## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

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.

To read the full-text of this research,

you can request a copy directly from the authors.

... 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 ...

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 ...

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]). ...

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]. ...

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: ...

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)). ...

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]). ...

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, ...

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]). ...

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. ...

The main purpose of this article is to provide alternative proofs of the metrizability of metric-like spaces like b-metric spaces, F-metric spaces, and θ-metric spaces. We improve upon the metrizability result of An et al. [Topology Appl. 185-186 (2015)] for b-metric spaces. Moreover, we provide two shorter proofs of the metrizability of F-metric spaces recently introduced by Jleli and Samet in 2018. Furthermore, we give a partial answer to an open problem regarding the openness of F-open balls in F-metric spaces. Finally, we give an alternative proof of the metrizability of θ-metric spaces.

... 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]). ...

Some complicated events can be modeled by systems of differential equations. On the other hand, inclusion systems can describe complex phenomena having some shocks better than the system of differential equations. Also, one of the interests of researchers in this field is an investigation of hybrid systems. In this paper, we study the existence of solutions for hybrid and non-hybrid k-dimensional sequential inclusion systems by considering some integral boundary conditions. In this way, we use different methods such as α-ψ contractions and the endpoint technique. Finally, we present two examples to illustrate our main results.
MSC: Primary 34A08; secondary 34A12

... 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. ...

In this paper, we introduce a new structure of the generalized multi-point thermostat control model motivated by its standard model. By presenting integral solution of this boundary problem, the existence property along with the uniqueness property are investigated by means of a special version of contractions named μ-ϕ-contractions and the Banach contraction principle. Then, on the given nonlinear generalized BVP of thermostat, the Bernstein polynomials are introduced and numerical solutions obtained by them are presented. At the end, three different structures of nonlinear thermostat models are designed and the results are examined.

In this research work, a newly-proposed multiterm hybrid multi-order fractional boundary value problem is studied. The existence results for the supposed hybrid fractional differential equation that involves Riemann–Liouville fractional derivatives and integrals of multi-orders type are derived using Dhage’s technique, which deals with a composition of three operators. After that, its stability analysis of Ulam–Hyers type and the relevant generalizations are checked. Some illustrative numerical examples are provided at the end to illustrate and validate our obtained results.

In this paper, we will study different kind of Ulam stability concepts for the zero point equation. Our approach is based on weakly Picard operator theory related to fixed point and coincidence point equations.

Starting with a stability problem posed by Ulam for group homomorphisms, we characterize the functions with values in a Banach space, which can be approximated by cubic mappings with a given error.

In this paper, we introduce a new concept of Ulam stability of fixed point equation with respected to a directed graph. Two fixed point theorems of Matkowski and of Jachymski are discussed further in the sense of this stability concept. Some examples about the validity of our notion are given. Finally, we discuss the vagueness of the recent stability results of Sintunavarat [Sintunavarat, W., A new approach to α-ψ-contractive mappings and generalized Ulam–Hyers stability, well-posedness and limit shadowing results, Carpathian J. Math., 31 (2015), 395–401].

In this manuscript, we deal with the nonlocal controllability results for the fractional evolution system of $1< r<2$ 1 < r < 2 in a Banach space. The main results of this article are tested by using fractional calculations, the measure of noncompactness, cosine families, Mainardi’s Wright-type function, and fixed point techniques. First, we investigate the controllability results of a mild solution for the fractional evolution system with nonlocal conditions using the Mönch fixed point theorem. Furthermore, we develop the nonlocal controllability results for fractional integrodifferential evolution system by applying the Banach fixed point theorem. Finally, an application is presented for drawing the theory of the main results.

Using the fixed point theorem [12, Theorem 1] in (2, β)-Banach spaces, we prove the generalized hyperstability results of the bi-Jensen functional equation 4f(( x + z)/2,( y + w ) /2) = f (x, y) + f (x, w) + f (z, y) + f (y, w). Our main results state that, under some weak natural assumptions, functions satisfying the equation approximately (in some sense) must be actually solutions to it. The method we use here can be applied to various similar equations in many variables.

The approximate controllability of second-order integro-differential evolution control systems using resolvent operators is the focus of this work. We analyze approximate controllability outcomes by referring to fractional theories, resolvent operators, semigroup theory, Gronwall’s inequality, and Lipschitz condition. The article avoids the use of well-known fixed point theorem approaches. We have also included one example of theoretical consequences that has been validated.

