convex solutions of polynomial like iterative equations in this paper convex solutions and concave solutions of polynomial like iterative equations are investigated a result for non monotonic solutions is given first and applied then to prove the existence of convex continuous solutions and concave ones furthermore another condition for convex solutions which is weaker in some aspects is also given the uniqueness and stability of those solutions are also discussed

another approach to the existence of value functions of stochastic differential games abstract the existence of value functions for general two player zero sum stochastic differential games has been obtained by fleming and souganidis in this paper we present a new approach to this problem we prove optimality inequalities of dynamic programming for viscosity sub and supersolutions of the associated bellman isaacs equations these inequalities are well known for deterministic differential games but are new for stochastic differential games it then easily follows that value functions are the unique viscosity solutions of the bellman isaacs equations and satisfy the principle of dynamic programming the results presented here are not the same as those of fleming and souganidis because we work with different reference spaces and the independence of value functions of the choice of reference spaces is not clear to us

a boundary element method for solving 2 d and 3 d static gradient elastic problems part i integral formulation abstract a boundary element formulation is developed for the static analysis of two and three dimensional solids and structures characterized by a linear elastic material behavior taking into account microstructural effects the simple gradient elastic theory of aifantis expressed in the framework of mindlin s general theory is used to model this material behaviour a variational statement is established to determine all possible classical and non classical due to gradient terms boundary conditions of the general boundary value problem the gradient elastic fundamental solution for both two and three dimensional cases is explicitly derived and used to construct the boundary integral representation of the solution with the aid of the reciprocal integral identity especially established for the gradient elasticity considered here it is found that for a well posed boundary value problem in addition to a boundary integral representation for the displacement a second boundary integral representation for its normal derivative is also necessary explicit expressions for interior displacements and stresses in integral form are also presented all the kernels in the integral equations are explicitly provided

an algebraic equation for linear operators let t be a linear operator on a vector space v possibly of infinite dimension over a general field k we solve the functional equation p t f where p k x and f an algebraic operator on v are given for nilpotent f we give an explicit linear system which determines the solutions by their similarity classes the method is based on a canonical decomposition theorem

two player zero sum stochastic differential games with regime switching abstract this paper is concerned with the two player zero sum stochastic differential game in a regime switching model with an infinite horizon the state of the system is characterized by a number of diffusions coupled by a continuous time finite state markov chain based on the dynamic programming principle dpp the lower and upper value functions are shown to be the unique viscosity solutions of the associated lower and upper hamilton jacobi bellman isaacs hjbi equations respectively moreover the lower and upper value functions coincide under the isaacs condition which implies that the game admits a value all the proofs in this paper are markedly different from those for the case when there is no regime switching

two person zero sum game in weak formulation and path dependent bellman isaacs equation in this paper we study a two person zero sum stochastic differential game in weak formulation unlike the standard literature which uses strategy type controls the weak formulation allows us to consider the game with control against control we shall prove the existence of game value under natural conditions another main feature of the paper is that we allow for non markovian structure and thus the game value is a random process we characterize the value process as the unique viscosity solution of the corresponding path dependent bellman isaacs equation a notion recently introduced by ekren et al ann probab 42 2014 pp 204 236 and ekren touzi and zhang stochastic process to appear preprint arxiv 1210 0006v2 preprint arxiv 1210 0007v2

analytic solutions of iterative functional equations abstract let r be a given positive number denote by d dr the closed disc in the complex plane c whose center is the origin and radius is r for any subset k of c and any integer m 1 write a d m k f f d m k is a continuous map and f dm is analytic suppose g a d n 1 c and h k a d k c k 2 n in this paper we study the iterative functional equation g z f z f2 h2 z f z fn hn z f z fn 1 z 0 and give some conditions for the equation to have a solution and a unique solution in a d d

asymptotic perron s method and simple markov strategies in stochastic games and control we introduce a modification of perron s method where semisolutions are considered in a carefully defined asymptotic sense with this definition we can show in a rather elementary way that in a zero sum game or a control problem with or without model uncertainty the value function over all strategies coincides with the value function over markov strategies discretized in time therefore there are always discretized markov varepsilon optimal strategies uniform with respect to the bounded initial condition with a minor modification the method produces a value and approximate saddle points for an asymmetric game of feedback strategies versus counterstrategies

maximum principle for nonzero sum stochastic differential game with delays in this technical note we discuss a nonzero sum stochastic differential game with delays not only the state variable but also control variables of players involve delays this kind of games are motivated by some interesting problems arising from economics and finance using anticipated backward stochastic differential equations we establish a necessary condition and a sufficient condition of maximum principle for the delayed game problem to explain theoretical results we apply them to an economic problem

