stability of the classification of returns to scale in fdh models this paper deals with the estimation of returns to scale rts in free disposal hull fdh models and provides some stability intervals for preserving the rts classification it has been shown that the proposed stability intervals can be obtained via a polynomial time algorithm based on the calculation of certain ratios of inputs and outputs without solving any mathematical programming problem the results of the study have been proved via some lemmas and theorems and have been illustrated by numerical examples and a real application

optimal control of evolution inclusions abstract we develop the method of discrete approximations for studying optimal control problems of the bolza type for evolution systems governed by generally nonconvex differential inclusions in infinite dimensional spaces the results obtained involve well posedness stability of discrete approximations necessary optimality conditions for general discrete time systems and necessary optimality conditions for continuous time inclusions established in the extended euler lagrange form our approach is based on the generalized differential constructions of variational analysis introduced earlier by the author

the solution to the conjecture on properness of weakly relaxed delayed controls in 1986 warga proposed a weak relaxation procedure applicable to fully nonlinear problems with delays in the control variables and showed that the resulting relaxed problem has a solution however in the event of commensurate delays several examples were found for which weakly relaxed controls cannot be approximated with original controls so that this extension fails to be proper although the case of commensurate delays was solved by the introduction of a strong model the question of how to properly relax noncommensurately delayed controls has remained open and a natural candidate has been precisely that of weakly relaxed controls in this paper a general counterexample is constructed which rules out the weak relaxation as a proper relaxation when there are two or more delays it is hoped that this result will provide some insight into the problem of finding a general representation of properly relaxed controls

variational analysis in nonreflexive spaces and applications to control problems with l 1 perturbations we provide a refined sensitivity analysis for finite and infinite horizon control problems where in both cases the perturbation space is l 1 our underlying technique relies on a recent sequential description of both the generalized gradient of clarke and of the approximate g subdifferential of functions defined on a smooth banach space we also show that the proximal limit formula for the generalized gradient and its l p analogue are direct consequences of these sequential formulas related characterizations of lipschitzness of a function on a smooth space are given

a representation of properly relaxed controls with two delays abstract the search of a proper relaxation procedure in the sense that relaxed minimizers can be approximated with ordinary controls for optimal control problems involving delays in the control variables has been the main topic of several recent papers three models of relaxation which we call the weak strong and d procedures have been proposed and in all cases the existence of minimizers has been established it has been proved that the d model is a proper relaxation procedure but determining the set of d relaxed controls for specific problems is very difficult and perhaps even a hopeless task thus there is a need to find more concrete characterizations of the closure of the space of ordinary delayed controls in the event of commensurate delays this is solved through the strong model the space of strongly relaxed controls coincides with the space of d relaxed controls for the noncommensurate case the problem of how to characterize d relaxed controls has remained unsolved and although a natural candidate had been the space of weakly relaxed controls now we know that for either commensurate or noncommensurate delays it may fail to be proper in this paper we are finally able to characterize d relaxed controls for two noncommensurate delays by the introduction of a new model of relaxation the p model which turns out to be a space strictly contained in that of weakly relaxed controls

the weighted russell measure of technical efficiency theoretical consideration of technical efficiency has existed since koopmans 10 defined it for production possibilities for which it is not possible to increase any output without simultaneously increasing any input ceteris paribus the nonparametric approach to efficiency measurement known as data envelopment analysis is based on the index of farrell 9 which measures radial reduction in all inputs consistent with observed output even after farrell efficiency is achieved however there may exist additional slack in individual inputs suggesting that the farrell index does not necessarily measure koopmans inefficiency to solve this problem the non radial russell measure was introduced this paper shows that problems may arise with the russell measure due to restrictive assumptions on the implicit weighting of inputs and outputs this paper develops a new measure the weighted russell measure that relaxes this assumption using simulated data the new measure is shown to be preferred to existing methods in addition the new method is applied to analyze the performance of new york state school districts

