the existence of countably many positive solutions for singular multipoint boundary value problems abstract in this paper we study the existence of countably many positive solutions for a singular multipoint boundary value problem by using fixed point index theory and the leggett williams fixed point theorem sufficient conditions for the existence of countably many positive solutions are established

existence of solutions to k n k 2 boundary value problems we provide very general existence criteria to solutions for a class of higher order boundary value problems our results supplement as well as improve several recent results established in the literature

the green s function for k n k conjugate boundary value problems and its applications abstract a new integral representation of the green s function for k n k conjugate boundary value problems is obtained its applications are also given

singular perturbations for third order nonlinear multi point boundary value problem abstract this paper is devoted to study the following third order multi point singularly perturbed boundary value problem x t f t x t x t x t 0 0 t 1 0 1 x 0 0 ax 0 bx 0 i 1 n 2 i x i a cx 1 dx 1 i 1 n 2 i x i b where a b c d 0 a b r a b 0 c d 0 i 0 i 0 i 1 2 n 2 0 1 2 n 2 1 and 0 1 2 n 2 1 the existence uniqueness and asymptotic estimates of solutions of the boundary value problem are give by using priori estimates differential inequalities technique and leray schauder degree theory

existence results for some n dimensional nonlocal boundary value problems abstract in this article we study an n th dimensional nonlocal boundary value problem for a second order differential equation of sturm liouville type by using the linear part of the boundary condition an equivalent integral equation is obtained and then the nonlinear alternative fixed point result is applied some special cases are also given

positive solutions of a nonlinear three point boundary value problem in this paper by using krasnoselskii s fixed point theorem in a cone we study the existence of single and multiple positive solutions to the three point boundary value problem bvp y t a t f y t 0 0

structured singular values with nondiagonal structures i characterizations the purpose of this two part series is to provide a robustness analysis framework for a class of problems with highly structured modeling uncertainties this framework is more general than that of the usual block diagonally structured uncertainties and it corresponds to a structure consisting of block by block matrix perturbations we study the structured singular value with respect to this structure and we establish a number of novel results for this notion this paper contains a study on the properties of the structured singular value we give an alternative characterization of this notion as the solution of a smooth optimization problem furthermore we show that under a certain circumstance the structured singular value reduces to a vector induced matrix norm

solvability of multipoint boundary value problems at resonance for higher order ordinary differential equations let f 0 1 x r n r be a continuous function and e l 1 0 1 let b j 1 j m 2 r 0 h 1 h 2 h m 2 1 be given this paper is concerned with the existence of solutions for the following n t h order multipoint boundary value problems at resonance case x n t f t x t x t x n 1 t e t email protected 0 1 x 0 x 0 x n 2 0 0 x 1 email protected email protected jx h j and x n t f t x t x t x n 1 t e t email protected 0 1 x 0 x 0 x n 2 0 0 x 1 email protected email protected jx h j some existence results are obtained by using the coincidence degree theory of mawhin

positive solutions for a three point boundary value problem the second order three point boundary value problem x t 2x t h t f t x t t 0 1 x 0 0 x x 1 is considered under some conditions concerning the first eigenvalue of the relevant linear operator where 0 1 is a constant h t is allowed to be singular at t 0 and t 1 the existence of positive solutions is studied by means of fixed point index theory

brief paper probabilistic bounds for l 1 uncertainty model validation this paper is concerned with a mixed deterministic probabilistic model validation problem which amounts to determining the probability for a given model to reproduce given experimental data we consider an additive uncertain model in which the modelling uncertainty is characterized in time domain by the 1 induced system norm the data available for validation are input output time series and are assumed to have been corrupted by a random noise for a given uncertainty norm bound our objective is to compute the probability for an uncertainty to exist so that it may satisfy the prescribed bound and match the input output measurements while the exact computation of this probability poses a formidable task we derive its upper and lower bounds this is carried out with respect to noise sequences with gaussian distribution and more generally when only such statistical information as the expectation and covariance of the noise are known

first order three point boundary value problems at resonance we consider three point boundary value problems for a system of first order equations in perturbed systems of ordinary differential equations at resonance we obtain new results for the above boundary value problems with nonlinear boundary conditions the existence of solutions is established by applying a version of brouwer s fixed point theorem which is due to miranda

model validation a connection between robust control and identification the gap between the models used in control synthesis and those obtained from identification experiments is considered by investigating the connection between uncertain models and data the model validation problem addressed is given experimental data and a model with both additive noise and norm bounded perturbations is it possible that the model could produce the observed input output data this problem is studied for the standard h sub infinity mu framework models a necessary condition for such a model to describe an experimental datum is obtained for a large class of models in the robust control framework this condition is computable as the solution of a quadratic optimization problem

