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

constructing tables of minimum aberration pn m designs fries and hunter 1980 presented a practical algorithm for selecting standard 2 n m fractional factorial designs based on a criterion they called minimum aberration in this article some simple results are presented that enable the fries hunter algorithm to be used for a wider range of n and m and for designs with factors at p levels where p 2 is prime examples of minimum aberration 2 n m designs with resolution r 4 are given for m n m 9 a matrix is given for generating 3 n m designs with m n m 6 which have or nearly have minimum aberration

choice of optimal blocking schemes in two level and three level designs blocking is commonly used in design of experiments to reduce systematic variation and increase precision of effect estimation for 2n k designs in 2p blocks others have studied the confounding pattern between blocking effects and treatment effects and proposed some criteria to choose optimal blocking schemes we propose alternative criteria that allow estimation of more lower order effects extensions to three level fractional factorial design are considered and tables of useful designs are given

minimum aberration designs with two level and four level factors summary for optimal assignment of factors to columns of a 4 2 n design we consider extensions of the minimum aberration criterion for the 2 level designs minimum aberration 412n and 422n designs in 16 and 32 runs are obtained and tabulated some results on the wordlength patterns and design classification are obtained

fixed sign solutions of second order difference equations with neumann boundary conditions in this paper we study the existence of solution of the nonlinear second order difference problem with neumann boundary conditions assuming the existence of a pair of ordered lower and upper solutions and we obtain optimal existence results for the case and even for to this end we do an exhaustive study of the values of the real parameters and for which the associated green s function to the linear operator l u u 2 2 u 1 u with neumann boundary conditions has fixed sign

the method of lower and upper solutions for systems of boundary value problems the method of lower and upper solutions for second order differential system x f t x with dirichlet boundary condition x 0 x 1 0 is used to give sufficient conditions for the existence of at least one solution

second order periodic boundary value problems on time scales in this paper we study the existence of solutions of periodic boundary value problem for second order nonlinear dynamic equations on time scales for this purpose we use two methods here first sign properties of the green s function are investigated and existence results of bounded solutions of nonlinear boundary value problems are established as a result of schauder fixed point theorem second the monotone method is discussed to assure the existence and uniqueness of solutions of such periodic boundary value problems on time scales

on the existence and uniqueness for higher order periodic boundary value problems abstract this paper discusses the existence and uniqueness for the n th order periodic boundary value problem l n u t f t u t 0 t 2 u i 0 u i 2 i 0 1 n 1 where l n u t u n t i 0 n 1 a i u i t is an n th order linear differential operator n 2 and f 0 2 r r is continuous in the case that l n has an even order derivative we present some new spectral conditions for the nonlinearity f t u to guarantee the existence and uniqueness these spectral conditions allow f t u to be a superlinear growth and are the extension for the spectral separation condition presented recently in y li existence and uniqueness for higher order periodic boundary value problems under spectral separation conditions j math anal appl 322 2 2006 530 539

simple c algebras with perforation abstract simple c algebras are constructed for which the ordered k 0 group is not weakly unperforated the algebras are approximately homogeneous and do not have slow dimension growth

positive periodic solutions for nonlinear difference equations with periodic coefficients abstract we prove the existence of positive solutions of second order nonlinear difference equations on a finite discrete segment with periodic boundary conditions obtained results yield positive periodic solutions of the equation on the whole discrete axis provided that the coefficients are periodic

multiplicity results for nonlinear periodic fourth order difference equations with parameter dependence and singularities abstract this paper is devoted to the study of nonlinear singular and non singular fourth order difference equations coupled with periodic boundary value conditions in the paper some existence and nonexistence results are given the results are deduced from krasnoselskii s fixed point theorems in cones an exhaustive study of the green s function related to a linear fourth order problem is done

existence and uniqueness results for discrete second order periodic boundary value problems abstract we prove existence and uniqueness results for solutions of second order nonlinear difference equations on a finite discrete segment with periodic boundary conditions the results are based on the notion of upper and lower solutions

