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An ingenious approach to generalize Banach contraction principle was adopted by Suzuki in his seminal papers (Proc Am Math Soc 136:1861–1869, 2008, Nonlinear Anal Theory Methods Appl 71:5313–5317, 2009). In this paper we prove certain common fixed point results for generalized Suzuki contractions in the set-up of b-metric spaces, where the b-metric function is not necessarily continuous. Finally, some examples are presented to verify the effectiveness and applicability of our main results.

In this paper, we present a study which is the completion of the problem left open in the appendix of the paper Nuno et al. (Eur J Appl Math 21:459–478, 2010). The goal of the paper is to offer a deeper analysis of the type of the ODE system featured in the paper above concerning the evolution of the total wealth and cheater population and, mainly, the purpose is to generalize this one to a Cellular Automata model introducing spatial and local effects. The work presented shows the original interaction between an Ordinary Differential Equations model and a Cellular Automata model.

This paper investigates properties of convergence of distances of $$p$$ -cyclic $$\alpha$$ - $$\psi$$ -type contractions on the union of the $$p$$ subsets of a space $$X$$ defining probabilistic metric spaces and Menger spaces. The paper also investigates the characterization of both Cauchy and $$G$$ -Cauchy sequences which are convergent, in particular, to best proximity points. On the other hand, the existence and uniqueness of fixed points and best proximity points of $$p$$ -cyclic $$\alpha$$ - $$\psi$$ -type contractions are also investigated. The fixed points of the $$p$$ -composite self-mappings, which are obtained from the $$p$$ -cyclic self-mapping restricted to each of the $$p$$ subsets in the cyclic disposal, are also investigated while a generalization and some illustrative examples are also given.

In this paper, we establish strong and $$\Delta$$ -convergence theorems for a relatively new iteration process generated by generalized nonexpansive mappings in uniformly convex hyperbolic spaces. The theorems presented in this paper generalizes corresponding theorems for uniformly convex normed spaces of Kadioglu and Yildirim (Approximating fixed points of nonexpansive mappings by faster iteration process, arXiv:1402.6530v1 [math.FA], 2014) and CAT(0)-spaces of Abbas et al. (J Inequal Appl 2014:212, 2014) and many others in this direction.

In this paper, we present a novel approach for approximating Hammerstein–Volterra delay integral equations (HVDIEs) by applying the universal approximation method through an artificial intelligence utility in a simple way. In this paper, neural network model (NNM) is applied as universal approximators for any nonlinear continuous functions. Here, neural network is considered as a part of large field called neural computing or soft computing. With this capability, the solution of Hammerstein–Volterra delay integral equation can be approximated by the appropriate NNM within an arbitrary accuracy.

Purpose: This paper investigates an analytical approximate solution of a fourth-order differential equation with nonlinear boundary conditions modeling beams on elastic foundations using iterative reproducing kernel method. Methods: The solution obtained using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel method can not be used directly to solve the problems since there is no method of obtaining a reproducing kernel satisfying nonlinear boundary conditions. The aim of this paper is to fill this gap. Results: Several illustrative examples are given to demonstrate the effectiveness of the present method. Conclusions: Results obtained using the scheme presented here show that the numerical scheme is very effective and convenient for the beam equation with third-order nonlinear boundary conditions.

In this paper, we introduce two iterative schemes for finding a common solution of a generalized vector equilibrium problem and relatively nonexpansive mappings in a real Banach space. We study the strong and weak convergence of the sequences generated by the proposed iterative schemes. The results presented in this paper are the supplement, extension, and generalization of the previously known results in this area.

Purpose: This paper studies the � -four equation that appears in relativistic quantum mechanics. The primary purpose of this paper is to address the numerical simulations of this model. Methods: The singular soliton solution is also obtained by the ansatz method. Results: The constraint conditions are indicated for the existence of the soliton. Numerical solutions are obtained by using the spectral method where rational Chebyshev functions are used as basis functions. Conclusions: Results are validated by finding error estimates.

