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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.

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 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 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.

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.

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.

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.

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.

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 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 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.

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, 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.

In this paper, we consider the optimal feedback control problems of a system governed by strongly nonlinear impulsive differential equations which contains monotone operators and nonlinear nonmonotone perturbations. Based on the existence of feasible pairs, an existence result of optimal control pairs for the Lagrange problem is presented.

The authors employ the method of upper and lower solution coupled with the monotone iterative technique to obtain results of existence and uniqueness for a nonhomogeneous boundary value problem of impulsive differential equations with delay.

In this paper, we consider a class of variational inequality problems for monotone type mappings. By using one coercive assumption which is weaker than the one used in [1], we establish some existence results of solutions to variational inequality problems with generalized pseudomonotone mappings, generalized pseudomonotone perturbation and T-pseudomonotone perturbation of maximal monotone mappings, respectively.

In this paper, by using a resolvent operator technique of maximal monotone mappings and the property of a fixed-point set of set-valued contractive mappings, we study the behavior and sensitivity of the solutions of the parametric completely generalized strongly nonlinear implicit quasi-variational inclusion in Hilbert space. Our results extend, improve, and unify the previously many results in this field.

This paper considers the existence of extreme solutions of the periodic boundary value problems for the second order functional differential equations. The novelty is that we introduce a new concept for lower and upper solutions and present that the method of lower and upper solutions coupled with monotone iterative technique is still valid. Meanwhile, we extend previous results.

In this paper, we study the existence of positive solution for two classes of nonlinear second-order three-point boundary value problems using some monotone iterative schemes. In both cases, such schemes start off with known constant functions and, therefore, are useful to computation purpose.

Consider the following separable nonlinear delay differential equation dy(t)dt=−∑i=0mai(t)gi(y(τi(t))),t≥t0y(t)=φ(t),t≤t0, where we assume that, there is a strictly monotone increasing function f(x) on (−∞, +∞) such that f(0)=0,0

This paper discusses the antiperiodic boundary value problem for first-order impulsive ordinary differential equations. We establish several existence results by using the Leray-Schauder alternative, the lower and upper solution method and the monotone iterative technique.

The quasilinearization method is successfully applied to differential problems with nonlinear boundary conditions of antiperiodic type. Given are sufficient conditions under which the monotone iterations converge quadratically to a unique solution of our problem. The advantage of this approach lies in the fact that the approximate solutions are obtained iteratively as solutions of the above-mentioned iterations.

We establish sufficient conditions for the existence of solutions (unique or extremal) of differential equations with nonlinear multipoint boundary conditions using the monotone iterative technique.

An accelerated monotone iterative scheme for numerical solutions of a class of nonlinear elliptic boundary value problems is presented. The mathematical analysis is devoted to a system of discretized equations of the elliptic boundary value problem by the finite-difference or finite-element method. It is shown that the sequence of iterations from a linear iteration process converges monotonically and quadratically to a unique solution in a sector between a pair of upper and lower solutions. This result is then used to show the quadratic convergence of the iterations to a maximal solution and a minimal solution when the nonlinear discrete system possesses multiple solutions. An application is given to a tabular reactor model from chemical engineering for numerical solutions, and the number of iterations are compared with that by the regular monotone iterative scheme.

This paper is concerned with parameterized boundary value problems (BVPs) for semilinear evolution equations in a general Banach space. The abstract monotone iterative schemes are constructed by combining the theory of semigroups of operators and the method of upper and lower solutions. Some existence results are established under the suitable conditions. Applications to periodic BVPs with a parameter and parabolic partial differential equations are also given to illustrate the main results.

The method of lower and upper solutions combined with monotone iterative techniques is used for ordinary differential equations with integral boundary conditions showing the existence of extremal solutions. Some existence results are also formulated using the numerical-analytic method. Sufficient conditions for existence are given.

By establishing a comparison result and using the monotone iterative technique, the author obtains the existence of maximal and minimal solutions of the initial value problem for a class of n th -order impulsive integrodifferential equations in a Banach space.

