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

a fourth order compact finite difference method for nonlinear higher order multi point boundary value problems a fourth order compact finite difference method is proposed for a class of nonlinear 2nth order multi point boundary value problems the multi point boundary condition under consideration includes various commonly discussed boundary conditions such as the three or four point boundary condition n 2 point boundary condition and 2 n m point boundary condition the existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term the convergence and the fourth order accuracy of the method are proved an efficient monotone iterative algorithm is developed for solving the resulting nonlinear finite difference systems various sufficient conditions for the construction of upper and lower solutions are obtained some applications and numerical results are given to demonstrate the high efficiency and advantages of this new approach

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

multiple positive solutions for nth order m point boundary value problems with all derivatives abstract we investigate the existence of positive solutions for the boundary value problem u n t f t u t u t u n 2 t u n 1 t 0 0 t 1 u 0 u 0 u n 2 0 0 u n 1 1 i 1 m 2 k i u n 1 i where f 0 1 r n r is continuous k i 0 i 1 2 m 2 0 i 1 m 2 k i 1 we give at first the associated green s function and some of its properties then imposing growth conditions on f we obtain the existence of at least three positive solutions for the boundary value problem by using the five functional fixed point theorem finally we give an example to demonstrate our result

triple positive solutions and dependence on higher order derivatives abstract in this paper we consider the lidstone boundary value problem y 2 m t f y t y 2 j t y 2 m 1 t 0 t 1 y 2 i 0 0 y 2 i 1 0 i m 1 where 1 m f 0 growth conditions are imposed on f and inequalities involving an associated green s function are employed which enable us to apply the leggett williams fixed point theorem to cones in ordered banach spaces this in turn yields the existence of at least three positive symmetric concave solutions the emphasis here is that f depends on higher order derivatives

existence and multiplicity of solutions of a kind of fourth order boundary value problem abstract in this paper the existence and multiplicity results of the solutions are obtained for the fourth order boundary value problem u 4 t f t u t for all t 0 1 subject to u 0 u 1 u 0 u 1 0 where f is continuous the monotone operator theory and critical point theory are employed to discuss this problem respectively

solving singular boundary value problems of higher even order in this paper we present a new algorithm to solve a kind of singular boundary value problems of higher even order based on the theory of numerical analysis in the reproducing kernel space w 2 2 m 3 0 1 singular boundary value problem of sixth order has been discussed for the problem u 6 x a x u 4 x b x u x c x u x f x 0

existence of multiple symmetric positive solutions of higher order lidstone problems this paper gives the sufficient conditions of the existence of at least three symmetric positive solutions for one type of higher order autonomous lidstone problem by applying the five functionals fixed point theorem the analogous result for higher order nonautonomous singular lidstone problem is also proved here

existence and multiplicity of solutions to 2mth order ordinary differential equations abstract in this paper the existence and multiplicity of solutions are obtained for the 2mth order ordinary differential equation two point boundary value problems 1 m u 2 m t i 1 m 1 m i a i u 2 m i t f t u t for all t 0 1 subject to dirichlet neumann mixed and periodic boundary value conditions respectively where f is continuous a i r for all i 1 2 m since these four boundary value problems have some common properties and they can be transformed into the integral equation of form u i 1 m a i t i u t m f u we firstly deal with this nonlinear integral equation by using the strongly monotone operator principle and the critical point theory we establish some conditions on f which are able to guarantee that the integral equation has a unique solution at least one nonzero solution and infinitely many solutions furthermore we apply the abstract results on the integral equation to the above four 2mth order two point boundary problems and successfully resolve the existence and multiplicity of their solutions

multiple positive solutions of a boundary value problem for a class of 2 nth order m point singular integro differential equations in banach spaces in this paper we obtain the existence of multiple positive solutions of a boundary value problem for 2nth order singular nonlinear integro differential equations in a banach space by means of fixed point index theory of completely continuous operators

bifurcation techniques for lidstone boundary value problems abstract by employing bifurcation techniques this paper investigates the existence of nontrivial solutions single and multiple for lidstone boundary value problems depending on higher order derivatives our results improve on those in the literature

a necessary and sufficient condition for 2nth order singular super linear m point boundary value problems abstract in this paper we study the existence of positive solutions for singular super linear m point boundary value problems of 2nth order ordinary differential equations a necessary and sufficient condition for the existence of c 2 n 2 0 1 positive solutions as well as c 2 n 1 0 1 positive solutions is given by means of the fixed point theorems on cones

