We consider a Clenshaw–Curtis–Filon-type method for highly oscillatory Bessel transforms. It is based on a special Hermite interpolation polynomial at the Clenshaw–Curtis points that can be efficiently evaluated using $O(NlogN)$ operations, where N is the number of Clenshaw–Curtis points in the interval of integration. Moreover, we derive corresponding error bounds in terms of the frequency and the approximating polynomial. We then show that this method yields an efficient numerical approximation scheme for a class of Volterra integral equations containing highly oscillatory Bessel kernels (a problem for which standard numerical methods fail), and it also allows the study of the asymptotics of the solutions. Numerical examples are used to illustrate the efficiency and accuracy of the Clenshaw–Curtis–Filon-type method for approximating these highly oscillatory integrals and integral equations.

This paper proposes two algorithms that are based on a subgradient and an inertial scheme with the explicit iterative method for solving pseudomonotone equilibrium problems. The weak convergence of both algorithms is well-established under standard assumptions on the cost bifunction. The advantage of these algorithms is that they did not require any line search procedure or any knowledge about bifunction Lipschitz-type constants for step-size evaluation. A practical explanation of this is that they use a sequence of step-size that are revised at each iteration based on some previous iteration. For a numerical experiment, we consider a well-known Nash-Cournot equilibrium model of electricity markets and also other test problems to assist the well-established convergence results and be able to see that our proposed algorithms have a competitive advantage over the time of execution and the number of iterations.