a cotlar type maximal function associated with fourier multipliers abstract we prove the l p boundedness of a maximal operator associated with a dyadic frequency decomposition of a fourier multiplier under a weak regularity assumption

positive solutions of some singular m point boundary value problems at non resonance this paper investigates the existence of positive solutions for second order singular m point boundary value problems at non resonance a necessary and sufficient condition for the existence of c 0 1 as well as c 1 0 1 positive solutions is given by constructing lower and upper solutions and with the maximal theorem our nonlinearity f t x may be singular at x t 0 and or t 1

musings around the geometry of interaction and coherence we introduce the danos regnier category dr m of a linear inverse monoid m as a categorical description of geometries of interaction goi inspired from the weight algebra the natural setting for goi is that of a so called weakly cantorian linear inverse monoid in which case dr m is a kind of symmetrized version of the classical abramsky haghverdi scott construction of a weak linear category from a goi situation it is well known that goi is perfectly suited to describe the multiplicative fragment of linear logic and indeed dr m will be a autonomous category in this case it is also well known that the categorical interpretation of the other linear connectives conflicts with goi interpretations we make this precise and show that dr m has no terminal object no cartesian product of any two objects and no exponential whatever m is unless m is trivial however a form of coherence completion of dr m a la hu joyal which for additives resembles a layered approach a la hughes van glabbeek provides a model of full classical linear logic as soon as m is weakly cantorian one finally notes that girard s notion of coherence is pervasive and instrumental in every aspect of this work

maximal theorems for the directional hilbert transform on the plane for a schwartz function f on the plane and a non zero v r 2 define the hilbert transform of f in the direction v to be h v f x p v r f x vy dy y let be a schwartz function with frequency support in the annulus 1 2 the l 2 estimate is sharp the method of proof is based upon techniques related to the pointwise convergence of fourier series indeed our main theorem implies this result on fourier series

existence theorems for a second order three point boundary value problem abstract letf 0 1 r 2 r be continuous ande c 0 1 let i 0 1 ai r all of theai s having the same sign i 1 2 m 2 0 x t f t x t x t e t t 0 1 e x 0 0 x 1 i 1 m 2 a i x i bc m we discuss existence theorems for the problem e bcm under some sign condition offabout the origin our analysis is based on a nonlinear alternative of leray schauder

positive solutions of operator equations on ordered banach spaces and applications this paper is concerned with an operator equation on ordered banach spaces the existence and uniqueness of its positive solutions is obtained by using the properties of cones and monotone iterative technique as applications we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for two classes of integral equations

positive solutions for semi positone three point boundary value problems using a fixed point theorem of generalized cone expansion and compression we present in this paper criteria which guarantee the existence of at least two positive solutions for semi positone three point boundary value problems with parameter l 0 belonging to a certain interval

square functions for bi lipschitz maps and directional operators abstract first we prove a littlewood paley diagonalization result for bi lipschitz perturbations of the identity map on the real line this result entails a number of corollaries for the hilbert transform along lines and monomial curves in the plane second we prove a square function bound for a single scale directional operator as a corollary we give a new proof of part of a theorem of katz on direction fields with finitely many directions

on the consequences of a mihlin hormander functional calculus maximal and square function estimates we prove that the existence of a mihlin hormander functional calculus for an operator l implies the boundedness on l p of both the maximal multiplier operators and the continuous square functions build on spectral multipliers of l the considered multiplier functions are finitely smooth and satisfy an integral condition at infinity in particular multipliers of compact support are admitted

positive solutions of a nonlinear three point boundary value problem in this paper by using krasnoselskii s fixed point theorem in a cone we study the existence of single and multiple positive solutions to the three point boundary value problem bvp y t a t f y t 0 0

uniformity and the taylor expansion of ordinary lambda terms we define the complete taylor expansion of an ordinary lambda term as an infinite linear combination with rational coefficients of terms of a resource calculus similar to boudol s lambda calculus with multiplicities or with resources in our resource calculus all applications are multi linear in the algebraic sense i e commute with linear combinations of the function or the argument we study the collective behaviour of the beta reducts of the terms occurring in the taylor expansion of any ordinary lambda term using in a surprisingly crucial way a uniformity property that they enjoy as a corollary we obtain the main part of a proof that this taylor expansion commutes with bohm tree computation syntactically

