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# A fixed point theorem and the Ulam stability in generalized dq-metric spaces

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## Abstract

We prove a fixed point theorem for function spaces, that is a very efficient and convenient tool for the investigations of various operator inequalities connected to Ulam stability issues, in classes of functions taking values in various spaces (e.g., in ultrametric spaces, dq-metric spaces, quasi-Banach spaces, and p-Banach spaces). The theorem is a natural generalization and extension of the classical Banach Contraction Principle and some other more recent results.

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... Our method in this paper is based on fixed point approaches. Also, we can find more ideas on fractional calculus and its applications in [3,[25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. ...
... The right-hand sides of (34) and (33) approach 0 as t * → t. Therefore relations (30), (31), (32), and (34) imply that ...
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This research is conducted for studying some qualitative specifications of solution to a generalized fractional structure of the standard snap boundary problem. We first rewrite the mathematical model of the extended fractional snap problem by means of the G-operators. After finding its equivalent solution as a form of the integral equation, we establish the existence criterion of this reformulated model with respect to some known fixed point techniques. Then we analyze its stability and further investigate the inclusion version of the problem with the help of some special contractions. We present numerical simulations for solutions of several examples regarding the fractional G-snap system in different structures including the Caputo, Caputo–Hadamard, and Katugampola operators of different orders.
... Therefore they have obtained importance due to their applications in science and engineering such as, physics, chemistry, mechanics, fluid dynamic, etc. [1,2]. Meanwhile, there have appeared many papers dealing with the existence of solutions for different types of fractional boundary value problems; see, for example, [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. ...
... is in the form of Eq. (15). Hence, Lemma 3.8 implies ...
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In this paper, we propose the conditions on which a class of boundary value problems, presented by fractional q -differential equations, is well-posed. First, under the suitable conditions, we will prove the existence and uniqueness of solution by means of the Schauder fixed point theorem. Then, the stability of solution will be discussed under the perturbations of boundary condition, a function existing in the problem, and the fractional order derivative. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.
... The main result in [9] can be formulated as follows (see [7] for an abstract generalization of it). ...
... The notion of Ulam stability (see [8,16,22] for details) has motivated several generalizations of Banach Contraction Principle for various function spaces (see, e.g., [3,4,6,7,9]). Let us recall for instance the main result in [9] (see also [4,Theorem 2]). ...
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We prove and discuss several fixed point results for nonlinear operators, acting on some classes of functions with values in a b-metric space. Thus we generalize and extend a recent theorem of Dung and Hang (J Math Anal Appl 462:131–147, 2018), motivated by several outcomes in Ulam type stability. As a simple consequence we obtain, in particular, that approximate (in some sense) eigenvalues of some linear operators, acting in some function spaces, must be eigenvalues while approximate eigenvectors are close to eigenvectors with the same eigenvalue. Our results also provide some natural generalizations and extensions of the classical Banach Contraction Principle.
... Impulsive differential equations are best tools to model a physical situation that contains abrupt changes at certain instants. These equations describe medicine, biotechnology process, population dynamics, biological systems, chemical energy, mathematical economy, pharmacokinetics, etc. [31][32][33][34][35][36][37][38][39][40][41][42][43]. ...
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In this article, we establish a new class of mixed integral fractional delay dynamic systems with impulsive effects on time scales. We investigate the qualitative properties of the considered systems. In fact, the article contains three segments, and the first segment is devoted to investigating the existence and uniqueness results. In the second segment, we study the stability analysis, while the third segment is devoted to investigating the controllability criterion. We use the Leray–Schauder and Banach fixed point theorems to prove our results. Moreover, the obtained results are examined with the help of an example.
