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This paper deals with a class of real-valued random-walk processes, observed over random epochs of time, that forms a delayed renewal process. The present model does not restrict this class to a merely monotone random walk, which is easier to analyze and find explicit form functional. The objective is to find the first passage of the process exiting a rectangular set and registering the value of the process at this time, thus generalizing past models where either the observed process was monotone or the first passage time reduced to the moment of the first drop. The joint transformation of the named random characteristics of the process are derived in a closed form. The paper concludes with examples, including numerical examples, demonstrating the use of the results as well as practical applications to finance.

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.

This paper presents a new class of projection and contraction methods for solving monotone variational inequality problems. The methods can be viewed as combinations of some existing projection and contraction methods and the method of shortest residuals, a special case of conjugate gradient methods for solving unconstrained nonlinear programming problems. Under mild assumptions, we show the global convergence of the methods. Some preliminary computational results are reported to show the efficiency of the methods.

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.

In the paper, we obtain the existence of monotone positive solutions and establish acorresponding iterative scheme for the following multipoint boundary value problem with p-Laplacian operator, (φp(u′))′(t)+q(t)f(t,u(t))=0,0

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.

This paper contains two contributions of the theory of T-monotone operators introduced by Chen. First, we prove a new fixed-point theorem for a discontinuous T-monotone mapping. After, we use this theory to obtain the solution of a classical continuous problem, for which the usual iterative methods fail. © 2004 Elsevier Ltd. All rights reserved.

Some new results are obtained for the bounded and monotone properties of solutions of second-order quasi-linear forced difference equations δ(P n −1(δY n −1) α )=q n Y n β +r n , n≥n 0 , where n 0 ϵ N = {0,1,2,3,…}, { p n } n = n 0 −1 ∞ is a positive sequence, { q n } n = n 0 ∞ is a nonnegative real sequence with q n ≢ 0, { r n } n = n 0 ∞ is a real sequence, and α and β are quotients of odd positive integers. Some errors in [1] are pointed out and addressed. Examples are given to illustrate the advantages of the new results.

Some sufficient conditions for the existence of fixed points of increasing operators in fuzzy real line R L are given. We also establish some coupled quasi-fixed-point theorems of mixed monotone operators in fuzzy real line R L.

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 is a survey of some recent developments of Numerov's method for solving nonlinear two-point boundary value problems. The survey consists of three different parts: the existence-uniqueness of a solution, computational algorithm for computing a solution, and some extensions of Numerov's method. The sufficient conditions for the existence and uniqueness of a solution are presented. Some of them are best possible. Various iterative methods are reviewed, including Picard's iterative method, modified Newton's iterative method, monotone iterative method, and accelerated monotone iterative method. In particular, two more direct monotone iterative methods are presented to save computational work. Each of these iterative methods not only gives a computational algorithm for computing a solution, but also leads to an existence (and uniqueness) theorem. The estimate on the rate of convergence of the iterative sequence is given. The extensions of Numerov's method to a coupled problem and a general problem are addressed. The numerical results are presented to validate the theoretical analysis.

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.

The augmented Lagrangian method (also referred to as an alternating direction method) solves a class of variational inequalities (VI) via solving a series of sub-VI problems. The method is effective whenever the subproblems can be solved efficiently. However, the subproblem to be solved in each iteration of the augmented Lagrangian method itself is still a VI problem. It is essentially as difficult as the original one, the only difference is that the dimension of the subproblems is lower. In this paper, we propose a new alternating direction method for solving a class of monotone variational inequalities. In each iteration, the method solves a convex quadratic programming with simple constrains and a well-conditioned system of nonlinear equations. For such ‘easier’ subproblems, existing efficient numerical softwares are applicable. The effectiveness of the proposed method is demonstrated with an illustrative example.

Let H be a real Hilbert space, K a nonempty closed convex subset of H and T : K → K a Lipschitz pseudocontraction with a nonempty fixed-point set. Weak and strong convergence theorems for the iterative approximation of fixed points of T are proved. Furthermore, if T : H → H is a Lipschitz monotone operator and f ∈ R(T), where R(T) denotes the range of T, weak and strong convergence theorems for the iterative approximation of solutions of the operator equation Tx = f are proved.

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.

In this paper, we suggest and analyze a new iterative method for solving general monotone variational inequalities by using the updating technique. The new method differs from the extragradient and other modified projection methods. The new method is versatile and easy to implement. Our proof of convergence is very simple.

In this paper, two existence theorems of solutions for a kind of quasi-variational inequalities with monotone type multivalued mapping are obtained.

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.

In this paper, we introduce a new class of generalized nonlinear implicit quasi-variational inclusions and prove its equivalence with a class of fixed point problems by making use of the properties of maximal monotone. We also prove the existence of solutions for this generalized nonlinear implicit quasi-variational inclusions and the convergence of iterative sequences generated by the perturbed algorithms.

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.

Comparison theorems and existence criteria are summarized and new ones are derived for the monotone solutions of a class of nonlinear difference equations which arises from Hardy's inequality.

Utilizing dynamic systems on time scales, the theory of monotone flows and fixed points is considered, which unifies the theory of continuous and discrete dynamic systems.