positive solutions of the three point boundary value problem for second order differential equations with an advanced argument abstract in this paper by using the fixed point index method we establish the existence of at least one or at least two positive solutions to the three point boundary value problem for a second order differential equation with an advanced argument u t a t f u h t 0 t 0 1 u 0 0 u u 1 where 0 1 0 and 1 0 as an application we also give an example to demonstrate our result

solvability of a three point nonlinear boundary value problem for a second order ordinary differential equation in case f x u u ps x p x u pjx u with pk 0 l 4 r locally integrable the boundary value problem l lt 1 2 was studied by kiguradze and lomtatidze i the purpose of this paper is to obtain existence and uniqueness theorems for the boundary value problem 1 1 t 1 2 under natural conditions on f using degree theoretic arguments we note that if u is a solution of 1 1 1 2 then there exists at least one 5 e q l such that u c 0 accordingly the boundary value problem

existence of sign changing solutions for some asymptotically linear three point boundary value problems abstract in this paper by using the topological degree theory and the fixed point index theory the existence of three kinds of solutions i e sign changing solutions positive solutions and negative solutions for asymptotically linear three point boundary value problems is obtained

a maximal function for families of hilbert transforms along homogeneous curves let h u be the hilbert transform along the parabola t ut 2 where u in mathbb r for a set u of positive numbers consider the maximal function mathcal h u f sup h u f u in u we obtain an essentially optimal result for the l p operator norm of mathcal h u when 2 p infty the results are proved for families of hilbert transforms along more general nonflat homogeneous curves

positive periodic solutions of first order singular systems abstract we establish a complete structure for the existence of positive periodic solutions of first order non autonomous singular system u i t a i t u i t b i t f i u t i 1 n the methods employed are fixed point index theorems leray schauder degree and lower and upper solutions method

reversibility in the higher order π calculus the notion of reversible computation is attracting increasing interest because of its applications in diverse fields in particular the study of programming abstractions for reliable systems in this paper we continue the study undertaken by danos and krivine on reversible ccs by defining a reversible higher order calculus called rho we prove that reversibility in our calculus is causally consistent and that the causal information used to support reversibility in rho is consistent with the one used in the causal semantics of the calculus developed by boreale and sangiorgi finally we show that one can faithfully encode rho into a variant of higher order substantially improving on the result we obtained in the conference version of this paper

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

proof nets and the call by value λ calculus this paper gives a detailed account of the relationship between a variant of the call by value lambda calculus and linear logic proof nets the presentation is carefully tuned in order to realize an isomorphism between the two systems every single rewriting step on the calculus maps to a single step on proof nets and vice versa in this way we obtain an algebraic reformulation of proof nets moreover we provide a simple correctness criterion for our proof nets which employ boxes in an unusual way and identify a subcalculus that is shown to be as expressive as the full calculus

multiple positive solutions of quasi linear boundary value problems for finite difference equations abstract in this paper we consider a discrete second order quasi linear boundary value problem which nonlinear term f is involved with the first order difference by using a generalization of the leggett williams fixed point theorem due to avery and peterson we provide sufficient conditions for the existence of at least three positive solutions

multiplicity results for positive solutions of some semi positone three point boundary value problems in this paper some multiplicity results for positive solutions of some singular semi positone three point boundary value problem be obtained by using the fixed point index method

positive solutions of three point boundary value problems for second order impulsive differential equations with advanced arguments in this paper we discuss three point boundary value problems for second order impulsive differential equations with advanced arguments we establish sufficient conditions under which such problems have positive solutions to obtain the results we use the fixed point index

multiple positive solutions of multi point boundary value problem for second order impulsive differential equations this paper is devoted to study the existence of multiple positive solutions for the second order multi point boundary value problem with impulse effects the arguments are based upon fixed point theorems in a cone an example is worked out to demonstrate the main results

positive solutions to a system of second order nonlocal boundary value problems abstract in this paper we study the existence of positive solutions to a system of second order nonlocal boundary value problems by using fixed point index theory in a cone our hypotheses imposed on nonlinearities are those which characterize systems of nonlocal boundary value problems and our boundary value conditions are expressed in terms of possibly nonlinear functions of riemann stieltjes integrals thus generalizing and unifying the boundary value conditions in the literature therefore our results cannot be routinely deduced from the ones for single nonlocal problem in the literature

