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

the bounds on the solutions of certain two dimensional delay dynamic systems on time scales by means of a new linear gronwall bellman inequality on time scales and an elementary inequality the bounds on the solutions of a class of new nonlinear two dimensional dynamic systems on time scales are obtained

about some new integral inequalities of wendroff type for discontinuous functions abstract in this article we obtain some new nonlinear integral inequalities for discontinuous functions of two independent variables wendroff type by including also inequalities with delay we deduce new generalizations of earlier results given by r p agarwal r bellman i bihari b k bondge v lakshmikantham s leela b g pachpatte for continuous and discrete functions furthermore generalizations of some results for integro sum inequalities are obtained as well

spiraling attractors and quantum dynamics for a class of long range magnetic fields abstract we consider the long time behavior of a quantum particle in a 2d magnetic field which is homogeneous of degree 1 if the field never vanishes above a certain energy the associated classical dynamical system has a globally attracting periodic orbit in a reduced phase space for that energy regime we construct a simple approximate evolution based on this attractor and prove that it completely describes the quantum dynamics of our system

comparison of classical and quantum spectra for a totally bound potential the quantal energy spectrum is compared with the classical motion for the totally bound potential 1 2 x2 y2 ax2y2 the classical phase space is filled with regular trajectories at lower energies but as the energy is increased both regular and irregular trajectories are observed to coexist at very high energies the classical phase space is almost totally filled with irregular trajectories the work reported here is similar to that performed by the authors on the henon heiles potential with the purpose of testing the prediction by percival 1973 that there is good agreement between the amount of classical irregular motion and the portion of energy eigenvalues sensitive to small changes in the perturbation parameter however the potential investigated has several computational advantages over the henon heiles potential as well as avoiding complications due to quantum mechanical tunnelling the results show good agreement with percival s predictions

on certain new nonlinear retarded integral inequalities in two independent variables and applications abstract the purpose of this paper is to establish some new non linear retarded integral inequalities in two independent variables which can be used as handy tools to study the boundedness of solutions of differential integral equations with the initial conditions an application is given to illustrate the usefulness of our results

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

some new generalized volterra fredholm type finite difference inequalities involving four iterated sums in this paper based on some new generalized volterra fredholm type finite difference inequalities involving four iterated sums we derive some new explicit bounds for the unknown functions concerned these inequalities are of more general forms than some existing results in the literature and are generalization of the main results of ma j comput appl math 233 2010 2170 2180 as application of the established results we present a volterra fredholm sum difference equation and research the qualitative as well as quantitative properties for it

state feedback stabilization of a class of uncertain nonlinear systems on non uniform time domains abstract this paper deals with the stabilization problem for a class of uncertain nonlinear systems on non uniform time domains some sufficient conditions are derived to design a state feedback controller for a class of uncertain nonlinear time varying systems under vanishing perturbations using some gronwall s integral inequalities the uniform exponential stability of the closed loop systems on arbitrary time scales is guaranteed then based on the lyapunov theory new sufficient conditions are proposed to derive the controller which ensures the practical stability of the closed loop time invariant nonlinear uncertain system under non vanishing perturbations some examples illustrate these results

high energy and smoothness asymptotic expansion of the scattering amplitude abstract we find an explicit expression for the kernel of the scattering matrix for the schrodinger operator containing at high energies all terms of power order it turns out that the same expression gives a complete description of the diagonal singularities of the kernel in the angular variables the formula obtained is in some sense universal since it applies both to short and long range electric as well as magnetic potentials

topological transversality application to an integrodifferential equation 2 k t s x f f t g s w x t s x g r r 3 q where f g h g c r r and w belongs to the class r of continuous q q nondecreasing functions on r with w r 0 for r 0 and satisfying q h dsrw s s 1 under assumptions 2 and 3 we show that 1 has a solution on rq and we give bounds on that solution our discussion is based on topological transversality arguments the last section is devoted to an application in the mathematical theory of water percolation phenomena

some pachpatte type inequalities on time scales in this paper using the comparison theorem we investigate some pachpatte type integral inequalities on time scales which provide explicit bounds on unknown functions our results extend some known dynamic inequalities on time scales unify and extend some continuous inequalities and their corresponding discrete analogues some applications of the main results are given in the end of this paper

