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

quasi solutions vector lyapunov functions and monotone method the notion of quasi solution which was introduced earlier has been developed in this paper further the notion of coupled quasi solutions and coupled maximal and minimal solutions has been introduced it is shown that the idea of quasi solutions leads to isolated subsystems and has obtained error estimates between solutions and quasi solutions also monotone iterative techniques have been developed to obtain coupled maximal and minimal quasi solutions

technical communique optimal evaluation of generalized euler angles with applications to control given two linearly independent matrices in so 3 z 1 and z 2 every rotation matrix x f so 3 can be written as the product of alternate elements from the one dimensional subgroups corresponding to z 1 and z 2 namely x f e z 1 t 1e z 2 t 2e z 1 t 3 e z 1 t s the parameters t i i 1 s are called generalized euler angles in this paper the minimum number of factors required for the factorization of x f so 3 as a function of x f is evaluated an algorithm is given to determine the generalized euler angles in the optimal factorization the results can be applied to the bang bang control with minimum number of switches of some classical and quantum systems

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

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

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

generalizing the continued fraction algorithm to arbitrary dimensions this paper presents for the first time a higher dimensional continued fraction algorithm abbreviated cfa that produces diophantine approximations of more than linear goodness on input x 1 cdots x n 1 in bf r it produces vectors p 1 k cdots p n 1 k q k in bf z n k 1 2 cdots such that max limits 1 leq i leq n 1 left x i frac p i k q k right leq frac x cdot text const n q k 1 1 2n n 1 by a theorem of dirichlet there is no algorithm that replaces the term 1 2n n 1 by a term bigger than 1 n 1 the higher dimensional cfa s analyzed so far do not achieve better than max 1 leq i leq n 1 x i p i k q k leq o 1 q k the o 1 term decreases with k but is not known to be related with q k other properties of the cfa are also generalized by the algorithm on input x 1 cdots x n 1 it starts with the standard basis of bf z n and then c

integral boundary value problems for first order integro differential equations with deviating arguments this paper deals with first order integro differential equations of mixed type with deviating arguments we investigate the existence of solutions of such problems with integral boundary conditions by establishing a comparison result and applying the monotone iterative technique to obtain corresponding results we also discuss first order differential inequalities with deviating arguments two examples demonstrate the application of our results

general methods to control right invariant systems on compact lie groups and multilevel quantum systems for a right invariant system on a compact lie group g i present two methods to design a control to drive the state from the identity to any element of the group the first method under appropriate assumptions achieves exact control to the target but requires estimation of the size of a neighborhood of the identity in g and solution of a nonlinear algebraic equation the second method does not involve any mathematical difficulty and obtains control to a desired target with arbitrary accuracy a third method is then given combining the main ideas of the previous methods this is also very simple in its formulation and turns out to be generically more efficient as illustrated by one of the examples i consider the methods described in the paper provide arbitrary constructive control for any right invariant system on a compact lie group in particular the results can be applied to the coherent control of general multilevel quantum systems to an arbitrary target

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

existence theorems of global solutions for nonlinear volterra type integral equations in banach spaces in this paper first a new fixed point theorem is established and then by the use of it the existence theorems of global solutions for nonlinear volterra type integral equations in banach spaces are investigated the results obtained in this paper generalize and improve the results corresponding to those obtained by others

controllability for a class of integro differential evolution equations involving non local initial conditions abstractthis paper deals with the exact controllability for a class of first order integro differential evolution equations involving non local initial conditions by using sadovskii fixed point theorem exact controllability results are obtained without assuming the compactness and lipschitz conditions for non local functions an example is given to illustrate the effectiveness of the abstract results

