Search

The stochastic approximation (SA) method stands as the foundational mathematical tool for modern large-scale optimization and machine learning. Therefore, gaining a theoretical understanding of SA algorithms is of fundamental interest. In their paper titled "A Lyapunov Theory for Finite-Sample Guarantees of Markovian Stochastic Approximation," Chen et al. present a unified Lyapunov framework for the finite-sample analysis of a Markovian SA algorithm under a contractive operator with respect to an arbitrary norm. The key novelty lies in the construction of a smooth Lyapunov function called the generalized Moreau envelope. The authors demonstrate the effectiveness of their SA results in the context of reinforcement learning (RL), specifically through popular algorithms such as variants of temporal difference (TD) learning and Q-learning. As byproducts, the results provide theoretical insights into the efficiency of bootstrapping in TD learning with eligibility traces and the bias-variance tradeoff in off-policy learning. This paper develops a unified Lyapunov framework for finite-sample analysis of a Markovian stochastic approximation (SA) algorithm under a contraction operator with respect to an arbitrary norm. The main novelty lies in the construction of a valid Lyapunov function called the generalized Moreau envelope. The smoothness and an approximation property of the generalized Moreau envelope enable us to derive a one-step Lyapunov drift inequality, which is the key to establishing the finite-sample bounds. Our SA result has wide applications, especially in the context of reinforcement learning (RL). Specifically, we show that a large class of value-based RL algorithms can be modeled in the exact form of our Markovian SA algorithm. Therefore, our SA results immediately imply finite-sample guarantees for popular RL algorithms such as n-step temporal difference (TD) learning, TD (λ) , off-policy V-trace, and Q-learning. As byproducts, by analyzing the convergence bounds of n-step TD and TD (λ) , we provide theoretical insight into the problem about the efficiency of bootstrapping. Moreover, our finite-sample bounds of off-policy V-trace explicitly capture the tradeoff between the variance of the stochastic iterates and the bias in the limit. Funding: This work was supported by RTX, the National Science Foundation [Grants 2019844, 2107037, 211247, 2112533, 2144316, and 2240982], and the Machine Learning Laboratory at University of Texas at Austin. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2022.0249. [ABSTRACT FROM AUTHOR]

Two of the most well-known belief contraction operators are partial meet contractions (PMCs) and kernel contractions (KCs). In this paper we propose two new classes of contraction operators, namely the class of generalized partial meet contractions (GPMC) and the class of generalized kernel contractions (GKC), which strictly contain the classes of PMCs and of KCs, respectively. We identify some extra conditions that can be added to the definitions of GPMCs and of GKCs, which give rise to some interesting subclasses of those classes of functions, namely the classes of extensional and of uniform GPMCs/GKCs. In the context of contractions on belief sets the classes of partial meet contractions, uniform GPMCs and extensional GPMCs are all identical. Nevertheless, when considered as operations on belief bases, the class of uniform GPMCs coincides with the class of partial meet contractions, but the extensional GPMCs constitute a new kind of belief base contraction functions whose characterizing postulate of irrelevance of syntax is extensionality—the same postulate of irrelevance of syntax which occurs in the classical axiomatic characterization of partial meet contractions for belief sets—rather than the postulate of uniformity—which is the irrelevance of syntax postulate used in the axiomatic characterization for partial meet contractions on belief bases. Analogous results are obtained regarding the classes of extensional and of uniform GKCs. We present the interrelations in the sense of inclusion among all the new classes of operators presented in this paper and several well known classes of PMCs and of KCs. [ABSTRACT FROM AUTHOR]

In this paper, we study a type of nonlinear hybrid Δ-difference equations of fractional-order. The main objective is to establish some stability criteria including the Ulam–Hyers stability, generalized Ulam–Hyers stability together with the Mittag-Leffler–Ulam–Hyers stability for the addressed problem. Prior to the stabilization processes, solvability criteria for the existence and uniqueness of solutions are considered. For this purpose, a hybrid fixed point theorem for triple operators and the Banach contraction mapping principle are applied, respectively. For the sake of illustrating the practical impact of the proposed theoretical criteria, we finish the paper with particular examples. [ABSTRACT FROM AUTHOR]

