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

multiple positive solutions of conjugate boundary value problems with singularities the existence of one or several positive solutions for some nth order differential equations with conjugate boundary conditions is obtained the approach is to employ the well known results on the existence of positive solutions for hammerstein integral equations obtained recently by the author this avoids utilizing the theory of fixed point index for compact maps defined on cones directly new properties on the kernels corresponding to the boundary value problems are provided and then employed to prove new properties of nonzero positive solutions for these boundary value problems and new inequalities on functions satisfying the conjugate boundary conditions our results improve and generalize many recent results

existence of positive solutions for non positive higher order bvps abstract we shall provide conditions on non positive function f t u 1 u n 1 so that the boundary value problem has at least one positive solution then we shall apply this result to establish several existence theorems which guarantee the multiple positive solutions

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

numerical methods for the solution of special and general sixth order boundary value problems with applications to benard layer eigenvalue problems a family of numerical methods is developed for the solution of special nonlinear sixth order boundary value problems methods with second fourth sixth and eighth order convergence are contained in the family the problem is also solved by writing the sixth order differential equation as a system of three second order differential equations a family of second and fourth order convergent methods is then used to obtain the solution a second order convergent method is discussed for the numerical solution of general nonlinear sixth order boundary value problems this method with modifications where necessary is applied to the sixth order eigenvalue problems associated with the onset of instability in a benard layer numerical results are compared with asymptotic estimates appearing in the literature

regularity of the free boundary for the porous medium equation with r m 1 m 1 when u 0 then f 0 and both of the above equations become degenerate this degeneracy results in the interesting phenomenon of the finite speed of propagation if the initial data u is compactly supported in 1ztn the solution u t will remain compactly supported for all time t in this work we will show that under rather general assumptions on the initial data the free boundary

on nonexistence of type ii blowup for a supercritical nonlinear heat equation in this paper we study blowup of radially symmetric solutions of the nonlinear heat equation ut 1 u u p 1 u either on r n or on a finite ball under the dirichlet boundary conditions we assume that the exponent p is supercritical in the sobolev sense that is p ps n 2 n 2 we prove that if ps p the above range of exponent p is optimal we will also derive various fundamental estimates for blowup that hold for any p ps and regardless of type of blowup among other things we classify local profiles of type i and type ii blowups in the rescaled coordinates w e then establish useful estimates for the so called incomplete blowup which r eveal that incomplete blowup solutions belong to nice function spaces even after the blowup time c 2004 wiley periodicals inc

complete blow up after tmax for the solution of a semilinear heat equation soit un sous ensemble ouvert borne de r n a frontiere lisse on considere le probleme u t u f u dans o t u 0 sur o t u x o u 0 x x ou f r r est localement de lipschitz non decroissante et f o 0

continuation of blowup solutions of nonlinear heat equations in several space dimensions the possible continuation of solutions of the nonlinear heat equation in r n r ut u m u p with m 0 p 1 after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p thus for m p 2 we find a phenomenon of nontrivial continuation where the regionfx u x t 1g is bounded and propagates with finite speed this we call incomplete blowup for n 3 and p m n 2 n 2 we find solutions that blow up at finite t t and then become bounded again for t t otherwise we find that blowup is complete for a wide class of initial data in the analysis of the behavior for large p a list of critical exponents appears whose role is described we also discuss a number of related problems and equations we apply the same technique of analysis to the problem of continuation after the onset of extinction for example for the equation ut u m u p m 0

upper estimate for positive solutions of the p n p conjugate boundary value problem abstract we consider the p n p conjugate boundary value problem a new upper estimate for positive solutions of the problem is obtained at the end of the paper we discuss the sharpness of this new upper estimate and a lower estimate from the literature

lidstone polynomials and boundary value problems abstract we define lidstone polynomials provide their explicit representations give their relations with bernoulli polynomials obtain best possible error estimates in lidstone interpolation and establish several best possible sharp inequalities which compare favourably with the known results next we use these results to study even order differential equations together with lidstone boundary conditions

lagrangian approach to the study of level sets application to a free boundary problem in climatology we study the dynamics and regularity of level sets in solutions of the semilinear parabolic equation u t delta u in a mathbb h u mu quad rm in quad q it omega times 0 t where omega subset mathbb r n is a ring shaped domain a and are given positive constants mathbb h cdot is the heaviside maximal monotone graph mathbb h s 1 if s 0 mathbb h 0 0 1 mathbb h s 0 if s 0 such equations arise in climatology the so called budyko energy balance model as well as in other contexts such as combustion we show that under certain conditions on the initial data the level sets it gamma mu mathbf x t u mathbf x t mu are n dimensional hypersurfaces in the x t space and show that the dynamics of is governed by a differential equation which generalizes the classical darcy law in filtration theory this differential equation expresses the velocity of advancement of the level surface through spatial derivatives of the solution u our approach is based on the introduction of a local set of lagrangian coordinates the equation is formally considered as the mass balance law in the motion of a fluid and the passage to lagrangian coordinates allows us to watch the trajectory of each of the fluid particles

