population models with diffusion strong allee effect and nonlinear boundary conditions abstract we consider a population model with diffusion a strong allee effect per capita growth function and constant yield harvesting in particular we focus our study on a population living in a patch r n with n 1 that satisfies a certain nonlinear boundary condition we establish our existence results by the method of sub supersolutions

on s shaped bifurcation curves where sz c r n 2 1 is bounded with smooth boundary aq readers interested in the derivation of the problem are referred to the work by bebernen l particularly section 2 relevant work on the analysis of this problem has been done by brown et al 2 shivaji 3 61 hastings and mcleod 7 sattinger 8 parks 9 and tam lo clearly u 0 is a solution of 1 l when 2 0 it follows from the contraction mapping theorem that l l always has a solution for a sufficiently small moreover let

a correction for a paper by j smoller and a wasserman abstract in j differential equations 39 1981 269 290 j smoller and a wasserman presented some rich ideas and results on the bifurcation of steady state solutions of a reaction diffusion equation in one space variable however the proof of their theorem 2 1 is incorrect the case a 0 and to correct it we find one additional hypothesis is necessary

consistency property of extended least squares parameter estimation for stochastic diffusion equation the extended least squares parameter estimate for stochastic heat diffusion equations is considered the unknown parameter is a heat diffusion coefficient which is a function of a spatial variable almost sure convergence for the estimated parameter is proved a numerical example is demonstrated for supporting the theoretical results developed here

integral inequalities for some special functions several integral inequalities for the classical hypergeometric confluent hypergeometric and confluent hypergeometric limit functions are given the related results for bessel and whittaker functions as well as for laguerre hermite and jacobi polynomials are discussed

rigorous analysis and estimates of s shaped bifurcation curves in a combustion problem with general arrhenius reaction rate laws we study the bifurcation of solutions of the combustion problem with general arrhenius reaction rate laws d 2 u d x 2 1 e u m e u 1 e u 0 1 x 1 u 1 u 1 0 for this two point boundary value problem we give a bifurcation analysis in the physically important ranges 0 m 1 and m 0 it is proved that the bifurcation curve is s shaped in certain parameter ranges and a lower bound is given for the frank kamenetskii transition value

quenching for solutions of a plasma type equation here 0 l the so called slow diffusion case the analysis will be carried out in the manner of 5 in which levine gave a complete qualitative study of problem i with m 1 the plan of our paper is as follows in the next section we present the comparison theorem and local existence of solutions for i in the third section we characterize the set of stationary solutions of i finally in the last section we establish stability and quenching results for problem i

exact multiplicity for degenerate two point boundary value problems with p convex nonlinearity the exact number of positive solutions of a degenerate quasilinear two point boundary value problem is investigated for the generalization of earlier results concerning the nondegenerate case with convex nonlinearity suitably defined p convex nonlinearities are considered strictly p convex c2 functions having a non negative root are classified according to the shape of the bifurcation diagram of positive solutions versus the length of the interval we have uniqueness when f 0 0 and 0 1 or 2 solutions when f 0 0 similarly to the non degenerate case provided that the number of solutions is finite however now there may also occur a continuum of solutions connected to a dead core type phenomenon the proof of our results relies on the shooting method for the characterization of the shape of the time map in contrast to the non degenerate case the shooting method does not determine directly the number of solutions owing to the lack of uniqueness of the corresponding ivp exact conditions on the uniqueness of the ivp and in the case of non uniqueness the number and types of its local solutions are given based on this all the positive solutions of the bvp can be compiled

global branches of non radially symmetric solutions to a semilinear neumann problem in a disk let d r 2 be a disk and let f c 3 we assume that there is a r such that f a 0a ndf a 0 in this article we are concerned with the neumann problem u f u 0i nd u 0o n d we show the following there are unbounded continuums consisting of non radially symmetric solutions emanating from the second and third eigenvalues if f u u u u p 1 a 1 or if f is of bistable type then the unbounded branches emanating from non principal eigenvalues are unbounded in the positive direction of the branch emanating from the second eigenvalue is unique near the bifurcation point up to rotation the main tool of this article is the zero level set the nodal curve of u and ux

exact structure of positive solutions for some p laplacian equations this paper establishes the exact multiplicities and properties of positive solutions for some second order differential equations involving p laplacian operator

a quadratically convergent algorithm for convex set constrained signal recovery this paper addresses the problem of recovering a signal that is constrained to lie in a convex set from linear measurements the current standard is the alternating projections paradigm pocs which has only first order convergence in general we present a quadratically convergent iterative algorithm newton algorithm for signal recovery from linear measurements and convex set constraints a new result on the existence and construction of the derivative of the projection operator onto a convex set is obtained which is used in the newton algorithm an interesting feature of the new algorithm is that each iteration requires the solution of a simpler subspace constrained reconstruction problem a computation and memory efficient version of the algorithm is also obtained by using the conjugate gradient algorithm within each newton iteration to avoid matrix inversion and storage from a computational point of view the computation per iteration of this algorithm is similar to the computation per iteration of the standard alternating projections algorithm the faster rate of convergence compared to alternating projections enables us to obtain a high resolution reconstruction with fewer computations the algorithm is thus well suited for large scale problems that typically arise in image recovery applications the algorithm is demonstrated in several applications