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

estimates on some power nonlinear volterra fredholm type discrete inequalities and their applications some explicit bounds on solutions to a class of new power 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 given

propagation estimates for schrodinger type operators on obtient des estimations de propagation pour un operateur de type schrodinger en utilisant des techniques de commutateur multiple on donne une nouvelle methode pour obtenir des estimations pour des puissances de la resolvante

some inequalities via probabilistic method in this paper we obtain some new inequalities by means of the mean inequalities of random variables which include generalizations of the greub rheinboldt inequality

period doubling bifurcations and universality in conservative systems we have studied numerically a sequence of period doubling bifurcations of periodic orbits of henon s conservative two dimensional mapping in agreement with other recent studies we also find evidence that such sequences possess universality properties similar to the ones observed for dissipative systems the corresponding universal constants however have significantly different values in each case we suggest that a possible explanation for this discrepancy may be the fact that conservative systems owing to their measure preserving property do not become asymptotically one dimensional as dissipative systems do

some new delay integral inequalities and their applications the aim of the present paper is to establish some new delay integral inequalities which provide explicit bounds on unknown functions the inequalities given here can be used as tools in the qualitative theory of certain delay differential equations and delay integral equations

on retarded integral inequalities in two independent variables and their applications the object of this paper is to establish some nonlinear retarded integral inequalities in two independent variables which can be used as handy tools in the theory of partial differential and integral equations with time delays

a retarded integral inequality and its applications we prove a retarded nonlinear integral inequality and present some applications of it to the global existence of solutions to differential equations with time delay

the scattering matrix for the schrodinger operator with a long range electromagnetic potential we consider the schrodinger operator h i a 2 v in the space l2 rd with long range electrostatic v x and magnetic a x potentials using the scheme of smooth perturbations we give an elementary proof of the existence and completeness of modified wave operators for the pair h0 h our main goal is to study spectral properties of the corresponding scattering matrix s we obtain its stationary representation and show that its singular part up to compact terms is a pseudodifferential operator with an oscillating amplitude which is an explicit function of v and a finally we deduce from this result that in general for each 0 the spectrum of s covers the whole unit circle

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

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

some new delay integral inequalities and their applications the main objective of this paper is to establish some new delay integral inequalities which provide explicit bounds on unknown functions the inequalities given here can be used as tools in the qualitative theory of certain delay differential equations and delay integral equations

the inequalities for some types of q integrals using the restriction of the q integral over a b to a finite sum and q integral of riemann type we establish new integral inequalities of q chebyshev type q gruss type q hermite hadamard type and cauchy buniakowsky type some inequalities which include the boundaries of functions are also indicated

on some integral inequalities and their applications abstract some variants of two dimensional integral inequalities so called inequalities of the volterra fredholm type are considered in particular generalizations of the gronwall inequality are obtained these results are applied to study the boundedness stability and uniqueness of the solutions of some integral equations and their systems

improvement of some ostrowski gruss type inequalities abstract in this paper we improve and further generalize some ostrowski gruss type inequalities involving bounded once and twice differentiable mappings

asymptotic properties of solutions to some nonlinear integral equations of convolution type for a class of nonlinear integral equations of convolution type we give necessary and sufficient conditions for the boundedness of nonnegative solutions moreover conditions for the solution to converge asymptotically to a determined limit are obtained

integral inequalities for retarded volterra equations integral inequalities are very useful in the qualitative analysis of differential and integral equations starting with o lipovan a retarded gronwall like inequality and its applications j math anal appl 252 2000 389 401 several recent investigations see o lipovan a retarded integral inequality and its applications j math anal appl 285 2003 436 443 b g pachpatte explicit bounds on certain integral inequalities j math anal appl 267 2002 48 61 b g pachpatte on some retarded integral inequalities and applications j inequal pure appl math 3 2002 article 18 b g pachpatte on a certain retarded integral inequality and its applications j inequal pure appl math 5 2004 article 19 b g pachpatte on some new nonlinear retarded integral inequalities j inequal pure appl math 5 2004 article 80 were devoted to retarded integral inequalities in this paper we consider the case of retarded volterra integral equations we establish bounds on the solutions and by means of examples we show the usefulness of our results in investigating the asymptotic behaviour of the solutions

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

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

some generalized integral inequalities and their applications in this paper we generalize some integral inequalities to more general situations we establish bounds on the solutions and by means of examples we show the usefulness of our results in investigating the global existence of the solutions of integral equations and differential equations

on retarded integral inequalities and their applications abstract the main results of j math anal appl 285 2003 436 443 are here generalized to the following retarded integral inequalities u m t c m m n m m n 0 t f s u n s w u s g s u n s d s and u m t c m m n m m n 0 t f s u n s w u s d s m m n 0 t g s u n s w u s d s where m n 0 are constants and t r 0 the results given here can be applied to the global existence of solutions to differential equations with time delay

