abel gontscharoff interpolation continuous discrete and time scale in this paper we shall discuss the abel gontscharoff interpolation problem in the continuous discrete and time scale cases let pn 1 denote the n 1 th degree abel gontscharoff interpolating polynomial of a given function x the interpolating conditions in the continuous discrete and time scale cases are respectively given by pn 1 i ai 1 x i ai 1 ipn 1 ai 1 ix ai 1 and p in 1 ai 1 0 i n 1 where ai 1 i n are the interpolating nodes in each of the three cases we shall present the best possible error bounds under different settings of ai s furthermore as an application of the error estimates obtained criteria are developed for the right disfocality as well as disconjugacy for higher order equations in each of the three cases

eigenvalue distribution of second order dynamic equations on time scales considered as fractals abstract let t a b be a time scale with a b t in this paper we study the asymptotic distribution of eigenvalues of the following linear problem u u with mixed boundary conditions u a u a 0 u b u b it is known that there exists a sequence of simple eigenvalues k k we consider the spectral counting function n t k k and we seek for its asymptotic expansion as a power of let d be the minkowski or box dimension of t which gives the order of growth of the number k t e of intervals of length e needed to cover t namely k t e e d we prove an upper bound of n which involves the minkowski dimension n t c d 2 where c is a positive constant depending only on the minkowski content of t roughly speaking its d volume although the minkowski content is not a measure we also consider certain limiting cases d 0 infinite minkowski content and we show a family of self similar fractal sets where n t admits two side estimates

the barrier transmission problem treated by the amplitude phase method and expressed in terms of an invariant of the ermakov lewis type transmission and reflection of a quantal particle by a single hump potential barrier are analysed by means of an amplitude phase decomposition of the wavefunction on both sides of the barrier the amplitude phase analysis of the wavefunction provides a particular invariant of the ermakov lewis type which originates in the matching process the transmission and reflection coefficients turn out to be simple functions of this invariant numerical calculations of the invariant for an eckart epstein potential barrier provide very accurate results

positive solutions for a class of boundary value problems on time scales by obtaining intervals of the parameter l this paper is concerned with the existence and nonexistence of positive solution of the second order nonlinear dynamic equation on time scales p t x d t q t x t lw t f t x ax r a bx d r a 0 cx b dx d b 0 for t a b t where t is a time scale a 0 c 0 b 0 d 0 with a c 0 the arguments are based upon fixed point theorems in a cone

on a criterion for discrete or continuous spectrum of p laplace eigenvalue problems with singular sign changing weights abstract in this paper we study the existence of spectrum for one dimensional p laplace eigenvalue problems with singular weights subject to dirichlet boundary condition for certain class of singular sign changing weights we prove the existence of discrete spectrum proofs are based on the c 1 0 1 regularity of solutions and construction of the first eigenvalue in a variational set up we also present an example of a weight for which the spectrum is continuous and any corresponding eigenfunction does not belong to c 1 0 1

the similarity problem for j nonnegative sturm liouville operators abstract sufficient conditions for the similarity of the operator a 1 r x d 2 d x 2 q x with an indefinite weight r x sgn x r x are obtained these conditions are formulated in terms of titchmarsh weyl m coefficients sufficient conditions for the regularity of the critical points 0 and of j nonnegative sturm liouville operators are also obtained this result is exploited to prove the regularity of 0 for various classes of sturm liouville operators this implies the similarity of the considered operators to self adjoint ones in particular in the case r x sgn x and q l 1 r 1 x d x we prove that a is similar to a self adjoint operator if and only if a is j nonnegative the latter condition on q is sharp i e we construct q 1 l 1 r 1 x d x such that a is j nonnegative with the singular critical point 0 hence a is not similar to a self adjoint operator for periodic and infinite zone potentials we show that j positivity is sufficient for the similarity of a to a self adjoint operator in the case q 0 we prove the regularity of the critical point 0 for a wide class of weights r this yields new results for forward backward diffusion equations

