Explicit bounds on some new nonlinear integral inequalities with delay

The purpose of the present note is to establish some new delay integral inequalities, which provide explicit bounds on unknown functions and generalize some results of Li et al. [Some new delay integral inequalities and their applications, J. Comput. Appl. Math. 180 (2005) 191–200]. The inequalities given here can be used to investigate the qualitative properties of certain delay differential equations and delay integral equations.     

Extending the applicability of the Gauss–Newton method under average Lipschitz–type conditions

We extend the applicability of the Gauss–Newton method for solving singular systems of equations under the notions of average Lipschitz–type conditions introduced recently in Li et al. (J Complex 26(3):268–295, 2010). Using our idea of recurrent functions, we provide a tighter local as well as semilocal convergence analysis for the Gauss–Newton method than in Li et al. (J Complex 26(3):268–295, 2010) who recently extended and improved earlier results (Hu et al. J Comput Appl Math 219:110–122, 2008; Li et al. Comput Math Appl 47:1057–1067, 2004; Wang Math Comput 68(255):169–186, 1999). We also note that our results are obtained under weaker or the same hypotheses as in Li et al. (J Complex 26(3):268–295, 2010). Applications to some special cases of Kantorovich–type conditions are also provided in this study.     

Abelian integrals and period functions for quasihomogeneous Hamiltonian vector fields

In this paper, we study the number of zeros of Abelian integrals and the monotonicity of period functions for planar quasihomogeneous Hamiltonian vector fields. The result for Abelian integrals extends the recent work of Li et al. [C. Li, W. Li, J. Llibre, Z. Zhang, Polynomial systems: A lower bound for the weakened 16th Hilbert problem, Extracta Math. 16 (3) (2001) 441–447] and Llibre and Zhang [J. Llibre, X. Zhang, On the number of limit cycles for some perturbed Hamiltonian polynomial systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 8 (2) (2001) 161–181].     

A nonlinear diffusion system coupled via nonlinear boundary flux

In this paper, we deal with the global existence and nonexistence of solutions to a nonlinear diffusion system coupled via nonlinear boundary flux. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of nonnegative solutions. The critical curve of Fujita type is conjectured with the aid of some new results, which extend the recent results of Wang et al. [Nonlinear Anal. 71 (2009) 2134-2140] and Li et al. [J. Math. Anal. Appl. 340 (2008) 876-883] to more general equations.     

A modified projection method with a new direction for solving variational inequalities

In this paper, we propose a new projection method for solving variational inequality problems, which can be viewed as an improvement of the method of Li et al. [M. Li, L.Z. Liao, X.M. Yuan, A modified projection method for co-coercive variational inequality, European Journal of Operational Research 189 (2008) 310-323], by adopting a new direction. Under the same assumptions as those in Li et al. (2008), we establish the global convergence of the proposed algorithm. Some preliminary computational results are reported, which illustrated that the new method is more efficient than the method of Li et al. (2008).     

Comment on the orthogonality of the Macdonald functions of imaginary order

Recently, Yakubovich [Opuscula Math. 26 (2006) 161-172] and Passian et al. [J. Math. Anal. Appl. 360 (2009) 380-390] have presented alternative proofs of an orthogonality relation obeyed by the Macdonald functions of imaginary order. In this note, we show that the validity of that relation may be also proven in a simpler way by applying a technique occasionally used in mathematical physics to normalize scattering wave functions to the Dirac delta distribution.     

Comments on “Model for cover cracking due to corrosion expansion and uniform stresses at infinity”

This paper presents a discussion on a model for cover cracking due to corrosion expansion and uniform stress at infinity developed by Li et al. [Shu-cai Li, Ming-bin Wang, Shu-chen Li, Model for cover cracking due to corrosion expansion and uniform stresses at infinity, Appl. Math. Modell., doi:10.1016/j.apm.2007.06.003]. The model for cover cracking by the authors uses an equation of the penetration depth of corrosion with time developed from the Faraday’s Law. It is however shown in the discussion that the equation of the penetration depth due to corrosion developed by Li et al. is incorrect. Consequently other equations developed by Li et al. to model cover cracking that use the equation of penetration depth are also incorrect.     