the optimal control related to riemannian manifolds and the viscosity solutions to hamilton jacobi bellman equations abstract in this paper we study the optimal stochastic control problem for stochastic differential equations on riemannian manifolds the cost functional is specified by controlled backward stochastic differential equations in euclidean space under some suitable assumptions we conclude that the value function is the unique viscosity solution to the associated hamilton jacobi bellman equation which is a fully nonlinear parabolic partial differential equation on riemannian manifolds

a closed loop saddle point for zero sum linear quadratic stochastic differential games with mean field type abstract this paper is concerned with a linear quadratic zero sum differential game with mean field type the notions of explicit and implicit strategy laws are proposed based on them a closed loop formulation for saddle points in the mixed strategy law form is established in order to construct the saddle point the classical entire completion of square argument is developed to a four step completion of square argument and a pair of optimality conditions is proposed

stochastic verification theorem of forward backward controlled systems for viscosity solutions abstract in this paper we investigate the controlled systems described by forward backward stochastic differential equations with the control contained in drift diffusion and generator of bsdes a new verification theorem is derived within the framework of viscosity solutions without involving any derivatives of the value functions it is worth pointing out that this theorem has wider applicability than the restrictive classical verification theorems as a relevant problem the optimal stochastic feedback controls for forward backward systems are discussed as well

iterative equations in banach spaces abstract let x be a banach space let k x f x x lipschitz f id sup and let p k x k x f k x by applying the banach contraction principle we prove that if p is sufficiently close in a certain sense to the identity then the equation p f f has a unique solution f as a corollary we obtain results on iterative equations of the types i a i f i x f x or i a i f i x f x with operator coefficients in banach spaces

maximum principle for differential games of forward backward stochastic systems with applications abstract this paper is concerned with a maximum principle for both zero sum and nonzero sum games the most distinguishing feature compared with the existing literature is that the game systems are described by forward backward stochastic differential equations this kind of games is motivated by linear quadratic differential game problems with generalized expectation we give a necessary condition and a sufficient condition in the form of maximum principle for the foregoing games finally an example of a nonzero sum game is worked out to illustrate that the theories may find interesting applications in practice in terms of the maximum principle the explicit form of an equilibrium point is obtained

on dynkin games with incomplete information in this paper we investigate a game of optimal stopping with incomplete information there are two players but only one is informed about the precise structure of the game observing the informed player the uninformed player is given the possibility of guessing the missing information we show that these games have a value which can be characterized as a viscosity solution to a fully nonlinear variational pde furthermore we derive a dual representation of the value function in terms of a minimization procedure this representation allows us under some additional assumptions to determine optimal strategies for the informed player

temporary removal on gradient enriched elasticity theories a reply to comment on on non singular crack fields in helmholtz type enriched elasticity theories and important theoretical aspects this article has been withdrawn at the request of the editors the publisher apologizes for any inconvenience this may cause the full elsevier policy on article withdrawal can be found at https www elsevier com about our business policies article withdrawal

leading coefficient problem for polynomial like iterative equations abstract because of difficulties in applying fixed point theorems most of known results on the polynomial like iterative equation j 1 n j f j x f x were given by assuming 1 0 and the existence of solutions under the most natural assumption n 0 is an interesting problem called leading coefficient problem for this problem locally expansive invertible c 1 solutions are obtained in the expansive case and the non hyperbolic case in w zhang on existence for polynomial like iterative equations results math 45 2004 185 194 and c 0 increasing solutions are constructed in b xu w zhang construction of continuous solutions and stability for the polynomial like iterative equation j math anal appl 325 2007 1160 1170 in this paper we discuss c 1 solutions for more combinations between expansive mappings and contractive ones and combinations between increasing mappings and decreasing ones

one kind of linear quadratic zero sum stochastic differential game with jumps this paper is concerned with one kind of two person zero sum linear quadratic stochastic differential game with jumps a closed loop formulation for the saddle points of the game is established in

transversal homoclinic points of a class of conservative diffeomorphisms abstract we develop a global graph transformation to obtain estimates for certain invariant manifolds of a class of area preserving diffeomorphisms with symmetries the existence of homoclinic or heteroclinic points and the analyticity of the angle between the invariant manifolds at them is proved for the henon and the standard maps a lower bound of this angle is given for a large set of values of the parameters

