a note on the second order non autonomous neutral stochastic evolution equations with infinite delay under caratheodory conditions abstract in this note we aim to study a class of second order non autonomous neutral stochastic evolution equations with infinite delay driven by a standard cylindrical wiener process and an independent cylindrical fractional brownian motion with hurst parameter h 1 2 1 in which the initial value belongs to the abstract space b we establish the existence and uniqueness of mild solutions for this kind of equations under some caratheodory conditions by means of the successive approximation the obtained result extends some well known results an example is proposed to illustrate the theory

almost automorphic solutions for stochastic differential equations driven by levy noise abstract in this paper we are concerned with almost automorphic solutions for semilinear stochastic differential equations driven by levy noise on the hilbert space we present a new variant of gronwall s lemma which improves the variant of gronwall s lemma in kamenskii 2015 based on this new variant of gronwall s lemma suitable conditions on the coefficients we prove the existence and uniqueness of almost automorphic solution in distribution for some stochastic differential equations driven by levy noise on the hilbert space which improves and generalizes the results in liu 2014 subsequently by using this new variant of gronwall s lemma we investigate the existence and uniqueness of almost automorphic solution in distribution for stochastic differential equations driven by levy noise with markov switching processes under some weaker conditions in the end two examples are given to illustrate the theoretical results obtained in this paper

stochastic controllability and minimum energy control of systems with multiple delays in control in recent years various controllability problems for different types of linear dynamical systems have been considered in many publications and monographs

approximate controllability of fractional stochastic differential equations driven by mixed fractional brownian motion via resolvent operators abstractin this manuscript we investigate the approximate controllability of a class of fractional stochastic differential equations driven by mixed fractional brownian motion in hilbert space with hurst parameter h 12 1 the main results are obtained by employing fractional calculus analytic resolvent operators and with the help of schaefer s fixed point theorem under the assumption that the corresponding linear system is approximately controllable finally in order to validate the obtained theoretical results an illustrative example is provided

impulses induced p exponential input to state stability for a class of stochastic delayed partial differential equations abstractthis paper proposes a new stability concept called p exponential input to state stability piss for impulsive stochastic delayed partial differential equations isdpdes by employing the

positive solutions for n order m point p laplacian operator singular boundary value problems in this paper we study the existence of positive solutions for the nonlinear n order m point p laplacian operator singular boundary value problems by using the method of defining operator by the reverse function of green function and the fixed point index theory the existence of positive solution and many positive solutions for nonlinear third order two point singular boundary value problem with p laplacian operator are obtained

estimation of object location from diffraction tomographic intensity data it is shown that an image of the log likelihood function for estimating the location of a known weakly inhomogeneous object can be generated directly from noisy intensity distributions measured in a set of scattering experiments this image is a close approximation to the exact log likelihood function generated from complete data i e from both the magnitude and phase of the transmitted wavefields as long as the measurements are performed sufficiently far from the object the performance of the procedure is measured through calculation of the cramer rao bound for the estimation error and is illustrated by a computer simulation study

weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations this paper is concerned with the existence and uniqueness of a weighted pseudo almost automorphic mild solution to the semilinear fractional equation d t au t au t d t a 1f t u t t r 1 a 2 in complex banach spaces with s p weighted pseudo almost automorphic coefficients where a is a linear densely defined operator of sectorial type on a complex banach space x moreover we present an application to a fractional wave equation

stability analysis for neutral stochastic differential equation of second order driven by poisson jumps this paper studies the existence uniqueness and exponential stability in mean square for the mild solution of neutral second order stochastic partial differential equations with infinite delay and poisson jumps by utilizing the banach fixed point theorem first the existence and uniqueness of the mild solution of neutral second order stochastic differential equations is established then the mean square exponential stability for the mild solution of the stochastic system with poisson jumps is obtained with the help of an established integral inequality

remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay in this paper we obtain some results on the existence and uniqueness of solutions to stochastic functional differential equations with infinite delay at phase space bc 0 r d which denotes the family of bounded continuous r d value functions f defined on 0 with norm f sup q 0 f q under non lipschitz condition with lipschitz condition being considered as a special case and a weakened linear growth condition the solution is constructed by the successive approximation

