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a b s t r a c t We study the global behavior of the price dynamics in a commodity market governed by a balance between demand and supply. While the dependence of demand on price is considered instantaneous, the supply term contains a delay, leading to a delay–differential equation. A discrete model is naturally defined as a limit case of this equation. We provide a thorough study of the discrete case, and use these results to get new sufficient conditions for the global convergence of the solutions to the positive equilibrium in the continuous case. For when the equilibrium is unstable, we provide some bounds for the amplitude of the oscillations that are quite sharp when the delay is large.

In this paper we consider the highly nonlinear model in finance proposed by Ait-Sahalia [Y. Ait-Sahalia, Testing continuous-time models of the spot interest rate, Rev. Finan. Stud. 9 (2) (1996) 385–426]. Both the drift and diffusion coefficients in this model do not obey the classical linear growth condition. To overcome the difficulties due to the highly nonlinear coefficients, we develop several new techniques to study the analytical properties of the model including the positivity and boundedness. In particular, we show that the Euler–Maruyama approximate solutions converge to the true solution in probability. The convergence result justifies clearly that the Monte Carlo simulations based on the Euler–Maruyama scheme can be used to compute the expected payoff of financial products e.g. options.

An R-order bound for the Halley method is obtained in this work, where an analysis of the convergence of the method is also presented under mild differentiability conditions. To do this, a new technique is developed, where the involved operator must satisfy some recurrence relations. © 2004 Elsevier Inc. All rights reserved.

In this paper, the inequality estimates of Bernstein basis functions and Meyer–Konig and Zeller basis functions are studied. Exact bounds for these two basis functions are obtained. Moreover, some application results of the new estimates in estimating the rate of convergence of Durrmeyer operators and Meyer–Koning and Zeller operators for functions of bounded variation are also given.

We introduce an iterative method for finding a common element of the set of solutions of an equilibrium problem and of the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem.

The paper describes a modification to the Wilson-Han-Powell sequential quadratic programming technique that is applicable to potentially infeasible nonlinear programs. The method is designed to locate stationary points of a nonlinear program in a generalized sense (Definition 2.8). These stationary points include the familiar Kuhn-Tucker and Fritz John points, but also include what we call external and strong-external stationary points. The external and strong external stationary points are not feasible. Thus the problem may be infeasible and yet have stationary points. Loosely speaking, an external stationary point is one that satisfies a certain first-order necessary optimality condition for being as close to feasibility as possible. The proposed technique is similar to those described by J. V. Burke and S-P. Han (“A Modified Sequential Quadratic Programming Method,” manuscript) and M. Sahba ( J. Optim. Theory Appl. , 52 (1987), 291–309); however, it is shown to possess substantially enhanced stability properties. The global convergence properties of the method are described along with some continuity results for the quadratic programming subproblems.

Thurston conjectured that the Riemann mapping function from a simply connected region onto the unit disk can be approximated by regular hexagonal packings. Schramm introduced circle patterns with combinatorics of the square grid (SG) and showed that SG circle patterns converge to meromorphic functions. He and Schramm proved that hexagonal disk packings converge in C ∞ to the Riemann mapping. In this paper we show a similar C ∞ -convergence for SG circle patterns.

In the present paper, an optimal control problem constrained by the tridomain equations in electrocardiology is investigated. The state equations consisting in a coupled reaction–diffusion system modeling the propagation of the intracellular and extracellular electrical potentials, and ionic currents, are extended to further consider the effect of an external bathing medium. The existence and uniqueness of solution for the tridomain problem and the related control problem is assessed, and the primal and dual problems are discretized using a finite volume method which is proved to converge to the corresponding weak solution. In order to illustrate the control of the electrophysiological dynamics, we present some preliminary numerical experiments using an efficient implementation of the proposed scheme.