In this paper we investigate the existence of a solution of Integral equations of fractional order via Hadamard's fractional integral by reducing the research to the search of the existence of fixed points of appropriate operators. The nonlinear alternative fixed point theorem for the sum of a completely continuous operator and a contraction one in Frechet spaces due to Avraniescu and a fractional version of Pachpatte's inequality are the main tools in this study. We demonstrate the existence of a solution with a suitable illustrative example. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new mapping namely L-contraction for fixed function and we also prove a novel fixed function theorem using L-contraction mapping. Moreover, we apply our result to solve integral equation. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following integral system u i (x) = ∫ ℝ n u i + 1 p i + 1 (y) | x − y | n − α d y , i = 1 , 2 , ... , m − 1 , u m (x) = ∫ ℝ n u 1 p 1 (y) | x − y | n − α d y , m ≥ 1 , n ≥ 1 , where u i > 0 , 0 < α < n , and p i > 1 (i = 1 , 2 , ... , m). When m ∈ { 1 , 2 } , such an integral system is associated with the best constants of the Hardy–Littlewood–Sobolev inequality. Chen, Li and their cooperators obtained optimal integrability intervals of the finite energy solutions by an argument of contraction and shrinking operators. This result is helpful to well understand the classification of the extremal functions of the Hardy–Littlewood–Sobolev inequality. The critical condition plays a key role in their work. In this paper, we study optimal integrability intervals when the positive solutions have some initial integrability. Now, the critical condition is not necessary, and we apply a weaker condition, the Serrin-type condition, to establish some important relations of exponents which come into play to lift the regularity. In addition, we also generalize this result to the case of m ∈ ℤ + . [ABSTRACT FROM AUTHOR]

The unique solvability of the inverse problem of recovering the source function in a nonuniformly parabolic equation with many independent variables in a bounded domain is proved. As an additional condition, the integral observation condition is given. The unknown source function can be found by iteration from the operator equation with contraction operator. An example of an the inverse problem for which the results proved in this paper are applicable is presented. The results obtained for the inverse problem are based on the preliminary study of the unique solvability of the corresponding direct problem, which is also of interest in itself. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a novel fixed point iterative scheme based on Green's function, called the Picard–Ishikawa–Green's iterative scheme and use it in approximating the solution of boundary value problems (BVPs). It is proved that Picard–Ishikawa–Green's scheme converges strongly for an integral operator which represents the solution of BVP and the scheme is stable. Moreover, we prove that the integral operator is a contraction. Furthermore, it is shown that the novel scheme converges faster than all of Ishikawa–Green's, Khan–Green's, and Mann–Green's schemes. Finally, numerical examples are given to substantiate the validity of our results for third‐order BVPs. Our results extend and generalize several other results in literature. [ABSTRACT FROM AUTHOR]

Telecommunications companies are constantly chasing continuous technological advances with their management methods of Operations & Maintenance (O&M) that still struggle to turn their eye toward innovation and simplification of processes. In a future that aims at fully fiber-optic networks, the objective of the research is to propose guidelines and provide data to support the Wholesale Operator for the definition of the acceptable attenuation threshold on the Passive Optical Network Fiber-To-The-Home (PON FTTH) infrastructure to determine the Service Level Agreements (SLA) to be contracted with Retail Operators. Following exceeding the threshold, the Retail Operator has the right to open a Trouble Ticket to request the resolution of the anomaly. Consequently, the definition of the attenuation threshold strongly affects the number of Trouble Tickets to be managed and the related costs the Wholesale Operator bears. This paper analyzes a specific case of the PON FTTH network of the Italian wholesale operator Open Fiber. The studied infrastructure, with a length of 11 km, allows a maximum attenuation of 37 dB without degrading the service. An economic model has been proposed to assess the cost impact of moving the attenuation threshold in the SLA. [ABSTRACT FROM AUTHOR]