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

spectral analysis of n body schrodinger operators for a large class of two body potentials we solve two of the main problems in the spectral analysis of multiparticle quantum hamiltonians explicitly we prove that the point spectrum lies in a closed countable set and describe that set in terms of the eigenvalues of hamiltonians of subsystems and that there is no singular continuous spectrum we accomplish this by extending mourre s work on three body problems to n body problems

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 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 some new integral inequalities and their applications the aim of the present paper is to establish some new integral inequalities in two independent variables which provide explicit bounds on unknown functions the inequalities given here can be used as tools in the qualitative theory of certain partial differential equations

some new integral inequalities of bellman bihari type with delay for discontinuous functions abstract in this article we investigate some integral functional inequalities of bellman bihari type for piecewise continuous functions with some fixed points of discontinuity we also prove a new analogy and generalization of results which were obtained by bellman and bihari to integro sum inequalities with delay and discontinuities that do not belong to lipschitz s type

the scattering amplitude for the schrodinger equation with a long range potential we consider the schrodinger operator with a long range potential v x in the space our goal is to study spectral properties of the corresponding scattering matrix and a diagonal singularity of its kernel the scattering amplitude it turns out that in contrast to the short range case the dirac function singularity of at the diagonal disappears and the spectrum of the scattering matrix covers the whole unit circle for an asymptotically homogeneous function v x of order we show that typically where the module w and the phase are asymptotically homogeneous functions as of orders and respectively leading terms of asymptotics of w and at are calculated in the case 1 our results generalize in the limit the well known formula of gordon and mott

radiation conditions and scattering theory for n particle hamiltonians the correct form of the angular part of radiation conditions is found in scattering problem forn particle quantum systems the estimates obtained allow us to give an elementary proof of asymptotic completeness for such systems in the framework of the theory of smooth perturbations

oscillation for nonlinear second order dynamic equations on a time scale abstract we obtain some oscillation criteria for solutions to the nonlinear dynamic equation x q t x p t f x 0 on time scales in particular no explicit sign assumptions are made with respect to the coefficients p t q t we illustrate the results by several examples including a superlinear emden fowler dynamic equation

some new volterra fredholm type dynamic integral inequalities on time scales in this paper some new explicit bounds on solutions to a class of new volterra fredholm type dynamic integral inequalities on time scales are established which can be used as effective tools in the study of certain dynamic equations some applications for dynamic equations are also indicated

on the mathematical theory of the aharonov bohm effect we consider the schr dinger operator h i a 2 in the space l2 2 with a magnetic potential a x a x2 x1 x 2 where a is an arbitrary function on the unit circle our goal is to study spectral properties of the corresponding scattering matrix s 0 we obtain its stationary representation and show that its singular part up to compact terms is a pseudodifferential operator of zero order whose symbol is an explicit function of a we deduce from this result that the essential spectrum of s does not depend on and consists of two complex conjugated and perhaps overlapping closed intervals of the unit circle finally we calculate the diagonal singularity of the scattering amplitude kernel of s considered as an integral operator in particular we show that for all these properties only the behaviour of a potential at infinity is essential the preceding papers on this subject treated the case a const and used the separation of variables in the schr dinger equation in the polar coordinates this technique does not of course work for arbitrary a from an analytical point of view our paper relies on some modern tools of scattering theory and well known properties of pseudodifferential operators

a model for the coexistence of diffusion and accelerator modes in a chaotic area preserving map a random walk model for the coexistence of diffusion and accelerator modes for a chaotic two dimensional area preserving map is constructed and solved analytically in order to explain the time behaviour of the numerically calculated diffusion coefficient for such maps

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

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

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

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

initial value problem for first order integro differential equation of volterra type on time scales abstract in this paper we combine the monotone iterative technique with fixed point theory to investigate the extremal solutions of nonlinear integro differential equations on time scales several important lemmas and existence criteria are obtained

generalized gronwall bellman type delay dynamic inequalities on time scales and their applications abstract in this paper we establish some new gronwall bellman type delay dynamic inequalities on time scales which can be used as a handy tool in the investigation of qualitative as well as quantitative analysis of solutions of delay dynamic equations on time scales our results generalize the main results of ou iang l ou iang the boundedness of solutions of linear differential equations y a t y 0 shuxue jinzhan 3 1957 409 415 jiang f c jiang f w meng explicit bounds on some new nonlinear integral inequality with delay j comput appl math 205 2007 479 486 and yuan z l yuan x w yuan f w meng some new delay integral inequalities and their applications appl math comput 208 2009 231 237

