the method of lower and upper solutions for 2nth order multi point boundary value problems abstract in this paper we consider the following 2 n th order multi point boundary value problems 1 n u 2 n t f t u t u t 1 i u 2 i t 1 n 1 u 2 n 2 t 0 t 1 with the 2 n m point boundary conditions u 2 j 0 u 2 j 1 0 j 0 1 2 m m n 2 u 2 j 0 a j u 2 j j 1 0 u 2 j 1 b j u 2 j j 2 0 j m 1 n 1 where a j 0 b j 0 0 j 1 j 2 1 j m 1 n 1 the existence of iterative solutions is obtained by using the lower and upper solution method for the above 2 n m point boundary value problems

on the existence of invariant absolutely continuous measures let be a measure space with normalized measure f a nonsingular transformation we prove there exists anf invariant normalized measure which is absolutely continuous with respect to if and only if there exist 0 and 0 1 such that e implies f k e for allk 0

existence of positive solutions for 2nth order singular sublinear boundary value problems abstract this paper investigates the existence of positive solutions for 2nth order n 1 singular sub linear boundary value problems a necessary and sufficient condition for the existence of c 2 n 2 0 1 as well as c 2 n 1 0 1 positive solutions is given by constructing lower and upper solutions and with the maximal theorem our nonlinearity f t x 1 x 2 x n may be singular at x i 0 i 1 2 n t 0 and or t 1

infinitely many mountain pass solutions on a kind of fourth order neumann boundary value problem in this paper the existence of infinitely many mountain pass solutions are obtained for the fourth order boundary value problem bvp u 4 t 2u t u t f u t 0r is continuous the study of the problem is based on the variational methods and critical point theory we prove the conclusion by using sub sup solution method mountain pass theorem in order intervals leray schauder degree theory and morse theory

existence of solutions for some higher order boundary value problems abstract in this paper we are concerned with the existence of solutions for the higher order boundary value problem in the form u 2 m 2 x f x u x u x u 2 m x x 0 1 u 2 i 0 u 2 i 1 0 0 i m where m is a given positive integer and f 0 1 r m 1 r is continuous we introduce a new maximum principle of higher order equations and develop a monotone method in the presence of lower and upper solutions for this problem

a monotone iterative technique for solving the bending elastic beam equations abstract this paper discusses the solvability of the fourth order boundary value problem u 4 f t u u 0 t 1 u 0 u 1 u 0 u 1 0 which models a statically bending elastic beam whose two ends are simply supported where f 0 1 r 2 r is continuous we build a maximum principle for the corresponding linear equation using this maximum principle we develop a monotone iterative technique in the presence of lower and upper solutions to solve the nonlinear equation and obtain some existence and uniqueness results

multiple sign changing solutions for fourth order m point boundary value problems abstract in this paper we study the existence and multiplicity of nontrivial solutions for the fourth order m point boundary value problems making use of the theory of the fixed point index in a cone and the leray schauder degree under general conditions on nonlinearity we prove that there exist at least six different nontrivial solutions for the fourth order m point boundary value problems furthermore if the nonlinearity is odd we obtain that there exist at least eight different nontrivial solutions

higher order lidstone boundary value problems for elliptic partial differential equations the aim of this paper is to show the existence and uniqueness of a solution for a class of 2nth order elliptic lidstone boundary value problems where the nonlinear functions depend on the higher order derivatives sufficient conditions are given for the existence and uniqueness of a solution it is also shown that there exist two sequences which converge monotonically from above and below respectively to the unique solution the approach to the problem is by the method of upper and lower solutions together with monotone iterative technique for nonquasimonotone functions all the results are directly applicable to 2nth order two point lidstone boundary value problems

the sub harmonic bifurcation of stokes waves steady periodic water waves on the free surface of an infinitely deep irrotational flow under gravity without surface tension stokes waves can be described in terms of solutions of a quasi linear equation which involves the hilbert transform and which is the euler lagrange equation of a simple functional the unknowns are a 2 periodic function w which gives the wave profile and the froude number a dimensionless parameter reflecting the wavelength when the wave speed is fixed and vice versa

existence of infinitely many mountain pass solutions for some fourth order boundary value problems with a parameter abstract in this paper for the fourth order boundary value problem bvp u 4 t u t 2 4 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 0 is a parameter the existence of infinitely many mountain pass solutions are obtained with the variational methods and critical point theory we prove the conclusion by combining sub sup solution method mountain pass theorem in order intervals leray schauder degree theory and morse theory