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

risk sensitive zero sum differential games we consider two player risk sensitive zero sum differential games rszsdgs in our problem setup both the drift term and the diffusion term in the controlled stochastic differential equation are dependent on the state and controls of both players and the objective functional is of the risk sensitive type first a stochastic maximum principle type necessary condition for an open loop saddle point of the rszsdg is established via nonlinear transformations of the adjoint processes of the equivalent risk neutral stochastic zero sum differential game in particular we obtain two variational inequalities namely the pair of saddle point inequalities of the rszsdg next we obtain the hamilton jacobi isaacs partial differential equation for the rszsdg which provides a sufficient condition for a feedback saddle point of the rszsdg using a logarithmic transformation of the associated value function finally we study the extended linear quadratic rszsdg lq rszsdg we show intractability of the extended lq rszsdg with the state and or controls of both players appearing in the diffusion term this unexpected intractability could lead to nonlinear open loop and feedback saddle points even if the problem itself is essentially lq and the isaacs condition holds

stochastic differential games and viscosity solutions of hamilton jacobi bellman isaacs equations in this paper we study zero sum two player stochastic differential games with the help of the theory of backward stochastic differential equations bsdes more precisely we generalize the results of the pioneering work of fleming and souganidis indiana univ math j 38 1989 pp 293 314 by considering cost functionals defined by controlled bsdes and by allowing the admissible control processes to depend on events occurring before the beginning of the game this extension of the class of admissible control processes has the consequence that the cost functionals become random variables however by making use of a girsanov transformation argument which is new in this context we prove that the upper and the lower value functions of the game remain deterministic apart from the fact that this extension of the class of admissible control processes is quite natural and reflects the behavior of the players who always use the maximum of available information its combination with bsde methods in particular that of the notion of stochastic backward semigroups introduced by peng bsde and stochastic optimizations in topics in stochastic analysis science press beijing 1997 allows us then to prove a dynamic programming principle for both the upper and the lower value functions of the game in a straightforward way the upper and the lower value functions are then shown to be the unique viscosity solutions of the upper and the lower hamilton jacobi bellman isaacs equations respectively for this peng s bsde method is extended from the framework of stochastic control theory into that of stochastic differential games

switching games of stochastic differential systems a two player zero sum switching game is formulated for general stochastic differential systems and is studied using a combined dynamic programming and viscosity solution approach the existence of the game value is proved for the proof of the related dynamic programming principle ddp for the lower and upper value functions the measurability problem of the same kind as mentioned in the paper of fleming and souganidis is also encountered and we are able to get around it via a delicate adaptation of their technique moreover the traditional direct method to prove the time continuity of lower and upper value functions also gives rise to a serious measurability problem to get around the new difficulty a subtle dynamic programming argument is developed to obtain the time continuity which in return is used to derive the ddp for random intermediate times from the ddp with deterministic intermediate times

leader follower stochastic differential game with asymmetric information and applications this paper is concerned with a leader follower stochastic differential game with asymmetric information where the information available to the follower is based on some sub algebra of that available to the leader such kind of game problem has wide applications in finance economics and management engineering such as newsvendor problems cooperative advertising and pricing problems stochastic maximum principles and verification theorems with partial information are obtained to represent the stackelberg equilibrium as applications a linear quadratic leader follower stochastic differential game with asymmetric information is studied it is shown that the open loop stackelberg equilibrium admits a state feedback representation if some system of riccati equations is solvable

zero sum stochastic differential games and backward equations abstract in this paper we are concerned with the applications to stochastic zero sum differential games of the results on backward stochastic differential equations using these techniques the existence of a saddle point in the bounded case is obtained if the isaacs condition holds furthermore this technique is also a very simple approach for the existence of an optimal strategy in the case of controlled diffusions

selection of a correlated equilibrium in markov stopping games abstract this paper deals with an extension of the concept of correlated strategies to markov stopping games the nash equilibrium approach to solving nonzero sum stopping games may give multiple solutions an arbitrator can suggest to each player the decision to be applied at each stage based on a joint distribution over the players decisions according to some optimality criterion this is a form of equilibrium selection examples of correlated equilibria in nonzero sum games related to the best choice problem are given several concepts of criteria for selecting a correlated equilibrium are used

solutions of equivariance for a polynomial like iterative equation using fixed point theorems we discuss continuous solutions of equivariance for a polynomial like iterative equation on the real line where is a closed subgroup of the general linear group gl r our main results guarantee the existence of solutions with certain kinds of symmetry we show that under restrictive hypotheses similar results can be proved in a higher dimensional case where the symmetry group is a topologically finitely generated subgroup of the group generated by rotations and dilations in n dimensional euclidean space

