on the existence and nonexistence of positive solutions for nonlinear sturm liouville boundary value problems abstract in this paper the existence and nonexistence results of positive solutions are obtained for sturm liouville boundary value problem p x u q x u f x u x 0 1 a u 0 b p 0 u 0 0 c u 1 d p 1 u 1 0 where p c 1 0 1 q c 0 1 p x 0 q x 0 for x 0 1 f c 0 1 r a b c d 0 are constants and satisfy a b c d 0 the discussion is based on the positivity estimation for the green s function of associated linear boundary value problem and the fixed point index theory in cones

theory and design of signal adapted fir paraunitary filter banks we study the design of signal adapted fir paraunitary filter banks using energy compaction as the adaptation criterion we present some important properties that globally optimal solutions to this optimization problem satisfy in particular we show that the optimal filters in the first channel of the filter bank are spectral factors of the solution to a linear semi infinite programming sip problem the remaining filters are related to the first through a matrix eigenvector decomposition we discuss uniqueness and sensitivity issues the sip problem is solved using a discretization method and a standard simplex algorithm we also show how regularity constraints may be incorporated into the design problem to obtain globally optimal in the energy compaction sense filter banks with specified regularity we also consider a problem in which the polyphase matrix implementation of the filter bank is constrained to be dct based such constraints may also be incorporated into our optimization algorithm therefore we are able to obtain globally optimal filter banks subject to regularity and or computational complexity constraints numerous experiments are presented to illustrate the main features that distinguish adapted and nonadapted filters as well as the effects of the various constraints the conjecture that energy compaction and coding gain optimization are equivalent design criteria is shown not to hold for fir filter banks

new existence theorems of positive solutions for sturm liouville boundary value problems by constructing available operators some new existence theorems of positive solutions are obtained for the sturm liouville boundary value problem p t u t email protected t f t u t 0 a 1u 0 email protected 1p 0 u 0 0 a 2u 1 email protected 2p 1 u 1 0 where a t 0 and l 0 is a parameter

granular neural networks with evolutionary interval learning to deal with different membership functions of the same linguistic term a new interval reasoning method using new granular sets is proposed based on yin yang methodology to make interval valued granular reasoning efficiently and optimize interval membership functions based on training data effectively a granular neural network gnn with a new high speed evolutionary interval learning is designed simulation results in nonlinear function approximation and bioinformatics have shown that the gnn with the evolutionary interval learning is able to extract interval valued granular rules effectively and efficiently from training data by using the new evolutionary interval learning algorithm

two dimensional filter bank design for optimal reconstruction using limited subband information in this correspondence we propose design techniques for analysis and synthesis filters of 2 d perfect reconstruction filter banks prfb s that perform optimal reconstruction when a reduced number of subband signals is used based on the minimization of the squared error between the original signal and some low resolution representation of it the 2 d filters are optimally adjusted to the statistics of the input images so that most of the signal s energy is concentrated in the first few subband components this property makes the optimal prfb s efficient for image compression and pattern representations at lower resolutions for classification purposes by extending recently introduced ideas from frequency domain principal component analysis to two dimensions we present results for general 2 d discrete nonstationary and stationary second order processes showing that the optimal filters are nonseparable particular attention is paid to separable random fields proving that only the first and last filters of the optimal prfb are separable in this case simulation results that illustrate the theoretical achievements are presented

invariant sets of descending flow in critical point theory with applications to nonlinear differential equations abstract properties of invariant sets of descending flow defined by a pseudogradient vector field of a functional in a banach space are studied in this way several critical points can be found by constructing different invariant sets on which the functional is bounded below under suitable conditions the existence of at least four critical points of a functional is proved each critical point being in a certain invariant set the theoretical results are applied to nonlinear elliptic boundary value problems and nonlinear systems of ordinary differential equations in variant cases at least four solutions are obtained for these equations

first order impulsive ordinary differential equations with anti periodic and nonlinear boundary conditions the monotone iterative technique is a powerful method that have been used to approximate the extremal solutions of several problems 1 2 on the other hand impulsive differential equations are a basic tool to study some problems of biology medicine engineering and physics 3 in this paper we deal with the following nonlinear boundary value problem for first order differential equation with impulses at fixed points