multiple positive solutions of multi point boundary value problem for second order impulsive differential equations this paper is devoted to study the existence of multiple positive solutions for the second order multi point boundary value problem with impulse effects the arguments are based upon fixed point theorems in a cone an example is worked out to demonstrate the main results

green s function and positive solutions for higher order ode abstract green s function for k n k conjugate boundary value problem 1 n k x n t t f x t a x i a 0 0 i k 1 x j b 0 0 j n k 1 is obtained and its integral representation is given as applications we obtain sufficient conditions for the existence of one two or three positive solutions

big islands in dispersing billiard like potentials abstract we derive a rigorous estimate of the size of islands in both phase space and parameter space appearing in smooth hamiltonian approximations of scattering billiards the derivation includes the construction of a local return map near singular periodic orbits for an arbitrary scattering billiard and for the general smooth billiard potentials thus universality classes for the local behavior are found moreover for all scattering geometries and for many types of natural potentials which limit to the billiard flow as a parameter 0 islands of polynomial size in appear this suggests that the loss of ergodicity via the introduction of the physically relevant effect of smoothening of the potential in modeling for example scattering molecules may be of physically noticeable effect

existence of solutions to functional boundary value problem of second order nonlinear differential equation abstract in this paper a functional boundary value problem is studied based on mawhin s coincidence degree theory some existence theorems are obtained in the case of non resonance and the cases of dim ker l 1 and dim ker l 2 at resonance the results not only generalize and improve some known results of multi point and integral boundary value problems but also give some existence theorems for boundary value problems that all their boundary value conditions are relied on both x and x

time domain identification for robust control abstract in this paper we are concerned with a problem of robust control oriented system identification in the time domain based on the well known schur takagi aak theorem we propose a linear algorithm to obtain the nominal model of the plant to be identified and the minimal bound of the uncertainty of the nominal model error which is measured by h norm it is also shown that in the model set defined by the nominal model and the uncertainty bound there exists at least one model which matches the prescribed input output data given in the time domain

optimal interval lengths for nonlocal boundary value problems associated with third order lipschitz equations abstract for the third order differential equation y f x y y y where f x y 1 y 2 y 3 is lipschitz continuous in terms of y i i 1 2 3 we obtain optimal bounds on the length of intervals on which there exist unique solutions of certain nonlocal three and four point boundary value problems these bounds are obtained through an application of the pontryagin maximum principle from the theory of optimal control

positive solutions for some second order four point boundary value problems abstract in this paper the second order four point boundary value problem x t h t f t x t 0 0 t 1 x 0 a x x 1 b x is studied where 0 1 0 a b 1 and h 0 1 0 f 0 1 0 0 are nonnegative continuous functions by the use of the property of the corresponding green s function fixed point index theory leray schauder degree and upper and lower solution method some existence nonexistence and multiplicity results of positive solutions are acquired

probabilistic estimates for mixed model validation problems with cal h _ infty type uncertainties a mixed deterministic probabilistic model validation problem is investigated in this technical note which consists in an additive uncertain model with model uncertainty characterized by the h norm the data available for validation are time domain experimental data corrupted by a random noise sequence our aim is to compute the probability for such an uncertain model to be validated by the data and our main results are bounds on this probability that are computable based on the distribution of chi square random variables when the noise is a gaussian variable and solvable as an lmi problem when only statistical information such as the expectation and covariance of the noise are known

positive solutions of a nonlinear three point boundary value problem in this paper by using the fixed point index theorems we study the existence of at least one or two positive solutions to the three point boundary value problem y t a t f y t 0 0 t 1 y 0 0 y 1 y where 0 1 0 1 as an application we also give some examples to demonstrate our results

lft uncertain model validation with time and frequency domain measurements in this paper the authors study a model validation problem pertaining to linear fractional uncertainty models they extend the previous validation approaches based upon either time or frequency measurements to one using simultaneously time and frequency domain data they show that this problem can be reduced to two independent convex feasibility tests each of which corresponds to the time or frequency domain data alone and each can be verified numerically using available algorithms and software programs the merit of such a mixed time and frequency domain approach therefore lies in that not only can it accommodate the two distinctively different types of measurement data simultaneously but also from a computational standpoint it will not suffer from a complexity growth beyond that associated with the existing time domain tests

