Solving non-linear functional equations by relaxed new iterative method

For solving various equations of the form $v=f+N (v)$, the new iterative method and the new algorithm proposed by V. Daftardar–Gejji et al. [Daftardar–Gejji V., Jafari H. J. Math. Anal. Appl. 316 (2), 753–763 (2006); Kumar M., Jhinga A., Daftardar–Gejji V. Int. J. Appl. Comp. Math. 6 (2), 26 (2020)] are been employed successfully and accurately.  Our aim in this paper is to present a relaxed new iterative method by introducing a controlled parameter $\omega$ in order to extend these methods.  According to the values of the parameter $\omega$, we discuss and provide the convergence analysis.  The proposed algorithm is fast, effective and simple to implement as compared to the existing one.  Numerous non-linear equations are solved to show the applicability and efficiency of the algorithm compared to the other methods.