existence and nonexistence of solutions of second order nonlinear boundary value problems abstract we study the nonlinear boundary value problem consisting of the equation y q t y w t f y on a b and a general separated homogeneous linear boundary condition by comparing this problem with a corresponding linear sturm liouville problem we obtain conditions for the existence and nonexistence of solutions of this problem more specifically let n n 0 1 2 be the n th eigenvalues of the corresponding linear sturm liouville problem then under certain assumptions the boundary value problem has a solution with exactly n zeros in a b if n is in the interior of the range of f y y y 0 and does not have any solution with exactly n zeros in a b if n is outside of the range of f y y y 0 these conditions become necessary and sufficient when f y y is monotone the existences of multiple and even an infinite number of solutions are derived as consequences we also discuss the changes of the number and the types of nontrivial solutions as the interval a b shrinks as q increases in a given direction and as the boundary condition changes

roughness of a fuzzy set an integration between the theories of fuzzy sets and rough sets has been attempted by providing a measure of roughness of a fuzzy set several properties of this new measure are established some of the possible applications for handling uncertainties in the field of pattern recognition are mentioned

variational methods to sturm liouville boundary value problem for impulsive differential equations abstract in this paper we study a linear sturm liouville impulsive problem and a nonlinear sturm liouville impulsive problem by applying a new approach via critical point theory and variational methods the existence results of positive solutions are obtained

on the optimality of ideal filters for pyramid and wavelet signal approximation the reconstructed lowpass component in a quadrature mirror filter qmf bank provides a coarser resolution approximation of the input signal since the outputs of the two qmf branches are orthogonal the transformation that provides the maximum energy compaction in the lowpass channel is also the one that results in the minimum approximation error this property is used as a common strategy for the optimization of qmf banks orthogonal wavelet transforms and least squares pyramids a general solution is derived for the qmf bank that provides the optimal decomposition of an arbitrary wide sense stationary process this approach is extended to the continuous case to obtain the minimum error approximation of a signal at a given sampling rate in particular it is shown that the sine wavelet transform is optimal for the representation at all scales of signals with non increasing spectral density

a shooting reproducing kernel hilbert space method for multiple solutions of nonlinear boundary value problems in this work an iterative method is proposed to predict and demonstrate the existence and multiplicity of solutions for nonlinear boundary value problems in addition the proposed method is capable of calculating analytical approximations for all branches of solutions this method is a combination of reproducing kernel hilbert space method and a shooting like technique which takes advantage of two powerful methods for solving nonlinear boundary value problems the formulation and implementation of this iterative method is discussed for nonlinear second order with two and three point boundary value problems also the convergence of the proposed method is proved to demonstrate the computational efficiency the mentioned method is implemented for some nonlinear exactly solvable differential equations including strongly nonlinear bratu equation and nonlinear reaction diffusion equation it is also applied successfully to two nonlinear three point boundary value problems with unknown exact solutions in the last example a new branch of solutions is found which shows the power of the method to search for multiple solutions and indicates that the method may be successful in cases where purely analytic methods are not

multidimensional paraunitary principal component filter banks in this correspondence the one dimensional 1 d principal component filter banks pcfb s derived by tsatsatsanis and giannakis 1995 are generalized to higher dimensions as presented by tsatsatsanis and giannakis pcfb s minimize the mean squared error mse when only q out of p subbands are retained previously two dimensional 2 d pcfb s were proposed by tirakis et al 1995 the work by tirakis et al was limited to 2 d signals and separable resampling operators the formulation presented here is general in that it can easily accommodate signals of arbitrary yet finite dimension and nonseparable sampling a major result presented in this paper is that in addition to minimizing mse when reconstructing from q out of p subbands the pcfb s result in maximizing theoretical coding gain tcg thereby performing optimally in a energy compaction sense

existence of sign changing solutions for some asymptotically linear three point boundary value problems abstract in this paper by using the topological degree theory and the fixed point index theory the existence of three kinds of solutions i e sign changing solutions positive solutions and negative solutions for asymptotically linear three point boundary value problems is obtained

multiple positive solutions of dirichlet boundary value problems for second order impulsive differential equations abstract this paper is devoted to study the existence of multiple positive solutions for the second order dirichlet boundary value problem with impulse effects the main results here is the generalization of liu and li l liu f y li multiple positive solution of nonlinear two point boundary value problems j math anal appl 203 1996 610 625 for ordinary differential equations existence is established via the theory of fixed point index in cones

