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Delay partial differential equations (PDEs) are widely utilized in many fields, such as climate prediction and epidemiology. But observation data in real world is often noisy and discrete. And in order to expand the applications of delay PDEs, we consider numerical Gaussian processes to solve these models. In this paper, numerical Gaussian processes for predicting the latent solution of a type of delay PDEs with multi-delays are investigated, and various delay PDEs are studied, including problems governed by variable-order fractional order operators and nonlinear operators, so as to adapt to the needs of practical applications. Numerical Gaussian processes are very good at fitting latent solution of PDEs, when all observation data is noisy and discontinuous. And the methodology can clearly quantify the uncertainty of the predicted solution. For complex boundaries controlled by ODEs, we consider mixed boundary conditions of delay PDEs in this paper. And we also apply Runge-Kutta methods to enhance the prediction accuracy of these problems. Finally, we design seven numerical examples to investigate the efficiency of NGPs and how the noisy data influences the solution of our studied problems. • Numerical Gaussian processes for solving delay PDEs models governed by variable-order fractional order operators and nonlinear operators. • Numerical Gaussian processes are very good at fitting latent solution of PDEs, when all observation data is noisy and discontinuous. • Runge-Kutta methods are applied to enhance the prediction accuracy of these problems. • Seven numerical examples to investigate the efficiency of NGPs and how the noisy data influences the solution of our studied problems. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper, we define and study a new three-step iterative scheme inspired by Suantai [J. Math. Anal. Appl. 311 (2005) 506–517]. This scheme includes many well-known iterations, for examples, modified Mann-type, modified Ishikawa-type iterative schemes, and the three-step iterative scheme of Xu and Noor. Several convergence theorems of this scheme are established for asymptotically nonexpansive mappings. Our results extend and improve the recent ones announced by Schu [J. Math. Anal. Appl. 158 (1991) 407–413; Bull. Aust. Math. Soc. 43 (1991) 153–159], Xu and Noor [J. Math. Anal. Appl. 267 (2002) 444–453], Suantai [J. Math. Anal. Appl. 311 (2005) 506–517], and many others. A misleading conclusion of Theorem 2.6 in Suantai’s paper is also corrected. [Copyright &y& Elsevier]

In this paper, we consider the semilocal convergence on a class of improved Chebyshev–Halley type methods for solving F ( x ) = 0 , where F : Ω ⊆ X → Y is a nonlinear operator, X and Y are two Banach spaces, Ω is a non-empty open convex subset in X . To solve the problems that F  ′′′( x ) is unbounded in Ω and it can not satisfy the whole Lipschitz or H o ¨ lder continuity, ‖ F  ′′′( x )‖ ≤ N is replaced by ∥ F ′ ′ ′ ( x 0 ) ∥ ≤ N ¯ , for all x ∈ Ω , where N , N ¯ ≥ 0 , x 0 is an initial point. Moreover, F  ′′′( x ) is assumed to be local H o ¨ lder continuous. So the convergence conditions are relaxed. We prove an existence-uniqueness theorem for the solution, which shows that the R -order of these methods is at least 5 + q , where q ∈ (0, 1]. Especially, when F  ′′′( x ) is local Lipschitz continuous, the R -order will become six. [ABSTRACT FROM AUTHOR]

Recently, Cui and Wang (2014) constructed an algorithm for demicontractive operators that converges weakly, under mild assumptions, to some solution of the split common fixed point problem. In this paper, based on Halpern’s type method (1967), we construct an algorithm for demicontractive operators that produces sequences that always converge strongly to a specific solution of the split common fixed point problem. Particular cases of directed operators and quasi-nonexpansive mappings are also considered. [ABSTRACT FROM AUTHOR]

George and Elmahdy (2012), considered an iterative method which converges quadratically to the unique solution x α δ of the method of Lavrentiev regularization, i.e., F ( x ) + α ( x - x 0 ) = y δ , approximating the solution x ˆ of the ill-posed problem F ( x ) = y where F : D ( F ) ⊆ X ⟶ X is a nonlinear monotone operator defined on a real Hilbert space X . The convergence analysis of the method was based on a majorizing sequence. In this paper we are concerned with the problem of expanding the applicability of the method considered by George and Elmahdy (2012) by weakening the restrictive conditions imposed on the radius of the convergence ball and also by weakening the popular Lipschitz-type hypotheses considered in earlier studies such as George and Elmahdy (2012), Mahale and Nair (2009), Mathe and Perverzev (2003), Nair and Ravishankar (2008), Semenova (2010) and Tautanhahn (2002). We show that the adaptive scheme considered by Perverzev and Schock (2005) for choosing the regularization parameter can be effectively used here for obtaining order optimal error estimate. In the concluding section the method is applied to numerical solution of the inverse gravimetry problem. [ABSTRACT FROM AUTHOR]