In this paper, a generalization of the fixed point theorem of the Reichtype mapping on a complete metric space (X,d) Xd is given. Continuous, injective and sub-sequentially convergent mapping T was used, as well as it is taken that function f is from the class Θ continuous monotonically nondecreasing functions: [0,+∞) f: [0,+∞) such that f-1(0)={0}, where it is additionally assumed that it is subadditive, i.e. f (x + y) ≤ f (x) + f (y), for each x, y ∊ [0,+∞). [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the optimal integrability, regularity and symmetry of integrable solutions for the following Wolff type integral systems: (1.1) { u (x) = R 1 (x) W β , γ (v q) (x) , u (x) > 0 , x ∈ R N , v (x) = R 2 (x) W β , γ (u p) (x) , v (x) > 0 , x ∈ R N , where γ > 2 , β > 0 , β γ < N , p , q > γ − 1 with γ − 1 p + γ − 1 + γ − 1 q + γ − 1 = N − β γ N , R 1 , R 2 are double bounded in R N and W β , γ (h) (x) : = ∫ 0 ∞ [ ∫ B t (x) h (y) d y t N − β γ ] 1 γ − 1 d t t. Firstly, we prove the optimal integrability and boundedness of solutions (u , v) ∈ L p + γ − 1 (R N) × L q + γ − 1 (R N) for system (1.1) , by constructing a nonlinear, contracting operator and applying the regularity lifting lemma of C. Ma, W. Chen and C. Li (2011) [21]. Moreover, we exploit the general regularity lifting theorem to derive the Lipschitz continuity of u and v when R 1 ≡ R 2 ≡ 1 in R N. These extend the above important results of C. Ma, W. Chen and C. Li to γ > 2. We also prove that u and v vanish at infinity. As the corollaries of the above results, we obtain the optimal integrability, boundedness and the property of vanishing at infinity of integrable solutions for the corresponding γ -Laplace and k -Hessian systems. Secondly, we use the method of moving planes in integral forms to prove the symmetry and monotonicity of solutions (u , v) ∈ L p + γ − 1 (R N) × L q + γ − 1 (R N) for system (1.1) when R 1 ≡ R 2 ≡ 1 in R N , which extends the useful result of W. Chen and C. Li (2011) [5] to γ > 2. In comparison with the above two papers, Minkowski's inequality is crucial in our proofs. We believe that our arguments can be used to prove similar results for other Wolff type integral systems when γ > 2. [ABSTRACT FROM AUTHOR]

In this paper, we extend and improve a common fixed point theorem of G. Jungck. We utilize the notions of weakly commuting and compatible mappings in probabilistic Banach spaces to prove some common fixed point theorems for improved type Jungck contractions. In addition, we present some examples which support our theorems. [ABSTRACT FROM AUTHOR]

The interval Newton method, which combines the Newton method with the interval arithmetic and branch-and-prune algorithm, has been applied to the problem of finding all direct current operating points of nonlinear circuits. The modified linear programming narrowing technique has also been proposed to improve the computational efficiency of the interval Newton method when applying to a circuit whose nonlinear elements are only diodes and bipolar junction transistors. In this paper, a contraction operator based on interval linearization has been applied to further improve the performances. In addition, the application of the proposed method to circuits containing MOSFET is presented. The numerical experimental results on some test nonlinear circuits indicate that the proposed method can further reduce the number of search boxes and improve the computation time. [ABSTRACT FROM AUTHOR]

In open quantum systems, the quantum Zeno effect consists in frequent applications of a given quantum operation, e.g., a measurement, used to restrict the time evolution (due, for example, to decoherence) to states that are invariant under the quantum operation. In an abstract setting, the Zeno sequence is an alternating concatenation of a contraction operator (quantum operation) and a C 0 -contraction semigroup (time evolution) on a Banach space. In this paper, we prove the optimal convergence rate O (1 n) of the Zeno sequence by proving explicit error bounds. For that, we derive a new Chernoff-type n -Lemma, which we believe to be of independent interest. Moreover, we generalize the convergence result for the Zeno effect in two directions: We weaken the assumptions on the generator, inducing the Zeno dynamics generated by an unbounded generator, and we improve the convergence to the uniform topology. Finally, we provide a large class of examples arising from our assumptions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an orthogonal L ⋆ -contraction map concept and prove the fixed-point theorem in an orthogonal complete Branciari metric space (OCBMS). We also provide illustrative examples to support our theorems. We demonstrated the existence of a uniqueness solution to the fourth-order differential equation using a more orthogonal L ⋆ contraction operator in OCBMS as an application of the main results. [ABSTRACT FROM AUTHOR]