on positive solutions of some pairs of differential equations this paper is a sequel to the author s paper 6 which studied positive solutions of a pair of differential equations introduced by conway gardner and smoller 4 their equations were equations for population problems in biology their significance is discussed further in 141 the purpose of the present paper is to obtain further results for the conwat gardner smoller equations and to show how the methods in 6 can be modified to decide exactly when the classical predator prey system as in blat and brown 2 has a strictly positive solution we also show how the methods can be applied to a number of other problems in particular our results answer some questions in conway s secondly we show how the asymptotic methods developed in 6 can be used to study the existence and uniqueness of strictly positive solutions of competing species problems our results suggest that this is a much more complicated problem than the predator prey problem we do obtain a condition which is except for a few special cases necessary and sufficient for the existence of a strictly positive solution however the condition is complicated and rather implicit thirdly we prove the uniqueness of the strictly positive solution in the conway smoller system if d is small and n 1 that is the case of ordinary differential equations this partially answers a question in 4 our proof is a local result and can be used in other situations for example it can be used to show that in certain cases of asocial nonlinearities where our notation follows 4 sect 4 there are exactly two strictly positive solutions for small d we also show how our method applies to more general predator prey problems in section 1 we find necessary and sufficient conditions for the existence of strictly positive solutions of predator prey systems in section 2 we study competing species models by iteration and asymptotic methods finally in section 3 we prove our uniqueness results in an appendix we briefly discuss how the methods of section 3 apply to more general models

a dual approach to linear inverse problems with convex constraints a simple constraint qualification is developed and used to derive an explicit solution to a constrained optimization problem in hilbert space a finite parameterization is obtained for the minimum norm element in the intersection of a linear variety of finite co dimension and a closed convex constraint set the result extends previous duality theorems for convex cone set constraints a fixed point iteration is presented for computing the parameters and yields a least squares solution when the variety and constraint set have empty intersection proofs rely on nearest point projections onto convex sets and the properties of monotone firmly nonexpansive and averaged mappings

proximity for sums of composite functions we propose an algorithm for computing the proximity operator of a sum of composite convex functions in hilbert spaces and investigate its asymptotic behavior applications to best approximation and image recovery are described

stationary solutions of chemotaxis systems the keller segel model is a system of partial differential equations modelling a mutual attraction of amoebae caused by releasing a chemical substance chemotaxis this paper analyzes the stationary solutions of the system with general nonlinearities via bifurcation techniques and gives a criterion for bifurcation of stable nonhomogeneous aggregation patterns examples are discussed with various kinds of nonlinearities modelling the sensitivity of the chemotaxis response

diffusive logistic equation with constant yield harvesting and negative density dependent emigration on the boundary abstract the structure of positive steady state solutions of a diffusive logistic population model with constant yield harvesting and negative density dependent emigration on the boundary is examined in particular a class of nonlinear boundary conditions that depends both on the population density and the diffusion coefficient is used to model the effects of negative density dependent emigration on the boundary our existence results are established via the well known sub super solution method

comparison theorems for radial solutions of semilinear elliptic equations on obtient des resultats sur les solutions radiales des equations elliptiques de la forme u f u 0

blowup in a partial differential equation with conserved first integral a reaction diffusion equation with a nonlocal term is studied the nonlocal term acts to conserve the spatial integral of the unknown function as time evolves such equations give insight into biological and chemical problems where conservation properties predominate the aim of the paper is to understand how the conservation property affects the nature of blowup the equation studied has a trivial steady solution that is proved to be stable existence of nontrivial steady solutions is proved and their instability established numerically blowup is proved for sufficiently large initial data by using a comparison principle in fourier space the nature of the blowup is investigated by a combination of asymptotic and numerical calculations

properties of the period function for some hamiltonian systems and homogeneous solutions of a semilinear elliptic equation abstract in this work we study the period function t of solutions to the conservative equation x t f x t 0 we present conditions on f that imply the monotonicity and convexity of t as a consequence we obtain the criterium established by c chicone and find conditions easier to apply we also get a condition obtained by cima gasull and manosas about monotonicity and following some of their calculations present results on the period function of hamiltonian systems where h x y f x n 1 y n using the monotonicity of t we count the homogeneous solutions to the semilinear elliptic equation u u 1 in two dimensions