The property that a nonlinear operator on a Banach space preserves an order relation, is subhomogeneous or order concave w.r.t. an order cone has profound consequences. In Nonlinear Analysis it allows to solve related equations by means of suitable fixed point or monotone iteration techniques. In Dynamical Systems the possible long term behavior of associate integrodifference equations is drastically simplified. This paper contains sufficient conditions for vector-valued Urysohn integral operators to be monotone, subhomogeneous or concave. It also provides conditions guaranteeing that these properties are preserved under spatial discretization of particularly Nyström type. This fact is crucial for numerical schemes to converge, or for simulations to reproduce the actual behavior and asymptotics. [ABSTRACT FROM AUTHOR]

Abstract: The fixed point theorems presented in this paper generalize recent results of Dutta and Choudhury [3], Branciari [2], Banach [1], Rhoades [9], Khan et al. [6], Kada [5] and Rhoades and Abbas [10]. [Copyright &y& Elsevier]

Abstract: In this paper, the upper and lower estimates of the radius of the convergence ball of the modified Newton’s method in Banach space are provided under the hypotheses that the Fréchet derivative of the nonlinear operator are center Hölder continuous for the initial point and the solution of the operator. The error analysis is given which matches the convergence order of the modified Newton’s method. The uniqueness ball of solution is also established. Numerical examples for validating the results are also provided, including a two point boundary value problem. [Copyright &y& Elsevier]

Abstract: In this paper we demonstrate that a number of fixed point iteration problems can be solved using a modified Krasnoselskij iteration process, which is much simpler to use than the other iteration schemes that have been defined. [Copyright &y& Elsevier]

Abstract: In the present paper, numerical methods are developed to approximate the solutions of some evolutionary nonlinear problems. The continuous problems are transformed into some Lyapunov type equations and then analysed for existence, uniqueness, convergence, stability and error estimates. The main idea consists in applying Fourier analysis and Von Neumann criterion acting translation and scaling parameter methods to obtain contractive operators leading to fixed point theory. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the existence of multiple positive solutions for some nonlinear singular four-point boundary value problems with p-Laplacian. Under appropriate assumptions on the nonlinear terms, by applying several known fixed point theorems, sufficient conditions for the existence of at least three positive solutions are established. [Copyright &y& Elsevier]

Abstract: In this paper, the problem of existence and attractivity of almost periodic solutions for delayed neural field (DNF) with variable coefficients is discussed. First, the model of DNF is established as a modified neural field model. Secondly, by combining the theory of the exponential dichotomy, employing Banach fixed point theory and using differential inequality technique, some conditions for the the existence and attractivity of almost periodic solutions for the DNF are proposed. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to generalize and to unify fixed point theorems of Das and Naik, Ćirić, Jungck and Ume in terms of a w-distance on complete metric space. [Copyright &y& Elsevier]

Abstract: In this paper, by using Krasnoselskii fixed point theorem, we study the existence of at least or two positive solutions to the nonlinear second order three-point boundary value problemwhere are given constants. As an application, we also give some examples to demonstrate our results. [Copyright &y& Elsevier]

Abstract: This paper presents the existence of positive solutions for the singular three-point boundary value problemwhere is allowed to be singular at and f may be singular at u =0. The existence criteria for positive solutions of the above problem is established by applying the fixed point index theorem under some weaker conditions concerning the first eigenvalue corresponding to the relevant linear operator. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce and consider a new class of Wiener–Hopf equations involving the nonlinear operator and nonexpansive operators. Essentially using the projection technique we establish the equivalence between the Wiener–Hopf equations and variational inequalities. Using this alternative equivalent formulation, we suggest and analyze an iterative method for finding the common element of the set of fixed points of nonexpansive mappings and the set of solutions of the variational inequalities. We also study the convergence criteria of iterative methods under some mild conditions. Our results include the previous results as special cases and may be considered as an improvement and refinement of the previously known results. [Copyright &y& Elsevier]

Abstract: In this paper a modified Adomian method is introduced to solve some systems of nonlinear differential equations. The method can be mechanized in Maple and a procedure is written to solve the approximate analytic solution of the systems. Some examples will be presented to illustrate the effectiveness of the method and the procedure. A comparison between the program and the original Adomian method’s program is drawn. [Copyright &y& Elsevier]