In this paper we investigate AGM belief contraction operators by using the tools of algebraic logic. We generalize the notion of contraction to arbitrary finitary propositional logics, and we show how to switch from a syntactic-based approach to a semantic one. This allows to build a solid bridge between the validity of AGM postulates in a propositional logic and specific algebraic properties of its intended algebraic counterpart. Such a connection deserves particular attention when we deal with maxichoice contractions, as studied in the final part of the paper. [ABSTRACT FROM AUTHOR]

Purpose – The main intention of the analysis was an attempt at defining the income gap and identifying the return from condo units in Poland. Design/methodology/approach – The article presents the subject of condo properties. Two main types of condo properties have been distinguished: condo apartments and condo hotel rooms. The fundamental differences are presented. The research uses case study approach. Findings – The research suggest that rent calculation mechanism is the decisive factor for the occurrence of an income gap in investments in condo property. This is affected by three aspects: The requirement to buy furniture/equipment and no rent charged for the invested capital. The requirement to buy a parking place, which is not included by the operator in the determination of yearly rent on the invested capital. The investor exercising their right to become a resident owner of the condo unit on the applicable terms. Research limitations: The analysis covers condo units in towns with significant tourism potential, and in Warsaw. The sample includes selected operator contracts examined for six condo properties in Poland. Research implications: The paper answers two vital questions that may be interesting to property investors, both institutional and individual: Which factors explain the annual retur from cond properties in Poland? What are the annual returns under specific contract terms in the circumstances of the Polish condo property segment? [ABSTRACT FROM AUTHOR]

Discussions of the issue of iterated belief revision are commonly accompanied by the presentation of three "concrete" operators: natural, restrained and lexicographic. This raises a natural question: What is so distinctive about these three particular methods? Indeed, the common axiomatic ground for work on iterated revision, the AGM and Darwiche-Pearl postulates, leaves open a whole range of alternative proposals. In this paper, we show that it is satisfaction of an additional principle of "Independence of Irrelevant Alternatives", inspired by the literature on Social Choice, that unites and sets apart our three "elementary" revision operators. A parallel treatment of iterated belief contraction is also given, yielding a family of elementary contraction operators that includes, besides the well-known "conservative" and "moderate" operators, a new contraction operator that is related to restrained revision. [ABSTRACT FROM AUTHOR]

This paper introduces a new iteration process for the approximation of fixed points of contraction mapping. We prove the comparison theorem to indicate that our iteration process converges faster than the other iteration processes and also prove data dependence, stability. and convergence results. Moreover, we give numerical examples to check the performance of our algorithm over the Piri, SM, CR, and K* iterations process for different types of functions. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the regularity of nonnegative solutions of Lane-Emden system {−Δu = vp −Δv = uq in Rn. By means of Kelvin transform, we turn this problem into estimating the local integrability of (u,v). Assume that (u,v) possesses some initial local integrability beforehand.(u,v) ∈ Lr0loc(Rn) × Ls0loc(Rn) for any suitable r0 and s0 under specified conditions. Then through a regularity lifting method by contracting operators, we prove that(u,v) ∈ Lrloc(Rn) × Lsloc(Rn) for r and s sufficiently large under twice regularity lifting if needed. Furthermore, we lift the regularity of solutions to L∞(Rn) × L∞(Rn). We believe that these new methods employed in this paper can be widely applied to study a variety of other problems with different spaces and linear or nonlinear problems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a common fixed point theory for a pair of multivalued operators in complete metric spaces. Existence and stability results for the common fixed point problem will be discussed. The approach is based on some fixed point results for multivalued graphical contractions and for multivalued operators of Feng-Liu type. Our results extend to the multivalued case some recent results in the literature. [ABSTRACT FROM AUTHOR]

