The purpose of this paper is to introduce some basic deﬁnitions about ﬁxed point and best proximity point in two classes of probabilistic metric spaces and to prove contrac- tion mapping principle and relevant best proximity point theorems. The ﬁrst class is the so-called S-probabilistic metric spaces. In S-probabilistic metric spaces, the **generalized** **contraction** mapping principle and **generalized** best proximity point theorems have been proved by authors. These results improve and extend the recent results of Su and Zhang []. The second class is the so-called Menger probabilistic metric spaces. In Menger prob- abilistic metric spaces, the **contraction** mapping principle and relevant best proximity point theorems have been proved by authors. These results also improve and extend the results of many authors. In order to get the results of this paper, some new methods have been used. Meanwhile some error estimate inequalities have been established.

Show more
20 Read more

equivalence between Moudafi’s viscosity approximation with contractions and Brow- der-type iterative processes (Halpern-type iterative processes); see [14] for more details. In this article, inspired by above result, we introduce and study the explicit viscosity iterative scheme for the **generalized** **contraction** f and a nonexpansive semigroup {T(t) : t ≥ 0}:

20 Read more

Fixed point theorems are proved for contraction maps on M-convex spaces... Fixed Point Theorems, Contraion principle, M-convex spaces..[r]

Fixed point theory is rapidly moving into the mainstream of Mathematics mainly because of its applications in diverse fields which include numerical methods like Newton-Raphson method, establishing Picard’s existence theorem, existence of solution of integral equations and a system of linear equations. In 1922, S. Banach [1], The first important and significant result was proved a fixed point theorem for **contraction** mappings in complete metric space and also called it Banach fixed point theorem / Banach **contraction** principle which is considered as the mile stone in fixed point theory. This theorem states that, A mapping : → where( , ) is a metric space, is said to be a **contraction** if there exists ∈ [0,1) such that

Show more
Poincare [17] was the first to work in this field Fixed Point Theory. Then Brouwer [4] in 1912, proved fixed point theorem for the solution of the equation ( ) = . He also proved fixed point theorem for a square, a sphere and their n-dimensional parts which means further extended by Kakutani [15] in 1922, Banach [5] proved that a **contraction** mapping in complete metric space possessed a unique fixed point.

generalizations of Banach **contraction** principle, some generalizations of **contraction** condition was obtained in ( [7] - [10], [13] ). Recently Hussain, Parvaneh, samet and Vetro [9] introduced a new **contraction** map, namely JS-**contraction** map and proved the existence and uniqueness of fixed points in complete metric spaces.

21 Read more

direction or/and generalization of ambiant spaces of the operator under consideration on the other. Banach **contraction** principle [5] which states that if (X , d) is complete metric space and T : X → X is a **contraction** map then T has a unique fixed point, is a fundamental result in this theory. Due to its importance and simplicity several authors have obtained many interesting extensions and generalizations of Banach **contraction** principle, some generalizations of con- traction condition was obtained ([6]-[9], [12]). Recently, Hussain, Parvanch, Samet and Vetro [8] introduced a new **contraction** map, namely JS-**contraction** map and proved the existence and uniqueness of fixed points in complete metric spaces.

Show more
34 Read more

The very ﬁrst contribution to ﬁxed point theory was due to Banach [3] in 1922. He con- ferred the celebrated result in his thesis, namely the Banach **contraction** principle. Later on, many researchers, fascinated by his idea, extended this result in various directions. One of these is by generalizing the metric. In this direction, Matthews [4] in 1994 pre- sented the idea of a partial metric by extending the concept of metric and proved a sup- plementary result of the Banach **contraction** principle in partial metric spaces. Thereafter, many results on the ﬁxed points in partial metric spaces were established (see [5–14] and the references therein).

Show more
22 Read more

Another recent direction of such generalizations, see [–], has been studied by weak- ening the contractive conditions and, in compensation, by simultaneously enriching the metric space structure with a partial order. Very recently, Su [] presented the deﬁnition of **generalized** altering distance function to prove the following new ﬁxed point theorem of **generalized** **contraction** mappings in a complete metric space endowed with a partial order.

25 Read more

In this paper we establish some coincidence point results for **generalized** weak contractions with discontinuous control functions. The theorems are proved in metric spaces with a partial order. Our theorems extend several existing results in the current literature. We also discuss several corollaries and give illustrative examples. We apply our result to obtain some coupled coincidence point results which eﬀectively generalize a number of established results.

21 Read more

In 2000, Branciari [5] introduced the following notion of a **generalized** metric space where the triangle inequality of a metric space had been replaced by an inequality involing three terms instead of two. Later, many authors worked on this interesting space (e.g. [6-11]).

In this paper we introduce the notion of probabilistic G-**contraction** and establish some ﬁxed point theorems in such settings. Our results generalize/extend some recent results of Jachymski and Sehgal and Bharucha-Reid. Consequently, we obtain ﬁxed point results for ( , δ )-chainable PM-spaces and for cyclic operators.

