a complete classification of bifurcation diagrams of a dirichlet problem with concave convex nonlinearities abstract we study the bifurcation diagrams of positive solutions of the two point boundary value problem u x f u x 0 1 u 1 u 1 0 where f u g u h u g h c 0 c2 0 and 0 is a bifurcation parameter we assume that functions g and h satisfy hypotheses h1 h3 under hypotheses h1 h3 we give a complete classification of bifurcation diagrams and we prove that on the u plane each bifurcation diagram consists of exactly one curve which is either a monotone curve or has exactly one turning point where the curve turns to the left hence the problem has at most two positive solutions for each 0 more precisely we prove the exact multiplicity of positive solutions

uniqueness results for the one dimensional p laplacian this paper is concerned with positive solutions of the boundary value problem y p 2y f y 0 y b 0 y b where p 1 b is a positive parameter assume that f is continuous on 0 changes sign from nonpositive to positive and f y yp 1 is nondecreasing in the interval of f 0 the uniqueness results are proved using a time mapping analysis

exact number of solutions of stationary reaction diffusion equations we study both existence and the exact number of positive solutions of the problem p l l u p 2u f u 0in 0 1 u 0 u 1 0 where l is a positive parameter p 1 the nonlinearity f is positive in 0 1 and f 0 f 1 0 assuming that f satisfies the condition lim s 1 f s 1 s email protected 0 where email protected 0 p 1 we study its behavior near zero and we obtain existence and exactness results for positive solutions we prove the results using the shooting method we show that there always exist solutions with a flat core for l sufficiently small as an application we prove the existence of a non negative solution for a class of singular quasilinear elliptic problems in a bounded domain in r n having a flat core in a ball

stationary profiles of degenerate problems with inhomogeneous saturation values abstract this work is concerned with the profiles of solutions for stationary problems between the degenerate diffusion of the p laplacian and a reaction with an inhomogeneous saturation value it is shown that if a parameter for the diffusion is sufficiently small then the solution attains the saturation value of the reaction in each region where the saturation value is constant the associated evolutional problem is also mentioned

quasinormal cones in banach spaces keywords quasinormality cone in banach space reference ana article 1983 001doi 10 1016 0362 546x 83 90028 7view record in web of science record created on 2008 12 10 modified on 2016 08 08

asymptotic behaviour of non linear parabolic equations with monotone principal part abstract we consider non linear parabolic equations with subdifferential principal part and give conditions under which they posses global attractors in spite of considering non lipschitz perturbations the case of globally lipschitz perturbations of a maximal monotone operator has been addressed in boll un mat ital b 8 2 2000 693 706 in the case of perturbations which are not globally lipschitz the main difficulty is the lack of uniqueness of solutions which at first does not even allow us to define attractors we overcome this difficulty for problems enjoying certain regularity and absorption properties that allow uniqueness of solutions after some time has been elapsed the results developed here are applied to the case when the subdifferential operator is the p laplacian to obtain existence of attractors and the existence of periodic solutions

numerical solution of systems of ordinary differential equations with the tau method an error analysis the recursive formulation of the tau method is extended to the case of systems of ordinary differential equations and an error analysis is given upper and lower error bounds are given in one of the examples considered the asymptotic behavior of the error compares in this case with that of the best approximant by algebraic polynomials for each of the components of the vector solution 1 introduction interest in the tau method see 2 3 5 for a long time regarded only as a tool for the construction of accurate approximations of a very restricted class of functions has been enhanced by the availability of software for its computer implementation and by the possibility of using it in the numerical solution of complex nonlinear differential equations over extended intervals the approxima tion of the solution of such type of equations is achieved as a result of finding tau approximants of a sequence of problems defined by linear differential equations details of this technique are given in 6 the subject of this paper is the extension of ortiz recursive formulation of lanczos tau method 5 to the case of systems of differential equations and more particularly to its error analysis for such systems our error estimation technique is applied to three model examples for which the exact solution is readily available it is discussed in general and with more detail when applied to the first of these examples for the second example we show how to get upper and lower error bounds we then compare these bounds with those given by meinardus 4 for the best uniform approximation of each of the components of the vector solution by algebraic polynomials to find that they are asymptotically equivalent the third example is a differential equation with variable coefficients and a nonempty subspace of residuals see 5 results of numerical experiments on the use of the tau method for the approxi mate solution of systems of ordinary differential equations with particular reference to stiff systems are reported in 8 the problems discussed in this paper can also be considered in the framework of simultaneous approximation of a function and its derivatives with the tau method see 1