measurement of returns to scale using a non radial dea model a range adjusted measure approach abstract this research theoretically explores the measurement of returns to scale rts using a non radial dea data envelopment analysis model a range adjusted measure ram is used as a representative of such non radial models historically a type of rts has been discussed within an analytical framework of radial models the radial based rts measurement is replaced by the non radial ram rts measurement in this study when discussing the non radial ram rts measurement this study finds a problem of multiple projections that cannot be found in the radial measurement in this research a new linear programming approach is proposed to identify all efficient dmus decision making units on a reference set the important feature of the proposed approach is that it can deal with a simultaneous occurrence of a multiple reference sets b multiple supporting hyperplanes and c multiple projections all of the three difficulties are handled by the proposed ram rts measurement in particular we discuss both when the three different types of multiple solutions occur on the ram rts measurement and how to deal with such difficulties our research results make it possible to measure not only the type of rts but also the magnitude of rts in the ram measurement

necessary and sufficient conditions for discrete and differential inclusions of elliptic type abstract this paper deals for the first time with the dirichlet problem for discrete p d discrete approximation problem on a uniform grid and differential p c inclusions of elliptic type in the form of euler lagrange inclusion necessary and sufficient conditions for optimality are derived for the problems under consideration on the basis of new concepts of locally adjoint mappings the results obtained are generalized to the multidimensional case with a second order elliptic operator

convex input and output projections of nonconvex production possibility sets in this paper we characterize the smallest production possibility set that contains a specified set of input output combinations in accordance with neoclassical production economics this possibility set has convex projections into the input and output spaces convex isoquants and it satisfies the assumption of free disposability we obtain it by means of a possibly infinite recursion which builds the possibility set as an ever larger union of convex sets we remark on the nature of the approximations obtained by truncating the recursion and we obtain a necessary and sufficient condition checkable in one iteration for the recursion to stop in the next for the case in which the recursion stops we provide a succinct characterization of the dominance relations among the constituent sets produced by the procedure finally we present examples of both finite and infinite cases the example for the finite case illustrates the construction of the possibility set along with its associated production and consumption sets

scale elasticity and returns to scale in the presence of alternative solutions scale elasticity se and returns to scale rts are important topics in performance analysis which help managers to make decisions about the expansion or contraction of the operation of decision making units under assessment in this paper some new results about these topics in the presence of alternative solutions regarding the concept of multifunction are provided at first some properties of some multifunctions functions defined with respect to the optimal solutions of dea models are established which help us in what follows in turn the relationships between the considered multifunctions and the concept of rts and se are studied finally an approach for the estimation of the rts classification of units is obtained which leads to an important corollary as an interesting result which introduces a connection between two concepts rts and se this is important from an applied point of view also from a technical point of view the proofs of theorems which give this corollary use a main lemma of convex analysis literature and give a constructive proof about rts

higher order conditions for conical controllability let q be a convex subset of a linear space mathcal u subset q mathcal z a topological vector space c subset mathcal z convex with a nonempty interior bar q in q varphi varphi 1 varphi 2 q to mathbb r m times mathcal z and varphi 2 bar q in c also let mathcal a u in mathcal u varphi 2 u in c this framework can be used to model many optimization problems including control problems with unilateral state inequality constraints sufficient conditions are derived for varphi 1 mathcal a to cover the interior of a convex set containing varphi 1 bar q and an open neighborhood of a point varphi 1 bar q w for w in mathbb r m these sufficient conditions can be used to provide information about the attainable sets of control problems and in some cases to show that varphi 1 mathcal a contains a neighborhood of varphi 1 bar q several illustrative examples are also discussed

single valued representation of set valued mappings ii application to differential inclusions we consider a set valued mapping q t x the first argument ranging through a measurable space and the second through a space with a different structure topological metric differentiable etc we are interested in representing q by a single valued mapping with properties compatible with the structures i e in the existence of a topological space z and a mapping f t x z measurable in t continuous in x z and depending on x as dictated by the structure of the space of x s say continuous lipschitz or differentiable in x etc and such that f t x z q t x for all t x approximate representations i e those for which the latter equality holds approximately of convex valued mappings are also considered and certain applications mainly to differential inclusions are discussed

the adjoint arc in nonsmooth optimization we extend the theory of necessary conditions for nonsmooth problems of bolza in three ways first we incorporate state constraints of the intrinsic type x t e x t for all t second we make no assumption of calmness or normality and third we show that a single adjoint function of bounded variation simultaneously satisfies the hamiltonian inclusion the euler lagrange inclusion and the weierstrass pontryagin maximum condition along with the usual transversality relations