a note on bifurcation from an interval we study the global bifurcation of nonlinear sturm liouville problems of the form pu qu lambda a x f u b 0 u 0 c 0 u 0 0 b 1 u 1 c 1 u 1 0 which are not linearizable in any neighborhood of the origin c 2005 published by elsevier ltd

on sign changing solutions for nonlinear operator equations abstract in this paper the existence of sign changing solutions for nonlinear operator equations is discussed by using the topological degree and fixed point index theory the main theorems are some new three solution theorems which are different from the famous amann s and leggett williams three solution theorems as well as the results in f li g han generalization for amann s and leggett williams three solution theorems and applications j math anal appl 298 2004 638 654 these three solutions are all nonzero one of them is positive another is negative and the third one is a sign changing solution furthermore the theoretical results are successfully applied to both integral and differential equations

principal component filter banks for optimal multiresolution analysis an important issue in multiresolution analysis is that of optimal basis selection an optimal p band perfect reconstruction filter bank prfb is derived in this paper which minimizes the approximation error in the mean square sense between the original signal and its low resolution version the resulting prfb decomposes the input signal into uncorrelated low resolution principal components with decreasing variance optimality issues are further analyzed in the special case of stationary and cyclostationary processes by exploiting the connection between discrete time filter banks and continuous wavelets an optimal multiresolution decomposition of l sub 2 r is obtained analogous results are also derived for deterministic signals some illustrative examples and simulations are presented

global attractivity of positive periodic solutions for an impulsive delay periodic model of respiratory dynamics in this paper we shall consider the following nonlinear impulsive delay differential equation x t v t x t xn t m n xn t m t a e t 0 t tk x tk 1 1 bk x tk k 1 2 where m and n are positive integers v t and t are positive periodic continuous functions with period 0 in the nondelay case m 0 we show that the above equation has a unique positive periodic solution x t which is globally asymptotically stable in the delay case we present sufficient conditions for the global attractivity of x t our results imply that under the appropriate periodic impulsive perturbations the impulsive delay equation shown above preserves the original periodic property of the nonimpulsive delay equation in particular our work extends and improves some known results

toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic there are three basic concepts that underlie human cognition granulation organization and causation informally granulation involves decomposition of whole into parts organization involves integration of parts into whole and causation involves association of causes with effects granulation of an object a leads to a collection of granules of a with a granule being a clump of points objects drawn together by indistinguishability similarity proximity or functionality for example the granules of a human head are the forehead nose cheeks ears eyes etc in general granulation is hierarchical in nature a familiar example is the granulation of time into years months days hours minutes etc modes of information granulation ig in which the granules are crisp c granular play important roles in a wide variety of methods approaches and techniques crisp ig however does not reflect the fact that in almost all of human reasoning and concept formation the granules are fuzzy f granular the granules of a human head for example are fuzzy in the sense that the boundaries between cheeks nose forehead ears etc are not sharply defined furthermore the attributes of fuzzy granules e g length of nose are fuzzy as are their values long short very long etc the fuzziness of granules their attributes and their values is characteristic of ways in which humans granulate and manipulate information

multiple positive solutions of singular two point boundary value problems for second order impulsive differential equations we study the existence behavior of positive solutions to a singular two point boundary value problem of second order impulsive equation with fixed moments proofs are based on combining a method of upper and lower solutions with fixed point index theorems on a cone

new approaches to fuzzy rough feature selection there has been great interest in developing methodologies that are capable of dealing with imprecision and uncertainty the large amount of research currently being carried out in fuzzy and rough sets is representative of this many deep relationships have been established and recent studies have concluded as to the complementary nature of the two methodologies therefore it is desirable to extend and hybridize the underlying concepts to deal with additional aspects of data imperfection such developments offer a high degree of flexibility and provide robust solutions and advanced tools for data analysis fuzzy rough set based feature fs selection has been shown to be highly useful at reducing data dimensionality but possesses several problems that render it ineffective for large datasets this paper proposes three new approaches to fuzzy rough fs based on fuzzy similarity relations in particular a fuzzy extension to crisp discernibility matrices is proposed and utilized initial experimentation shows that the methods greatly reduce dimensionality while preserving classification accuracy

