asymptotic behavior of nonlinear difference systems in this paper we consider two dimensional nonlinear difference systems of the email protected n a nf y n dy n 1 b ng x n we classify their solutions according to asymptotic behavior and give some necessary and sufficient conditions for the existence of solutions of such classes

the oscillation and asymptotically monotone solutions of second order quasilinear differential equations abstract we offer sufficient conditions for the oscillation of all solutions of the perturbed quasilinear differential equation a t y 1 y q t y p t y y as well as for the existence of a positive monotone solution of the damped differential equation a t y 1 y b t y 1 y h t y 0 where 0 examples that dwell upon the importance of our results are also included

existence of uncountably many bounded nonoscillatory solutions and their iterative approximations for second order nonlinear neutral delay difference equations in this paper the banach fixed point theorem is employed to establish several existence results of uncountably many bounded nonoscillatory solutions for the second order nonlinear neutral delay difference email protected n dx n b n x email protected email protected n x h 1 n x h 2 n x h k n f n x f 1 n x f 2 n x f k n c n n n 0 where t email protected n n email protected n 0 a b c n n 0 r with a n 0 for email protected n n 0 h f n n 0xr k r and h l f l n n 0 z withlimn h l n limn f l n email protected 1 2 k a few mann type iterative approximation schemes with errors are suggested and the errors estimates between the iterative approximations and the nonoscillatory solutions are discussed seven nontrivial examples are given to illustrate the advantages of our results

the pspace completeness of black white pebbling the complexity of the black white pebbling game has remained an open problem for 30 years in this paper we show that the black white pebbling game is pspace complete

periodic solutions of a class of nonlinear difference equations via critical point method in this paper we obtain some new sufficient conditions for the existence of nontrivial m periodic solutions of the following nonlinear difference equation d p n dx n 1 d f n x n n z by using the critical point method where f zxr r is continuous in the second variable m 2 is a given positive integer p n m p n for any n z and f t m z f t z for any t z zxr 1 d 1 and d 0

oscillation and nonoscillation of two dimensional difference systems this paper is concerned with the oscillation and nonoscillation behavior of solution of the nonlinear two dimensional difference system xn ang yn yn 1 f n xn n n n0 n0 n0 1 some necessary and sufficient conditions are given for the system to admit the existence of oscillatory and nonoscillatory solutions with special asymptotic properties some important results in the literatures are generalized examples to illustrate the result are included

oscillation for second order nonlinear neutral delay difference equations in this paper we obtain sufficient conditions for the oscillation of all solutions of the second order nonlinear delay difference email protected a n d x n p nx g n a f n x s n 0 where dx n x n 1 x n is the forward difference operator p n is a sequence of nonnegative real numbers and a 1 is a quotient of positive odd integers our results extend and improve the known results in the literature an example that dwell up the advantage of our results is included

an existence theorem for a nonlinear difference equation where p 1 and s p is a real sequence we establish conditions under which 1 1 has a positive nondecreasing solution here a solution of 1 1 is a real sequence y y which satisfies 1 1 since 1 1 is a recurrence relation given real initial values y and y it is clear that we can inductively obtain y y since p is a real number greater than 1 the sequence yk may not be real that is not a solution note however that any constant multiple of a solution is again a solution the existence question originated in l where it was shown that hardy s inequality for a series 2 can be viewed as a necessary condition for the existence of a positive nondecreasing solution of 1 1 our existence theorem reduces to a nonoscillation criterion in the case p 2 since then 1 1 changes to a second order linear difference equation see e g 3 a brief outline of the paper is as follows section 2 introduces a riccati type transformation and then develops various necessary conditions for the existence of a positive nondecreasing solution section 3 discusses some of the functional analytic background which is needed in order to apply schauder s theorem section 4 contains the proof of our existence theorem and section 5 describes a comparison theorem which allows us to drop the assumption that the coefficients sk in 1 1 are nonnegative

oscillation theorems for certain second order nonlinear difference equations in this paper discrete inequalities are used to offer sufficient conditions for the oscillation of all solutions of the difference equation delta bigl a n delta y n sigma bigr q n 1 f y n 1 r n qquad n ge 0 where 0 p qwithp qodd integers orpeven andqodd integers several examples which dwell upon the importance of our results are also included

on existence of positive solutions and bounded oscillations for neutral difference equations abstract a criterion for the existence of a positive solution for the first order difference equation xn cxn m pnxn k 0 pn 0 is established results are also obtained for the oscillation and nonoscillation of solutions of a second order difference equation 2 xn cxn m pnxn k where the pn are either of constant sign or are oscillatory some of the results are sharp