A newly developed method in the mixed space–wavenumber domain is proposed for 3D DC resistivity modeling in anisotropic structures. Unlike the traditional method for numerical simulation of DC resistivity, this method performs a 2D Fourier transform directly on the partial differential equation of DC abnormal potential along the horizontal direction, so that the horizontal direction is converted into the wavenumber domain while the vertical direction remains in the spatial domain. In this way, the 3D partial differential equation in the spatial domain is transformed into a 1D ordinary differential equation in different wave numbers. Therefore, a large-scale 3D numerical simulation problem is decomposed into many 1D numerical simulation problems. The 1D finite element method is used to solve the equations, and a contraction operator is used for iterative calculation to obtain a highly accurate solution. The accuracy of the algorithm is verified by comparison with the goal-oriented adaptive finite element method. Compared with the COMSOL Multiphysics commercial finite element simulation software, the algorithm in this paper has high computational efficiency. To identify the electrical anisotropy, we propose a new measurement and then use it to study the response characteristics of a half-space model with two prisms. The test results show that this measurement can accurately reflect the underground electrical anisotropy. Finally, the code is applied to the numerical simulation of a vertical-borehole survey and shows that the significant information on the hydraulically induced fracturing system can be reflected well by the electric anisotropy. [ABSTRACT FROM AUTHOR]

The bounded linear operators T on a Hilbert space which have 2-isometric liftings S on are investigated in this paper. We refer to the structure of those operators in the case of general liftings, as well as for a more special type of such liftings S, namely for which is invariant to the orthogonal projection onto the kernel of S∗S − I. In this special case we describe the canonical triangulation of T induced by S, with entries expressed by operators close to contractions or isometries. Also, in this case we refer to some asymptotic properties of T and of its adjoint T∗ obtained by means of asymptotic limits associated to T∗ and T, relative to a lifting S for T. The canonical triangulations of T in this setting are described in detail, and as an application, the Wold type decomposition of a concave operator is obtained. [ABSTRACT FROM AUTHOR]

Nowadays, the issue of traffic flow assignment has become an interdisciplinary topic that concerns multiple research areas and branches of science. This work is focussed on the mathematical and computational aspects of the equilibrium traffic assignment problem in the case when route flows are considered to be decision variables. Firstly, we obtain a fixed-point mapping with the explicit contraction operator, which is proven to equilibrate the journey times on feasible routes between a single origin-destination pair of nodes with the quadratic rate. Remarkable that from mathematical perspectives, the developed operator generalizes most path-equilibration operators already exploited by researchers. Secondly, we use the obtained fixed-point procedure to run the sequential path-equilibration algorithm for traffic flow assignment on well-known test urban road networks with arc-additive travel time functions. Our computational results appear to demonstrate higher accuracy of user-equilibrium traffic assignment solutions than the best ones known to us. In other words, developed within this paper sequential path-equilibration algorithm leads to solutions with less goal function values compared to the best solutions for today, according to our knowledge. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a new three-step fixed point iteration called SNIA-iteration (Naveen et al. iteration) which converges faster than the original iteration. It is possible by introducing a contraction operator on the existing iteration algorithm where the coefficients of the new iterative process are chosen in (1 2 , 1) instead of [0, 1]. The strong convergence result for the SNIA-iteration method is also proved by showing the convergence of this iteration method towards its fixed point. Further, the coefficients involved in SNIA-iteration have been interchanged, and then the result is compared with the original iteration. It is found that the new result is better than the previous one in the sense of its rate of convergence for contraction mappings. [ABSTRACT FROM AUTHOR]

This paper proves a new q-Hermite-Hadamard inequality for convex functions using quantum integrals. We also prove some new midpoint-type inequalities for q-differentiable convex functions. Moreover, we present some examples to illustrate our established results, supplemented with graphs. [ABSTRACT FROM AUTHOR]

