Search

Collage Theorem provides a bound for the distance between an element of a given space and a fixed point of a self-map on that space, in terms of the distance between the point and its image. We give in this paper some results of Collage type for Reich mutual contractions in b-metric and strong b-metric spaces. We give upper and lower bounds for this distance, in terms of the constants of the inequality involved in the definition of the contractivity. Reich maps contain the classical Banach contractions as particular cases, as well as the maps of Kannan type, and the results obtained are very general. The middle part of the article is devoted to the invertibility of linear operators. In particular we provide criteria for invertibility of operators acting on quasi-normed spaces. Our aim is the extension of the Casazza-Christensen type conditions for the existence of inverse of a linear map defined on a quasi-Banach space, using different procedures. The results involve either a single map or two operators. The latter case provides a link between the properties of both mappings. The last part of the article is devoted to study the construction of fractal curves in Bochner spaces, initiated by the first author in a previous paper. The objective is the definition of fractal curves valued on Banach spaces and Banach algebras. We provide further results on the fractal convolution of operators, defined in the same reference, considering in this case the nonlinear operators. We prove that some properties of the initial maps are inherited by their convolutions, if some conditions on the elements of the associated iterated function system are satisfied. In the last section of the paper we use the invertibility criteria given before in order to obtain perturbed fractal spanning systems for quasi-normed Bochner spaces composed of Banach-valued integrable maps. These results can be applied to Lebesgue spaces of real valued functions as a particular case. [ABSTRACT FROM AUTHOR]

Impressed with the very recent developments of noncoercive complementarity problems and the use of recession sets in complementarity problems, here, we discuss mixed generalized complementarity problems in Hausdorff topological vector spaces. We used the Tikhonov regularization procedure, as well as arguments from the recession analysis, to establish the existence of solutions for mixed generalized complementarity problems without coercivity assumptions in Banach spaces. [ABSTRACT FROM AUTHOR]

Since the topological properties of a b-metric space (which generalizes the concept of a metric space) seem sometimes counterintuitive due to the fact that the "open" balls may not be open sets, we review some aspects of these spaces concerning compactness, metrizability, continuity and fixed points. After doing so, we introduce new types of contractivities that extend the concept of Banach contraction. We study some of their properties, giving sufficient conditions for the existence of fixed points and common fixed points. Afterwards, we consider some iterative schemes in quasi-normed spaces for the approximation of these critical points, analyzing their convergence and stability. We apply these concepts to the resolution of a model of integral equation of Urysohn type. In the last part of the paper, we refine some results about partial contractivities in the case where the underlying set is a strong b-metric space, and we establish some relations between mutual weak contractions and quasi-contractions and the new type of contractivity. [ABSTRACT FROM AUTHOR]

In this paper, an inverse scattering problem for discrete Sturm–Liouville equation with eigenparameter‐dependent boundary condition is investigated. In quest of finding scattering function and the main equation of this problem, the uniqueness of the kernel is proven. Also, an appropriate Levinson‐type formula based on the continuity of scattering function is given. [ABSTRACT FROM AUTHOR]

Inspired by the ideas of R. Kannan, we define the new concepts of mutual Kannan contractivity and mutual contractivity between two self-maps on a metric space that generalize the concepts of the Kannan map and contraction. We give some examples and deduce the properties of the operators satisfying this type of condition; in particular, we study the case where the space is normed, and the maps are linear. Then we generalize some theorems proposed by this author on the existence of a fixed point of one operator or a common fixed point for two operators. Our results first prove the existence of a common fixed point of a set of self-maps of any cardinal number (countable or uncountable) satisfying the conditions of Kannan type in metric spaces. The same is proved for a set of maps satisfying the mutual relations of classical contractivity. We prove in both cases the convergence of iterative schemes involving operators with mutual relations of contractivity, proposing sufficient conditions for the iteration of the operators on any element of the space to converge to the common fixed point when a different operator is taken in each step. The results obtained are applied to operators acting on real functions, coming from the fractal convolution with the null function. [ABSTRACT FROM AUTHOR]

We consider the fractal convolution of two maps f and g defined on a real interval as a way of generating a new function by means of a suitable iterated function system linked to a partition of the interval. Based on this binary operation, we consider the left and right partial convolutions, and study their properties. Though the operation is not commutative, the one-sided convolutions have similar (but not equal) characteristics. The operators defined by the lateral convolutions are both nonlinear, bi-Lipschitz and homeomorphic. Along with their self-compositions, they are Fréchet differentiable. They are also quasi-isometries under certain conditions of the scale factors of the iterated function system. We also prove some topological properties of the convolution of two sets of functions. In the last part of the paper, we study stability conditions of the dynamical systems associated with the one-sided convolution operators. [ABSTRACT FROM AUTHOR]