Purpose: In this paper, we consider the maximum traveling salesman problem, a variation of the usual traveling salesman problem, in which the objective is to maximize the cost of a tour of the salesman. The main purpose of this paper is to develop a hybrid genetic algorithm (GA) for obtaining a heuristically optimal solution to the problem. Methods: First, a simple GA and then a hybrid GA have been proposed to solve the problem. As crossover operator plays a vital role in GAs, we modify the sequential constructive crossover operator for our simple GA to solve the problem. To improve the quality of the solution obtained by the crossover operator, restricted 2-opt search is applied. Then a hybrid GA is developed by incorporating a new local search algorithm to the simple GA in order to obtain a heuristic solution to the problem. Results: We compare the efficiency of our hybrid GA against an existing heuristic algorithm for symmetric traveling salesman problem library (TSPLIB) instances. Finally, we present solutions to the problem for asymmetric TSPLIB instances. Since, to the best of our knowledge, no literature presents solution for asymmetric instances, hence, we could not carry out any comparative study to show the efficiency of our hybrid GA for the asymmetric instances. Conclusions: The comparative study shows the effectiveness of our hybrid GA.

Purpose: In this paper, we extend the concept of covering dimension of general topological spaces to L-topological spaces using α-Q-covers and quasi-coincidence relation. Methods: Dimension theory is a branch of topology devoted to the definition and study of the notion of dimension in certain classes of topological spaces. The dimension of a general topological space X can be defined in three different ways: the small inductive dimension indX, the large inductive dimension IndX, and the covering dimension dimX. The covering dimension dim behaves somewhat better than the other two dimensions, i.e., that for the dimension dim, a large number of theorems of the classical theory can be extended to general topological spaces. Also, there is a substantial theory of covering dimension for normal spaces. Results: A characterization of covering dimension in the weakly induced L-topological spaces is obtained. Moreover, a characterization of covering dimension for fuzzy normal spaces is also obtained. Conclusions: Finally, This paper provides some brief sketches regarding the topics covering dimension in L-topological spaces and covering dimension for fuzzy normal spaces. The neighborhood structure used for the investigations is the quasi-coincident neighborhood structure.

It is more appropriate that many industrial products be evaluated and qualified by an imprecise (fuzzy) quality. By this idea the products could be evaluated using two membership functions for specification limits rather than two real numbers used in classical quality control. This idea leads the researchers to be able to deal with the vague process capability indices modeled as triangular fuzzy numbers. In this paper, we discuss on such fuzzy qualities and review some fuzzy process capability indices. Then we will bring them up to analyze several educational systems, such as comparing capability indices of two or more teachers, schools, and so on. The idea of this paper could be applied in other similar evaluation schemes as well.

Purpose: Our aim in this paper is to study generalized composition operators on α- Bloch and QK,ω(p, q) spaces. Methods: By the help of generalized composition operators, we act between several classes of weighted function spaces. Some important results obtained by using modified Nevanlinna counting function. Results: The boundedness and compactness of the generalized composition operator C g φ acting between two different M ¨ obius invariant spaces QK1 (p, q) and QK2 (p, q) are studied. Conclusions: Our results in this paper extend, generalize and improve a lot of previous results.

This paper proposes the use of different analytical methods in obtaining approximate solutions for nonlinear differential equations with oscillations. Three methods are considered in this paper: Lindstedt-Poincare method, the Krylov-Bogoliubov first approximate method, and the differential transform method. Figures that are given in this paper give a strong evidence that the proposed methods are effective in handling nonlinear differential equations with oscillations. This study reveals that the differential transform method provides a remarkable precision compared with other perturbation methods.

methods Our purpose in this paper is researching about characteristics of convergent in probability and almost surely convergent in Serstnev space. We prove that if two sequences of random variables are convergent in probability (almost surely), then, sum, product and scalar product of them are also convergent in probability (almost surely). Meanwhile, we will prove that each continuous function of every sequence convergent in probability sequence is convergent in probability too. Finally, we represent that for independent random variables, every almost surely convergent sequence is convergent in probability. In this paper, we conclude results in Serstnev space are similar to probability space.

Purpose: Our purpose in this paper is studying the strong convergence of implicit and explicit iterative schemes for approximating solutions of some variational inequalities on the sets of common fixed points for a semigroup of asymptotically nonexpansive mappings. Methods: We prove strong convergence theorems of such iterative scheme in a uniformly convex and uniformly smooth Banach spaces under certain conditions. Results: We obtain two convergence theorems of such iterative scheme for approximating a common fixed point, which is a unique solutions of the variational inequality. Conclusions: Our results extend and improve those of Zegeye et al., Sunthrayuth and Kumam, and Li et al. The methods of the proof given in this paper are also different.