faedo galerkin approximate solutions of a neutral stochastic fractional differential equation with finite delay abstract in this work we study a class of neutral stochastic fractional differential equation in an arbitrary separable hilbert space h we obtain an associated integral equation and then consider a sequence of approximate integral equations obtained by projection of considered associated integral equation onto finite dimensional space the existence and uniqueness of solutions to every approximate integral equation are obtained by using banach fixed point theorems and analytic semigroup theory we show the convergence of the solutions by using faedo galerkin approximations and give an example to show the effectiveness of the main theory finally we provide the conclusion at the end

singular nonlinear boundary value problems for second order ordinary differential equations abstract the boundary value problem y f x y 0 y 0 y 0 0 y 1 y 1 0 is studied under conditions which are modeled on the problem y a x y p 0y 0 y 1 0 or y 0 y 1 0 which occurs in a number of applications the technical arguments involve properties of solutions i e concavity and the use of iterative techniques a new fixed point theorem for cones for decreasing mappings is proved and used it is of interest in its own right since most such theorems are for monotone increasing mappings

weighted pseudo almost automorphic functions and applications to abstract differential equations abstract this paper is concerned with weighted pseudo almost automorphic functions which are more general and complicated than weighted pseudo almost periodic functions using these properties we establish an existence and uniqueness theorem for pseudo almost automorphic mild solutions to semilinear differential equations in a banach space which extends known results

existence uniqueness and stability of mild solutions for second order neutral stochastic evolution equations with infinite delay and poisson jumps in this paper we study a class of second order neutral stochastic evolution equations with infinite delay and poisson jumps snseeips in which the initial value belongs to the abstract space b we establish the existence and uniqueness of mild solutions for snseeips under non lipschitz condition with lipschitz condition being considered as a special case by means of the successive approximation furthermore we give the continuous dependence of solutions on the initial data by means of a corollary of the bihari inequality an application to the stochastic nonlinear wave equation with infinite delay and poisson jumps is given to illustrate the theory

existence and uniqueness attraction for stochastic age structured population systems with diffusion and poisson jump in this paper the problems about the existence and uniqueness attraction for strong solution of stochastic age structured population systems with diffusion and poisson jump are considered under the non lipschitz condition with the lipschitz condition being considered as a special case the existence and uniqueness for such systems is firstly proved by using the burkholder davis gundy inequality b d g inequality and ito s formula and then by using a novel inequality technique some sufficient conditions ensuring the existence for the domain of attraction are established as another by product the exponential stability in mean square moment of strong solution for such systems can be also discussed

existence of almost automorphic mild solutions to non autonomous neutral stochastic differential equations abstract in this paper we consider the existence and uniqueness of square mean and weighted pseudo almost automorphic mild solutions to a class of nonautonomous stochastic neutral differential equations with infinite delay in real separable hilbert spaces a new set of sufficient conditions are established for obtaining the required result to prove the main result we use banach contraction principle together with evolution family of operators some known results are generalized and improved an application to the stochastic nonlinear heat equation with infinite delay is provided to illustrate the obtained theory

approximate controllability for semilinear second order stochastic evolution systems with infinite delay in this work we study the approximate controllability for a class of semilinear second order stochastic evolution systems with infinite delay the main technique is the fundamental solution theory

positive solutions for n 1 1 three point boundary value problems with coefficient that changes sign abstract in this paper we establish existence results for positive solutions for the n 1 1 three point boundary value problems consisting of the equation u n a t f u t 0 t 0 1 with one of the following boundary value conditions u 0 u u 1 u u i 0 0 for i 1 2 n 2 and u n 2 0 u n 2 u n 2 1 u n 2 u i 0 0 for i 0 1 n 3 where 0 1 0 0 and a 0 1 r may change sign and r f 0 0 0 is a parameter our approach is based on the leray schauder degree theory this paper is motivated by eloe and henderson nonlinear anal 28 1997 1669 1680

on backward stochastic evolution equations in hilbert spaces and optimal control abstract in this paper a new result on the existence and uniqueness of the adapted solution to a backward stochastic evolution equation in hilbert spaces under a non lipschitz condition is established the applicability of this result is then illustrated in a discussion of a concrete backward stochastic partial differential equation furthermore a stochastic maximum principle for optimal control problems of stochastic systems governed by backward stochastic evolution equations in hilbert spaces is obtained