eigencurves for two parameter sturm liouville equations this paper concerns two parameter sturm liouville problems of the form p x y q x y lambda r x mu y quad a leqslant x leqslant b with self adjoint boundary conditions at a and b the set of lambda mu in bf r 2 for which there exists a nontrivial y satisfying the differential equation and the boundary conditions turns out to be a countable union of graphs of analytic functions our focus is on these graphs which are termed eigencurves in the literature although eigencurves have been used in a variety of ways for about a century they seem comparatively underdeveloped in their own right our plan is to give motivation for the topic elementary properties of eigencurves illustrations on a simple example first studied by richardson in 1918 and since then by several authors and some natural questions which may whet the reader s appetite some of these questions lead to new types of inverse sturm liouville problems

oscillation problem of left definite sturm liouville problems with coupled bcs this paper studies oscillation problems of left definite regular self adjoint slps with real coupled bcs for a given eigenvalue l of the left definite slps with real coupled bcs we conclude that l is an eigenvalue of the slps with certain separated bcs we also establish some inequalities among the eigenvalues for the slps with separated bcs and coupled bcs with the help of the obtained results we give an estimate of the index of eigenvalues and the number of zeros of eigenfunction for the slps with real coupled bcs

lyapunov inequalities for discrete linear hamiltonian systems abstract in this paper we present some lyapunov type inequalities for discrete linear scalar hamiltonian systems when the coefficient c t is not necessarily nonnegative valued and when the end points are not necessarily usual zeros but rather generalized zeros applying these inequalities we obtain some disconjugacy and stability criteria for discrete hamiltonian systems

matrix riccati equations and matrix sturm liouville problems abstract the asymptotic behavior of determinants of unitary solutions of matrix riccati differential equations containing a large parameter is determined the result leads to theorems on existence and asymptotic distribution of eigenvalues of indefinite matrix sturm liouville problems

nonresonance conditions for generalised ϕ laplacian problems with jumping nonlinearities abstract we consider the boundary value problem 0 1 x u x u x f x u x u x a e x 0 1 0 2 c 00 u 0 c 01 u 0 c 10 u 1 c 11 u 1 where c j 0 c j 1 0 for each j 0 1 and f 0 1 r 2 r are caratheodory functions with suitable additional properties the differential operator generated by the left hand side of 0 1 together with the boundary conditions 0 2 is a generalisation of the usual p laplacian and also of the so called laplacian which corresponds to x s t t with an odd increasing homeomorphism for the p laplacian problem and more particularly the semilinear case p 2 nonresonance conditions which ensure the solvability of the problem 0 1 0 2 have been obtained in terms of either eigenvalues for non jumping f or the fucik spectrum or half eigenvalues for jumping f of the p laplacian in this paper under suitable growth conditions on and f we extend these conditions to the general problem 0 1 0 2

multiple positive solutions of n point boundary value problems for p laplacian impulsive dynamic equations on time scales in this paper we study the n point boundary value problems for p laplacian impulsive dynamic equations on time scales by using the leray schauder fixed point theorem and the nonlinear alternative of leray schauder type we get the existence of at least one positive solution we also consider the existence of at least three positive solutions by using a new fixed point theorem as an application we give an example to demonstrate our results

global bifurcation phenomena for singular one dimensional p laplacian in this paper we present global existence results for the following problem p p u t h t f u t 0 a e in 0 1 u 0 u 1 0 where p x x p 2x p 1 a positive parameter and h a nonnegative measurable function on 0 1 which may be singular at t 0 and or t 1 and f c r r with r 0 by applying the global bifurcation theorem and figuring the shape of unbounded subcontinua of solutions we obtain many different types of global existence results of positive solutions we also obtain existence results of sign changing solutions for p when f is an odd symmetric function

the intersection of the spectra of two sturm liouville equations this paper studies the intersection of the spectra for two sturm liouville equations with general separated bcs and real coupled bcs under certain conditions we proved that a two dimensioned vectorial slps with separated bcs only has finitely many double eigenvalues and obtain a bound m q depending on q x and its eigenvalues which are larger than m q are all simple finally with the help of the obtained results we conclude that the number of the same eigenvalue is finite for two one dimensioned slps with general separated bcs or real coupled bcs