... There is a strong relation between fixed point theory and Ulam stability; see, e.g., [36][37][38][39][40][41]. is called generalized Ulam stable if for each ε > 0 and s ∈ M, there exists n(s) ∈ {1, 2, ...} such that for any v ∈ M satisfying the inequality: ...
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In this paper, in the setting of Δ -symmetric quasi-metric spaces, the existence and uniqueness of a fixed point of certain operators are scrutinized carefully by using simulation functions. The most interesting side of such operators is that they do not form a contraction. As an application, in the same framework, the Ulam stability of such operators is investigated. We also propose some examples to illustrate our results.
... Recently, Brzdek et al. [19] introduced a notion of generalized d q metric which can be defined as: a function d : X × X → R + satisfying following axioms for all a, b, c ∈ X, (B 1 ) if d(a, b) = 0 and d(b, a) = 0, then a = b; (B 2 ) there exist a mapping µ : R + × R + → R + which is nondecreasing with respect to each variable such that d(a, c) ≤ µ(d(a, b), d(b, c)). ...
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A new proper generalization of metric called as θ-metric is introduced by Khojasteh et al. (Mathematical Problems in Engineering (2013) Article ID 504609). In this paper, first we prove the Caristi type fixed point theorem in an alternative and comparatively new way in the context of θ-metric. We also investigate two θ-metrics on CB(X) (family of nonempty closed and bounded subsets of a set X). Furthermore, using the obtained θ-metrics on CB(X), we prove two new fixed point results for multi-functions which generalize the results of Nadler and Lim type in the context of such spaces. In order to illustrate the usability of our results, we equipped them with competent examples.
... The concept of a b-metric space was nominated by Czerwik in [20]. Later, many interesting results about the existence of fixed points in b-metric spaces have been acquired (see, [2,[21][22][23][24][25][26][27][28][29][30][31][32][33]). ...
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In this paper, we present the concept of Θ − ( σ , ξ ) Ω -contraction mappings and we nominate some related fixed point results in ordered p-metric spaces. Our results extend several famous ones in the literature. Some examples and an application are given in order to validate our results.
... see[9, Theorems 12,13,16,19,22, 25 & 28]. Heidarpour[36] proved the superstability of n-ring homomorphisms on C [36, Theorems 2.1 & 2.3] and established Ulam-Hyers stability of n-ring homomorphisms in p-Banach algebras [36, Theorems 3.1-3.3, ...
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In this chapter, we give a survey on Ulam-Hyers stability of functional equations in quasi-β-Banach spaces, in particular in p-Banach spaces, quasi-Banach spaces and (β, p)-Banach spaces.
... for all x, y, z ∈ Y . The notion of a dq-metric space is a natural generalization of the usual definitions of metric, quasi-metric, partial metric, and metriclike spaces and plays crucial roles in computer science and cryptography (see, e.g., [2,4,11,13,15,[20][21][22]25,26]). ...
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We prove a fixed point result for nonlinear operators, acting on some classes of functions with values in a dq-metric space, and show some applications of it. The result has been motivated by some issues arising in Ulam stability. We use a restricted form of a contraction condition.
... In 2010, Khamsi and Hussain [13] reconsidered the concept of a b-metric under the name metric-type spaces, where they had considered the coefficient to be K > 0. To avoid confusion, the metric-type in the sense of Khamsi and Hussain [13] will be called b-metric in this short note. For related results and applications in b-metric spaces see also [4][5][6] and [17]. Before going further, we like to recall the definition of a b-metric space from [13] as follows: Definition 1.1. ...
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... In recent years, many researchers have become interested in the field of fractional calculus. There are various techniques in this way such as fixed point theory to investigate the existence of solutions for fractional differential equations (see, for example, [1][2][3][4][5][6][7][8][9][10]). One can find different techniques or ideas in [11][12][13] or many applied ideas in this area (see, for example, [14][15][16]). ...
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... The Bernstein polynomials are one of the strongest numerical techniques which possess some important properties such as the unity partition and continuity on [0, 1] [41]. In recent years, due to the importance and accuracy of this technique, the numerical solutions of a wide range of linear\nonlinear BVPs have been obtained for Riccati type FDEs, Bessel FDE, Lane-Emden equations, etc., [11][12][13][42][43][44]. We here use these polynomials to find approximate solution of our given multi-point FBVP introduced in the sequel. ...
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