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

optimal path in weak and strong disorder abstract we generate optimal paths between two given sites on a lattice representing a disordered energy landscape by applying the dijkstra algorithm we study the geometrical and energetic scaling properties of the optimal path under two different energy distributions that yield the weak and strong disorder limits our numerical results for both two and three dimensions suggest that the optimal paths in weak disorder are in the same universality class as the directed polymers and in the strong disorder limit are fractals with exponents similar to that found by cieplak et al phys rev lett 72 1994 2320 76 1996 3754

positive solutions for a class of 2nth order singular boundary value problems abstract we establish sufficient and necessary conditions for guaranteeing the existence of c 2 n 2 0 1 as well as c 2 n 1 0 1 positive solutions for 2 n th order singular boundary value problems the arguments are based upon the fixed point theorem an the maximal principle as well as constructing lower and upper 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 meanwhile an example is worked out to demonstrate the main results

positive solutions for 2nth order singular sub linear m point boundary value problems abstract this paper investigates the existence of positive solutions for 2nth order singular sub linear m point boundary value problems firstly we establish a comparison theorem then we define a partial ordering in c2n 2 a b c2n a b and construct lower and upper solutions to give a necessary and sufficient condition for the existence of c2n 2 0 1 as well as c2n 1 0 1 positive solutions our nonlinearity f t x1 x2 xn may be singular at xi 0 i 1 2 n t 0 and or t 1

multiple solutions of some fourth order boundary value problems abstract for some fourth order boundary value problems several new existence theorems on multiple positive negative and sign changing solutions are obtained the critical point theory and the supersolution and subsolution method are employed to discuss this problem

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

symmetric positive solutions of nonlinear boundary value problems abstract we study the nonlinear boundary value problem u 2 m f t u u u 2 m 2 t 0 1 u 2 i 0 u 2 i 1 0 i 0 m 1 the existence of symmetric positive solutions of the above problem is discussed sufficient conditions are obtained for the problem to have one any finite number and a countably infinite number of such solutions our results extend some recent work in the literature on boundary value problems of ordinary differential equations we illustrate our results by two examples none of which can be handled using the existing results

perturbation theory without diagrams the polaron case higher order perturbative calculations in quantum field theory suffer from the factorial increase of the number of individual diagrams here i describe an approach which evaluates the total contribution numerically for finite temperature from the cumulant expansion of the corresponding observable followed by an extrapolation to zero temperature this method originally proposed by bogolyubov and plechko is applied to the calculation of higher order terms for the ground state energy of the polaron using state of the art multidimensional integration routines two new coefficients are obtained corresponding to a four and five loop calculation several analytical and numerical procedures have been implemented which were crucial for obtaining reliable results

positive solutions of a 2nth order boundary value problem involving all derivatives via the order reduction method this paper is mainly concerned with the existence multiplicity and uniqueness of positive solutions for the 2nth order boundary value problem 1 nu 2 n f t u u 1 i 2 u i 1 n 1u 2 n 1 u 2 i 0 u 2 i 1 1 0 i 0 1 n 1 where n 2 and f c 0 1 xr 2 n r r 0 we first use the method of order reduction to transform the above problem into an equivalent initial value problem for a first order integro differential equation and then use the fixed point index theory to prove the existence multiplicity and uniqueness of positive solutions for the resulting problem based on a priori estimates achieved by developing spectral properties of associated parameterized linear integral operators finally as a byproduct our main results are applied for establishing the existence multiplicity and uniqueness of symmetric positive solutions for the lidstone problem involving all derivatives

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

solutions of two point boundary value problems for even order differential equations abstract the existence of solutions of the two point boundary value problems consisting of the even order differential equations x 2 n t f t x t x t x 2 n 2 t r t 0 t 1 and the boundary value conditions i x 2 i 0 i x 2 i 1 0 0 i x 2 i 1 i x 2 i 1 1 0 i 0 1 n 1 is studied sufficient conditions for the existence of at least one solution of above bvps are established it is interesting that the nonlinearity f in the equation depends on all lower derivatives especially odd order derivatives and the growth conditions imposed on f are allowed to be super linear the degrees of phases variables are allowed to be greater than 1 if it is a polynomial the results are different from known ones since we do not apply the green s functions of the corresponding problem and the method to obtain a priori bounds of solutions is different from known ones examples that cannot be solved by known results are given to illustrate our theorems