sharp error bounds for the derivatives of lidstone spline interpolation ii abstract in this paper we shall derive explicit error estimates in l norm between a given function f x pc n a b 4 n 6 and its quintic lidstone spline interpolate the results obtained are then used to establish precise error bounds for the approximated and biquintic lidstone spline interpolates we also include applications to integral equations and boundary value problems as well as sufficient numerical examples which dwell upon the sharpness of the obtained results

iterative methods for a fourth order boundary value problem abstract we consider a fourth order nonlinear ordinary differential equation together with two point boundary conditions and provide a priori error estimates on the length of the interval b a so that the picard s iterative method the approximate picard s iterative method and the quasilinear iterative method convergence to the solution of the problem

monotone convergence of iterative methods for right focal point boundary value problems abstract we provide sufficient conditions so that the picard iterative scheme for the right focal point boundary value problems converges monotonically an example which dwells upon the usefulness of this technique is also included

sign patterns that allow diagonalizability abstract an n n sign pattern a allows diagonalizability if there exists a real matrix b in the qualitative class q a of a such that b is diagonalizable the question of characterizing sign patterns that allow diagonalizability is open in this paper we obtain some sufficient conditions for a sign pattern allowing diagonalizability in particular it is proved that the combinatorially symmetric sign patterns a allow diagonalizability we give also two counterexamples for eschenbach johnson s conjecture in linear algebra appl 190 1993 169 mr 94i 15003

existence of positive solutions for a system of generalized lidstone problems in this paper we study the existence of positive solutions for the system of generalized lidstone problems for ordinary differential equations 1 mu 2 m f 1 t u u 1 m 1u 2 m 2 v v 1 n 1v 2 n 2 1 nv 2 n f 2 t u u 1 m 1u 2 m 2 v v 1 n 1v 2 n 2 a 0u 2 i 0 b 0u 2 i 1 0 a 1u 2 i 1 b 1u 2 i 1 1 0 i 0 1 m 1 a 0v 2 j 0 b 0v 2 j 1 0 a 1v 2 j 1 b 1v 2 j 1 1 0 j 0 1 n 1 here f 1 f 2 c 0 1 xr m n r r 0 our hypotheses imposed on the nonlinearities f 1 and f 2 are formulated in terms of nonnegative matrices we use fixed point index theory to establish our main results based on a priori estimates of positive solutions

computational methods for discrete boundary value problems on considere des methodes constructives pour les problemes aux valeurs limites discrets y t 1 g t y t t i n f y 0 ou y et g t y sont des vecteurs a n dimensions i n est l ensemble des points discrets 0 1 n f y est un operatuer de b i n dans r n et b i n est l espace de toutes les fonctions n vectorielles reelles definies sur i n

b spline collocation for solution of two point boundary value problems a numerical method based on b spline is developed to solve the general nonlinear two point boundary value problems up to order 6 the standard formulation of sextic spline for the solution of boundary value problems leads to non optimal approximations in order to derive higher orders of accuracy high order perturbations of the problem are generated and applied to construct the numerical algorithm the error analysis and convergence properties of the method are studied via green s function approach o h 6 global error estimates are obtained for numerical solution of these classes of problems numerical results are given to illustrate the efficiency of the proposed method results of numerical experiments verify the theoretical behavior of the orders of convergence

existence of solutions for third order boundary value problems on a time scale abstract let t be a closed subset of r with inf t and sup t for the third order nonlinear equation u 3 f t u u u t t where f t r3 r is continous we assume solutions of initial value problems are unique and extend to t we consider questions of the uniqueness of solutions implying the existence of solutions for conjugate boundary problems on t

uniqueness and error estimates for solutions to higher order boundary value problems abstract little work has been done for the error estimates of the homotopy analysis method for general 2 n th order linear and nonlinear differential equations with lidstone boundary conditions we obtain sharp upper bounds for the absolute error of the approximations given by the homotopy analysis method to achieve this goal the existence and uniqueness of solutions to these differential equations are considered numerical results confirm the theoretical predictions

global integrability related to anisotropic operators we consider boundary value problems of the form i 1ndi ai x du x i 1ndifi x in u x u x on where ai rn rai rn r i 1 2 ni 1 2 n are caratheodory functions satisfying i 1n ai x z pi c1 i 1n zi pi g1 x and c2 i 1n zi pi g2 x i 1nai x z zi for some positive constants c1c1 c2c2 and some functions g1g1 g2g2 we also consider minimizers u u w01 pi of the integral functional i u f x du x dx where the integrand f x z rn 0 f x z rn 0 satisfies c3 i 1n j 1n zj pj pi 2pi zi 2 g3 x f x z c4 i 1n j 1n zj pj pi 2pi zi 2 g4 x for some positive constants c3c3 c4c4 and some functions g3g3 g4g4 we show by a different method from the classical ones that higher integrability of the boundary datum u u forces u to have higher integrability as well similar results are also obtained for obstacle problems