multiple solutions of impulsive sturm liouville boundary value problem via lower and upper solutions and variational methods abstract in this paper we prove the existence of multiple solutions for second order sturm liouville boundary value problem l u f x u x 0 1 r 1 u 0 r 2 u 0 where l u p x u q x u is a sturm liouville operator r 1 u u 0 u 0 r 2 u u 1 u 1 the technical approach is fully based on lower and upper solutions and variational methods the interesting point is that the existence of four solutions and seven solutions is given

signal adapted multiresolution transform for image coding the authors consider the problem of designing multiresolution transforms that are adapted to the given image signal in the sense that they maximize the coding gain at each resolution level a simple alternating optimization algorithm is derived for solving this problem in the framework of the lattice realization of para unitary quadrature mirror filters the resulting large coding scheme is discussed in some detail and its performance is compared with that of the discrete cosine transform jpeg technique and with that of some nonadapted multiresolution transforms

theory and design of optimum fir compaction filters the problem of optimum fir energy compaction filter design for a given number of channels m and a filter order n is considered the special cases where n m and n spl infin have analytical solutions that involve eigenvector decomposition of the autocorrelation matrix and the power spectrum matrix respectively we deal with the more difficult case of m n spl infin for the two channel case and for a restricted but important class of random processes we give an analytical solution for the compaction filter that is characterized by its zeros on the unit circle this also corresponds to the optimal two channel fir filter bank that maximizes the coding gain under the traditional quantization noise assumptions with a minor extension this can also be used to generate optimal wavelets for the arbitrary m channel case we provide a very efficient suboptimal design method called the window method the method involves two stages that are associated with the above two special cases as the order increases the suboptimality becomes negligible and the filter converges to the ideal optimal solution we compare the window method with a previously introduced technique based on linear programming

the upper and lower solution method for some fourth order boundary value problems abstract in this paper we are concerned with the fourth order two point boundary value problem u i v t f t u t u t u t u t 0 t 1 u 0 u 1 u 0 u 1 0 by placing certain restrictions on the nonlinear term f we obtain the existence results for the fourth order two point boundary value problem via the lower and upper solution method in particular a new truncating technique and an appropriate nagumo type condition are introduced and employed

theory and design of m channel maximally decimated quadrature mirror filters with arbitrary m having the perfect reconstruction property based on the concept of losslessness in digital filter structures this paper derives a general class of maximally decimated m channel quadrature mirror filter banks that lead to perfect reconstruction the perfect reconstruction property guarantees that the reconstructed signal hat x n is a delayed version of the input signal x n i e hat x n x n n 0 it is shown that such a property can be satisfied if the alias component matrix ac matrix for short is unitary on the unit circle of the z plane the number of channels m is arbitrary and when m is two the results reduce to certain recently reported 2 channel perfect reconstruction qmf structures a procedure based on recently reported fir cascaded lattice structures is presented for optimal design of such fir m channel filter banks design examples are included

periodic boundary value problem for non lipschitzian impulsive functional differential equations we consider a class of first order impulsive functional differential equations where the functional dependence is not necessarily a lipschitzian function the new maximum principle improves and extends previous results and uniqueness of solution between a lower and an upper solution for a particular nonlinear problem is presented we give conditions for existence of extremal solutions in an interval delimited by a lower and an upper solution

fuzzy rough granular neural networks fuzzy granules and classification we introduce a fuzzy rough granular neural network frgnn model based on the multilayer perceptron using a back propagation algorithm for the fuzzy classification of patterns we provide the development strategy of the network mainly based upon the input vector initial connection weights determined by fuzzy rough set theoretic concepts and the target vector while the input vector is described in terms of fuzzy granules the target vector is defined in terms of fuzzy class membership values and zeros crude domain knowledge about the initial data is represented in the form of a decision table which is divided into subtables corresponding to different classes the data in each decision table is converted into granular form the syntax of these decision tables automatically determines the appropriate number of hidden nodes while the dependency factors from all the decision tables are used as initial weights the dependency factor of each attribute and the average degree of the dependency factor of all the attributes with respect to decision classes are considered as initial connection weights between the nodes of the input layer and the hidden layer and the hidden layer and the output layer respectively the effectiveness of the proposed frgnn is demonstrated on several real life data sets

