Algorithm for solving a new class of general mixed variational inequalities in Banach spaces

In this paper, a new concept of η-proximal mapping for a proper subdifferentiable functional (which may not be convex) on a Banach space is introduced. An existence and Lipschitz continuity of the η-proximal mapping are proved. By using properties of the η-proximal mapping, a new class of general mixed variational inequalities is introduced and studied in Banach spaces. An existence theorem of solutions is established and a new iterative algorithm for solving the general mixed variational inequality is suggested. A convergence criteria of the iterative sequence generated by the new algorithm is also given.     

Asymptotic constancy for a differential equation with multiple state-dependent delays

In this paper, we investigate the asymptotic behavior of solutions to a differential equation with multiple state-dependent delays. It is shown that every bounded solution of such an equation tends to a constant as t→∞. Our results improve and extend some corresponding ones already known.     

Gale duality and Koszul duality

Given a hyperplane arrangement in an affine space equipped with a linear functional, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories of our algebras, which are in many ways analogous to integral blocks of category O.     

A cone complementarity linearization approach to robust controller design for continuous-time piecewise linear systems with linear fractional uncertainties

This paper investigates the robust H_∞ control problem for uncertain continuous-time piecewise systems by using the piecewise continuous Lyapunov function. The uncertainties of the systems under consideration are expressed in a linear fractional form. A strict linear matrix inequality approach is developed to obtain stability condition and H_∞ performance. The H_∞ controller design problem is solved by exploiting the cone complementarity linearization (CCL) method, which can be cast into an iterative minimization problem subject to LMI constraints. Finally two examples are given to illustrate the application of the proposed approach.     

A new exclusion test for finding the global minimum

Exclusion algorithms have been used recently to find all solutions of a system of nonlinear equations or to find the global minimum of a function over a compact domain. These algorithms are based on a minimization condition that can be applied to each cell in the domain. In this paper, we consider Lipschitz functions of order α and give a new minimization condition for the exclusion algorithm. Furthermore, convergence and complexity results are presented for such algorithm.     

Successive iteration and positive symmetric solution for a Sturm–Liouville-like four-point boundary value problem with a -Laplacian operator

In this paper, we consider the following Sturm–Liouville-like four-point p-Laplacian boundary value problem with the nonlinear term f depending on the first-order derivative subject to the boundary conditions By using a monotone iterative technique, the existence of symmetric positive solutions and corresponding iterative schemes are obtained.     

Two new modified Gauss–Seidel methods for linear system with -matrices

In 2002, H. Kotakemori et al. proposed the modified Gauss–Seidel (MGS) method for solving the linear system with the preconditioner [H. Kotakemori, K. Harada, M. Morimoto, H. Niki, A comparison theorem for the iterative method with the preconditioner () J. Comput. Appl. Math. 145 (2002) 373–378]. Since this preconditioner is constructed by only the largest element on each row of the upper triangular part of the coefficient matrix, the preconditioning effect is not observed on the nth row. In the present paper, to deal with this drawback, we propose two new preconditioners. The convergence and comparison theorems of the modified Gauss–Seidel methods with these two preconditioners for solving the linear system are established. The convergence rates of the new proposed preconditioned methods are compared. In addition, numerical experiments are used to show the effectiveness of the new MGS methods.     

Existence of periodic solutions for a fourth-order -Laplacian equation with a deviating argument

In this paper, we study the existence of periodic solutions for a fourth-order p-Laplacian differential equation with a deviating argument as follows:

[φ_p(u^″(t))]^″+f(u^″(t))+g(u(t−τ(t)))=e(t).

Under various assumptions, the existence of periodic solutions of the problem is obtained by applying Mawhin’s continuation theorem.     

An iterative algorithm based on -proximal mappings for a system of generalized implicit variational inclusions in Banach spaces

In this paper, we give the notion of M-proximal mapping, an extension of P-proximal mapping given in [X.P. Ding, F.Q. Xia, A new class of completely generalized quasi-variational inclusions in Banach spaces, J. Comput. Appl. Math. 147 (2002) 369–383], for a nonconvex, proper, lower semicontinuous and subdifferentiable functional on Banach space and prove its existence and Lipschitz continuity. Further, we consider a system of generalized implicit variational inclusions in Banach spaces and show its equivalence with a system of implicit Wiener–Hopf equations using the concept of M-proximal mappings. Using this equivalence, we propose a new iterative algorithm for the system of generalized implicit variational inclusions. Furthermore, we prove the existence of solution of the system of generalized implicit variational inclusions and discuss the convergence and stability analysis of the iterative algorithm.     

