existence results for dirichlet problems with q p laplacian abstract we prove an existence result for a class of nonlinear systems involving a duality mapping defined on a product spaces as an application we obtained an existence result for a system with q p laplacian

three solutions for a dirichlet boundary value problem involving the p laplacian abstract in this paper we prove the existence of at least three weak solutions for the problem p u f x u a x u p 2 u in u 0 on our main tool is a recent three critical points theorem of b ricceri on a three critical points theorem arch math basel 75 2000 220 226

multiplicity results for a class of gradient systems depending on two parameters abstract in this paper we prove the existence of at least three solutions for a class of two point boundary value systems of the type u i u i f u i x u 1 u n g u i x u 1 u n u i a u i b 0 for 1 i n the approach is fully based on a recent three critical points theorem of ricceri b ricceri a three critical points theorem revisited nonlinear anal 70 9 2009 3084 3089

multiple solutions for an eigenvalue problem involving p laplacian type operators abstract in this paper we establish the existence of two nontrivial weak solutions of some eigenvalue problems involving a general elliptic operator in divergence form first we study a one parameter problem under homogeneous dirichlet boundary conditions in bounded domains and then a two parameters problem under inhomogeneous nonlinear robin boundary conditions in regular bounded domains these results complete in several directions some recent papers where a ricceri type critical point theorem was used the main argument of this paper is based on two recent theorems due to arcoya and carmona 2007 in 2 which we use in a slightly modified version see section 2

existence of three positive solutions for some second order boundary value problems in this paper a new fixed point theorem of functional type in a cone is established with using the new fixed point theorem and imposing growth conditions on the nonlinearity the existence of three positive solutions for the boundary value problemx t f t x t x t 0 0 0 is continuous finally an example is given to illustrate the importance of results obtained

on multiple fixed sign solutions of a discrete system with hermite boundary conditions abstract we consider the discrete system m u i k p i k u 1 k u 2 k u n k k 0 1 n 1 i n together with hermite boundary conditions j u i k 0 j 0 m 1 1 r 1 i n where r 2 m 1 for 1 r 1rm m and k s are integers such that 0 k1

multiple fixed sign solutions for a system of difference equations with sturm liouville conditions we consider the following system of difference equations d mu i k p i k u 1 k u 2 k u n k 0 k 0 1 n i 1 2 ntogether with sturm liouville boundary conditions d ju i 0 0 0 2 n m 1 z 0 w 0 h 0 d w z w n 1 z d h w 0 by using two different fixed point theorems we develop criteria for the existence of three solutions of the system which are of fixed signs on 0 1 n m examples are also included to illustrate the results obtained

a singular boundary value problem for the one dimensionalp laplacian abstract the singular boundary value problem formula is studied in this paper whereg s s p 2s p 1 the singularity may appear atu 0 and att 0 ort 1 and the functionfmay be discontinuous the authors prove that for anyp 1 and for any positive nonincreasing functionfand nonnegative measurable functionkwith some integrability conditions the above mentioned problem has a unique solution also the properties of the solution are discussed in the paper

the weak solution of a functional differential equation in a general banach space abstract the existence of a unique weak lipschitz continuous solution of the abstract functional differential equation u t a t u t g t u t t 0 t fde u t t t r 0 is obtained by a fixed point argument here x is a general banach space a t d a t x x is m accretive t 0 t and g is lipschitz continuous in both variables stability properties of this weak solution are given and a first order hyperbolic pde to which the theory can be applied is considered

