Browsing by Author "Agarwal, R.P."

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Citation - Scopus: 8
Analytic Solution of Space Time Fractional Advection Dispersion Equation With Retardation for Contaminant Transport in Porous Media
(Natural Sciences Publishing, 2019) Agarwal, R.; Yadav, M.P.; Agarwal, R.P.; Baleanu, D.; 56389; Matematik
Motivated by recent applications of fractional calculus, in this paper, we derive analytical solutions of fractional advection-dispersion equation with retardation by replacing the integer order partial derivatives with fractional Riesz-Feller derivative for space variable and Caputo fractional derivative for time variable. The Laplace and Fourier transforms are applied to obtain the solution in terms of the Mittag-Leffler function. Some interesting special cases of the time-space fractional advection-dispersion equation with retardation are also considered. The composition formulas for Green function has been evaluated which enables us to express the solution of the space time fractional advection dispersion equation in terms of the solution of space fractional advection dispersion equation and time fractional advection dispersion equation. Furthermore, from this representation we derive explicit formulae, which enable us to plot the probability densities in space for the different values of the relevant parameters. © 2019 NSP Natural Sciences Publishing Cor.
Citation - Scopus: 4
Cirić And Meir-Keeler Fixed Point Results In Super Metric Spaces
(Biemdas Academic Publishers, 2022) Agarwal, R.P.; Karapinar, E.; Khojasteh, F.; 19184; Matematik
In this paper, we consider Meir Keeler and Ćirić contractions in the setting of super metric spaces which is an interesting generalization of standard metric space. We investigate the existence and uniqueness of fixed points for these operators in this new structure. ©2022 Applied Set-Valued Analysis and Optimization.
Citation - Scopus: 5
Fixed Point Theorems in Partial Metric Spaces
(Springer Nature, 2022) Karapınar, E.; Agarwal, R.P.; Matematik
In this chapter, we recollect some crucial and exciting fixed point theorems in the context of partial metric space. In addition, we underline the importance of the partial metric space in fixed point theory. Matthews [210] not only introduced the partial metric spaces but also obtained the first fixed point theorem in this new setting. More precisely, Matthews [210] showed that the famous Banach fixed point theory is valid in the framework of complete partial metric space. After this pioneering result of Matthews [210] in fixed point theory, a considerable number of researchers have investigated the partial metric spaces, and remarkable number of fixed point results in the context of partial metrics have appeared in the literature (see e.g. [1, 2, 7, 27–30, 34, 38–40, 42, 46, 49–53, 86, 92, 94, 102, 120, 125, 130, 136, 137, 146, 154, 155, 157, 164–166, 173, 175, 176, 178, 181, 188, 189, 202, 204, 210, 211, 222, 225, 230, 243, 245–250, 255, 258, 261, 262, 265, 266, 272–278] and the reference therein). © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.
Fixed Point Theorems on B-Metric Spaces
(Springer Nature, 2022) Karapınar, E.; Agarwal, R.P.; Matematik
In this chapter, we collect some important fixed point theorems. Fixed point theorems in b-metric spaces have been studied by many author, e.g. [11, 12, 20, 23, 36, 37, 45, 57, 59, 71, 72, 74, 76, 77, 79, 117, 119, 161, 182, 217] © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.
Citation - Scopus: 8
Fixed Point Theory in Generalized Metric Spaces
(Springer Nature, 2022) Karapınar, E.; Agarwal, R.P.; Matematik
Generalization of Metric Spaces
(Springer Nature, 2022) Karapınar, E.; Agarwal, R.P.; Matematik
In this chapter, we discuss some of the interesting generalizations and extensions of the notion of the metric. Roughly speaking, the notion of metric can be considered as an axiomatic form of the “distance”. For this reason, the metric and all generalizations and extensions of the metric can be called “distance function”. One of the well-known examples of the metric is due to Euclid which is known as Euclidean metric. © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.
Metric Fixed Point Theory
(Springer Nature, 2022) Karapınar, E.; Agarwal, R.P.; Matematik
The aim of this chapter is to give a brief history of metric fixed point theory. In this section, we discuss the pioneer metric fixed point theorem that was given by Banach [56]. This outstanding result is known as the contraction mapping principle or the Banach contraction mapping principle. The main advantage of Banach’s metric fixed point theorem is the following property: This theorem not only guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces but also indicates how to construct an iterative sequence that provides the desired fixed points. It is worth mentioning that this famous fixed point theorem was formulated in his thesis in 1920 and published in 1922 in the setting of normed linear spaces (not metric spaces). As we mentioned in the previous section, Banach Contraction Mapping Principle is not the first fixed point theorem in the literature but the most interesting fixed point theorem in the context of metric fixed point theory. Indeed, Brouwer gave the first result, which only guarantees the existence of the fixed point. Unfortunately, Brouwer’s fixed point theorem does not explain how to get the guaranteed fixed point and how to ensure the uniqueness of this mentioned fixed point. © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.
Metric Spaces
(Springer Nature, 2022) Karapınar, E.; Agarwal, R.P.; Matematik
The notion of the metric can be considered as a generalization of two point distance that was contrived systematically first by Euclid. In the modern mathematical set-up, Maurice René Frechét [116] is the first mathematician who axiomatically formulated the notion of metric space, under the name of L-space. On the other hand, first Felix Hausdorff [129] used the term “metric space” although he mainly focused on the role of point-sets within abstract set theory. Throughout the book, all sets are presumed nonempty. A distance function over a set X, namely, d: X× X→ [ 0, ∞), is called metric, or usual metric or standard metric if (d1) d(x, y) = d(y, x) = 0 ⟹ x= y ; (d2) d(x, x) = 0 ; (d3) d(x, y) = d(y, x) ; (d4) d(x, z) ≤ d(x, y) + d(y, z) ; for each x, y, z∈ X. In particular, the pair (X, d) is called metric space or usual metric space or standard metric space. © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.