solvability of some fourth and higher order singular boundary value problems abstract existence results for a variety of singular fourth order boundary value problems of the form yiv f t y y y are given here our nonlinear term f may be singular at t 0 t 1 y 0 and or y 0 for example some singularities of the type y a and y b are included also we discuss and treat the extension of these results to nth order boundary value problems

on the lower and upper solutions method for the problem of elastic beam with hinged ends abstract we establish the existence theorem for solutions of fourth order nonlinear equations with the lidstone boundary conditions which model deflections of an elastic beam curve with the hinged ends the new lower and upper solutions method is formulated and then applied to a class of singularly perturbed problems and here is shown that the solutions uniformly converge to the solution of a reduced problem in the interior points of elastic curve and formula for an approximate solution without the use of numerical or iterative methods is derived

the method of lower and upper solutions for fourth order two point boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u iv f x u x u x u 0 u 1 u 0 u 1 0 where f 0 1 r r r is continuous

fourth order problems with nonlinear boundary conditions we develop a new method of lower and upper solutions for a fourth order nonlinear boundary value problem where the differential equation has dependence on all lower order derivatives our boundary conditions are nonlinear we will assume the functions that define the nonlinear boundary conditions are either monotone or nonmonotone as a result we obtain existence principles which improve recent results in the literature

numerical solution of fourth order problems with separated boundary conditions in this paper the fourth order linear and nonlinear differential equations with separated boundary conditions is converted into an optimal control problem then a convergent approximate solution is constructed such that the exact boundary conditions are satisfied the analysis is accompanied by numerical examples the obtained results demonstrate reliability and efficiency of the proposed scheme

eigenvalues of fourth order singular sturm liouville boundary value problems abstract in this paper by establishing a new comparison theorem and constructing upper and lower solutions some sufficient conditions of existence of positive solutions for the following nonlinear fourth order singular sturm liouville eigenvalue problem 1 p t p t u t f t u t 0 1 u 0 u 1 0 u 0 lim t 0 p t u t 0 u 1 lim t 1 p t u t 0 are established due to the schauder s fixed point theorem for large enough where 0 0 f and p can be singular at t 0 and or 1 moreover f can also be singular at u 0 in addition some peculiar cases are discussed and some further results are obtained

positive solutions of even higher order singular superlinear boundary value problems abstract this paper investigates 2 m th order m 1 superlinear singular two point boundary value problems and obtains some necessary and sufficient conditions for existence of c 2 m 1 or c 2 m 1 positive solutions on the closed interval

feller processes on nonlocally compact spaces we consider feller processes on a complete separable metric space x satisfying the ergodic condition of the form lim sup n 1 n n i 1 p i x o 0 for some x x where o is an arbitrary open neighborhood of some point z e x and p is a transition function it is shown that e chains which satisfy the above condition admit an invariant probability measure some results on the stability of such processes are also presented

multiple solutions for fourth order boundary value problem abstract in this paper we study the existence and multiplicity of nontrivial solutions for the fourth order two point boundary value problems making use of the theory of fixed point index in cone and leray schauder degree under general conditions on nonlinearity we prove that there exist at least six different nontrivial solutions for the fourth order two point boundary value problems furthermore if the nonlinearity is odd we obtain that there exist at least eight different nontrivial solutions

upper and lower solution methods for fully nonlinear boundary value problems abstract sufficient conditions are given for the existence of a solution of a fourth order nonlinear boundary value problem with nonlinear boundary conditions the conditions assume the existence of a strong upper solution lower solution pair a concept that is defined in the paper the differential equation has nonlinear dependence on all lower order derivatives of the unknown in particular appropriate nagumo conditions are obtained and employed

positive solutions of fourth order boundary value problems with two parameters abstract in this paper the existence results of positive solutions are obtained for fourth order boundary value problem u 4 u u f t u 0 where f 0 1 r r is continuous r and satisfy

the regularity and local bifurcation of steady periodic water waves steady periodic water waves on infinite depth satisfying exactly the kinematic and dynamic boundary conditions on the free surface with or without surface tension are given by solutions of a rather tidy nonlinear pseudo differential operator equation for a 2 periodic function of a real variable being an euler lagrange equation this formulation has the advantage of gradient structure but is complicated by the fact that it involves a non local operator namely the hilbert transform and is quasi linear

