initial value problems for systems of integrodifferential equations in banach spaces in this paper by the use of new comparison results and mixed monotone iterative techniques the existence of solutions of initial value problems for systems of integrodifferential equations in banach spaces is discussed the results obtained in this paper generalize and improve the results corresponding to those obtained by others our result is new even in finite dimensional spaces

existence results for urysohn integral inclusions involving the henstock integral this paper presents two existence results for urysohn integral inclusions in banach spaces the set valued integral involved being the henstock integral in particular the existence of continuous solutions for integral inclusions of volterra and of hammerstein type is obtained our results extend the existence theorems given in literature in the single or set valued case under bochner or pettis integrability hypothesis the compactness of solutions set is also studied

improved delay dependent stability criteria for uncertain lur e systems with sector and slope restricted nonlinearities and time varying delays abstract this paper deals with the absolute stability analysis for uncertain time delay lur e systems that have time varying delays and sector and slope restricted nonlinearities new delay dependent stability criteria are derived via linear matrix inequality lmi formulation that can be easily solved by various optimization techniques sector bounds and slope bounds are employed to the lyapunov krasovskii functional through convex representation of the nonlinearities so that less conservative stability conditions are obtained numerical examples show effectiveness of the proposed stability condition

a quantitative central limit theorem for the excursion area of random spherical harmonics over subdomains of s2 in recent years considerable interest has been drawn by the analysis of geometric functionals for the excursion sets of random eigenfunctions on the unit sphere spherical harmonics in this paper we extend those results to proper subsets of the sphere s2 i e spherical caps focussing in particular on the excursion area precisely we show that the asymptotic behaviour of the excursion area is dominated by the so called second order chaos component and we exploit this result to establish a quantitative central limit theorem in the high energy limit these results generalize analogous findings for the full sphere their proofs however require more sophisticated techniques in particular a careful analysis of some independent interest for smooth approximations of the indicator function for spherical cap subsets

on the area of excursion sets of spherical gaussian eigenfunctions the high frequency behaviour for random eigenfunctions of the spherical laplacian has been recently the object of considerable interest also because of strong motivation arising from physics and cosmology in this paper we are concerned with the high frequency behaviour of excursion sets in particular we establish a uniform central limit theorem for the empirical measure i e the proportion of spherical surface where spherical gaussian eigenfunctions lie below a level z our proofs borrow some techniques from the literature on stationary long memory processes in particular we expand the empirical measure into hermite polynomials and establish a uniform weak reduction principle entailing that the asymptotic behaviour is asymptotically dominated by a single term in the expansion as a result we establish a functional central limit theorem the limiting process is fully degenerate

fixed point theorems for set valued maps and existence principles for integral inclusions new fixed point theorems of monch type are presented for set valued maps these theorems are then used to establish general existence principles for hammerstein integral inclusions in banach spaces

decomposition of besov and triebel lizorkin spaces on the sphere abstract a discrete system of almost exponentially localized elements needlets on the n dimensional unit sphere s n is constructed it shown that the needlet system can be used for decomposition of besov and triebel lizorkin spaces on the sphere as an application of besov spaces on s n a jackson estimate for nonlinear m term approximation from the needlet system is obtained

solvability of inclusions of hammerstein type abstract we establish a universal rule for solving operator inclusions of hammerstein type in lebesgue bochner spaces with the aid of some recently proven continuation theorem of leray schauder type for the class of so called admissible multimaps examples illustrating the legitimacy of this approach include the initial value problem for perturbation of m accretive multivalued differential equation the nonlocal cauchy problem for semilinear differential inclusion abstract integral inclusion of fredholm and volterra type and the two point boundary value problem for nonlinear evolution inclusion

on nonlinear functionals of random spherical eigenfunctions we prove central limit theorems and stein like bounds for the asymptotic behaviour of nonlinear functionals of spherical gaussian eigenfunctions our investigation combines asymptotic analysis of higher order moments for legendre polynomials and in addition recent results on malliavin calculus and total variation bounds for gaussian subordinated fields we discuss applications to geometric functionals like the defect and invariant statistics e g polyspectra of isotropic spherical random fields both of these have relevance for applications especially in an astrophysical environment

