This paper is concerned with the asymptotic properties of solutions of third-order nonlinear dynamic equations on time scales. Some sufficient conditions for oscillation and nonoscillation of solutions as well as the boundedness of the solutions are established.

The objective of this article is to solve the unconstrained optimization problem by using Marquardt method together with MQ-N (modified quasi-Newton) method. The Hessian matrix will be computed numerically by using modified Broyden-Flechert-Goldfarb-Shanno (BFGS) update ( H-version) after convert it to the B-version by using Sherman-Morisson-Woodburge (SMW) formula which guarantee the two important properties SPD (symmetric and positive definite), and hence the exact second derivative of OF (objective function) does not be needed to compute. The line search technique is very important to accelerate the method to terminate at the minimum value of OF, so in this article the line search technique is instead by used the search technique of Marquardt method together with MQ-N method (especially modified BFGS method where the step size is equal one) to solve the unconstrained optimization problem. Test problems are solved by Matlab software to prove the effective of this new technique so called the Marquardt extended technique (MET).

In this research we introduced a new update of the Hessian matrix or we updating only the diagonal elements of Hessian matrix, and make the non-diagonal elements always equal to zero and in this case we can preserve the sparse property so called the Diagonal Update.

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