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We present a monotone finite volume element scheme for the diffusion problem on triangular grids, which stems from some idea about the two-point flux in the literature [15] . A new nonlinear two-point flux formulation is obtained for some part of one control volume edge, and this straight edge crosses through some common edge of two neighboring triangular elements where two inner points from these neighboring elements are chosen and linked so that it does not cross through other triangle elements even if the meshes are severely distorted. The approximation of κ∇u in some quadrilateral region including this straight control volume edge, are obtained by some weighted average with the gradient of some finite element functions, and the weighted coefficients are determined by the nonlinear two-point flux idea. Furthermore, we prove that the new scheme is monotone under some conditions that one modified barycenter dual partition Ω h ⁎ is chosen for some given partition Ω h and the exact solution u ∈ C 1 ( Ω ) , and the corresponding coefficient matrix is of M-matrix. Finally, we carry on some typical experiments. Numerical results verify the monotonicity of this new scheme, and also confirm the accuracy and stability.

Complex symmetric Sylvester matrix equations appear in many applications, such as the numerical solution of the complex Helmholtz equations. In this paper, by designing a global complex symmetric M -Lanczos process we develop a global variant of the conjugate A-orthogonal conjugate residual method (Gl-COCR) for solving the Sylvester matrix equation A X + X B = C with complex symmetric coefficient matrices. To obtain the smooth and monotone convergence behavior, we also propose a smoothed Gl-COCR method, denoted by SGl-COCR. Finally, numerical examples are given to illustrate the performances of our methods.

This paper introduces a novel method for deducing high-order finite difference schemes for the three-dimensional Helmholtz equation, Δ p ± λ 2 p = f . The new method can reach an eighth-order of accuracy. It is based on an implicit formulation by computing simultaneously the unknown variable and its corresponding derivatives. The scheme is obtained from Taylor series expansion and plane wave theory analysis, and it is constructed from a few modifications to the standard centered finite differences. A classical dispersion analysis and the error between the numerical wavenumber and the exact wavenumber is given. Numerical experiments are presented to verify the feasibility and accuracy of the proposed methods for a broad range of λ -values. Moreover, the numerical method is capable to solve with great precision the monotone and oscillatory Helmholtz problem with variable λ -values. Thus, the proposed formulation results in an attractive method, easy to implement, with high order of accuracy but nearly the same computational cost as those of an explicit formulation.

This paper is devoted to the study of mono-stable and bistable traveling waves for monotone or non-monotone equations with nonlocal time delays in a cylinder. When the time delay is finite, the existence of the traveling wave solutions is established by appealing to the theory of monotone semi-flows for the monotone equation and the fixed point theory coupled with upper–lower solutions method for the non-monotone equation. When the time delay is infinite, the finite-delay approximations method is used to obtain the existence of the traveling waves.

We consider an abstract system with Lagrange multipliers which consists of a hemivariational inequality and a variational inequality. The hemivariational inequality is governed by a hemicontinuous, generalized monotone, possibly nonlinear operator as well as by a bilinear form. This bilinear form also governs the variational inequality. We are looking for a pair solution in a subset of a product of two real reflexive Banach spaces. In order to illustrate the applicability of the abstract results, two examples in terms of PDEs are delivered. Each example is related to the weak solvability of a boundary value problem arising in contact mechanics.

In this work, we present an improvement of the Lagrangian–Eulerian space–time tracking forward scheme to deal with balance laws and related applications. This extended algorithm is shown in the most simple setting and it is a result of our previous works. We describe and explain a new strategy of discretization of conservation laws, starting from the scalar case in one space dimension, extending it to systems and to the multi-dimensional setting. The computations are fast, accurate and stable with good resolution. This algorithm is very easy to implement in a computer to address the delicate well-balancing between the first-order hyperbolic flux and the source term. We do not use approximate or exact Riemann solvers, nonlinear reconstructions, or upwind source term discretizations. The scheme is written into the classical theory of monotone schemes, which produces a scheme that converges to entropy solutions linked to the purely hyperbolic counterpart. This method can produce well-balanced approximations of solutions for nonlinear balance laws. Numerical experiments also demonstrate the robustness of the forward tracking to solve related problems involving systems and two-dimensional models.