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

stable methods for vortex sheet motion in the presence of surface tension boundary integral techniques provide a convenient way to study the evolution of an interface between inviscid liquids several studies have revealed that standard numerical approximations tend to lead to unstable methods and various remedies have been introduced and tested in this paper we conduct a stability analysis of the linearized equations with a particular objective in mind the determination of how the discrete system fails to capture the physical dispersion relation precisely for the available discrete modes we discover two reasons for the typical failure in numerical discretizations one is the inability of the mesh to represent the vorticity created by surface tension effects on the finest scale and the other is the inaccuracies in the evaluation of the boundary integral for the velocity with the insight gained from our linear analysis we propose a new method that is spectrally accurate and linearly stable further the exact dispersion relation is obtained for all the available discrete modes numerical tests suggest that the method is also stable in the nonlinear regime however our method runs into difficulties generic to methods based on lagrangian motion the markers accumulate near a stagnation point on the interface forcing us to use an ever decreasing time step in our explicit method we introduce a redistribution of markers to overcome this difficulty when we redistribute according to equal arclength we find excellent agreement with a method based on preserving equal spacing in arclength

a monotone iterative scheme for nonlinear reaction diffusion systems having nonmonotone reaction terms abstract a monotone scheme is proposed for the solution of weakly coupled systems of reaction diffusion equations without any monotonicity property of the nonlinear reaction terms the considerations are taken within the framework of weak solutions

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

monotone iterative technique for the elastic systems with structural damping in banach spaces in this paper we use a monotone iterative technique in the presence of lower and upper solutions to discuss the existence of mild solutions to a second order evolution equation initial value problem in an ordered banach space which can model an elastic system with structural damping under monotone condition and the noncompactness measure condition on nonlinearity we obtain the existence of extremal mild solutions and positive mild solutions moreover some applications are given to illustrate our theoretical results

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

axisymmetric vortex sheet motion accurate evaluation of the principal value integral this paper concerns the accurate evaluation of the principal value integral governing axisymmetric vortex sheet motion previous quadrature rules for this integral lose accuracy near the axis of symmetry an approximation by de bernadinis and moore dbm that converges pointwise at the rate of o h3 has maximal errors near the axis that are o h as a result the discretization error is not smooth it contains high wavenumber frequencies that make it difficult to resolve the vortex sheet motion this paper explains the reason for the degeneracy near the axis and proposes a modified quadrature rule that is uniformly o h3 the results are based on an analytic approximation of the integrand whose integral can be precomputed the modification is implemented at negligible additional cost per timestep as an example it is applied to compute the evolution of an initially spherical vortex sheet

positive solutions for nonlinear operator equations and several classes of applications in this paper we study a class of nonlinear operator equations x ax x 0 on ordered banach spaces where a is a monotone generalized concave operator using the properties of cones and monotone iterative technique we establish the existence and uniqueness of solutions for such equations in particular we do not demand the existence of upper lower solutions and compactness and continuity conditions as applications we study first order initial value problems and two point boundary value problems with the nonlinear term is required to be monotone in its second argument in the end applications to nonlinear systems of equations and to nonlinear matrix equations are also considered

application of adaptive quadrature to axi symmetric vortex sheet motion studies of the formation of fine structures on free surfaces in liquids such as curvature singularities or interface pinching demand that the motion of the interface must be computed very accurately boundary integral techniques are a popular choice in such studies because they reduce the dimension of the problem by one on the other hand the boundary integrals are singular and their accurate evaluation can prove quite challenging in two dimensional motion the interface is just a curve when this curve is closed or periodic the singularity in the integrand may be removed and the trapezoidal rule may be applied with spectral accuracy unfortunately the nature of the singularity in the integrand for three dimensional motion is much more difficult to treat in this paper we present an accurate adaptive quadrature to compute the motion of a vortex sheet in axi symmetric flow the technique is based on a vector potential formulation which offers some computational advantages over other methods based on the biot savart integral direct numerical computations show that our technique is much more accurate and efficient than existing techniques we apply our technique to study the evolution of an initially spherical vortex sheet we present evidence of the formation of a 3 2 power singularity in the curvature of the vortex sheet

a monotone method for the equation uxyz f comparison theorems for the nonlinear boundary value problem u x y z f x y z u u x u y u z u x y u y z u z x with the boundary conditions u o y z a y z u x 0 z b z x u x y 0 c x y a y 0 c 0 y a 0 z b z 0 b 0 x c x 0 are established under certain conditions then it is shown that there exist sequences of lower and upper solutions which converge uniformly and monotonically to minimal and maximal solutions respectively