on the characterization of properly relaxed delayed controls abstract in previous papers warga and zhu have proven the conjecture of rosenblueth and vinter that a particular abstract model characterizes the set r of relaxed controls that are limits of ordinary controls with possibly time dependent delays rosenblueth and vinter have also provided a simple and concrete characterization of r for the case of constant delays that are consecutive multiples of a common divisorh in the present paper we provide a concrete characterization of r for the case of delays that are arbitrary and not necessarily consecutive multiples of someh we also exhibit a simple compactification of the set of solutions of controlled differential equations with such delays in control

using input output orientation model for determining most productive scale size in dea one of the existing approaches for determining mpss is email protected model proposed in cooper thompson and thrall s article annals of operations research 66 1996 3 the direct solving of that model is rather difficult because the objective function is fractional hence in this paper besides introducing a model with output input orientation the previous model difficulties will be removed because the new objective function is linear and also it is proved that both models are the same in determining mpss furthermore by solving this model it is not necessary to scaling the data as it is necessary in ccr model so that figurative dmu move in the joint frontier of bcc and ccr models see introduction to the theory and applications of data envelopment analysis a foundation text with integrated software 2001 european journal of operational research 17 1984 35 so we can consider this as another merit for this model in determining mpss we obtain the smallest and the most mpss corresponding with the evaluating dmu and also we present necessary and sufficient condition for boundedness of this model by dual model and numerical example confirms validity of this model as a means for determining mpss

periodic p laplacian with nonlocal terms abstract in this paper we study the existence of non trivial periodic solutions for a periodic p laplacian with nonlocal terms based on the theory of leray schauder degree the key step is dealing with the degeneracy of the p laplacian and the logistic type terms arising in the right hand side of the equation

on periodic solutions of superlinear parabolic problems on etudie l existence des solutions periodiques non triviales positives des problemes paraboliques semilineaires on demontre des resultats de bifurcation

problems with relaxed shifted controls and the pointwise maximum principle abstract in previous work j warga a proper relaxation of controls with variable shifts j math anal appl 196 1995 783 793 it has been shown that controls with k variable and unrelated time shifts form a set whose closure in the weak star topology of l 1 t c k 1 is a convex and compact space this provides a basis for establishing the existence of an optimal solution of the shifted control problem by essentially copying the assumptions and arguments used for control problems without shifted controls similarly the necessary optimality conditions for original and relaxed minima that define an extremal previously studied for differential and functional equations apply directly to such problems with shifted controls except that the maximum principle follows directly in integral form only in the present note we first establish the form of dependence of the relaxed shifted controls on the shifts and then use the integral form of the maximum principle to derive a corresponding pointwise maximum principle for optimal ordinary or relaxed shifted controls acting through ordinary differential and certain functional differential equations

the instability of optimal control problems to time delay this paper considers a general optimal control problem with a small time delay delta in both the state and control the objective is to find a correct way to neglect the delay or in other words to construct a well posed instantaneous approximation of the delayed problem the natural way to do so is by merely ignoring the delay in the model it is shown that this way is almost always incorrect it gives rise to an error that does not vanish as delta to 0 and so results in an ill posed model a proper approximation of the delayed problem is offered

estimation of returns to scale using data envelopment analysis abstract generalization of the measure of returns to scale from a single number to an interval permits extension of the concept to dea data domains with multiple inputs and multiple outputs the key new approach is a partition of the optimal frontier into three parts corresponding respectively to increasing constant and decreasing returns to scale these parts are characterized in terms of optimal primal solutions and optimal dual solutions for both the original charnes cooper rhodes model 1978 and the later banker charnes cooper model 1984 and relying on concepts developed by r d banker 1984 and r m thrall 1988

hamiltonian necessary conditions for a multiobjective optimal control problem with endpoint constraints this paper discusses hamiltonian necessary conditions for a nonsmooth multiobjective optimal control problem with endpoint constraints involving a general preference the transversality condition in our necessary conditions is stated in terms of normal cones to the level sets of the preference examples involving a number of useful preferences are discussed

on return to scale of fully efficient dmus in data envelopment analysis under interval data recently cooper et al manage sci 45 1999 597 proposed imprecise data envelopment analysis idea a situation in which some or all data are unknown the data in idea could be interval data ordinal data or data with bounded ratio usually in such cases the corresponding data envelopment analysis models becomes non linear when data assume to be interval decision making units dmus can divide to three classes as follows i efficient in any cases ii efficient in maximal sense iii always inefficient in this paper we investigate to the return to scale of dmus that are locate in first class also we introduce models that use exact data and present a combination if any which via that the dmu becomes a most productive scale size