14 Read more

Abstract. In this paper, we establish the existence of some fixed point results for **generalized** (α, β, F )-Geraghty **contraction** in metric-like spaces. We provide an example in order to support our results where some consequence applications of such result will be considered in this article. The obtained results improve and extend some well-known common fixed point results in the literature.

15 Read more

This paper is organized as follows: We define ‘dilation asymmetry’ and ‘erosion asymmetry’ in the next section. These terms are not mentioned in classical mathematical morphology and are defined in this paper to elaborate smoothing of an image by expansion and **contraction** respectively. After this, we generalize these terms for multiscale smoothing and show their oneness respectively with ridge and skeleton. This is followed by a discussion on shape-size complexities of image background and foreground along with their ‘degree of stability’, respectively via multiple close- hulls and open-skulls, under the influence of increasing cycles of morphological closing and opening. We also propose algorithms to obtain critical scales corresponding to multiple close-hulls and open-skulls of an image along with their degree of stability. Afterwards, we demonstrate experimental results of these algorithms on deterministic and random binary Koch quadric fractals. As an outcome of analysis of these algorithms, we define systematic quantization techniques to quantify image and its background into zonal fragments with open-skulls and close-hulls res- pectively being the quantifiers. There exist various techniques to estimate morphology based fractal dimen- sion (e.g. Maragos and Sun, 1993; Radhakrishnan et al., 2004). We compute a scale invariant but shape- dependent morphological quantitative index and correlate this with analytical fractal dimension of binary fractal. The paper ends with conclusion of the results of our study.

Show more
13 Read more

In the same vein probabilistic **generalized** metric spaces have been introduced by Zhou et al. [] wherein they also proved certain primary ﬁxed point results in these spaces. An interesting class of problems in probabilistic ﬁxed point theory was addressed in re- cent times by use of gauge functions. These are control functions which have been used to extend the Sehgal **contraction** in probabilistic metric spaces. Some examples of these applications are in [, , , , , ]. One of such gauge functions was introduced by Fang []. Here we use the gauge function used by Fang [] to obtain a ﬁxed point result in probabilistic G-metric spaces. Our result is supported with an example.

Show more
The purpose of this paper is to modify the Ćirić quasi-contractions. In this paper we introduce D-admissible mappings and establish the ﬁxed point theorem for Ćirić quasi- contractions with the help of D-admissible mappings. This article includes an example of a D-**generalized** metric space to show that a sequence in this setting may be convergent without being a Cauchy sequence. We also investigate the existence and uniqueness of a ﬁxed point for the mappings satisfying nonlinear rational **contraction** and Wardowski type F-**contraction**, where the function F is taken from a more general class of functions than that known in the existing literature.

Show more
14 Read more

In 1992, the notion of partial metric space was introduced by Matthews [7] as a part of the study of denotational semantics of dataflow networks. Henceforward, many authors made ef- forts to study various fixed point theorems and obtained a lot of perfect results. Ravi P Agarwal [10] proved some fixed point results for **generalized** cyclic **contraction** on partial metric spaces. M ˘ ad ˘ alina and Ioan [3] proved a Maia type fixed point theorem for cyclic ϕ -**contraction**, which extended the results of W.A. Kirk [9]. Z. Qingnian and S. Yisheng [12] proved fixed point re- sults for single-valued hybrid **generalized** ϕ-weak contractions. Very recently, many new fixed point results were established by Yi Zhang, Jiang Zhu [8] and L.N. Mishra [11], etc.

Show more
17 Read more

[10]. Moradi S. and Farajzadeh A., On the fixed point of ( 𝜓 − 𝜑 ) - weak and **generalized** (𝜓 − 𝜑) - weak **contraction** mappings, Applied Mathematics Letters 25 (2012), 1257-1262. [11]. Nadler,S.B.JR. sequence of contractions and

Let X be an arbitrary nonempty set. A ﬁxed point for a self-mapping T : X → X is a point x ∈ X such that Tx = x. The applications of ﬁxed point theory are very important in diverse disciplines of mathematics, statistics, chemistry, biology, computer science, engineering, and economics. One of the very popular tools of ﬁxed point theory is the Banach con- traction principle, which ﬁrst appeared in . It states that if (X, d) is a complete metric space and T : X → X is a **contraction** mapping (i.e., d(Tx, Ty) ≤ αd(x, y) for all x, y ∈ X, where α ∈ [, )), then T has a unique ﬁxed point. It has been **generalized** in diﬀerent ways by mathematicians over the years (see [–]). However, almost all such results relate to the existence of a ﬁxed point for self-mappings.

Show more
16 Read more

We introduce the class of **generalized** ψ, ϕ-weak contractive set-valued mappings on a metric space. We establish that such mappings have a unique common end point under certain weak conditions. The theorem obtained generalizes several recent results on single-valued as well as certain set-valued mappings.