controllability of fractional integro differential evolution equations with nonlocal conditions this paper concerns the controllability for a class of fractional integro differential evolution equations with nonlocal initial conditions by using the fractional calculus measure of noncompactness and the monch fixed point theorem we obtain a controllability result for the nonlocal cauchy problem of the fractional integro differential evolution equations involving noncompact semigroups and the nonlocal functions without lipschitz continuity an example is given to illustrate the effectiveness of the abstract results

robust stabilization of discrete time nonlinear lur e systems with sector and slope restricted nonlinearities abstract in this paper we present a novel method for the robust control problem of uncertain nonlinear discrete time linear systems with sector and slope restrictions the nonlinear function considered in this paper is expressed as convex combinations of sector and slope bounds then the equality constraint is derived by using convex properties of the nonlinear function a stabilization criterion for the existence of the state feedback controller is derived in terms of linear matrix inequalities lmis by using finsler s lemma the proposed method is demonstrated by a system with saturation nonlinearity

on the distribution of the nodal sets of random spherical harmonics we study the volume of the nodal set of eigenfunctions of the laplacian on the m dimensional sphere it is well known that the eigenspaces corresponding to en n n m 1 are the spaces en of spherical harmonics of degree n of dimension n we use the multiplicity of the eigenvalues to endow en with the gaussian probability measure and study the distribution of the m dimensional volume of the nodal sets of a randomly chosen function the expected volume is proportional to en one of our main results is bounding the variance of the volume to be o en n in addition to the volume of the nodal set we study its leray measure we find that its expected value is n independent we are able to determine that the asymptotic form of the variance is const n

controllability of measure driven evolution systems with nonlocal conditions this paper investigates complete controllability of semilinear measure driven differential systems with nonlocal conditions without assuming the compactness of the evolution system related to the linear part of the measure system some sufficient conditions for controllability are established by using the measure of noncompactness and the monch fixed point theorem the results obtained here improve and generalize many known results

a kyp lemma and invariance principle for systems with multiple hysteresis non linearities absolute stability criteria for systems with multiple hysteresis non linearities are given in this paper it is shown that the stability guarantee is achieved with a simple two part test on the linear subsystem if the linear subsystem satis es a particular linear matrix inequality and a simple residue condition then as is proven the non linear system will be asymptotically stable the main stability theorem is developed using a combination of passivity lyapunov and popov stability theories to show that the state describing the linear system dynamics must converge to an equilibrium position of the non linear closed loop system the invariant sets that contain all such possible equilibrium points are described in detail for several common types of hystereses the class of non linearities covered by the analysis is very general and includes multiple slope restricted memoryless non linearities as a special case simple numerical examples are used to demonstrate the e ectiveness of the new analysis in comparison to other recent results and graphically illustrate state asymptotic stability

absolute stability criteria for multiple slope restricted monotonic nonlinearities absolute stability criteria such as the classical popov criterion guarantee stability for a class of sector bounded nonlinearities although the sector restriction bounds the admissible class of the nonlinearities the local slope of the nonlinearity may be arbitrarily large in this paper the authors derive absolute stability criteria for multiple slope restricted time invariant monotonic nonlinearities like the popov criterion in the single input single output case the authors results provide a simple graphical interpretation involving a straight line in a modified popov plane

nodal length fluctuations for arithmetic random waves using the spectral multiplicities of the standard torus we endow the laplace eigenspaces with gaussian probability measures this induces a notion of random gaussian laplace eigenfunctions on the torus arithmetic random waves we study the distribution of the nodal length of random eigenfunctions for large eigenvalues and our primary result is that the asymptotics for the variance is nonuniversal our result is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy

weakly singular functional integral equation in infinite dimensional banach spaces the existence of l p w e e to be a banach space and w be a bounded closed subset in the euclidean space r n of a weakly singular functional integral equation has been established the notation of measure of noncompactness and monch s fixed point theorem are the main tools in our analysis

fixed points of discontinuous multivalued increasing operators in banach spaces with applications by using relatively weakly compactness conditions we obtain existence theorems of fixed points for discontinuous multivalued increasing operators in banach spaces as an application we utilize the results obtained in section 2 to lead to the existence principles for integral inclusions of hammerstein type multivalued maps