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

global bifurcation of steady state solutions in this paper we study the bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation a f u 0 on the interval l 0 or 0 for some nonzero function c c a for the neumann problem we use entirely different techniques to prove that s is never critical for any cubic polynomial j this implies at once that the neumann problem can have at most one nonconstant solution having a given number of maxima or minima this solution is necessarily strongly nondegenerate in the sense that zero is not contained in the spectrum of the linearized operator see 2 for the dirichlet problem the situation is far more complicated and the bifurcation diagrams undergo qualitative changes depending on the positions of the roots off for example fig 1 shows the bifurcation diagram for the positive solutions for two different cubic functions of the form f u a u u b u c it is interesting to note however that in every case we study there are at most three solutions to the dirichlet problem for each

on the number of critical points of the period in the study of subharmonic bifurcations from a planar hamiltonian system it is often assumed that the period of a one parameter family of periodic orbits of the unperturbed system is a monotone function of the parameter see e g hale and taboas s chow and hale es however this hypothesis is in many instances difficult to verify for example it is well known that every orbit in the first quadrant of the vollterra lotka equation is periodic it is only in recent years that the monotonicity of the periods of these periodic orbits has been shown hsu 9 and waldvogel 151 such problems also occur in the study of bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation u f u o uer 1 1 on some interval in 14 smoller and wasserman showed that if f u u u u u cc

existence uniqueness and nondegeneracy of positive solutions of semilinear elliptic equations we study positive solutions of the dirichlet problem a u x f u x 0 x d u x o x e ad where d is an n ball we find necessary and sufficient conditions for solutions to be nondegenerate we also give some new existence and uniqueness theorems in this paper we study positive solutions of the dirichlet problem au x f u x o x f2

representation and asymptotic formulas for 1 dimensional linearized eigenvalue problems with dirichlet boundary condition abstract in this article nontrivial solutions to a nonlinear dirichlet boundary value problem with a small parameter and the corresponding linearized eigenvalue problems are investigated in a typical case of nonlinearity asymptotic formulas of all eigenvalues and asymptotic profiles of all eigenfunctions are proved proofs are based on the method of representation equation and asymptotic results in the corresponding neumann problem obtained in the recent papers by the author and s yotsutani

bifurcation and stability of stationary solutions of the fitz hugh nagumo equations the fitz hugh nagumo equations have been of some interest to both mathematicians and theoretical biologists for several years the reason for this stems from the fact that they can be considered as a simpler model for the celebrated hodgkin huxley equations in that they exhibit many of the features of this latter system indeed mathematicians have studied them because their structure is different from the usually encountered equations in physics and they therefore admit solutions with less familiar properties homoclinic travelling waves threshold effects etc the equations can be written as

a singularity theory approach to a semilinear boundary value problem where a b 3 are real parameters with 0 a b 1 this problem arises in population genetics with the unknown function u corresponding to the relative frequency of a gene the parameter a is given by a combination of the death rates of the various genotypes involved b gives the frequency of the gene on the boundary of the region and 1 is the reciprocal of the diffusion rate of the gene see aronson and weinberger 1 for further details solutions of 1 1 correspond to possible steady states of the gene populations we are interested mainly in how the bifurcation diagram changes as a and b vary and we investigate this question by exploiting singularity theory for bifurcation problems as developed by golubitsky and schaeffer c51 equation 1 1 is a simple example of a semilinear elliptic boundary value problem and a standard method of investigating the bifurcation of

exact multiplicity results for boundary value problems with nonlinearities generalizing cubic using techniques of bifurcation theory we present two exact multiplicity results for boundary value problems of the type the first result concerns the case when the nonlinearity is independent of x and behaves like a cubic in u the second one deals with a class of nonlinearities with explicit x dependence

inequalities for formal power series and entire functions we present several integral and exponential inequalities for formal power series and for both arbitrary entire functions of exponential type and generalized borel transforms they are obtained through certain limit procedures which involve the multiparameter binomial inequalities integral inequalities for continuous functions and weighted norm inequalities for analytic functions some applications to the confluent hypergeometric functions bessel functions laguerre polynomials and trigonometric functions are discussed also some generalizations are given

global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications we study the global bifurcation and exact multiplicity of positive solutions of u x lambda f epsilon u 0 1 f epsilon u epsilon u 3 sigma u 2 tau u rho where lambda epsilon 0 are two bifurcation parameters and sigma rho 0 tau 0 are constants by developing some new time map techniques we prove the global bifurcation of bifurcation curves for varying epsilon 0 more precisely we prove that for any sigma rho 0 tau 0 there exists epsilon 0 such that on the lambda parallel to u parallel to infinity plane the bifurcation curve is s shaped for 0 epsilon we also prove the global bifurcation of bifurcation curves for varying lambda 0 thus we are able to determine the exact number of positive solutions by the values of epsilon and lambda we give an application to prove a long standing conjecture for global bifurcation of positive solutions for the problem u x lambda epsilon u 3 u 2 u 1 0 1 u 1 u 1 0 which was studied by crandall and rabinowitz arch rational mech anal 52 1973 p 177 in addition we give an application to prove a conjecture of smoller and wasserman j differential equations 39 1981 p 283 lines 2 3 on the maximum number of positive solutions of a positone problem