global solutions for some nonlinear parabolic equations with nonmonotonic peturbations publisher summary this chapter discusses the existence uniqueness and decay property of the solutions of two problems it also presents the sobolev s lemma and gagliardo nirenberg inequality

periodic solutions of non newtonian polytropic filtration equations with nonlinear sources in this paper the authors first consider the dirichlet boundary value problem to the non newtonian polytropic filtration equation of the form u t div u m p 2 u m h x t u a in wxrwith strong nonlinear sources the existence of nontrivial periodic solutions is established based on topological degree theory the authors also studied the dirichlet boundary value problem to the equation in the form u t div u m 1u p 2 u m 1u b x t u f x t in wxrwith weak nonlinear sources the existence is treated with leray schauder fixed point theory

strongly and weakly relaxed controls for time delay systems in recent articles two different relaxation procedures for time delay problems in optimal control have been proposed a weak procedure due to warga nonadditively coupled delayed controls privately circulated 1986 for which the existence of minimizers is assured applies to fully nonlinear problems with delays in the state and control variables nevertheless an example is found for which the effect of weak relaxation reduces the infimum cost the example involves two delays one being twice the value of the other and it belongs to a class of problems that is called commensurate where the quotient of any two delays is rational for these problems a strong procedure is proposed and it is proved that the extended problem has a solution and the extension is proper i e the infimum costs coincide in this paper it is shown through several examples why in general the weak relaxation technique may fail to provide a proper extension and for noncommensurate delay problems an approximation r

a decoupling principle in the calculus of variations abstract we present a technique for approximating a variational problem by another in which the endpoint cost and the running cost are decoupled and in which the state and the velocity are also severed

optimization and controllability without differentiability assumptions let mathcal x and mathcal y be real normed vector spaces k subset mathcal x convex and compact c subset mathcal y a convex body mathcal u subset k and phi phi k to mathbb r m times mathcal y a function that can be appropriately approximated by functions phi i phi i whose compositions with linear maps of small finite dimensional simplices are c 1 we derive sufficient conditions for phi to be locally controllable on mathcal u subject to the restriction phi u in c and obtain as a corollary corresponding necessary conditions for a related restricted minimum these conditions are formulated in terms of directional derivate containers which are a type of set valued derivatives of phi phi and they improve on and extend previously obtained results they are used elsewhere to obtain new conditions for controllability and restricted minimum in nonsmooth optimal control problems defined by differential or functional integral equatio

periodic solutions of the evolutionary p laplacian with nonlinear sources abstract this paper is concerned with the evolutionary p laplacian with nonlinear and periodic sources we will give a rather complete characterization in terms of the parameter p and the exponent q of the source of whether or not the positive periodic solutions exist

contingent cones to reachable sets of control systems high order necessary conditions for optimality for an optimal control problem are studied via properties of contingent cones to reachable sets along the optimal trajectory it is shown that the adjoint vector of pontryagin s maximum principle is normal to the set of variations of reachable sets results are applied to study optimal control problems for dynamical systems described by 1 closed loop control systems 2 nonlinear implicit systems 3 differential inclusions 4 control systems with jumps

an alternative approach in the estimation of returns to scale under weight restrictions abstract this paper discusses the issue of returns to scale rts under weight restrictions in data envelopment analysis dea we first review tone s method k tone on returns to scale under weight restrictions in data envelopment analysis journal of productivity analysis 16 2001 31 47 for estimating returns to scale under weight restrictions then a new approach is introduced for this task based upon the sum of the optimal lambda values in the weighted ccr model the equivalence of this method and tone s method is proved we then apply the method to a real world data set

a discussion of testing dmus returns to scale abstract the issue of using banker s most productive scale size mpss concept to characterize dmus returns to scale rts is studied in this paper it is shown that the mpss concept can always be used to estimate rts without any adjustments unless a set of efficient dmus exhibit linear dependency i e it is the dmu itself that causes the mpss concept not to work the mpss concept itself is independent of assuming a linear production function in the ccr model as a result chang and ghu s conclusions are erroneous also several other errors in chang and ghu s proof and illustration of proof are listed the linear dependency condition in the current paper corresponds to the nonunique lambda solution situation in banker and thrall meanwhile a remedy is developed for the mpss concept so that it can still work under linear dependency it is also shown that those dmus which naturally exhibit increasing and decreasing returns to scale have nothing to do with linear dependency