on the stability of the linear mapping in banach modules abstract we prove the generalized hyers ulam rassias stability of the linear mapping in banach modules over a unital banach algebra

small divisor problem for an analytic q difference equation abstract similar to the problem of linearization the small divisor problem also arises in the discussion of invertible analytic solutions of a class of q difference equations in this paper we give the existence of such solutions under the brjuno condition and prove that the equation may not have a nontrivial analytic solution when the brjuno condition is violated these results are applied to discussing a nonlinear iterative equation

analytic solutions of a nonlinear iterative equation near neutral fixed points and poles in this paper existence of analytic solutions of a nonlinear iterative equations is studied when given functions are all analytic and when given functions have poles as well as in many previous works we reduce this problem to finding analytic solutions of a functional equation without iteration of the unknown function f for technical reasons in previous works an indeterminate constant related to the eigenvalue of the linearized f at its fixed point o is required to fulfill the diophantine condition that o is an irrationally neutral fixed point of f in this paper the case of rationally neutral fixed points is also discussed where the diophantine condition is not required

approximate solutions of continuous time stochastic games the paper is concerned with a zero sum continuous time stochastic differential game with a dynamics controlled by a markov process and a terminal payoff the value function of the original game is estimated using the value function of a model game the dynamics of the model game differs from the original one the general result applied to differential games yields the approximation of the value function of the differential game by the solution of countable system of odes

a second strain gradient elasticity theory with second velocity gradient inertia part i constitutive equations and quasi static behavior abstract a multi cell homogenization procedure with four geometrically different groups of cell elements respectively for the bulk the boundary surface the edge lines and the corner points of a body is envisioned which is able not only to extract the effective constitutive properties of a material but also to assess the surface effects produced by the boundary surface on the near bulk material applied to an unbounded material in combination with the thermodynamics energy balance principles this procedure leads to an equivalent continuum constitutively characterized by ordinary double and triple generalized stresses and momenta also applying this procedure to a finite body suitably modelled as a simple material cell system in association with the principle of the virtual power pvp for quasi static actions an equivalent structural system is derived featured by a macro scale pvp having the typical format as for a second strain gradient material model due to the surface effects the latter model does work as a combination of two subsystems i e the bulk material behaving as a cauchy continuum and the boundary surface operating as a membrane like boundary layer each subsystem being in local and global equilibrium by its own further the applied ordinary boundary traction splits into two response dependent parts i e the cauchy traction transmitted to the bulk material and the gurtin murdoch traction acting together with all other boundary tractions upon the boundary layer the role of the boundary layer as a two dimensional manifold enclosing a cauchy continuum is elucidated also with the aid of a discrete model a strain gradient elasticity theory is proposed which includes a minimum total potential energy principle featuring the relevant boundary value problem for quasi static loads and its unique solution a simple application is presented two appendices are included one reports the proof of the global equilibrium of the boundary layer the other is concerned with double and triple stresses the paper is complemented by a companion part ii one on dynamics previous findings by the author polizzotto c 2012 a gradient elasticity theory for second grade materials and higher order inertia int j solids struct 49 2121 2137 are improved and extended

decreasing solutions and convex solutions of the polynomial like iterative equation abstract in this paper we prove the existence and uniqueness of decreasing solutions for the polynomial like iterative equation so as to answer problem 2 in j zhang l yang w zhang some advances on functional equations adv math china 24 1995 385 405 or problem 3 in w zhang j a baker continuous solutions of a polynomial like iterative equation with variable coefficients ann polon math 73 2000 29 36 furthermore we completely investigate increasing convex or concave solutions and decreasing convex or concave solutions of this equation so that the results obtained in w zhang k nikodem b xu convex solutions of polynomial like iterative equations j math anal appl 315 2006 29 40 are improved

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

cooperative equilibria in differential games abstract this paper deals with a class of equilibria for differential games in which memory strategies are allowed this class permits the inclusion of a threat in a cooperative strategy the threat can be used by one player as soon as the other player does not control the system according to the agreed upon cooperative control we characterize the set of such equilibria

linear quadratic mean field stochastic zero sum differential games abstract in this paper we consider the two player linear quadratic mean field stochastic differential game lq mf szsdg where the expected values of state and players control variables are included in the corresponding mean field stochastic system and the objective functional we characterize the explicit condition under which the feedback saddle point equilibrium for the lq mf szsdg exists the results are obtained by using the completion of squares method and the ordered interchangeability property of multiple saddle points we show that the corresponding saddle point equilibrium is linear in state which is characterized by the two coupled riccati differential equations crdes we also discuss the solvability conditions of the crdes numerical results are presented to verify the theoretical results of the paper