Let C be a closed convex subset of a real Hilbert space H and assume that T is a κ-strict pseudo-contraction on C. Consider Mann's iteration algorithm given by ∀ x 0 ∈ C , x n + 1 = α n x n + ( 1 − α n ) T x n , n ⩾ 0 . It is proved that if the control sequence { α n } is chosen so that κ α n 1 and ∑ n = 0 ∞ ( α n − κ ) ( 1 − α n ) = ∞ , then lim n → ∞ ‖ x n − T x n ‖ = d ( 0 , R ( A ) ¯ ) , where A = I − T and d ( 0 , D ) denotes the distance between the origin and the subset set D of H. As a consequence of this result, we prove that if T has a fixed point in C, then { x n } converges weakly to a fixed point of T. Also, we extend a result due to Reich to κ-strict pseudo-contractions in the Hilbert space setting. Further, by virtue of hybridization projections, we establish a strong convergence theorem for Lipschitz pseudo-contractions. The results presented in this paper improve or extend the corresponding results of Browder and Petryshyn [F.E. Browder, W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl. 20 (1967) 197–228], Rhoades [B.E. Rhoades, Fixed point iterations using infinite matrices, Trans. Amer. Math. Soc. 196 (1974) 162–176] and of Marino and Xu [G. Marino, H.-K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (1) (2007) 336–346].

A stabilized implicit fractional-step method for numerical solutions of the time-dependent Navier–Stokes equations is presented in this paper. The time advancement is decomposed into a sequence of two steps: the first step has the structure of the linear elliptic problem; the second step can be seen as the generalized Stokes problem. The two problems satisfy the full homogeneous Dirichlet boundary conditions on the velocity. On the other hand, a locally stabilized term is added in the second step of the schemes. It allows one to enhance the numerical stability and efficiency by using the equal-order pairs. Convergence analysis and error estimates for the velocity and pressure of the schemes are established via the energy method. Some numerical experiments are also used to demonstrate the efficiency of this new method.

In this paper we show the weak lower semicontinuity of some classes of functionals, using the concentration-compactness principle of P. L. Lions. These functionals involve an integral term, and we do not know whether it can be handled by the De Giorgi theorem. The semicontinuity result allows us to prove several existence results for quasilinear elliptic equations.

We solve a convection–diffusion–sorption (reaction) system on a bounded domain with dominant convection using an operator splitting method. The model arises in contaminant transport in groundwater induced by a dual-well, or in controlled laboratory experiments. The operator splitting transforms the original problem to three subproblems: nonlinear convection, nonlinear diffusion, and a reaction problem, each with its own boundary conditions. The transport equation is solved by a Riemann solver, the diffusion one by a finite volume method, and the reaction equation by an approximation of an integral equation. This approach has proved to be very successful in solving the problem, but the convergence properties where not fully known. We show how the boundary conditions must be taken into account, and prove convergence in L 1 , loc of the fully discrete splitting procedure to the very weak solution of the original system based on compactness arguments via total variation estimates. Generally, this is the best convergence obtained for this type of approximation. The derivation indicates limitations of the approach, being able to consider only some types of boundary conditions. A sample numerical experiment of a problem with an analytical solution is given, showing the stated efficiency of the method.

International audience; We consider a Fisher-KPP equation with density-dependent diffusion and advection, arising from a chemotaxis-growth model. We study its behavior as a small parameter, related to the thickness of a diffuse interface, tends to zero. We analyze, for small times, the emergence of transition layers induced by a balance between reaction and drift effects. Then we investigate the propagation of the layers. Convergence to a free-boundary limit problem is proved and a sharp estimate of the thickness of the layers is provided.

The paper deals with the convergence of positive solutions for almost-periodic competition diffusion systems. The asymptotic almost periodicity of a positive solution for such a system is described by the almost periodicity of the ω-limit set of the corresponding positive motion in the associated skew-product flow. In the framework of the skew-product flow, it will be proved that the ω-limit set of any spatially homogeneous positive motion contains at most two minimal sets which are both almost automorphic. It will also be proved that if each spatially homogeneous positive solution is asymptotically almost periodic and each spatially homogeneous positive almost periodic solution is lower (upper) asymptotically Lyapunov stable, then every positive solution converges to a spatially homogeneous almost periodic solution. Several important special cases are described where every positive solution converges to a spatially homogeneous almost-periodic solution.

We prove a weak–strong convergence result for functionals of the form ∫RNj(x,u,Du)dx on W1,p, along equiintegrable sequences. We will then use it to study cases of equality in the extended Polya–Szegö inequality and discuss applications of such a result to prove the symmetry of minimizers of a class of variational problems including nonlocal terms under multiple constraints.

