Abstract: We present converse theorems for the principal Lyapunov results concerning partial stability properties of general dynamical systems defined on metric spaces. We consider the general case of partial stability (y-stability) with respect to disturbances in all of the initial state variables x/sub 0/=(y/sub 0/, z/sub 0/) as well as the case of y-stability under arbitrary z-perturbations.

Abstract: In this paper a first order analytical system of difference equations is considered. For an asymptotically stable fixed point x 0 of the system a gradual approximation of the domain of attraction (DA) is presented in the case when the matrix of the linearized system in x 0 is a contraction. This technique is based on the gradual extension of the ”embryo” of an analytic function of several variables. The analytic function is a Lyapunov function whose natural domain of analyticity is the DA and which satisfies an iterative functional equation. The equation permits to establish an ”embryo” of the Lyapunov function and a first approximation of the DA. The ”embryo” is used for the determination of a new ”embryo” and a new part of the DA. In this way, computing new ”embryos” and new domains, the DA is gradually approximated. Numerical examples are given for polynomial systems.

Abstract: The paper deals with the dynamical analysis of a non-linear model of detritus based ecosystem involving detritivores and predator of detritivores. We have first derived the conditions for the existence of steady-states and then discussed with the boundedness and persistence of the model system. Next we have dealt with the condition for local stability and Hopf-bifurcation. We have then considered the effect of detritus-diffusion on the stability and bifurcation property of the model system.

Abstract: In this paper we modify an eco-epdemiological model proposed by Chattopadhyay and Bairagi ( Ecological Modelling, 136, pp103-112, 2001) by taking into account non-linear incidence rate instead of bilinear incidenc rate as considered by Chattopadhyay and Bairagi We observe that limit cycle oscillations around the positive interior equilibrium may be controlled by suitable choice of non-linear incidence rates The value for which the disease will be spread and also the system around positive equilibrium enters into Hopf-bifurcation are obtained It is also observed that the bifurcating branches are super-critical To subtantiate the analytical finding numerical simulations are also performed for a hypothetical set of parameter values

Abstract: By means of the Riccati transformation techniques, we will establish some new oscillation criteria for hyperbolic nonlinear neutral delay difference equations. Our results can be considered as the discrete analogues of the Kamenev-type and Philos-type oscillation criteria.

Abstract: We extend the generalized quasilinearization method of Lakshmikantham for an initial value problem for an ordinary differential equation to the case of a sum of two right hand functions. The first member will satisfy a convexity condition proposed by Lakshmikantham, while the second member will satisfy a corresponding weaker form of convexity condition. Monotone and quadratic convergence are preserved.

Abstract: We present a homotopic invariance result for set valued generalized contractive maps with closed values. In addition we present a fuzzy version of this result.

Abstract: In this work we consider an initial one-point boundary value problem to the heat equation with the Bessel operator ut − (uxx + 1 xux) = |u| p−2u. We first prove a local existence result. Then we show that the solution blows up in finite time.

Abstract: Ekeland's Variational Principle is used in the framework of Orlicz- Sobolev spaces to obtain the existence and uniqueness of the weak solution to a class of nonlinear elliptic boundary value problems with slowly increasing non- polynomial coecient functions. Dependence of the solution on the data is also examined.