We present a model and discretization that couples the Ericksen model of liquid crystals with variable degree of orientation to the Allen–Cahn equations with a mass constraint. The coupled system models liquid crystal droplets with anisotropic surface tension effects due to the liquid crystal molecular alignment. The total energy consists of the Ericksen energy, phase-field (Allen–Cahn) energy, and a weak anchoring energy that couples the liquid crystal to the diffuse interface. We describe our discretization of the total energy along with a method to compute minimizers via a discrete gradient flow algorithm which has a strictly monotone energy decreasing property. Numerical experiments are given in three dimensions that illustrate a wide variety of droplet shapes that result from their interaction with defects.

The present study is concerned with the nontrivial solutions for fractional Schrodinger–Poisson system with the Bessel operator. Under certain assumptions on the nonlinearity f , a nontrivial nonnegative solution is obtained by perturbation method for the given problem. In particular, the Ambrosetti–Rabinowitz type condition or the monotone assumption on the nonlinearity is unnecessary.

In this paper, we present a semismooth Newton method for a kind of HJB equation. By suitably choosing the initial iterative point, the method is proved to have monotone convergence. Moreover, the semismooth Newton method has local superlinear convergence rate. Some simple numerical results are reported.

In this paper, we present a projection method to solve monotone nonlinear equations with convex constraints. This method can be viewed as an extension of CG_DESCENT method which is one of the most effective conjugate gradient methods for solving unconstrained optimization problems. Because of derivative-free and low storage, the proposed method can be used to solve large-scale nonsmooth monotone nonlinear equations. Its global convergence is established under some appropriate conditions. Preliminary numerical results show that the proposed method is effective and promising. Moreover, we also successfully use the proposed method to solve the sparse signal reconstruction in compressive sensing.

In this paper we focus on the numerical solution of nonlinear Black–Scholes equation modeling illiquid markets. Two monotone unconditionally stable splitting methods, ensuring positive numerical solution and avoiding unstable oscillations, are applied to solve nonlinear Black–Scholes equation modeling illiquid markets. These numerical methods are based on the LOD methods which allow us to solve the discrete equation explicitly. The properties of these methods are analyzed. The numerical results for vanilla call option are compared to the local Crank–Nicolson scheme. The numerical results for European butterfly spread are also provided.

Based on a classical multislope MUSCL (monotone upstream scheme for conservation law) method, this paper addresses a new multislope MUSCL method which is used to extrapolate the values of flow variables at the midpoints of cell edges. The main merit of the new method is its more straightforward implementation than that of the classical one. The new method is incorporated into a two-dimensional unstructured cell-centered Godunov-type finite volume model to simulate shallow water flows. The performance of the model with the new multislope MUSCL method is demonstrated by simulating three test cases including both theoretical and practical ones. It is verified that the new method is more efficient than the classical one and is able to produce slightly better or similar numerical results.

In this paper, we study numerical methods for a class of two-dimensional semilinear elliptic boundary value problems with variable coefficients in a union of rectangular domains. A compact finite difference method with an anisotropic mesh is proposed for the problems. The existence of a maximal and a minimal compact difference solution is proved by the method of upper and lower solutions, and two sufficient conditions for the uniqueness of the solution are also given. The optimal error estimate in the discrete L ∞ norm is obtained under certain conditions. The error estimate shows the fourth-order accuracy of the proposed method when two spatial mesh sizes are proportional. By using an upper solution or a lower solution as the initial iteration, an "almost optimal" Picard type of monotone iterative algorithm is developed for solving the resulting nonlinear discrete system efficiently. Numerical results are presented to confirm our theoretical analysis.

