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The main objective of this paper is to present a reproducing kernel Hilbert space method for computing solutions of nonlinear third-order ordinary differential equations under multipoint boundary conditions. Some other aspects of the reproducing kernel Hilbert space method, such as convergence analysis and error estimate, are also discussed. The proposed method is a well-performance technique for calculating the best approximate solution of nonlinear boundary value problems. Our numerical experiments validate our theoretical findings and demonstrate the desirable performance of the algorithm in terms of simplicity, accuracy, and efficiency.

The dynamical behavior of the Morse oscillator is investigated primarily by means of the Lyapunov exponent and bifurcation diagrams. Then, the problem of controlling chaos for this oscillator is studied using a new method introduced by Behnia and Akhshani, which is based on the construction of slave-master feedback. In the control model based on slave-master feedback, the oscillator as the slave system is coupled with a dynamical system as the master, so its implementation becomes quite simple and similar statements can be made for the high dimensional cases. The validity of this method is veried by numerical simulations. The obtained results show the eectiveness of the proposed control model.

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In this paper we introduce ω-proximal quasi contraction mapping and best ωproximity point in modular metric spaces. In fact, we show that every ω-proximal quasi contraction mapping has unique best ω-proximity point in modular metric spaces. Finally, we give an example to illustrate the applications of our results. MSC(2010): 47H09; 46A80; 46S50.