denseness of the generalized eigenvectors of an h s discrete operator let t be a closed densely defined linear operator in a hilbert space h and assume there exists 0 p t such that r 0 t is a hilbert schmidt operator the operator t is a special type of discrete operator a so called h s discrete operator which is shown to be a fredholm operator in h with fredholm set equal to the whole complex plane let t i i 1 where the operator t i has ascent mi let pi i 1 2 be the projection of h onto the generalized eigenspace n t i mi along r t i mi and let s and m be the subspaces of h consisting of all x h such that x i 1 pix and such that pix 0 for i 1 2 respectively sufficient conditions are introduced which guarantee that s h and m 0 these conditions require that r t be bounded on certain rays in the complex plane and r t 0 as on at least one of the rays but specific decay rates for r t are not necessary

fixed points and approximate solutions for nonlinear operator equations we study a nonlinear operator equation and obtain a sequence of approximate solutions by extending the monotone iterative technique to new situations some examples and applications are presented

coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces abstract we introduce the concept of a mixed g monotone mapping and prove coupled coincidence and coupled common fixed point theorems for such nonlinear contractive mappings in partially ordered complete metric spaces presented theorems are generalizations of the recent fixed point theorems due to bhaskar and lakshmikantham t g bhaskar v lakshmikantham fixed point theorems in partially ordered metric spaces and applications nonlinear anal tma 65 2006 1379 1393 and include several recent developments

new fixed point theorems and applications of mixed monotone operator in this paper we study the uniqueness and existence of fixed point of mixed monotone operators in the partially ordered banach space the results extend and improve recent related results

ironing sweeping and multidimensional screening we provide existence proofs and characterization results for the multidimensional version of the multiproduct monopolist problem of mussa and rosen 1978 these results are also directly applicable to the multidimensional nonlinear pricing problems studied by wilson 1993 and armstrong 1996 we establish that bunching is robust in these multidimensional screening problems even with very regular distributions of types this comes from a strong conflict between participation constraints and second order incentive compatibility conditions we consequently design a new technique the sweeping procedure for dealing with bunching in multidimensional contexts this technique extends the ironing procedure of mussa and rosen 1978 to several dimensions we illustrate it on two examples we first solve a linear quadratic version of the bidimensional nonlinear pricing problem where consumers types are exponentially distributed the solution involves pure bundling for consumers with low demands the second example is the bidimensional version of the mussa and rosen problem when consumers types are uniformly distributed on a square the solution is such that the seller offers a full differentiation of products in the upper part of the qualities spectrum but only limited choice for lower qualities this seems to be a quite general pattern for multidimensional screening problems the sweeping procedure is potentially applicable to other multidimensional screening problems

coupled fixed points for mixed monotone condensing operators and an existence theorem of the solutions for a class of functional equations arising in dynamic programming abstract in this paper the existence and iterative approximation of coupled fixed points for mixed monotone condensing set valued operators are discussed the results presented here extend and improve the corresponding results of d guo and v lakshmikantham coupled fixed points of nonlinear operators 1 2 as an application we utilize the results obtained in section 2 to study the existence problem of solutions for a class of functional equations arising in dynamic programming

fixed point theorems for generalized contractions in ordered metric spaces abstract the purpose of this paper is to present some fixed point results for self generalized contractions in ordered metric spaces our results generalize and extend some recent results of a c m ran m c reurings a c m ran m c reurings a fixed point theorem in partially ordered sets and some applications to matrix equations proc amer math soc 132 2004 1435 1443 j j nieto r rodriguez lopez j j nieto r rodriguez lopez contractive mapping theorems in partially ordered sets and applications to ordinary differential equations order 22 2005 223 239 j j nieto r rodriguez lopez existence and uniqueness of fixed points in partially ordered sets and applications to ordinary differential equations acta math sin engl ser 23 2007 2205 2212 j j nieto r l pouso r rodriguez lopez j j nieto r l pouso r rodriguez lopez fixed point theorem theorems in ordered abstract sets proc amer math soc 135 2007 2505 2517 a petrusel i a rus a petrusel i a rus fixed point theorems in ordered l spaces proc amer math soc 134 2006 411 418 and r p agarwal m a el gebeily d o regan r p agarwal m a el gebeily d o regan generalized contractions in partially ordered metric spaces appl anal in press as applications existence and uniqueness results for fredholm and volterra type integral equations are given