theory of optimal orthonormal subband coders the theory of the orthogonal transform coder and methods for its optimal design have been known for a long time we derive a set of necessary and sufficient conditions for the coding gain optimality of an orthonormal subband coder for given input statistics we also show how these conditions can be satisfied by the construction of a sequence of optimal compaction filters one at a time several theoretical properties of optimal compaction filters and optimal subband coders are then derived especially pertaining to behavior as the number of subbands increases significant theoretical differences between optimum subband coders transform coders and predictive coders are summarized finally conditions are presented under which optimal orthonormal subband coders yield as much coding gain as biorthogonal ones for a fixed number of subbands

periodic solutions for ordinary differential equations with sublinear impulsive effects abstract the continuation method of topological degree is used to investigate the existence of periodic solutions for ordinary differential equations with sublinear impulsive effects the applications of the abstract approach include the generalizations of some classical nonresonance theorem for impulsive equations for instance the existence theorem for asymptotically positively homogeneous differential systems and the existence theorem for second order equations with landesman lazer conditions

periodic boundary value problems for second order impulsive integrodifferential equations of mixed type in banach spaces abstract in this paper the author obtains an existence theorem of minimal and maximal solutions for the periodic boundary value problems of second order impulsive integrodifferential equations of mixed type in banach space by means of the monotone iterative technique and cone theory based on a comparison result

the monotone iterative technique for impulsive hybrid set valued integro differential equations abstract we develop the monotone method for impulsive hybrid set integro differential equations in all its generality some interesting observations are presented

nontrivial solutions of singular sublinear sturm liouville problems abstract the singular superlinear sturm liouville problems l x h x f x 0 x 1 r 1 1 0 1 0 0 r 2 2 1 2 1 0 are considered under some conditions concerning the first eigenvalues corresponding to the relevant linear operators where l x p x x q x x and h x is allowed to be singular at both x 0 and x 1 in particular f is not necessary to be nonnegative the existence results of nontrivial solutions and positive solutions are given by means of the topological degree theory

periodic boundary value problems of second order impulsive differential equations abstract this paper discusses periodic boundary value problems of second order impulsive differential equations by using schaeffer s fixed point theorem and monotone iterative technique some existence results are obtained

fucik spectrum sign changing and multiple solutions for semilinear elliptic boundary value problems with resonance at infinity in this paper we use fucik spectrum ordinary differential equation theory of banach spaces to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity especially with resonance at infinity and obtain new results on the existence of multiple solutions and sign changing solutions with a weakening of the p s condition in one case we get up to seven nontrivial solutions the techniques have independent interest

fir principal component filter banks two dimensional 2 d principal component filter banks pcfbs of finite impulse response fir are proposed for 2 d signals among all uniform paraunitary fir analysis synthesis filter banks the fir pcfbs have the most energy compaction and maximize the arithmetic mean to geometric mean ratio am gm ratio of subband variances which is the theoretic coding gain tcc of the systems under proper assumptions the theoretic proof and design techniques are provided several special cases are discussed experimental results show the potential power of the fir pcfbs

periodic boundary value problems for a class of second order impulsive integro differential equations in banach spaces abstract this paper investigates periodic boundary value problems for a class of second order nonlinear impulsive integro differential equations of mixed type in a banach space by establishing a comparison result criteria on the existence of maximal and minimal solutions are obtained

multiple nonnegative solutions for second order impulsive differential equations we study the existence of single and multiple solutions to a second order impulsive equation with fixed moments our analysis relies on combining a nonlinear alternative of leray schauder type with krasnoselskii s fixed point theorem in a cone

rough sets and boolean reasoning abstract in this article we discuss methods based on the combination of rough sets and boolean reasoning with applications in pattern recognition machine learning data mining and conflict analysis

sign changing solutions of kirchhoff type problems via invariant sets of descent flow we obtain sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems using variational methods and invariant sets of descent flow

sign changing and multiple solutions of kirchhoff type problems without the p s condition abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper we use minimax methods and invariant sets of descent flow to prove two interesting existence theorems for the following 4 superlinear kirchhoff type problems without the p s condition one concerning the existence of a nontrivial solution and the other one concerning the existence of sign changing solutions and multiple solutions a b u 2 u f x u in u 0 on