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

the benjamin lighthill conjecture for steady water waves revisited the two dimensional nonlinear problem of steady gravity waves on water of finite depth is considered the benjamin lighthill conjecture is proved for these waves provided bernoulli s constant attains near critical values in fact this is a consequence of the following more general results if bernoulli s constant is near critical then all corresponding waves have sufficiently small heights and slopes moreover for every near critical value of bernoulli s constant there exist only the following waves a solitary wave and the family of stokes waves having their crests strictly below the crest of this solitary wave this family is parametrised by wave heights which increase from zero to the height of the solitary wave all these waves are unique up to horizontal translations most of these results were proved in our previous paper kozlov and kuznetsov in arch rational mech anal 197 433 488 2010 in which it was supposed that wave slopes are bounded a priori here we show that the latter condition is superfluous by proving the following theorem if any steady wave has the free surface profile of a sufficiently small height then the slope of this wave is also small

bernoulli free boundary problems in strip like domains and a property of permanent waves on water of finite depth we study weak solutions for a class of free boundary problems which includes as a special case the classical problem of travelling gravity waves on water of finite depth we show that such problems are equivalent to problems in fixed domains and study the regularity of their solutions we also prove that in very general situations the free boundary is necessarily the graph of a function

on the existence of positive solutions for the bending elastic beam equations abstract this paper discusses the existence of positive solutions of the fourth order boundary value problem u 4 f t u u 0 t 1 u 0 u 1 u 0 u 1 0 which models a statically bending elastic beam whose two ends are simply supported where f 0 1 r r r is continuous the essential conditions on f guaranteeing the existence of positive solution are presented the discussion is based on the fixed point index theory in cones

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

positive solutions for a fourth order p laplacian boundary value problem abstract in this paper we study the existence multiplicity and uniqueness of positive solutions for the fourth order p laplacian boundary value problem u p 1 u f t u u 2 i 0 u 2 i 1 0 i 0 1 here p 0 and f c 0 1 r r r 0 based on a priori estimates achieved by utilizing properties of concave functions we use fixed point index theory to establish our main results

upper and lower solution method for fourth order four point boundary value problems this paper is concerned with the fourth order ordinary differential equation u 4 t f t u t u t 0 0 some new existence results are obtained by developing the upper and lower solution method and the monotone iterative technique

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 and uniqueness of fixed points for markov operators and markov processes this paper concerns a markov operator t on a space l i and a markov process p which defines a markov operator on a space m of finite signed measures for t the paper presents necessary and suf ic ient conditions for a the existence of invariant probability densities ipds b existence of strictly positive ipds and c existence and uniqueness of ipds similar results on invariant probability measures for p are presented the basic approach is to pose a fixed point problem as the problem of solving a certain linear equation in a suitable banach space and then obtain necessary and sufficient conditions for this equation to have a solution

existence of positive solutions for 2nth order singular superlinear boundary value problems this paper investigates the existence of positive solutions for 2nth order n 1 singular superlinear boundary value problems a necessary and sufficient condition for the existence of c 2 n 2 0 1 as well as c 2 n 1 0 1 positive solutions is given by constructing a special cone and with the e norm

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

fourth order two point boundary value problems estimates by two sided bounds on presente une methode simple pour transformer des problemes aux deux points limites d ordre 4 en problemes d ordre 2

nontrivial solutions on a kind of fourth order neumann boundary value problems abstract nontrivial solutions for the fourth order neumann boundary value problem u 4 t 2 u t u t f t u t for all t 0 1 and subject to u 0 u 1 u 0 u 1 0 are obtained via morse theory to compute the critical groups at infinity of the relevant functional we propose an approach by combining the homotopy and reduction methods

searching the least value method for solving fourth order nonlinear boundary value problems this paper obtains a searching least value slv method for a class of fourth order nonlinear boundary value problems is investigated the argument is based on the reproducing kernel space w 5 0 1 the approximate solutions u n x and u n k x are uniformly convergent to the exact solution u x and u k x k 1 2 3 4 respectively numerical results are verified that the method is quite accurate and efficient for this kind of problem

multiple solutions of some boundary value problems with parameters abstract in this paper we study the existence and multiplicity of nontrivial solutions for the following second order dirichlet nonlinear boundary value problem with odd order derivative u t a u t b u t f t u t for all t 0 1 with u 0 u 1 0 where a b r 1 f c 1 0 1 r 1 r 1 by using the morse theory we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution two nontrivial solutions and infinitely many solutions separately

fundamental bounds for steady water waves the two dimensional free boundary problem of steady gravity waves on water of finite depth is considered bounds on the free surface profiles and on the values of bernoulli s constant are obtained under minimal assumptions about properties of solutions to the problem