stochastic recursive optimal control problem of reflected stochastic differential systems in this paper we study one kind of stochastic recursive optimal control problem in which the control system is stochastic differential equations reflected in a domain and the cost functional is de

dislocations in gradient elasticity revisited in this paper we consider dislocations in the framework of first as well as second gradient theory of elasticity using the fourier transform rigorous analytical solutions of the two dimensional bi helmholtz and helmholtz equations are derived in closed form for the displacement elastic distortion plastic distortion and dislocation density of screw and edge dislocations in our framework it was not necessary to use boundary conditions to fix constants of the solutions the discontinuous parts of the displacement and plastic distortion are expressed in terms of two dimensional as well as one dimensional fourier type integrals all other fields can be written in terms of modified bessel functions

the general exact bijective continuous solution of feigenbaum s functional equation solutionsf of feigenbaum s functional equationf f x 1 f x where 0 is a fixed real number account for many of the fascinating properties of the behaviour of successive iterates of one parameter families of nonlinear maps in connection with the phenomenon of intermittency interesting families of exact solutions have recently been found for 0 these solutions can all be derived from continuous bijective solutions which are topologically equivalent to translations in this paper thegeneral exact continuous bijective solution is found for any 0 positive or negative in particular it is shown that forany 0 there are solutions which areinequivalent to translations and it is shown that bijective solutions equivalent to translations exist only when 0 1 these results considerably enlarge the stock of available exact solutions of feigenbaum s equation

a pontryagin s maximum principle for non zero sum differential games of bsdes with applications this technical note is concerned with a maximum principle for a new class of non zero sum stochastic differential games the most distinguishing feature compared with the existing literature is that the game systems are described by backward stochastic differential equations bsdes this kind of games are motivated by some interesting phenomena arising from financial markets and can be used to characterize the players with different levels of utilities we establish a necessary condition and a sufficient condition in the form of maximum principle for open loop equilibrium point of the foregoing games respectively to explain the theoretical results we use them to study a financial problem

construction of continuous solutions and stability for the polynomial like iterative equation abstract most of known results such as existence uniqueness and stability for polynomial like iterative equations were given under the assumption that the coefficient of the first order iteration term does not vanish the existence with a non zero leading coefficient was therefore raised as an open problem it was positively answered for local c 1 solutions later in this paper this problem is answered further by constructing c 0 solutions moreover we discuss the stability of those c 0 solutions which consequently implies a result of the stability for iterative roots

stochastic differential switching game in infinite horizon abstract we study a zero sum stochastic differential switching game in infinite horizon we prove the existence of the value of the game and characterize it as the unique viscosity solution of the associated system of quasi variational inequalities with bilateral obstacles we also obtain a verification theorem which provides an optimal strategy of the game finally some numerical examples with two regimes are given

hyersulam stability of the iterative equation with a general boundary restriction hyersulam stability has played an important role not only in the theory of functional equations but also in a variety of branches of mathematics such as differential equations integral equations and linear operators in the present paper we will discuss the hyersulam stability of the iterative equation with a general boundary restriction by the construction of a uniformly convergent sequence of functions we prove that for every approximate solution of such an equation there exists an exact solution near it

on zero sum stochastic differential games with jump diffusion driven state a viscosity solution framework a zero sum differential game with controlled jump diffusion driven state is considered and studied by a combination of dynamic programming and viscosity solution techniques we prove under certain conditions that the value of the game exists moreover the value function is shown to be the unique viscosity solution of a fully nonlinear integro partial differential equation in addition we formulate and prove a verification theorem for such games within the viscosity solution framework for nonlocal equations

non monotonic solutions and continuously differentiable solutions of conjugacy equations in this paper solutions of conjugacy equation f f x g f x for a strictly decreasing continuous given function f and a continuous given function g maybe non monotonic are constructed by piecewise defining we determine the conditions for piecewise continuously differentiable solutions of conjugacy equations with a strictly decreasing continuously differentiable given function f and a continuously differentiable given function g finally the recursive algorithm is implemented in matlab software and two examples are respectively presented for a non monotonic solution and a continuously differentiable one

analytic solutions of a general nonlinear functional equations near resonance existence of analytic solutions of a general class of nonlinear functional equations is discussed this general class includes some specific functional equations studied recently moreover we can generalize this problem to finding analytic solutions of a general class of iterative equations