Let f:B(x0,R)⊆X→Y be an operator, with X and Y Banach spaces, and f′ be Holder continuous with exponent θ. The convergence of the sequence of Newton–Kantorovich approximations xn=xn−1−f′(xn−1)−1f(xn−1),n∈N, is a classical tool to solve the equation f(x)=0. The convergence of xn is often reduced to the study of the majorizing sequence rn defined by r0=0,r1=a,rn+1=rn+bk(rn−rn−1)1+θ(1+θ)(1−bkrnθ),n∈N, with a,b,k parameters related to f and f′. In the paper [F. Cianciaruso, E. De Pascale, Estimates of majorizing sequences in the Newton–Kantorovich method, submitted for publication] we proved that, if ξ:=aθbk⩽1(1+θθ1−θ)1−θ(θ1+θ)θ, then the following estimates for rn hold rn⩽(bk)−1θ(1+θθ1−θ)1−θθ(1−1(1+θ)n),∀n∈N. In the present paper we give a stronger (at least asymptotically) estimates on rn under a weaker condition on ξ. The techniques employed in the paper are similar to the ones used in [F. Cianciaruso, E. De Pascale, Estimates of majorizing sequences in the Newton–Kantorovich method, submitted for publication]. Finally, we make a comparison with previous results.

In this paper we establish the equivalence between the general variational inequalities and tangent projection equations. This equivalence is used to discuss the local convergence analysis of a wide class of iterative methods for solving the general variational inequalities. We show that some existing methods can identify the optimal face after finitely many iterations under the degenerate assumption.

In this paper we develop the method suggested by S. Pehlivan and M. A. Mamedov (“Statistical Cluster Points and Turnpike,” submitted), where it was proved that under some conditions optimal paths have the same unique stationary limit point–stationary cluster point. This notion was introduced by J. A. Fridy (1993, Proc. Amer. Math. Soc.118, 1187–1192) and it turns out to be a very useful and interesting tool in turnpike theory. The purpose of this paper is to avoid the convexity conditions. Here the turnpike theorem is proved under conditions that are quite different from those of Pehlivan and Mamedov and may be satisfied for the mappings with nonconvex images and for nonconcave functions in the definition of functionals.

It is shown that the extremal solutions of fixed duration Mayer control problems with implicit terminal constraints can be interpreted as fixed points of certain function-valued operators F constructed by solving pairs of initial value problems in tandem. A class of simple recursive averaging processes is proposed for approximating the fixed points of F . Results from the theory of monotone Hilbert space operators are used to establish the convergence of the averaging processes for a general linear-quadratic curve follower problem with unbounded control inputs, and for a simple second order bounded control input problem.

Convergence theorems for approximation of common fixed points of strictly pseudocontractive mappings of Browder–Petryshyn type are proved in Banach spaces using a new composite implicit iteration scheme with errors. The results presented in this paper generalize and improve the corresponding results of M.O. Osilike [M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl. 294 (2004) 73–81].

At the beginning of each period a system is in a certain state. Depending on the state, choice of an action determines the income for that period and the transition probabilities for moving to the next state. The problem is to choose the action at the beginning of each period which will maximize future total discounted income. A modified dynamic programming method for this problem is described. The method gives improved error control by showing how to compute upper and lower bonds on the optimal return which are, respectively, monotone decreasing and monotone increasing, and converge to the optimal return. The convergence appears to be quite rapid.

The Borel exceptional value and the exponents of convergence of poles, zeros and fixed points of finite order transcendental meromorphic solutions for difference Painleve I and II equations are estimated. And the forms of rational solutions of the difference Painleve II equation and the autonomous difference Painleve I equation are also given. It is also proved that the non-autonomous difference Painleve I equation has no rational solution.

Equivalent conditions for a trajectory of a C 0 -semigroup T ( ⋅ ) (resp. cosine function C ( ⋅ ) ) of operators to have the growth order O ( t α ) or o ( t α ) are expressed in terms of Cesaro and Abel means of the norm of the trajectory. We then deduce characterizations of growth order and stability for T ( ⋅ ) and C ( ⋅ ) . It is also shown that under some Tauberian condition the uniform boundedness (resp. strong convergence) of T ( ⋅ ) is equivalent to the uniform boundedness (resp. strong convergence) of its Abel means.

Extension of results of Huang and Thomee (Math. Comput.37, 1981, 327-346) to optimal order error estimates for general types of linear nonhomogeneous parabolic evolution problems is developed.