Approximation theorems for total quasi-?-asymptotically nonexpansive mappings with applications

The purpose of this article is to prove some approximation theorems of common fixed points for countable families of total quasi-?-asymptotically nonexpansive mappings which contain several kinds of mappings as its special cases in Banach spaces. In order to get the approximation theorems, the hybrid algorithms are presented and are used to approximate the common fixed points. Using this result, we also discuss the problem of strong convergence concerning the maximal monotone operators in a Banach space. The results of this article extend and improve the results of Matsushita and Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theor. 134 (2005) 257-266], Plubtieng and Ungchittrakool [S. Plubtieng, K. Ungchittrakool, Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces, J. Approx. Theor. 149 (2007) 103-115], Li, Su [H. Y. Li, Y. F. Su, Strong convergence theorems by a new hybrid for equilibrium problems and variational inequality problems, Nonlinear Anal. 72(2) (2010) 847-855], Su, Xu and Zhang [Y.F. Su, H.K. Xu, X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010) 3890-3960], Wang et al. [Z.M. Wang, Y.F. Su, D.X. Wang, Y.C. Dong, A modified Halpern-type iteration algorithm for a family of hemi-relative nonexpansive mappings and systems of equilibrium problems in Banach spaces, J. Comput. Appl. Math. 235 (2011) 2364-2371], Chang et al. [S.S. Chang, H.W. Joseph Lee, Chi Kin Chan, A new hybrid method for solving a generalized equilibrium problem solving a variational inequality problem and obtaining common fixed points in Banach spaces with applications, Nonlinear Anal. 73 (2010) 2260-2270], Chang et al. [S.S. Chang, C.K. Chan, H.W. Joseph Lee, Modified block iterative algorithm for quasi-?-asymptotically nonexpansive mappings and equilibrium problem in Banach spaces, Appl. Math. Comput. 217 (2011) 7520-7530], Ofoedu and Malonza [E.U. Ofoedu, D.M. Malonza, Hybrid approximation of solutions of nonlinear operator equations and application to equation of Hammerstein-type, Appl. Math. Comput. 217 (2011) 6019-6030] and Yao et al. [Y.H. Yao, Y.C. Liou, S.M. Kang, Strong convergence of an iterative algorithm on an infinite countable family of nonexpansive mappings, Appl. Math. Comput. 208 (2009) 211-218].     

Rank functions of strict cg-matroids

A matroid-like structure defined on a convex geometry, called a cg-matroid, is defined by Fujishige et al. [Matroids on convex geometries (cg-matroids), Discrete Math. 307 (2007) 1936-1950]. A cg-matroid whose rank function is naturally defined is called a strict cg-matroid. In this paper, we give characterizations of strict cg-matroids by their rank functions.     

A Wide Class of Ultrabornological Spaces of Measurable Functions

Our main result states that a bornological locally convex space having a suitable Boolean algebra of projections is ultrabornological. This general theorem, whose proof is a variation of the sliding-hump techniques used in [Díaz et al., Arch. Math. (Basel)60 (1993), 73-78; Díaz et al., Resultate Math.23 (1993), 242-250; Drewnowski el al., Proc. Amer. Math. Sec.114 (1992), 687-694; Drewnowski et al., Atti. Sem. Mat. Fis. Univ. Modena41 (1993), 317-329], is applied to prove that some non-complete normed spaces such as the spaces of Dunford, Pettis, or McShane integrable functions, as well as other interesting spaces of weakly or strongly measurable functions, are ultrabornological. We also give applications to vector-valued sequence spaces; in particular, we prove that ℓ^p{X} (1 ≤ p < ∞) is an ultrabornological DF-space when X is.     