optimal abel gontscharoff interpolation error bounds on measure chains in this paper we offer the best possible error bounds for abel gontscharoff interpolation on a measure chain our results unify the error analysis performed on continuous and discrete abel gontscharoff interpolation furthermore as an application of the error estimates obtained criteria are developed for the right disfocality as well as disconjugacy for a higher order equation on a measure chain

oscillation of second order damped dynamic equations on time scales abstract the study of dynamic equations on time scales has been created in order to unify the study of differential and difference equations the general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so called time scale which may be an arbitrary closed subset of the reals this way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained in this paper by employing the riccati transformation technique we will establish some oscillation criteria for second order linear and nonlinear dynamic equations with damping terms on a time scale t our results in the special case when t r and t n extend and improve some well known oscillation results for second order linear and nonlinear differential and difference equations and are essentially new on the time scales t h n h 0 t q n for q 1 t n 2 etc some examples are considered to illustrate our main results

oscillation criteria for second order nonlinear neutral delay dynamic equations in this paper we will establish some oscillation criteria for the second order nonlinear neutral delay dynamic equation r t y t p t y t f t y t 0 on a time scale t here 0 is a quotient of odd positive integers with r t and p t real valued positive functions defined on t to the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales so this paper initiates the study

variational and non variational eigenvalues of the p laplacian it is well known that all the eigenvalues of the linear eigenvalue problem u q r u in rn can under appropriate conditions on q r and be characterized by minimax principles but it has been a long standing question whether that remains true for analogous equations involving the p laplacian p it will be shown that there are corresponding nonlinear eigenvalue problems pu q r u p 1sgnu in rn with 1 0 on for which not all eigenvalues are of variational type as far as we know this is the first observation of such a phenomenon and examples will be given for one and higher dimensional equations the question of exactly which eigenvalues are variational is also discussed when n 1

oscillation of second order delay differential equations on time scales abstract by means of riccati transformation technique we establish some new oscillation criteria for a second order delay differential equation on time scales in terms of the coefficients

comparison of eigenvalues for lidstone boundary value problems on a measure chain preprint the theory of u0 positive operators with respect to a cone in a banach space is applied to the class of even order differential equation on measure chains x 2m t 0m 1 pi t x 2i t t 0 1 satisfying the lidstone boundary conditions x 2i 0 0 0 i m 1 x 2j 2 1 0 0 j m 1 the existence of a smallest positive eigenvalue is obtained and then a theorem is established comparing the smallest positive eigenvalues for two problems of this type

oscillation and interlacing for various spectra of the p laplacian abstract we consider the eigenvalue problem p u r q u p 1 sgn u a e on 0 b where b 0 p 1 p is the p laplacian q r l 1 0 b r 0 and r a variety of boundary conditions will be imposed at 0 and b but the main focus is on dirichlet neumann and periodic anti periodic conditions in the linear case p 2 there are well known oscillation and interlacing results for the eigenvalues of the above problem with these boundary conditions and we explore similarities and differences between this case and the nonlinear case with p 2 we will see that several new phenomena occur in the periodic anti periodic problems for example with separated boundary conditions the structure of the spectrum when p 2 is the same as when p 2 but this is not true in the periodic anti periodic cases we also consider the set of half eigenvalues and the fucik spectrum of the problem these concepts are useful for solving jumping nonlinearity problems and we show that new phenomena also appear here in addition to analogues of those occurring for the usual eigenvalues