a generalized upper and lower solutions method and multiplicity results for nonlinear first order ordinary differential equations abstract we introduce a generalized upper and lower solutions method for the solvability of first order ordinary differential equations u t t u t u 0 u 1 in order to cover the case when the function satisfies caratheodory conditions this method is then applied to get multiplicity results when the nonlinearity interacts with the real eigenvalue of the linearized problem our proofs are based on differential inequalities and classical leray schauder degree

green functions and positive solutions for boundary value problems of third order difference equations in this paper we investigate the boundary value problems for the third order nonlinear difference equation d 3u t 2 a t f u t 0 t 2 t 2 satisfying five kinds of different boundary value conditions the green functions are constructed carefully and their bounds are evaluated some verifiable criteria for the existence of at least one positive solution and two positive solutions are obtained by using the green functions and the evaluations of bounds some examples are given to demonstrate the applications of the theorems obtained

twin monotone positive solutions to a singular nonlinear third order differential equation abstract this paper discusses the existence of at least one or two nondecreasing positive solutions for the following singular nonlinear third order differential equation x t t f t x t 0 a t b x a x a x b 0 where 0 is a parameter c a b r f c a b 0 r t may be singular at t a b and f t s may be singular at s 0 green s function and the fixed point theorem of cone expansion and compression type are utilized to illustrate the results four nontrivial examples are included

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

existence multiplicity and infinite solvability of positive solutions to a nonlinear fourth order periodic boundary value problem abstract the existence of n and infinitely many positive solutions is proved for the nonlinear fourth order periodic boundary value problem u 4 t u t u t f t u t 0 t 1 u i 0 u i 1 i 0 1 2 3 where n is an arbitrary natural number and 2 2 0 1 2 2 2 2 4 2 1 0 this kind of fourth order boundary value problems usually describes the equilibrium state of an elastic beam with periodic boundary condition the main results show that the problem may have n or infinitely many positive solutions provided the growth rates of nonlinear term f t l are appropriate on some bounded subsets of its domain

nearly orthogonal arrays with mixed levels and small runs in running a factorial experiment it may be desirable to use an orthogonal array with different mixed numbers of factor levels because of the orthogonality requirement such arrays may have a large run size by slightly sacrificing the orthogonality requirement we can obtain nearly orthogonal arrays with economic run size some general methods for constructing such arrays are given for 12 18 20 and 24 runs many orthogonal arrays and nearly orthogonal arrays with mixed levels are constructed and tabulated effects of near orthogonality on estimation efficiency and analysis are studied

existence and approximation of solutions for first order discontinuous difference equations with nonlinear global conditions in the presence of lower and upper solutions abstract this paper is devoted to the study of the existence of solutions of first order difference equations verifying nonlinear conditions that involve the global behavior of the solution we prove that the existence of lower and upper solutions warrants the existence of such solutions lying in the sector formed by the mentioned functions we also can prove that some classical results for differential equations are not true in general for this case

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

existence of solutions to nonlinear first order pbvps on time scales abstract in this paper we are concerned with the following nonlinear first order periodic boundary value problem on time scales x t p t x t g t x t t 0 t t x 0 x t some new existence criteria of at least one solution are established by using novel inequalities and the well known schaefer fixed point theorem

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

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

multiple solutions for higher order difference equations abstract this paper establishes the existence of twin nonnegative solutions to i conjugate and ii n p higher order discrete boundary value problems

a monotone method for constructing extremal solutions to second order periodic boundary value problems abstract in this paper we describe a constructive method which yields two monotone sequences that converge uniformly to extremal solutions to the periodic boundary value problem u t f t u u t t 0 2 u 0 u 2 u 0 u 2 in the presence of an upper solution and a lower solution with

existence of solutions for a class of third order nonlinear boundary value problems abstract in this paper we are concerned with the following third order ordinary differential equation x t f t x t x t x t 0 0 with the nonlinear boundary conditions x 0 0 g x 0 x 0 a h x 1 x 1 b where a b r f 0 1 r 3 r is continuous g h r 2 r are continuous the existence result is given by using a priori estimate nagumo condition upper and lower solutions and leray schauder degree and we give an example to demonstrate our result