solutions of 2nth lidstone boundary value problems and dependence on higher order derivatives abstract in this paper we consider the lidstone boundary value problem 1 n u 2n x f x u x u x u 2 n 1 x 0 u 2i 0 u 2i 1 0 0 i n 1 some monotone conditions are imposed on f which enable us to apply a new maximum principle to deduce the existence of solution the emphasis here is that f depends on higher order derivatives two examples are also included to dwell upon the importance of the results obtained

classification of type i and type ii behaviors for a supercritical nonlinear heat equation we study blow up of radially symmetric solutions of the nonlinear heat equation ut u u p 1u either on rn or on a finite ball under the dirichlet boundary conditions we assume p ps n 2n 2 and that the initial data is bounded possibly sign changing our first goal is to establish various characterizations of type i and type ii blow ups among many other things we show that the following conditions are equivalent a the blow up is of type ii b the rescaled solution w y s converges to either y or y as s where denotes the singular stationary solution c u x t x tends to 1 as x 0 where t is the blow up time our second goal is to study continuation beyond blow up among other things we show that if a blow up is of type i and incomplete then its limit l1 continuation becomes smooth immediately after blow up and that type i blow up implies type i regularization that is t t 1 p 1 u t l is bounded as t t we also give various criteria for complete and incomplete blow ups

multiple positive solutions of some boundary value problems abstract we study the existence of multiple positive solutions of the equations u t u subject to linear boundary conditions we show that there are at least two positive solutions if t u is superlinear at one end point zero or infinity and sublinear at the other applications of these results are provided to yield multiple positive solutions of some elliptic boundary value problems on an annulus

bifurcation techniques and positive solutions of discrete lidstone boundary value problems abstract let n 1 t n 5 be two positive integers and t n n n 1 t we consider the existence of solutions of the discrete lidstone boundary value problem 1 n 2 n u t n f t u t 2 u t 1 2 n 2 u t n 1 t t n 2 i u n i 1 2 i u t i 1 0 i 0 1 n 1 where f t n r n r is continuous

an l blow up estimate for a nonlinear heat equation on decrit une borne superieure au taux d explosion des solutions de l equation de la chaleur non lineaire u t u u p ou u 0 p 1

existence and uniqueness of solutions for singular k n k conjugate boundary value problems by mixed monotone method the existence and uniqueness are established for singular k n k conjugate boundary value problems the theorems obtained are very general and complement previous known results

positive solutions of 2nth order ordinary differential equations with multi point boundary conditions in this paper we study the existence of positive solutions for 2nth order n 1 boundary value problems 1 nu 2n t f t u 0 t 1 subject to the 2 n m point boundary value conditions u 2j 0 u 2j 1 0 j 0 1 2 m m n 2 u 2j 0 bju 2j j 1 0 u 2j 1 dju 2j j 2 0 j m 1 n 1 where bj 0 dj 0 0 j 1 j 2 1 j m 1 n 1 by means of the fixed point theorems and the lower and upper solution method some results on the existence of positive solutions to the above 2 n m point boundary value problems are obtained

on eigenvalue intervals and twin eigenfunctions of higher order boundary value problems abstract in this paper we shall consider the boundary value problem y n q t y y 1 y n 2 p t y y 1 y n 2 n 2 t 1 1 y i 0 0 0 i n 3 y n 2 0 y n 1 0 0 yy n 2 1 y n 1 1 0 where 0 and are constants satisfying 0 0 0 and 0 intervals of are determined to ensure the existence of a positive solution of the boundary value problem for 1 we shall also offer criteria for the existence of two positive solutions of the boundary value problem in addition upper and lower bounds for these positive solutions are obtained for special cases several examples are included to dwell upon the importance of the results obtained

energy non collapsing and refined blowup for a semilinear heat equation abstract refined structures of blowup for non collapsing maximal solution to a semilinear parabolic equation u t u u p 1 u with p 1 are studied we will prove that the blowup set is empty for non collapsing blowing up in subcritical case and all finite time non collapsing blowing up must be refined type ii in critical case when p p s n 2 n 2 for n 3 the hausdorff dimension of the blowup set for maximal solution whose energy is non collapsing is shown to be no greater than n 2 4 p 1 which answers a question proposed in 7 positively at the end of this paper we also present some new examples of collapsing and non collapsing blowups