application of a maximum likelihood estimator in an experimental study in ultrasonic diffraction tomography a recently proposed approach to the inverse problem of detecting the presence and estimating the location of a known object from data collected in a set of diffraction tomographic experiments is evaluated experimental data are used to validate of the filtered backpropagation algorithms used and their robustness to modeling errors and to severe limitations in the angular coverage of the tomographic data is demonstrated a potential application to medical imaging of soft tissue is illustrated

exponential stabilization of non autonomous delayed neural networks via riccati equations this paper concerns with the problem of exponential stabilization for a class of non autonomous neural networks with mixed discrete and distributed time varying delays two cases of discrete time varying delay namely i slowly time varying and ii fast time varying are considered by constructing an appropriate lyapunov krasovskii functional in case i and utilizing the razumikhin technique in case ii we establish some new delay dependent conditions for designing a memoryless state feedback controller which stabilizes the system with an exponential convergence of the resulting closed loop system the proposed conditions are derived through solutions of some types of riccati differential equations applications to control a class of autonomous neural networks with mixed time varying delays are also discussed in this paper some numerical examples are provided to illustrate the effectiveness of the obtained results

impulsive neutral stochastic functional integro differential equations with infinite delay driven by fbm in this paper we study a class of impulsive neutral stochastic functional integro differential equations with infinite delay driven by a standard cylindrical wiener process and an independent cylindrical fractional brownian motion fbm with hurst parameter h 1 2 1 in the hilbert space we prove the existence and uniqueness of the mild solution for this kind of equations with the coefficients satisfying some non lipschitz conditions which include the classical lipschitz conditions as special case an example is provided to illustrate the theory some well known results are generalized and extended

weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations abstract this paper is concerned with the existence and uniqueness of weighted pseudo almost automorphic mild solution to the semilinear fractional equation d t u t au t d t 1 f t u t bu t t r 1 2 where a is a linear densely defined operator of sectorial type on a complex banach space x and b is a bounded linear operator defined on x under the assumption of uniformly continuity on f we establish a composition of weighted pseudo almost automorphic in a general banach space and obtain existence results by means of banach contraction mapping the results obtained are utilized to study the existence and uniqueness of a weighted pseudo almost automorphic solution to fractional diffusion wave equation with dirichlet conditions

approximate controllability of second order damped mckean vlasov stochastic evolution equations in this paper sufficient conditions for the approximate controllability of a class of second order nonlinear stochastic functional differential equations of mckean vlasov type are derived the nonlinearities at a given time t considered depend not only on the state of the solution at time t but also on the corresponding probability distribution at time t an example is given to illustrate the theory

weighted stepanov like pseudo almost automorphy and applications abstract in this paper we propose a new class of functions called weighted stepanov like pseudo almost automorphic functions which generalize in a natural fashion the concept of almost automorphy and its various extensions we systematically explore the properties of the weighted stepanov like pseudo almost automorphic functions in general banach space including a composition result as an application we establish some sufficient criteria for the existence uniqueness of the weighted pseudo almost automorphic solution to a class of partial neutral functional differential equations and also to a class of nonlinear volterra integral equations of convolution type with infinite delay in banach space some interesting examples are presented to illustrate the main findings

the existence of countably many positive solutions for a system of nonlinear singular boundary value problems with the p laplacian operator abstract in this paper we study the existence of countable many positive solutions for a class of nonlinear singular boundary value systems with p laplacian operator p 1 u a 1 t f u v 0 0 t 1 p 2 v a 2 t g u v 0 0 t 1 1 p 1 u 0 1 p 1 u 0 0 1 p 1 u 1 1 p 1 u 1 0 2 p 2 v 0 2 p 2 v 0 0 2 p 2 v 1 2 p 2 v 1 0 where p i s s p i 2 s p i 1 f g are lower semi continuous functions a i t has countable many singularities on 0 1 2 i 1 2 by using the fixed point theorem of cone expansion and compression of norm type the existence of countable many positive solutions for nonlinear singular boundary value system with p laplacian operator are obtained

a note on exponential stability for impulsive neutral stochastic partial functional differential equations in this note the problem on the exponential stability in mean square moment of mild solution to impulsive neutral stochastic partial functional differential equations is considered by employing the inequality technique some sufficient conditions are established for the concerned problem and some existing results are generalized and improved finally an illustrative example is given to demonstrate the effectiveness of the obtained result

existence results for an impulsive neutral stochastic fractional integro differential equation with infinite delay abstract in this work we consider an impulsive neutral stochastic fractional integro differential equation with infinite delays in an arbitrary separable hilbert space the existence of mild solution is obtained by using resolvent operator and fixed point theorems an example is considered to illustrate the theory and conclusion is also provided at the end of the manuscript