In this paper, we define fractal bases and fractal frames of L²(I x J), where I and J are real compact intervals, in order to approximate two-dimensional square-integrable maps whose domain is a rectangle, using the identification of L²(I x J) with the tensor product space L²(I) ⊗ L²(J). First, we recall the procedure of constructing a fractal perturbation of a continuous or integrable function. Then, we define fractal frames and bases of L²(I x J) composed of product of such fractal functions. We also obtain weaker families as Bessel, Riesz and Schauder sequences for the same space. Additionally, we study some properties of the tensor product of the fractal operators associated with the maps corresponding to each variable. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of mixed variational-like inequalities with respect to generalized weakly relaxed η-α monotone mappings, involving nonlinear bifunctions, in finitely continuous topological space, in short FC space. Existence of the solution to the problem is established relaxing convexity and linearity condition by using generalized RKKM theorem. We have proposed a proximal iterative scheme using auxiliary principle technique. Solvability of the auxiliary variational inequality problem is established. Finally convergence of the iterates to the exact solution is proved. Some results with application to equilibrium problem are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we have proved certain kinds of Tauberian theorems for Nörlund-Cesáro summability methods in m-normed space X in statistically sense. We give necessary and sufficient Tauberian condition for this method of summability, in statistically sense. Also, we prove that Tauberian theorems under these summability methods are valid under oscillating slow condition of Schmidt-type and Hardy-type conditions. [ABSTRACT FROM AUTHOR]

In the present paper, we study the mixed summation integral type operators having Baskakov and Szász basis functions in summation and integration form, respectively. We also give here the alternate form of the operators in terms of hypergeometric functions and estimated moments of these operators using hypergeometric series. In the last section, we present some results for operators related to convergence. [ABSTRACT FROM AUTHOR]

Abstract: In this paper the SIR and SIS epidemic models in biology are solved by means of an analytic technique for nonlinear problems, namely the homotopy analysis method (HAM). Both of the SIR and SIS models are described by coupled nonlinear differential equations. A one-parameter family of explicit series solutions are obtained for both models. This parameter has no physical meaning but provides us with a simple way to ensure convergent series solutions to the epidemic models. Our analytic results agree well with the numerical ones. This analytic approach is general and can be applied to get convergent series solutions of some other coupled nonlinear differential equations in biology. [Copyright &y& Elsevier]

This paper deals with the degree of approximation by an operator based on the partial sum of the Fourier series of a function of class L p , p ≥ 1. We use modulus of continuity of the function satisfying weaker conditions than those used by Deǧer and Bayindir [On the degree of approximation by the Woronoi-Nörlund and Riesz type means in the GHM, Azerb. J. Math., 8 (2018), 122-141]. The results of our paper also improve those of Singh and Sonker [Degree of approximation of function in H p w in generalized Hölder metric by matrix means, Mathematical modelling and scientific computation, 1-10, Commun. Comput. Inf. Sci., 283, Springer, Heidelberg, 2012]. [ABSTRACT FROM AUTHOR]

Generalized inverses of tensors play increasingly important roles in computational mathematics and numerical analysis. It is appropriate to develop the theory of generalized inverses of tensors within the algebraic structure of a ring. In this paper, we study different generalized inverses of tensors over a commutative ring and a noncommutative ring. Several numerical examples are provided in support of the theoretical results. We also propose algorithms for computing the inner inverses, the Moore–Penrose inverse, and weighted Moore–Penrose inverse of tensors over a noncommutative ring. The prowess of some of the results is demonstrated by applying these ideas to solve an image deblurring problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solutions for mixed equilibrium problems associated with a set-valued operator in the general setting of vector spaces in duality, and in particular in Banach spaces. We use a Galerkin-type method and the notion of pseudomonotonicity in the sense of Brézis for bifunctions. As application, we study the existence of solutions for quasi-hemivariational inequalities governed by a set-valued mapping and perturbed with a nonlinear term. Our main results can be applied to differential inclusions, evolution equations and evolution hemivariational inequalities. [ABSTRACT FROM AUTHOR]

