In this paper, we have established some new Hermite–Hadamard–Mercer type of inequalities by using $\kappa$–Riemann–Liouville fractional integrals. Moreover, we have derived two new integral identities as auxiliary results. From the applied identities as auxiliary results, we have obtained some new variants of Hermite–Hadamard–Mercer type via $\kappa$–Riemann–Liouville fractional integrals. Several special cases are deduced in detail and some know results are recaptured as well. In order to illustrate the efficiency of our main results, some applications regarding special means of positive real numbers and error estimations for the trapezoidal quadrature formula are provided as well.

In this paper, we consider a new fractional dynamical system for variational inequalities using the Wiener Hopf equations technique. We show that the fractional Wiener-Hopf dynamical system is exponentially stable and converges to its unique equilibrium point under some suitable conditions. We also discuss some special cases, which can be obtained from our main results.

In this paper, we aimed to investigate the error inequality of the open method, known as Euler-Maclaurin's inequality, which is similar to Simpson's rule. We intended to explore some novel Maclaurin-like inequalities involving functions having convexity properties. To further accomplish this task, we built an identity and demonstrated new inequalities. With the help of a new auxiliary result and some well-known ones, like Hölder's, the power mean, improved Hölder, improved power mean, convexity, and bounded features of the function, we obtained new bounds for Euler-Maclaurin's inequality. From an applicable perspective, we developed several intriguing applications of our results, which illustrated the relationship between the means of real numbers and the error bounds of quadrature schemes. We also included a graphical breakdown of our outcomes to demonstrate their validity. Additionally, we constructed a new iterative scheme for non-linear equations that is cubically convergent. Afterwards, we provided a comparative study between the proposed algorithm and standard methods. We also discussed the proposed algorithm's impact on the basins of attraction.