spectral decomposition of a hilbert space by a fredholm operator abstract let t be a fredholm operator in a hilbert space h with spectrum t i i 1 let t i t i m i where m i is the ascent of the operator t i i i t and let p i denote the projection of h onto the generalized eigenspace n t i along r t i in this paper it is shown that s s and h s m topological direct sum iff there exists m 0 such that i 1 n p i m n 1 2 where s x h i 1 x i x i n t i and m is the zero or infinite dimensional subspace i 1 r t i this result is then applied to a differential operator l in l 2 0 1 showing that s s l 2 0 1 for this l also an example with m 0 is presented

spectral theory of two point differential operators determined by d2 i spectral properties abstract the necessary tools for the complete resolution of the spectral theory for all linear two point differential operators l in l2 0 1 determined by d2 and by boundary values b1 b2 are developed all work is based on six numerical parameters defined in terms of the coefficients of b1 b2 these quantities are utilized in the development of the characteristic determinant of l and the green s function of i l and in the development of the projections of l2 0 1 onto the generalized eigenspaces of l they are used in the study of the decay rates of the resolvent operator r l as and they are used to identify the 13 cases treated in developing the spectral theory for d2 all results pertaining to the spectrum l of l to the algebraic multiplicities of the eigenvalues l to the ascents of the operators i l to the boundedness of the family of all finite sums of the projections associated with l to the denseness of the generalized eigenfunctions and to bases consisting of generalized eigenfunctions are summarized in a table the proofs of these results are the subject of the sequel part ii

eigenfunction expansions for the nonspectral differential operators determined by d2 abstract for a nonspectral differential operator l determined by d2 sufficient conditions are established for the l2 expansion of a function u l2 0 1 in terms of the eigenfunctions of l in some cases both necessary and sufficient conditions are derived

fixed point theorems in partially ordered metric spaces for operators with ppf dependence abstract in this paper fixed point theorems of a nonlinear operator are developed where the domain space c a b e is different from the range e which is a metric space with partial order an application of the fixed point theorem to a periodic boundary value problem with delay is also given

fixed point theorems for mixed monotone operators and applications to integral equations abstract the purpose of this paper is to present some coupled fixed point theorems for a mixed monotone operator in a complete metric space endowed with a partial order by using altering distance functions we also present an application to integral equations

fixed point theorems for weakly contractive mappings in partially ordered sets abstract the purpose of this paper is to present some fixed point theorems for weakly contractive maps in a complete metric space endowed with a partial order

fixed points of multivalued operators in ordered metric spaces with applications abstract the purpose of this paper is to present some new hybrid fixed point theorems involving multivalued operators which satisfy weakly generalized contractive conditions in an ordered complete metric space an example of the existence of solutions for a perturbed impulsive hyperbolic differential inclusion with variable times is given to illustrate the usability of our results

fixed point theorems in partially ordered metric spaces and applications abstract we prove a fixed point theorem for a mixed monotone mapping in a metric space endowed with partial order using a weak contractivity type of assumption besides including several recent developments our theorem can be used to investigate a large class of problems as an application we discuss the existence and uniqueness of solution for a periodic boundary value problem