on the initial value problem of fractional evolution equations with noncompact semigroup in this paper we study the initial value problem of fractional semilinear evolution equations with noncompact semigroup in banach spaces the existence of saturated mild solutions and global mild solutions is obtained under weaker conditions particularly we find that the results obtained are also valid for the initial value problem of fractional ordinary differential equations in banach spaces the theorems proved in this paper improve and extend some related conclusions on this topic an example is given to illustrate that our results are valuable

solvability of nonlinear impulsive volterra integral inclusions and functional differential inclusions abstract this paper presents sufficient conditions for the existence of solutions to nonlinear impulsive volterra integral inclusions and initial value problems for second order impulsive functional differential inclusions in banach spaces our results are obtained via a fixed point theorem due to hong j math anal appl 282 2003 151 162 for discontinuous multivalued increasing operators

a unified existence theory for evolution equations and systems under nonlocal conditions abstract we investigate the effect of nonlocal conditions expressed by linear continuous mappings over the hypotheses which guarantee the existence of global mild solutions for functional differential equations in a banach space a progressive transition from the volterra integral operator associated to the cauchy problem to fredholm type operators appears when the support of the nonlocal condition increases from zero to the entire interval of the problem the results are extended to systems of equations in a such way that the system nonlinearities behave independently as much as possible and the support of the nonlocal condition may differ from one variable to another

on the connection of degree theory and 0 epi maps abstract we show that in a large class of continuous operators on jordan domains the 0 epi maps are precisely those maps with nonzero degree the class contains in particular perturbations of the identity by countably 1 2 condensing operators this implies that the example of z ding 1996 j math anal appl 199 458 468 contains a mistake

extension of the leray schauder degree for abstract hammerstein type mappings abstract we introduce a new extension of the classical leray schauder topological degree in a real separable reflexive banach space the new class of mappings for which the degree will be constructed is obtained essentially by replacing the compact perturbation by a composition of mappings of monotone type it turns out that the class contains the leray schauder type maps as a proper subclass the new class is not convex thus preventing the free application of affine homotopies however there exists a large class of admissible homotopies including subclass of affine ones so that the degree can be effectively used we shall construct the degree and prove that it is unique we shall generalize the borsuk theorem of the degree for odd mappings and show that the principle of omitted rays remains valid to illuminate the use of the new degree we shall briefly consider the solvability of abstract hammerstein type equations and variational inequalities

gaussian random measures generated by berry s nodal sets we consider vectors of random variables obtained by restricting the length of the nodal set of berry s random wave model to a finite collection of possibly overlapping smooth compact subsets of mathbb r 2 our main result shows that as the energy diverges to infinity and after an adequate normalisation these random elements converge in distribution to a gaussian vector whose covariance structure reproduces that of a homogeneous independently scattered random measure a by product of our analysis is that when restricted to rectangles the dominant chaotic projection of the nodal length field weakly converges to a standard wiener sheet in the banach space of real valued continuous mappings over a fixed compact set an analogous study is performed for complex valued random waves in which case the nodal set is a locally finite collection of random points

global bifurcation of the p laplacian and related operators abstract we prove some global bifurcation theorems of rabinowitz type for a large class of operators the results apply in particular to equations involving the p laplacian

asymptotics for spherical needlets we investigate invariant random fields on the sphere using a new type of spherical wavelets called needlets these are compactly supported in frequency and enjoy excellent localization properties in real space with quasi exponentially decaying tails we show that for random fields on the sphere the needlet coefficients are asymptotically uncorrelated for any fixed angular distance this property is used to derive clt and functional clt convergence results for polynomial functionals of the needlet coefficients here the asymptotictheory is considered in the high frequency sense our proposals emerge from strong empirical motivations especially in connection with the analysis of cosmological data sets 1 introduction over the last two decades wavelets have emerged as one of the most interesting tools of statistical investigation in this paper we give an application to the statistical analysis of data sets indexed by the unit sphere s 2 this is motivated mostly by the analysis of the cosmic

semi fredholm operators and periodic solutions for linear functional differential equations abstract we deal with the inhomogeneous linear periodic equation with infinite delay of the formdx dt ax t b t xt f t whereais the generator of ac0 semigroup on a banach space assuming that it has a bounded solution we obtain several criteria on the existence and the uniqueness of periodic solutions for the equation in the general phase space b and in the concrete phase space b ucg the key of our approach is the employment of the perturbation theory of semi fredholm operators to show that the period map satisfies the condition of the fixed point theorem by chow and hale funkcial ekvac 17 1974 31 38