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

harmonic analysis on product spaces rn x r and commute with the dilations p33 8 x2 x2 61x1 62x2 for x1 e rn x2 e rm and 61 82 0 then the corresponding operators are related to many problems in the theory of multiple fourier integrals see zygmund 21 the maximal operator which commutes with the p31382 is the jessen marcinkiewicz and zygmund 14 strong maximal operator ms defined by 1 ms f x1 x2 sup ri a y1 y2 1 dy1 dy2 xl x2 gr jr1

reversible irreversible and optimal l machines abstract lambda calculus is the core of functional programming and many different ways to evaluate lambda terms have been considered one of the nicest from the theoretical point of view is head linear reduction we compare two ways of implementing that specific evaluation strategy krivine s abstract machine and the interaction abstract machine runs on those machines stand in a relation which can be accurately described using the call return symmetry discovered by asperti and laneve

second order nonlinear singular sturm liouville problems with integral boundary conditions this paper is concerned with the second order singular sturm liouville integral boundary value problems u t lh t f t u t 00 h is allowed to be singular at t 0 1 and f t x may be singular at x 0 by using the fixed point theory in cones an explicit interval for l is derived such that for any l in this interval the existence of at least one positive solution to the boundary value problem is guaranteed our results extend and improve many known results including singular and non singular cases

positive solutions for second order four point boundary value problems with alternating coefficient abstract in this paper we investigate the existence of positive solutions for a class of nonlinear second order four point boundary value problems with alternating coefficient our approach relies on the krasnosel skii fixed point theorem the result of this paper is new and extent the previously known result

on maximal functions for mikhlin hormander multipliers abstract given mikhlin hormander multipliers m i i 1 n with uniform estimates we prove an optimal log n 1 bound in l p for the maximal function sup i f 1 m i f and related bounds for maximal functions generated by dilations these improve the results in m christ l grafakos p honzik a seeger maximal functions associated with multipliers of mikhlin hormander type math z 249 2005 223 240

multiplicity results for some second order four point boundary value problems abstract in this paper by using a fixed point theorem of functional type in a cone some new results for the multiplicity of positive solutions of second order four point boundary value problems are obtained the emphasis is put on the nonlinear term involved with the first order derivative

positive solutions for third order sturm liouville boundary value problems with p laplacian in this paper by using the fixed point index method we establish the existence of at least one or at least two positive solutions for the third order sturm liouville boundary value problem with p laplacian f p u t f t u t 0 email protected 0 1 au 0 email protected 0 0 cu 1 email protected 1 0 u 0 0 where f p s s p 2s p 1 as an application an example is given to demonstrate the main result

the bilinear hilbert transform in umd spaces we prove l p bounds for the bilinear hilbert transform acting on functions valued in intermediate umd spaces such bounds were previously unknown for umd spaces that are not banach lattices our proof relies on bounds on embeddings from bochner spaces l p mathbb r x into outer lebesgue spaces on the time frequency scale space mathbb r 3

positive solutions for fourth order differential equations with deviating arguments and integral boundary conditions abstract in this paper we investigate integral boundary value problems for fourth order differential equations with deviating arguments we discuss our problem both for advanced or delayed arguments we establish sufficient conditions under which such problems have positive solutions to obtain the existence of multiple at least three positive solutions we use a fixed point theorem due to avery and peterson an example is also included to illustrate that corresponding assumptions are satisfied the results are new

light logics and optimal reduction completeness and complexity typing of lambda terms in elementary and light affine logic eal and lal respectively has been studied for two different reasons on the one hand the evaluation of typed terms using lal eal respectively proof nets admits a guaranteed polynomial elementary respectively bound on the other hand these terms can also be evaluated by optimal reduction using the abstract version of lamping s algorithm the first reduction is global while the second one is local and asynchronous we prove that for lal eal respectively typed terms lamping s abstract algorithm also admits a polynomial elementary respectively bound we also give a proof of its soundness and completeness for eal and lal with type fixpoints by using a simple geometry of interaction model context semantics