spectral theory of two point differential operators determined by d2 ii analysis of cases abstract in part i the necessary tools are developed for the complete resolution of the spectral theory for all linear two point differential operators l in l 2 0 1 determined by d 2 and by boundary values b 1 b 2 the development emphasizes six numerical parameters defined in terms of the coefficients of b 1 b 2 in this paper we state and prove all results pertaining to the spectrum l of l to the algebraic multiplicities of the eigenvalues l and the ascents of the operators i l to the boundedness of the family of all finite sums of the projections associated with l to the denseness of the generalized eigenfunctions and to the existence of bases consisting of generalized eigenfunctions we utilize the tools developed in part i to establish these results in particular this paper provides either the explicit or the asymptotic form for the eigenvalues of l provides explicit numbers for the algebraic multiplicities and ascents provides the explicit or the asymptotic form for the projections and provides explicit bounds on the family of all finite sums of these projections or shows that the family is unbounded

coupled fixed points of nonlinear operators with applications a partir de quasi solutions couplees des problemes aux valeurs initiales pour des equations differentielles ordinaires on introuit le concept de points fixes couples abstraits pour certains operateurs on donne des theoremes d existence

new fixed point theorems of mixed monotone operators and applications in this paper a class of mixed monotone operators with convexity and concavity are investigated and existence and uniqueness theorems of fixed points are obtained moreover some applications to nonlinear integral equations on unbounded regions and differential equations in banach spaces are given

on a new method for solving irregular problems abstract we suggest a new approach to the investigation of boundary value problems with a parameter in the case when the parameter appears in both the equation and the boundary conditions the method is applied to the regge problem in spite of the fact that the regge problem is birkhoff irregular see g d birkhoff trans amer math soc 9 1908 373 395 the problem is coercive but with a defect with respect to the spectral parameter we also prove 2 fold completeness of root functions of the regge problem

spectral theory for a differential operator characteristic determinant and green s function abstract for a two point differential operator l in l2 a b it is shown that the green s function has the representation g t s h t s d for belonging to the resolvent set l where d is the characteristic determinant and h t s is an entire function in the variable admitting a power series expansion about any point 0 c this representation is given several applications first to calculate the coefficient operators in the laurent series for the resolvent r l about each point 0 in the spectrum l and second to relate the algebraic multiplicity v 0 of an eigenvalue 0 to the ascent m0 of the operator 0i l

the spectral theory of second order two point differential operators i a priori estimates for the eigenvalues and completeness this paper is the first part in a four part series which develops the spectral theory for a two point differential operator l in l 2 0 1 determined by a second order formal differential operator l d 2 pd q and by independent boundary values b 1 b 2 the differential operator l is classified as belonging to one of five cases cases 1 5 according to conditions satisfied by the coefficients of b 1 b 2 for cases 1 4 it is shown that if 2 is any eigenvalue of l with sufficiently large then lies in the interior of a horizontal strip cases 1 3 or the interior of a logarithmic strip case 4 and in each of these cases the generalised eigenfunctions of l are complete in l 2 0 1

generalized contractions in partially ordered metric spaces and applications to ordinary differential equations the purpose of this paper is to present some fixed point theorems in a complete metric space endowed with a partial order by using altering distance functions we also present some applications to first and second order ordinary differential equations

a coupled coincidence point result in partially ordered metric spaces for compatible mappings abstract in this paper we introduce the notion of compatibility of mappings in a partially ordered metric space and use this notion to establish a coupled coincidence point result our work extends the work of bhaskar and lakshmikantham t gnana bhaskar v lakshmikantham fixed point theorems in partially ordered metric spaces and applications nonlinear anal tma 65 2006 1379 1393 an example is also given

optimal dynamic auctions and simple index rules a monopolist seller has multiple units of an indivisible good to sell over a discrete finite time horizon buyers with unit demand arrive over time and each buyer privately knows her arrival time her value for a unit and her deadline we study whether the seller s optimal allocation rule is a simple index rule each buyer is assigned an index and the allocation rule is calculated by a dynamic knapsack algorithm using those indices simple indicates that the index of a buyer depends only on local information i e the distribution information for that time period if buyer deadlines are public such simple index rules are optimal if the standard increasing hazard rate condition on the distribution of valuations holds and given two buyers with the same deadline the later arriving one has a lower hazard rate implying stochastically higher valuations when buyer deadlines are private this condition is neither sufficient nor necessary if the rule we identify is not feasible then the optimal allocation rule is not a simple index rule and cannot be calculated by backward induction