This paper investigates the relationship between approximation error and complexity. A variety of complexity measures are used, including: the number of alternating strictly monotone segments; computation time; and for piecewise linear approximations, the number of linear segments. The results apply to piecewise monotone functions and to finite maps from reals to reals, i.e., real data. We provide a theoretical framework expressing the exact relationship between approximation error and the number of alternating strictly monotone segments. We provide a linear-time algorithm taking an error bound as input and returning a minimal segmentation of the approximated function's domain such that there exists an approximation, alternatingly strictly monotone on the segments, with error less than the given bound. For real data, we provide a suboptimal tradeoff between approximation error and number of linear segments in piecewise linear approximation. The results are obtained by extending the theory of best piecewise monotone approximation to piecewise monotone functions, and by application of a new concept, scale-dependent monotonicity.

Triangulation of polygons is a classical problem in computational geometry. For an arbitrary polygon, the triangulation may depend on its shape. In this paper, we describe a new geometric classification of triangulations of a convex polygon and then derive expressions for counting the number of nonisomorphic triangulations in each class.

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.

Consider a nondecreasing nonnegative integrable function f on [0, 1]. Draw an independent sample ( X 1 , Y 1 ), …, ( X n , Y n ) of size n from the uniform distribution under f , and let N n be the number of records in the sample, where ( X i , Y i ) is a record when, for all j ≠ i , either X j > X i or Y j Y i . We study the dependence upon f of the constant C in the asymptotic formula E N n ∼ C √ n , and show that whenever ∫ 0 1 f ′ > 0, N n /E N n → 1 in probability. The results are related to the expected time analysis of algorithms for finding the collection of all records (i.e., the maximal layer).

This article considers a modification of the secretary problem with random number of applicants discussed by Presman and Sonin [1]. Our problem also allows both uncertainty of employment and restriction of offering chances, that is, an offer of acceptance is declined by the applicant with a fixed known probability 1 − p , (0≤ p ≤ 1) and the offering chances, before the decision maker gets one applicant, are at most M times. Section 2 gives the sufficient condition for the problem tto be monotone and the optimal strategy of the monotone problem. In Section 3 we give the examples for uniform distribution of random number of applicants.

The stochastic approximation method of Robbins and Monro, after some modifications, is shown to be a powerful technique for obtaining numerical solutions of deterministic problems. Both algebraic equations and two point boundary value problems in ordinary differential equations are solved. The convergence rate is shown to be reasonably fast and results are surprisingly accurate. Our proposed procedure is more general than the original approach. Deterministic problems with multiple roots can be solved whereas the original Robbins-Monro method is restricted to a monotone increasing regression function with a single root.

Methods of maximal monotone operators are used in order to study, from a general point of view, duality numerical algorithms for solving variational inequalities. With classical algorithms, such as Uzawa's method for the standard and augmented Lagrangian, this paper presents some new algorithms, which appear to have very good numerical performances.

Publisher Summary This chapter discusses the solution of dynamic linear rationalexpectations models. A number of techniques have been developed to deal with the problem of multiplicity of equilibria in linear dynamic rational expectations models. The behavior of K-step-ahead prediction error variance minimizing solutions of dynamic linear rational expectations models, when the driving process is serially correlated, has been explored. The properties of the sequence of K-period-ahead forecast error variance minimizing forward weights have been studied. It has been shown that under certain conditions the sequence is monotone increasing and bounded above by the Taylor minimum variance forward weight for the white noise case. The Taylor solution is obtained by minimizing unconditional variance, which is equivalent to minimizing the K-period-ahead prediction error variance.

Constructive methods by obtaining multiple solutions of first and second order nonlinear hyperbolic periodic boundary value problems are given. Appropriate partial differential inequalities are developed, and are then employed to derive the iterative schemes which yield the desired solutions. The advantage of these methods is that the solutions of the given nonlinear problems are constructed as uniform limits of monotone iterates. These iterates being solutions of linear problems, are explicitly computable.

In this note we derive optimal error estimates for finite element approximations of a restricted class of one-dimensional nonlinear elliptic problems. The operators involved are monotone and differentiable, and it is assumed that the solutions are smooth.

This paper establishes necessary and sufficient conditions for operators to be H-monotone. Based on these conditions, we introduce a new iterative algorithm for solving a class of variational inclusions. Strong convergence of this algorithm is established under appropriate assumptions on the parameters. Estimate of its convergence rate is also provided.

We set up a class of parallel nonlinear AOR method in the sense of matrix multi-splitting for solving the large scale system of nonlinear equations Ax + ϕ(x) = b with A ∈ L(Rn) nonsingular, b ∈ Rn and ϕ : Rn → Rn being continuously diagonal. The global as well as the monotone convergence of this method is proved.

In this paper, the authors prove some existence results for solutions for a new class of generalized bi-quasi-variational inequalities (GBQVI) for quasi-pseudo-monotone type I and strongly quasi-pseudo-monotone type I operators defined on non-compact sets in locally convex Hausdorff topological vector spaces. In obtaining these results on GBQVI for quasi-pseudo-monotone type I and strongly quasi-pseudo-monotone type I operators, we shall use Chowdhury and Tan’s generalized version of Ky Fan’s minimax inequality as the main tool.