necessary conditions for multiobjective optimal control problems with free end time necessary conditions of optimality are derived for multiobjective optimal control problems with free end time in which the dynamics constraint is modeled as a nonconvex differential inclusion the obtained results cover some previous results on necessary conditions for multiojective and single objective optimal control problems

aggregation of scale efficiency abstract in this article we generalize the aggregation theory in efficiency and productivity analysis by deriving solutions to the problem of aggregation of individual scale efficiency measures primal and dual into aggregate primal and dual scale efficiency measures of a group e g industry the new aggregation result is coherent with aggregation framework and solutions that were earlier derived for other related efficiency measures and can be used in practice for estimation of scale efficiency of an industry or other groups of firms within it

existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with nonlocal terms abstract in this paper the authors establish the existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with nonlocal terms by using the theory of leray schauder s degree

the measurement of returns to scale under a simultaneous occurrence of multiple solutions in a reference set and a supporting hyperplane this research theoretically explores the measurement of rts returns to scale under a possible occurrence of multiple solutions in dea data envelopment analysis in this study the occurrence of multiple solutions is classified into type i and type ii type i is an occurrence of multiple solutions in a reference set type ii is an occurrence of multiple solutions on a supporting hyperplane passing on the reference set both types i and ii are very well known among dea researchers but previous research has not sufficiently explored a simultaneous occurrence of type i and type ii in the rts measurement the two types of multiple solutions influence a degree of rts in the dea measurement such a quantitative issue on rts is examined from the perspective of the type i and type ii problems to deal with such difficulties a new linear programming approach is proposed to identify all efficient dmus decision making units that consist of a reference set even if multiple solutions occur on the reference set based upon the research result we can identify when multiple solutions of type i and or type ii occur on the rts measurement and how to deal with such difficulties our research result makes it possible to measure a degree of scale economies rts under the simultaneous occurrence of type i and type ii

maximum likelihood consistency and data envelopment analysis a statistical foundation this paper provides a formal statistical basis for the efficiency evaluation techniques of data envelopment analysis dea dea estimators of the best practice monotone increasing and concave production function are shown to be also maximum likelihood estimators if the deviation of actual output from the efficient output is regarded as a stochastic variable with a monotone decreasing probability density function while the best practice frontier estimator is biased below the theoretical frontier for a finite sample size the bias approaches zero for large samples the dea estimators exhibit the desirable asymptotic property of consistency and the asymptotic distribution of the dea estimators of inefficiency deviations is identical to the true distribution of these deviations this result is then employed to suggest possible statistical tests of hypotheses based on asymptotic distributions

a dual simplex based method for determination of the right and left returns to scale in dea abstract this study reviews the concept of the right and the left returns to scale rts in data envelopment analysis dea and a dual simplex based method for determining these two notions in rts is proposed which has computational advantages as compared to the customary method

locally adjoint mappings and optimization of the first boundary value problem for hyperbolic type discrete and differential inclusions abstract the present paper deals with discrete approximation on a uniform grid of the first boundary value problem p c for differential inclusions of hyperbolic type in the form of euler lagrange inclusions necessary and sufficient conditions for optimality are derived for the discrete p d and continuous p c problems on the basis of new concepts of locally adjoint mappings the results obtained are generalized to the multidimensional case with a second order elliptic operator

a topological index condition for conical controllability covers the interior of a convex set containing e q and a neighborhood of a y w we say that p1 is controllable at 4 if it is w conically controllable at w 0 these concepts are motivated by problems of the optimal control theory and are closely related to the investigation of attainable sets certain higher order sufficient conditions for controllability and conical controllability were derived in l 21 these conditions also ensure that controllability respectively conical controllability is preserved when 9 p2 is replaced by v e pz ez for all higher order perturbations et ej in the present paper we generalize the basic result of 2 by replacing a homeomorphism condition with a more general nonzero index condition let v be an open subset of ky h i r continuous and w e r h av we refer to the elements of g w as w points of g let 71 be a bounded connected component of h w admitting an open neighborhood u whose closure is contained in i and is disjoint from other components of h w we refer to such a component rc as a bounded isolated w patch of h and denote by i h rrw w the topological degree d h u w for any choice of u as described above 3 we refer to i h rr w as the index of h at w with respect to rc we also denote by jh x the jacobian determinant det h x