By using Aoki‐Rolewicz Theorem on p‐normalizing a quasi‐normed space, we prove stability results for Euler‐Lagrange quadratic functional equations in quasi‐Banach spaces. These results improve stability results and give the answer to Kim‐Rassias's question.

A list of 22 definitions of stability of a functional equation and 5 definitions of stability of the alternation of two functional equations is given as well as some simple examples.

Given a vector space X, we investigate the solutions f: R → X of the linear functional equation of third order f (x) = p f (x 1) + q f (x 2) + r f (x - 3), which is strongly associated with a well-known identity for the Fibonacci numbers. Moreover, we prove the Hyers-Ulam stability of that equation.

This is a survey paper concerning stability results for the linear functional equation in single variable. We discuss issues that have not been considered or have been treated only briefly in other surveys concerning stability of the equation. In this way, we complement those surveys.

We prove the Hyers-Ulam stability of the generalized Fibonacci functional equation F(x)-g(x)F(h(x))=0, where g and h are given functions.

The aim of this paper is to investigate some fixed point results in dislocated quasi metric (dq-metric) spaces. Fixed point results for different types of contractive conditions are established, which generalize, modify and unify some existing fixed point theorems in the literature. Appropriate examples for the usability of the established results are also given. We notice that by using our results some fixed point results in the context of dislocated quasi metric spaces can be deduced.
MSC:
47H10, 54H25.

The fixed point method has been applied for the first time, in proving the stability results for functional equations, by Baker (1991); he used a variant of Banach’s fixed point theorem to obtain the stability of a functional equation in a single variable. However, most authors follow the approaches involving a theorem of Diaz and Margolis. The main aim of this survey is to present applications of different fixed point theorems to the theory of stability of functional equations, motivated by a problem raised by Ulam in 1940.

This is a monograph on fixed point theory, covering the purely metric aspects of the theory-particularly results that do not depend on any algebraic structure of the underlying space. Traditionally, a large body of metric fixed point theory has been couched in a functional analytic framework. This aspect of the theory has been written about extensively. There are four classical fixed point theorems against which metric extensions are usually checked. These are, respectively, the Banach contraction mapping principal, Nadler's well known set-valued extension of that theorem, the extension of Banach's theorem to nonexpansive mappings, and Caristi's theorem. These comparisons form a significant component of this book. This book is divided into three parts. Part I contains some aspects of the purely metric theory, especially Caristi's theorem and a few of its many extensions. There is also a discussion of nonexpansive mappings, viewed in the context of logical foundations. Part I also contains certain results in hyperconvex metric spaces and ultrametric spaces. Part II treats fixed point theory in classes of spaces which, in addition to having a metric structure, also have geometric structure. These specifically include the geodesic spaces, length spaces and CAT(0) spaces. Part III focuses on distance spaces that are not necessarily metric. These include certain distance spaces which lie strictly between the class of semimetric spaces and the class of metric spaces, in that they satisfy relaxed versions of the triangle inequality, as well as other spaces whose distance properties do not fully satisfy the metric axioms. © Springer International Publishing Switzerland 2014. All rights are reserved.

We discuss topological structure of 𝑏-metric-like spaces and demonstrate a fundamental lemma for the convergence of sequences.
As an application we prove certain fixed point results in the setup of such spaces for different types of contractivemappings. Finally,
some periodic point results in 𝑏-metric-like spaces are obtained. Two examples are presented in order to verify the effectiveness
and applicability of our main results.

In this paper we prove a fixed-point theorem for a class of operators with suitable properties, in very general conditions. Also, we show that some recent fixed-points results in Brzdęk et al., (2011) and Brzdęk and Ciepliński (2011) can be obtained directly from our theorem. Moreover, an affirmative answer to the open problem of Brzdęk and Ciepliński (2011) is given. Several corollaries, obtained directly from our main result, show that this is a useful tool for proving properties of generalized Hyers-Ulam stability for some functional equations in a single variable.

In this paper we obtain a result on Hyers-Ulam stability of the linear
functional equation in a single variable $f(\varphi(x)) = g(x) \cdot f(x)$ on a
complete metric group.