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

asymptotic behavior of solutions of a third order nonlinear dynamic equation on time scales in this paper we will establish some sufficient conditions which guarantee that every solution of the third order nonlinear dynamic equation c t a t x t q t f x t 0 t t0 oscillates or converges to zero

one dimensional p laplacian with a strong singular indefinite weight i eigenvalue abstract in this paper we prove the existence of eigenvalues for the problem p u t h t p u t 0 a e in 0 1 u 0 u 1 0 where p s s p 2 s p 1 is a real parameter and the indefinite weight h is a nonnegative measurable function on 0 1 which may be singular at 0 and or 1 and h 0 on any compact subinterval in 0 1 we derive similar properties of eigenvalues as known in linear case p 2 or continuous case h c 0 1 if h satisfies 0 1 t p 1 1 t p 1 h t d t when 1 p 2 and 0 1 2 p 1 s 1 2 h d d s 1 2 1 p 1 1 2 s h d d s when p 2 respectively for the result we establish the c 1 regularity of all solutions at the boundary for the above problem as well as the following problem p u t h t f u t 0 a e in 0 1 u 0 u 1 0 where f c r r s f s 0 for s 0 f is odd and f s p s is bounded above as s 0

can milne s method work well for the coulomb like potentials we extend the improved milne s milne spline method for obtaining eigenvalues and eigenfunctions to the cases of long range and singular potentials for which we have conjectured that it is difficult to apply the method contrary to our conjecture it turned out that the method is valid also for coulomb potential and repulsive 1 xn n 2 3 type potential further we applied the method for two cases for which the solutions are not known in order to investigate the stability of the multi dimensional universe it has been shown that the extra dimensional internal space of our universe is not stable in classical einstein gravity as well as canonically quantized one two possibilities for stabilization were investigated i noncanonically quantized einstein gravity and ii canonically quantized higher curvature gravity it has been suggested that the space is stable by qualitative and approximate methods exact analytical treatment is very difficult so that numerical investigation is highly desirable numerical investigation shows that the space is stable with sufficient reliability

interlacing and oscillation for sturm liouville problems with separated and coupled boundary conditions an extension is given to the known inequalities which interlace the eigenvalues corresponding to separated and coupled boundary conditions for the problem py qy ry on a b assuming 1 p q and r l1 a b the key is a new interlacing principle for intervals of eigenvalues corresponding to one parameter sets of boundary conditions application is given to eigenfunction oscillation

calculus of variations on time scales weak local piecewise crd1 solutions with variable endpoints abstract a nonlinear calculus of variations problem on time scales with variable endpoints is considered the space of functions employed is that of piecewise rd continuously differentiable functions c1prd for this problem the euler lagrange equation the transversality condition and the accessory problem are derived as necessary conditions for weak local optimality assuming the coercivity of the second variation a corresponding second order sufficiency criterion is established

a local bifurcation theorem for degenerate elliptic equations with radial symmetry abstract in this work we provide local bifurcation results for equations involving the p laplacian in balls we analyze the continua c n of radial solutions emanating from n p 0 n p being the radial eigenvalues of p first we show that the only nontrivial solutions close to n p 0 lie on a continuous curve thus extending the crandall rabinowitz theorem second it is proved that c n n p 0 splits into two unbounded connected pieces characterized by their nodal properties thus sharpening previous results

half eigenvalues of periodic sturm liouville problems we consider the nonlinear sturm liouville problem 1 pu qu au bu u in 0 2 2 u 0 u 2 pu 0 pu 2 where 1 p q l1 0 2 with p 0 a e on 0 2 a b l1 0 2 is a real parameter and u t max u t 0 for t 0 2 values of for which 1 2 has a non trivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions the set of half eigenvalues will be denoted by h we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having certain nodal properties and we obtain certain spectral and degree theoretic properties associated with h these properties yield results on the existence and non existence of solutions of the problem 3 pu qu f t u h in 0 2 together with 2 where h l1 0 2 f 0 2 r r is a caratheodory function and the limits a t lim f t b t lim f t exist for a e t 0 2 when the functions a and b are constant the set h is closely related to the fucik spectrum of the problem and equivalent solvability results are obtained using the two approaches however when a and b are not constant the half eigenvalue approach yields stronger results

a homotopic deformation along p of a leray schauder degree result and existence for u p 2u ƒ t u 0 u 0 u t 0 p 1 soit f 0 1 r r une fonction continue et soit o p r r definie par o p s s p 2 s pour p fixe 1 on considere le probleme de dirichlet suivant o p u f t u 0 u 0 u t 0 ou t est un nombre reel positif