positive solutions for nonlinear operator equations and several classes of applications in this paper we study a class of nonlinear operator equations x ax x 0 on ordered banach spaces where a is a monotone generalized concave operator using the properties of cones and monotone iterative technique we establish the existence and uniqueness of solutions for such equations in particular we do not demand the existence of upper lower solutions and compactness and continuity conditions as applications we study first order initial value problems and two point boundary value problems with the nonlinear term is required to be monotone in its second argument in the end applications to nonlinear systems of equations and to nonlinear matrix equations are also considered

positive solutions of fourth order four point boundary value problems with p laplacian operator in this paper we consider the existence of positive solutions for the singular fourth order p laplacian equation p u t f t u t 0 t 1 with the four point boundary conditions u 0 0 u 1 au u 0 0 u 1 bu where p t t p 2t p 1 0 1 f c 0 1 0 0 may be singular at t 0 and or 1 and u 0 by using the upper and lower solution method and fixed point theorems the existence of positive solutions to the above the boundary value problem is obtained

maximal operators and hilbert transforms along variable non flat homogeneous curves we prove that the maximal operator associated with variable homogeneous planar curves t ut t r 1 positive is bounded on lp r2 for each p 1 under the assumption that u r2 r is a lipschitz function furthermore we prove that the hilbert transform associated with t ut t r 1 positive is bounded on lp r2 for each p 1 under the assumption that u r2 r is a measurable function and is constant in the second variable our proofs rely on stationary phase methods tt arguments local smoothing estimates and a pointwise estimate for taking averages along curves

fixed point theorems for τ φ concave operators and applications in this paper by introducing t f concave operators and using the properties of cones and monotone iterative technique some new existence and uniqueness theorems of fixed points for increasing operators with such concavity are established under more extensive conditions in ordered banach spaces as an application we study the existence and uniqueness of positive solutions of generalized three point boundary value problems for second order differential equations the results obtained here improve and generalize many known results

solutions and green s functions for some linear second order three point boundary value problems in this paper we consider the green s functions for a second order linear ordinary differential equation with some three point boundary conditions we give exact expressions of the unique solutions for the linear three point boundary problems by the green s functions as applications we study the iterative solutions for some nonlinear singular second order three point boundary value problems

multiplicity of positive solutions for second order three point boundary value problems we study the multiplicity of positive solutions for the second order three point boundary value problem u h t f u 0 t 0 1 u 0 0 u u 1 where 0 1 0 1 the methods employed are fixed point index theorems and leray schauder degree and upper and lower solutions

existence of positive solution for singular semi positone k n k conjugate m point boundary value problem this paper is about a class of singular semi positone k n k conjugate m point boundary value problems by using the fixed point index theory we get a sufficient condition for the existence of positive solution to this problem

new fixed point theorems for mixed monotone operators and local existence uniqueness of positive solutions for nonlinear boundary value problems abstract in this article we present a new fixed point theorem for a class of general mixed monotone operators which extends the existing corresponding results moreover we establish some pleasant properties of nonlinear eigenvalue problems for mixed monotone operators based on them the local existence uniqueness of positive solutions for nonlinear boundary value problems which include neumann boundary value problems three point boundary value problems and elliptic boundary value problems for lane emden fowler equations is proved the theorems for nonlinear boundary value problems obtained here are very general

multiple positive solutions to a three point boundary value problem with p laplacian abstract sufficient conditions are obtained that guarantee the existence of at least two positive solutions for the equation g u t a t f u 0 subject to boundary conditions by a simple application of a new fixed point theorem due to avery and henderson

positive solutions to semi positone second order three point problems on time scales using a fixed point theorem of generalized cone expansion and compression we establish the existence of at least two positive solutions for the nonlinear semi positone three point boundary value problem on time scales u d t email protected t u t 0 u a 0 au h u t here email protected a t t where t is a time scale a 0 email protected a r t t a h a 0 belongs to a certain interval these results are new for difference equations as well as for general time scales an example is provided for differential difference and q difference equations