computation of axisymmetric suction flow through porous media in the presence of surface tension abstract the effect of small surface tension on a class of axisymmetric flows with suction is studied numerically the dynamic evolution of a blob of incompressible viscous fluid surrounded by air and drawn into an interior sink is considered the velocity field of the viscous fluid is assumed to satisfy darcy s law and thus the motion is that of a flow through porous media the fluid interface motion is computed using a highly accurate boundary integral method this method combines recent numerical techniques to achieve efficient and high order space and time discretizations for this type of flow through accurate computations it is shown that in the presence of small surface tension the dynamic behavior of the axisymmetric flows is very similar to that of the two dimensional hele shaw counterparts a long finger develops in the fluid interface and forms a cone singularity when it reaches the sink before all the fluid is sucked out the finger bulges and develops a neck for sufficiently small surface tension the bulge neck formation is enhanced by the additional component of the mean curvature in the three dimensional flow however its effect is not strong enough to cause the interface to pinch off at the neck for the data considered here

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

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

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

vortex blob methods applied to interfacial motion we develop a boundary integral method for computing the motion of an interface separating two incompressible inviscid fluids the velocity integral is regularized so that the vortex sheet on the interface is replaced by a sum of blobs of vorticity the regularization allows control of physical instabilities we design a class of high order blob methods and analyze the errors numerical tests suggest that the blob size should be scaled with the local spacing of the interfacial markers for a vortex sheet in one fluid with a first order kernel we obtain a spiral roll up similar to krasny j comput phys 65 1986 292 but the higher order kernels lead to more detailed structure we verify the accuracy of the new method by computing a liquid gas interface with rayleigh taylor instability we then apply the method to the more difficult case of rayleigh taylor flow separating two fluids of positive density a case for which the regularization appears to be essential as found by kerr and tryggvason both j comput phys 76 1988 48 75 1988 253 we use a blob regularization in certain local terms in the evolution equations as well as in the velocity integral we find strong evidence that improved spatial resolution with fixed blob size leads to a converged regularized solution without numerical instabilities however it is not clear that there is a weak limit as the regularization is decreased

strong solutions for differential equations in abstract spaces abstract let e f be a locally convex space we denote the bounded elements of e by e b x e x f sup f x in this paper we prove that if b e b is relatively compact with respect to the f topology and f i e b e b is a measurable family of f continuous maps then for each x 0 e b there exists a norm differentiable i e differentiable with respect to the f norm local solution to the initial valued problem u t t f t u t u t 0 x 0 all of this machinery is developed to study the lipschitz stability of a nonlinear differential equation involving the hardy littlewood maximal operator

picard and weakly picard operators technique for nonlinear differential equations in banach spaces abstract in this paper we study nonlocal cauchy problems for differential equations in banach spaces using picard and weakly picard operators technique and suitable bielecki norms some existence uniqueness and data dependence results are obtained under some mild conditions as an application we also discuss a class of impulsive cauchy problems by adapting the same methods

notions of controllability for bilinear multilevel quantum systems in this note we define four different notions of controllability of physical interest for multilevel quantum mechanical systems these notions involve the possibility of driving the evolution operator as well as the state of the system we establish the connections among these different notions as well as methods to verify controllability

a note on the theorem of lakshmikantham and leela we study the initial value problem u f t u u 0 u 0 in a real banach space using the fixed point principle of daher and the theorem of monch and von harten we obtain a theorem which is an extension of that established by lakshmikantham and leela in 1

monotone iterative technique for delay differential equations in abstract cones a monotone iterative technique is developed for delay differential equations in a banach space by utilizing the method of upper and lower solutions