approximate controllability of second order semilinear stochastic system with nonlocal conditions in this paper the approximate controllability of second order semilinear stochastic system involving nonlocal conditions is studied by using sadovskii s fixed point theorem with stochastic analysis theory we derive a new set of sufficient conditions for the approximate controllability of second order semilinear stochastic system with nonlocal conditions under the assumption that the corresponding linear system is approximately controllable finally an application to a second order semilinear stochastic system with nonlocal initial condition is provided to illustrate the obtained theory

on the existence of positive solutions for semilinear elliptic equations in the annulus abstract we study the existence of positive radial solutions of u g x u 0 in annuli with dirichlet dirichlet neumann boundary conditions we prove that the problems have positive radial solutions on any annulus if is sublinear at 0 and

existence of pseudo almost automorphic mild solutions to stochastic fractional differential equations abstract fractional stochastic differential equations have gained considerable importance due to their application in various fields of science and engineering this paper is concerned with the square mean pseudo almost automorphic solutions for a class of fractional stochastic differential equations in a hilbert space the main objective of this paper is to establish the existence and uniqueness of square mean pseudo almost automorphic mild solutions to a linear and semilinear case of these equations a new set of sufficient conditions is obtained to achieve the required result by using the stochastic analysis theory and fixed point strategy finally an example is provided to illustrate the obtained theory

higher order nonlinear diffraction tomography inversion of the rytov series nonlinear tomographic reconstruction algorithms are developed for inversion of data measured in scattering experiments in which the complex phase of the wavefields is modeled by an arbitrarily large possibly infinite number of terms in the rytov series the algorithms attain the form of a volterra series of nonlinear operators with the usual filtered backpropagation algorithm of diffraction tomography as the leading linear term a computer simulation study is included to illustrate the performance of the algorithms for the case of scattering objects with cylindrical symmetry

existence uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay in this paper we obtain the existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay at phase space bc 0 r d which denotes the family of bounded continuous r d value functions f defined on 0 with norm email protected sup q 0 f q under non lipschitz condition with lipschitz condition being considered as a special case and a weakened linear growth condition the solution is constructed by the successive approximation furthermore we give the continuous dependence of solutions on the initial value by means of the corollary of bihari inequality

green s function for a third order generalized right focal problem abstract we determine the green s function for the third order three point generalized right focal boundary value problem x t 0 t 1 t t 3 x t 1 x t 2 0 x t 3 x t 3 0 where 0 and determine conditions on the coefficients and boundary points to ensure its positivity then we use the krasnoselskii and leggett and williams fixed point theorems to prove the existence of solutions to the nonlinear problem x t f t x t t 1 t t 3 x t 1 x t 2 0 x t 3 x t 3 0 for certain constraints placed on nonnegative f

approximate controllability for stochastic evolution inclusions of clarke s subdifferential type in this paper we deal with the approximate controllability of stochastic evolution inclusions of clarke s subdifferential type firstly by using stochastic analysis nonsmooth analysis theory of operator semigroups and fixed point theorems of multivalued maps we show the existence of mild solutions for the stochastic evolution inclusions then we provide a sufficient condition to guarantee the approximate controllability of the stochastic evolution inclusions actually our results cover a broader class of inclusion problems involving time depending operators finally an example is included to illustrate the applicability of the main results

on controllability of second order nonlinear impulsive differential systems abstract many dynamical systems in physics chemistry biology and engineering sciences have impulsive dynamical behaviors due to abrupt jumps at certain instants during the dynamical processes the mathematical models of such processes are called differential systems with impulse effect this paper studies the exact controllability issue of certain types of second order nonlinear impulsive control differential systems sufficient conditions are formulated and proved for the exact controllability of such systems without imposing a compactness condition on the cosine family of operators we establish controllability results by using a fixed point analysis approach finally an example is presented to illustrate the utility of the proposed result the results improve some recent results

square mean pseudo almost automorphic process and its application to stochastic evolution equations abstract this paper introduces the concept of the square mean pseudo almost automorphy for a stochastic process also it introduces the properties on the completeness and the composition of the space that consists of such processes with appropriate settings and by virtue of the theory of the semigroups of the operators to an evolution family the banach fixed point theorem and the stochastic analysis techniques this paper investigates the existence the uniqueness and the global stability of the square mean pseudo almost automorphic solutions for a general class of stochastic evolution equations finally this paper provides an illustrative example to justify the practical usefulness of the established theoretical results