We give some theorems on the stability of the equation of homomorphism, of Lobacevski’s equation, of almost Jensen’s equation,
of Jensen’s equation, of Pexider’s equation, of linear equations, of Schröder’s equation, of Sincov’s equation, of modified
equations of homomorphism from a group (not necessarily commutative) into a
\mathbbQ{\mathbb{Q}}-topological sequentially complete vector space or into a Banach space, of the quadratic equation, of the equation of a generalized
involution, of the equation of idempotency and of the translation equation. We prove that the different definitions of stability
are equivalent for the majority of these equations. The boundedness stability and the stability of differential equations
and the anomalies of stability are considered and open problems are formulated too.

Let X be a Banach space over the field Ã¢Â„Â or Ã¢Â„Â‚, a1,Ã¢Â€Â¦,apÃ¢ÂˆÂˆÃ¢Â„Â‚, and (bn)nÃ¢Â‰Â¥0 a sequence in X. We investigate the Hyers-Ulam stability of the linear recurrence xn+p=a1xn+pÃ¢ÂˆÂ’1+Ã¢Â‹Â¯+apÃ¢ÂˆÂ’1xn+1+apxn+bn, nÃ¢Â‰Â¥0, where x0,x1,Ã¢Â€Â¦,xpÃ¢ÂˆÂ’1Ã¢ÂˆÂˆX.

We discuss on the generalized Ulam-Hyers stability for functional equations in a single variable, including the nonlinear functional equations, the linear functional equations, and a generalization of functional equation for the square root spiral. The stability results have been obtained by a fixed point method. This method introduces a metrical context and shows that the stability is related to some fixed point of a suitable operator.

We give an answer to a problem formulated by Th. M. Rassias in 1991 concerning stability of the Cauchy equation; we also disprove a conjecture of Th. M. Rassias and J. Tabor. In particular, we present a new method for proving stability results for functional equations.

In this paper we solve a problem posed by John M. Rassias in 1992, concerning the stability of Euler-Lagrange equation in the Ulam sense.

The issue of Ulam's type stability of an equation is understood in the following way: when a mapping which satisfies the equation approximately (in some sense), it is "close" to a solution of it. In this expository paper, we present a survey and a discussion of selected recent results concerning such stability of the equations of homomorphisms, focussing especially on some conditional versions of them.

The fixed-point method, which is the second most popular technique of proving the Hyers-Ulam stability of functional equations, was used for the first time in 1991 by J. A. Baker in [Proc. Am. Math. Soc. 112, No. 3, 729–732 (1991; Zbl 0735.39004)], who applied a variant of Banach’s fixed-point theorem to obtain the stability of a functional equation in a single variable. However, most authors follow V. Radu’s [Fixed Point Theory 4, No. 1, 91–96 (2003; Zbl 1051.39031)] approach and make use of a theorem of J. B. Diaz and B. Margolis [Bull. Am. Math. Soc. 74, 305–309 (1968; Zbl 0157.29904)]. The main aim of this survey is to present applications of different fixed-point theorems to the theory of the Hyers-Ulam stability of functional equations.

If (X,d) is a so-called b-metric space, then in the space CL(X) of all nonempty closed subsets of X it induces the generalized Hausdorff b-metric. We prove that if (X,d) is complete, then CL(X) is complete too. Moreover, some fixed point theorems for nonlinear set-valued contraction mappings are presented.

Shadowing Near an Invariant Set: Basic Definitions. Shadowing Near a Hyperbolic Set for a Diffeomorphism. Shadowing for Mappings of Banach Spaces. Limit Shadowing. Shadowing for Flows.- Topologically Stable, Structurally Stable, and Generic Systems: Shadowing and Topological Stability. Shadowing in Structurally Stable Systems. Shadowing in Two-Dimensional Diffeomorphisms. C0-Genericity of Shadowing for Homeomorphisms.- Systems with Special Structure: One-Dimensional Systems. Linear and Linearly Induced Systems. Lattice Systems. Global Attractors for Evolution Systems.- Numerical Applications of Shadowing: Finite Shadowing. Periodic Shadowing for Flows. Approximation of Spectral Characteristics. Approximation of the Morse Spectrum. Discretizations of PDEs.- References.- Index.