oscillation of certain second order nonlinear difference equations abstract in this paper we consider the second order nonlinear difference equation formula where y n y n 1 y n is the forward difference operator f c r r f is nondecreasing and uf u 0 as u 0 q n is a real sequence some new sufficient conditions for the oscillation of all solutions of 1 are obtained

global solvability for a second order nonlinear neutral delay difference equation this paper studies the global existence of solutions of the second order nonlinear neutral delay difference equation d a email protected x n bx n t f n x n d 1 n x n d 2 n x n d k n c n n n 0 with respect to all email protected r a few results on global existence of uncountably many bounded nonoscillatory solutions are established for the above difference equation several nontrivial examples which dwell upon the importance of the results obtained in this paper are also included

oscillation criteria for difference equations with damping terms by means of generalized riccati transformation techniques we establish some new oscillation criteria for second order nonlinear difference equations with damping

oscillatory and asymptotic behavior of second order nonlinear difference equations in this paper we are dealing with the oscillatory and asymptotic behaviour of solutions of second order nonlinear difference equations of the form rn xn f n xn 0 n n n0 1 we obtain the following results a if k n0 l rk any nonoscillatory solution of 1 must belong to one of the following four types k k k 0 k 0 b if k n0 l rk any nonoscillatory solution of 1 must belong to one of the following three types k0 k k0 c necessary and sufficient conditions for 1 to have a nonoscillatory solution which belongs to k k k 0 k0 or k are given depending on whether f is a superlinear or sublinear function all these results include and improve b szmanda s results in bull polish acad sci math 34 nos 3 4 1986 133 141

the treewidth of proofs abstract so called ordered variants of the classical notions of pathwidth and treewidth are introduced and proposed as proof theoretically meaningful complexity measures for the directed acyclic graphs underlying proofs ordered pathwidth is roughly the same as proof space and the ordered treewidth of a proof is meant to serve as a measure of how far it is from being treelike length space lower bounds for k dnf refutations are generalized to arbitrary infinity axioms and strengthened in that the space measure is relaxed to ordered treewidth

oscillation of a class of partial difference equations with unbounded delay abstract this paper is concerned with the partial difference equation where and n z are nondecreasing n and n are strictly less than n for n n lim n n lim n n a m n b m n and p m n are three real double sequences some oscillation criteria for this equation are obtained

nonexistence of positive solution of a class of nonlinear difference equation sufficient conditions are derived for the nonexistence of positive solution of a class of nonlinear difference equations the conditions are based on nonexistence of positive characteristic roots of the characteristic equation of the nonlinear difference equation

oscillation criteria for nonlinear partial difference equations with delays abstract we offer sufficient conditions for the oscillation of all solutions of the partial difference equations y m 1 n m n y m n 1 m n y m n p m n y m k n l q m n y m k n l and y m 1 n m n y m n 1 m n y m n i 1 p i m n y m k i n l i i 1 q i m n y m k i n l i several examples which dwell upon the importance of our results are also included

oscillation of the emden fowler difference systems abstract in this paper we are concerned with some new criteria for the oscillation of the discrete emden fowler system

oscillation and nonoscillation criteria for linear dynamic systems on time scales in this paper we establish oscillation and nonoscillation criteria for the linear dynamic system u d pv v d qu s here we assume that p and q are nonnegative rd continuous functions on a time scale t such that supt indeed we extend some known oscillation results for differential systems and difference systems to the so called dynamic systems

oscillation theorems for general quasilinear second order difference equations the authors consider difference equations of the form where an 0 qn 0 f and are continuous real valued functions and uf u 0 for u 0 they give oscillation results for equation e examples are included to illustrate the results

oscillations and nonoscillations of half linear difference equations generated by deviating arguments for the half linear difference equation y k 1 y k i 1npi k y gi k 1y gi k k a where 0 we shall offer sufficient conditions for the oscillation of all solutions as well as necessary and sufficient conditions for the existence of both bounded and unbounded nonoscillatory solutions several examples which dwell upon the importance of our results are also included

oscillation criteria for two dimensional difference systems of first order linear difference equations sufficient conditions are established for the oscillation of the linear two dimensional difference system dx n p ny n dy n 1 q nx n n n n 0 n 0 n 0 1 where p n q n are nonnegative real sequences our results extend the results in the literature

oscillation of certain two dimensional nonlinear difference systems abstract several new oscillation criteria for two dimensional nonlinear difference systems are established examples which dwell upon the importance of our results are also included

