on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

homoclinic solutions for a class of the second order hamiltonian systems abstract we study the existence of homoclinic orbits for the second order hamiltonian system q v q t q f t where q r n and v c 1 r r n r v t q k t q w t q is t periodic in t a map k satisfies the pinching condition b 1 q 2 k t q b 2 q 2 w is superlinear at the infinity and f is sufficiently small in l 2 r r n a homoclinic orbit is obtained as a limit of 2 kt periodic solutions of a certain sequence of the second order differential equations

preconditioned gmres methods for discretization equation of nonsymmetric and indefinite elliptic problem in this paper a class of preconditioners for gmres method are proposed for solving linear systems arising from discretization second order nonsymmetric and indefinite elliptic problems by finite element method the convergence of gmres method is verified and the rate of convergence is shown the generality of our result enables us to apply not only any known preconditioners designed for symmetric positive definite problems but also many other preconditioners based on original operator to nonsymmetric and indefinite problems without losing optimality

a preconditioned gmres method for nonsymmetric or indefinite problems a preconditioning technique is proposed for nonsymmetric or indefinite linear systems of equations the main idea in our theory roughly speaking is first to use some coarser mesh space to correct the nonpositive portion of the eigenvalues of the underlying operator and then switch to use a symmetric positive definite preconditioner the generality of our theory allows us to apply any known preconditioners that were orginally designed for symmetric positive definite problems to nonsymmetric or indefinite problems without losing the optimality that the original one has some numerical experiments based on gmres are reported

infinitely many homoclinic solutions for nonautonomous p t laplacian hamiltonian systems by using the symmetric mountain pass theorem we establish some existence criteria which guarantee that the second order ordinary p t laplacian systems of the form ddt email protected t p t email protected t a t u t p t 2u t email protected w t u t 0 have infinitely many homoclinic solutions where email protected r email protected r n email protected c r r and p t 1 email protected c r r and email protected c 1 rxr n r are non periodic in t

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

existence and multiplicity of solutions for p x laplacian equations in rn abstract this paper investigates the existence and multiplicity of solutions for p x laplacian equations in r n

multivariable aumann integrals and controlled wave equations abstract the paper is concerned with the closure convexity and other properties of joint marginal integrals of multifunctions in several independent variables the results are then applied to a controlled wave equation obtaining a bang bang theorem and an existence theorem for optimal distributed controls without the usual convexity assumptions

homoclinic solutions for a class of second order hamiltonian systems abstract some new results for the existence of homoclinic orbits are obtained for the second order hamiltonian systems u t v t u t f t where t r u r n and v c 1 r r n r v t x k t x w t x is t periodic with respect to t t 0 and f r r n is a continuous and bounded function our assumptions on k and f are rather loose

homoclinic solutions for a class of second order hamiltonian systems abstract a new result for existence of homoclinic orbits is obtained for the second order hamiltonian systems u t l t u t w 1 t u t w 2 t u t f t where t r u r n and w 1 w 2 c 1 r r n r and f c r r n are not necessary periodic in t this result generalizes and improves some existing results in the literature

homoclinic orbits for a class of hamiltonian systems consider the second order hamiltonian system where q n and v c 1 n is t periodic in t suppose v q t 0 0 0 is a local maximum for v t and v t x x under these and some additional technical assumptions we prove that hs has a homoclinic orbit q emanating from 0 the orbit q is obtained as the limit as k of 2 kt periodic solutions i e subharmonics q k of hs the subharmonics q k are obtained in turn via the mountain pass theorem

new existence of homoclinic orbits for a second order hamiltonian system abstract by using the mountain pass theorem and the symmetric mountain pass theorem we establish some existence criteria to guarantee that the second order hamiltonian system u t a t u t p 2 u t w t u t 0 has at least one or infinitely many homoclinic orbits where t r u r n a c r r and w c 1 r r n r are not periodic in t our conditions on the potential w t x are rather relaxed

a pointwise constrained version of the liapunov convexity theorem for vectorial linear first order control systems we generalize the liapunov convexity theorem s version for vectorial control systems driven by linear odes of first order p 1p 1 in any dimension d nd n by including a pointwise state constraint more precisely given a x wp 1 a b rd solving the convexified p th order differential inclusion lpx t co u0 t u1 t um t a e consider the general problem consisting in finding bang bang solutions i e lpx t u0 t u1 t um t a e under the same boundary data x k a x k a x k b x k b k 0 1 p 1k 0 1 p 1 but restricted moreover by a pointwise state constraint of the type x t x t t a b e g 1 0 0 1 0 0 yielding x 1 t x 1 t previous results in the scalar d 1d 1 case were the pioneering amar cellina paper dealing with lpx x followed by cerf mariconda results who solved the general case of linear differential operators lplp of order p 2p 2 with c0 a b coefficients this paper is dedicated to focus on the missing case p 1p 1 i e using lpx x a x generalize the dimension of x x from the scalar case d 1d 1 to the vectorial d nd n case weaken the coefficients from continuous to integrable so that a a now becomes a d dd d integrable matrix and allow the directional vector to become a moving ac function previous vectorial results had constant no matrix i e a 0a 0 and considered constant control vertices amar mariconda and more recently integrable control vertices ourselves

existence of solutions for p x laplacian dirichlet problem this paper presents several sufficient conditions for the existence of solutions for the dirichlet problem of p x laplacian div u p x 2 u f x u x u 0 x especially an existence criterion for infinite many pairs of solutions for the problem is obtained the discussion is based on the theory of the spaces lp x and w01 p x

homoclinic solutions of some second order non autonomous systems abstract in this paper we investigate the existence of homoclinic solutions for the following second order non autonomous system q a q l t q w q t q 0 ds where a is an antisymmetric constant matrix l c r r n 2 is a symmetric and positive definite matrix for all t r w t q a t v q such that a r r is a continuous function and v c 1 r n r assuming that v q is subquadratic as q and some technical assumptions on a and l we establish two existence criteria to guarantee that ds has at least one nontrivial homoclinic solution by using a standard minimizing argument besides that in some particular case for the first time the uniqueness of homoclinic solutions of ds is also obtained recent results in the literature are generalized and significantly improved