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

piecewise complementary lidstone interpolation and error inequalities the purpose of this paper is to develop piecewise complementary lidstone interpolation in one and two variables and establish explicit error bounds for the derivatives in l and l 2 norms

iterative methods for solving right focal point boundary value problems abstract we provide necessary and sufficient conditions for the existence of a solution of n th order differential equation satisfying right focal point boundary conditions for the linear problems we propose forward forward and backward backward one pass practical shooting methods which convert the boundary value problems to its equivalent initial value problems for the nonlinear problems iterative methods are discussed and a priori conditions for the convergence are obtained several examples are also illustrated

existence uniqueness and iterative methods for third order boundary value problems abstract in this paper we shall provide necessary and sufficient conditions for the existence and uniqueness of solutions of third order nonlinear differential equations satisfying three point boundary conditions for the linear case we propose a constructive method which is a variation of the method of chasing for the nonlinear problems sufficient conditions are provided to ensure the convergence of a general class of iterative methods several examples are also included

explicit error bounds for the derivatives of piecewise lidstone interpolation abstract the purpose of this paper is to provide explicit error bounds for the derivatives of piecewise lidstone interpolation in l and l2 norms the results obtained are best possible sharp and supplement those established in earlier papers

sign patterns that require repeated eigenvalues abstract motivated by the question of which sign patterns allow a diagonalizable matrix we relate a number of properties to that of requiring repeated eigenvalues these are then used to make several observations about sign patterns that allow diagonalizability the only barrier to diagonalizability is a required nontrivial jordan structure associated with zero we note that the question of sign patterns that require diagonalizability is also open

local regularity for solutions of anisotropic obstacle problems abstract this paper deals with anisotropic obstacle problems for the a harmonic equation div a x u x 0 a local regularity result is given under suitable assumptions

uniqueness and existence for bounded boundary value problems the existence and uniqueness of solutions for the boundary value problems with general linear point evaluation boundary conditions is established we assume that f is bounded and that there is uniqueness on a homogeneous problem and on the linear variational problems

estimates for fully anisotropic elliptic equations with a zero order term abstract integral estimates for weak solutions to a class of dirichlet problems for nonlinear fully anisotropic elliptic equations with a zero order term are obtained by symmetrization techniques the anisotropy of the principal part of the operator is governed by a general n dimensional young function of the gradient which is not necessarily of polynomial type and need not satisfy the 2 condition

boundary value problems for higher order integro differential equations where r t x t x t x j t and a is a continuous operator map c l al a c ar a the function f is continuous in all of its arguments the problem l l 1 2 includes several particular cases for example the boundary value problems for the differential equations if a 0 considered in 7 8 and the references therein for the integro differential equations of the volterra type if

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

boundary value problems for second order differential equations of sobolev type abstract in this paper we provide necessary and sufficient conditions for the existence and uniqueness of solutions of second order differential equations of sobolev type satisfying some boundary conditions we also give sufficient conditions so that the picard and approximate picard method converge to the unique solution of these boundary value problems

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

eigenvalues and the one dimensional p laplacian we consider the boundary value problem p u f t u 0 with p 1 t 0 1 u 0 u 1 0 and with 0 the value of is chosen so that the boundary value problem has a positive solution in addition we derive an explicit interval for such that for any in this interval the existence of a positive solution to the boundary value problem is guaranteed in addition the existence of two positive solutions for in an appropriate interval is also discussed

nontrivial solutions for higher order multi point boundary value problems in the case where a nonlinearity may change sign and contains higher derivatives we consider the existence of nontrivial solutions for a class of higher order multi point boundary value problems some sufficient conditions for the existence of nontrivial solutions are established under certain suitable growth conditions our proof is based on leray schauder nonlinear alternative