an approximate method for one dimensional structures with strong nonlinear and nonhomogenous boundary conditions abstract there is a lack of analytic methods for vibration analysis of structures with strong nonlinear and non homogenous boundary conditions this paper proposes such a method a distinct feature of the method is to introduce the generalized coordinates for nonlinear and nonhomogeneous boundary conditions the total solution of the structure consists of a sum of a spatial particular solution for the boundary conditions and a field solution satisfying linear and homogeneous boundary conditions the field solution can be obtained in terms of the modal expansion the modal coefficients together with the generalized coordinates for the boundary conditions satisfy a set of nonlinear ordinary differential equations which are solved with the harmonic balance method two examples are presented to study the convergence and accuracy of the solution the first example is the longitudinal vibration of a bar subject to a base excitation on the left and restrained by a damper on the right the wave method and the differential quadrature method dqm are compared with the proposed method and are found in good agreement the second example is a beam with nonlinear torsional boundary absorbers the method shows a good accuracy to deal with the complex boundary conditions as compared with the differential quadrature element method dqem the proposed method is also compared with the multi scale method msm and shows an advantage to deal with strong nonlinearities the proposed method is highly promising for vibration analysis of complex structures with nonlinear and nonhomogeneous boundary conditions

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

on 2nth order lidstone boundary value problems this paper is concerned with the solutions of a class of 2nth order lidstone boundary value problems sufficient conditions for the existence and uniqueness of a solution are given a monotone iteration is developed so that the iteration sequence converges monotonically to a maximal or a minimal solution the approach to the problem is by the method of upper and lower solutions with a new maximum principle

existence and location results for hinged beam equations with unbounded nonlinearities abstract this work presents some existence non existence and location results for the problem composed by the fourth order fully nonlinear equation u 4 x f x u x u x u x u x s p x for x 0 1 where f 0 1 r 4 r and p 0 1 r are continuous functions and s is a real parameter with the lidstone boundary conditions u 0 u 1 u 0 u 1 0 this problem models several phenomena such as the bending of an elastic beam simply supported at the endpoints the arguments used apply a lower and upper solutions technique a priori estimations and topological degree theory in this paper we replace the usual bilateral nagumo condition by some one sided conditions which enables us to consider unbounded nonlinearities

a fourth order compact finite difference method for higher order lidstone boundary value problems a compact finite difference method is proposed for a general class of 2nth order lidstone boundary value problems the existence and uniqueness of the finite difference solution is investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term a monotone iteration process is provided for solving the resulting discrete system efficiently and a simple and easily verified condition is obtained to guarantee a geometric convergence of the iterations the convergence of the finite difference solution and the fourth order accuracy of the proposed method are proved numerical results demonstrate the high efficiency and advantages of this new approach

existence of positive solutions for fourth order boundary value problem with variable parameters abstract in this paper by using the fixed point theorem and operator spectral theorem the author establishes the theorem on the existence of positive solutions for the fourth order boundary value problem with variable parameters as follows u 4 b t u a t u f t u 0 t 1 u 0 u 1 u 0 u 1 0 where a t b t c 0 1 and f t u 0 1 0 0 is continuous

general lidstone problems multiplicity and symmetry of solutions abstract for the 2mth order lidstone boundary value problem y 2 m t f y t y t y 2 i t y 2 m 1 t t 0 1 y 2 i 0 y 2 i 1 0 0 i m 1 where 1 mf r m 0 thinsp is continuous growth conditions are imposed on f which yield the existence of at least three symmetric positive solutions this generalizes earlier papers which have applied avery s generalization of the leggett williams theorem to lidstone problems we then prove the analogous result for difference equations

monotone iterative for fourth order p laplacian boundary value problems with impulsive effects sufficient conditions are given for the existence of a solution of fourth order p laplacian boundary value problems with impulsive effects the conditions assume the nonlinear function in equation is monotone and the existence of an upper lower solution pair the properties of p laplacian operator are employed

the method of lower and upper solutions for a bending of an elastic beam equation abstract in this paper by combining a new maximum principle and the theory of the two parameter linear eigenvalue problem we develop the monotone method in the presence of lower and upper solutions for the problem u 4 x f x u x u x 0 x 1 u 0 u 1 u 0 u 1 0 where f 0 1 r r r is continuous

the benjamin lighthill conjecture for near critical values of bernoulli s constant in 1954 benjamin and lighthill made a conjecture concerning the classical nonlinear problem of steady gravity waves on water of finite depth according to this conjecture a point of some cusped region on the r s plane r and s are the non dimensional bernoulli s constant and the flow force respectively corresponds to every steady wave motion described by the problem conversely at least one steady flow corresponds to every point of the region in the present paper this conjecture is proved for near critical flows when r attains values close to one under the assumption that the slopes of wave profiles are bounded another question studied here concerns the uniqueness of solutions and it is proved that for every near critical value of r only the following waves do exist i a unique up to translations solitary wave ii a family of stokes waves unique up to translations which is parametrised by the distance from the bottom to the wave crest the latter parameter belongs to the interval bounded below by the depth of the subcritical uniform stream and above by the distance from the bottom to the crest of solitary wave corresponding to the chosen value of r