erratum to non atomicity and the darboux property for fuzzy and non fuzzy borel baire multivalued set functions fuzzy sets and systems 160 2009 1308 1317

new oscillation criteria for second order nonlinear neutral delay difference equations in this paper we will establish some new oscillation criteria for the second order nonlinear neutral delay difference equation d a n dx n p nx n t c f n x n s 0 n 0 1 2 where c 0 is a quotient of odd positive integers

oscillation for a second order neutral differential equation with impulses we consider a certain type of second order neutral delay differential systems and we establish two results concerning the oscillation of solutions after the system undergoes controlled abrupt perturbations called impulses as a matter of fact some particular non impulsive cases of the system are oscillatory already thus we are interested in finding adequate impulse controls under which our system remains oscillatory

comparison theorems for second order nonlinear difference equations abstract new oscillation results are obtained for the second order nonlinear difference equation r n f x n 1 g n x n 0 and its functional form r n f x n 1 g n x n 0 the role played by the argument n on the oscillation of the functional equation is explored in particular we characterize a class of sequences n which have a harmless effect on the oscillation of this type of equations some of our results generalize improve or unify known fundamental oscillation results for several particular cases of the above equations

existence of bounded positive solutions for a system of difference equations abstract this paper deals with the second order nonlinear neutral delay system of difference equations a n x n b n x n f n x h 1 n x h kn y w 1 n y w kn c n n n 0 p n y n q n y n g n x s 1 n x s kn y t 1 n y t kn r n n n 0 by virtue of the banach fixed point theorem and some natural modifications of the techniques in the literature we prove the existence results of uncountably many bounded positive solutions for the system of difference equations seven examples are constructed to explain the main results presented in this paper

oscillation of difference systems of the neutral type the authors consider the nonlinear difference systems d x n a nx s n p ng y n dy n dq nf x t n n n n 0 where d 1 a n p n and q n are real sequences s n and t n are real non negative sequences of integers and f g r r are continuous with uf u 0 and ug u 0 for u 0 the authors obtain sufficient conditions for all solutions of the system to be oscillatory examples to illustrate the results are included

on oscillation of difference equations with bounded phi laplacian oscillation of second order difference equations with bounded phi laplacian is studied by comparing it with the discrete emden fowler difference equation analogues and discrepancies in behavior of solutions of these equations are also presented and a comparison with the continuous case is given

oscillation criteria for second order neutral delay difference equations abstract in this paper we establish some oscillation criteria of the second order delay difference equation a n 1 x n 1 p n 1 x n 1 q n f x n 0 where and are nonnegative constants an and pn qn are nonnegative sequences and f c

oscillation of second order perturbed nonlinear difference equations using the riccati transformation techniques we will establish some new oscillation criteria for the second order perturbed nonlinear difference equation d a n 1 dx n 1 c f n x n g n x n dx n n 1where c 0 is a quotient of odd positive integers some comparison between our theorems and those previously known results are indicated examples are interested in the text to illustrate the relevance of our results

classification schemes for nonoscillatory solutions of two dimensional nonlinear difference systems abstract classification schemes for nonoscillatory solutions of a class of nonlinear two dimensional nonlinear difference systems are given in terms of their asymptotic magnitudes and necessary as well as sufficient conditions for the existence of these solutions are also provided

a new kind of tradeoffs in propositional proof complexity we exhibit an unusually strong tradeoff in propositional proof complexity that significantly deviates from the established pattern of almost all results of this kind namely restrictions on one resource width in our case imply an increase in another resource tree like size that is exponential not only with respect to the complexity of the original problem but also to the whole class of all problems of the same bit size more specifically we show that for any parameter k e k n there are unsatisfiable k cnfs that possess refutations of width o k but such that any tree like refutation of width n1 e k must necessarily have doubly exponential size exp n k this means that there exist contradictions that allow narrow refutations but in order to keep the size of such a refutation even within a single exponent it must necessarily use a high degree of parallelism our construction and proof methods combine in a non trivial way two previously known techniques the hardness escalation method based on substitution formulas and expansion this combination results in a hardness compression approach that strives to preserve hardness of a contradiction while significantly decreasing the number of its variables

oscillation of second order nonlinear neutral differential equations consider the second order nonlinear neutral delay differential equation r t x t p t x email protected a f t x email protected 0 e where t and s are nonnegative constants and a is a quotient of positive odd integers some new oscillatory criteria for eq e are established several examples which dwell upon the importance of our results are also illustrated