quasilinearization and approximate quasilinearization for multipoint boundary value problems quasilinearization technique has been applied to a general nonlinear lidstone boundary value problem for the construction of a sequence of its approximate solutions xn t sufficient conditions

error and stability of monotone method for numerical solutions of fourth order semilinear elliptic boundary value problems this paper is concerned with the error and stability analysis of the monotone method for numerical solutions of fourth order semilinear elliptic boundary value problems a comparison result among the various monotone sequences is given the global error is analyzed and some sufficient conditions are formulated to guarantee a geometric rate of convergence the stability of the monotone method is proved some numerical results are presented

the existence and the uniqueness of symmetric positive solutions for a fourth order boundary value problem the purpose of this paper is to investigate the existence and the uniqueness of symmetric positive solutions for a class of fourth order boundary value problem y 4 t f t y t t 0 1 y 0 y 1 y 0 y 1 0 by using the fixed point index method we establish the existence of at least one or at least two symmetric positive solutions for the above boundary value problem further by using a fixed point theorem of general a concave operators we also present criteria which guarantee the existence and uniqueness of symmetric positive solutions for the above boundary value problem

integrability and boundedness of solutions to some anisotropic problems abstract in this article we give a series of results on the improved integrability and boundedness of solutions to several anisotropic problems in particular we consider the nonhomogeneous dirichlet problem for anisotropic elliptic second order equations moreover we study anisotropic variational inequalities with unilateral and bilateral obstacles as well as gradient constraints finally variational problems for integral functionals with anisotropic integrands are considered

disconjugacy disfocality and differentiation with respect to boundary conditions in particular under certain assumptions concerning the disconjugacy or disfocality of 1 we will consider the differentiability of solutions of 1 with respect to boundary conditions both boundary values and boundary points in so doing we will prove an analogue of a theorem of peano s which appears in hartman s text 1 theorem 3 1 p 951 with the consideration therein being with the differentiation of solutions of 1 with respect to initial conditions in this setting given a solution y x of 1 we are also interested in solutions of the linear nth order differential equation

sharp regularity for functionals with p q growth we prove regularity theorems for minimizers of integral functionals of the calculus of variations f x du x dx with non standard growth conditions of p q type z p f x z l z q 1 p q in particular we find that a sufficient condition for minimizers to be regular is qp n n where the function f x z is holder continuous with respect to the x variable and x rn this condition is also sharp we include results in the setting of orlicz spaces moreover we treat certain relaxed functionals too finally we address a problem posed by marcellini in 43 showing a minimizer with an isolated singularity

liapunov type inequalities for third order half linear equations and applications to boundary value problems abstract we derive liapunov type inequalities for third order half linear differential equations these inequalities utilize integrals of both q t and q t rather than those of q t as in most papers in the literature for higher order liapunov type inequalities furthermore by combining these inequalities with the uniqueness implies existence theorems by several authors we establish the uniqueness and hence existence uniqueness for several classes of boundary value problems for third order linear equations we believe that this is the first time for liapunov type inequalities to be used to deal with boundary value problems and expect that this approach can be further applied to study general higher order boundary value problems

existence of solutions of right focal point boundary value problems for ordinary differential equations y f x y y y 1 1 with respect to l two conditions which we will assume are a f x y 7 yz y is continuous on a b x r b solutions of initial value problems for 1 are unique motivated by the idea of right disfocality of a linear nth order differential operator given by muldowney 11 we shall be concerned with uniqueness of solutions of 1 which satisfy boundary conditions of the form y l x yi 1 i n where x1 x q a x for related results in the linear case where x1 x x x see for example nehari 2 peterson 3 and elias 4 5 1 to be more definite we define what is meant by a right m 1 m focal point boundary value problem for 1 on an interval i

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 solutions of k 2 point nonlocal boundary value problems for ordinary differential equations abstract sufficient conditions are given under which the existence of solutions of 4 point nonlocal boundary value problems for n th order nonlinear ordinary differential equations yields the existence of unique solutions of k 2 point boundary value problems for 1 k n 1 the results are motivated by a henderson and jackson paper involving relationships between 2 point and n point conjugate boundary value problems

a reliable algorithm for solving boundary value problems for higher order integro differential equations the aim of this paper is to present an efficient analytical and numerical procedure for solving boundary value problems for higher order integro differential equations the modified form of adomian decomposition method is found to be fast and accurate the analysis is accompanied by numerical examples the results demonstrate reliability and efficiency of the proposed algorithm