resolution limits in signal recovery resolution analysis for the problem of signal recovery from finitely many linear measurements is the subject of this paper the classical rayleigh limit serves only as a lower bound on resolution since it does not assume any recovery strategy and is based only on observed data we show that details finer than the rayleigh limit can be recovered by simple linear processing that incorporates prior information we first define a measure of resolution based on allowable levels of error that is more appropriate for current signal recovery strategies than the rayleigh definition in the practical situation in which only finitely many noisy observations are available we have to restrict the class of signals in order to make the resolution measure meaningful we consider the set of bandlimited and essentially timelimited signals since it describes most signals encountered in practice for this set we show how to precompute resolution limits from knowledge of measurement functionals signal to noise ratio passband energy concentration regions energy concentration factor and a prescribed level of error tolerance in the process we also derive an algorithm for high resolution signal recovery we illustrate the results with examples in one and two dimensions

classes of analytic functions associated with the generalized hypergeometric function using the generalized hypergeometric function we introduce and study a class of analytic functions with negative coefficients coefficients estimates distortion theorems extreme points and the radii of convexity and starlikeness for this class are given relevant connections of these results with those in several earlier investigations are indicated

on the evolution and qualitative behaviors of bifurcation curves for a boundary value problem abstract we study the evolution and qualitative behaviors of bifurcation curves of positive solutions for u x u 1 sin u u p 1 x 1 u 1 u 1 0 where 0 is a bifurcation parameter and p 1 is an evolution parameter on the u plane we prove that the bifurcation curve has exactly one turning point where the curve turns to the left for p 2 it is a monotone curve for p 2 it has at least two turning points for 1 p p 1 2 cos 1 sin 1 1 2 cos 1 sin 1 1 629 and it has infinitely many turning points for p 1 hence we are able to determine the exact number of positive solutions in particular we give complete descriptions of the structure of bifurcation curves when p 2 we also give some numerical simulations of bifurcation curves for p 1

weighted norm inequalities for analytic functions we present a weighted norm inequality involving convolutions of arbitrary analytic functions and certain confluent hypergeometric functions this result implies a family of weighted norm inequalities both for entire functions of exponential type and for generalized hypergeometric series the approach is based on author s general inequality for continuous functions and some hypergeometric transformations

low dimensional qualitative and numerical approximation of bifurcation in a semilinear elliptic problem abstract a nonlinear boundary value problem exhibiting a nonlocal bifurcation in its solution set is considered it is shown that for some values of the problem parameters there is only the zero solution while for other values of these parameters two nontrivial solutions also appear these nontrivial solutions do not lie in small neighborhoods of the zero function so that the bifurcation analysis is not local to the origin the method of reduction to alternative finite dimensional problems is applied to give a numerical description of the bifurcation and in particular to describe a surface of bifurcating solutions in addition the solution set of the 1 dimensional galerkin approximation is also shown to have the same qualitative nature as the solution set of the full problem

verification of bifurcation diagrams for polynomial like equations the results of our recent paper p korman y li t ouyang computing the location and the direction of bifurcation math res lett 12 2005 933 944 appear to be sufficient to justify computer generated bifurcation diagram for any autonomous two point dirichlet problem here we apply our results to polynomial like nonlinearities

an extragradient method for solving split feasibility and fixed point problems the purpose of this paper is to introduce and analyze an extragradient method with regularization for finding a common element of the solution set c of the split feasibility problem and the set fix s of fixed points of a nonexpansive mapping s in the setting of infinite dimensional hilbert spaces combining the regularization method and the extragradient method due to nadezhkina and takahashi n nadezhkina w takahashi weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings j optim theory appl 128 2006 191 201 we propose an iterative algorithm for finding an element of fix s email protected we prove that the sequences generated by the proposed algorithm converge weakly to an element of fix s email protected under mild conditions

bifurcation diagrams of population models with nonlinear diffusion we develop analytical and numerical tools for the equilibrium solutions of a class of reaction diffusion models with nonlinear diffusion rates such equations arise from population biology and material sciences we obtain global bifurcation diagrams for various nonlinear diffusion functions and several growth rate functions

stabilization of solutions of nonlinear and degenerate evolution equations on etudie le comportement asymptotique des solutions bornees des equations de la forme u t u f x u dans r u x o u 0 x dans u x t u 1 x t dans r ou est un domaine borne de r n avec de classe c 2 0 1