eigenvalues of regular sturm liouville problems abstract the eigenvalues of sturm liouville sl problems depend not only continuously but smoothly on the problem an expression for the derivative of an eigenvalue with respect to a given parameter an endpoint a boundary condition a coefficient or the weight function is found

p laplacian problems with jumping nonlinearities we consider the p laplacian boundary value problem 1 p u x f x u x u x a e x 0 1 2 c00u 0 c01u 0 c10u 1 c11u 1 where p 1 is a fixed number p s s p 2s s r and for each j 0 1 cj0 cj1 0 the function f 0 1 r2 r is a caratheodory function satisfying for x s t 0 1 r2 x p s e x s t f x s t x p s e x s t s 0 where l1 0 1 and e has the form e x s t x e s t with l1 0 1 0 e 0 and limr e r r1 p 0 this condition allows the nonlinearity in 1 to behave differently as u such a nonlinearity is often termed jumping related to 1 2 is the problem 3 p u a p u b p u p u in 0 1 together with 2 where a b l1 0 1 r and u x max u x 0 for x 0 1 this problem is positively homogeneous and jumping values of for which 2 3 has a nontrivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having certain nodal properties and we obtain certain spectral and degree theoretic properties of the set of half eigenvalues these properties lead to existence and nonexistence results for the problem 1 2 we also consider a related bifurcation problem and obtain a global bifurcation result similar to the well known rabinowitz global bifurcation theorem this then leads to a multiplicity result for 1 2 when the functions a and b are constant the set of half eigenvalues is closely related to the fucik spectrum of the problem and equivalent solvability results are obtained using the two approaches however when a and b are not constant the half eigenvalue approach yields stronger results

mountain pass solutions to euler lagrange equations with general anisotropic operator abstract using the mountain pass theorem we show that the problem d d t l v t u t u t l x t u t u t for a e t a b u a u b 0 has a solution in anisotropic orlicz sobolev space we consider lagrangian l f t x v v t x f t x with growth conditions determined by anisotropic g function and some geometric conditions of ambrosetti rabinowitz type

oscillation criteria of second order half linear dynamic equations on time scales in this paper by using the riccati transformation technique chain rule and inequalitya l lab l 1 l 1 b l 0 l 1 where a and b are positive constants we will establish some oscillation criteria for the second order half linear dynamic equation p t x d t c d q t x c t 0 t a b on time scales where c 1 is an odd positive integer our results not only unify the oscillation of half linear differential and half linear difference equations but can be applied on different types of time scales and improve some well known results in the difference equation case some examples are considered here to illustrate our main results

the spectrum of the periodic p laplacian we consider one dimensional p laplacian eigenvalue problems of the form pu q u p 1sgnu on 0 b together with periodic or separated boundary conditions where p 1 p is the p laplacian q c1 0 b and b 0 r it will be shown that when p 2 the structure of the spectrum in the general periodic case that is with q 0 and periodic boundary conditions can be completely different from those of the following known cases i the general periodic case with p 2 ii the periodic case with p 2 and q 0 and iii the general separated case with any p 1

bifurcation phenomena associated to the p laplace operator on determine la structure de l ensemble des solutions u de u x p 2 u x x f u u p 2 u sur 0 1 telles que u 0 u 1 0 ou p 1 et r on demontre que les solutions avec k zeros sont uniques quand 1 2

relations among eigenvalues of left definite sturm liouville problems with coupled bcs and separated bcs this paper deals with left definite regular self adjoint slps with coupled boundary conditions it is obtained that for a given eigenvalue l of left definite regular slps with coupled bcs there are some separated bcs also having l as an eigenvalue with the same index or indices and then we can construct an algorithm for computing the index of a given eigenvalue for the left definite slps with coupled boundary conditions

