analytic continuation and boundary continuity of functions of several complex variables this note treats some questions about analytic continuation in several variables the first theorem in effect determines the envelops of holomorphy of certain domains in n the second main result is a continuity theorem if a bounded holomorphic function f on a convex domain in n has boundary values that are continuous on the complement in b of a set of the form int b where is a real hyperplane in n that misses then f is continuous on in addition we obtain what may be regarded as a local version of the theorem in our earlier paper concerning the one dimensional extension property our methods depend on hartogs theorem n 3 and directly on the bochnermartinelli formula n 2

nonlinear discrete inequalities with two variables and their applications abstract an error in the proof of theorem 1 by zheng et al k zheng y wu s zhong discrete nonlinear integral inequalities in two variables and their applications appl math comput 207 2009 140 147 is reported this paper gives the right proof under some additional condition application examples to show boundedness and uniqueness of solutions of a volterra type difference equation are also given

some new retarded nonlinear integral inequalities and their applications in differential integral equations in this paper we discuss some new retarded nonlinear gronwall bellman pachpatte integral inequalities and give upper bound estimation of the unknown function by integral inequality technique our results generalize those of abdeldaim and yakout a abdeldaim m yakout on some new integral inequalities of gronwall bellman pachpatte type appl math comput 217 2011 7887 7899 to some inequalities with both retarded argument and nonlinear integrand an application to estimate the solutions of differential integral equations is also given

on some new discrete generalizations of gronwall s inequality abstract this paper derives new discrete generalizations of the gronwall bellman integral inequality these generalizations should have wide application in the study of finite difference equations and numerical analysis the main result theorem 3 concerns a very general form of linear bellman type discrete inequalities in one independent variable it is a discrete analogue of an integral inequality obtained by the present author in j math anal appl 103 1984 184 197 theorem 4 and it extends many discrete inequalities of agarwal and thandapani pachpatte and sugiyama two nonlinear extensions of theorem 3 are also established here

on some class of sets with multi sheeted envelope of holomorphy let m be a smooth cr manifold of dimension 2n 2 contained in the unit sphere open image in new window we prove that if m is the finite union of boundaries of n 1 dimensional analytic varieties and m is globally minimal then the envelope of holomorphy of open image in new window is multi sheeted in some cases we prove that the number of sheets is exactly two it is also proved for 4 dimensional cr manifolds contained in open image in new window that the property of being globally minimal even being of finite type at each point is generic

a generalized gronwall bellman ou iang type inequality for discontinuous functions and applications to bvp abstract in this paper we investigate a generalized gronwall bellman ou iang type inequality for piecewise continuous functions by studying the unknown function in different continuous regions kj of we give the complete estimate for the unknown function we apply our result to a boundary value problem of a partial impulsive differential equation for estimation of its solution

henry gronwall type retarded integral inequalities and their applications to fractional differential equations with delay abstract this paper presents retarded integral inequalities of henry gronwall type applying these inequalities we study certain properties of solutions to fractional differential equations with delay

a generalized gronwall inequality and its application to a fractional differential equation this paper presents a generalized gronwall inequality with singularity using the inequality we study the dependence of the solution on the order and the initial condition of a fractional differential equation

some new nonlinear volterra fredholm type discrete inequalities and their applications some new explicit bounds on solutions to a class of new nonlinear volterra fredholm type discrete inequalities are established which can be used as effective tools in the study of certain sum difference equations application examples are also indicated

explicit bounds on certain integral inequalities abstract in this paper explicit bounds on retarded gronwall bellman and bihari like integral inequalities and their two independent variable generalizations are established some applications are also given to illustrate the usefulness of one of our results