optimal robust classifiers qualitatively a filter is said to be robust if its performance degradation is acceptable for distributions close to the one for which it is optimal that is the one for which it has been designed this paper adapts the signal processing theory of optimal robust filters to classifiers the distribution class conditional distributions to which the classifier is to be applied is parameterized by a state vector and the principle issue is to choose a design state that is optimal in comparison to all other states relative to some measure of robustness a minimax robust classifier is one whose worst performance over all states is better than the worst performances of the other classifiers defined at the other states a bayesian robust classifier is one whose expected performance is better than the expected performances of the other classifiers the state corresponding to the bayesian robust classifier is called the maximally robust state minimax robust classifiers tend to give too much weight to states for which classification is very difficult and therefore our effort is focused on bayesian robust classifiers whereas the signal processing theory of robust filtering concentrates on design with full distributional knowledge and a fixed number of observation variables features design via training from sample data and feature selection are so important for classification that robustness optimality must be considered from these perspectives in particular for small samples in this context for a given sample size we will be concerned with the maximally robust state feature pair all definitions are independent of the classification rule however applications are only considered for linear and quadratic discriminant analysis for which there are parametric forms for the optimal discriminants

a minimax inequality and its applications to ordinary differential equations the aim of this paper is to investigate the minimax inequality which plays a fundamental role in the critical points theorem of b ricceri below equivalent formulations are shown and characterization is proved in particular for a special class of functionals as an application a multiplicity result for an ordinary dirichlet problem is emphasized

multiplicity results for a differential inclusion problem with non standard growth abstract in this paper we examine the multiplicity of solutions of a differential inclusion problem involving p x laplacian of the type p p x u v x u p x 2 u j 1 x u x j 2 x u x in u n 0 on by using the nonsmooth version of ricceri variational principle we get three critical points of the corresponding energy motreanu panagiotopoulos type functional which are the solutions of p

on positive radial entire solutions of second order quasilinear elliptic systems abstract we study positive radial entire solutions of second order quasilinear elliptic systems of the form p u f x u v q v g x u v x r n sufficient and or necessary conditions of f and g are obtained for to have positive radial entire solutions with prescribed asymptotic behavior at infinity we first study the upper and lower estimates for the nonlinear integral operator which is inverse of one dimensional polar form of the n dimensional m laplace operator

multiplicity theorems for the dirichlet problem involving the p laplacian abstract in this paper we establish some results on the existence of at least three weak solutions for dirichlet problems involving the p laplacian by a variational approach

a three critical points theorem revisited abstract in this paper the following result is proved let x be a reflexive real banach space i r an interval x r a sequentially weakly lower semicontinuous c 1 functional bounded on each bounded subset of x whose derivative admits a continuous inverse on x j x r a c 1 functional with compact derivative assume that lim x x j x for all i and that there exists r such that sup i inf x x x j x inf x x sup i x j x then there exist a non empty open set a i and a positive real number r with the following property for every a and every c 1 functional x r with compact derivative there exists 0 such that for each 0 the equation x j x x 0 has at least three solutions in x whose norms are less than r

a semigroup approach to nonlinear nonautonomous neutral functional differential equations abstract a class of nonlinear nonautonomous neutral functional differential equations is studied by associating with it an evolution equation in the space of initial data the space w1 1 existence and regularity results are proved

multiple solutions for a neumann system involving subquadratic nonlinearities abstract in this paper we consider the model semilinear neumann system n u a x u c x f u u v in v b x v c x f v u v in u v 0 on where r n is a smooth open bounded domain denotes the outward unit normal to 0 is a parameter a b c l 0 and f c 1 r 2 r 0 is a nonnegative function which is subquadratic at infinity two nearby numbers are determined in explicit forms and with 0 such that for every 0 system n has only the trivial pair of solution while for every system n has at least two distinct nonzero pairs of solutions

three solutions for a class of quasilinear elliptic systems involving the p q laplacian abstract the existence of at least three weak solutions is established for a class of quasilinear elliptic systems involving the p q laplacian with dirichlet boundary condition our technical approach is based on the three critical points theorem obtained by b ricceri

multiple positive solutions for some p laplacian boundary value problems this paper deals with the existence of multiple positive solutions for the one dimensional p laplacian p x t q t f t x t x t 0 t 0 1 subject to one of the following boundary conditions p x 0 p x 0 0 p x 1 p x 1 0 or x 0 g1 x 0 0 x 1 g2 x 1 0 where p s s p 2s p 1 by means of a fixed point theorem due to avery and peterson sufficient conditions are obtained that guarantee the existence of at least three positive solutions the interesting point is the nonlinear term f is involved with the first order derivative explicitly