coupled fixed point theorems for mixed monotone mappings and an application to integral equations in this paper we extend the coupled fixed point theorems for a mixed monotone mapping f xxx x in partially ordered metric spaces established by bhaskar and lakshmikantham t gnana bhaskar v lakshmikantham fixed point theorems in partially ordered metric spaces and applications nonlinear anal tma 65 2006 1379 1393 an application to nonlinear integral equations is also given to illustrate our results

coupled common fixed point theorems for w compatible mappings in ordered cone metric spaces abstract we establish coupled coincidence point results for mixed g monotone mappings under general contractive conditions in partially ordered cone metric spaces over solid cones we also present results on existence and uniqueness of coupled common fixed points our results generalize extend and unify several well known comparable results in the literature to illustrate our results and to distinguish them from the earlier ones we equip the paper with examples

the nonspectral birkhoff regular differential operators determined by d2 the class of nonspectral birkhoff regular differential operators l in l2 0 1 determined by d2 and by boundary values b1 b2 is reexamined for the family of projections p0k k 1m pk k k0 p k k k0 which map l2 0 1 onto the generalized eigenspaces of l the norms of the pk and p k tend to while in sharp contrast the family of all finite sums of the projections p0k k 1m pk p k k k0 is uniformly bounded in norm in addition the expansion u k 1m p0k u k k0 pk p k u is valid for all u l2 0 1 even though there exist u in l2 0 1 for which both series k k0 pku and k k0 p ku diverge

a fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations we present a fixed point theorem for generalized contraction in partially ordered complete metric spaces as an application we give an existence and uniqueness for the solution of a periodic boundary value problem

coupled fixed point theorems for a generalized meir keeler contraction in partially ordered metric spaces abstract let x be a non empty set and f x x x be a given mapping an element x y x x is said to be a coupled fixed point of the mapping f if f x y x and f y x y in this paper we consider the case when x is a complete metric space endowed with a partial order we define generalized meir keeler type functions and we prove some coupled fixed point theorems under a generalized meir keeler contractive condition some applications of our obtained results are given the presented theorems extend and complement the recent fixed point theorems due to bhaskar and lakshmikantham t gnana bhaskar v lakshmikantham fixed point theorems in partially ordered metric spaces and applications nonlinear anal 65 2006 1379 1393

coupled fixed points in partially ordered metric spaces and application abstract in this paper we prove some coupled fixed point theorems for mappings having a mixed monotone property in partially ordered metric spaces the main results of this paper are generalizations of the main results of bhaskar and lakshmikantham t gnana bhaskar v lakshmikantham fixed point theorems in partially ordered metric spaces and applications nonlinear anal tma 65 2006 1379 1393 as an application we discuss the existence and uniqueness for a solution of a nonlinear integral equation

the dynamic and stochastic knapsack problem with deadlines in this paper a dynamic and stochastic model of the well known knapsack problem is developed and analyzed the problem is motivated by a wide variety of real world applications objects of random weight and reward arrive according to a stochastic process in time the weights and rewards associated with the objects are distributed according to a known probability distribution each object can either be accepted to be loaded into the knapsack of known weight capacity or be rejected the objective is to determine the optimal policy for loading the knapsack within a fixed time horizon so as to maximize the expected accumulated reward the optimal decision rules are derived and are shown to exhibit surprising behavior in some cases it is also shown that if the distribution of the weights is concave then the decision rules behave according to intuition

optimal auction design this paper considers the problem faced by a seller who has a single object to sell to one of several possible buyers when the seller has imperfect information about how much the buyers might be willing to pay for the object the seller s problem is to design an auction game which has a nash equilibrium giving him the highest possible expected utility optimal auctions are derived in this paper for a wide class of auction design problems