higher order conditions with and without lagrange multipliers let q be a convex subset of a vector space mathcal u subset q mathcal s a topological vector space c a convex subset of mathcal z with a nonempty interior phi phi 1 phi 2 q to mathbb r m times mathcal z bar q in q and phi 2 bar q in c we assume that phi has a pth order taylor approximation at bar q when it is restricted to an arbitrary finite dimensional simplex in q with a vertex at bar q in the case when mathcal u is a proper subset of q we also assume that qis a uniform space phi continuous and mathcal u abundant we establish a number of higher order sufficient conditions not involving any lagrange multipliers for the existence of neighborhoods g 1 and g 2 of the origins such that phi 1 bar q g 1 subset phi 1 u u in mathcal u phi 2 u g 2 subset c these sufficient conditions involving nonconvex sets of variations are shown by an example to be stronger than those in the literature we

necessary optimality conditions for nonconvex differential inclusions with endpoint constraints in this paper we derive necessary optimality conditions for optimization problems defined by non convex differential inclusions with endpoint constraints we do this in terms of parametrizations of the convexified form of the differential inclusion and under additional assumptions in terms of the inclusion itself

periodic doubly degenerate parabolic equation with nonlocal terms this paper is a remark on gao and wang 2007 4 under some weaker conditions than 4 we establish the existence of nontrivial nonnegative periodic solutions for a doubly degenerate parabolic equation with nonlocal terms the key is the use of de giorgi iteration technique and the theory of leray schauder degree

quasi uniformity for the abstract neumann antimaximum principle and applications with a priori estimates abstract in this paper we prove a general result giving the maximum and the antimaximum principles in a unitary way for linear operators of the form l i provided that 0 is an eigenvalue of l with associated constant eigenfunctions to this purpose we introduce a new notion of quasi uniform maximum principle named k uniform maximum principle it holds for belonging to certain neighborhoods of 0 depending on the fixed positive multiplier k 0 which selects the good class of right hand sides our approach is based on a l l p estimate for some related problems as an application we prove some generalization and new results for elliptic problems and for time periodic parabolic problems under neumann boundary conditions

data envelopment analysis in the absence of convexity the first purpose of this study is to present new dea data envelopment analysis approaches that distinguish the efficiency status efficient or inefficient of decision making units dmus and estimate right and left hand returns to scale rts of efficient dmus in free disposal hull fdh models the second one is to estimate most productive scale size mpss in fdh models the proposed rts method has three preferences with respect to the previous rts methods in this context first one is that it is capable of identifying all fdh efficient dmus whereas the previous methods may not be able to specify all fdh efficient dmus besides the second one is to estimate right and left hand rts of fdh efficient dmus while the former methods are only capable of estimating rts of fdh efficient dmus moreover the third one is that it is always able to estimate right and left hand rts correctly whereas the previous methods are sometimes capable of estimating rts correctly these issues are the motivations for creating this current research finally a numerical example and an empirical application are exhibited for illustrating and comparing the proposed method with the former methods then some consequences are provided based upon the obtained results and also some suggestions for future works are presented proofs of some theorems and lemmas are given in the appendix a dea method for identifying fdh efficient dmus two dea models for the estimation of right and left rts of fdh efficient dmus a dea method for determining mpss in fdh models presentation of two examples to illustrate the use of our proposed approach making comparisons among our rts method and other rts methods in fdh models

a proper relaxation of shifted and delayed controls abstract we consider optimal control problems defined by functional integral equations with variable shifts in the controls and with the shifted controls nonadditively coupled as a special case we obtain differential equations with delays and or advances in both the state and the control functions we prove that if the original controls are measurable functions u with values in a compact metric space then the k 1 tuples u u u h 1 u h k of shifted controls have relaxed limits that form a compact and convex set this enables us to prove an existence theorem to approximate any relaxed solution with original solutions and to derive necessary conditions generalizeing pontryagin s maximum principle in integral form even for nonsmooth problems we also discuss the dependence on the shifts hj of the set of original controls u of the set of their relaxed limits and of the set of the corresponding solutions of the functional integral equation our results include the proof that the d model of rosenblueth and vinter is proper