a characterization of 0 epi maps with a degree abstract a map of the form i f with a strictly condensing operator f is 0 epi if and only if it has nonzero degree on a component of its domain of definition this proves a conjecture of the second author in m vath j math anal appl 257 2001 223 237

fractional cauchy problems with almost sectorial operators this paper concerns the abstract cauchy problem of fractional evolution equations with almost sectorial operators the suitable mild solutions of evolution equations with riemann liouville derivative and caputo derivative are introduced respectively the existence theorems of mild solutions of cauchy problems are established the results obtained here improve and generalize some known results finally an example is given for demonstration

lmi approach to stability of a class of discrete time systems using two s complement arithmetic abstract ystematic and computationally tractable i e linear matrix inequality based approach to the global asymptotic stability of a class of discrete time systems implemented with fixed point arithmetic utilizing a very strong nonlinearity which is produced by employing two s complement overflow arithmetic is presented application of the approach to second order digital filter is discussed two third order examples illustrating the effectiveness of the approach are given

absolute stability analysis for negative imaginary systems this paper deals with absolute stability of a lur e system with positive feedback where the linear subsystem exhibits negative imaginary frequency response and the nonlinearity connected in feedback is time invariant memoryless and slope restricted the proposed absolute stability criterion requires the linear subsystem to belong to the strongly strict negative imaginary class along with that positive definiteness of a symmetric matrix needs to be ensured where the symmetric matrix is obtained by subtracting the dc gain matrix of the linear subsystem from a strictly positive diagonal matrix with elements indicating the reciprocal of the maximum slope bounds of the nonlinearities the stability criterion is proved using a lur e postnikov type lyapunov function numerical examples are presented to demonstrate the proposed results

nodal intersections for random eigenfunctions on the torus we investigate the number of nodal intersections of random gaussian laplace eigenfunctions on the standard two dimensional flat torus arithmetic random waves with a fixed smooth reference curve with nonvanishing curvature the expected intersection number is universally proportional to the length of the reference curve times the wavenumber independent of the geometry our main result prescribes the asymptotic behavior of the nodal intersections variance for smooth curves in the high energy limit remarkably it is dependent on both the angular distribution of lattice points lying on the circle with radius corresponding to the given wavenumber and the geometry of the given curve in particular this implies that the nodal intersection number admits a universal asymptotic law with arbitrarily high probability

augmented lyapunov function approach to l2 gain analysis for discrete time systems with saturation nonlinearities abstract this paper presents a method for l 2 gain analysis of discrete time systems with saturation nonlinearity by using an augmented lyapunov functional by using sector and slope properties of the saturation nonlinearity the l 2 gain performance criterion is obtained via lmi formulation the proposed lyapunov function have augmented quadratic form with states as well as the sector bounded nonlinear function numerical examples show that the proposed method is less conservative than the existing ones

novel criterion for stability of discrete time systems in a state space realization utilizing saturation nonlinearities abstract a new criterion for the global asymptotic stability of discrete time systems in a state space realization utilizing saturation nonlinearities is presented the form of the criterion is unconventional in that it offers the flexibility of using even an unsymmetric lyapunov matrix p a recently reported criterion is shown to be a special case of the present criterion an example shows the effectiveness of the new criterion

an infinite dimensional 0 epi mapping with degree zero abstract letxbe a real banach space and letd xbe an open and bounded set a mappingt d xwith 0 t d is called 0 epi mapping ondif the equationtx g x is solvable indfor every compact continuous mappingg d xwhich vanishes identically on d we show that there exists a mappingt l2 d l2such that the mappingi tis 0 epi but the degree deg i t g d 0 is well defined and equals zero for any such functiong this says that the degree theory cannot be applied directly toi t gin order to guarantee the solvability of i t x g x indfor any mappinggas above this fact provides a good justification for the study of 0 epi mappings by m furi m martelli and a vignoli ann mat pura appl 124 1980 321 343 in infinite dimensional spaces

a central limit theorem and higher order results for the angular bispectrum the angular bispectrum of spherical random fields has recently gained an enormous importance especially in connection with statistical inference on cosmological data in this paper we analyze its moments and cumulants of arbitrary order and we use these results to establish a multivariate central limit theorem and higher order approximations the results rely upon combinatorial methods from graph theory and a detailed investigation for the asymptotic behavior of coefficients arising in matrix representation theory for the group of rotations so 3

