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In this paper, we propose a new modified subgradient extragradient method for solving equilibrium problems involving pseudomonotone and Lipchitz-type bifunctions in Hilbert spaces. We establish the strong convergence of the proposed method under several suitable conditions. In addition, the linear convergence is obained under strong pseudomonotonicity assumption. Our results generalize and extend some related results in the literature. Finally, we provide numerical experiments to illustrate the performance of the proposed algorithm.

In this article, we introduce a general splitting method with linearization to solve the split feasibility problem and propose a way of selecting the stepsizes such that the implementation of the method does not need any prior information about the operator norm. We present the constant and adaptive relaxation parameters, and the latter is “optimal” in theory. These ways of selecting stepsizes and relaxation parameters are also practised to the relaxed splitting method with linearization where the two closed convex sets are both level sets of convex functions. The weak convergence of two proposed methods is established under standard conditions and the linear convergence of the general splitting method with linearization is analyzed. The numerical examples are presented to illustrate the advantage of our methods by comparing with other methods.

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.

In this paper, we introduce and solve the following additive (ρ1,ρ2)-functional inequalities $$\matrix{{\left\| {f(x + y + z) - f(x) - f(y) - f(z)} \right\|} \hfill \cr {\;\;\;\;\;\; \le \left\| {{\rho _1}(f(x + z) - f(x) - f(z))} \right\| + \left\| {{\rho _2}(f(y + z) - f(y) - f(z))} \right\|,} \hfill \cr} $$ where ρ1 and ρ2 are fixed nonzero complex numbers with ∣ρ1∣ + ∣ρ2∣ < 2. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the above additive (ρ1,ρ2)-functional inequality in complex Banach spaces. Furthermore, we prove the Hyers-Ulam stability of hom-derivations in C*-ternary algebras.

In the present paper, we establish the quantitative estimates in terms of weighted modulus of continuity, for the differences of Baskakov operators with Baskakov–Szasz operators, Baskakov–Kantorovich operators and Genuine Baskakov–Durrmeyer type operators. Also, we find the estimates for the mutual differences of these operators. Finally, we provide an estimate for the difference of operators by direct method.

A more accurate half-discrete Hilbert-type inequality in the whole plane with multi-parameters is established by the use of Hermite–Hadamard’s inequality and weight functions. Furthermore, some equivalent forms and some special types of inequalities and operator representations as well as reverses are considered.

By the use of weight coefficients and techniques of real analysis, we establish a new Hardy–Mulholland-type inequality with a mixed kernel and a best possible constant factor in terms of the hypergeometric function. Equivalent forms, an operator expression with the norm and reverses are also considered.

Using weight functions, we establish a few equivalent statements of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel in the whole plane. The constant factors related to the extended Hurwitz-zeta function are proved to be the best possible. In the form of applications, we deduce some special cases involving homogeneous kernel. We additionally consider some particular inequalities and operator expressions.

In the present paper by applying the series method we prove the Hyers–Ulam stability of the homogeneous hypergeometric differential equation in a subclass of analytic functions.

By the use of the methods of real analysis and the weight functions, a few equivalent conditions of a Hilbert-type integral inequality with the nonhomogeneous kernel in the whole plane are obtained. The best possible constant factor is related to the extended Riemann zeta function. As applications, a few equivalent conditions of a Hilbert-type integral inequality with the homogeneous kernel in the whole plane are deduced. We also consider the operator expressions and a few particular cases.

By using techniques of real analysis and weight functions, a few equivalent statements of a Hilbert-type integral inequality with the nonhomogeneous kernel in the whole plane are obtained. The constant factor related the extended Hurwitz zeta function is proved to be the best possible. As applications, a few equivalent statements of a Hilbert-type integral inequality with the homogeneous kernel in the whole plane are deduced. We also consider the operator expressions and some corollaries.

We consider cotangent sums associated to the zeros of the Estermann zeta function considered by the authors in their previous paper (Maier and Rassias, Generalizations of a cotangent sum associated to the Estermann zeta function, 2014). We settle a question on the rate of growth of the moments of these cotangent sums left open in Maier and Rassias (Generalizations of a cotangent sum associated to the Estermann zeta function, 2014), and obtain a simpler proof of the equidistribution of these sums.

In this paper we consider properties and power expressions of the functions $$f:(-1,1)\rightarrow \mathbb{R }$$ and $$f_L:(-1,1)\rightarrow \mathbb{R }$$ , defined by $$\begin{aligned} f(x;\gamma )=\frac{1}{\pi }\int \limits _{-1}^1 \frac{(1+xt)^\gamma }{\sqrt{1-t^2}}\,\mathrm{d}t \quad \text{ and}\quad f_L(x;\gamma )=\frac{1}{\pi }\int \limits _{-1}^1 \frac{(1+xt)^\gamma \log (1+x t)}{\sqrt{1-t^2}}\,\mathrm{d}t, \end{aligned}$$ respectively, where $$\gamma $$ is a real parameter, as well as some properties of a two parametric real-valued function $$D(\,\cdot \,;\alpha ,\beta ) :(-1,1) \rightarrow \mathbb{R }$$ , defined by $$\begin{aligned} D(x;\alpha ,\beta )= f(x;\beta )f(x;-\alpha -1)- f(x;-\alpha )f(x;\beta -1),\quad \alpha ,\beta \in \mathbb{R }. \end{aligned}$$ The inequality of Turan type $$\begin{aligned} D(x;\alpha ,\beta )>0,\quad -1 0$$ is proved, as well as an opposite inequality if $$\alpha +\beta