relaxation procedures for time delay systems abstract despite the attention that optimal control problems involving time delays have received over the last two decades little attention has been paid to the important question of how such problems should be relaxed in order to assure existence of minimizers recently warga has proposed a relaxation procedure for fully nonlinear problems with delays in the state and control variables and showed that the resulting relaxed problem has a solution nevertheless the effect of this relaxation may reduce the infimum cost as we show through an example which belongs to a class of problems with commensurate delays i e the quotient of any two delays is rational for problems of this nature we provide a new relaxation procedure for which the extension is proper i e the infimum costs coincide we also present an abstract relaxation procedure applicable to problems with possibly noncommensurate delays

the maximum principle for an optimal solution to a differential inclusion with end points constraints we derive pontryagin s maximum principle for a general optimal control problem using the set valued version of variational equation we achieve this aim by exploiting an adequate differential calculus of set valued maps furthermore the calmness condition is replaced by a surjectivity condition involving reachable sets of the set valued linearization of the initial control problem duality then provides both the adjoint differential inclusion and the maximum principle

global directional controllability this paper derives sufficient conditions for the image of a function varphi 1 q to mathbb r to cover an open segment originating at varphi 1 bar q and with direction w or even a neighborhood of such an interval subject to restrictions of the form varphi 2 q in c or varphi 2 q g subset c here varphi 2 q to mathcal z q is an arbitrary set c a convex subset of the topological vector space mathcal z and g a neighborhood of 0 in mathcal z also derived are similar directional controllability conditions subject to the additional restriction q in mathcal u subset q for an abundant subset mathcal u of q these conditions are applicable to unilateral problems of control theory and of infinite dimensional mathematical programming these results are global and apply to problems defined by functions whose restrictions to certain finite dimensional sets are differentiable but not necessarily c 1 or are locally uniform limits of ifferentiable fu

characteristics and structures of weak efficient surfaces of production possibility sets this paper studies the characteristics and structure of the weak surface of the production possibility set we apply techniques and methods of transferring a polyhedral cone from its intersection form to its sum form identify an intersection representation of the production possibility set we give the structure theorem of weak surface of the production possibility set which includes three complementary slackness conditions we define the input weak efficiency and output weak efficiency for different dea models according to the representation of the intersection form it investigates the characteristics of the weak surfaces and proves the structure theorems of input weak dea efficiency and output weak dea efficiency the structure theorems establish weighted combination of inputs and outputs that are weak dea efficient numerical examples are provided for illustration

global variation of outputs with respect to the variation of inputs in performance analysis generalized rts returns to scale rts is an important topic in performance analysis which helps managers to make decisions about the expansion or contraction of the operation of the dmu under assessment but the rts classification of dmus gives only partial information because it is a local notion in this paper we extend the concept of rts and we seek the precise relation between the proportional variation of outputs and the proportional variation of inputs an approach is provided which is able to determine this relation based on the parametric analysis and perturbation in linear programming

an extension to data envelopment analysis with preference structure for estimating overall inefficiency this paper extends data envelopment analysis dea with preference structure by fully considering the substitution effects among different inputs or outputs when the unit cost and price information on inputs and outputs are available the generalized weighted ccr gwccr models proposed in this paper can provide some scalar values for measuring the overall inefficiency it is found that the gwccr models focus on the relative aspects of overall inefficiency instead of the absolute aspects focused on by the weighted additive dea model

positive periodic solutions for a system of anisotropic parabolic equations abstract we consider a system of two degenerate parabolic equations with nonlocal terms and dirichlet boundary conditions more precisely the degeneracy in each equation of the system is of the type r x laplacian where r x is a function depending on x where is a bounded smooth domain of r n the system models the diffusion and the interaction between two different biological species sharing the same territory the paper provides conditions on the parameters of the problem that guarantee the coexistence of a t periodic non negative solution u v with both non trivial u v

periodic solutions of evolutionp laplacian equations with nonlinear sources abstract in this paper we are concerned with the time periodic solutions to the evolution p laplacian equations of the form formula and formula with the dirichlet boundary value condition where p 2 0 is a bounded convex domain in r n with smooth boundary m x t is continuous on r periodic in t with period and positive in r the existence of nontrivial nonnegative periodic solutions is established provided that n 1 and p 1 p 1 p n