Scopus İndeksli Yayınlar Koleksiyonu

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  • Book Part
    Citation - Scopus: 8
    Advanced Topics in Fractional Differential Equations a Fixed Point Approach
    (Springer Nature, 2023) Benchohra, M.; Karapınar, Erdal; Karapınar, E.; Lazreg, J.E.; Salim, A.; Matematik
  • Editorial
    Preface
    (Springer Nature, 2023) Benchohra, M.; Karapinar, E.; Lazreg, J.E.; Salim, A.; Hristov, Jordan; Anastassiou, George A.; Baleanu, Dumitru; Singh, Jagdev; Cattani, Carlo; Kumar, Devendra; Dutta, Hemen; Matematik
  • Book Part
    Introduction
    (Springer Nature, 2022) Karapınar, Erdal; Agarwal, Ravi P.
    Fixed point theory can be described as a framework for researching and investigating the existence of the solution of the equation f(p) = p for a certain self-mapping f that is defined on a non-empty set X. As is expected, here, p is called the fixed point of the mapping f. On the other side, we may re-consider the fixed point equation f(p) = p as T(p) = f(p) - p= 0 and, accordingly, finding the zeros of the mapping T and finding the fixed point of f becomes an equivalent statement. This equivalence, not only enriches the fixed point theory but also, opens the doors to a wide range of potential applications in the setting of almost all quantitative sciences. For example, let us consider one of the classical open problems of number theory, finding perfect numbers: Let p be a self-mapping on a natural number such that p(n) is the sum of all divisors of n for n> 1. Thus, any fixed points of the function p give a perfect number. In particular, 6 is the smallest perfect numbers, and 2 74207280× (2 74207281- 1 ), with 44, 677, 235 digits, is the biggest known perfect number. © 2022 Elsevier B.V., All rights reserved.
  • Book Part
    Preliminary Background
    (Springer Nature, 2023) Benchohra, M.; Karapınar, E.; Lazreg, J.E.; Salim, A.
    In this chapter, we discuss the necessary mathematical tools, notations, and concepts we need in the succeeding chapters. We look at some essential properties of fractional differential operators. We also review some of the basic properties of measures of noncompactness and fixed point theorems which are crucial in our results regarding fractional differential equations. © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.
  • Book Part
    Introduction
    (Springer Nature, 2023) Benchohra, M.; Karapınar, E.; Lazreg, J.E.; Salim, A.
    Fractional calculus is a field in mathematical analysis which is a generalization of integer differential calculus that involves real or complex order derivatives and integrals [10–14, 25, 28, 43, 50–52]. There is a long history of this concept of fractional differential calculus. One might wonder what meaning could be attributed to the derivative of a fractional order, that is dnydxn, where n is a fraction. Indeed, in correspondence with Leibniz, L’Hopital considered this very possibility. © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.
  • Book Part
    Citation - Scopus: 15
    Fixed Point Theory in Generalized Metric Spaces
    (Springer Nature, 2022) Karapınar, E.; Agarwal, R.P.
  • Book Part
    Introduction
    (Springer Nature, 2023) Benchohra, M.; Karapınar, E.; Lazreg, J.E.; Salim, A.
    Fractional calculus is an area of mathematical analysis that extends the concepts of integer differential calculus to involve real or complex order derivatives and integrals. © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.
  • Book Part
    Citation - Scopus: 1
    Metric Fixed Point Theory
    (Springer Nature, 2022) Karapınar, E.; Agarwal, R.P.
    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.
  • Book Part
    Metric Spaces
    (Springer Nature, 2022) Karapınar, E.; Agarwal, R.P.
    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.
  • Book Part
    Generalization of Metric Spaces
    (Springer Nature, 2022) Karapınar, E.; Agarwal, R.P.
    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.