Inverse eigenproblem for -symmetric matrices and their approximation

Let be a nontrivial involution, i.e., R=R⁻¹≠±I_n. We say that is R-symmetric if RGR=G. The set of all -symmetric matrices is denoted by . In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors in and a set of complex numbers , find a matrix such that and are, respectively, the eigenvalues and eigenvectors of A. We then consider the following approximation problem: Given an n×n matrix , find such that , where is the solution set of IEP and is the Frobenius norm. We provide an explicit formula for the best approximation solution by means of the canonical correlation decomposition.     

Highly connected coloured subgraphs via the regularity lemma

For integers , n≥k and r≥s, let m(n,r,s,k) be the largest (in order) k-connected component with at most s colours one can find in any r-colouring of the edges of the complete graph K_n on n vertices. Bollobás asked for the determination of m(n,r,s,k).Here, bounds are obtained in the cases s=1,2 and k=o(n), which extend results of Liu, Morris and Prince. Our techniques use Szemerédi’s Regularity Lemma for many colours.We shall also study a similar question for bipartite graphs.     

Shape preserving approximation on the real line with exponential weights

The well-known “Bernstein’s weighted problem” deals with the possibility of weighted approximation on the whole real line. In this paper, we show the possibility of k-monotone approximation on the real line with Freud’s weight .     

Reduction of the Gibbs phenomenon for smooth functions with jumps by the -algorithm

Recently, Brezinski has proposed to use Wynn's ε-algorithm in order to reduce the Gibbs phenomenon for partial Fourier sums of smooth functions with jumps, by displaying very convincing numerical experiments. In the present paper we derive analytic estimates for the error corresponding to a particular class of hypergeometric functions, and obtain the rate of column convergence for such functions, possibly perturbed by another sufficiently differentiable function. We also analyze the connection to Padé–Fourier and Padé–Chebyshev approximants, including those recently studied by Kaber and Maday.     

Characterizations and construction of Poisson/symplectic and symmetric multi-revolution implicit Runge–Kutta methods of high order

This paper deals with the characterizations and construction of Poisson/symplectic and (φ⁻¹)-symmetric implicit high-order multi-revolution Runge–Kutta methods (MRRKMs). The basic tool is a modified W-transformation based on quadrature formulas and orthogonal polynomials. Two sufficient conditions can be obtained under which MRRKMs are Poisson/symplectic or (φ⁻¹)-symmetric. We construct two classes of high order implicit MRRKMs by using these sufficient conditions. Our results can be considered as an extension of related results of the standard Runge–Kutta methods in some references.     

Stability and asymptotic stability of -methods for nonlinear stiff Volterra functional differential equations in Banach spaces

This paper is concerned with the stability and asymptotic stability of θ-methods for the initial value problems of nonlinear stiff Volterra functional differential equations in Banach spaces. A series of new stability and asymptotic stability results of θ-methods are obtained.     

Embedding fillings of contact 3-manifolds

In this survey article, we describe different ways of embedding fillings of contact 3-manifolds into closed symplectic 4-manifolds.     

Existence results for -point boundary value problem of second order ordinary differential equations

This study concerns the existence of positive solutions to the boundary value problemwhere ξ_i(0,1) with 0<ξ₁<ξ₂<<ξ_n-2<1, a_i, b_i[0,∞) with and . By applying the Krasnoselskii's fixed-point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of at least one positive solution or at least two positive solutions are established for the above general n-point boundary value problem.     

Second-order multiobjective symmetric duality with cone constraints

In this paper, we formulate Wolfe and Mond–Weir type second-order multiobjective symmetric dual problems over arbitrary cones. Weak, strong and converse duality theorems are established under η$\eta$-bonvexity/η$\eta$-pseudobonvexity assumptions. This work also removes several omissions in definitions, models and proofs for Wolfe type problems studied in Mishra [9]. Moreover, self-duality theorems for these pairs are obtained assuming the function involved to be skew symmetric.     

Anti-periodic solutions for a class of nonlinear th-order differential equations with delays

In this paper, we use the Leray–Schauder degree theory to establish new results on the existence and uniqueness of anti-periodic solutions for a class of nonlinear nth-order differential equations with delays of the form

x⁽ⁿ⁾(t)+f(t,x⁽ⁿ⁻¹⁾(t))+g(t,x(t−τ(t)))=e(t).