This paper concerns a Lotka–Volterra competition reaction–diffusion system with nonlinear diffusion effects. We first briefly discuss the stability of trivial solutions by spectrum analysis. Based on the boundedness of the solutions, the existence of the positive steady state solution is also investigated by the monotone iteration method.

In this paper, we use a monotone iterative technique in the presence of lower and upper solutions to discuss the existence of mild solutions to a second order evolution equation initial value problem in an ordered Banach space, which can model an elastic system with structural damping. Under monotone condition and the noncompactness measure condition on nonlinearity, we obtain the existence of extremal mild solutions and positive mild solutions. Moreover, some applications are given to illustrate our theoretical results.

The pattern of some real phenomenon can be described by compartmental in-series models. Nevertheless, most of these processes are characterized by their variability, which produces that the exact values of the model parameters are uncertain, although they can be bounded by intervals. The aim of this paper is to compute tight solution envelopes that guarantee the inclusion of all possible behaviors of such processes. Current methods, such as monotonicity analysis, enable us to obtain guaranteed solution envelopes. However, if the model includes nonmonotone compartments or parameters, the computation of solution envelopes may produce a significant overestimation. Our proposal consists of performing a change of variables in which the output is unaltered, and the model obtained is monotone with respect to the uncertain parameters. The monotonicity of the new system allows us to compute the output bounds for the original system without overestimation. These model transformations have been developed for linear and non-linear systems. Furthermore, if the conditions are not completely satisfied, a novel method to compute tight solution envelopes is proposed. The methods exposed in this paper have been applied to compute tight solution envelopes for two different models: a linear system for glucose modeling and a non-linear system for an epidemiological model., This work was partially supported by the Spanish Ministerio de Ciencia e Innovacion through Grant DPI-2010-20764-C02-01, and by the Generalitat Valenciana through Grant GV/2012/085.

We investigate nonnegativity of exact and numerical solutions to a generalized Keller-Segel model. This model includes the so-called ''minimal'' Keller-Segel model, but can cover more general chemistry. We use maximum principles and invariant sets to prove that all components of the solution of the generalized model are nonnegative. We then derive numerical methods, using finite element techniques, for the generalized Keller-Segel model. Adapting the ideas in our proof of nonnegativity of exact solutions to the discrete setting, we are able to show nonnegativity of discrete solutions from the numerical methods under certain standard assumptions. One of the numerical methods is then applied to the minimal Keller-Segel model. Recalling known results on the qualitative behavior of this model, we are able to choose parameters that yield convergence to a nonhomogeneous stationary solution. While proceeding to exhibit these stationary patterns, we also demonstrate how naive choices of numerical methods can give physically unrealistic solutions, thereby justifying the need to study positivity preserving methods.

In this paper, we consider the existence of generalized solutions for fractional functional integro-differential equations of mixed type with nonlinear boundary value conditions. By establishing a new comparison theorem and applying the monotone iterative technique, we show the existence of extremal generalized solutions.

In this paper, we consider the existence of generalized solutions for fractional differential systems with nonlinear boundary value conditions. We first establish a new comparison theorem. By applying the monotone iterative technique and the method of lower and upper generalized solutions, we obtain sufficient conditions for existence of extremal generalized solutions.

The purpose of this paper is to introduce and analyze an extragradient method with regularization for finding a common element of the solution set @C of the split feasibility problem and the set Fix(S) of fixed points of a nonexpansive mapping S in the setting of infinite-dimensional Hilbert spaces. Combining the regularization method and the extragradient method due to Nadezhkina and Takahashi [N. Nadezhkina, W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 128 (2006) 191-201], we propose an iterative algorithm for finding an element of Fix(S)@[email protected] We prove that the sequences generated by the proposed algorithm converge weakly to an element of Fix(S)@[email protected] under mild conditions.

