Superprocesses with spatial interactions in a random medium

We construct a class of interactive measure-valued diffusions driven by a historical super-Brownian motion and an independent white noise by solving a certain stochastic equation. In doing so, we show that the approach of Perkins (2002) [3] can be used to study the problem examined by Dawson et al. (2001) [1]. This unifies and extends both Dawson et al. (2001) [1] and Perkins (2002) [3] and establishes a new class of measure-valued diffusions. The existence and pathwise uniqueness of the solutions are proved, and the solutions are shown to satisfy the natural martingale problem.     

Some weak self-adjoint Hamilton-Jacobi-Bellman equations arising in financial mathematics

In Gandarias (2011) [12] one of the present authors has introduced the concept of weak self-adjoint equations. This definition generalizes the concept of self-adjoint and quasi self-adjoint equations that were introduced by Ibragimov (2006) [11]. In this paper we find a class of weak self-adjoint Hamilton-Jacobi-Bellman equations which are neither self-adjoint nor quasi self-adjoint. By using a general theorem on conservation laws proved in Ibragimov (2007) [9] and the new concept of weak self-adjointness (Gandarias, 2011) [12] we find conservation laws for some of these partial differential equations.     

Kantorovich-type semilocal convergence analysis for inexact Newton methods

We provide a new semilocal convergence analysis for generating an inexact Newton method converging to a solution of a nonlinear equation in a Banach space setting. Our analysis is based on our idea of recurrent functions. Our results are compared favorably to earlier ones by others and us (Argyros (2007, 2009) [5] and [6], Argyros and Hilout (2009) [7], Guo (2007) [15], Shen and Li (2008) [18], Li and Shen (2008) [19], Shen and Li (2009) [20]). Numerical examples are provided to show that our results apply, but not earlier ones [15], [18], [19] and [20].     

Approximations for nonlinear mappings by the hybrid method in Hilbert spaces

The hybrid method in mathematical programming was introduced by Haugazeau (1968) [1] and he proved a strong convergence theorem for finding a common element of finite nonempty closed convex subsets of a real Hilbert space. Later, Bauschke and Combettes (2001) [2] proposed some condition for a family of mappings (the so-called coherent condition) and established interesting results by the hybrid method. The authors (Nakajo et al., 2009) [10] extended Bauschke and Combettes’s results. In this paper, we introduce a condition weaker than the coherent condition and prove strong convergence theorems which generalize the results of Nakajo et al. (2009) [10]. And we get strong convergence theorems for a family of asymptotically κ-strict pseudo-contractions, a family of Lipschitz and pseudo-contractive mappings and a one-parameter uniformly Lipschitz semigroup of pseudo-contractive mappings.     

On the global regularity of generalized Leray-alpha type models

We generalize Leray-alpha type models studied in Cheskidov et al. (2005) [1] and Linshiz and Titi (2007) [4] via fractional Laplacians and employ Besov space techniques to obtain global regularity results with the logarithmically supercritical dissipation.     

On the Toeplitzness of the adjoint of composition operators

Building on techniques developed by Cowen (1988)  [3] and Nazarov–Shapiro (2007)  [10], it is shown that the adjoint of a composition operator, induced by a unit disk-automorphism, is not strongly asymptotically Toeplitz. This result answers Nazarov–Shapiro’s question in Nazarov and Shapiro (2007)  [10].     

Permanence of nonautonomous Lotka–Volterra delay differential systems

In this work, applying the results offered by S. Ahmad and A.C. Lazer [On a property of nonautonomous Lotka–Volterra competition model, Nonlinear Anal. 37 (1999) 603–611] and the recent work of R. Redheffer [Mean values and the nonautonomous May–Leonald equations, Nonlinear Anal. Real World Appl. 4 (2003) 301–306] to an nonautonomous Lotka–Volterra differential system with finite delays, we establish sufficient conditions for the permanence of the system.     

On pricing arithmetic average reset options with multiple reset dates in a lattice framework

We develop a straightforward algorithm to price arithmetic average reset options with multiple reset dates in a Cox et al. (CRR) (1979) [10] framework. The use of a lattice approach is due to its adaptability and flexibility in managing arithmetic average reset options, as already evidenced by Kim et al. (2003) [9]. Their model is based on the Hull and White (1993) [5] bucketing algorithm and uses an exogenous exponential function to manage the averaging feature, but their choice of fictitious values does not guarantee the algorithm’s convergence (cfr., Forsyth et al. (2002) [11]). We propose to overcome this drawback by selecting a limited number of trajectories among the ones reaching each node of the lattice, where we compute effective averages. In this way, the computational cost of the pricing problem is reduced, and the convergence of the discrete time model to the corresponding continuous time one is guaranteed.     

On the Secant method

We present a new semilocal convergence analysis for the Secant method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis is based on the weaker center-Lipschitz concept instead of the stronger Lipschitz condition which has been ubiquitously employed in other studies such as Amat et al. (2004)  [2], Bosarge and Falb (1969)  [9], Dennis (1971)  [10], Ezquerro et al. (2010)  [11], Hernández et al. (2005, 2000)   and , Kantorovich and Akilov (1982)  [14], Laasonen (1969)  [15], Ortega and Rheinboldt (1970)  [16], Parida and Gupta (2007)  [17], Potra (1982, 1984–1985, 1985)  ,  and , Proinov (2009, 2010)   and , Schmidt (1978) [23], Wolfe (1978)  [24] and Yamamoto (1987)  [25] for computing the inverses of the linear operators. We also provide lower and upper bounds on the limit point of the majorizing sequences for the Secant method. Under the same computational cost, our error analysis is tighter than that proposed in earlier studies. Numerical examples illustrating the theoretical results are also given in this study.     