successive iteration and positive solution for nonlinear second order three point boundary value problems in this paper we study the existence of positive solution for two classes of nonlinear second order three point boundary value problems using some monotone iterative schemes in both cases such schemes start off with known constant functions and therefore are useful to computation purpose

and variants of multiparameter calderon zygmund theory we consider calderon zygmund operators on product domains under certain weak conditions on the kernel a singular integral operator can be proved to be bounded on h p r r r 0 p 1 if its behaviour on l 2 and on certain scalar valued and vector valued rectangle atoms is known another result concerns an extension of the authors results on l p variants of calderon zygmund theory 1 23 to the product domain setting as an application one obtains estimates for fourier multipliers and pseudo differential operators

discrete third order three point boundary value problem we are concerned with the discrete boundary value problemlx t email protected 3x t f t x t 1 t 1

a sharper condition for the solvability of a three point second order boundary value problem letf 0 1 r2 rbe a function satisfying caratheodory s conditions ande t l1 0 1 let 0 1 r 1 1 be given this paper is concerned with the problem of existence of a solution for the three point boundary value problem formula omitted this problem was studied earlier by gupta ntouyas and tsamatos when 1 and when 1 with 1 in the general case this problem was studied by gupta as a multi point boundary value problem in this paper sharper existence conditions are obtained for the solvability of the above boundary value problem in the general case

some inequalities for singular convolution operators in spaces suppose that a bounded function m satisfies a localized multiplier condition supt o ii0m tp imp 0 the purpose of this paper is to find easily verified conditions that 1 implies me mp the condition llmll 0 then we ask under which conditions supt 0 qm tp 1mp oc implies m c mp throughout this paper x will always be chosen as in the following definition x c co rn satisfies a tauber condition with respect to the dilations tp if for every x 0 there is a tx such that q tpx 0 sometimes we need special bump functions of the following kind let p e c rn be a p homogeneous distance function this means that p tpx tp x received by the editors august 21 1986 1980 mathermatics subject lassijication 1985 revision primary 42b15 42b20 42b25

positive solutions to a type of nonlinear three point boundary value problem with sign changing nonlinearities in this paper by using krasnoselskii s fixed point theorem in a cone we study the existence of positive solutions for the second order three point boundary value problem u t a t u t lf t u t 0 0

existence of positive solutions to second order four point impulsive differential problems with deviating arguments in this paper we discuss four point boundary value problems for impulsive second order differential equations with deviating arguments we investigate separately cases when arguments are of delayed or advanced types we formulate sufficient conditions under which our problems have at least one or two positive solutions to obtain our results we apply the fixed point index

triple solutions for second order three point boundary value problems abstract we establish the existence of at least three positive solutions to the second order three point boundary value problem u f t u 0 u 0 0 u u 1 where 0 lt f 0 1 0 0 is continuous we accomplish this by making growth assumptions on f which can apply to many more cases than the sublinear and superlinear ones discussed in recent works

existence of a non zero fixed point for nondecreasing operators proved via krasnoselskii s fixed point theorem abstract in this paper we use krasnoselskii s fixed point theorem on cone expansions to prove a new fixed point theorem for nondecreasing operators on ordered banach spaces moreover we apply this abstract result to prove the existence of a positive periodic solution for a nonlinear boundary value problem

positive solutions of second order three point boundary value problems with change of sign we study the existence of positive solutions of the second order three point boundary value problems t a t f y t 0 0 t 1 y 0 0 y 1 y n where 0 1 0 1 we show the existence of at least one positive solution by applying the krasnoselskii fixed point theorem in a cone here a t is allowed to changes sign on 0 1 applications of our results are provided to yield positive radial solutions of some elliptic boundary value problems on an annulus

a sum operator equation and applications to nonlinear elastic beam equations and lane emden fowler equations article history this paper is concerned with an operator equation ax bx cx x on ordered banach spaces where a is an increasing concave operator b is an increasing sub homogeneous operator and c is a homogeneous operator the existence and uniqueness of its positive solutions is obtained by using the properties of cones and a fixed point theorem for increasing general concave operators as applications we utilize the fixed point theorems obtained in this paper to study the existence and uniqueness of positive solutions for two classes nonlinear problems which include fourth order two point boundary value problems for elastic beam equations and elliptic value problems for lane emden fowler equations