In this paper, some sufficient conditions for the oscillation of all solutions of second order dynamic equations with a negative sub-linear neutral term are established. The obtained results provide a unified platform that adequately covers both discrete and continuous equations. Furthermore, it covers a wide range of equations by utilizing different time scales. Illustrative examples are provided. [ABSTRACT FROM AUTHOR]

In this research paper, we investigate some new identifies for Sarıkaya fractional integrals which introduced by Sarıkaya and Ertugral in [20]. The fractional integral operators also have been applied to Hermite-Hadamard type integral inequalities to provide their generalized properties. Furthermore, as special cases of our main results, we present several known inequalities such as Simpson, Bullen, trapezoid for convex functions. [ABSTRACT FROM AUTHOR]

In this paper, our focus is to acquaint with the Suzuki-type mappings to establish some fixed point results using the new w-interpolative approach. We present some results for interpolative contraction operators via the w-admissible maps which satisfy the Kannan, Ćirić–Reich–Rus, and Hardy–Rogers contractions in quasi-partial b-metric space. Further, the outcomes so obtained are affirmed with relevant examples. [ABSTRACT FROM AUTHOR]

Automatic text processing is now a mature discipline in computer science, and so attempts at advancements using quantum computation have emerged as the new frontier, often under the term of quantum natural language processing. The main challenges consist in finding the most adequate ways of encoding words and their interactions on a quantum computer, considering hardware constraints, as well as building algorithms that take advantage of quantum architectures, so as to show improvement on the performance of natural language tasks. In this paper, we introduce a new framework that starts from a grammar that can be interpreted by means of tensor contraction, to build word representations as quantum states that serve as input to a quantum algorithm. We start by introducing an operator measurement to contract the representations of words, resulting in the representation of larger fragments of text. We then go on to develop pipelines for the tasks of sentence meaning disambiguation and question answering that take advantage of quantum features. For the first task, we show that our contraction scheme deals with syntactically ambiguous phrases storing the various different meanings in quantum superposition, a solution not available on a classical setting. For the second task, we obtain a question representation that contains all possible answers in equal quantum superposition, and we implement Grover's quantum search algorithm to find the correct answer, agnostic to the specific question, an implementation with the potential of delivering a result with quadratic speedup. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define generalized (α-η)-Θ contraction and to extend the results of Jleli and Samet [M. Jleli, B. Samet, J. Inequal. Appl., 2014 (2014), 8 pages] by applying a simple condition on the function Θ. We also deduce certain fixed and periodic point results for orbitally continuous generalized Θ-contractions and certain fixed point results for integral inequalities are derived. Finally, we provide an example to show the significance of the investigation of this paper. [ABSTRACT FROM AUTHOR]

This paper consists of Hankel results of ω-order reversing partial contraction as a semigroup of linear operator. Spectral mapping theorem was investigated and it was established that ω-order reversing partial contraction mapping is an eventually uniform continuous semigroup in which we showed that the proof of the extension of T(t) to a C0-semigroup ˆ T(t) on a space ˆX containing X isometrically bounded. The space is constructed in such a way that the spectrum of the generator A of T(t) coincides with the spectrum of the generator Aˆ of Tˆ(t) and the approximate point spectrum of A coincides with the point spectrum of Aˆ. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some common fixed point theorems for interpolative contraction operators using Perov operator which satisfy Suzuki type mappings. Further, some results are given. These results generalize several new results present in the literature. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of impulsive stochastic partial integrodifferential equations (ISPIEs) with delays and Poisson jumps. First, using the resolvent operator technique and contraction mapping principle, we can directly prove the existence and uniqueness of the mild solution for the system mentioned above. Then we develop a new impulsive integral inequality to obtain the global, both pth moment exponential stability and almost surely exponential stability of the mild solution is established with sufficient conditions. Also, a numerical example is provided to validate the theoretical result. [ABSTRACT FROM AUTHOR]