Two modified Dai-Yuan nonlinear conjugate gradient methods

In this paper, we propose two modified versions of the Dai-Yuan (DY) nonlinear conjugate gradient method. One is based on the MBFGS method (Li and Fukushima, J Comput Appl Math 129:15–35, 2001) and inherits all nice properties of the DY method. Moreover, this method converges globally for nonconvex functions even if the standard Armijo line search is used. The other is based on the ideas of Wei et al. (Appl Math Comput 183:1341–1350, 2006), Zhang et al. (Numer Math 104:561–572, 2006) and possesses good performance of the Hestenes-Stiefel method. Numerical results are also reported. This work was supported by the NSF foundation (10701018) of China.     

Properties of (θ,s)-continuous functions

Joseph and Kwack [Proc. Amer. Math. Soc. 80 (1980) 341–348] introduced the notion of (θ,s)-continuous functions in order to investigate S-closed spaces due to Thompson [Proc. Amer. Math. Soc. 60 (1976) 335–338]. In this paper, further properties of (θ,s)-continuous functions are obtained and relationships between (θ,s)-continuity, contra-continuity and regular set-connectedness defined by Dontchev et al. [Internat. J. Math. Math. Sci. 19 (1996) 303–310 and elsewhere] are investigated.     

Uniqueness of entire function sharing a small function with its derivative

This paper is devoted to studying the relationship between an entire function and its derivative when they share one small function. We generalize some previous results of Gundersen and Yang [G. Gundersen, L.Z. Yang, Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl. 223 (1998) 85–95], Chang and Zhu [J. Chang, Y. Zhu, Entire functions that share a small function with their derivatives, J. Math. Anal. Appl. 351 (2009) 491–496].     

Strong convergence theorems for countable families of asymptotically relatively nonexpansive mappings with applications

The purpose of this paper is by using the hybrid iterative method to prove some strong convergence theorems for approximating a common element of the set of solutions to a system of generalized mixed equilibrium problems and the set of common fixed points for two countable families of closed and asymptotically relatively nonexpansive mappings in Banach space. The results presented in the paper improve and extend the corresponding results of Su et al. [Y.F. Su, H.K. Xu, X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010) 3890-3906], Li and Su [H.Y. Li, Y.F. Su, Strong convergence theorems by a new hybrid for equilibrium problems and variational inequality problems, Nonlinear Anal. 72 (2) (2010) 847-855], Chang et al. [S.S. Chang, H.W. Joseph Lee, Chi Kin Chan, A new hybrid method for solving a generalized equilibrium problem solving a variational inequality problem and obtaining common fixed points in Banach spaces with applications, Nonlinear Anal. TMA 73 (2010) 2260-2270], Kang et al. [J. Kang, Y. Su, X. Zhang, Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications, Nonlinear Anal. HS 4 (4) (2010) 755-765], Matsushita and Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theory 134 (2005) 257-266], Tan et al. [J.F. Tan, S.S. Chang, M. Liu, J.I. Liu, Strong convergence theorems of a hybrid projection algorithm for a family of quasi-?-asymptotically nonexpansive mappings, Opuscula Math. 30 (3) (2010) 341-348], Takahashia and Zembayashi [W. Takahashi, K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. 70 (2009) 45-57] and Wattanawitoon and Kumam [K. Wattanawitoon, P. Kumam, Strong convergence theorems by a new hybrid projection algorithm for fixed point problem and equilibrium problems of two relatively quasi-nonexpansive mappings, Nonlinear Anal. Hybrid Systems 3 (2009) 11-20] and others.     

Value sharing results for shifts of meromorphic functions

In this paper, we deal with the value distribution of difference products of entire functions, and present some result on two difference products of entire functions sharing one value with the same multiplicities. The research findings also include an analogue for shift of a well-known conjecture by Brück. Our theorems improve the results of I. Laine and C.C. Yang [I. Laine, C.C. Yang, Value distribution of difference polynomials, Proc. Japan Acad. Ser. A Math. Sci. 83 (2007) 148-151], K. Liu and L.Z. Yang [K. Liu, L.Z. Yang, Value distribution of the difference operator, Arch. Math. 92 (2009) 270-278], and J. Heittokangas et al. [J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, J.L. Zhang, Value sharing results for shifts of meromorphic function, and sufficient conditions for periodicity, J. Math. Anal. Appl. 355 (2009) 352-363]. Moreover, we show by illustrating a number of examples that our results are best possible in certain senses.     