The idea of almost convergence for double sequences was introduced by Moricz and Rhoades [Math. Proc. Cambridge Philos. Soc. 104 (1988) 283–294] and they also characterized the four dimensional strong regular matrices. In this paper we define and characterize the almost strongly regular matrices for double sequences and apply these matrices to establish a core theorem.

We first prove characterizations of common fixed points of one-parameter nonexpansive semigroups. We next present convergence theorems to common fixed points.

In this paper we investigate the existence, uniqueness, and asymptotic behavior of a solution for a class of coupled nonlinear parabolic equations in a general unbounded domain that includes the whole space R n, the exterior of a bounded domain, and a half space in R n. The asymptotic behavior of the solution is with respect to a pair of quasi-solutions of the corresponding elliptic system, and when these two quasi-solutions coincide the solution of the parabolic system converges to a unique solution of the elliptic system.

Absolute continuity, s-boundedness, and extensions are studied, in the context of the so-called RD -convergence, for set functions taking values in Dedekind complete l-groups. Subsequently, we obtain results of uniform s-boundedness for RD -convergent sequences of measures (Vitali–Hahn–Saks–Nikodým theorem) and deduce a Schur-type theorem for measures defined on P ( N *).

The polar cone of the limit inferior of a sequence of cones of a Banach space is shown to be the Cesari's limit superior of the sequence of polar cones in the bounded weak ∗ topology. In a similar way the lower semicontinuity of a not necessarily countable family of closed convex cones is characterized in terms of Castaing's notion of pseudo-upper semicontinuity. More general results are given within the framework of convergence spaces and the pseudo-upper semicontinuity of a closed-valued multifunction is characterized as closedness with respect to a new convergence. Applications to the Young-Fenchel correspondence are pointed out.

In this paper we define an integral of Kurzweil-Henstock type over multidimensional unbounded intervals having the property that a multiple series converges in the sense of Hardy-Moricz if and only if the associated step function is integrable in this sense. We show that this integral has good properties and in particular Hake's property characterizing the integrability in terms of some convergence property of the associated indefinite integral, and a divergence theorem on unbounded intervals. (C) 1997 Academic Press

Convergence to equilibrium for every positive semi-orbit of strongly increasing discrete-time semigroups on strongly ordered spaces is proved. The domain of the semigroup is assumed to be slightly more general than a closed order interval in a strongly ordered Banach space which is continuously imbedded into a Banach lattice. The semigroup is assumed to be order-compact, and every positive semi-orbit is assumed to be order-bounded. The crucial hypothesis is the Ljapunov stability of all equilibria. It is also proved that the set of equilibria is a simply ordered arc. The key tools are invariant d -hypersurfaces.

Developments of the Hull elastoplastic numerical method lead to nonconservative versions, which in the case of shock waves involve multiplications of distributions of the type of powers of the Dirac delta function. In the one dimensional case of the shock wave equation u t + uu x = 0, the numerical solutions will converge to the solution of a different equation, if the convergence and the latter equation are considered within the nonlinear theory of generalized functions introduced recently by the second author. The study of this phenomenon, presented here in one of its relevant particular cases, offers for the first time a rigorous understanding of important similar situations encountered in industrial applications, when numerical solutions may show either agreement with, or deviations from the expected solutions.

In this paper, we suggest and analyze some new classes of three-step iterative algorithms for solving multivalued quasi variational inclusions by using the resolvent equations technique. New iterative algorithms include the Ishikawa, Mann, and Noor iterations for solving variational inclusions (inequalities) and optimization problems as special cases. The results obtained in this paper represent an improvement and a significant refinement of previously known results.

For a given nonlinear discrete system together with nonlinear boundary conditions we use Newton's iterative scheme and provide sufficient conditions for its convergence to the unique solution. Then, the computational aspects of this iterative method on a floating point system are discussed. This includes necessary and sufficient conditions for the convergence of the Approximate Newton's method, an oscillatory state, and the stopping criterion. To make the analysis widely applicable all the results are proved component-wise.

We prove Tauberian theorems for Jp-summability methods of power series type with respect to ordinary convergence. Under a o-Tauberian condition on the underlying sequence our Tauberian conclusion is valid for all Jp-methods. Relaxing the Tauberian assumption to a O-type condition, a related result is shown for methods with “moderate” weight sequences.

We show that solutions of a two-phase model involving a non-local interactive term become more regular immediately after the moment they separate from the pure phases. This result allows us to prove stronger convergence to equilibria. A new proof of the separation property is also given.