regularity properties for general hjb equations a backward stochastic differential equation method in this work we investigate regularity properties of a large class of hamilton jacobi bellman hjb equations with or without obstacles which can be stochastically interpreted in the form of a stochastic control system in which nonlinear cost functional is defined with the help of a backward stochastic differential equation bsde or a reflected bsde more precisely we prove that first the unique viscosity solution v t x of an hjb equation over the time interval 0 t with or without an obstacle and with terminal condition at time t is jointly lipschitz in t x f ort running any compact subinterval of 0 t second for the case that v solves an hjb equation without an obstacle or with an upper obstacle it is shown under appropriate assumptions that v t x i s jointly semiconcave in t x these results extend earlier ones by buckdahn cannarsa and quincampoix nonlinear differential equations appl 17 2010 pp 715 728 our approach embeds their idea of time change into a bsde analysis we also provide an elementary counterexample which shows that in general for the case that v solves an hjb equation with a lower obstacle the semiconcavity doesn t hold true

a weak dynamic programming principle for zero sum stochastic differential games with unbounded controls we analyze a zero sum stochastic differential game between two competing players who can choose unbounded controls the payoffs of the game are defined through backward stochastic differential equations we prove that each player s priority value satisfies a weak dynamic programming principle and thus solves the associated fully nonlinear partial differential equation in the viscosity sense

stochastic target games and dynamic programming via regularized viscosity solutions we study a class of stochastic target games where one player tries to find a strategy such that the state process almost surely reaches a given target no matter which action is chosen by the opponent our main result is a geometric dynamic programming principle which allows us to characterize the value function as the viscosity solution of a non linear partial differential equation because abstract measurable selection arguments cannot be used in this context the main obstacle is the construction of measurable almost optimal strategies we propose a novel approach where smooth supersolutions are used to define almost optimal strategies of markovian type similarly as in verification arguments for classical solutions of hamilton jacobi bellman equations the smooth supersolutions are constructed by an extension of krylov s method of shaken coefficients we apply our results to a problem of option pricing under model uncertainty with different interest rates for borrowing and lending

gradient elastic stress analysis near cylindrical holes in a plane under bi axial tension fields abstract this paper is devoted to a gradient elastic stress analysis of an infinite plate weakened by a cylindrical hole and subjected to two perpendicular and independent uni axial tensions at infinity the problem setting can be considered as an extension and generalization of the well known kirsch problem of the classical elasticity theory which is here extended with respect to the external loadings and generalized with respect to the continuum framework a closed form solution in terms of displacements is derived for the problem within the strain gradient elasticity theory on plane stress strain assumptions the main characters of the total and cauchy stress fields are analyzed near the circumference of the hole for different combinations of bi axial tensions and for different parameter values for the original kirsch problem concerning a uni axially stretched plate the analytical solution fields for stresses and strains are compared to numerical results these results are shown to be in a full agreement with each other and in particular they reveal a set of new qualitative findings about the scale dependence of the stresses and strains provided by the gradient theory not common to the classical theory based on these findings we finally consider the physicalness of the concepts total and cauchy stress appearing in the strain gradient model

relationship between maximum principle and dynamic programming for stochastic differential games of jump diffusions this paper is concerned with the relationship between maximum principle and dynamic programming for zero sum stochastic differential games of jump diffusions under the assumption that the value function is smooth enough relations among the adjoint processes the generalised hamiltonian function and the value function are given a portfolio optimisation problem under model uncertainty in an incomplete financial market is discussed to show the applications of our result

value in mixed strategies for zero sum stochastic differential games without isaacs condition in the present work we consider 2 person zero sum stochastic differential games with a nonlinear pay off functional which is defined through a backward stochastic differential equation our main objective is to study for such a game the problem of the existence of a value without isaacs condition not surprising this requires a suitable concept of mixed strategies which to the authors best knowledge was not known in the context of stochastic differential games for this we consider nonanticipative strategies with a delay defined through a partition pi of the time interval 0 t the underlying stochastic controls for the both players are randomized along pi by a hazard which is independent of the governing brownian motion and knowing the information available at the left time point t j 1 of the subintervals generated by pi the controls of players 1 and 2 are conditionally independent over t j 1 t j it is shown that the associated lower and upper value functions w pi and u pi converge uniformly on compacts to a function v the so called value in mixed strategies as the mesh of pi tends to zero this function v is characterized as the unique viscosity solution of the associated hamilton jacobi bellman isaacs equation

