For an asymptotically non-expansive self-mapping T, we will prove the strong convergence of {xn} defined by xn+1 = (1−an−bn)xn+anu+bnTnxn, xn+1 = anu+(1−an)Tnxn, whenever {bn}, {an}Ì (0,1) satisfy (C1) limn®¥ an=0, (C2) S¥n=0 =¥ (C3) limn®¥ Kn-1/an = 0 or (C4) S¥n=0 (kn-1)<+¥ application, we also establish the strong convergence of the viscosity approximation schemes with a contraction ¦ given by xn+1 = anf (xn) + (1 −an)Tnxn and xn+1 = (1 −an− bn)xn+an f(xn) +bnTnxn.

In [1], introduce a MODIFIED variational ITERATION method (MVIM) for a special kind of nonlinear differential equations. In this paper, we consider variational ITERATION method (VIM) and MVIM (introduced in [1]) to obtain exact solution to Fisher’s equation. It is shown in this paper the application of VIM to Fisher’s equation leads to calculation of unneeded terms and more time consumed in repeated calculations for series solutions. Therefore, we use MVIM for Fisher’s equation to overcome these disadvantages.

We prove a strong convergence theorem for the MODIFIED Noor ITERATIONs in the framework of CAT (0) spaces. Our results extend and improve the corresponding results of X. Qin, Y. Su and M. Shang, T.H. Kim and H.K. Xu and S. Saejung and some others.

An effective MODIFIED cation of the Picard ITERATION method (PIM) is presented for solving linear and nonlinear fractional optimal control problems (FOCP) in the Caputo sense. Here, the control function is first approximated by a finite series with unknown coefficients. Then the modi, ed PIM is utilized to simulate the resulting fractional equations. Finally, the unknown coefficients could be computed by ap-plying an optimization procedure. Some test examples are given to show the accuracy and validity of the method.

A universal approach by Laplace transform to the variational ITERATION method for fractional derivatives with the nonsingular kernel is presented; in particular, the Caputo-Fabrizio fractional derivative and the Atangana-Baleanu fractional derivative with the non-singular kernel is considered. The analysis elaborated for both non-singular kernel derivatives is shown the necessity of considering the MODIFIED Caputo-Fabrizio fractional derivative and the analogous modifications for the Atangana-Baleanu fractional derivative with non-singular Mittag-Leffler kernel in order to satisfy the initial conditions for some fractional differential equations.

In this article, a simple and new algorithm is proposed, namely the MODIFIED variational ITERATION algorithm-I (mVIA-I), for obtaining numerical solutions to different types of fifth-order Korteweg de-Vries (KdV) equations. In order to verify the precision, accuracy and stability of the mVIA-I method, generated numerical results are compared with the Laplace decomposition method, Adomian decomposition method, Homotopy perturbation transform method and the MODIFIED Adomian decomposition method. Comparison with the mentioned methods reveals that the mVIA-I is computationally attractive, exceptionally productive and achieves better accuracy than the others.