Based on the notions of both contractibility and local contractibility, many works were done in fixed point theory. The present paper concerns a relation between digital contractibility and the existence of fixed points of digitally continuous maps. In this paper, establishing a new digital homotopy named by a K-homotopy in the category of Khalimsky topological spaces, we prove that in digital topology, whereas contractibility implies local contractibility, the converse does not hold. Furthermore, we address the following problem, which remains open. Let X be a Khalimsky ( K- for short) topological space with K-contractibility. Then we may pose the following question: does the space X have the fixed point property ( FPP)? In this paper, we prove that not every K-topological space with K-contractibility has the FPP. [ABSTRACT FROM AUTHOR]

The first purpose of this paper is to prove an existence and uniqueness result for the multivariate fixed point of a contraction type mapping in complete metric spaces. The proof is based on the new idea of introducing a convenient metric space and an appropriate mapping. This method leads to the changing of the non-self-mapping setting to the self-mapping one. Then the main result of the paper will be applied to an initial-value problem related to a class of differential equations of first order. The second aim of this paper is to prove strong and weak convergence theorems for the multivariate fixed point of a N-variables nonexpansive mapping. The results of this paper improve several important works published recently in the literature. [ABSTRACT FROM AUTHOR]

Two interesting extensions of Banach contraction principle to mappings that do not to be continuous, are Kannan and Chatterjea's theorems. Before this, in the cyclical form, extensions of these two theorems and Banach contraction principle were produced. But so far, these theorems have not been studied in the noncyclical form. In this paper, we answer the question whether there are versions of these theorems for non-cyclic mappings, also we give generalizations of existing results. For this purpose, in the setting of metric spaces we introduce the notions of cyclic and noncyclic contraction of Fisher type. We establish the existence of fixed points for these mappings and iterative algorithms are furnished to determine such fixed points. As a result of our results we give new theorems for cyclic orbital contractions. [ABSTRACT FROM AUTHOR]

In this paper, the notion of fuzzy (θ,L)-weak contraction in G--metric space is introduced, and sufficient conditions for the existence of fuzzy fixed points for such mappings are investigated. Relevant illustrative examples are constructed to support the assumptions of our established theorems. It is observed that the principal ideas obtained herein extend and subsume some well-known results in the corresponding literature. A few of these special cases of our results are noted and discussed as corollaries. [ABSTRACT FROM AUTHOR]

As an important emission reduction source for the transportation industry, biofuel has received strong support from the Chinese government. However, the development of the biofuel industry is still struggling. The high degree of uncertainty makes the development of the industry face huge challenges. Kitchen waste, as a biodiesel raw material with a large yield, has good development prospects. Reuse of kitchen waste can solve public health and safety problems. This paper proposes a two-stage stochastic programming model under supply disturbance to optimize the supply chain from the perspective of contract. Then current three main flow directions of kitchen waste are analysed and the reasonable price for biodiesel operators to purchase is determined. By signing contracts with the biodiesel operators, restaurant is guaranteed and encouraged to provide a certain percentage of kitchen waste to meet the demand for biodiesel production. Using actual case in the Yangtze River Delta region, the performance of the stochastic programming model under disturbance was compared. Through the sensitivity analysis of different parameters, this paper determines the influence of its supply chain network design and expected total system cost. Through the optimization of the waste cooking oil (WCO) for biodiesel supply chain, this paper can effectively improve the efficiency of the supply chain, reduce system costs, increase the profits of biofuel operators, and promote the sustainable development of the biofuel industry. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse boundary value problem for a partial differential equation of third order with nonlocal boundary conditions, including an integral condition. Using analytical and operator-theoretic methods, as well as the Fourier method, the existence and uniqueness of the classical solution of this problem is proved. [ABSTRACT FROM AUTHOR]

In this paper, we prove coincidence and common fixed points results under nonlinear contractions on a metric space equipped with an arbitrary binary relation. Our results extend, generalize, modify and unify several known results especially those are contained in Berzig [J. Fixed Point Theory Appl. 12, 221-238 (2012))] and Alam and Imdad [To appear in Filomat (arXiv:1603.09159 (2016))]. Interestingly, a corollary to one of our main results under symmetric closure of a binary relation remains a sharpened version of a theorem due to Berzig. Finally, we use examples to highlight the accomplished improvements in the results of this paper. [ABSTRACT FROM AUTHOR]