strain gradient solution for eshelby s ellipsoidal inclusion problem eshelby s problem of an ellipsoidal inclusion embedded in an infinite homogeneous isotropic elastic material and prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is analytically solved the solution is based on a simplified strain gradient elasticity theory ssget that contains one material length scale parameter in addition to two classical elastic constants the fourth order eshelby tensor is obtained in analytical expressions for both the regions inside and outside the inclusion in terms of two line integrals and two surface integrals this non classical eshelby tensor consists of a classical part and a gradient part the former involves poisson s ratio only while the latter includes the length scale parameter additionally which enables the newly obtained eshelby tensor to capture the inclusion size effect unlike its counterpart based on classical elasticity the accompanying fifth order eshelby like tensor relates the prescribed eigenstrain gradient to the disturbed strain and has only a gradient part when the strain gradient effect is not considered the new eshelby tensor reduces to the classical eshelby tensor and the eshelby like tensor vanishes in addition the current eshelby tensor for the ellipsoidal inclusion problem includes those for the spherical and cylindrical inclusion problems based on the ssget as two limiting cases the non classical eshelby tensor depends on the position and is non uniform even inside the inclusion which differ from its classical counterpart for homogenization applications the volume average of the new eshelby tensor over the ellipsoidal inclusion is analytically obtained the numerical results quantitatively show that the components of the newly derived eshelby tensor vary with both the position and the inclusion size unlike their classical counterparts when the inclusion size is small it is found that the contribution of the gradient part is significantly large it is also seen that the components of the averaged eshelby tensor change with the inclusion size the smaller the inclusion the smaller the components moreover these components are observed to approach the values of their classical counterparts from below when the inclusion size becomes sufficiently large

on the dynamic programming principle for uniformly nondegenerate stochastic differential games in domains and the isaacs equations we prove the dynamic programming principle for uniformly nondegenerate stochastic differential games in the framework of time homogeneous diffusion processes considered up to the first exit time from a domain in contrast with previous results established for constant stopping times we allow arbitrary stopping times and randomized ones as well there is no assumption about solvability of the the isaacs equation in any sense classical or viscosity the zeroth order coefficient and the free term are only assumed to be measurable in the space variable we also prove that value functions are uniquely determined by the functions defining the corresponding isaacs equations and thus stochastic games with the same isaacs equation have the same value functions

linear exponential quadratic stochastic differential games in this technical note a two person noncooperative stochastic differential game described by a linear stochastic equation with a brownian motion and a payoff that is the exponential of a quadratic functional in the state and the control strategies of the two players is explicitly solved the optimal strategies form a nash equilibrium the value of the game is also determined explicitly the solution of this problem is naturally related to a linear quadratic stochastic differential game and a linear exponential quadratic control problem

on a theory of nonlocal elasticity of bi helmholtz type and some applications abstract a theory of nonlocal elasticity of bi helmholtz type is studied we employ eringen s model of nonlocal elasticity with bi helmholtz type kernels to study dispersion relations screw and edge dislocations the nonlocal kernels are derived analytically as green functions of partial differential equations of fourth order this continuum model of nonlocal elasticity involves two material length scales which may be derived from atomistics the new nonlocal kernels are nonsingular in one two and three dimensions furthermore the nonlocal elasticity of bi helmholtz type improves the one of helmholtz type by predicting a dispersion relationship with zero group velocity at the end of the first brillouin zone new solutions for the stresses and strain energy of screw and edge dislocations are found

strain gradient solution for a finite domain eshelby type anti plane strain inclusion problem abstract a solution for the finite domain eshelby type inclusion problem of a finite elastic body containing an anti plane strain inclusion of arbitrary cross sectional shape prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory ssget the formulation is facilitated by an extended betti s reciprocal theorem and an extended somigliana s identity based on the ssget and suitable for anti plane strain problems the disturbed displacement field is obtained in terms of the ssget based green s function for an infinite anti plane strain elastic body the solution reduces to that of the infinite domain anti plane strain inclusion problem when the boundary effect is not considered the problem of a circular cylindrical inclusion embedded concentrically in a finite cylindrical elastic matrix undergoing anti plane strain deformations is analytically solved by applying the general solution with the eshelby tensor and its average over the circular cross section of the inclusion obtained in closed forms this eshelby tensor being dependent on the position inclusion size matrix size and a material length scale parameter captures the inclusion size and boundary effects unlike existing ones it reduces to the classical linear elasticity based eshelby tensor for the circular cylindrical inclusion in an infinite matrix if both the strain gradient and boundary effects are suppressed numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is small and that the boundary effect can dominate when the inclusion volume fraction is high however the inclusion size effect is diminishing with the increase of the inclusion size and the boundary effect is vanishing as the inclusion volume fraction becomes sufficiently low