on the capacity of vector gaussian channels with bounded inputs the capacity of a deterministic multiple input multiple output channel under the peak and average power constraints is investigated for the identity channel matrix the approach of shamai et al is generalized to the higher dimension settings to derive the necessary and sufficient conditions for the optimal input probability density function this approach prevents the usage of the identity theorem of the holomorphic functions of several complex variables which seems to fail in the multi dimensional scenarios it is proved that the support of the capacity achieving distribution is a finite set of hyper spheres with mutual independent phases and amplitude in the spherical domain subsequently it is shown that when the average power constraint is relaxed if the number of antennas is large enough the capacity has a closed form solution and constant amplitude signaling at the peak power achieves it moreover it will be observed that in a discrete time memoryless gaussian channel the average power constrained capacity which results from a gaussian input distribution can be closely obtained by an input where the support of its magnitude is a discrete finite set finally we investigate some upper and lower bounds for the capacity of the non identity channel matrix and evaluate their performance as a function of the condition number of the channel

on the asymptotic behavior of solutions of certain second order nonlinear differential equations on considere l equation differentielle py qy g t y y ou p p t est une fonction positive continument differentiable sur i a a telle que p a 1 q q t est une fonction continue sur i a et g g t u v est une fonction continue sur i a r 2 qui verifie certaines conditions

hilfer fractional advection diffusion equations with power law initial condition a numerical study using variational iteration method abstract we propose a hilfer advection diffusion equation of order 0 1 and type 0 1 and find the power series solution by using variational iteration method power series solutions are expressed in a form that is easy to implement numerically and in some particular cases solutions are expressed in terms of mittag leffler function absolute convergence of power series solutions is proved and the sensitivity of the solutions is discussed with respect to changes in the values of different parameters for power law initial conditions it is shown that the hilfer advection diffusion pde gives the same solutions as the caputo and riemann liouville advection diffusion pde to leading order the fractional solution compared to the non fractional solution increases rapidly with for 0 7 at a given time t but for 0 7 this factor is weakly sensitive to we also show that the truncation errors arising when using the partial sum as approximate solutions decay exponentially fast with the number of terms n used we find that for 0 7 the number of terms needed is weakly sensitive to the accuracy level and to the fractional order n 20 but for 0 7 the required number of terms increases rapidly with the accuracy level and also with the fractional order

on a certain inequality arising in the theory of differential equations abstract in the present paper we establish some new inequalities related to a certain integral inequality arising in the theory of differential equations the inequalities that we propose here can be used as handy tools in the theory of certain new classes of differential and integral equations

on wedge extendability of cr meromorphic functions in this article we consider metrically thin singularities e of the solutions of the tangential cauchy riemann operators on a mathcal c 2 alpha smooth embedded cauchy riemann generic manifold m cr functions on m backslash e and more generally we consider holomorphic functions defined in wedgelike domains attached to m backslash e our main result establishes the wedge and the l1 removability of e under the hypothesis that the dim m 2 dimensional hausdorff volume of e is zero and that m and m backslash e are globally minimal as an application we deduce that there exists a wedgelike domain attached to an everywhere locally minimal m to which every cr meromorphic function on m extends meromorphically

the hartogs type extension theorem for meromorphic maps into compact kahler manifolds extension theorems of this type are often called hartogs type extension theorems the above result was conjectured by griffiths see problem 1 in gr who proved it in the special case when d p o i n t and f d x is holomorphic important contributions towards the proof of this conjecture were made by b shiffman y t siu n sibony see sh sil sb3 in particular y t siu proved a thullen type extension theorem for meromorphic maps into compact kfihler manifolds see w 1 the above theorem can be applied to the study of covering maps of zariski open subsets in compact kfihler manifolds recall that a subdomain d of a complex manifold m is called levi pseudoconvex if for any peod there exists a neighbourhood u of p in m such that u c d is stein we say that d covers a complex manifold x if there is a subgroup g c aut d acting properly discontinuously and freely on d with d g x

on some new inequalities related to certain inequalities in the theory of differential equations in the present paper we establish some new integral inequalities related to a certain inequality arising in the theory of differential equations the discrete analogues of the main results are also given the analysis used in the proofs is elementary and the results established here provide new estimates for these types of inequalities

