивна пресликавања. У петом поглављу разматрају се конусни метрички простори. Доказане су две теореме као аналогони одговарајућих резултата из обичних метричких простора. Изведена је и једна потпуно нова теорема која за последицу има Banach-ову теорему у конусним метричким просторима над конусом који не мора бити нормалан. Доказана је и генерализација Fischer-ове теореме у конусним метричким просторима са регуларним конусом. Већина резултата снабдевена је одговарајућим примерима који показују у чему се ти резултати разликују од већ познатих., This dissertation is the contribution to the Metric fixed point theory, the area that has recently been rapidly developing. It contains five chapters. The first chapter gives the proof of one already known lemma. This lemma is used in the proof of Banach’s theorem for orbital complete metric spaces. The second chapter contains the proofs of eight theorems, which generalize some known results from the theory of fixed points in metric spaces (Boyd-Wong’s, ´ Ciri´c ’ s, Pant’ s, and other). Some of these theorems are modifications of the known ones, while three are completely new. Three theorems are proven in the third chapter. They generalize the result of fixed point of mapping defined in compact metric space given by Nemytzki, as well as one generalization of Edelstein’s theorem. The proofs and stated corollaries of some theorems are original. Chapter four discusses b-metric spaces as a generalization of metric spaces. The generalization of Zamfirescu’s theorem of b-metric spaces is presented as well as some of its applications. A new result concerning weakly almost contractive mappings is also determined. Chapter five contains some new results in cone metric spaces. Two theorems are presented as the analogue of the same theorems in the setting of standard metric spaces. A completely new theorem is established which results in the Banach’s theorem in cone metric spaces whereby the cone does not need to be normal. A generalization of Fisher’s theorem in cone metric spaces over a regular cone is also proven. Almost all results in this dissertation are confirmed by corresponding examples, which explain how these results differ from the already known results. Key Words: Cone metric space, normal cone, b-complete metric space, Banach’s contraction, generalized contraction, fixed point, weakly compatible, weakly semi compatible, weakly commutative, the sequence of bounded variation

In this paper, we prove a common fixed point theorem for weakly compatible mappings using contractive condition of integral type in an intuitionistic fuzzy metric space. Our result generalizes many known results and explore the possibility of the notion of reciprocal continuity and weakly compatible maps to the problem of finding common fixed point of four mappings.

On partially ordered set equipped with a partial metric, we study the sufficient conditions for existence of common fixed points of various mappings satisfying generalized weak contractive conditions. These results unify several comparable results in the existing literature. We also study the existence of nonnegative solution of implicit nonlinear integral equation. Furthermore, we study the fractal of finite family of generalized contraction mappings defined on a partial metric space.

In this paper we prove a common fixed point theorem for weakly compatible mappings satisfies certain  contractive condition in non- empty closed subset of a separable Hilbert Space. Our results generalize and extend the result Chauhan [7].

We prove some common fixed point theorems of mappings on cone metric-type spaces in which the cone does not need to be normal. These theorems generalize the recent results of Jungck et al. [10] and references therein. The examples are given to illustrate the usability of our results.

In this paper we extend and generalize common fixed point theorems for six self-maps of Singh and Jain [B. Singh, S. Jain, A fixed point theorem in Menger space through weak compatibility, J. Math. Anal. Appl. 301 (2005) 439–448] from Menger and metric spaces to cone metric spaces. We also extend the notions of compatible and weakly compatible mappings from the setting of Menger and metric spaces to the setting of cone metric spaces. We do not impose the normality property on the cone, but suppose only that the cone P, in the ordered Banach space E, has a nonempty interior. Examples are given to illustrate the results.

One of the main problems in phylogenetics is to find good approximations of metrics by weighted trees. As an aid to solving this problem, it could be tempting to consider optimal realizations of metrics—the guiding principle being that, the (necessarily unique) optimal realization of a tree metric is the weighted tree that realizes this metric. And, although optimal realizations of arbitrary metrics are, in general, not trees, but rather weighted networks, one could still hope to obtain a phylogenetically informative representation of a given metric, maybe even more informative than the best approximating tree. However, optimal realizations are not only difficult to compute, they may also be non-unique. Here we focus on one possible way out of this dilemma: hereditarily optimal realizations. These are essentially unique, and can be described in a rather explicit way. In this paper, we recall what a hereditarily optimal realization of a metric is and how it is related to the 1-skeleton of the tight span of that metric, and we investigate under what conditions it coincides with this 1-skeleton. As a consequence, we will show that hereditarily optimal realizations for consistent metrics, a large class of phylogentically relevant metrics, can be computed in a straight-forward fashion.

Fixed point theorems for four self-maps on a metric space have been obtained by altering distances from which several known results in the literature can be deduced.