OSCILLATION THEOREMS FOR DELAY AND NEUTRAL DELAY DIFFERENTIAL EQUATIONS

Oscillation criteria for the delay differential equationx′(t)+p(t)x(t-т(t))=0,where p, т are non-negative real-valued continuous functions are investigated inthe case when the numbers k=integral from n=t-т(t) to t(p(s)ds),L=integral from n=t-т(t) to t(p(s)ds)satisfy 0≤k<1/e and 1/e≤L<1. The present result improves almost allresults of the literature concerning it. Furthermore, it is established that allsolutions of the odd-order neutral delay differential equation(x(t)-px(t-т))~(n)+Q(t)x(t-σ)=0,where 0≤p<1,т,σ∈(0,∞)and Q(t)≥0,are oscillatory ifintegral from n=t-σ to t ((t-s)~(n-1)Q(s)ds>(1/e)(1-p)(n-1))!.This result generalizes a theorem of Gopalsamy et. al (Czech. Math. J., 42(1992),313-323)and also extends a very well-known result of Ladas (ApplicableAnalysls, 9(1979),93-98).     

Uniqueness and value-sharing of entire functions

In this paper, we study the uniqueness problems on entire functions sharing one value with the same multiplicities. We generalize and unify some previous results of Fang and Hua [M.L. Fang, X.H. Hua, Entire functions that share one value, J. Nanjing Univ. Math. Biquarterly 13 (1) (1996) 44-48], Yang and Hua [C.C. Yang, X.H. Hua, Uniqueness and value-sharing of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 22 (2) (1997) 395-406] and Fang [M.L. Fang, Uniqueness and value-sharing of entire functions, Comput. Math. Appl. 44 (2002) 828-831].     

Complete Convergence for Randomly Weighted Sums of Random Variables Satisfying Some Moment Inequalities

For random variables and random weights satisfying Marcinkiewicz-Zygmund and Rosenthal type moment inequalities, we establish complete convergence results for randomly weighted sums of the random variables. Our results generalize those of(Thanh et al. SIAM J. Control Optim., 49,106–124(2011), Han and Xiang J. Ineq. Appl., 2016, 313(2016), Li et al. J. Ineq. Appl., 2017, 182(2017), and Wang et al. Statistics, 52, 503–518(2018).)     

Global relaxed modulus-based synchronous block multisplitting multi-parameters methods for linear complementarity problems

Recently, Bai and Zhang [Numerical Linear Algebra with Applications, 20(2013):425439] constructed modulus-based synchronous multisplitting methods by an equivalent reformulation of the linear complementarity problem into a system of ?xed-point equations and studied the convergence of them; Li et al. [Journal of Nanchang University (Natural Science), 37(2013):307-312] studied synchronous block multisplitting iteration methods; Zhang and Li [Computers and Mathematics with Application, 67(2014):1954-1959] analyzed and obtained the weaker convergence results for linear complementarity problems. In this paper, we generalize their algorithms and further study global relaxed modulus-based synchronous block multisplitting multi-parameters methods for linear complementarity problems. Furthermore, we give the weaker convergence results of our new method in this paper when the system matrix is a block H+?matrix. Therefore, new results provide a guarantee for the optimal relaxation parameters, please refer to [A. Hadjidimos, M. Lapidakis and M. Tzoumas, SIAM Journal on Matrix Analysis and Applications, 33(2012):97-110, (dx.doi.org/10.1137/100811222)], where optimal parameters are determined.     

整函数与其微分多项式IM分担两个值

设 f(Z)为非常数整函数 ,L (f )为 f (Z)的具有小函数系数的线性微分多项式 ,本文研究当 f(Z)与 L (f ) IM分担两个有穷复数时的唯一性问题 ,给出了 f (Z)与L (f )完全确定的表达式 ,改进了李平、杨重骏最近获得的一个结果 ,从而也改进了先前的一些结果 .