gronwall bellman type nonlinear delay integral inequalities on time scales abstract in this paper some gronwall bellman type nonlinear delay integral inequalities on time scales are established which provide a handy tool in deriving boundedness of solutions of certain delay dynamic equations on time scales our results generalize some of the main results in lipovan 2006 1 pachpatte 2000 2 ferreira and torres 2009 3 zhang and meng 2008 4 cheung and ren 2006 5 kim 2009 6 and some of our results unify continuous and discrete analysis in the literature

growth in an impulsive integral inequality abstract in this paper we study a finite sum integral inequality with a sequence of impulses making a sequential monotonization we give recursively defined functions to estimate solutions of the inequality piecewise in order to use the estimate to study boundedness and asymptotics of solutions for differential equations we regard the estimating function as a solution of a nonautonomous difference equation simplify the equation with composition of operators and reduce our discussion to qualitative analysis of the difference equation so that we give results on monotonicity boundedness and weighted boundedness for solutions of the inequality as applications we study boundedness and asymptotics of solutions for two impulsive differential equations

some new finite difference inequalities abstract in this paper we establish some new finite difference inequalities related to a certain integral inequality arising in the theory of differential equations the inequalities obtained here can be used as handy tools in the theory of some new classes of finite difference and sum difference equations

asymptotic expansions of the gamma function related to windschitl s formula we develop windschitl s approximation formula for the gamma function to two asymptotic expansions and provide a formula for determining the coefficients of each asymptotic expansion

generalization of a retarded gronwall like inequality and its applications abstract this paper generalizes a lipovan s result of gronwall like inequalities j math anal appl 252 2000 389 to a new type of retarded inequalities which includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals from our result bihari s result and pinto s result can be deduced as some special cases our result can be applied to give the global existence and estimate solutions we also apply it to improve an estimation for almost periodicity of an invariant manifold

inequalities asymptotic expansions and completely monotonic functions related to the gamma function abstract in this paper we present some completely monotonic functions and asymptotic expansions related to the gamma function based on the obtained expansions we provide new bounds for x 1 x 1 2 and x 1 2

some new generalized gronwall bellman type discrete fractional inequalities some new gronwall bellman type discrete fractional inequalities are established explicit bounds for unknown functions in fractional inequalities are derived properties of solutions of one certain fractional difference equation are researched in this paper we present some new generalized gronwall bellman type discrete fractional inequalities including fractional sum inequalities and fractional difference inequalities these inequalities are of new forms compared with the existing results so far in the literature and can be used to provide explicit bounds for unknown functions concerned for illustrating the effectiveness of the presented results we apply them to research qualitative and quantitative properties for the solutions of an ivp with respect to a fractional difference equation

on generalized of certain retarded nonlinear integral inequalities and its applications in retarded integro differential equations in this paper some new nonlinear retarded integral inequalities of gronwall bellman pachpatte type are investigated some applications are also presented to illustrate the usefulness of some of our results in estimation of solution of certain retarded integro differential equations with the initial conditions

comparison theorems for neutral stochastic functional differential equations abstract the comparison theorems under wu and freedman s order are proved for neutral stochastic functional differential equations with finite or infinite delay whose drift terms satisfy the quasimonotone condition and diffusion term is the same

on some generalized retarded integral inequalities and the qualitative analysis of integral equations abstract in this paper we obtain estimates on solutions of some retarded integral inequalities which include retarded gronwall like inequalities and retarded volterra fredholm type inequalities some particular cases with weakly singular kernels are discussed we also apply our results to investigate the boundedness uniqueness and continuous dependence of solutions of some integral equations

yosida approximation and nonlinear hyperbolic equation we are concerned with the initial value problem for nonlinear evolution equations of the form u t m a 1 2 u t 2 au t u t f t on 0 u 0 u o u o u 1 here a is a nonnegative selfadjoint operator in a real hilbert space h 0 is a constant and m is a c 1 class function satisfying m r m o 0 for r 0 with m o constant the equation has its motivation in the mathematical description of small amplitude vibrations of an elastic string