oscillation theorems for some nonlinear difference equations abstract some new oscillation theorems for the difference equations of the form r n u n q n f u n k 0 are established

classification and existence of positive solutions of systems of volterra nonlinear difference equations a classification scheme for the eventually positive solutions of a class of two dimensional volterra nonlinear difference equations is given in terms of their asymptotic magnitudes and necessary as well as sufficient conditions for the existence of such solutions are provided

size space tradeoffs for resolution we investigate tradeoffs of various basic complexity measures such as size space and width we show examples of formulas that have optimal proofs with respect to any one of these parameters but optimizing one parameter must cost an increase in the other these results have implications to the efficiency or rather inefficiency of some commonly used sat solving heuristics our proof relies on a novel connection of the variable space of a proof to the black white pebbling measure of an underlying graph

nonexistence of unbounded nonoscillatory solutions of partial difference equations abstract we develop criteria for the nonexistence of eventually positive negative and nondecreasing nonincreasing solutions of the partial difference equation m n y m n p m n y m k n l q m n y m k n l and m n y m n i 1 p i m n y m k i n l i i 1 q i m n y m k i n l i several examples which dwell upon the importance of our results are also included

continuity properties and alexandroff theorem in vietoris topology in several previous papers we introduced and studied various continuity properties in the set valued case precisely we established results concerning increasing decreasing convergence exhaustivity order continuity and regularity in hausdorff topology for closed set valued monotone set multifunctions on a real normed space in this paper taking as starting point of a previous work we introduce and study from another point of view corresponding continuity properties in vietoris topology for more general monotone set multifunctions on a hausdorff linear topological space thus we obtain generalizations of some previous results which in turn generalize to the set valued case well known results from classical fuzzy i e monotone measure theory an alexandroff theorem in vietoris topology and its converse are also established

oscillatory criteria for second order quasilinear neutral delay difference equations consider the second order quasilinear neutral difference equation da n d x n p nx g n a email protected x n p nx g n f n x s n 0 the sufficient conditions are established for oscillation of the solution of this eq these generalize and improve some known results about both neutral and delay difference equations

oscillation criteria for second order nonlinear difference equations with damped term abstract several oscillation criteria are established for the second order damped nonlinear difference equation a n 1 y n 1 p n 1 y n 1 q n y n 0 n n 0 0 where 0 is any quotient of odd integers p n and q n are real sequences and f c r r such that x f x 0 for x 0 several examples which dwell upon the importance of our results are also included

non atomicity and the darboux property for fuzzy and non fuzzy borel baire multivalued set functions in this paper we establish results concerning non atomicity and the darboux property for multisubmeasures fuzzy multivalued set functions defined on the baire respectively borel d ring b 0 respectively b of a locally compact hausdorff space

oscillation of solutions for second order nonlinear difference equations with nonlinear neutral term abstract some riccati type difference inequalities are given for the second order nonlinear difference equations with nonlinear neutral term and using these inequalities we obtain some oscillation criteria for the above equation

oscillation criteria for a nonlinear difference equation abstract essential and nontrivial improvements and extensions of several recent oscillation criteria are obtained for the nonlinear difference equation 1 below

an application of discrete inequality to second order nonlinear oscillation abstract by using some simple discrete inequalities oscillation criteria are provided for the second order difference equations 2 y n a n 1 y n 1 0 n n where the operator is defined by yn yn 1 yn an is a real sequence the function is such that u u 0 for u 0 and u v g u v u v for u v 0 for some nonnegative function g

oscillation for system of delay difference equations abstract consider the system of delay difference equations x i t x i t k 1 l j 1 n p ijk x j t k 0 where p ijk and k 0 i j 1 2 n k 1 2 l sufficient conditions are obtained for all solutions of the system to be oscillatory

oscillation and asymptotic behavior of two dimensional difference systems this paper is concerned with the oscillation behavior of solutions of the nonlinear two dimesional difference system dx n b ng y n dy n 1 a nf x n n n n 0 n 0 n 0 1 some necessary and sufficient conditions are given for the oscillation of all solution of the system our result improve and generalize some results in the literature examples to illustrate the result are included

classification schemes for positive solutions of a second order nonlinear difference equation classification schemes for eventually positive solutions of a class of second order nonlinear difference equations are given in terms of their asymptotic magnitudes and necessary as well as sufficient conditions for the existence of these solutions are also provided

oscillatory and asymptotic behavior of second order neutral difference equations with maxima abstract in this paper the asymptotic behavior of the non oscillatory solutions of neutral difference equations with maxima is considered sufficient conditions for oscillation of all solutions are also obtained