oscillation of nonlinear dynamic equations on time scales by means of riccati transformation techniques we establish some new oscillation criteria for the second order nonlinear dynamic equation p t x d t d q t f x s t 0 email protected a b on time scales in terms of the coefficients and the graininess function we illustrate our results with a nontrivial example

boundary problems of the second order with an indefinite weight function on considere des problemes aux limites de la forme au wu ou est un domaine borne de r n u w 1 2 soit a q soit a d p x d dx dx q x pour n 1 on suppose que w est une fonction de signe indefinie dans tandis que a a un spectre negatif

on green s functions and positive solutions for boundary value problems on time scales in this paper we offer a form of self adjoint differential equations on time scales so that the associated green s function is found symmetric in the usual sense for this purpose together with the delta derivative we employ the nabla derivative as well we introduce the concepts of lebesgue delta and nabla integrals on time scales next sign properties of the green s function are investigated and existence results for positive solutions of nonlinear boundary value problems are established upper and lower bounds for these positive solutions also are given

p laplace equations with singular weights we study a class of p laplacian dirichlet problems with weights that are possibly singular on the boundary of the domain and obtain nontrivial solutions using morse theory in the absence of a direct sum decomposition we use a cohomological local splitting to get an estimate of the critical groups

integration on time scales in this paper we study the process of riemann and lebesgue integration on time scales the relationship of the riemann and lebesgue integrals is considered and a criterion for riemann integrability is established

coupled radial schrodinger equations written as dirac type equations application to an amplitude phase approach the classical amplitude phase method due to milne wilson young and wheeler in the 1930s is known to be a powerful computational tool for determining phase shifts and energy eigenvalues in cases where a sufficiently slowly varying amplitude function can be found the key for the efficient computations is that the original single state radial schr dinger equation is transformed to a nonlinear equation the milne equation such an equation has solutions that may or may not oscillate depending on boundary conditions which then requires a robust recipe for locating the optimal almost constant solutions for its use in the method for scattering problems the solutions of the amplitude equations always approach constants as the radial distance r tends to infinity and there is no problem locating the optimal amplitude functions from a low order semiclassical approximation in the present work the amplitude phase approach is generalized to two coupled schr dinger equations similar to an earlier generalization to radial dirac equations the original scalar amplitude then becomes a vector quantity and the original milne equation is generalized accordingly numerical applications to resonant electron atom scattering are illustrated

a prufer angle approach to semidefinite sturm liouville problems with coupling boundary conditions abstract it is shown how to reduce the periodic antiperiodic sturm liouville problems to analysis of the prufer angle this provides an alternative to the more usual approaches via operator theory or the hill discriminant in the definite case and leads to new results in the semidefinite case an extension to more general coupling boundary conditions is also given

estimates on the non real eigenvalues of regular indefinite sturm liouville problems regular sturm liouville problems with indefinite weight functions may possess finitely many non real eigenvalues in this paper we prove explicit bounds on the real and imaginary parts of these eigenvalues in terms of the coefficients of the differential expression

spectral theory for systems of ordinary differential equations with distributional coefficients abstract we study the spectral theory for the first order system j u q u w f of differential equations on the real interval a b when j is a constant invertible skew hermitian matrix and q and w are matrices whose entries are distributions of order zero with q hermitian and w non negative also we do not pose the definiteness condition often required for the coefficients of the equation specifically we construct minimal and maximal relations and study self adjoint restrictions of the maximal relation for these we determine green s function and prove the existence of a spectral or generalized fourier transformation we have a closer look at the special cases when the endpoints of the interval a b are regular as well as the case of a 2 2 system two appendices provide necessary details on distributions of order zero and the abstract spectral theory for relations

continuity in weak topology and extremal problems of eigenvalues of the p laplacian we will study the dependence of eigenvalues of the one dimensional p laplacian on potentials or weights two results are obtained one is the continuity of eigenvalues in potentials with respect to the weak topologies of l spaces 1 and the other is the continuous differentiability of eigenvalues in potentials with respect to l norms as applications we will study some extremal problems of eigenvalues by developing some analytical methods

