It is well known that general variational inequalities provide us with a unified, natural, novel and simple framework to study a wide class of unrelated problems, which arise in pure and applied sciences. In this paper, we present a number of new and known numerical techniques for solving general variational inequalities and equilibrium problems using various techniques including projection, Wiener-Hopf equations, dynamical systems, the auxiliary principle and the penalty function. General variational-like inequalities are introduced and investigated. Properties of higher order strongly general convex functions have been discussed. The auxiliary principle technique is used to suggest and analyze some iterative methods for solving higher order general variational inequalities. Some new classes of strongly exponentially general convex functions are introduced and discussed. Our proofs of convergence are very simple as compared with other methods. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems. Since the general variational inequalities include (quasi) variational inequalities and (quasi) implicit complementarity problems as special cases, these results continue to hold for these problems. Some numerical results are included to illustrate the efficiency of the proposed methods. Several open problems have been suggested for further research in these areas.

Abstract. In this paper we obtain a result on Hyers-Ulam stability of the linear functional equation in a single variable f (ϕ(x)) = g(x) · f (x) on a complete metric group.

Cotangent sums are associated to the zeros of the Estermann zeta function. They have also proven to be of importance in the Nyman-Beurling criterion for the Riemann Hypothesis. The main result of the paper is the proof of the existence of a unique positive measure µ on R, with respect to which certain normalized cotangent sums are equidistributed. Improvements as well as further generalizations of asymptotic formulas regarding the relevant cotangent sums are obtained. We also prove an asymptotic formula for a more general cotangent sum as well as asymptotic results for the moments of the cotangent sums under consideration. We also give an estimate for the rate of growth of the moments of order 2k, as a function of k.

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Abstract. We prove uniqueness theorems concerning the functional inequalities in connection with an n -dimensional cubic-quadratic-additive equationby applying the direct method.Mathematics subject classification (2010): 39B82, 39B52.

Given a vector spaceX, we investigate the solutionsf:R→Xof the linear functional equation of third orderfx=pfx-1+qfx-2+rf(x-3), which is strongly associated with a well-known identity for the Fibonacci numbers. Moreover, we prove the Hyers-Ulam stability of that equation.