discrete nonlinear integral inequalities in two variables and their applications in this paper some discrete integral inequalities in two variables with more than one nonlinear term are established our results are also applied to show the boundedness of certain boundary value problem for difference equations

asymptotic expansions for the gamma function in terms of hyperbolic functions abstract the smith s approximation formula for the gamma function is given by x 1 2 2 x e x 2 x tanh 1 2 x x 2 1 o 1 x 5 as x in this paper by means of a little known power series we develop this formula to an asymptotic expansions x 1 2 2 x e x 2 x tanh 1 2 x x 2 exp n 3 2 n 2 n 1 2 2 n 1 1 2 1 2 n 2 n 2 n 1 2 n b 2 n x 2 n 1 as x and give an estimate of remainder in the above asymptotic series where b 2 n is the bernoulli number moreover as relevant results ramanujan type asymptotic expansions for the gamma function x 1 2 are obtained an asymptotic expansion for wallis ratio and an estimate of the remainder are established

on an integral inequality and its discrete analogue in this note we given the continuous as well as discrete versions of a gronwall type inequality this inequality originated in the works of ou iang and dafermos and was recently studied by pachpatte

wedge removability of metrically thin sets and application to the cr meromorphic extension we give a wedge removability theorem for metrically thin sets of two codimensional hausdorff null measure following 22 this removability theorem combined with the wedge removability theorem of 21 for closed subsets of two codimensional manifolds gives a cr meromorphic extension theorem in the greater codimensional case

some nonlinear gronwall bellman type retarded integral inequalities with power and their applications abstract the purpose of this paper is to establish some new retarded integral inequalities of gronwall bellman type which generalizes some known integral inequalities we use two lemmas to simplify the complex inequalities and give an upper bound estimation of the unknown function the inequalities given here can be used in the analysis of the qualitative properties of certain classes of differential equations and integral equations we apply our results to the boundedness of the solutions of integral equations and differential system

on some finite difference inequalities in two independent variables in this article we establish some new finite difference inequalities in two independent variables which can be used as tools in the study of certain classes of finite difference and sum difference equations

stability and the nonexistence of blowing up solutions of nonlinear delay systems with linear parts defined by permutable matrices abstract sufficient conditions for the exponential stability of the trivial solution of nonlinear differential equations with delay and with linear parts of the form a x t b x t 0 where a b b a are proved a result on the nonexistence of blowing up solutions is also proved

inequalities applicable in the theory of finite difference equations in the present paper we establish some new nonlinear finite difference inequalities applicable in the theory of certain classes of finite difference equations in various different situations we also present applications of some of the inequalities to show their importance to the literature

summability experiments with a class of divergent inverse kontorovich lebedev transforms abstract the computation of inverse kontorovich lebedev transforms is examined with the help of the extrapolative m w transformation first presented by sidi in user friendly form in sidi 1988 recently published improved values for the macdonald function of purely imaginary order have enabled this strategy to give 10 digit accuracy in the inversion including an example with competing oscillations having different phase functions moreover the method is extended with similar accuracy to cases where the integral diverges but is summable in the sense of abel further probing with divergent integrals that are not abel summable but instead summable in the sense defined by jones 1980 is also yielding encouraging results and some analysis culminating in a formal theorem of convergence in that sense is presented to support the interpretation and computation of these

on some new nonlinear discrete inequalities and their applications in this paper some new discrete inequalities in two independent variables which provide explicit bounds on unknown functions are established the inequalities given here can be used as handy tools in the qualitative theory of certain finite difference equations

integral inequalities of gronwall bellman bihari type and asymptotic behavior of certain second order nonlinear differential equations abstract several integral inequalities similar to gronwall bellmann bihari inequalities are obtained these inequalities are used to discuss the asymptotic behavior of certain second order nonlinear differential equations

gronwall bellman and pachpatte type integral inequalities with applications abstract in this article a number of two dimensional nonlinear retarded integral inequalities of gronwall bellman type which can be used as ready and powerful tools in the analysis of various classes of partial differential integral and integro differential equations are presented applications of nonlinear retarded integral inequalities are also presented