In this paper, we consider a diffusional food-limited population model incorporating a discrete time-delay. For the case when the delay is small, we show that monotone travelling front solutions exist connecting the two uniform steady states of the model. The effect of a larger delay is studied numerically, suggesting that in this case travelling fronts still exist but lose their monotonicity.

This paper deals with the finite element approximation of Hamilton-Jacobi-Bellman equations. We establish a convergence and a quasi-optimal L∞-error estimate, involving a weakly coupled systems of quasi-variational inequalities for the solution of which an interative scheme of monotone kind is introduced and analyzed.

This paper is concerned with iterative methods for numerical solutions of a class of nonlocal reaction-diffusion-convection equations under either linear or nonlinear boundary conditions. The discrete approximation of the problem is based on the finite-difference method, and the computation of the finite-difference solution is by the method of upper and lower solutions. Three types of quasi-monotone reaction functions are considered and for each type, a monotone iterative scheme is obtained. Each of these iterative schemes yields two sequences which converge monotonically from above and below, respectively, to a unique solution of the finite-difference system. This monotone convergence leads to an existence-uniqueness theorem as well as a computational algorithm for the computation of the solution. An error estimate between the computed approximations and the true finite-difference solution is obtained for each iterative scheme. These error estimates are given in terms of the strength of the reaction function and the effect of diffusion-convection, and are independent of the true solution. Applications are given to three model problems to illustrate some basic techniques for the construction of upper and lower solutions and the implementation of the computational algorithm.

In this paper, we introduce and study a new class of quasi-variational inequalities, which is called the generalized nonlinear set-valued mixed quasi-variational inequality. Using the resolvent operator technique for maximal monotone mapping, we construct some new iterative algorithms for solving this class of generalized nonlinear set-valued mixed quasi-variational inequalities. We prove the existence of solution for this kind of generalized nonlinear set-valued mixed quasivariational inequalities without compactness and the convergence of iterative sequences generated by the algorithms. We also discuss the convergence and stability of perturbed iterative algorithm for solving a class of generalized nonlinear mixed quasi-variational inequalities.

An accelerated monotone iterative method for a boundary value problem of second-order discrete equations is presented. This method leads to an existence-comparison theorem as well as a computational algorithm for the solutions. The monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration, and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function. Some numerical results are presented to illustrate the monotone convergence of the iterative sequences and the rate of convergence of the iterations.

We find necessary and sufficient conditions for the curvature of a quadratic rational Bezier curve to be monotone in [0, 1], to have a unique local minimum, to have a unique local maximum, and to have both extrema in (0, 1), and we also visualize them in figures. As an application, we present a necessary and sufficient condition for the offset curve to be regular and to have the same tangent direction with the given quadratic rational Bezier curve, and give a simple algorithm to find it.

A new concept of a pair of upper and lower solutions is introduced for a class of discrete boundary value problems without monotone nonlinearities. Some comparison results are established. An existence and enclosing theorem for the solutions is given in terms of upper and lower solutions. A new monotone iterative scheme is proposed. The numerical example is given.

A new comparison theorem on the monotone convergence rates of the parallel nonlinear multisplitting accelerated overrelaxation (AOR) method for solving the large scale nonlinear complementarity problem is established. Thus, the monotone convergence theory of this class of method is completed.

The improvement in convergence by means of accurate functional elimination in the context of the monotone Newton theorem is further analyzed and extended to discrete approximations of the Newton method. © 1995. Fil:Milaszewicz, J.P. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.

We construct discrete interpolating polynomials, provide explicit representations of discrete Green's functions, give several identities and inequalities for these Green's functions, use the explicit forms of the interpolating polynomials and that of Green's functions to establish several maximum principles. Further, we obtain error bounds in discrete polynomial interpolation and use them to study existence and uniqueness of the discrete boundary value problems. These bounds are also used to provide sufficient conditions for the convergence of the Picard's method, the approximate Picard's method, quasilinearization and the approximate quasilinearization. The monotone convergence of the Picard's iterative method is also analysed.