This paper considers ranking of generalised fuzzy numbers with different left height and right height using integral values. With the advances in new type of fuzzy number (generalised fuzzy number with different left height and right height) methods should be developed to compare them. Keeping this in view a new modified method has been proposed.

In this paper, an analytical technique, namely the new iterative method (NIM), is applied to obtain an approximate analytical solution of the fractional Fornberg–Whitham equation. The obtained approximate solutions are compared with the exact or existing numerical results in the literature to verify the applicability, efficiency, and accuracy of the method.

In present paper, we prove unique fixed point theorems for contractive maps in \(N\)-cone metric spaces. Our results extend and generalize some well-known results of (Banach, Fund Math 3:133–181 1992; Chatterjee, Rend Acad Bulgare Sci 25:727–730 1972; Kannan, Bull Calcutta Math Soc 60:71–76 1968; Rezapour and Hamlbarani, J Math Anal Appl 345:719–724 2008) in the setting of \(N\)-cone metric spaces.

This paper proposes a pseudo-spectral scheme for obtaining approximate optimal control and state. This computational scheme represents the solution of the optimal control problem (OCP) by an mth degree Lagrange interpolating polynomial, using Legendre nodes. Then, OCP of non-linear Volterra integral equation is transformed into an optimization problem. Directed tabu search (DTS) method is utilized to derive the solutions of the optimal control and state as well as the optimal value of the objective function. In the DTS method, two neighborhood-local search strategies based on Nelder–Mead method and adaptive pattern search are applied. In addition, a tabu list with anti-cycling rules, the so-called tabu regions and semi-tabu regions are used. In addition, diversification and intensification search schemes are employed. DTS is able to converge to the global optimum solutions of a set of numerical examples. Therefore, some good balance between the diversification and intensification ensures a faster and efficient convergence to get quality solutions. Numerical examples are provided which confirm the reliability and efficiency of the proposed method. Moreover, a comparison is made with optimal solutions obtained by the other numerical methods in the literature.

In this paper, we proved common coupled fixed point results for mixed g-monotone and w-compatible mappings in ordered cone b-metric spaces. Our results extend and generalize several well-known comparable results in literature.

In this paper, the concept of contraction via the measure of non-compactness on a Banach space is investigated by generalizing some results which have been previously discussed in literatures. Furthermore, to validity of the theorems and homotopy perturbation method (HPM), as a technical solution, they are applied on some nonlinear singular integral equations.

In this paper, we consider multivalued nonself weak contractions on convex metric-like spaces and we establish the existence of fixed point of such mappings. We provide some examples making effective our obtained result.

The asymptotic equilibrium problems of ordinary differential equations in a Banach space have been considered by several authors. In this paper, we investigate the asymptotic equilibrium of the integro-differential equations with infinite delay in a Hilbert space.

In this paper, we propose some analytical solutions of stochastic differential equations related to Martingale processes. In the first resolution, the answers of some stochastic differential equations are connected to other stochastic equations just with diffusion part (or drift free). The second suitable method is to convert stochastic differential equations into ordinary ones that it is tried to omit diffusion part of stochastic equation by applying Martingale processes. Finally, solution focuses on change of variable method that can be utilized about stochastic differential equations which are as function of Martingale processes like Wiener process, exponential Martingale process and differentiable processes.

Evaluating profit efficiency measure needs input costs and output prices as well as input–output quantities. It is important to rank production units based on relative or absolute values of their production when costs and prices are available. Ranking production units in data envelopment analysis constitutes ranking individual units based on their profit efficiency ratio measures. Two novel models are presented in this paper for evaluating units based on a profit-dominance criterion. Models consider not only self-appraisal DEA optimal weights, but also all feasible input and output weights. A novel ratio-based and complete ranking method is introduced that is based on computing upper and lower boundaries for profit performance of observed units. An illustrative application of the models is then presented and results are discussed.

This paper analyzes the laminar, incompressible natural convective transport inside vertical channel in an electrically conducting fluid saturated porous medium. In addition, this model incorporates the combined effects of Hall current, Soret and Joule heating. The non-linear governing equations with their corresponding boundary conditions are initially cast into a dimensionless form by using suitable transformations and Adomian decomposition method has been used to solve the system of equations. To explore the influence of various parameters on fluid flow properties, quantitative analysis is exhibited graphically and shown in the tabular form.