some remarks on a three critical points theorem abstract some remarks on a strict minimax inequality which plays a fundamental role in ricceri s three critical points theorem are presented as a consequence some recent applications of ricceri s theorem to nonlinear boundary value problems are revisited by obtaining more precise conclusions

nonlinear difference equations investigated via critical point methods abstract the multiplicity of solutions for nonlinear difference equations involving the p laplacian is investigated the approach is based on critical point theorems in the setting of finite dimensional banach spaces

calculus of variations for a boundary value problem of differential system on the half line boundary value problem of second order differential equations on the half line is investigated in this paper the method mainly depends on the calculus of variations to the classical functionals sufficient conditions are obtained for the existence of the solutions

existence of two local minima for functionals on reflexive banach spaces abstract in this paper we establish an existence theorem of two local minima for functionals on reflexive banach space as a consequence we obtain an existence theorem of three critical points which improves the conclusion of a result in a recent paper of g bonanno concerning some remarks on a three critical points theorem established by b ricceri

functional evolution equations abstract of concern is the functional evolution equation du dt bu f t u t t 0 u 0 s c 1 r 0 x where ut s u t s r 0 s r 0 and x is a real banach space it is shown by the method of lines combined with the crandall pazy theorem that for the initial data u0 in a generalized domain this equation has a limit solution which is lipschitz continuous in t and that this limit solution is a unique strong one if further assumptions on b are imposed and that the zero solution is asymptotically stable an application is given to a class of partial functional differential boundary value problems

existence of multiple solutions for quasilinear systems via fibering method existence and multiplicity results for a general class of radial quasilinear equations are proved using the fibering method introduced and developed by s i pohozaev

a critical points approach for the existence of multiple solutions of a dirichlet quasilinear system abstract in this paper we establish the existence of at least three classical solutions for the dirichlet quasilinear elliptic system p i 1 u i x p i 2 u i x f u i x u 1 u n g u i x u 1 u n h i x u i x a b u i a u i b 0 for i 1 n our main tool is a recent three critical points theorem of b ricceri a three critical points theorem revisited nonlinear anal 70 2009 3084 3089

determination of the optimal number of features for quadratic discriminant analysis via the normal approximation to the discriminant distribution given the joint feature label distribution increasing the number of features always results in decreased classification error however this is not the case when a classifier is designed via a classification rule from sample data typically for fixed sample size the error of a designed classifier decreases and then increases as the number of features grows the problem is especially acute when sample sizes are very small and the potential number of features is very large to obtain a general understanding of the kinds of feature set sizes that provide good performance for a particular classification rule performance must be evaluated based on accurate error estimation and hence a model based setting for optimizing the number of features is needed this paper treats quadratic discriminant analysis qda in the case of unequal covariance matrices for two normal class conditional distributions the qda classifier is determined according to a discriminant the standard plug in rule estimates the discriminant from a feature label sample to obtain an estimate of the discriminant by replacing the means and covariance matrices by their respective sample means and sample covariance matrices the unbiasedness of these estimators assures good estimation for large samples but not for small samples our goal is to find an essentially analytic method to produce an error curve as a function of the number of features so that the curve can be minimized to determine an optimal number of features we use a normal approximation to the distribution of the estimated discriminant since the mean and variance of the estimated discriminant will be exact these provide insight into how the covariance matrices affect the optimal number of features we derive the mean and variance of the estimated discriminant and compare feature size optimization using the normal approximation to the estimated discriminant with optimization obtained by simulating the true distribution of the estimated discriminant optimization via the normal approximation to the estimated discriminant provides huge computational savings in comparison to optimization via simulation of the true distribution feature size optimization via the normal approximation is very accurate when the covariance matrices differ modestly the optimal number of features based on the normal approximation will exceed the actual optimal number when there is large disagreement between the covariance matrices however this difference is not important because the true misclassification error using the number of features obtained from the normal approximation and the number obtained from the true distribution differ only slightly even for significantly different covariance matrices