Abstract: In this paper, a new algorithm for calculating Adomian polynomials for nonlinear operators will be established by parametrization. The algorithm requires less formula than the previous method developed by Adomian [Nonlinear Stochastic Operator Equations, Academic Press, 1986, G. Adomian, R. Rach, On composite nonlinearities and decomposition method. J. Math. Anal. Appl. 113 (1986) 504–509, G. Adomian, Applications of Nonlinear Stochastic Systems Theory to Physics, Kluwer, 1988]. Many forms of nonlinearity will be studied to illustrate the new algorithm. The new algorithm will be extended to calculate Adomian polynomials for nonlinearity of several variables. [Copyright &y& Elsevier]

Bugajewski [Math. Comput. Modell. 32 (2000) 1457] extend the Baillion fixed point theorem from [Contemporary Math. Am. Math. Soc. 72 (1988) 11]. In this paper, we proved a version of Bugajewski''s theorem for multivalued maps defined on a bounded hyperconvex subset of a normed space. [Copyright &y& Elsevier]

This paper is devoted to the existence and global attractivity of almost periodic solution for a class of cellular neural network with distributed delays and variable coefficients. Some sufficient conditions ensuring the existence and global attractivity of almost periodic solution are derived by employing Banach fixed point theory and using differential inequality technique. The results allow for the consideration of all unbounded neuron signal functions (but not necessarily surjective). Thus, these conditions obtained have highly important significance in designs and applications of the networks. We extend and improve previously known results. An example is also worked out to demonstrate the advantages of our results. [Copyright &y& Elsevier]

In this paper we introduce a new method for calculating Adomian polynomials. Implementing the new method is very similar to using derivative operator which is familiar to everyone. [Copyright &y& Elsevier]

In this paper, a symbolic implementation code is developed of a technique proposed by Wazwaz [Appl. Math. Comput. 111 (2000) 53] for calculating Adomian polynomials for nonlinear operators. The algorithm proposed by him [Appl. Math. Comput. 111 (2000) 53] offers a promising approach for calculating Adomian polynomials for all forms of nonlinearity, but it is not easy to implement due to its huge size of algebraic calculations, complicated trigonometric terms, and unique summation rules. It is well known that the algebraic manipulation language such as Mathematica is useful to facilitate such a hard computational work. Pattern-matching capabilities peculiar feature of Mathematica are used in index regrouping which is a key role in constructing Adomian polynomials. The computer algebra software Mathematica is used to collect terms to their order and to simplify the terms. The symbolic implementation code author developed (appearing at appendix) has the flexibility that may easily cover any length of Adomian polynomial for many forms of nonlinear cases. A nonlinear evolution equation is investigated in order to justify the availability of symbolic implementation code. [Copyright &y& Elsevier]

In this paper, the author will introduce the Sobolev space W12(Ωn) firstly and then give the reproducing properties of the space W12(Ωn). As these results of those, a class of the nonlinear operator equation ∑ni=1∏mij=1(Aiju)=f is transformed to the n-dimensional linear operator equation Au=f. Finally, we also give the exact solution of this nonlinear operator equation. [Copyright &y& Elsevier]

In this paper, we first discuss the problem of how to solve a continuous linear operator equation in separable Hilbert space. We give the necessary and sufficient condition for the existence of solution, and the method to decide the existence for solutions of the equation is also given. If the solution exists, we give the representation and approxima of the minimal normal solution of the equation and formula are obtained. We conduct some research of how to solve the operator equation AuBu+Cu=f. We give the factorization method and characteristic value method to solve such an equation. [Copyright &y& Elsevier]

This paper discusses some important block iterative methods for solving variational inequalities with a nonlinear operator and proves their monotone convergence properties and comparison theorems. These results extend C. V. Pao''s research work in Numer. Math. (72) (1995) 239 to variational inequalities. [Copyright &y& Elsevier]

• Fractional Tikhonov regularization (FTR) model for non-linear operator equations is studied. • The model is analyzed in the setting of Hilbert scales. • With Holder type source condition, an optimal error estimate is obtained. • The numerical verification is validated using various examples. Fractional Tikhonov regularization method (FTRM) for linear ill-posed operator equations in the setting of Hilbert scales is being studied in this paper. Using a general Holder type source condition, we obtain an error estimate. A new parameter choice strategy is being proposed for choosing the regularization parameter in FTRM in the setting of Hilbert scales. Also, the proposed method is applied to the well known examples in the setting of Hilbert scales. [ABSTRACT FROM AUTHOR]