Scopus İndeksli Yayınlar Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8651
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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: 15Fixed 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: 1Metric 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.Book Part Fixed Point Theorems on B-Metric Spaces(Springer Nature, 2022) Karapınar, E.; Agarwal, R.P.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.Book Part Citation - Scopus: 15Fixed Point Theorems in Partial Metric Spaces(Springer Nature, 2022) Karapınar, E.; Agarwal, R.P.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.