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The article presents a study on the hyperbolic singular perturbation problems for integrodifferential equations. The authors have obtained new convergence theorems for singular perturbation problems, which generalize some results by H. O. Fattorini and J. Liu before. The idea of hyperbolic singular perturbation problem came from Fattorini's work, wherein the inhomogenous hyperbolic perturbation problem emerge from problems of traffic flow. Some basic assumptions and results of Fattorin's work was used in the study.

We obtain an asymptotic formula for the mean value of the function τ( n), which is the number of solutions of the equation x ... x = n in natural numbers x, ..., x, in some special sequences of natural numbers. [ABSTRACT FROM AUTHOR]

The article examines the existence of solutions for weighted p(x)-Laplacian ordinary boundary value problem. The method used in this study is based on Leray-Schauder degree. As an application, the existence of weak solutions for p(x)-Laplacian partial differential equations is given. On the p(x)-Laplacian problems, there is a possibility that it does not have the first eigenvalue and the first eigenfunction. Several sub-super-solution theorems for the existence of solutions for weighted p(x)-Laplacian equation with Dirichlet, Robin, and Periodic boundary value conditions are established.

The article analyzes the boundary asymptotic behavior of solutions for weighted p(x)-Laplacian equations that take infinite value on a bounded domain. It was found that the boundary asymptotic of solutions of the Laplacian function is continuous and positive. The p(x)-Laplacian possesses more complicated inhomogeneous nonlinearities, thus, some special techniques are needed. Its main difficulty arises from the lack of compactness. The uniqueness and asymptotic behavior of solutions for problem is a nonnegative function which is allowed to vanish on the boundary.

The article presents a study on the mathematical equation of two-dimensional linear differential systems with deviating arguments using Kneser type solutions. It presents an equation with positive coefficient and two linearly independent solutions. The article also evaluates the existence and uniqueness of these types of solutions. It proves that Kneser type solution is equivalent to nonvanishing of the Wronskian fundamental system and positivity of Green's function of the problem.

The article presents a research on the discontinuous quasilinear elliptic variational-hemivariational inequalities which involves the ρ-Laplacian. It also notes on the extension of the theory for discontinuous problems with the use of the method of sub- and supersolutions and compared on the results of S. Carl. It also aims to prove the existence of extremal solutions within a given order interval of sub- and supersolutions. It also gives example of the construction of sub- and supersolutions.

The article presents a research on the iterative methods for generalized von Foerster equations with functional dependence. It examines the meeting of a natural iterative method to the exact solution of a differential-functional von Foerster-type equation that notes a single population depending on its total size, past time and states densities. It also assumes either Perron comparison conditions and some monotonicity. It also investigates on the natural conditions which ensures convergence of iterative methods.