positive solutions of multi point boundary value problems at resonance abstract we establish new results on the existence of positive solutions for some multi point boundary value problems at resonance our results are based on a recent leggett williams norm type theorem due to o regan and zima we also derive a new result for a three point problem previously studied by several authors

classification of bifurcation diagrams of a p laplacian dirichlet problem with examples abstract we study bifurcation diagrams of positive solutions of the p laplacian dirichlet problem p u x f u x 0 1 x 1 u 1 u 1 0 where p y y p 2 y p u is the one dimensional p laplacian and p 1 and 0 are two bifurcation parameters assume that f u g u h u where g h c 0 c 2 0 satisfy hypotheses h1 h5 presented herein for different values p with 1 p 2 and with p 2 we give a classification of totally six different bifurcation diagrams we prove that on the u plane each possible bifurcation diagram consists of exactly one curve with exactly one turning point where the curve turns to the right hence we are able to determine the exact multiplicity of positive solutions in addition for 1 p 2 and for p 2 we give interesting examples f u k u p 1 u q u r satisfying r q p 1 and k 0 and show complete evolution of bifurcation diagrams as evolution parameter k varies from 0 to

on the shape of the nonnegative solutions to a singularly perturbed quasilinear dirichlet problem abstract we study the shape of the nontrivial nonnegative solutions for a singularly perturbed quasilinear dirichlet problem with non lipschitz nonlinearity we find that there exists at least one nontrivial nonnegative solution of the problem which consists of some bubbles the profile of each bubble is discussed

exact multiplicity for some quasilinear elliptic dirichlet problems where the non linearity changes sign abstract the exact multiplicity of positive solutions is obtained for a class of quasilinear elliptic dirichlet problems where the non linearity changes sign meanwhile the uniqueness of positive ground state on r n is also obtained

leggett williams theorems for coincidences of multivalued operators abstract we study the existence of positive solutions to the operator equation l x n x where l is a linear fredholm mapping of index zero and n is a nonlinear multivalued operator

continuity properties on p for p laplacian parabolic problems abstract in this work we obtain some continuity properties on the parameter p at p 2 for the takeuchi yamada problem which is a degenerate p laplacian version of the chafee infante problem we prove the continuity of the flows and the equilibrium sets and the upper semicontinuity of the global attractors

coincidence sets in quasilinear elliptic problems of monostable type abstract this paper concerns the formation of a coincidence set for the positive solution of the boundary value problem e p u u q 1 f a x u in with u 0 on where e is a positive parameter p u div u p 2 u 1 q p f s s 1 s s 0 for some 0 and a x is a positive smooth function satisfying p a 0 in with inf a 0 it is proved in this paper that if 0 1 the coincidence set o e x u e x a x has a positive measure for small e and converges to with order o e 1 p as e 0 moreover it is also shown that if 1 then o e is empty for any e 0 the proofs rely on comparison theorems and the energy method for obtaining local comparison functions

leggett williams norm type fixed point theorems for multivalued mappings this paper presents several norm type versions of the leggett williams fixed point theorem for multivalued maps with nonempty convex compact values

generalized jacobian elliptic functions and their application to bifurcation problems associated with p laplacian abstract the jacobian elliptic functions are generalized and applied to bifurcation problems associated with p laplacian the values of bifurcation parameter and the corresponding solutions are represented in terms of common parameters and a complete description of the bifurcation diagram and a closed form representation of the corresponding solutions are obtained as a by product of the representation it turns out that a kind of solution is also a solution of an eigenvalue problem of p 2 laplacian

