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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.

We consider the second-order advanced functional difference equation Δ(a(n)Δχ(n)) + p(n)χ(g(n)) = 0 , where a ( n ) > 0, ∑ ∞ s=n 0 ( 1 a(s) ) = ∞ , p ( n ) ≥ 0, p ( n ) ≡ 0, g ( n ) ≥ n + 1, { g ( n )} is a monotone increasing integer sequence. We obtain some new oscillation criteria through an appropriate Riccati equation.

In this paper, we present a self-adaptive projection and contraction (SAPC) method for solving symmetric linear variational inequalities. Preliminary numerical tests show that the proposed method is efficient and effective and depends only slightly on its initial parameter. The global convergence of the new method is also addressed.

A new comparison theorem about the parallel nonlinear AOR method [1] is set up, which describes in detail the influence of either the multisplitting of the coefficient matrix or the pair of the relaxation parameters on the convergence rate of this method.

For difference equations which satisfy a strict monotonicity property a comparison principle is shown. This is used to prove relative stability for all solutions which implies, in particular, that all solutions must converge to a limit provided there is at least one bounded nonzero solution.

Using the method of upper and lower solutions and its associated monotone iterative, we present an existence theorem for a nonlinear fractional differential equation with nonlinear boundary conditions.

In this paper, we obtain a successively iterative scheme of positive solution for the nonlinear third-order two-point boundary value problem u‴+q(u″)f(t,u)=0,u(0)=A,u(1)=B,u″(0)=C, where f is a Carathéodory function, f and q satisfy some additional monotone conditions. The iterative scheme starts off with zero function and is therefore useful for computation purpose. The main tool is monotone iterative technique on Banach space. Moreover, the iterative scheme is independent of the existence of lower and upper solutions.

We consider monotone arithmetic circuits with restricted depths to compute monotone multivariate polynomials such as the elementary symmetric functions, convolution of several vectors and raising a matrix to a power. We develop general lower- and upper-bound techniques that seem to generate almost-matching bounds for all the functions considered. These results imply exponential lower bounds for circuits of bounded depths which compute any of these functions. We also obtain several examples for which negation can reduce the size exponentially.

This article presents algorithms of linear time complexity (O(n)) for computation of optimal solutions to the two problems of convex and monotone approximation where data points are approximated, respectively, by convex and monotone (nondecreasing) functions on a grid of (n + 1) points. For the convex approximation case, the algorithms are based on a linear programming approach which exploits the structure of matrices involved and uses a special pivoting procedure to obtain a “maximal” optimal solution. Analogously, in the monotone approximation case, the algorithms compute “maximal” and “minimal” optimal solutions which also “enclose” any other optimal solution between them. Computational results are presented.

In this paper, we introduce and study a new system of variational inclusions involving (H, η)-monotone operators in Hilbert space. Using the resolvent operator associated with (H, η)-monotone operators, we prove the existence and uniqueness of solutions for this new system of variational inclusions. We also construct a new algorithm for approximating the solution of this system and discuss the convergence of the sequence of iterates generated by the algorithm.

In this paper, we introduce iterative schemes based on the extragradient method for finding a common element of the set of solutions of a generalized mixed equilibrium problem and the set of fixed points of an infinite (a finite) family of nonexpansive mappings and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. We obtain some weak convergence theorems for the sequences generated by these processes in Hilbert spaces. The results in this paper generalize, extend and unify some well-known weak convergence theorems in the literature.

In this paper, we introduce a new concept of (A, η)-accretive mappings, which generalizes the existing monotone or accretive operators. We study some properties of (A, η)-accretive mappings and define resolvent operators associated with (A, η)-accretive mappings. By using the new resolvent operator technique, we also construct a new perturbed iterative algorithm with mixed errors for a class of nonlinear relaxed Cocoercive variational inclusions involving (A, η)-accretive mappings and study applications of (A, η)-accretive mappings to the approximation-solvability of this class of nonlinear relaxed Cocoercive variational inclusions in q-uniformly smooth Banach spaces. Our results improve and generalize the corresponding results of recent works.

In this paper we shall offer comparison results as well as monotone iterative schemes for the construction of solutions to a very general class of discrete boundary value problems. The discrete system we consider includes in particular the n th order prototype systems, finite as well as infinite discrete delay equations, and discrete integral equations. Further, the boundary conditions we consider include the initial, terminal, periodic and transport type problems. Numerical examples illustrating the usefulness of the proposed schemes to a variety of boundary value problems are also included.

A Boolean response to a random binary input of length n can be modeled as a {;0, 1}- valued function v defined on a discrete probability space Ω of all subsets of a finite set of size n. An ω ∈ Ω represents the locations of 1's in the input. For a particular jth location, 1 ≤ j ≤ n, we assume that 1 appears with probability ρj independently of other locations. Then, for ρ = (ρ1,…, ρn), we will investigate Pρ(v = 1) as a function of ρ. Using the sharp version of the Khinchin inequality, we give an upper estimate for the ℓ2 norm of the gradient of Pρ(v = 1) evaluated at ρ = (12,…, 12) (cf. (5) below). For monotone functions, the estimate applies also to vector of influences of Boolean functions. We also provide a handy expansion of P(·)(v = 1) based on a Fourier expansion of v (cf. (4) below).Numerical analysis of the bounds leads to the conjecture about the sharp bound that depends on cardinality of the underlying set; the sharp version of the Khinchin inequality is also conjectured.

This paper discusses anti-periodic boundary value problems of second order impulsive differential equations. By using the method of upper and lower solutions coupled with the monotone iterative technique, new existence results of coupled solutions and uniqueness of problems are obtained.