We investigate strong convergence for Bregman strongly nonexpansive mappings by Halpern’s iteration in the framework of reflexive Banach spaces. Using the obtained result, convergence for a family of Bregman strongly nonexpansive mappings is also discussed. As applications, we apply our main result to problems of finding zeros of maximal monotone operators and equilibrium problems in reflexive Banach spaces.

Let H be a real Hilbert space. For each i=1,2,...,m, let F"i,K"i:H->H be bounded and monotone mappings. Assume that the generalized Hammerstein equation u+@?"i"="1^mK"iF"iu=0 has a solution in H. We construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the generalized Hammerstein equation.

In this paper, we prove some coupled fixed point theorems involving a (@j,@f)-weakly contractive condition for mapping having the mixed monotone property in ordered partial metric spaces. These results are analogous to theorems of Van Luong and Xuan Thuan (2011) [10] on the class of ordered partial metric spaces. Also, an application is given to support our results.

In this paper, we extend the coupled fixed point theorems for a mixed monotone mapping F:X×X→X in partially ordered metric spaces established by Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65 (2006) 1379–1393]. An application to nonlinear integral equations is also given to illustrate our results.

This paper is concerned with an existing compact finite difference ADI method, published in the paper by Liao et al. (2002) [3], for solving systems of two-dimensional reaction–diffusion equations with nonlinear reaction terms. This method has an accuracy of fourth-order in space and second-order in time. The existence and uniqueness of its solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear reaction terms. The convergence of the finite difference solution to the continuous solution is proved. An efficient monotone iterative algorithm is presented for solving the resulting discrete system, and some techniques for the construction of upper and lower solutions are discussed. An application using a model problem gives numerical results that demonstrate the high efficiency and advantages of the method.

We introduce a modified alternating direction method for structured monotone variational inequalities by performing an additional projection step at each iteration and another optimal step length is employed to reach substantial progress in each iteration. This method only needs functional values for given variables in the solution process and avoids the task of estimating the co-coercive modulus. All the computing process are easily implemented and the global convergence is also presented under mild assumptions. Some preliminary computational results are given.

In this study, using fixed point index theorems and monotone iterative techniques, we present iterative schemes as well as the existence of positive solutions to the m-point boundary value problem on a half-line.

[EN] This paper deals with the numerical analysis and computing of a nonlinear model of option pricing appearing in illiquid markets with observable parameters for derivatives. A consistent monotone finite difference scheme is proposed and a stability condition on the stepsize discretizations is given. © 2010 Elsevier Ltd. All rights reserved., This paper has been supported by the Spanish Department of Science and Education grant DPI2010-C02-01.

In this paper, the Resolvent–Projection algorithm for solving the variational inclusion 0∈M(x) involving an A-monotone set-valued operator M is constructed in Hilbert spaces. The convergence of the iterative sequence generated by the algorithm is proved also.

In this paper, we propose a globally and quadratically convergent Newton-type algorithm for solving monotone second-order cone complementarity problems (denoted by SOCCPs). This algorithm is based on smoothing and regularization techniques by incorporating smoothing Newton’s method. Many Newton-type methods with smoothing and regularization techniques have been studied for solving nonlinear complementarity problems (NCPs) and box constrained variational inequalities (BVIs). Our algorithm is regarded as an extension of those methods to SOCCP. However, it is different from the existing methods, because we solve SOCCP by treating both the smoothing parameter μ and the regularization parameter ε as independent variables. In addition, numerical experiments indicate that the proposed method is quite effective.

This paper illustrates that the conditions and the main proof of two main theorems of Verma [R.U. Verma, The over-relaxed proximal point algorithm based on H-maximal monotonicity design and applications, Computers and Mathematics with Applications 55 (2008) 2673–2679] concerning the strong convergence of the over-relaxed proximal point algorithm for H-maximal monotone mappings in Hilbert spaces are incorrect.