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

nonconvex variational problems related to a hyperbolic equation we first prove a new lyapunov type theorem which will yield existence of solutions to nonconvex minimum problems involving some hyperbolic equations on rectangular domains with darboux boundary conditions some problems with obstacle and bang bang results are also considered

infinite periodic solutions to a class of second order sturm liouville neutral differential equations abstract by means of variational structure and z 2 group index theory we obtain infinite periodic solutions to a class second order sturm liouville neutral delay equations p t x t s q t x t s f t x t x t x t 2 x t 2 s 0

homoclinic solutions for a class of non autonomous subquadratic second order hamiltonian systems abstract in this paper we consider the existence of homoclinic solutions for the following second order non autonomous hamiltonian system hs q l t q w q t q 0 where l t c r r n 2 is a symmetric and positive definite matrix for all t r w t q a t q with a t r r is a positive continuous function and 1 2 is a constant adopting some other reasonable assumptions for l and w we obtain a new criterion for guaranteeing that hs has one nontrivial homoclinic solution by use of a standard minimizing argument in critical point theory recent results from the literature are generalized and significantly improved

some new error estimates for ritz galerkin methods with minimal regularity assumptions new uniform error estimates are established for finite element approximations u h of solutions u of second order elliptic equations lu f using only the regularity assumption u 1 c f 1 using an aubin nitsche type duality argument we show for example that for arbitrary fixed e sufficiently small there exists an h 0 such that for 0 h h 0 u u h 0 e u u h 1 here s denotes the norm on the sobolev space h s other related results are established

notes on periodic solutions of subquadratic second order systems some existence theorems are obtained for periodic solutions of the subquadratic second order systems by the minimax methods in critical point theory

existence of periodic solutions for p t laplacian systems abstract in this paper by using minimax methods in critical point theory we obtain some existence theorems for periodic solutions of p t laplacian system u t p t 2 u t f t u t 0 u 0 u t u 0 u t 0 and our results generalize some existed theorems

a preconditioned gmres method in this paper we show the convergence of gmres method for nonsymmetric problems preconditioned with a multigrid method based on the original operator this is simpler and more natural to implement than using a preconditioner based on the principal part as suggested by cai and xu

a knobloch type result for p t laplacian systems we consider the periodic boundary value problem of ordinary differential systems with p t laplacian of the form u p t 2u f t u u 0 u t u 0 u t 0 where p c r r is a t periodic function and p t 1 for t r f c r rn rn and f t u is t periodic with respect to t we prove that if there exists some r 0 such that f t u u 0 for t r and u rn with u r then the problem has at least one solution u satisfying u t r for t r this is a generalization of the results obtained by knobloch and mawhin under the case of p t 2 and p t p 1 respectively

some results on connecting orbits for a class of hamiltonian systems abstract the existence of various kinds of connecting orbits is established for a certain hamiltonian system as well as its time dependent analogue for the autonomous case our main assumption is that v has a global maximum e g at x o and we find a various kinds of orbits terminating at o for the time dependent case v has a local but not global maximum at x o and we find a homoclinic orbit emanating from and terminating at o keywords periodic solution

new estimates for multilevel algorithms including the v cycle the purpose of this paper is to provide new estimates for certain multilevel algorithms in particular we are concerned with the simple additive multilevel algorithm discussed recently together with j xu and the standard v cycle algorithm with one smoothing step per grid we shall prove that these algorithms have a uniform reduction per iteration independent of the mesh sizes and number of levels even on nonconvex domains which do not provide full elliptic regularity for example the theory applies to the standard multigrid vcycle on the l shaped domain or a domain with a crack and yields a uniform convergence rate we also prove uniform convergence rates for the multigrid v cycle for problems with nonuniformly refined meshes finally we give a new multigrid approach for problems on domains with curved boundaries and prove a uniform rate of convergence for the corresponding multigrid v cycle algorithms

uniform convergence of multigrid v cycle iterations for indefinite and nonsymmetric problems in this paper an analysis of a multigrid method for nonsymmetric and or indefinite elliptic problems is presented in this multigrid method various types of smoothers may be used one type of smoother considered is defined in terms of an associated symmetric problem and includes point and line jacobi and gauss seidel iterations smoothers based entirely on the original operator are also considered one smoother is based on the normal form that is the product of the operator and its transpose other smoothers studied include point and line jacobi and gauss seidel it is shown that the uniform estimates of j h bramble and j e pasciak math comp 60 1993 pp 447 471 for symmetric positive definite problems carry over to these algorithms more precisely the multigrid iteration for the nonsymmetric and or indefinite problem is shown to converge at a uniform rate provided that the coarsest grid in the multilevel iteration is sufficiently fine but not dependent on the number of multigrid levels

compact imbedding theorems with symmetry of strauss lions type for the space w1 p x ω abstract in this paper we extend some compact imbedding theorems of strauss lions type to the space w 1 p x when the domain has some symmetric properties and p x satisfies some conditions

some existence results on periodic solutions of ordinary p laplacian systems some existence theorems are obtained for periodic solutions of ordinary p laplacian systems by the minimax methods in critical point theory