We investigate the Hyers-Ulam stability of the functional equation f(phi(x)) = phi(x)f(x) + psi(x) and the stability in the sense of R. Ger of the functional equation f (phi(x)) = phi(x)f(x) in the following two settings: parallel tog(phi(x)) - phi(x)g(x) - psi(x)parallel to less than or equal to epsilon(x) and \g(phi(x)) /phi(x)g(x)\ < epsilon(x).

We present a survey of some selected recent developments (results and methods) in the theory of Ulam's type stability. In particular we provide some information on hyperstability and the fixed point methods.

By a metric-like space, as a generalization of a partial metric space, we mean a pair By a metric-like space, as a generalization of a partial metric space, we mean a pair (X,σ), where X is a nonempty set and σ:X×X→ℝ satisfies all of the conditions of a metric except that σ(x,x) may be positive for x∈X. In this paper, we initiate the fixed point theory in metric-like spaces. As an application, we derive some new fixed point results in partial metric spaces. Our results unify and generalize some well-known results in the literature.

In this note, we prove a simple fixed point theorem for a special class of complete metric spaces (namely, complete non-Archimedean metric spaces which are connected with some problems coming from quantum physics, p-adic strings and superstrings). We also show that this theorem is a very efficient and convenient tool for proving the Hyers–Ulam stability of a quite wide class of functional equations in a single variable.

The aim of this paper is to prove the stability (in the sense of Ulam) of the functional equation: f(t) = alpha(t) + beta(t)f(phi(t)), where alpha and beta are given complex valued functions defined on a nonempty set S such that sup{\beta(t)\:t is-an-element-of S} < 1 and phi is a given mapping of S into itself.

In this short paper the core of the direct method for proving stability of functional equations is described in a clear way and in a quite general form.

We give an answer to a problem of G. Isac and Th. M. Rassias concerning Hyers–Ulam–Rassias stability of the linear mappings. A new characterization of ψ-additive mappings is also given.

We prove a simple fixed point theorem for some (not necessarily linear) operators and derive from it several quite general results on the stability of a very wide class of functional equations in single variable.

This paper discusses Hyers–Ulam stability for functional equations in single variable, including the forms of linear functional equation, nonlinear functional equation and iterative equation. Surveying many known and related results, we clarify the relations between Hyers–Ulam stability and other senses of stability such as iterative stability, continuous dependence and robust stability, which are used for functional equations. Applying results of nonlinear functional equations we give the Hyers–Ulam stability of Böttcher's equation. We also prove a general result of Hyers–Ulam stability for iterative equations.

Consideration of the Associativity Equation,x (y z) = (x y) z, in the case where:I I I (I a real interval) is continuous and satisfies a cancellation property on both sides, provides a complete characterization of real continuous cancellation semigroups, namely that they are topologically order-isomorphic to addition on some real interval: ( – ,b), ( – ,b], –, +), (a, + ), or [a, + ) — whereb = 0 or –1 anda = 0 or 1. The original proof, however, involves some awkward handling of cases and has defied streamlining for some time. A new proof is given following a simpler approach, devised by Ples and fine-tuned by Craigen.

A shadow is an exact solution to a set of equations that remains close to a numerical solution for a long time. Shadowing can thus be used as a form of backward error analysis for numerical solutions to ordinary differential equations. This survey introduces the reader to shadowing with a detailed tour of shadowing algorithms and practical results obtained over the last 15 years.

- Y Benyamini
- J Lindenstrauss

Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Vol. 1, Colloq. Publ., vol. 48, Amer. Math.
Soc., Providence, 2000.

- P Găvruţa

P. Găvruţa, An answer to a question of John M. Rassias concerning the stability of Cauchy equation, in: Advances in
Equations and Inequalities, Hadronic Math. Ser., 1999, pp. 67-71.

- F Q Gouvêa

F.Q. Gouvêa, p-adic Numbers. An Introduction, Universitext, Springer-Verlag, Berlin, 1997.

- Z.-Ch Lin
- M.-R Zhou

Z.-Ch. Lin, M.-R. Zhou, Perturbation Methods in Applied Mathematics, Jiangsu Education Press, Nanjing, PRC, 1995.

- A M Robert

A.M. Robert, A Course in p-adic Analysis, Graduate Texts in Mathematics, vol. 198, Springer-Verlag, New York, 2000.

- I A Rus

I.A. Rus, Remarks on Ulam stability of the operatorial equations, Fixed Point Theory 10 (2009) 305-320.