uniqueness implies existence on time scales abstract the uniqueness of solutions of initial value boundary value problems along with some uniqueness conditions on two point boundary value problems imply the existence of solutions for the same boundary value problems for the second order nonlinear dynamic equation y f t y y on a time scale this existence is established using shooting methods

an approximate method for solving a class of weakly singular volterra integro differential equations abstract in this paper we present a new approach to resolve linear and nonlinear weakly singular volterra integro differential equations of first or second order by first removing the singularity using taylor s approximation and then transforming the given first or second order integro differential equations into an ordinary differential equation such as the well known legendre degenerate hypergeometric euler or abel equations in such a manner that adomian s asymptotic decomposition method can be applied which permits convenient resolution of these equations some examples with closed form solutions are studied in detail to further illustrate the proposed technique and the results obtained demonstrate this approach is indeed practical and efficient

a subdivision approach to the construction of approximate solutions of boundary value problems with deviating arguments abstract using the ideas employed in the construction of subdivision algorithms we offer here a high accuracy algorithm to compute numerical solutions for two point boundary value problems of differential equations with deviating arguments numerical examples are included to demonstrate the fast convergence and high accuracy of the algorithm this paper is a further development to our previous works for solving various types of boundary value problems

finite difference methods for boundary value problems of differential equations with deviating arguments abstract we propose a convergent finite difference method to construct an approximate solution of the boundary value problems with deviating arguments the method developed takes into account the continuity limitations of the solutions of these problems to demonstrate the effectiveness of the proposed method numerical evidence is also included

discrete polynomial interpolation green s functions maximum principles error bounds and boundary value problems abstract we construct discrete interpolating polynomials provide explicit representations of discrete green s functions give several identities and inequalities for these green s functions use the explicit forms of the interpolating polynomials and that of green s functions to establish several maximum principles further we obtain error bounds in discrete polynomial interpolation and use them to study existence and uniqueness of the discrete boundary value problems these bounds are also used to provide sufficient conditions for the convergence of the picard s method the approximate picard s method quasilinearization and the approximate quasilinearization the monotone convergence of the picard s iterative method is also analysed

is every nonsingular matrix diagonally equivalent to a matrix with all distinct eigenvalues abstract it is shown that a 2 2 complex matrix a is diagonally equivalent to a matrix with two distinct eigenvalues iff a is not strictly triangular it is established in this paper that every 3 3 nonsingular matrix is diagonally equivalent to a matrix with 3 distinct eigenvalues more precisely a 3 3 matrix a is not diagonally equivalent to any matrix with 3 distinct eigenvalues iff det a 0 and each principal minor of a of order 2 is zero it is conjectured that for all n 2 an n n complex matrix is not diagonally equivalent to any matrix with n distinct eigenvalues iff det a 0 and every principal minor of a of order n 1 is zero

on the right focal point boundary value problems for integro differential equations abstract for the nth order nonlinear integro differential equations subject to right focal point boundary conditions we provide necessary and sufficient conditions for the existence uniqueness and convergence of an approximate iterative method

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

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

sinc galerkin method for solving nonlinear boundary value problems the sinc galerkin method is used to approximate solutions of nonlinear problems involving nonlinear second fourth and sixth order differential equations with homogeneous and nonhomogeneous boundary conditions the scheme is tested on four nonlinear problems the results demonstrate the reliability and efficiency of the algorithm developed

uniqueness of solutions of right focal problems for third order differential equations on considere l unicite des solutions quand elles existent de certains problemes aux valeurs limites pour l equation differentielle d ordre 3 y f x y y y on suppose que f a b r 3 r est continue

boundary data smoothness for solutions of nonlocal boundary value problems for second order differential equations abstract under certain conditions solutions of the boundary value problem y f x y y y x 1 y 1 and y x 2 i 1 m r i y i y 2 are differentiated with respect to boundary conditions where a x 1 1 m x 2 b r 1 r m r and y 1 y 2 r

numerical method for the solution of special nonlinear fourth order boundary value problems the sinc galerkin method is used to approximate solution of nonlinear problems this work deals with the sinc galerkin method for solving nonlinear fourth order differential equations with homogeneous and nonhomogeneous boundary conditions the scheme is tested on three nonlinear problems and a comparison with finite difference methods and fourth order multiderivative method is made it is shown that the sinc galerkin method yields better results

regularity for entropy solutions to degenerate elliptic equations abstract we derive estimates for entropy solutions to degenerate elliptic equations of the form div a x u x u x f x x u x 0 x where the caratheodory function a r r n r n satisfies degenerate coercivity condition a x s p 1 s and controllable growth condition a x s p 1 for 1 p n 0 p 1 and 0 and f lies in marcinkiewicz spaces we use generalized stampacchia lemma to prove the main theorem counterexample shows that one of the results in this paper is optimal