existence of solutions for nth order integro differential equations in banach spaces abstract this paper discusses the existence of solutions of initial value problems for n th order nonlinear integro differential equations of mixed type on an infinite interval in a banach space

iterative method for solutions and coupled quasi solutions of nonlinear integro differential equations of mixed type in banach spaces we consider in this paper the existence of solutions coupled minimal and maximal quasi solutions and iterative approximation of the unique solution for the following initial value problems ivp of the nonlinear integro di erential equations of mixed type in ordered banach spaces e u f t u u tu su t i u t0 x0 1 where i t0 t0 a a 0 x0 e f c i e e e e e and

stability criteria of sector and slope restricted lur e systems presents stability criteria of sector and slope restricted lur e systems in terms of linear matrix inequalities by fully exploiting inherent properties of sector and slope restrictions in the time domain interpreting the time domain criteria in the frequency domain furthermore supplies simpler expressions several examples show excellent performances of these criteria

concepts of strict positive realness and the absolute stability problem of continuous time systems after the conditions for absolute stability and positive realness are reviewed it is shown that the current definition of positive realness is inadequate to establish less conservative criteria for absolute stability in this paper the concepts of wide positive realness wpr and wide strict positive realness wspr are proposed both wpr and wspr functions may have poles in the open left and right half planes and wpr functions may have single poles on the imaginary axis including j using the concepts of wpr and wspr less conservative criteria including algebraic and frequency domain criteria are presented for absolute stability of slope restricted multiple nonlinearity feedback systems refined results for non slope restricted nonlinearity feedback systems are also proposed it is pointed out that there is no spectral factorization for wpr and wspr functions if they have poles at both j and j this can be used to explain the deficiencies of algebraic criteria presented in previous publications

stein malliavin approximations for nonlinear functionals of random eigenfunctions on s d abstract we investigate stein malliavin approximations for nonlinear functionals of geometric interest for random eigenfunctions on the unit d dimensional sphere s d d 2 all our results are established in the high energy limit i e as the corresponding eigenvalues diverge in particular we prove a quantitative central limit theorem for the excursion volume of gaussian eigenfunctions this goal is achieved by means of several results of independent interest concerning the asymptotic analysis for the variance of moments of gaussian eigenfunctions the rates of convergence in various probability metrics for hermite subordinated processes and quantitative central limit theorems for arbitrary polynomials of finite order or general square integrable nonlinear transforms some related issues were already considered in the literature for the 2 dimensional case s 2 our results are new or improve the existing bounds even in these special circumstances proofs are based on the asymptotic analysis of moments of all order for gegenbauer polynomials and make extensive use of the recent literature on so called fourth moment theorems by nourdin and peccati

the asymptotic stability of nonlinear lur e systems with multiple slope restrictions the authors present the analysis of the asymptotic stability of multiple slope restricted nonlinear lur e systems by providing a lyapunov function they obtain a matrix language criterion in terms of algebraic riccati equations and linear matrix inequalities which are discussed at the point of computational issues additionally they consider the frequency domain interpretation of the result

absolute stability analysis of discrete time systems with composite quadratic lyapunov functions a generalized sector bounded by piecewise linear functions was introduced in a previous paper for the purpose of reducing conservatism in absolute stability analysis of systems with nonlinearity and or uncertainty this paper will further enhance absolute stability analysis by using the composite quadratic lyapunov function whose level set is the convex hull of a family of ellipsoids the absolute stability analysis will be approached by characterizing absolutely contractively invariant aci level sets of the composite quadratic lyapunov functions this objective will be achieved through three steps the first step transforms the problem of absolute stability analysis into one of stability analysis for an array of saturated linear systems the second step establishes stability conditions for linear difference inclusions and then for saturated linear systems the third step assembles all the conditions of stability for an array of saturated linear systems into a condition of absolute stability based on the conditions for absolute stability optimization problems are formulated for the estimation of the stability region numerical examples demonstrate that stability analysis results based on composite quadratic lyapunov functions improve significantly on what can be achieved with quadratic lyapunov functions