second order neutral impulsive stochastic evolution equations with delay in this paper we study the second order neutral stochastic evolution equations with impulsive effect and delay snseeids we establish the existence and uniqueness of mild solutions to snseeids under non lipschitz condition with lipschitz condition being considered as a special case by the successive approximation furthermore we give the continuous dependence of solutions on the initial data by means of corollary of the bihari inequality an application to the stochastic nonlinear wave equation with impulsive effect and delay is given to illustrate the theory

the existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional brownian motion abstract in this paper we investigate the existence uniqueness and exponential asymptotic behavior of mild solutions to stochastic delay evolution equations perturbed by a fractional brownian motion b q h t d x t a x t f t x t d t g t d b q h t with hurst parameter h 1 2 1 we also consider the existence of weak solutions

existence and controllability results for stochastic fractional evolution hemivariational inequalities in this paper we study a class of stochastic evolution hemivariational inequalities with fractional derivative the existence of mild solutions and controllability results are obtained by applying fractional calculation stochastic analysis techniques a fixed point theorem of multivalued maps and properties of generalized clarke subdifferential an example is included to show the applicability of our results

periodic solutions for differential equations with state dependent delay and positive feedback existence of a nontrivial periodic orbit and a homoclinic orbit connecting 0 to the periodic orbit is shown for the differential equation x t x t f x t r r r x t with state dependent delay and positive feedback here 0 f and r are smooth real functions with r 0 1 and f 0

exponential stability for second order neutral stochastic differential equations with impulses in this paper we investigate the problem on the exponential stability of mild solution for the second order neutral stochastic partial differential equations with impulses by utilising the cosine function theory a set of novel sufficient conditions is derived by establishing an impulsive integral inequality as a final point an example is given to illustrate the effectiveness of the obtained theory

image restoration with multiplicative noise incorporating the sensor nonlinearity a linear minimum mean square error deconvolution filter in the presence of multiplicative noise is derived the importance of incorporating the nonlinear sensor characteristics into the restoration of noisy and blurred scanned photographic images is discussed it is proposed to restore images in the exposure domain where a linear convolutional relationship between the original and the observed images can be established results are presented on restoring photographic blurred images using the proposed filter in the exposure domain whereas the use of the classical wiener filter in the density domain with additive noise assumption does not yield any visible improvement

a note on exponential stability for second order neutral stochastic partial differential equations with infinite delays in the presence of impulses in this work the problem on the exponential stability for second order neutral stochastic partial differential equations with infinite delays is considered in the presence of impulses under some conditions by employing the new integral inequality technique some algebraic criteria of stability are established for the concerned problem and some existing results are generalized and improved finally an illustrative example is given to demonstrate the effectiveness of the obtained results

the existence of positive solutions for nonlinear singular boundary value system with p laplacian in this paper we study the existence of positive solutions for the following nonlinear singular boundary value problem with p laplacian f p u a t f u t 0 01 f is lower semi continuous functions by using the fixed point theorem of cone expansion and compression of norm type the existence of positive solution and infinitely many positive solutions for nonlinear singular boundary value problem p laplacian are obtained

a class of second order mckean vlasov stochastic evolution equations driven by fractional brownian motion and poisson jumps abstract this paper focuses on a class of second order mckean vlasov stochastic evolution equations driven by a fractional brownian motion and poisson jumps specifically we allow nonlinearities and the jump term to depend not only of the state of the solution but also on the corresponding probability law of the state the global existence and uniqueness of mild solutions is established under various growth conditions and a related stability result is discussed an example is presented to illustrate the applicability of the theory

square mean almost automorphic mild solutions to non autonomous stochastic differential equations in hilbert spaces in this paper we first refine the definition of square mean almost automorphic functions introduced in fu and liu 2010 11 then we prove the existence and uniqueness of square mean almost automorphic mild solutions for a class of non autonomous stochastic differential equations in a real separable hilbert space some additional properties of square mean almost automorphic functions are also provided to prove our main result we use the banach contraction mapping principle