a generalization of retarded integral inequalities in two independent variables and their applications the object of this paper is to establish some nonlinear retarded integral inequalities in two independent variables which can be used as handy tools in the theory of partial differential and integral equations with time delays

on quadratic derivative schrodinger equations in one space dimension we consider the schrodinger equation with derivative perturbation terms in one space dimension for the linear equation we show that the standard strichartz estimates hold under specific smallness requirements on the potential as an application we establish existence of local solutions for quadratic derivative schrodinger equations in one space dimension with small and rough cauchy data

ulam s type stability of higher order nonlinear delay differential equations via integral inequality of gronwall bellman bihari s type abstract in this paper we prove the hyers ulam stability and the hyers ulam rassias stability of a class of higher order nonlinear delay differential equations with multiple bounded variable delays on a compact interval result of existence and uniqueness of solution is obtained by fixed point approach meanwhile for the first time integral inequality of gronwall bellman bihari s type with delay is applied to prove the stability theorem which made our results more generalized and interesting

dynamic equations on time scales a survey the study of dynamic equations on time scales which goes back to its founder stefan hilger 1988 is an area of mathematics that has recently received a lot of attention it has been created in order to unify the study of differential and difference equations in this paper we give an introduction to the time scales calculus we also present various properties of the exponential function on an arbitrary time scale and use it to solve linear dynamic equations of first order several examples and applications among them an insect population model are considered we then use the exponential function to define hyperbolic and trigonometric functions and use those to solve linear dynamic equations of second order with constant coefficients finally we consider self adjoint equations and more generally so called symplectic systems and present several results on the positivity of quadratic functionals

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

spectral and scattering theory for differential and hankel operators abstract we consider a class of hankel operators h realized in the space l 2 r as integral operators with kernels h t s where h t p ln t t 1 and p x x n p n 1 x n 1 p 0 is an arbitrary real polynomial of degree n this class contains the classical carleman operator when n 0 we show that a hankel operator h in this class can be reduced by an explicit unitary transformation essentially by the mellin transform to a differential operator a v q d v in the space l 2 r here q x x n q n 1 x n 1 q 0 is a polynomial determined by p x and v 1 2 cosh 1 2 is the universal function then the operator a v q d v reduces by the generalized liouville transform to the standard differential operator b d n b n 1 x d n 1 b 0 x with the coefficients b m x m 0 n 1 decaying sufficiently rapidly as x this allows us to use the results of spectral theory of differential operators for the study of spectral properties of generalized carleman operators in particular we show that the absolutely continuous spectrum of h is simple and coincides with r if n is odd and it has multiplicity 2 and coincides with 0 if n 2 is even the singular continuous spectrum of h is empty and its eigenvalues may accumulate to the point 0 only as a by product of our considerations we develop spectral theory of a new class of degenerate differential operators a v q d v where q x is an arbitrary real polynomial and v is an almost arbitrary real function decaying at infinity

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

application of the theory of hill s equation to the study of the stability of periodic classical orbits the theory of hill s equation is applied to periodic orbits in a classical model of the magnetic hydrogen atom it is shown how an infinite hill determinant may be approximated and computed thus giving the discriminant of the relevant whittaker hill equation this discriminant supplies information on the stability of the above periodic orbits which is much more easily obtained than by the numerical integrations of the orbits appearing in previous publications the method is also applicable to other classical systems of current interest

about some new generalizations of bellman bihari results for integro functional inequalities with discontinuous functions and applications abstract in this article we present some new nonlinear bellman bihari type integral inequalities with delay for discontinuous functions integro sum inequalities impulse integral inequalities some applications of the results are included conditions of boundedness uniformly stability by lyapunov uniformly practical stability by chetaev uniformly for the solutions of impulsive differential and integro differential systems of ordinary differential equations

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 integral inequalities and their applications in studying the stability of nonlinear integro differential equations with time delay abstract in this paper we generalize some integral inequalities to more general situations and the inequalities of pachpatte type are corollaries of our s we establish bounds on the solutions and we show the usefulness of our results in investigating the asymptotic behavior and the stability on the solutions of integral equations differential equations and integro differential equations with time delay