evaluation of the modified bessel function of the third kind of imaginary orders the evaluation of the modified bessel function of the third kind of purely imaginary order kia x is discussed we also present analogous results for the derivative the methods are based on the use of maclaurin series nonoscillatory integral representations asymptotic expansions and a continued fraction method depending on the ranges of x and a we discuss the range of applicability of the different approaches considered and conclude that power series the continued fraction method and the nonoscillatory integral representation can be used to accurately compute the function kia x in the range 0 a 200 0 x 100 using a similar scheme the derivative k ia x can also be computed within these ranges

branches of positive and free boundary solutions for some singular quasilinear elliptic problems we study the existence and multiplicity of solutions strictly positive or nonnegative having a free boundary the boundary of the set where the solution vanishes of some one dimensional quasilinear problems of eigenvalue type with possibly singular nonlinear terms

multiplicity result for a degenerate elliptic equation with logistic reaction abstract a degenerate elliptic equation pu uq 1 1 ur 0 with zero dirichlet boundary condition where is a positive parameter 2 0 is studied it has been known that there exists a positive number such that if then the problem has no positive solution if then there exists a maximal solution and for sufficiently small

infinitely many nodal solutions for generalized logistic equations without odd symmetry on reaction abstract we consider nonlinear parametric logistic type equations driven by a nonlinear nonhomogeneous differential operator the reaction term is a caratheodory function which is not assumed to be odd the subdiffusive equidiffusive and superdiffusive cases are all treated we prove the existence of infinitely many nodal solutions we also prove a bifurcation type result describing the dependence of the set of constant sign solutions on the parameter in the superdiffusive case our approach uses variational methods together with upper lower solutions and truncation techniques and flow invariance arguments

bifurcation phenomena associated to the p laplace operator on determine la structure de l ensemble des solutions u de u x p 2 u x x f u u p 2 u sur 0 1 telles que u 0 u 1 0 ou p 1 et r on demontre que les solutions avec k zeros sont uniques quand 1 2

on p laplacian differential inclusions global existence compactness properties and asymptotic behavior abstract in this work we consider coupled systems of p laplacian differential inclusions we obtain the global existence of solutions and good compactness properties in such way that the set of all possible solutions compose a generalized semiflow which has a global attractor for each pair of positive diffusion coefficients we also prove that the attractors are upper semicontinuous on positive finite diffusion parameters

canonical vector polynomials for the computation of complex order bessel functions with the tau method abstract a modification of the lanczos tau method for the approximate solution of second order differential equations with polynomial coefficients was proposed by the author earlier this modification chooses a minimum perturbation among all possible on the right hand side of the corresponding volterra integral equation a tau method computational scheme is applied to the approximate solution of a system of differential equations related to the differential equation of hypergeometric type various vector perturbations are discussed our choice of the perturbation term is a shifted chebyshev polynomial with a special form of selected transition and normalization the minimality conditions for the perturbation term are found for one equation they are sufficiently simple for verification in a number of important cases ortiz s recursive approach to the tau method is extended to canonical vector polynomial coefficients and to the coefficients of the linear combination that generate an approximate solution the advantages of the proposed approach are shown for modified bessel functions computation seven digit tables of modified bessel function k 1 2 i x have been computed and published the computer programs for the calculations may be found in the state fund of algorithms and programs and in the united library of numerical analysis of the moscow state university

a variational inequality theory for demicontinuous s contractive maps with applications to semilinear elliptic inequalities abstract a variational inequality theory for demicontinuous s contractive maps in hilbert spaces is established by employing the ideas of granas topological transversality such a variational inequality theory has many properties similar to those of fixed point theory for demicontinuous weakly inward s contractive maps and to those of fixed point index for condensing maps the variational inequality theory will be applied to study the existence of positive weak solutions and eigenvalue problems for semilinear second order elliptic inequalities with nonlinearities which satisfy suitable lower bound conditions involving the critical sobolev exponent there has been little discussion for such elliptic inequalities involving the critical sobolev exponent in the literature