multiply sign changing solutions for fourth order nonlinear boundary value problems abstract this paper deals with the existence of sign changing solutions for the fourth order nonlinear boundary value problem u 4 f u u 0 t 1 u 0 u 1 u 0 u 1 0 where f r 2 r is a continuously odd function some conditions on f guaranteeing the existence and multiplicity of sign changing solutions are presented the existence and multiplicity conditions concern the eigenline of the associated two parameter linear eigenvalue problem the discussion is based on the fixed point index theory in cones and an anti symmetrical extension method of solution

existence and uniqueness results for the bending of an elastic beam equation at resonance etude d un probleme non lineaire aux valeurs limites pour la flexion d une poutre elastique en appui simple aux deux extremites dans le cas de resonance theoremes d existence et d unicite

two parameter nonresonance condition for the existence of fourth order boundary value problems abstract in this paper we discuss the existence of the fourth order boundary value problem u 4 f t u u 0 t 1 u 0 u 1 u 0 u 1 0 where f 0 1 r r r is continuous and partly solve the del pino and manasevich s conjecture on the nonresonance condition involving the two parameter linear eigenvalue problem we also present a two parameter nonresonance condition described by circle

on positive solutions of some nonlinear fourth order beam equations abstract the existence uniqueness and multiplicity of positive solutions of the following boundary value problem is considered u 4 t f t u t 0 for 0 where 0 is a constant f 0 1 0 0 is continuous

existence of solutions for 2n order boundary value problem in this paper we discuss the existence of solutions for the 2n order boundary value problem 1 nu 2 n t f t u t u t u 2 n 2 t 0r is continuous

positive solutions of some nonlocal fourth order boundary value problem by the use of the krasnosel skii s fixed point theorem the existence of one or two positive solutions for the nonlocal fourth order boundary value problemu 4 t email protected t email protected t u t u t 00 email protected c 0 1 x 0 x 0 0

existence of n solutions and or positive solutions to a semipositone elastic beam equation abstract the existence of n solutions and or positive solutions is established for a nonlinear fourth order two point boundary value problem where n is an arbitrary natural number and the nonlinear term has a nonpositive lower bound in mechanics the problem describes deformation of the elastic beam whose two ends are simply supported the main results show that the problem has at least n solutions and or positive solutions provided that the heights of the nonlinear term are appropriate on some bounded sets

on fourth order elliptic boundary value problems with nonmonotone nonlinear function this paper is concerned with the fourth order elliptic boundary value problems with nonmonotone nonlinear function the existence and uniqueness of a solution is proven by the method of upper and lower solutions a monotone iteration is developed so that the iteration sequence converges monotonically to a maximal solution or a minimal solution depending on whether the initial iteration is an upper solution or a lower solution

a class of 2nth order singular boundary value problems abstract this paper investigates the existence of positive solutions for 2 n th order n 1 singular sub linear boundary value problems first of all we establish the maximal principle and some important lemmas then we define a partial ordering in c 2 n 2 0 1 c 2 n 0 1 and construct lower and upper solutions to give a necessary and sufficient condition for the existence of c 2 n 2 0 1 as well as c 2 n 1 0 1 positive solutions our nonlinearity f t x 1 x 2 x n may be singular at x i 0 i 1 2 n t 0 and or t 1

eigenvalue of fourth order m point boundary value problem with derivatives this paper deals with the existence of positive solutions for the fourth order nonlinear ordinary differential equation x t lg t f t x t x t 0 0 are constants such that r a d a c b d 0 and x i 0 1 h i 0 i 1 2 m 2 by means of a fixed point theorem due to krasnaselskii some new existence results of positive solutions for the above multi point boundary value problem are obtained which improve the main results of graef et al j r graef c qian b yang a three point boundary value problem for nonlinear fourth order differential equations j math anal appl 287 2003 217 233 an example is given to demonstrate the main results of this paper

the method of lower and upper solutions for second third fourth and higher order boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u n t t u t u i a u i b i r i 0 n 1 where is a caratheodory function we obtain necessary and sufficient conditions in to guarantee the existence of solutions between a lower solution and an upper solution for n 2 if n 3 either or and n 4 if furthermore we obtain sufficient conditions in for n 2 k 6 when