existence and controllability results for second order impulsive stochastic evolution systems with state dependent delay in this paper we discuss a kind of impulsive second order stochastic evolution systems with state dependent delay in a real separable hilbert space the results concerning the existence and controllability of mild solutions have been addressed by means of the fixed point techniques some sufficient conditions are formulated as well as an application involving partial differential equation with impulses is presented

existence results for an impulsive abstract partial differential equation with state dependent delay in this paper we establish the existence of mild solutions for a class of impulsive abstract partial functional differential equation with state dependent delay

green s function and positive solutions of nth order m point boundary value problem abstract in this paper we study the expression and properties of green s function for nth order m point boundary value problem u n t a t f u 0 t 0 1 u 0 u 0 u n 2 0 0 u 1 i 1 m 2 i u i where 0 0 with i 1 m 2 i i n 1 1 furthermore we obtained the existence of positive solutions by meas of fixed point index theory

almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations abstract in this paper applying the theory of semigroups of operators to evolution family and banach fixed point theorem we prove the existence and uniqueness of an a almost automorphic weighted pseudo almost automorphic mild solution of the semilinear evolution equation x t a t x t f t x t in banach space under conditions

a composition theorem for weighted pseudo almost automorphic functions and applications abstract in this paper we establish a composition theorem for weighted pseudo almost automorphic functions under a weaker lipschitz condition our composition theorem generalizes some known results moreover the existence and uniqueness of pseudo almost automorphic solutions for abstract semilinear evolution equations are studied

existence and iteration of positive solutions for a generalized right focal boundary value problem with p laplacian operator abstract in the paper we obtain the existence of positive solutions and establish a corresponding iterative scheme for the following third order generalized right focal boundary value problem with p laplacian operator p u t q t f t u t 0 t 1 u 0 i 1 m i u i u 0 u 1 i 1 n i u i the main tool is the monotone iterative technique

maximum likelihood estimation of object location in diffraction tomography the problem is formulated within the context of diffraction tomography where the complex phase of the diffracted wavefield is modeled using the rytov approximation and the measurements consist of noisy renditions of this complex phase at a single frequency the log likelihood function is computed for the case of additive zero mean gaussian white noise and shown to be expressible in the form of the filtered backpropagation algorithm of diffraction tomography in this form however the filter function is no longer the rho filter appropriate to least square reconstruction but is now the generalized projection propagation of the object centered at the origin onto the line s parallel to the measurement line s but passing through the origin this result allows the estimation problem to be solved via a diffraction tomographic imaging procedure where the noisy data is filtered and backpropagated in a first step and the point of maximum value of the resulting image is then the maximum likelihood ml estimate of the object s location the authors include a calculation of the cramer rao bound for the estimation error and a computer simulation study illustrating the estimation procedure

approximate mild solutions of fractional stochastic evolution equations in hilbert spaces in this paper we study the approximate mild solutions of a class of fractional stochastic evolution equations in hilbert spaces by constructing picard type approximate sequences we present the approximate results under the non lipschitz and linear growth conditions

asymptotic stability of second order neutral stochastic differential equations in this paper we study the existence and asymptotic stability in pth moment of mild solutions to second order nonlinear neutral stochastic differential equations further this result is extended to establish stability criterion for stochastic equations with impulsive effects with the help of fixed point strategy stochastic analysis technique and semigroup theory a set of novel sufficient conditions are derived for achieving the required result finally an example is provided to illustrate the obtained result

asymptotic behavior for second order stochastic evolution equations with memory abstract in this paper a general second order integro differential evolution equation with memory driven by multiplicative noise is considered we prove the existence of global mild solution and asymptotic stability of the zero solution using lyapunov function techniques moreover we discuss three examples to show that the asymptotic stability results can be applied to various partial differential equations

a note on the neutral stochastic functional differential equation with infinite delay and poisson jumps in an abstract space in this note we prove the existence and uniqueness of the solution to neutral stochastic functional differential equation with infinite delay and poisson jumps in the phase space b with non lipschitz coefficients the known results that appeared in the studies by ren and xia appl math comput 210 72 2009 ren and xiaappl math comput 214 457 2009 and xu and hu acta appl math unpublished are generalized and improved an example is provided to illustrate the theory

a multiplicity result for a boundary value problem with infinitely many singularities abstract we consider the second order boundary value problem u t a t f u t 0 where a t lp 0 1 for some p 1 and has countably many singularities in 0 1 2 we show that there exist countably many positive solutions using holder s inequality and krasnosel skii s fixed point theorem for operators on a cone