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

an approach to the construction of asymmetrical orthogonal arrays abstract use of asymmetrical orthogonal arrays oa s for planning industrial experiments with mixed levels has become increasingly popular in addition to applications to experimental investigations in many disciplines they can also be used in the balanced repeated replications method of inference from general stratified survey samples in this article we propose a general approach to the construction of asymmetrical oa s with economic run size and flexibility in the choice of factor levels as applications of this approach we construct several general classes of arrays that include numerous existing classes of arrays as special cases a large number of known asymmetrical oa s can be reproduced by our approach in a unified and simple manner many new arrays are obtained a list of asymmetrical oa s with moderate run size is given our approach consists of three steps we first construct an oa as the generalized kronecker sum of an oa and some difference matrices we then add to the constructed oa co

dimension and stable rank in the k theory of c algebras in topological k theory which can be viewed as the algebraic side of the theory of vector bundles some of the interesting properties which one investigates are for example the conditions under which bundles must possess trivial direct summands or the extent to which the cancellation property for direct whitney sums of bundles holds such properties turn out to be controlled in part by the dimension of the base space and results describing the nature of this control are among what are frequently called stability results 17 during the past few years there has been a sudden flowering of a theory for c algebras see surveys 35 36 37 since c algebras are profitably thought of as non commutative locally compact spaces with the finitely generated projective modules being the appropriate generalization of vector bundles according to swan s theorem 40 it would be natural to look for stability results for c algebras but until now the theory has been focussed on the c groups of c algebras and there has been little discussion of stability properties presumably in part for lack of an appropriate concept of dimension for c algebras one of the main objectives of this paper is to introduce a concept of dimension for c algebras which directly generalizes the classical concept of dimension for compact spaces 16 20 22 and to develop some techniques for calculating it notably for c algebras which are obtained as crossed product algebras for an action of the group of integers in algebraic k theory which is the generalization of topological c theory to rings a substantial number of stability properties have been discovered 2 39 42 48 within that theory the concept which has played a role most analogous to that of dimension is the bass stable rank but despite its many successes the bass stable rank has often been difficult to calculate even for relatively uncomplicated rings another main objective of the present paper is to make accessible for application to c algebras some of the stability results from algebraic c theory this comes about from the fact that the concept of dimension which we introduce dominates the bass stable rank i was led to seek stability results for c algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation c algebras discussed in 30 which as elliott 11 has suggested can appropriately be called 2 dimensional non commutative tori in more concrete terms the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent the present paper provides the general theory which is needed for the affirmative answer which i obtained in a subsequent paper i will derive the

lifting kk elements asymptotic unitary equivalence and classification of simple c algebras let a and c be two unital simple c algebras with tracial rank zero suppose that c is amenable and satisfies the universal coefficient theorem denote by kke c a the set of those in kk c a for which k0 c 0 k0 a 0 and 1c 1a suppose that kke c a we show that there is a unital monomorphism c a such that suppose that c is a unital ah algebra and is a continuous affine map for which t p t p for all projections p in all matrix algebras of c and any tt a where t a is the simplex of tracial states of a and is the convex set of faithful tracial states of c we prove that there is a unital monomorphism c a such that induces both and suppose that h c a is a unital monomorphism and hom k1 c aff a we show that there exists a unital monomorphism c a such that h in kk c a t t h for all tracial states t and the associated rotation map can be given by denote by kkt c a the set of compatible pairs where kle c a and is a continuous affine map from t a to together with a result on asymptotic unitary equivalence in h lin asymptotic unitary equivalence and asymptotically inner automorphisms arxiv math 0703610 2007 this provides a bijection from the asymptotic unitary equivalence classes of unital monomorphisms from c to a to where is a subgroup related to vanishing rotation maps as an application combining these results with a result of w winter w winter localizing the elliott conjecture at strongly self absorbing c algebras arxiv 0708 0283v3 2007 we show that two unital amenable simple stable c algebras are isomorphic if they have the same elliott invariant and the tensor products of these c algebras with any uhf algebra have tracial rank zero in particular if a and b are two unital separable simple stable c algebras with unique tracial states which are inductive limits of c algebras of type i then they are isomorphic if and only if they have isomorphic elliott invariants