oscillation theory for sturm liouville problems with indefinite coefficients oscillation and related results are given for the problem under separated end conditions assuming 1 p q and r l 1 i attention focuses on the two cases i p 0 with r indefinite and ii p indefinite with r 0

the fucik spectrum of general sturm liouville problems abstract consider the boundary value problem pu qu u u in 0 c 00 u 0 c 01 u 0 0 c 10 u c 11 u 0 where u max u 0 the set of points r 2 for which this problem has a non trivial solution is called the fucik spectrum when p 1 q 0 and either dirichlet or periodic boundary conditions are imposed the fucik spectrum is known explicitly and consists of a countable collection of curves with certain geometric properties in this paper we show that similar properties hold for the general problem above and also for a further generalization of the fucik spectrum we also discuss some spectral type properties of a positively homogeneous half linear problem and use these results to consider the solvability of a nonlinear problem with jumping nonlinearities

nonuniform nonresonance of semilinear differential equations abstract consider the dirichlet problem of nonlinear differential equations with the principal part the p laplacian when the nonlinearity satisfies some semilinearity conditions the usual nonuniform nonresonance conditions are obtained by comparing nonlinear equations with the classical eigenvalues in this article we will introduce some weighted eigenvalues the nonuniform nonresonance conditions proved in this article using weighted eigenvalues will improve the usual ones significantly

existence and asymptotics of eigenvalues of indefinite systems of sturm liouville and dirac type abstract existence and asymptotic behavior of eigenvalues of indefinite sturm liouville and dirac systems with integrable coefficients are investigated using prufer angle analysis the results generalize those previously obtained by atkinson mingarelli gohberg and krein and are based on a new theorem that determines the asymptotic behavior of the solution of a riccati type equation containing a large parameter

bifurcation of sign changing solutions for one dimensional p laplacian with a strong singular weight p sublinear at abstract in this paper we consider one dimensional p laplacian problems with a singular weight which may not be in l 1 using the properties of eigenfunctions and the global bifurcation theory we prove existence uniqueness nonexistence and multiplicity results of positive solutions as well as sign changing solutions specially when the nonlinear term is p linear near 0 and p sublinear at

on a disconjugacy criterion for second order dynamic equations on time scales in this paper a sufficient condition for disconjugacy of second order differential equations on time scales is given for this purpose a mean value result on time scales is employed

relations among eigenvalues of sturm liouville problems with different types of leading coefficient functions abstract for any sturm liouville problem with a separable boundary condition and whose leading coefficient function changes sign exactly once we first give a geometric characterization of its eigenvalues n using the eigenvalues of some corresponding problems with a definite leading coefficient function consequences of this characterization include simple proofs of the existence of the n s their prufer angle characterization and a way for determining their indices from the zeros of their eigenfunctions then interlacing relations among the n s and the eigenvalues of the corresponding problems are obtained using these relations a simple proof of asymptotic formulas for the n s is given

higher order dynamic equations on measure chains wronskians disconjugacy and interpolating families of functions this paper introduces generalized zeros and hence disconjugacy of nth order linear dynamic equations which cover simultaneously as special cases among others both differential equations and difference equations we also define markov fekete and descartes interpolating systems of functions the main result of this paper states that disconjugacy is equivalent to the existence of any of the above interpolating systems of solutions and that it is also equivalent to a certain factorization representation of the operator the results in this paper unify the corresponding theories of disconjugacy for nth order linear ordinary differential equations and for nth order linear difference equations

sturm liouville eigenvalue problems on time scales for sturm liouville eigenvalue problems on time scales with separated boundary conditions we give an oscillation theorem and establish rayleigh s principle our results not only unifly the corresponding theories for differential and difference equations but are also new in the discrete case

asymptotics of the number of zeros and of the eigenvalues of general weighted sturm liouville problems soit n le nombre de zeros d une solution de l equation de sturm liouville ponderee generale p x y r x q x y 0 sur 0 b ou 0 b p q r 0 b r et 1 p q r l 0 b on donne une demonstration directe de l estimation asymptotique de n