a generalized retarded gronwall like inequality in two variables and applications to bvp abstract in this paper we establish a generalized retarded integral inequality of gronwall like type in two variables which includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals without assumption of monotonicity using our result we can solve both the integral inequality in w s cheung some new nonlinear inequalities and applications to boundary value problems nonlinear anal 64 2006 2112 2128 and the one in r p agarwal s deng w zhang generalization of a retarded gronwall like inequality and its applications appl math comput 165 2005 599 612 we apply our result to a boundary value problem of a partial differential equation for boundedness uniqueness and continuous dependence

a retarded gronwall like inequality and its applications we prove a gronwall like inequality and present some of its applications to the qualitative study of retarded differential equations the problems of global continuation of the solutions and the existence of nonoscillatory solutions are considered by means of examples we show the usefulness of our results

some new nonlinear difference inequalities and their applications in this paper we establish some new nonlinear difference inequalities in two independent variables which can be used as handy tools in the study of qualitative properties of solutions of certain classes of difference equations

on some new integrodifferential inequalities of the gronwall and wendroff type abstract in this paper some new integrodifferential inequalities of the gronwall and wendroff type in several independent variables are established these inequalities are useful in the study of many qualitative as well as quantitative properties of solutions of partial differential and integral equations

some new nonlinear inequalities and applications to boundary value problems abstract in this paper we establish some new nonlinear integral inequalities of the gronwall bellman ou iang type in two variables these on the one hand generalizes and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of differential equations we illustrate this by applying our new results to certain boundary value problem

several integral inequalities and their applications in nonlinear differential systems in this paper several new integral inequalities are presented which are effective in dealing with the integro differential inequalities whose variable exponents are greater than one compared with existing integral inequalities those proposed here can be applied to more complicated differential equations the notions of uniform lipschitz stability are generalized and the relations between these notions are analyzed several sufficient conditions about uniform lipschitz asymptotic stability of nonlinear systems are established by the proposed integral inequalities these sufficiently conditions can be similarly generalized to linearly perturbed differential systems that appear in the literature finally an example of uniform lipschitz asymptotic stability of nonlinear differential systems is shown

estimation of unknown function of a class of integral inequalities and applications in fractional integral equations in this paper we establish a class of iterated integral inequalities which includes a nonconstant term outside the integrals the upper bound of the embedded unknown function is estimated explicitly by adopting novel analysis techniques such as change of variable amplification method differential and integration the derived result can be applied in the study of qualitative properties of solutions of fractional integral equations

discrete non linear inequalities and applications to boundary value problems in this paper we generalize some existing discrete gronwall bellman ou iang type inequalities to more general situations these are in turn applied to study the boundedness uniqueness and continuous dependence of solutions of certain discrete boundary value problem for difference equations

on some new nonlinear delay integral inequalities abstract some new nonlinear delay integral inequalities of ou iang type are obtained which generalise some results of b g pachpatte 1995 period math hungar 31 229 234 and e h yang 1997 ann differential equations 13 180 188 an application example is also indicated

on the asymptotic representation of the euler gamma function by ramanujan abstract the problem of approximation to the euler gamma function on the basis of some ramanujan s formulas is considered the function h x g x 6 8x3 4x2 x where g x e x x 1 x is studied it is proved that on the interval 1 the function h x is increasing monotonically from h 1 0 0111976 to h 1 30 0 0333

on some inequalities for the gamma and psi functions we present new inequalities for the gamma and psi functions and we provide new classes of completely monotonic star shaped and superadditive functions which are related to and

multiple positive solutions for discrete nonlocal boundary value problems in this paper we investigate a second order nonlinear difference equation with sign changing nonlinearity subject to two different sets of nonlocal boundary conditions the explicit expressions of the associated green s functions are presented by using a recently developed fixed point theorem we establish sufficient conditions for the existence of multiple positive solutions of the boundary value problem