a homotopic deformation along p of a leray schauder degree result and existence for u p 2u ƒ t u 0 u 0 u t 0 p 1 soit f 0 1 r r une fonction continue et soit o p r r definie par o p s s p 2 s pour p fixe 1 on considere le probleme de dirichlet suivant o p u f t u 0 u 0 u t 0 ou t est un nombre reel positif

existence and multiplicity of solutions for fourth order boundary value problems with parameters abstract in this paper we study the existence and multiplicity of the solutions for the fourth order boundary value problem bvp u 4 t u t u t f t u t 0 t 1 u 0 u 1 u 0 u 1 0 where f 0 1 r r is continuous r and r are parameters by means of the idea of the decomposition of operators shown by chen w y chen a decomposition problem for operators xuebao of dongbei renmin university 1 1957 95 98 see also m krasnosel skii topological methods in the theory of nonlinear integral equations gostehizdat moscow 1956 and the critical point theory we obtain that if the pair is on the curve 2 4 satisfying 2 2 then the above bvp has at least one two three and infinitely many solutions for being in different interval respectively

existence of three solutions for a quasilinear two point boundary value problem abstract in this paper we investigate a quasilinear second order differential equation with dirichlet boundary conditions the existence of an open interval of parameters which ensures this problem admits at least three solutions is determined by using the critical point theory

existence of three solutions for p q biharmonic systems abstract in this paper we establish the existence of at least three solutions to a navier boundary problem involving the p q biharmonic systems the technical approach is mainly based on a three critical points theorem of b ricceri

multiple positive solutions for some p laplacian boundary value problems this paper deals with the existence of multiple positive solutions for the one dimensional p laplacian f p x t q t f t x t x t 0 t 0 1 subject to the following boundary value conditions x 0 i 1n a ix x i x 1 i 1n b ix x i where f p s s p 2 s p 1 by means of a fixed point theorem due to avery and peterson sufficient conditions are obtained that guarantee the existence of at least three positive solutions to the above boundary value problem

three solutions for a p laplacian boundary value problem with impulsive effects in this paper a p laplacian boundary value problem with impulsive effects is considered multiplicity of solutions is obtained by three critical points theorem an example is presented to illustrate main result

three symmetric solutions for a differential inclusion system involving the p x q x laplacian in rn abstract in this paper we consider a differential inclusion system involving the p x q x laplacian in r n applying a version of the nonsmooth three critical points theorem we obtain the existence of three pairs of radially symmetric solutions of the problem in w 1 p x r n w 1 q x r n

existence of three solutions for a class of quasilinear elliptic systems involving the p x q x laplacian abstract we study the solutions of the p x q x laplacian equations with dirichlet boundary condition on a bounded domain and obtain three solutions under appropriate hypotheses our technical approach is based on the general three critical points theorem obtained by b ricceri

effects of sample size in classifier design the effect of finite sample size on parameter estimates and their subsequent use in a family of functions are discussed general and parameter specific expressions for the expected bias and variance of the functions are derived these expressions are then applied to the bhattacharyya distance and the analysis of the linear and quadratic classifiers providing insight into the relationship between the number of features and the number of training samples because of the functional form of the expressions an empirical approach is presented to enable asymptotic performance to be accurately estimated using a very small number of samples results were experimentally verified using artificial data in controlled cases and using real high dimensional data

singular nonlinear parabolic boundary value problems in one space dimension global existence and uniqueness are established for the mixed initial boundary problem for the nonlinear parabolic equation u t x u x 2u x2 0 x 1 t 0 where x 0 x 0 for 0 x 1 and 01 0 x 1 dx the boundary conditions can be either linear e g dirichlet neumann or periodic or nonlinear in which case they take the form 1 j u j t j ux j t for j 0 1 where j is a maximal monotone graph in r r containing the origin