The session initiation protocol is used for communication purposes in a client-server environment, where for each time session the client and server agree upon a shared secret session key through an authentication system. After establishing the connection between client and server for a session, both the parties use a session key to encrypt/decrypt messages for communicating within that session securely. Therefore, authentication plays a vital role in sharing the secret session key. Recently, Chaudhry et al. proposed an authentication scheme; yet this paper shows that Chaudhry et al.'s scheme has inefficient login and password change phases, respectively, and does not take care of the users' anonymity. Therefore, this study proposes an enhanced scheme, referred to as the secure authentication scheme for session initiation protocol (SAS-SIP) to eliminate the drawbacks of the scheme proposed by Chaudhry et al. In addition, the proposed SAS-SIP uses fuzzy extractors to incorporate biometric data along with the password to enhance the degree of security in the authentication system. After performing the security analysis through a random oracle model, this paper concludes that SAS-SIP is secure from secret information retrieval of communicators to the attacker. Furthermore, it has a better trade-off among several measurement costs along with security. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the new concept of mutual Reich contraction that involves a pair of operators acting on a distance space. We chose the framework of strong b-metric spaces (generalizing the standard metric spaces) in order to add a more extended underlying structure. We provide sufficient conditions for two mutually Reich contractive maps in order to have a common fixed point. The result is extended to a family of operators of any cardinality. The dynamics of iterative discrete systems involving this type of self-maps is studied. In the case of normed spaces, we establish some relations between mutual Reich contractivity and classical contractivity for linear operators. Then, we introduce the new concept of mutual functional contractivity that generalizes the concept of classical Banach contraction, and perform a similar study to the Reich case. We also establish some relations between mutual functional contractions and Banach contractivity in the framework of quasinormed spaces and linear mappings. Lastly, we apply the obtained results to convolutional operators that had been defined by the first author acting on Bochner spaces of integrable Banach-valued curves. [ABSTRACT FROM AUTHOR]

In Albayrak & Pehlivan (2013), we generalized the concepts of pointwise convergence, uniform convergence and α-convergence for sequences of functions on metric spaces by using the filters on N. In this work, we define the concepts of limit function, F-limit function and F-cluster function respectively for each of these three types of convergence, where F is a filter on N. We investigate some topological properties of the sets of F-pointwise cluster functions, F-α-cluster functions and F-uniform cluster functions by using pointwise and uniform convergence topologies. [ABSTRACT FROM AUTHOR]

Abstract: In the case that a frame is prescribed for applications and erasures occur in the process of data transmissions, we examine optimal dual frames for the recovery from single erasures. In contrast to earlier papers, we consider the spectral radius of the error operator instead of its operator norm as a measure of optimality. This notion of optimality is natural when the Neumann series is used to recover the original data in an iterative manner. We obtain a complete characterization of spectrally one-erasure optimal dual frames in terms of the redundancy distribution of the prescribed frame. Our characterization relies on the connection between erasure optimal frames and the linear connectivity property of the frame. We prove that the linear connectivity property is equivalent to the intersection dependent property, and is also closely related to the well-known concept of a k-independent set. Additionally, we also establish several necessary and sufficient conditions for the existence of an alternate dual frame to make the iterative reconstruction work. [Copyright &y& Elsevier]

Abstract: We extend the Casazza–Christensen general perturbation theorem for Hilbert space frames to modular frames in Hilbert -modules. In the Hilbert space setting, under the same perturbation condition, the perturbation of any Riesz basis remains to be a Riesz basis. However, this result is no longer true for Riesz bases in Hilbert -modules. We obtain a necessary and sufficient condition under which the perturbation (under Casazza–Christensen’s perturbation condition) of Riesz bases of Hilbert -modules remains to be Riesz bases. [Copyright &y& Elsevier]

Abstract: This paper deals with the local approximation properties of a certain variant of the Bernstein–Durrmeyer operators. Firstly, we obtain an estimate on the rate of convergence of by means of the decomposition technique for functions of bounded variation. It will be shown that our estimate in its most convenient form (Corollary 2) is asymptotically optimal. Furthermore, we derive the complete asymptotic expansion for the sequence of the operators as tends to infinity. [Copyright &y& Elsevier]

Abstract: The present paper deals with the Beta approximation operators. We obtain an estimate on the rate of convergence for functions of bounded variation by means of the decomposition technique. Furthermore we derive the complete asymptotic expansion of the sequence for smooth functions as n tends to infinity. [Copyright &y& Elsevier]

Abstract: We investigate dual frames of modular frames and Riesz bases in Hilbert -modules. In Hilbert -modules, a Riesz basis may have many dual modular frames, and it may even admit two different dual modular frames both of which are Riesz bases. We characterize those Riesz bases that have unique duals. In addition, we obtain a necessary and sufficient condition for a dual of a Riesz basis to be again a Riesz basis, and prove some new related results. [Copyright &y& Elsevier]