structure of the solution set to volterra integral inclusions and applications abstract the topological and geometric structure of the solution set to volterra integral inclusions in banach spaces is investigated it is shown that the set of solutions in the sense of the aumann integral is nonempty compact acyclic in the space of continuous functions or is even an r set provided some appropriate conditions on the banach space are imposed applications to the periodic problem for this type of inclusions are given

solvability of the cauchy problem for infinite delay equations abstract in this paper we study the solvability of the cauchy problem for functional differential equations with infinite delay in a banach space x which are more general than those in many previous publications we undertake our study under suitable hypotheses based on noncompactness measures and kamke type functions and in a general setting of admissible phase space p including the space lp 0 x therefore our results extend and unify many existing results for finite and infinite delay equations as a sample of possible applications we analyze the solvability of the cauchy problem for a schrodinger type partial functional integro differential equation in l1 r

mixed monotone iterative technique for a class of semilinear impulsive evolution equations in banach spaces abstract this paper deals with the existence of mild l quasi solutions to the initial value problem for a class of semilinear impulsive evolution equations in an ordered banach space e under a new concept of upper and lower solutions a new monotone iterative technique on the initial value problem of impulsive evolution equations has been established the results improve and extend some relevant results in ordinary differential equations and partial differential equations an example is also given

monotone iterative technique for addressing impulsive integro differential equations in banach spaces abstract in this paper we use a monotone iterative technique in the presence of lower and upper solutions to discuss the existence of solutions for the initial value problem of the impulsive integro differential equation of volterra type in a banach space e u t f t u t t u t t j t t k u t t k i k u t k k 1 2 m u 0 x 0 where f c j e e e j 0 a 0 t 1 t 2 t m a and i k c e e k 1 2 m under wide monotone conditions and the noncompactness measure condition of nonlinearity f we obtain the existence of extremal solutions and a unique solution between lower and upper solutions our result improves and extends some relevant results in abstract differential equations

a stability inequality for nonlinear feedback systems with slope restricted nonlinearity using lyapunov s direct method a stability inequality in frequency domain is derived for nonlinear feedback systems with slope restricted nonlinearity in the present approach a transformed system that involves the slope of the nonlinearity is considered thus leading to a stability inequality that incorporates the slope information but not the sector information of the nonlinearity the approach is essentially an extension of a recent idea of the author

approximate controllability of nonlocal and impulsive neutral integro differential equations using the resolvent operator theory and an approximating technique abstract our purpose in this paper is to study the approximate controllability of nonlocal and impulsive neutral integro differential equations using the resolvent operator theory the approximating technique semigroup theory and the theory of fixed point we also derive a variation of constants formula for representing a solution of the given system by resolvent operators and then study the existence of a mild solution of the system we prove the main results by taking the impulsive functions to be continuous only and taking the nonlocal initial condition function to be lipschitz continuous in the first case and continuous only in the second case finally the main results are illustrated using examples

monotone iterative technique for second order delayed periodic problem in banach spaces in this paper we deal with the existence of periodic solutions for second order functional differential equation with delay in e u t f t u t u t t r where e is an ordered banach space f r e e e is a continuous function which is periodic in t and 0 is a constant we first build a new maximum principle for the periodic solutions of the corresponding linear equation with delay with the aid of this maximum principle under the assumption that the nonlinear function is quasi monotonicity we study the existence of the minimal and maximal periodic solutions for abstract delayed equation by combining perturbation method and monotone iterative technique of the lower and upper solutions

high resolution asymptotics for the angular bispectrum of spherical random fields in this paper we study the asymptotic behavior of the angular bispectrum of spherical random fields here the asymptotic theory is developed in the framework of fixed radius fields which are observed with increasing resolution as the sample size grows the results we present are then exploited in a set of procedures aimed at testing non gaussianity for these statistics we are able to show convergence to functionals of standard brownian motion under the null hypothesis analytic results are also presented on the behavior of the tests in the presence of a broad class of non gaussian alternatives the issue of testing for non gaussianity on spherical random fields has recently gained enormous empirical importance especially in connection with the statistical analysis of cosmic microwave background radiation

comparison results for initial value problems of second order impulsive integro differential inclusions abstract in this paper we present new maximum and minimum principles for the initial value problem for a class of second order nonlinear impulsive integro differential inclusions with discontinuous multivalued functions on the right hand side in a real banach space this allows us to lead on to the development of the comparison principle and monotone iterative technique for a nonlinear second order impulsive integro differential inclusion