This paper considers the existence of solutions for a class of integral boundary value problems with causal operators. We build a new comparison theorem. By utilizing the monotone iterative technique and the method of lower and upper solutions, we formulate sufficient conditions under which such problems have extremal or quasisolutions in a corresponding sector.

In this paper we study the variational inequality problems with a particular splitting structure, in which the mapping F does not have an explicit form and only its function values can be employed in the numerical methods for solving such problems. Studies and applications of such problems can be found in Fukushima (1992) [3], Glowinski (1984) [4], Glowinski (1989) [5] and Xu (2007) [15]. The paper He et al. (2006) [6] presents an efficient prediction–correction method for such problems. Based on the predictor method in the latter, this paper presents two classes of correction methods which are more convenient to be carried out than that in this reference. Numerical experiments show that the new method is effective.

In this paper, we introduce and study a general iterative method with strongly positive operators for finding solutions of a general variational inequality problem with inverse-strongly monotone mapping in a real Hilbert space. The explicit and implicit iterative algorithms are proposed by virtue of the general iterative method with strongly positive operators. Under two sets of quite mild conditions, we prove the strong convergence of these explicit and implicit iterative algorithms to the unique common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general variational inequality problem, respectively.

In this paper, we introduce a new class of accretive operators-(H(@?,@?),@h)-accretive operators, which generalize many existing monotone or accretive operators. The resolvent operator associated with an (H(@?,@?),@h)-accretive operator is defined and its Lipschitz continuity is presented. By using the new resolvent operator technique, we also introduce and study a new class of set-valued variational inclusions involving (H(@?,@?),@h)-accretive operators and construct a new algorithm for solving this class of set-valued variational inclusions. These results are new, and improve and generalize many known corresponding results.

In this paper, we introduce a new class of generalized vector variational-type inequalities with set-valued mappings (GVVTI). First, the solvability of generalized vector variational-type inequalities without monotone assumption is considered in Banach spaces by using Brouwer’s fixed point theorem. Second, by the introduction of the concepts of pseudomonotone, v-coercive and η-hemicontinuous, the solvability of (GVVTI) is obtained with the assumption of η-hemicontinuous and pseudomonotone.

In this paper, by introducing τ–φ-concave operators and using the properties of cones and monotone iterative technique, some new existence and uniqueness theorems of fixed points for increasing operators with such concavity are established under more extensive conditions in ordered Banach spaces. As an application, we study the existence and uniqueness of positive solutions of generalized three-point boundary value problems for second order differential equations. The results obtained here improve and generalize many known results.

The convergence of the multiplicative multisplitting-type method for solving the linear complementarity problem with an H-matrix is discussed using classical and new results from the theory of splitting. This directly results in a sufficient condition for guaranteeing the convergence of the multiplicative multisplitting method. Moreover, the multiplicative multisplitting method is applied to the H-compatible splitting and the multiplicative Schwarz method, separately. Finally, we establish the monotone convergence of the multiplicative multisplitting method under appropriate conditions.

The purpose of this paper is to develop a generalized matrix Riccati technique for the self-adjoint matrix Hamiltonian system U′=A(x)U+B(x)V,V′=C(x)U−A∗(x)V. Together with the integral averaging technique and monotone functionals, new oscillation and interval oscillation criteria are established for the system. These criteria extend, improve, complement a number of existing results, and handle some cases not covered by known criteria. In particular, two interesting examples are included to illustrate the versatility of our results.

This paper deals with the existence of positive solutions for some second-order multi-point boundary value problem on the half-line. Our approach is based on the fixed point theorem and the monotone iterative technique. Without the assumption of the existence of lower and upper solutions, we obtain not only the existence of positive solutions for the problems, but also establish iterative schemes for approximating the solutions.

In this paper, we employ partial order method, cone theory and monotone iterative technique to prove several new theorems on the existence and uniqueness of fixed points for decreasing operators without compactness. Some applications are also given.