robust performance against time varying structured perturbations in this paper we consider the problem of robust performance analysis for some nominal system m z against bounded linear time varying structured feedback perturbations we introduce a natural input output notion of rate of variation for a linear time varying operator we then exhibit upper and lower bounds on the maximum rate of variation of these perturbations against which robust performance is achievable using these bounds we show that the existence of frequency dependent d scales that render the norm of m z less than one is necessary and sufficient for robust performance against arbitrarily slowly varying structured linear perturbations of norm less than one this result suggests that it is natural and well justified to deal with the frequency dependent upper bound for the spl mu norm rather than spl mu itself this is reassuring given that the upper bound is easily and reliably computable while computation of complex spl mu appears difficult

stable regions and singular trajectories in chaotic soft wall billiards abstract we present numerical and experimental results for the development of islands of stability in atom optics billiards with soft walls as the walls are soften stable regions appear near singular periodic trajectories in converging focusing and dispersing billiards and are surrounded by areas of stickiness in phase space the size of these islands depends on the softness of the potential in a very sensitive way

positive solutions for k n k conjugate boundary value problems in banach spaces abstract the existence and multiplicity of positive solutions for a integral equation in banach spaces are obtained using these results we investigate the existence of multiple positive solutions for k n k conjugate boundary value problems the arguments are based upon a specially constructed cone and the fixed point theory in a cone for strict set contraction operators this work gives an unified approach applicable to many boundary value problems

validation of linear fractional uncertain models solutions via matrix inequalities a time domain approach is provided in this paper to tackle the problem of model validation pertaining to uncertain models described by linear fractional transforms both unstructured and structured dynamic uncertainties are considered it is shown that in the first case the problem can be solved by finding a feasible solution to a convex optimization problem while in the second case it amounts to solving a biaffine matrix inequality problem to which only a convex optimization based necessary condition is given

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

model invalidation in spl lscr sub 1 using frequency domain data in this note we study the problem of invalidating uncertain models with an additive uncertainty the problem is to check the existence of an uncertainty and a measurement noise which fit to the given model structure and the uncertainty noise description as well as the experimental data used for invalidation we consider a mixed setting in which the uncertainty is characterized in time domain by the spl lscr sub 1 induced system norm while the available data are frequency response samples of the system we show that this problem which by formulation poses an infinite dimensional primal optimization problem can be solved in a dual finite dimensional space with finitely many constraints

multi point boundary value problems on time scales at resonance abstract we study the nonlinear second order differential equation on a time scale u t f t u t u t e t t 0 1 t subject to the multi point boundary conditions u 0 0 u 1 i 1 m i u i and u 0 0 u 1 i 1 m i u i where t is a time scale such that 0 1 t i 0 1 t i 1 m and f is a continuous function satisfying a caratheodory type growth condition and e is a lebesgue integrable function our existence results are obtained by applying a coincidence degree theorem due to mawhin

on computational complexity of invalidating structured uncertainty models abstract in this paper we study a class of uncertainty model validation invalidation problems for linear discrete time systems we consider models described by a linear fractional transformation lft of modeling uncertainties which are structured time invariant or time varying and are bounded in h or l 1 norms the experimental data available for use in invalidation are either input output time series observations or frequency response measurements we analyze the computational complexity associated with these problems while earlier work showed that for lft models with an unstructured uncertainty the invalidation problems can be reduced to one of solving linear matrix inequalities and hence may be tackled readily using well developed numerical methods in this paper we provide a simple proof to show that in the presence of structured uncertainties they are all np hard with respect to the number of uncertainty blocks suggesting that the problems are inherently intractable from a computational standpoint additionally we also demonstrate that the problems do become tractable when the lft model description is changed to an additive one leading to lmi based invalidation tests similar to those obtained elsewhere these results indicate that the main source of the computational difficulty lies in the lft description together with the structured nature of the uncertainties

frequency domain tests for validation of linear fractional uncertain models a frequency domain approach is adopted in this paper to tackle the problem of validating uncertainty models described by linear fractional transforms this problem amounts to verifying the consistency of certain given mathematical models to experimental information obtained from a physical plant using either input output frequency domain measurements or frequency samples of the plant linear fractional models with both unstructured and structured uncertainties are considered the problem is resolved in the former case and solved approximately in the latter both results lead to tests that are readily computable via convex optimization methods and can be implemented using standard algorithms in comparison to previously available algorithms based on time domain information the main advantage of these tests is that they have a considerably lower level of computational complexity it is shown that the validation problem reduces to one of nevanlinna pick boundary interpolation and it can be solved by computing independently a sequence of convex programs of a lower dimension each of which corresponds to only one frequency sample