on the existence of positive solutions for semilinear elliptic equations in the annulus abstract we study the existence of positive radial solutions of u g x u 0 in annuli with dirichlet dirichlet neumann boundary conditions we prove that the problems have positive radial solutions on any annulus if is sublinear at 0 and

existence of positive solution for singular semi positone k n k conjugate m point boundary value problem this paper is about a class of singular semi positone k n k conjugate m point boundary value problems by using the fixed point index theory we get a sufficient condition for the existence of positive solution to this problem

the singular set of stationary solution for semilinear elliptic equations with supercritical growth abstract in this paper we are concerned with the partial regularity of the stationary solution to semilinear elliptic equations for almost all supercritical exponents we improve the upper bounds of the hausdorff dimension of the singular set which are derived by pacard to the optimal ones

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

numerical methods for unilateral problems abstract the theory of variational inequalities besides being elegant and synthetic also provides a unified general framework for applying the numerical methods for solving many unrelated free boundary value problems in this paper we describe numerical experience on the use of variational inequalities and pade approximants to obtain approximate solutions to a class of unilateral boundary value problems of elasticity like those describing the equilibrium configuration of an elastic beam stretched over an elastic obstacle in the case of a known obstacle these problems can be alternatively formulated as nonlinear boundary value problems without constraints for which the technique of pade approximants can be successfully employed the variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution

a generalization of the cone expansion and compression fixed point theorem and applications abstract the classical krasnosel skii fixed point theorem concerning cone compression and expansion of norm type is extended in it the norm is replaced with some convex functional as an application the existence of positive solutions for the singular second order m point boundary value problem is considered

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

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

lagrangian approach to the study of level sets ii a quasilinear equation in climatology we study the dynamics and regularity of the level sets in solutions of the semilinear parabolic equation u t delta p u f is an element of ah u mu in q omega x 0 t p is an element of 1 infinity where omega subset of r n is a ring shaped domain delta p u is the p laplace operator a and mu are given positive constants and h is the heaviside maximal monotone graph h s 1 if s 0 h 0 0 1 h s 0 if s 0 the mathematical models of this type arise in climatology the case p 3 was proposed and justified by p stone in 1972 we establish the conditions on the initial data which guarantee that the level sets gamma mu t x u x t mu are hypersurfaces study the regularity of gamma mu t and derive the differential equation that governs the dynamics of gamma mu t the analysis is based on the introduction of a system of lagrangian coordinates that transforms the moving surface gamma mu t into a stationary one

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

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

boundary value problems occurring in plate deflection theory abstract the paper deals with several iterative methods for solving some boundary value problems occurring in plate deflection theory making it possible to avoid the lengthy problem preparation which is required to solve these problems by finite difference methods

finite difference methods for twelfth order boundary value problems the effect of rotation acting on a layer of fluid heated from below is similar to the effect of a magnetic field acting under the same conditions they both inhibit the onset of instability and they both elongate the cells which appear at marginal stability however it must not be assumed that acting together rotation and magnetic field reinforce each other in fact they have conflicting tendencies when acting together the effect of rotation and magnetic field together leads to a twelfh order eigenvalue problem chandrasekhar 1961 which will be addressed in a future paper experience in solving high order boundary value problems has shown that considerable insight may be obtained by solving the special problem first moreover beside the mathematical interest the computational aspects of the special problem need to be considered to this end finite difference methods of order two and global extrapolation on two grids are proposed to solve the following special nonlinear twelfth order boundary value problem w xii x f x w a x b w 2i a a2i w 2i b b2i i 0 1 5 a simple example is carried out to illustrate the results given by the methods

three symmetric solutions of lidstone boundary value problems for difference and partial difference equations we consider the boundary value problem 2my k m f y k 2y k 1 2iy k 1 2 m 1 y k m 1 k a 1 b 1 2iy a 1 m 2iy b 1 m 2i 0 0 i m 1 where m 1 and 1 m f rm 0 is continuous by using amann and leggett williams fixed point theorems we develop growth conditions on f so that the boundary value problem has triple positive symmetric solutions the results obtained are then applied in the investigation of radial solutions for certain partial difference equation subject to lidstone type conditions

a sixth order multiderivative method for two beam problems a sixth order method is developed and analysed for the numerical solution of the linear fourth order boundary value problem for which the boundary conditions are given in terms of functional values and i first order derivatives the clamped clamped beam problem or ii second order derivatives the simple simple beam problem the method is tested on a number of problems from the literature

upper and lower estimates for positive solutions of the higher order lidstone boundary value problem we consider the higher order lidstone boundary value problem new upper and lower estimates for positive solutions of the problem are obtained a discussion of the sharpness of the estimates is included