A compact finite difference method is proposed for a general class of 2nth-order Lidstone boundary value problems. The existence and uniqueness of the finite difference solution is investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. A monotone iteration process is provided for solving the resulting discrete system efficiently, and a simple and easily verified condition is obtained to guarantee a geometric convergence of the iterations. The convergence of the finite difference solution and the fourth-order accuracy of the proposed method are proved. Numerical results demonstrate the high efficiency and advantages of this new approach.

This paper discusses the periodic boundary value problem for a class of first-order impulsive functional differential equations. We establish several existence results under weaker hypotheses by using the Leray–Schauder alternative, the lower and upper solution method and the monotone iterative technique. The corresponding results in the literature are improved or extended, some examples are also given to illustrate the advantage of the results.

In this paper, for a second-order boundary value problem on time scales, a method of generalized quasilinearization, under coupled upper and lower solutions, is discussed. An attempt for the method is to establish sufficient conditions for generating monotone iterative schemes whose elements converge rapidly to the unique solution of the given problem. Furthermore, the convergence is of order k(k≥2), which is even. Finally, two examples are provided to illustrate our results.

In this paper, we are concerned with the third order two-point generalized right focal boundary value problem x‴(t)+f(t,x(t))=0,a

In this paper, we propose a hybrid nonlinear decomposition–projection method for solving a class of monotone variational inequality problems. The algorithm utilizes the problems’ structure conductive to decomposition and a projection step to get the next iterate. To make the method more practical, we allow solving of the subproblems approximately and adopt the constructive accuracy criterion developed recently by Solodov and Svaiter for classical proximal point algorithm and by the author for generalized proximal point algorithm. The Fejér monotonicity to the solution set of the problem is obtained by only assuming the underlying mapping is monotone and the solution set is nonempty. The parameter is allowed to vary in a larger interval than that of Auslender and Teboulle, and we also propose some improved self-adaptive strategies to choose the sequence of parameters, which makes the algorithm more flexible.

Generalized nonlinear set-valued mixed quasi-variational inequalities with maximal monotone mappings have been studied in the literature. Since the maximal monotone operators are known as special cases of the H-monotone operators, we generalize the study for these inequality problems with H-monotone mappings in this paper. Precisely, we establish the existence of solutions for the generalized nonlinear set-valued mixed quasi-variational inequality problem with H-monotone mappings. An iterative method for finding their approximate solutions are proposed. Furthermore, a new convergence criteria for this iterative method is imposed.

The aim of this paper is to establish the equivalence of convergence for H-monotone mappings of variational inclusions between the one-step iteration algorithm of the paper [Y.P. Fang, N.J. Huang, H-monotone opertor and resolvent operator technique for variational inclusions, Appl. Math. Comput. 145 (2003) 795-803] and the two-step iteration algorithm of the paper [L.C. Zeng, S.M. Guu, J.C. Yao, Characterization of H-monotone operators with applications to variational inclusions, Comput. Math. Appl. 50 (2005) 329-337]. It is clear from this paper that the one-step iteration algorithm with easy implement and small computational workload has more advantages than the two-step iteration algorithm.

In this paper, three new classes of generalized monotone operators are introduced: the relaxed μ pseudomonotone operator, relaxed μ quasimonotone operator and densely relaxed pseudomonotone operator. The relations between these three classes and the relations between them and some other well known classes of generalized monotone operators are discussed in detail. Various existence results for variational inequalities in normed spaces are derived. The results generalize some known results.

Since proximal point algorithms (abbreviated as PPA) are attractive for solving monotone variational inequality problems, various approximate versions of PPA (APPA) are developed for practical applications. In this paper, we make a comparison between two different versions of APPAs (APPA I and APPA II) in the literature which share some common properties. Both of the algorithms use the same inexactness restriction and the same step length. The only difference is that they use different search directions. Through theoretical analysis and numerical experiment, we can see that APPA II usually performs better than APPA I.