In this paper, we are concerned with the following fractional functional differential equations with nonlocal initial conditions in Banach space $$\begin{aligned} \hbox {D}^{\alpha }x(t)=Ax(t)+f(t,x(t),x_{t}),\ \ t\in [0, T], \ \ x(0)=\phi +g(x). \end{aligned}$$ By virtue of the theory of measure of noncompactness associated with Darbo’s fixed point theorem, upon making some suitable assumptions, some existence results of mild solutions are obtained. Moreover the results obtained are utilized to study the existence of solutions to fractional parabolic equations as an illustrative example to show the practical usefulness of the analytical results.

This paper derives a new family of Burr-type distributions as new Burr distribution. This particular skewed distribution that can be used quite effectively in analyzing lifetime data. It is observed that the new distribution has modified unimodal hazard function. Various properties of the new Burr distribution, such that moments, quantile functions, hazard function, and Shannon’s entropy are obtained. The exact form of the probability density function and moments of $$i{\mathrm{th}}$$ -order statistics in a sample of size n from new Burr distribution are derived. Estimation of parameters and change-point of hazard function by the maximum likelihood method are discussed. Change-point of hazard function is usually of great interest in medical or industrial applications. The flexibility of the new model is illustrated with an application to a real data set. In addition, a goodness-of-fit test statistic based on the Renyi Kullback–Leibler information is used.

In this paper, we establish quadruple fixed point theorems for compatible type mappings in partially ordered -metric spaces without the mixed -monotone property under some conditions. Also, an example is given to show our results are real generalization of known results in quadruple fixed point theory.

Ranking of intuitionistic fuzzy numbers is a difficult task. Many methods have been proposed for ranking of intuitionistic fuzzy numbers. In this paper we have ranked both trapezoidal intuitionistic fuzzy numbers and triangular intuitionistic fuzzy numbers using the centroid concept. Some of the properties of the ranking function have been studied. Also, comparative examples are given to show the effectiveness of the proposed method.

In this paper, we extend some estimates of the right-hand side of a Hermite–Hadamard–Fejer type inequality for functions whose first derivatives’ absolute values are convex. The results presented here would provide extensions of those given in earlier works.

In this paper, first, we obtained two new $$s$$ -Godunova–Levin type inequalities about “the mean value Theorem for integrals”. Second, some inequalities were proved for mappings $$q$$ th powers of first derivatives belonging to class $$Q_{s}(I)$$ using the Cebysev’s inequality, H older inequality, Power mean inequality, and some other classical inequalities. Finally, some error estimates for the Trapezoidal formula are also given.The results obtained are consistent with literature.

In this paper, we use a general identity for generalized hypergeometric series to obtain some new applications. The first application is a hypergeometric-type decomposition formula for elementary special functions and the second one is a generalization of the well-known Euler identity \(e^{i\,x} = \cos x + i\,\sin x\) and an extension of hyperbolic functions in the sequel. Applying the mentioned identity on classical hypergeometric orthogonal polynomials and deriving summation formulae for some classical summation theorems are two further applications of this identity.

In this study, a new class of polynomial functions although equipped with a parameter is introduced. This class can be employed for computational solution of linear or non-linear functional equations, including ordinary differential equations or integral equations. The extra parameter permits us to obtain more accurate results. In the present paper, a number of numerical examples show the ability of this class of polynomial functions.

In this paper, we prove that coupled and tripled coincidence point theorems under $$(F, g)$$ -invariant sets for weakly contractive mappings defined on a $$G$$ -metric space are immediate consequences of corresponding results via rectangular $$G$$ - $$\alpha $$ -admissible mappings. This idea can also be applied to obtain coupled and tripled fixed point theorems in other spaces under various contractive conditions which reduces the proof considerably.

In this paper, we present a direct computational method to solve Volterra integral equations. The proposed method is a direct method based on approximate functions with the Bernstein Multiscaling polynomials. In this method, using operational matrices, the integral equation turns into a system of equations. Our approach can solve nonlinear integral equations of the first kind and the second kind with piecewise solution. The computed operational matrices in this article are exact and new. The comparison of obtained solutions with the exact solutions shows that this method is acceptable. We also compared our approach with two direct and expansion–iterative methods based on the block-pulse functions. Our method produces a system, which is more economical, and the solutions are more accurate. Moreover, the stability of the proposed method is studied and analyzed by examining the noise effect on the data function. The appropriateness of noisy solutions with the amount of noise approves that the method is stable.

