Abstract: In this paper, we establish sufficient conditions for the existence of solutions of nonlinear functional integral equations, via a fixed point theorem of Krasnoselskii-Schaefer type.

Abstract: In this paper an autonomous analytical system of ordinary differential equations is considered For an asymptotically stable steady state 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 diagonalizable This technique is based on the gradual extension of the "embryo" of an analytic function of several complex variables The analytic function is the transformed of a Lyapunov func- tion whose natural domain of analyticity is the DA and which satisfies a linear non-homogeneous partial differential equation The equation permits to establish an "embryo" of the transformed function and a first approximation of 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 grad- ually approximated Numerical examples are given for polynomial systems For systems considered recently in the literature the results are compared with those obtained with other methods

Abstract: An abstract unified theory of both monotone iterative and generalized quasilinearization methods is presented for operator equations of coincidence type in ordered Banach spaces. Applications are given for semilinear problems in $C(\overline{\Omega};\mathbb{R}^k)$ and $L^p(\Omega;\methbb{R}^k)$.

Abstract: In this paper we give weaker conditions for well-posedness of impulsive problems for nonlinear parabolic equations, including existence, uniqueness and continuous dependence, even in case their corresponding Cauchy problems do not have global solutions. With some concrete examples of PDEs we show when their impulsive problems are well-posed and how the well-posedness is related to life-span of PDEs. We also indicate that some assumptions in [6] and [9] are unnecessary.

Abstract: We present results for partial stability of invariant sets and boundedness of motions for discontinuous dynamical systems (DDS) defined on metric space. We first establish a general comparison theory for DDS using stability preserving mappings. Next, we specialize these results by utilizing in particular vector Lyapunov functions as stability preserving mappings. For the scalar case, these results reduce to the Principal Lyapunov Results for partial stability and boundedness of general motions of DDS defined on metric space. We demonstrate the applicability of our results by considering a special class of finite dimensional dynamical systems subjected to impulse effects. We show that for this particular class of DDS, our results are less conservative than existing results. The present results are applicable to a much larger class of DDS than existing results on partial stability and boundedness (including to DDS that cannot be characterized by the usual classical equations and inequalities). Also, in contrast to existing results which pertain primarily to the analysis of equilibria, the present results apply to invariant sets of DDS (including equilibria as a special case). Finally, some of the present results are less conservative than existing ones. The results of the present paper are in the same spirit as our earlier work concerning the partial stability of invariant sets and boundedness of motions, for continuous dynamical systems defined on metric space.

Abstract: The objective of this paper is to give a control technique for the constant-control rolling maneuver. For this purpose, the real steady states of the equations of motion of a flying vehicle are organized in branches. Using Liapunov stability and non stability theorems as well as the stable-manifold theorem, the real steady states of a branch are classified in asymptotically stable, partially asymptotically stable (i.e. for the linearized system in this point there exist eigenvalues with negative real part as well as eigenvalues with positive real part) and repulsive steady states. Constant-control rolling maneuvers are found which lead the vehicle from a real steady state to an asymptotically stable or partially asymptotically stable real steady state. For the "Automatic Flight Experiment" (ALFLEX) model plane [3]-[4], which is a reduced scale model of an unmanned orbiting spacecraft and has the same magnitude of $i_3$ as that of the F100A fighter, numerical results and maneuver simulations are presented.

Abstract: The aim of the present paper is to establish a stability result for the so called Mountain Pass type solutions of the following class of semilinear elliptic variational inequalities (Pn)  un ∈ H 0 (Ω), un ≤ ψn in Ω 〈Anun, v − un〉 − λ ∫ Ω un(x)(v − un)(x)dx ≥ ∫ Ω pn(x, un(x))(v − un)(x)dx ∀v ∈ H 0 (Ω), v ≤ ψn in Ω, where Ω is an open bounded subset of R (N ≥ 1) with a sufficiently smooth boundary and λ is a real parameter. Moreover, for any n ∈ N, An is a uniformly elliptic operator, ψn belongs to H(Ω), (ψn)|∂Ω ≥ 0 and pn is a continuous real function which satisfies some general superlinear and subcritical growth conditions at zero and at infinity.

Abstract: In this paper we introduce the notion of controlled modulus condition as a generalization of the quasiextremal distance (or QED) condition introduced by Gehring and Martio We show that domains satisfying the controlled modulus condition enjoy a lot of properties that QED domains have, such as the linear local connectivity and the extendability and Lipschitz continuity of quasiconformal maps defined on such domains On the other hand, we also establish some properties for domains with controlled modulus, such as the quasimobius invariance, the controlled modulus condition But the converse remains open We also prove that the controlled modulus condition is equivalent to the Loewner condition recently introduced by Heinonen and Koskela in their study of quasiconformality in general metric spaces

Abstract: In this paper we give sufficient conditions for the existence of nonnegative solutions of the periodic problem for first order ordinary differential equations with impulses at fixed moments. For some of the results our method of proof makes use of Landesman-Lazer type conditions.

Abstract: Via the direct method of Liapunov, this paper presents a convergence criterion for Hopfield-type artificial neural networks with time-varying external inputs. Also, in the presence of such inputs, it is shown, via the concept of practical stability, that the boundedness of the neuron activation functions is all that is required to ensure boundedness of solutions.

Abstract: In this paper sufficient conditions for the existence of almost periodic solutions of differential equations with perturbations on the linear part are obtained. The impulses take place at fixed moments.

Abstract: We consider the stochastic differential equation $$ du = \alpha (u, \xi)dt + \sigma (u) dp \tag{1}$$ on a separable Banach space. We give sufficient conditions for the existence of an invariant measure for the semigroup $ \{ P^t \}_{t \ge 0} $ corresponding to the stochastic differential equation (1). We show that the existence of an invariant measure for a Markov operator $\overline{P} $ corresponding to the change of measures from jump to jump implies the existence of an invariant measure for the semigroup $ \{ P^t \}_{t \ge 0} $ describing the evolution of measures along trajectories.

Abstract: Consider a full second order nonlinear scalar differential equations where the nonlinearities is negative, associated to some linear Sturm-Liouville boundary conditions with their coefficients not always positive. Existence results of positive (and monotonous at some cases) solutions of above BVPs are given, under superlinear and/or sublinear growth in f. The approach is based on an analysis of the corresponding vector field on the face-plane and Kneser's property of solutions funnel.

Abstract: This paper presents a Newton-type iterative scheme for finding the zero of the sum of a differentiable function and a multivalued maximal monotone function. Local and semi-local convergence results are proved for the Newton scheme, and an analogue of the Kantorovich theorem is proved for the associated modified scheme that uses only one Jacobian evaluation for the entire iteration. Applications in variational inequalities are discussed, and an illustrative numerical example is given. Abbreviated title: Generalized Newton method. Mathematics Subject Classification (1991): 47H15, 65H10, 65K05, 65K10, 49J40, 47H19

Abstract: The global uniform exponential stability independent of delay (g.u.e.s.i.d.) is investigated for a wide class of time-delay systems that may involve both point and distributed delays on finite intervals as well as infinitely distributed Volterra integro-differential dynamics. The stability problem is considered as a robust stability one with respect to an auxiliary system which may be defined very freely. The proposed method allows a very important generalisation related to the usual problem statement in the literature when the auxiliary system is defined by deleting the whole delayed dynamics. Conditions are established that ensure that the Laplace operator characterising the system has a bounded inverse on the closed complex right-half plane. The analysis is slightly modified for investigating uniform stability dependent of delay.