An efficient solution of the three-dimensional Helmholtz equation is known to be crucial in many applications, especially geophysics. In this paper, we present and test two preconditioning approaches for the discrete problem resulting from the second order finite-difference discretization of this equation. The first approach combines shifted-Laplacian preconditioner with inversion of a separable matrix, corresponding to the horizontally-layered velocity model, using fast Fourier based transforms. The second approach is novel and involves a special transformation resulting in a preconditioner with a contraction operator (CO preconditioner). The two approaches have near the same arithmetical complexity; however, the second approach, developed in this paper, provides a faster convergence of an iterative solver as illustrated by numerical experiments and analysis of the spectral properties of the preconditioned matrices. Our numerical experiments involve parallel modeling of highly heterogeneous lossy and lossless media at different frequencies. We show that the CO-based solver can tackle problems with hundreds of millions of unknowns on a conventional cluster node. The CO preconditioned solver demonstrates a very moderate increase of iteration count with the frequency. We have conducted a comparison of the performance of the developed method versus open-source parallel sweeping preconditioner. The results indicate that, the CO solver is several times faster with respect to the wall-clock time and consumes substantially less memory than the code based on the sweeping preconditioner at least in the example we tested. [ABSTRACT FROM AUTHOR]

In the present paper, we define an analogue of Hardy-Rogers type contraction in A-metric spaces and prove a fixed point theorem for such mappings under appropriate conditions in such spaces. Also, we give some results for Kannan, Chatterjea, Reich, Ciric type contraction mappings in A-metric spaces. Our results generalize many known results in the fixed point theory. [ABSTRACT FROM AUTHOR]

Let (X, p) be the universal covering space of a compact metrizable space X, which is compact and locally path connected. In this paper, we show that there eXist metrics d and d 'for X andX, respectively, such that any contractive map f: X → X induces a contractive map on X. As an application, it is obtained that every iterated function system(IFS) on the space X with attractor K, induces an IFS on X with attractor K, such that p(K) = K. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear neutral fractional difference equations. By applying Krasnoselskii's fixed point theorem, sufficient conditions for the existence of solutions are established. Also, the uniqueness of a solution is given. As an application of the main theorems, we provide the existence and uniqueness of the discrete fractional Lotka-Volterra model of neutral type. Our main results extend and generalize the results that are obtained in [6]. [ABSTRACT FROM AUTHOR]

In this paper, by developing Lieb’s methods, we establish uniqueness of the positive radial solutions to the following solitary Choquard type equation This, together with Ma and Zhao’s classification results, enables us to show the uniqueness of the positive solutions when. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of pseudo almost periodic solutions for shunting inhibitory cellular neural networks with leakage delays and complex deviating arguments. Applying the contraction mapping fixed point theorem and inequality techniques, some sufficient conditions are presented to ensure the existence of pseudo almost periodic solutions of this model, which are new and supplement some previously known ones. [ABSTRACT FROM AUTHOR]

The question if a polynomially bounded operator is similar to a con- traction was posed by Halmos and was answered in the negative by Pisier. His counterexample is an operator of infinite multiplicity, while all its restrictions on invariant subspaces of finite multiplicity are similar to contractions. In the paper, cyclic polynomially bounded operators which are not similar to contractions and are quasisimilar to C0-contractions or to isometries are constructed. The construction is based on a perturbation of the sequence of finite dimensional operators which is uniformly polynomially bounded, but is not uniformly completely polynomially bounded, constructed by Pisier. [ABSTRACT FROM AUTHOR]

This paper deals with a size-structured population model consisting of a quasi-linear first-order partial differential equation with nonlinear boundary condition. The existence and uniqueness of solutions are firstly obtained by transforming the system into an equivalent integral equation such that the corresponding integral operator forms a contraction. Furthermore, the existence of global attractor is established by proving the asymptotic smoothness and eventual compactness of the nonlinear semigroup associated with the solutions. Finally, we discuss the uniform persistence and existence of compact attractor contained inside the uniformly persistent set. [ABSTRACT FROM AUTHOR]