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

regularity for anisotropic elliptic equations with degenerate coercivity abstract this paper deals with boundary value problems of the form i 1 n d i a i x u x d u x f x x u x 0 x assume that there exist c 1 0 such that for almost all x and all s z r r n a i x s z c 1 1 z i p i 1 i 1 n and i 1 n z i p i 1 s i 1 n a i x s z z i we let f l m and we derive regularity results for weak solutions

uniqueness existence and optimality for fourth order lipschitz equations for each t y y y y4 t zi z2 z3 z e a 6 x r4 in particular we will characterize optimal length subintervals of a b in terms of the lipschitz coefficients k i 1 2 3 4 on which certain two three and four point boundary value problems for 1 have unique solutions the techniques we employ here involve applications of the pontryagin maximum principle 14 p 3141 in conjunction with uniqueness implies existence results for solutions of boundary value problems for 1 these techniques are motivated by works of melentsova 15 and melentsova and mil shtein 16 171 and most notably by the two papers by jackson ll 121 furthermore the results contained herein can be considered as extensions of a recent paper by henderson 6 dealing with boundary value problems for third order equations in relating the results of this paper to previous works we will formulate the boundary value problems in terms of the nth order differential equation

focal boundary value problems for nonlinear difference equations ii on considere l unicite des solutions impliquant l existence des solutions de problemes aux valeurs limites de type focal pour l equation aux differences non lineaire d ordre n u t n f t u t u t n 1 ou f a r n r est continue et l equation u n 1 f t u 1 u n peut etre resolue pour u 1 comme fonction continue des variables t u 2 u u 1

integrability for solutions to some anisotropic elliptic equations abstract we consider the boundary value problem i 1 n d i a i x d u x 0 x u x u x x we show that higher integrability of the boundary datum u forces solutions u to have higher integrability as well assumptions on a i x z are suggested by the euler equation of the anisotropic functional d 1 u p 1 d 2 u p 2 d n u p n d x

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

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

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

uniqueness implies existence and uniqueness conditions for nonlocal boundary value problems for nth order differential equations abstract for the nth order differential equation y n f x y y y n 1 we consider uniqueness implies existence results for solutions satisfying certain nonlocal k 2 point boundary conditions 1 k n 1 uniqueness of solutions when k n 1 is intimately related to uniqueness of solutions when 1 k n 2 these relationships are investigated as well

existence and uniqueness of positive solutions for a higher order boundary value problem the goal of this paper is to study the existence and uniqueness of positive solutions for the 2nth order boundary value problem 1 nu 2 n f t u u 1 n 1u 2 n 2 a 0u 2 i 0 email protected 0u 2 i 1 0 0 i 0 1 n 1 a 1u 2 i 1 email protected 1u 2 i 1 1 0 i 0 1 n 1 the krein rutman theorem and the krasnoselskii zabreiko fixed point theorem are the main tools that have been used to develop our work

method of chasing for multipoint boundary value problems the method of chasing developed by gel fand and lokutsiyevskii is completely formulated here for nth order ordinary linear differential equations subject to separated multipoint boundary conditions

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

right focal point boundary value problems for ordinary differential equations and variational equations where x g x such problems are called right focal point boundary value problems in the last few years a number of papers have appeared dealing with such boundary value problems most notable in the case when 1 is linear are papers by elias 11 muldowney 7 8 and peterson 19 io and for nonlinear considerations the papers by henderson 3 4 jackson 5 and peterson 11 121 in those papers several analogues of known results for conjugate type boundary value problems are considered for eq l we will have need of the hypotheses

global integrability for weak solutions to some anisotropic elliptic equations abstract we consider the boundary value problem i 1 n d i a i x d u x 0 x u x u x x we show that higher integrability of the boundary datum u forces solutions u to have higher integrability as well assumptions on a i x z are suggested by euler equation of the anisotropic functional i 1 n 2 d i u 2 d i u sin d i u p i 2