on certain second order ordinary differential equations associated with sobolev poincare type inequalities on considere l equation e i 1 n u x i p 2 u x i x i u q 2 u 0 dans u x 0 sur on donne les proprietes des solutions de e pour le cas n 1 et a b mais pour p q 1 avec p q regularite existence et unicite de solutions positives

structures and evolution of bifurcation diagrams for a one dimensional diffusive generalized logistic problem with constant yield harvesting abstract we study the one dimensional diffusive generalized logistic problem with constant yield harvesting u x g u 0 1 x 1 u 1 u 1 0 where 0 we assume that nonlinearity g satisfies g 0 g 1 0 g u 0 on 0 1 and g either is concave on 0 1 or is concave convex on 0 1 and satisfies a certain condition we prove that for any fixed 0 on the u plane the bifurcation diagram consists of a shaped curve and then we study the structures and evolution of bifurcation diagrams for varying 0 we also prove that for any fixed 2 4 g 0 on the u plane the bifurcation diagram consists of a reversed shaped curve and then we study the structures and evolution of bifurcation diagrams for varying 2 4 g 0

multiple positive solutions of the one dimensional p laplacian in this work we investigate the existence of positive solutions of the p laplacian using the quadrature method we prove the existence of multiple solutions of the one dimensional p laplacian for 0 and determine their exact number for 0 2005 elsevier inc all rights reserved

lap number properties for p laplacian problems investigated by lyapunov methods abstract in this work we consider a one dimensional quasilinear parabolic equation and we prove that the lap number of any solution cannot increase through orbits as the time passes if the initial data is a continuous function we deal with the lap number functional as a lyapunov function and apply lap number properties to reach an understanding on the asymptotic behavior of a particular problem

a variational inequality index for condensing maps in hilbert spaces and applications to semilinear elliptic inequalities a variational inequality index for condensing maps is established in hilbert spaces new results on existence of nonzero positive solutions of variational inequalities for such maps are proved by using the theory of variational inequality index applications of such a theory are given to existence of nonzero positive weak solutions for semilinear second order elliptic inequalities where previous results of variational inequalities for s contractive maps cannot be applied

numerical solution of stiff and singularly perturbed boundary value problems with a segmented adaptive formulation of the tau method this paper concerns the application of ortiz recursive formulation of the tau method to the construction of piecewise polynomial approximations to the solution of linear and nonlinear boundary value problems for ordinary differential equations a practical error estimation technique related to the concept of correction in zadunaisky s sense is considered and used in the design of an adaptive approach to the tau method it proves efficient in the numerical treatment of problems with rapid functional variations stiff and singularly per turbed problems a technique of increased accuracy at matching points of segmented tau approximants is also discussed and successfully applied to several problems numerical examples show that for a given degree of approximation our segmented tau approximant gives an accuracy comparable to that of the best segmented approximation of the exact solution by means of algebraic polynomials 1 introduction we discuss the use of ortiz recursive formulation of the tau method 23 25 in the numerical solution of boundary value problems for linear and nonlinear differential equations defined over an interval a x b we con sider global approximations over a b with a single polynomial expression and segmented forms based on a step by step formulation of the tau method considered by ortiz in 26 the tau approximate solution of a differential problem defined by a differential operator d is represented in terms of the elements of a sequence q of canonical polynomials such a sequence is uniquely determined by d it is independent of the specific boundary conditions of the problem and of the particular interval a b in which the solution is required these properties make possible the use of segmenta tion within the framework and with the software 32 designed for the recursive formulation of the tau method the concept of correction in zadunaisky s sense 37 see also stetter 36 is discussed in the context of the tau method and related to a practical error estimation technique this technique based on tau estimators introduced here is systematically applied to all examples linear or nonlinear it is