A parallel monotone iterative relaxation method for a class of two-dimensional discrete boundary value problems is established, and the sequence of iterations is shown to converge monotonically either from above or below to a solution of the problem. This monotone convergence results yields a parallel computational algorithm as well as an existence-comparison result for the solutions. To compute the sequence of iterations, the Thomas algorithm can be used in the same fashion as for one-dimensional problem. The existence and comparison results of the upper and lower solutions are given. The local as well as global existence-uniqueness of the solution are obtained. The global convergence of the iterations is investigated, and the influence of the parameters on the rate of convergence of the iterations is analyzed. Numerical results are given to corroborate the analytical results.

This paper is devoted to a regularization approach for variational inequalities. A general method based on the theory of linear continuous and linear compact operators is developed for the stable approximation of ill-posed variational inequalities. Variational inequalities are considered for noncoercive monotone and pseudo-monotone operators. The approach is shown to be successful in handling variational inequalities formulated in reflexive, as well as in nonreflexive spaces.

This paper is motivated by recent interests in space discrete Nagumo equations and is concerned with the existence of solutions of a nonlinear discrete boundary value problem. Monotone methods are used to derive the existence theorems. These methods, as is well known, provide constructive schemes for calculating the solutions.

There are considered elliptic and parabolic equations of arbitrary dimension with alternating coefficients at mixed derivatives. For such equations, monotone difference schemes of the second order of local approximation are constructed. Schemes suggested satisfy the principle of maximum. A priori estimates of stability in the norm C without limitation on the grid steps τ and h α, α = 1,2,…, p are obtained (unconditional stability).

A new relaxed algorithmic procedure based on the notion of A-maximal relaxed monotonicity is developed, and then the convergence analysis in the context of solving a general class of nonlinear inclusion problems is examined. Furthermore, some results involving A-maximal relaxed monotone mappings in a Hilbert space setting are included.

A general design for the Eckstein–Bertsekas proximal point algorithm, using the notion of the A-maximal monotonicity, is developed. Convergence analysis for the generalized Eckstein–Bertsekas proximal point algorithm in the context of solving a class of nonlinear inclusion problems is explored. Some auxiliary results of interest involving A-maximal monotone mappings are also included. The obtained results generalize investigations on general maximal monotonicity and beyond.

This paper is concerned with the existence and uniqueness of monotone positive solutions for an elastic beam equation with nonlinear boundary conditions. The proof of our main results is based upon a new fixed point theorem of generalized concave operators.

We consider and analyze some new projection-splitting algorithms for solving monotone variational inequalities by using the technique of updating the solution. Our modification is in the spirit of the extragradient method. The modified methods converge for monotone continuous operators. The new iterative method differs from the existing projection methods.

This paper deals with the numerical analysis and simulation of nonlinear Black–Scholes equations modeling illiquid markets where the implementation of a dynamic hedging strategy affects the price process of the underlying asset. A monotone difference scheme ensuring nonnegative numerical solutions and avoiding unsuitable oscillations is proposed. Stability properties and consistency of the scheme are studied and numerical simulations involving changes in the market liquidity parameter are included.

In Bor (2011) [5], we proved a main theorem dealing with absolute Nörlund summability factors of infinite series. In the present paper, this result has been further generalized under weaker conditions by using δ-quasi-monotone sequences

A nonlinear difference equation of the order n+m in two independent variables, ΔnxΔmyu(x, y, 〈 u 〉), together with appropriate initial and periodic boundary conditions, is considered. Several comparison results for these problems are established. These results are then used to provide sufficient conditions for the convergence of monotone iterative methods. Numerical illustrations dwelling upon the importance of our results are also included.

The authors consider the quasilinear difference equation δ(p n−1 (δy n−1 )α)=q n y β n +r n and obtain results on the asymptotic behavior of solutions of (∗) including sufficient conditions for all solutions to be bounded or unbounded. Some results on the existence and behavior of nonincreasing solutions of (∗) are also obtained. Examples are inserted to illustrate the results.

In this paper, we are concerned with a two-competition model described by a reaction–diffusion system with nonlocal delays which account for the drift of individuals to their present position from their possible positions at previous times. By using the iterative technique recently developed in Wang et al. (2006) [14], the sufficient conditions are established for the existence of travelling wave solutions connecting the semi-trivial steady state to the coexistence steady state of the considered system. When the domain is bounded, we investigate the global attractivity of the coexistence steady state of the system under homogeneous Neumann boundary conditions as well. The approach used is the upper–lower solutions and monotone iteration technique.

This paper is concerned with the existence of extreme solutions of multi-point boundary value problem for a class of second-order functional differential equations. We introduce a new concept of lower and upper solutions. By using the method of upper and lower solutions and monotone iterative technique, we obtain the existence of extreme solutions.

The approximate controllability for the nonlinear control system with nonlinear monotone hemicontinuous and the nonlinear perturbations is studied. The existence, uniqueness, and a variation of solutions of the system are also given.

In this paper, we propose some improvements on a new gradient-type method for solving large-scale unconstrained optimization problems, in which we use data from two previous steps to revise the current approximate Hessian. The new method which we considered, resembles to that of Barzilai and Borwein (BB) method. The innovation features of this approach consist in using approximation of the Hessian in diagonal matrix form based on the modified weak secant equation rather than the multiple of the identity matrix in the BB method. Using this approach, we can obtain a higher order accuracy of Hessian approximation when compares to other existing BB-type method. By incorporating a simple monotone strategy, the global convergence of the new method is achieved. Practical insights into the effectiveness of the proposed method are given by numerical comparison with the BB method and its variant.

In this note, we show some new inequalities of the Huygens type for circular functions, hyperbolic functions, and the reciprocals of circular and hyperbolic functions by using a monotone form of l’Hospital’s rule.