The adaptive stochastic filtering problem for Gaussian processes is considered. The self-tuning synthesis procedure is used to derive two algorithms for this problem. Almost sure convergence for the parameter estimate and the filtering error will be established. The convergence analysis is based on an almost-supermartingale convergence lemma that allows a stochastic Lyapunov-like approach.

The aim of this paper is to prove the a.e. convergence of sequences of the Fejer means of the trigonometric Fourier series of two variable integrable functions. That is, let a = ( a 1 , a 2 ) : N → N 2 such that a j ( n + 1 ) ⩾ α sup k ⩽ n a j ( k ) ( j = 1 , 2 , n ∈ N ) for some α > 0 and a 1 ( + ∞ ) = a 2 ( + ∞ ) = + ∞ . Then for each integrable function f ∈ L 1 ( T 2 ) we have the a.e. relation lim n → ∞ σ a ( n ) f = f . It will be a straightforward and easy consequence of this result the historical cone restricted a.e. convergence result with respect to the two-dimensional Fejer means of integrable functions due to Marcinkiewicz and Zygmund (1939) [7] .

Let { X n , n ⩾ 1 } be a sequence of i.i.d. random vectors taking values in a 2-smooth separable Banach space, and set S n = X 1 + ⋯ + X n . For 0 p 2 and r ⩾ 1 ∨ p , put f ( e ) = ∑ n ⩾ 1 n r / p − 2 P ( ‖ S n ‖ ⩾ e n 1 / p ) , e > 0 . Jain (1975) [4] proved that f ( e ) ∞ , e > 0 , if and only if E ‖ X 1 ‖ r ∞ and E X 1 = 0 . We strengthen this result by showing that, except for the case p = r = 1 , which is treated separately, ∫ δ ∞ f ( e ) d e ∞ , δ > 0 , if and only if E ‖ X 1 ‖ r ∞ and E X 1 = 0 .

We present a new method for solving the Hausdorff moment problem which makes use of Pollaczek polynomials. This problem is ill-posed in the sense of Hadamard and therefore it is unstable from a numerical viewpoint. We shall prove that the expansion, which we obtain, is asymptotically convergent in the L2-norm when the data (i.e., the moments) are perturbed by noise and finite in number. This result is largely used in the numerical analysis of the problem.

This paper studies the dynamics of a system of retarded functional differential equations (i.e., RFDEs), which generalize the Hopfield neural network models, the bidirectional associative memory neural networks, the hybrid network models of the cellular neural network type, and some population growth model. Sufficient criteria are established for the globally exponential stability and the existence and uniqueness of pseudo almost periodic solution. The approaches are based on constructing suitable Lyapunov functionals and the well-known Banach contraction mapping principle. The paper ends with some applications of the main results to some neural network models and population growth models and numerical simulations.

In this paper, we prove the almost sure limit theorem of the maxima for a kind of strongly dependent stationary Gaussian vector sequences.

The complexity of a differentiable function can be measured according to its differentiability properties, the integrability properties of its derivative, or the convergence properties of its Fourier series. This produces three natural ordinal ranks. The relationships between these ranks are investigated. In particular it is shown that the differentiability rank dominates the integrability rank. Some related results and unsolved problems are also mentioned.

The authors investigated in Boos and Leiger (2008) [5] the ‘duality’ of the Nikodym property (NP) of the set of all null sets of the density defined by any nonnegative matrix and the Hahn property (HP) of the strong null domain of it. In this paper, the investigation of the intimated duality is continued by considering densities defined by sequences of nonnegative matrices. These considerations are motivated by the known result that the ideal of the null sets of the uniform density has NP. In this context the general notion of S -convergence of double sequences (cf. Drewnowski, 2002 [8] ) containing Pringsheim convergence, Hardy convergence and uniform convergence of double sequences is used.

In this work wome connections are pursued between weak and strong convergence in the spaces Cm (m-times continuously differentiable functions on Rn). Let fn, f ϵ Cm + 1, where n = 1, 2,…, and m is a nonnegative integer. Suppose that the sequence {fn} converges to f relative to the weak topology of Cm + 1. It is shown that this implies the convergence of {fn} to f with respect to the strong topology of Cm. Several corollaries to this theorem are established; among them is a sufficient condition for uniform convergence. A stronger result is shown to exist when the sequence constitutes an output sequence of a linear weakly continuous operator.