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

operateurs conjugues et proprietes de propagation we use an algebraic criteria based on local positivity of a commutator which asserts the existence of a direction of propagation for the flowe iht associated to a self adjoint operatorh

stability analysis of x or y periodic orbits in a 1 2 q xy 2 potential a combination of analytic and numerical methods is used to show the existence of simple stable orbits in the 1 2 q xy 2 potential

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

estimates on solutions of some new nonlinear retarded volterra fredholm type integral inequalities abstract some new explicit bounds on solutions to a class of new nonlinear retarded volterra fredholm type integral inequalities are established which can be used as effective tools in the study of certain integral equations applications examples are also indicated

absence of singular continuous spectrum for certain self adjoint operators we give a sufficient condition for a self adjoint operator to have the following properties in a neighborhood of a pointe of its spectrum a its point spectrum is finite b its singular continuous spectrum is empty c its resolvent satisfies a class of a priori estimates

some new nonlinear volterra fredholm type dynamic integral inequalities on time scales in this paper some new explicit bounds on solutions to a class of new nonlinear volterra fredholm type dynamic integral inequalities on time scales are established which can be used as effective tools in the study of certain dynamic equations some applications for dynamic equations are also indicated

quantum scattering at low energies for a class of negative slowly decaying potentials including v x x with 0 2 we study the quantum mechanical scattering theory in the low energy regime using appropriate modifiers of the isozaki kitada type we show that scattering theory is well behaved on the whole continuous spectrum of the hamiltonian including the energy 0 we show that the modified scattering matrices s are welldefined and strongly continuous down to the zero energy threshold similarly we prove that the modified wave matrices and generalized eigenfunctions are norm continuous down to the zero energy if we use appropriate weighted spaces these results are used to derive oscillatory asymptotics of the standard short range and dollard type s matrices for the subclasses of potentials where both kinds of s matrices are defined for potentials whose leading part is x we show that the location of singularities of the kernel of s experiences an abrupt change from passing from positive energies to the limiting energy 0 this change corresponds to the behaviour of the classical orbits under stronger conditions one can extract the leading term of the asymptotics of the kernel of s at its singularities

remez type inequalities and their applications abstract the remez inequality gives a sharp uniform bound on 1 1 for real algebraic polynomials p of degree at most n if the lebesgue measure of the subset of 1 1 where p is at most 1 is known remez type inequalities give bounds for classes of functions on a line segment on a curve or on a region of the complex plane given that the modulus of the functions is bounded by 1 on some subset of prescribed measure this paper offers a survey of the extensive recent research on remez type inequalities for polynomials generalized nonnegative polynomials exponentials of logarithmic potentials and muntz polynomials remez type inequalities play a central role in proving other important inequalities for the above classes the paper illustrates the power of remez type inequalities by giving a number of applications

some delay integral inequalities on time scales in this paper using gronwall s inequality we investigate some delay integral inequalities on time scales which provide explicit bounds on unknown functions our results unify and extend some delay integral inequalities and their corresponding discrete analogues the inequalities given here can be used as tools in the qualitative theory of certain classes of delay dynamic equations on time scales

integral inequalities in two independent variables for retarded volterra equations in this paper we obtain some retarded volterra integral inequalities in two independent variables which can be used as tools in the theory of partial differential and integral equations with time delays

some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations abstract some new weakly singular integral inequalities of gronwall bellman type are established which generalized some known weakly singular inequalities and can be used in the analysis of various problems in the theory of certain classes of differential equations integral equations and evolution equations some applications to fractional differential and integral equations are also indicated

a generalization of discrete gronwall inequality and its application to weakly singular volterra integral equation of the second kind this paper presents a new discrete gronwall inequality using the inequality we prove convergence and error estimate of the numerical solutions of the second weakly singular volterra integral equation where discrete equation is derived by novot s quadrature formula

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

on certain integral inequalities in two independent variables for retarded equations in this paper we establish some new retarded integral inequalities in two independent variables which can be used as tools in the theory of integral equations with time delays applications are given to illustrate the usefulness of the inequalities

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

markov tree model of transport in area preserving maps abstract transport in an area preserving map with a mixture of regular and chaotic regions is described in terms of the flux through invariant cantor sets called cantori a model retaining a discrete set of cantori approaching a boundary circle gives the markov chain description of hanson cary and meiss the inclusion of cantori surrounding island chains and islands about islands etc gives a markov tree model with a slower decay rate the survival probability distribution is shown to decay asymptotically as a power law the decay exponent agrees reasonably well with the computations of karney and of chirikov and shepelyanski