computational complexity of spl mu calculation the structured singular value spl mu measures the robustness of uncertain systems numerous researchers over the last decade have worked on developing efficient methods for computing spl mu this paper considers the complexity of calculating spl mu with general mixed real complex uncertainty in the framework of combinatorial complexity theory in particular it is proved that the spl mu recognition problem with either pure real or mixed real complex uncertainty is np hard this strongly suggests that it is futile to pursue exact methods for calculating spl mu of general systems with pure real or mixed uncertainty for other than small problems

escape probability for classically chaotic systems abstract a statistical approach to study the escape of an ensemble of particles from classical chaotic systems is proposed the universal kinetic decay laws exponential and algebraic are found through the velocity angle distribution and shown to be specific of distinct motion regimes in the billiards with a small opening on the basis of the particular case of the dispersing sinai billiard the escape probability is given in explicit form and the temporal and geometrical conditions are enunciated to make controlled observations of decay dynamics in numerical experiments

robust stability with time varying structured uncertainty considers the problem of assessing robust stability in the presence of block diagonally structured time varying dynamic uncertainty it is shown that robust stability holds only if there exist constant scalings which lead to a small gain condition the notion of stability here is finite gain stability over finite energy signals in sharp contrast to the case of time invariant dynamic uncertainty this result is not limited by the number uncertainty blocks these results parallel previous results regarding finite gain stability over persistent bounded signals

positive solutions to semipositone k n k conjugate eigenvalue problems abstract in this paper we consider the existence of positive solutions to the following semipositone k n k conjugate eigenvalue problems scep 1 n k u n t f t u t 0 t 1 u i 0 0 0 i k 1 u j 1 0 0 j n k 1 where n 2 1 k n 1 and 0 is positive parameter the nonlinear function f may change sign for 0 t 1 i e we allow that the nonlinear term f is both semipositone and lower unbounded without making any monotone type assumptions by using the fixed point index theory we derive an explicit interval of such that for any in this interval the existence of at least one positive solution to the boundary value problem is guaranteed and the existence of at least two solutions for in an appropriate interval is also discussed

existence of multiple positive solutions for sturm liouville like four point boundary value problem with p laplacian abstract in this paper we consider the sturm liouville like four point boundary value problem with p laplacian p u t f t u t 0 t 0 1 u 0 u 0 u 1 u 0 where p s s p 2 s p 1 by means of a fixed point theorem for operators on a cone the existence of multiple at least three positive solutions to the above boundary value problem is obtained

a note on multi point boundary value problems abstract multi point boundary value problems for a second order ordinary differential equation are considered in this note an existence result is obtained with the help of coincidence degree theory

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

singular nonlinear k n k conjugate boundary value problems abstract for 1 k n 1 solutions are obtained for the boundary value problem 1 n k y n f x y y i 0 0 0 i k 1 and y j 1 0 0 j n k 1 where f x y is singular at y 0 an application is made of a fixed point theorem for operators that are decreasing with respect to a cone

a time domain approach to model validation in this paper we offer a novel approach to control oriented model validation problems the problem is to decide whether a postulated nominal model with bounded uncertainty is consistent with measured input output data our approach directly uses time domain input output data to validate uncertainty models the algorithms we develop are computationally tractable and reduce to generally nondifferentiable convex feasibility problems or to linear programming problems in special cases we give analytical solutions to these problems

positive solutions for p n p conjugate boundary value problems a new existence result is presented for the p n p conjugate boundary value problem 1 n py n h t y 0 t 1 withy i 0 0 0 i p 1 andy i 1 0 0 i n p 1 our nonlinearityhmay be singular aty 0 t 0 and ort 1

eigenvalues of lidstone boundary value problems we consider the following boundary value problem 1 ny 2 n lf t y n 1 t 0 1 y 2 i 0 y 2 i 1 0 0 0 the values of l are characterized so that the boundary value problem has a positive solution in addition we derive explicit intervals of l such that for any l in the interval existence of a positive solution of the boundary value problem is guaranteed several examples are also included to dwell upon the importance of the results obtained

solvability of an m point boundary value problem for second order ordinary differential equations let 0 1 r2 r be a function satisfying caratheodory s conditions and e t l1 0 1 let i 0 1 ai r all of the ai s having the same sign i 1 2 m 2 0 1 2 m 2 1 be given this paper is concerned with the problem of existence of a solution for the m point boundary value problem x t t x t x t e t t 0 1 e formula conditions for the existence of a solution for the above boundary value problem are given using the leray schauder continuation theorem

positive solutions to a type of nonlinear three point boundary value problem with sign changing nonlinearities in this paper by using krasnoselskii s fixed point theorem in a cone we study the existence of positive solutions for the second order three point boundary value problem u t a t u t lf t u t 0 0