a novel feature selection methodology for automated inspection systems this paper proposes a new feature selection methodology the methodology is based on the stepwise variable selection procedure but instead of using the traditional discriminant metrics such as wilks lambda it uses an estimation of the misclassification error as the figure of merit to evaluate the introduction of new features the expected misclassification error rate mer is obtained by using the densities of a constructed function of random variables which is the stochastic representation of the conditional distribution of the quadratic discriminant function estimate the application of the proposed methodology results in significant savings of computational time in the estimation of classification error over the traditional simulation and cross validation methods one of the main advantages of the proposed method is that it provides a direct estimation of the expected misclassification error at the time of feature selection which provides an immediate assessment of the benefits of introducing an additional feature into an inspection classification algorithm

three periodic solutions for p hamiltonian systems abstract the existence of at least three periodic solutions is established for a class of p hamiltonian systems our technical approach is based on two general three critical points theorems obtained by ricceri and averna bonanno

existence of three solutions for a non homogeneous neumann problem through orlicz sobolev spaces abstract the aim of this paper is to establish a multiplicity result for an eigenvalue non homogeneous neumann problem which involves a nonlinearity fulfilling a nonstandard growth condition precisely a recent critical points result for differentiable functionals is exploited in order to prove the existence of a determined open interval of positive eigenvalues for which the problem admits at least three weak solutions in an appropriate orlicz sobolev space

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

three solutions for a navier boundary value problem involving the p biharmonic the existence of at least three weak solutions is established for a class of navier boundary value problem involving the p biharmonic our technical approach is based on the three critical points theorem obtained by b ricceri

on the existence of solutions for functional evolution equations soit x un espace de banach de norme soit pc l espace des fonctions continues par morceaux r 0 x pour un r fixe 0 pour pc on pose pc sup r 0 on considere l equation d evolution fonctionnelle non lineaire abstraite g t u t u t a t u t u 0 t 0 t

three solutions for a neumann type differential inclusion problem involving the p x laplacian abstract in this paper we consider neumann type differential inclusion problems involving the p x laplacian of the type div u p x 2 u u p x 2 u f x u in u 0 on applying a version of the non smooth three critical points theorem we obtain the existence of three solutions of the problem in w 0 1 p x

multiplicity theorems for a class of dirichlet quasilinear elliptic systems involving the p1 pn laplacian abstract in this paper we prove the existence of at least three weak solutions for the quasilinear elliptic systems p 1 u 1 f u 1 x u 1 u 2 u n 0 in p 2 u 2 f u 2 x u 1 u 2 u n 0 in p n u n f u n x u 1 u 2 u n 0 in u i 0 for 1 i n on our main tool is a recent three critical points theorem of averna and bonanno d averna g bonanno a three critical points theorem and its applications to the ordinary dirichlet problem topol methods nonlinear anal 22 2003 93 104

multiple solutions for p laplacian type equations abstract in this paper we establish the existence of three weak solutions of an equation which involves a general elliptic operator in divergence form in particular a p laplacian operator while the nonlinearity has a p 1 sublinear growth at infinity this result completes some recent papers where mountain pass type solutions were obtained providing the nonlinear term via a p 1 superlinear growth at infinity fulfilling an ambrosetti rabinowitz type condition in our case an abstract critical point result is applied proved by g bonanno g bonanno some remarks on a three critical points theorem nonlinear analysis 54 2003 651 665

a method of lines for a nonlinear abstract functional evolution equation let x be a real banach space with x uniformly convex a method of lines is introduced and developed for the abstract functional problem lipschitzian like function in its two variables further conditions are given for the convergence of the method to a strong solution of e recent results for perturbed abstract ordinary equations are substantially improved the method applies also to large classes of functional parabolic problems as well as problems of integral perturbations the method is straightforward because it avoids the introduction of the operators a t and the corresponding use of nonlinear evolution operator theory 1 introduction preliminaries let x be a real banach space with norm 11 let