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

multiplicity results for sturm liouville boundary value problems multiplicity results for sturm liouville boundary value problems are obtained proofs are based on variational methods

the multiplicity of solutions for perturbed second order hamiltonian systems with impulsive effects abstract in this paper we study the existence of multiple solutions for a class of second order impulsive hamiltonian systems we give some new criteria for guaranteeing that the impulsive hamiltonian systems with a perturbed term have at least three solutions by using a variational method and some critical points theorems of b ricceri we extend and improve on some recent results finally some examples are presented to illustrate our main results

a multiplicity result for p lapacian boundary value problems via critical points theorem abstract in this paper we deal with the existence of at least three classical solutions for the following two point boundary value problem p u f t u 0 1 u a 2 u a 0 1 u b 2 u b 0 where p s s p 2 s p 1 is a constant is a positive parameter a b r a b our main tool is a recent three critical point theorem of ricceri our emphasis here is constructing an appropriate separable and reflexive real banach space

new comparison results for impulsive integro differential equations and applications we prove some new maximum principles for ordinary integro differential equations this allows us to introduce a new definition of lower and upper solutions which leads to the development of the monotone iterative technique for a periodic boundary value problem related to a nonlinear first order impulsive integro differential equation

generalized fuzzy rough sets determined by a triangular norm the theory of rough sets has become well established as an approach for uncertainty management in a wide variety of applications various fuzzy generalizations of rough approximations have been made over the years this paper presents a general framework for the study of t fuzzy rough approximation operators in which both the constructive and axiomatic approaches are used by using a pair of dual triangular norms in the constructive approach some definitions of the upper and lower approximation operators of fuzzy sets are proposed and analyzed by means of arbitrary fuzzy relations the connections between special fuzzy relations and the t upper and t lower approximation operators of fuzzy sets are also examined in the axiomatic approach an operator oriented characterization of rough sets is proposed that is t fuzzy approximation operators are defined by axioms different axiom sets of t upper and t lower fuzzy set theoretic operators guarantee the existence of different types of fuzzy relations producing the same operators the independence of axioms characterizing the t fuzzy rough approximation operators is examined then the minimal sets of axioms for the characterization of the t fuzzy approximation operators are presented based on information theory the entropy of the generalized fuzzy approximation space which is similar to shannon s entropy is formulated to measure uncertainty in t generalized fuzzy rough sets a notion of fuzziness is introduced some basic properties of this measure are examined for a special triangular norm t min it is proved that the measure of fuzziness of the generalized fuzzy rough set is equal to zero if and only if the set is crisp and definable

attribute selection with fuzzy decision reducts rough set theory provides a methodology for data analysis based on the approximation of concepts in information systems it revolves around the notion of discernibility the ability to distinguish between objects based on their attribute values it allows to infer data dependencies that are useful in the fields of feature selection and decision model construction in many cases however it is more natural and more effective to consider a gradual notion of discernibility therefore within the context of fuzzy rough set theory we present a generalization of the classical rough set framework for data based attribute selection and reduction using fuzzy tolerance relations the paper unifies existing work in this direction and introduces the concept of fuzzy decision reducts dependent on an increasing attribute subset measure experimental results demonstrate the potential of fuzzy decision reducts to discover shorter attribute subsets leading to decision models with a better coverage and with comparable or even higher accuracy

the existence of positive solutions for the sturm liouville boundary value problems abstract for the sturm liouville boundary value problem p t u t f t u t 0 0 t 1 1 u 0 1 p 0 u 0 0 2 u 1 2 p 1 u 1 0 where 0 we shall use a fixed point theorem in a cone to obtain the existence of positive solutions for on a suitable interval

impulsive periodic boundary value problems of first order differential equations in this paper by using schaeffer s theorem we prove new existence theorems for a nonlinear periodic boundary value problem of first order differential equations with impulses our results improve and generalize some known results

sign changing and multiple solutions of the sturm liouville boundary value problem via invariant sets of descending flow abstract in this paper we prove the existence of sign changing and multiple solutions for the second order sturm liouville boundary value problem l u f x u x 0 1 r 1 u 0 r 2 u 0 where l u p x u q x u is the sturm liouville operator r 1 u u 0 u 0 and r 2 u u 1 u 1 the technical approach is fully based on minimax methods and invariant sets of descending flow

multiple solutions of some fourth order boundary value problems abstract for some fourth order boundary value problems several new existence theorems on multiple positive negative and sign changing solutions are obtained the critical point theory and the supersolution and subsolution method are employed to discuss this problem