multiplicity results for singular conjugate focal and n p problems abstract a new technique is presented which enables us to discuss the existence of one or more solutions to singular conjugate focal and n p boundary value problems our nonlinearity f t y may be singular at y 0 t 0 and or t 1

on an m point boundary value problem for second order ordinary differential equations where q e 0 1 is given we obtain conditions for the existence and uniqueness of a solution for the boundary value problem 2 using the leray schauder continuation theorem 2 we give an example of a three point boundary value problem where the existence condition is not satisfied and no solution exists gupta 3 recently studied the boundary value problem 2 when q 1 our results on the three point boundary value problem 2 extend the results of gupta 3 to the case of general cy see also 4 51 we use the classical spaces c o 11 c o 11 lk o 11 and l o l of continuous k times continuously differentiable measurable real valued functions whose kth power of the absolute value is lebesgue integrable on 0 11 or measurable functions that are essentially bounded

positive solutions of three point boundary value problems for systems of nonlinear second order ordinary differential equations abstract in this paper we study the three point boundary value problems for systems of nonlinear second order ordinary differential equations of the form u f t v t 0 1 v g t u t 0 1 u 0 v 0 0 u u 1 v v 1 under some conditions we show the existence and multiplicity of positive solutions of the above problem by applying the fixed point index theory in cones

an extension of mawhin s continuation theorem and its application to boundary value problems with a p laplacain abstract in order to study boundary value problems bvps with a p laplacian an extension of mawhin s continuation theorem is given then some existence theorems for bvps with a p laplacian at resonance are proved

on the existence of positive solutions for 2 n order singular boundary value problems by applying upper and lower solutions method the existence of c2n 2 0 1 and c2n 1 0 1 positive solution for a class of 2n order singular differential equations boundary value problems is obtained

necessary and sufficient conditions for the existence of solution to three point bvp abstract we study the existence of a solution of the three point boundary value problem p x x g x h x 0 0 x x we give sufficient and necessary conditions on g and h in order for p to have a real solution our conditions are of landesman lazer type and depend on the real values of and

positive solutions of a nonlinear m point boundary value problem let ai 0 for i 1 m 3 and am 2 0 let i satisfy 0 1 2 m 2 1 and m 2i 1 ai i 1 we study the existence of positive solutions to the boundary value problem where a c 0 1 0 and f c 0 0 we show the existence of at least one positive solution if f is either superlinear or sublinear by applying a fixed point theorem in cones

positive solutions for semipositone k n k conjugate boundary value problems for 1 k n 1 positive solutions are obtained for the boundary value problem 1 n k y n fx y x 0 1y i 0 0 0 i k 1 y j 1 0 0 j n k 1 where f x y m and m is a positive constant we show the existence of positive solutions by using a fixed point theorem in cones

positive solutions of m point boundary value problems the second order m point boundary value problem x h x f x 0 0 x 1 0 0 1 i 1m 2ai i is considered under some conditions concerning the first eigenvalue of the relevant linear operator where i 0 1 with 0 1 2 m 2 1 ai 0 with i 1m 2ai 1 h x is allowed to be singular at x 0 and x 1 the existence of positive solutions and multiple positive solutions is obtained by means of fixed point index theory similarly conclusions hold for some other m point boundary value conditions

on the existence of positive solution for a kind of multi point boundary value problem at resonance abstract this paper concerns the existence of positive solution for a class of second order m point boundary value problems under different resonant conditions by using the leggett williams norm type theorem due to o regan and zima we obtain the existence of positive solution an example is given to demonstrate the main results

positive solutions of singular three point boundary value problems for second order differential equations in this paper we study the existence of positive solutions of a singular three point boundary value problem for the following second order differential equation y ma t f t y t 0 t 0 1 y 0 by 0 0 y 1 ay h where m 0 is a parameter b 0 0 0 by constructing an available integral operator and combining fixed point index theory with properties of green s function under some conditions concerning the first eigenvalues corresponding to the relevant linear operator the sufficient conditions of the existence of positive solutions for the boundary value problems are established the interesting point of the results is that the term a t may be singular at t 0 and or t 1 moreover the nonlinear f t x is also allowed to have singularity at x 0

solvability of a second order multi point boundary value problem at resonance abstract based on the coincidence degree theory of mawhin we get a general existence result for the following second order multi point boundary value problem at resonance x t f t x t x t e t t 0 1 x 0 i 1 m i x i x 1 j 1 n j x j where f 0 1 r 2 r is a caratheodory function e l 1 0 1 0 1 2 m 1 i r i 1 2 m m 2 and 0 1 n 1 j r j 1 n n 1 in this paper both of the boundary value conditions are responsible for resonance