Coderivatives of normal cone mappings and Lipschitzian stability of parametric variational inequalities

In this paper, we provide a comprehensive study of coderivative formulas for normal cone mappings. This allows us to derive necessary and sufficient conditions for the Lipschitzian stability of parametric variational inequalities in reflexive Banach spaces. Our development not only gives an answer to the open questions raised in Yao and Yen (2009) [11], but also establishes generalizations and complements of the results given in Henrion et al. (2010) [4] and Yao and Yen (2009) [11] and [12].     

Regularity for the Harmonic-Einstein equation

We establish a regularity theorem for the Harmonic-Einstein equation. As a byproduct of the local regularity, we also have a compactness theorem on the Harmonic-Einstein equation. The method is mainly the Moser iteration technique which has been used and developed by Bando et al. (1989) [4], Tian (1990) [22], Tian and Viaclovsky (2005) [23] and others.     

On a recent result of Cazenave,Dickstein and Weissler

In this note, we present an estimate from below of the blow-up time of blowing-up solutions of a non-linear non-local in the time evolution equation recently introduced by Cazenave et al. (2008) [1]. Moreover, we give an alternative proof of one of the results of Cazenave et al. (2008) [1] which gives more precise information on the hypotheses than those in the paper Cazenave et al. (2008) [1].     

Fixed point theorems for some generalized multivalued nonexpansive mappings

In this paper, we introduce a condition on multivalued mappings which is a multivalued version of condition (C_λ) defined by Garcia-Falset et al. (2011) [3]. It is shown here that some of the classical fixed point theorems for multivalued nonexpansive mappings can be extended to mappings satisfying this condition. Our results generalize the results in Lim (1974), Lami Dozo (1973), Kirk and Massa (1990), Garcia-Falset et al. (2011), Dhompongsa et al. (2009) and Abkar and Eslamian (2010) [4], [5], [6], [3], [7] and [8] and many others.     

An improved monotone algorithm for scheduling related machines with precedence constraints

We answer an open question posed by Krumke et al. (2008) [6] by showing how to turn the algorithm of Chekuri and Bender for scheduling related machines with precedence constraints into an O(logm)-approximation algorithm that is monotone in expectation. This significantly improves on the previously best known monotone approximation algorithms for this problem, from Krumke et al. [6] and Thielen and Krumke (2008) [8], which have an approximation guarantee of O(m^2/3).     

Existence of multidimensional phase transitions in a non-isothermal van der Waals fluid

In this paper, we prove the existence of multidimensional subsonic phase transitions in a non-isothermal van der Waals fluid. The argument is based on the result of the existence of travelling waves given in [S.-Y. Zhang, Existence of travelling waves in non-isothermal phase dynamics, J. Hyperbolic Differ. Equ. 4 (3) (2007) 391–400] and the result of multidimensional stability given in [S.-Y. Zhang, Stability of multidimensional subsonic phase transition in a non-isothermal van der Waals fluid, Preprint]. An iteration technique [A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 281 (1983) 1–93] is applied to achieve the result.     

Erratum to “Generalized Newton’s method based on graphical derivatives” [Nonlinear Anal. TMA 75 (2012) 1324–1340]

We point out and discuss two erroneous statements in the paper [T. Hoheisel, C. Kanzow, B.S. Mordukhovich, H. Phan, Generalized Newton’s method based on graphical derivatives, Nonlinear Analysis TMA 75 (2012) 1324–1340] that were brought to our attention by Bernd Kummer. Furthermore, the correction of the corresponding global convergence result is given in this note.     

A representation theorem for generators of BSDEs with continuous linear-growth generators in the space of processes

In this paper, on the basis of some recent works of Fan, Jiang and Jia, we establish a representation theorem in the space of processes for generators of BSDEs with continuous linear-growth generators, which generalizes the corresponding results in Fan (2006, 2007) [10] and [11] and Fan and Hu (2008) [9].     

A mathematical modeling for the lookback option with jump-diffusion using binomial tree method

The binomial tree method (BTM), first proposed by Cox et al. (1979) [4] in diffusion models and extended by Amin (1993) [9] to jump-diffusion models, is one of the most popular approaches to pricing options. In this paper, we present a binomial tree method for lookback options in jump-diffusion models and show its equivalence to certain explicit difference scheme. We also prove the existence and convergence of the optimal exercise boundary in the binomial tree approximation to American lookback options and give the terminal value of the genuine exercise boundary. Further, numerical simulations are performed to illustrate the theoretical results.     

Ramsey numbers for a disjoint union of good graphs

We give the Ramsey number for a disjoint union of some G-good graphs versus a graph G generalizing the results of Stahl (1975) [5] and Baskoro et al. (2006) [1] and the previous result of the author Bielak (2009) [2]. Moreover, a family of G-good graphs with s(G)>1 is presented.