on the controllability of quantum mechanical systems the systems theoretic concept of controllability is elaborated for quantum mechanical systems sufficient conditions being sought under which the state vector can be guided in time to a chosen point in the hilbert space h of the system the schrodinger equation for a quantum object influenced by adjustable external fields provides a state evolution equation which is linear in and linear in the external controls thus a bilinear control system for such systems the existence of a dense analytic domain d in the sense of nelson together with the assumption that the lie algebra associated with the system dynamics gives rise to a tangent space of constant finite dimension permits the adaptation of the geometric approach developed for finite dimensional bilinear and nonlinear control systems conditions are derived for global controllability on the intersection of d with a suitably defined finite dimensional submanifold of the unit sphere sh in h several soluble examples are presented to illuminate th

monotone iterative technique for differential equations in a banach space iffis continuous steps i and iii are standard and straightforward it is step ii that deserves attention this in turn leads to three possibilities namely to show that the sequence of approximate solutions is a a cauchy sequence b relatively compact so that one can appeal to ascoli s theorem and c a monotone sequence in a cone the first two possibilities are well known and are discussed in 2 31 this paper is devoted to the investigation of c which leads to the development of a monotone iterative technique in an arbitrary cone

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

numerical simulations of inviscid capillary pinchoff inviscid capillary pinchoff is studied numerically for an axisymmetric model problem in which a bubble pinches at two points on the symmetry plane breaking into two symmetric end bubbles and a satellite bubble in between results are presented for a range of density jumps across the bubble the numerical method uses a formulation in terms of arclength and tangent angle and incorporates a new procedure to redistribute the computational points dynamically in order to maintain resolution in regions of high curvature the results are compared with alternative computations by lepinnen and lister phys fluids 15 2003 568 where available new results include details about the cone crater structure near pinchoff and the dependence of satellite bubble volume and pinchoff time on the density ratio

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

high order quadratures for the evaluation of interfacial velocities in axi symmetric stokes flows we propose new high order accurate methods to compute the evolution of axi symmetric interfacial stokes flow the velocity at a point on the interface is given by an integral over the surface quadrature rules to evaluate these integrals are developed using asymptotic expansions of the integrands both locally about the point of evaluation and about the poles where the interface crosses the axis of symmetry the local expansions yield methods that converge to the chosen order pointwise for fixed evaluation point the pole expansions yield corrections that remove maximal errors of low order introduced by singular behaviour of the integrands as the evaluation point approaches the poles an interesting example of roundoff error amplification due to cancellation is also addressed the result is a uniformly accurate fifth order method second order pointwise fifth order and uniform fifth order methods are applied to compute three sample flows each of which presents a different computational difficulty an initially bar belled drop that pinches in finite time a drop in a strain flow that approaches a steady state and a continuously extending drop in each case the fifth order methods significantly improve the ability to resolve the flow the examples furthermore give insight into the effect of the corrections needed for uniformity we determine conditions under which the pointwise method is sufficient to obtain resolved results and others under which the corrections significantly improve the results

boundary value problems for systems of nonlinear integro differential equations with deviating arguments by establishing a comparison result and using the method of upper and lower solutions and the monotone iterative technique we investigate the systems of nonlinear mixed integro differential equations with deviating arguments and mixed boundary value conditions sufficient conditions are established for the existence of extremal system of solutions of the given problem

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

an algorithm for approximate solving of differential equations with maxima an algorithm for constructing two sequences of successive approximations of the solution of the initial value problem for nonlinear differential equations with maxima is given this algorithm is based on the monotone iterative technique it is proved that both sequences are monotonically convergent each term of the constructed sequences is a solution of an initial value problem for linear differential equations with maxima and it is a lower upper solution of the given problem both the scalar case and the multidimensional case are studied an example solved by computer realization of the suggested algorithm illustrates the practical application of the method

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

uniform finite generation of compact lie groups abstract consider a compact connected lie group g and the corresponding lie algebra l let x1 xm be a set of generators for the lie algebra l we prove that g is uniformly finitely generated by x1 xm this means that every element k g can be expressed as k ext1ext2 extl where the indeterminates x are in the set x1 xm t i r i 1 l and the number l is uniformly bounded this extends a previous result by f lowenthal in that we do not require the connected one dimensional lie subgroups corresponding to the x i i 1 m to be compact we link the results to the existence of universal logic gates in quantum computing and discuss the impact on bang bang control algorithms for quantum mechanical systems