positive solutions to nonlinear first order pbvps with parameter on time scales abstract in this paper we consider the following nonlinear first order periodic boundary value problem with parameter on a time scale t x t p t x t f x t t 0 t t x 0 x t where 0 is a parameter for suitable 0 some existence multiplicity and nonexistence criteria of positive solutions are established by using well known results from the fixed point index

a generalization of concavity for finite differences abstract the concept of concavity is generalized to discrete functions u satisfying the n th order difference inequality 1 n k n u m 0 m 0 1 n and the homogeneous boundary conditions u 0 u k 1 0 u n k 1 u n n 0 for some k 1 n 1 a piecewise polynomial is constructed which bounds u below the piecewise polynomial is employed to obtain a positive lower bound on u m for m k n k where the lower bound is proportional to the supremum of u an analogous bound is obtained for a related green s function

twin positive solutions of nonlinear first order boundary value problems on time scales abstract in this paper we consider the following nonlinear first order boundary value problem on time scale t x t f x t t 0 t t x 0 x t where 0 1 some existence criteria for at least two positive solutions are established by using the well known twin fixed point theorem due to avery and henderson

existence of positive solutions of bvps for second order nonlinear difference systems this paper is concerned with the following system d 2u i k f i k u 1 k u 2 k u n k 0 k 0 t i 1 2 nwith the dirichlet boundary conditionu i 0 u i t 2 0 i 1 2 n some new results are obtained for the existence and multiplicity of positive solutions to the above system by using nonlinear alternative of leray schauder type krasnosel skii s fixed point theorem in a cone and leggett williams fixed point theorem in particular it proves that the above system has n positive solutions under suitable conditions where n is an arbitrary integer

existence of multiple positive periodic solutions for delay differential equation whose order is a multiple of 4 abstract this article shows the existence of positive periodic solutions for retarded advanced neutral and ordinary differential equations whose order is a multiple of 4 first we find a positive green s function explicitly then assuming that the coefficient of the unknown in the linear part of the equation is bounded above and below by positive constants we find one and then three solutions by applying the krasnoselskii and legget william fixed point theorems

comparison results for first order linear operators with reflection and periodic boundary value conditions abstract this work is devoted to the study of the first order operator x t m x t coupled with periodic boundary value conditions we describe the eigenvalues of the operator and obtain the expression of its related green s function in the non resonant case we also obtain the range of the values of the real parameter m for which the integral kernel which provides the unique solution has constant sign in this way we automatically establish maximum and anti maximum principles for the equation some applications to the existence of nonlinear periodic boundary value problems are shown

functionally induced priors for the analysis of experiments this work develops the idea of using functional priors for the design and analysis of three level and higher level experiments developing a prior distribution for model parameters is challenging because a factor can be qualitative or quantitative we propose appropriate correlation functions and coding schemes so that the prior distribution is simple and the results are interpretable the prior incorporates well known principles such as effect hierarchy and effect heredity which helps resolve the aliasing problems in fractional designs almost automatically the usefulness of the new approach is illustrated through the analysis of some real experiments

double positive solutions of n p boundary value problems for higher order difference equations abstract we shall provide existence criteria for double positive solutions of the n p boundary value problem n y f k y y n 2 y g k y y n 1 y n 1 k n i y 0 0 0 i n 2 p y n n p 0 where n 2 and 0 p n 1 is fixed upper and lower bounds for the two positive solutions are also established for a particular boundary value problem when n 2 several examples are included to dwell upon the importance of the results obtained