The nonlinear partial differential equation of Harry Dym is generalized by replacing the first-order time derivative by a fractional derivative of order \(\alpha ,\; 0 \le \alpha \le 2\). The aim of the present paper is to obtain an approximate solution of time fractional generalized Harry Dym equation using Adomian Decomposition Method (ADM). The fractional derivative is described in the Caputo sense. Numerical examples are given to show the application of the present technique. The results show that the solution of ADM is in good agreement with the exact solution when \(\alpha =1\), also reveal that the method is very simple and effective.

In this paper, a new nonpolynomial cubic spline method is developed for solving two-parameter singularly perturbed boundary value problems. Convergence analysis is briefly discussed. Numerical examples and computational results illustrate and guarantee a higher accuracy by this technique. Comparisons are made to confirm the reliability and accuracy of the proposed technique.

In this paper, new periodic fractional trigonometric functions with the period 2πα are presented. We have generalized the Floquet system to the fractional Floquet system. The fractional derivatives are described with the use of modified Riemann–Liouville derivative. Moreover, the stability analysis of fractional Floquet system is introduced.

In this paper, we complement the result of Zhu and Chang (J Ineq Appl 2013:146, 2013) by proving strong convergence theorems for approximation of a fixed point of a left Bregman strongly relatively nonexpansive mapping which is also a solution to a finite system of equilibrium problems in the framework of reflexive real Banach spaces using the Halpern–Mann’s iterations used in Zhu and Chang (J Ineq Appl 2013:146, 2013). We also discuss the approximation of a common fixed point of a family of left Bregman strongly nonexpansive mappings which is also solution to a finite system of equilibrium problems in reflexive real Banach spaces. Our results complement many known recent results in the literature.

In this paper, the two-dimensional Bernoulli wavelets (BWs) with Ritz–Galerkin method are applied for the numerical solution of the time fractional diffusion-wave equation. In this way, a satisfier function which satisfies all the initial and boundary conditions is derived. The two-dimensional BWs and Ritz–Galerkin method with satisfier function are used to transform the problem under consideration into a linear system of algebraic equations. The proposed scheme is applied for numerical solution of some examples. It has high accuracy in computation that leads to obtaining the exact solutions in some cases.

In this paper, we study an electrostatic model for zeros of polynomials orthogonal with respect to inner product

Sufficient conditions guaranteeing the existence of three Heteroclinic solutions of a class of bilateral difference systems are established using a fixed point theorem. It is the purpose of this paper to show that the approach to get Heteroclinic solutions of BVPs using multi-fixed-point theorems can be extended to treat the bilateral difference systems with the nonlinear operators and .

In this paper we consider compactification spaces of ideal extension for topological semigroups. As a consequence, we characterize compactification spaces for Brandt λ-extension of topological semigroups.

The aim of this paper is to present some common fixed point results for six selfmappings satisfying generalized weakly (ψ, ϕ)-contractive condition in the setup of partially ordered G-metric spaces. Our results extend and generalize the comparable results in the work of Abbas from the context of ordered metric spaces to the setup of ordered G-metric spaces. Also, our results are supported by an example.

Purpose The purpose of this paper is to develop a set of identities for Euler type sums of products of harmonic numbers and reciprocal binomial coefficients.

Purpose The purpose of this paper is to study the reflection of plane periodic wave’s incident on the surface of generalized thermoelastic micropolar transversely isotropic medium.

This paper presents the following definition which is a natural combination of the definitions for asymptotically equivalent λ-statistical convergence and σ -convergence of fuzzy numbers. Two sequences X and Y of fuzzy numbers are said to be asymptotically λ-invariant statistical equivalent of multiple L provided that for every e> 0, lim n 1 λn k ∈ In : − d X σ k (m) Y σ k(m) , L ≥ e = 0, uniformly in m denoted by X S L ,λ (F) ∼ Y and simply asymptotically λ-invariant statistical equivalent if L = 1.

Purpose The purpose of this paper is to study a common fixed point theorem for two pairs of weakly compatible maps in dislocated metric spaces.