This paper extends the notion of weak convergence in metric spaces to the case of S-metric spaces. Moreover, some results on the weak convergence of fixed point iterations of Banach's, Kannan's, Chatterjee's, Reich's, Hardy and Roger's types of contractions on S-metric spaces are obtained. In addition, an example is presented to demonstrate our primary result.

In the paper, we introduce a new class of noncyclic -contractions as a generalization of the class of noncyclic contractions which was first introduced in the paper [R. Esp´ınola, M. Gabeleh, On the structure of minimal sets of relatively nonexpansive mappings, Numerical Functional Analysis and Optimization 34 (8), 845-860, 2013] and study the existence, uniqueness and convergence of a fixed point for such class of noncyclic mapping in the framework of uniformly convex Banach spaces. We obtain existence results of the best proximity points for cyclic -contractions as a consequence of our main theorems.

Let y"=f(x,y,y') be a 2nd order ODE. By Cartan equivalence method, we will study the local equivalence problem under the transformations group of time-fixed coordinates. We are going to solve this problem by an applicable method which has been recognized by R. Gardner, and classify them.

Zero point problems of the sum of two monotone mappings and fixed point problems of a strictly pseudocontractive mapping are investigated. A strong convergence theorem for the common solutions of the problems is established in the framework of Hilbert spaces.

In this paper, a new smooth second order sliding mode control is proposed. This algorithm is a modified form of Super Twisting algorithm. The Super Twisting guarantees the asymptotic stability, but the finite time stability of proposed method is proved with introducing a new particular Lyapunov function. The Proposed algorithm which is able to control nonlinear systems with matched structured uncertainty, is able to guarantee the finite time stability. The main advantage of this second order sliding mode control is reaching to sliding surface with high precision without chattering in control signal. In simulation section, the proposed algorithm is compared with the boundary layer sliding mode control and then is applied to designing a finite time nonlinear guidance law that is robust with respect to target maneuvers. Simulation results show that the control input in this algorithm is smooth and has no chattering and by applying this method, sliding variables will converge to zero in a given desired finite time.