generalized minimum aberration for asymmetrical fractional factorial designs by studying treatment contrasts and anova models we propose a generalized minimum aberration criterion for comparing asymmetrical fractional factorial designs the criterion is independent of the choice of treatment contrasts and thus model free it works for symmetrical and asymmetrical designs regular and nonregular designs in particular it reduces to the minimum aberration criterion for regular designs and the minimum g 2 aberration criterion for two level nonregular designs in addition by exploring the connection between factorial design theory and coding theory we develop a complementary design theory for general symmetrical designs which covers many existing results as special cases

extremal solutions for the difference φ laplacian problem with nonlinear functional boundary conditions abstract in this paper we study the existence and approximation of the extremal solutions for the laplacian problem lying between a pair of a lower solution and an upper solution such that we consider general boundary functional conditions that include classical ones as separated or periodic

on the stable rank of simple c algebras in 16 m a rieffel introduced the notion of stable rank for c algebras for a a unital c algebra the stable rank denoted by sr a is the least integer n such that the set of n tuples over a which generate a as a left ideal is dense in an if no such n exists we set sr a oo for unital c algebras the stable rank was shown to coincide with the bass stable rank which is defined for rings see 12 the stable rank can also be defined for nonunital c algebras but we shall only consider the unital case stable rank is related to nonstable k theory in 16 theorem 10 12 and 3 corollary 7 14 it was shown that if a is a simple unital c algebra of stable rank one then the natural map u a u a o 4 ki a is an isomorphism see also 17 2 and 18 it is a natural question which values occur as the stable rank of simple c algebras suppose that a is simple and infinite then by 7 proposition 1 5 a contains two isometries with orthogonal ranges and so by 16 proposition 6 5 sr a oo in the case of finite simple c algebras the following is known whenever a is simple and stably finite and b is a uhf algebra the tensor product a0 b has stable rank one see 19 this is also the case for every simple approximately homogeneous ah algebra with slow dimension growth see 4 and 8 so in short sr oo for every infinite simple c algebra and sr 1 for various classes of finite simple c algebras in this paper we answer the question above by showing that every n c n u oo is the stable rank of some simple finite c algebra more precisely it is shown that for every n 2 3 4 oo there exists a simple separable and unital ah algebra a of stable rank n in particular the group of invertible elements in a is not dense and like the examples in 20 a does not have slow dimension growth and is not approximately divisible see 5 i thank marius dadarlat for suggesting that i study ah algebras with connecting homomorphisms as described in the following section and juliana erlijman for drawing my attention to some of the powerful tools of differential topology and i thank george elliott for my stay at the fields institute where this paper was written

existence and uniqueness for higher order periodic boundary value problems under spectral separation conditions abstract this paper deals with the existence and uniqueness for the nth order periodic boundary value problem l n u t f t u t 0 t 2 u i 0 u i 2 i 0 1 n 1 where l n u t u n t i 0 n 1 a i u i t is an nth order linear differential operator and f 0 2 r r is continuous we present some spectral conditions for the nonlinearity f t u to guarantee the existence and uniqueness these spectral conditions are the generalization for nonresonance condition of duffing equation

existence and multiplicity of positive solutions to nonlinear first order pbvps on time scales in this paper we consider the following nonlinear first order periodic boundary value problems on time scales x d t p t x s t f x t email protected 0 t t x 0 x s t some new existence and multiplicity criteria of positive solutions are established by using several well known fixed point theorems

blocked regular fractional factorial designs with minimum aberration this paper considers the construction of minimum aberration ma blocked factorial designs based on coding theory the concept of minimum moment aberration due to xu 2003 for unblocked designs is extended to blocked designs the coding theory approach studies designs in a row wise fashion and therefore links blocked designs with nonregular and supersaturated designs a lower bound on blocked wordlength pattern is established it is shown that a blocked design has ma if it originates from an unblocked ma design and achieves the lower bound it is also shown that a regular design can be partitioned into maximal blocks if and only if it contains a row without zeros sufficient conditions are given for constructing ma blocked designs from unblocked ma designs the theory is then applied to construct ma blocked designs for all 32 runs 64 runs up to 32 factors and all 81 runs with respect to four combined wordlength patterns