positive solutions for three point one dimensional p laplacian boundary value problems with advanced arguments this paper considers the existence of positive solutions for advanced differential equations with one dimensional p laplacian to obtain the existence of at least three positive solutions we use a fixed point theorem due to avery and peterson

a structural approach to reversible computation reversibility is a key issue in the interface between computation and physics and of growing importance as miniaturization progresses towards its physical limits most foundational work on reversible computing to date has focussed on simulations of low level machine models by contrast we develop a more structural approach we show how high level functional programs can be mapped compositionally i e in a syntax directed fashion into a simple kind of automata which are immediately seen to be reversible the size of the automaton is linear in the size of the functional term in mathematical terms we are building a concrete model of functional computation this construction stems directly from ideas arising in geometry of interaction and linear logic but can be understood without any knowledge of these topics in fact it serves as an excellent introduction to them at the same time an interesting logical delineation between reversible and irreversible forms of computation emerges from our analysis

on logarithmic bounds of maximal sparse operators given sparse collections of measurable sets mathcal s k k 1 2 ldots n in a general measure space x mathfrak m mu let lambda mathcal s k be the sparse operator corresponding to mathcal s k we show that the maximal sparse function lambda f max 1 le k le n lambda mathcal s k f satisfies begin aligned vert lambda vert l p x mapsto l p infty x lesssim log n cdot vert m mathcal s vert l p x mapsto l p infty x 1 le p infty lambda l p x mapsto l p x lesssim log n max 1 1 p 1 cdot vert m mathcal s vert l p x mapsto l p x 1 p infty end aligned where m mathcal s is the maximal function corresponding to the collection of sets mathcal s cup k mathcal s k as a consequence one can derive norm bounds for maximal functions formed from taking measurable selections of one dimensional calderon zygmund operators in the plane prior results of this type had a fixed choice of calderon zygmund operator for each direction

asymptotic solutions for a second order differential equation with three point boundary conditions abstract in this paper we are concerned with the following singularly perturbed second order three point boundary value problem 1 1 y 2 q 1 x q 2 x y 0 1 1 2 y 0 a y 1 y where 0

maximal functions associated with fourier multipliers of mikhlin hormander type we show that maximal operators formed by dilations of mikhlin hormander multipliers are typically not bounded on lp d we also give rather weak conditions in terms of the decay of such multipliers under which lp boundedness of the maximal operators holds

positive solutions of nonlinear third order m point bvp for an increasing homeomorphism and homomorphism with sign changing nonlinearity in this paper several existence theorems of positive solutions are established for a nonlinear m point boundary value problem for the following third order differential equations f u a t f t u t 0 email protected 0 1 f u 0 email protected i 1m 2a email protected u x i u 0 0 u 1 email protected i 1m 2b iu x i where f email protected r is an increasing homeomorphism and homomorphism and f 0 0 the nonlinear term f may change sign as an application an example to demonstrate our results is given

nonnegative solutions to nonlocal boundary value problems for systems of second order differential equations dependent on the first order derivatives abstract in this paper we consider nonlocal boundary value problems for systems of second order differential equations with dependence on the first order derivatives and deviating arguments by using a fixed point theorem due to avery and peterson sufficient conditions are obtained that guarantee the existence of at least three nonnegative solutions to such problems we investigate our problem both for delayed and advanced arguments i i and also for the case when i t i t t t 0 1 in all cases arguments i i can change the character on 0 1 so in some subinterval i of 0 1 they can be delayed in i and advanced in 0 1 i some remarks concern also the case when differential equations do not depend on the first order derivatives examples illustrate some results

an even order three point boundary value problem on time scales abstract we study the even order dynamic equation 1 nx n t h t f x t t a c satisfying the boundary conditions x i a 0 and x i c x i b for 0 i n 1 the three points a b c are from a time scale t where 0 0 f is a positive function and h is a nonnegative function that is allowed to vanish on some subintervals of a c of the time scale

existence of three positive solutions for boundary value problems with p laplacian abstract by using fixed point theorem we study the following equation g u t a t f u 0 subject to boundary conditions where g v v p 2 v with p 1 the existence of at least three positive solutions is proved