bayesian inspired minimum aberration two and four level designs summary motivated by a bayesian framework we propose a new minimum aberration type criterion for designing experiments with two and four level factors the bayesian approach helps in overcoming the ad hoc nature of effect ordering in the existing minimum aberration type criteria the approach is also capable of distinguishing between qualitative and quantitative factors numerous examples are given to demonstrate its advantages

on the construction of asymmetrical orthogonal arrays general techniques for the construction of asymmetrical orthogonal arrays of strength 2 are presented these are then applied to special cases to obtain new families of such arrays among these are saturated main effect plans based on s m runs with factors at s nu i levels i 0 1 ldots r where m geq v r v 0 1 v i 1 divides v i i 1 2 ldots r and s is a prime power

discrete third order three point right focal boundary value problems we are concerned with the discrete right focal boundary value problem 3x t f t x t 1 x t1 x t2 2x t3 0 and the eigenvalue problem 3x t a t f x t 1 with the same boundary conditions where t1 t2 t3 under various assumptions on f a and we prove the existence of positive solutions of both problems by applying a fixed point theorem

uniqueness comparison and existence results for third order initial boundary value problems abstract we derive maximum principles for a nonlinear third order differential operator and prove uniqueness comparison and existence results for different types of third order initial boundary value problems including existence results for functional problems

extremality and comparison results for discontinuous implicit third order functional initial boundary value problems extremality results are derived for implicit third order functional initial boundary value problems differential equations initial conditions and boundary conditions of problems may involve discontinuous nonlinearities

fractional resolution and minimum aberration in blocked 2 n k designs systematic sources of variation can be reduced in fractional factorial experiments by grouping the runs into blocks this is accomplished through the use of blocking factors the concepts of resolution and minimum aberration design optimization criteria ordinarily used to rank unblocked fractional factorial designs are extended to such blocked fractional factorial designs by treating the treatment and blocking factors differently in terms of their contribution to word length in the defining contrast subgroup some limited theoretical results are derived and tables of minimumaberration blocked two level fractional factorial designs are presented and considered the relationship between clear effects effects that are estimable when higher order effects are assumed negligible and minimum aberration in the presence of blocking is discussed

construction of 2 m4 n designs via a grouping scheme on developpe une methode pour grouper les 2 k 1 effets factoriels dans un plan factoriel a deux niveaux dans des ensembles disjoints mutuellement s t st ou st est l interaction generalisee des effets s et t comme application on construit des tableaux orthogonaux oa 2 k 2 m 4 n 2 de taille 2 k m contraintes avec 2 niveaux et n contraintes avec 4 niveaux satisfaisant m 3n 2 k 1 et de puissance 2 le nombre maximum de contraintes avec 4 niveaux dans la construction ne peut plus etre ameliore dans ce sens le schema de groupement qu on donne est optimal on discute les avantages de la presente approche par rapport a d autres methodes de construction

extremal solutions and green s functions of higher order periodic boundary value problems in time scales abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u n t j 1 n 1 m j u j t f t u t t a b u i a u i b i 0 n 1 here f a b r r is such that f x is rd continuous in i for every x r and f t is continuous in r uniformly at t i m j r are given constants and a b t n for an arbitrary bounded time scale t we obtain sufficient conditions in f to guarantee the existence and approximation of solutions lying between a pair of ordered lower and upper solutions and to this end given m 0 we study some maximum principles related with operators t n m u t u n t j 1 n 1 m j u j t mu t in the space of periodic functions

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

existence and comparison results for difference φ laplacian boundary value problems with lower and upper solutions in reverse order abstract this paper is devoted to the study of the existence and comparison results for nonlinear difference laplacian problems with mixed dirichlet neumann and periodic boundary value conditions we deduce existence of extremal solutions of periodic and neumann boundary value problems lying between a pair of lower and upper solutions given in reverse order we prove the optimality of some assumptions in