dislocation based fracture mechanics within nonlocal and gradient elasticity of bi helmholtz type part i antiplane analysis abstract this paper is the sequel of a companion part i paper devoted to dislocation based antiplane fracture mechanics within nonlocal and gradient elasticity of bi helmholtz type in the present paper the inplane analysis is carried out to study cracks of modes i and ii generalized continua including nonlocal elasticity of bi helmholtz type and gradient elasticity of bi helmholtz type second strain gradient elasticity offer nonsingular frameworks for the discrete dislocations consequently the dislocation based fracture mechanics within these frameworks is expected to result in a regularized fracture theory by distributing the climb and glide edge dislocations modes i and ii cracks are modeled distinctive features are captured for crack solutions within second grade theories nonlocal and gradient elasticity of bi helmholtz type comparing with solutions within first grade theories nonlocal and gradient elasticity of helmholtz type as well as classical elasticity other than the total stress tensor all of the field quantities are regularized within second grade theories while first grade theories give singular double stress and dislocation density and classical elasticity leads to singularity in the stress field and dislocation density similar to gradient elasticity of helmholtz type first strain gradient elasticity crack tip plasticity is captured in gradient elasticity of bi helmholtz type without any assumption of the cohesive zone

a probabilistic representation for the value of zero sum differential games with incomplete information on both sides we prove that for a class of zero sum differential games with incomplete information on both sides the value admits a probabilistic representation as the value of a zero sum stochastic differential game with complete information where both players control a continuous martingale a similar representation as a control problem over discontinuous martingales was known for games with incomplete information on one side see cardaliaguet rainer 8 and our result is a continuous time analog of the so called splitting game introduced in laraki 20 and sorin 27 in order to analyze discrete time models it was proved by cardaliaguet 4 5 that the value of the games we consider is the unique solution of some hamilton jacobi equation with convexity constraints our result provides therefore a new probabilistic representation for solutions of hamilton jacobi equations with convexity constraints as values of stochastic differential games with unbounded control spaces and unbounded volatility

brief paper soft constrained stochastic nash games for weakly coupled large scale systems in this paper we discuss infinite horizon soft constrained stochastic nash games involving state dependent noise in weakly coupled large scale systems first we formulate linear quadratic differential games in which robustness is attained against model uncertainty it is noteworthy that this is the first time conditions for the existence of robust equilibria have been derived based on the solutions of sets of cross coupled stochastic algebraic riccati equations csares after establishing an asymptotic structure with positive definiteness for csares solutions we derive a recursive algorithm by means of newton s method so that it can be used to obtain solutions for csares as another important feature we propose a high order approximate nash strategy based on iterative solutions finally we provide a numerical example to verify the efficiency of the proposed algorithms

stability of functional equations in single variable abstract this paper discusses hyers ulam stability for functional equations in single variable including the forms of linear functional equation nonlinear functional equation and iterative equation surveying many known and related results we clarify the relations between hyers ulam stability and other senses of stability such as iterative stability continuous dependence and robust stability which are used for functional equations applying results of nonlinear functional equations we give the hyers ulam stability of bottcher s equation we also prove a general result of hyers ulam stability for iterative equations

the risk sensitive maximum principle for controlled forward backward stochastic differential equations abstract in this paper we consider the risk sensitive optimal control problem for forward backward stochastic differential equations fbsdes we consider two different cases case 1 the drift and diffusion terms of the forward part are independent of the backward part not fully coupled fbsde and the running cost does not include the backward part case 2 the fully coupled fbsde with terminal and initial costs in both cases the risk sensitive objective functional is considered the corresponding control domain is not necessarily convex and the diffusion term of the forward part depends on control and state variables in both cases by using the nonlinear transformations of the equivalent risk neutral problems we obtain two different risk sensitive maximum principles which are characterized in terms of the variational inequalities we show that the risk sensitive maximum principle of case 1 consists of the two coupled adjoint equations with an additional scalar adjoint equation which are backward stochastic differential equations bsdes the risk sensitive maximum principle of case 2 consists of the two coupled adjoint equations which are coupled fbsdes with an additional scalar adjoint equation and optimality conditions in both cases the additional scalar adjoint equations are bsdes which are induced due to the nonlinear transformations of the adjoint processes in the equivalent risk neutral problems through the application of results we consider the linear quadratic risk sensitive optimal control problem for the linear fbsde and by using the risk sensitive maximum principle of case 1 we characterize an explicit optimal solution

exponentially growing solutions of parabolic isaacs equations abstract this paper contributes to the literature on unbounded viscosity solutions of fully nonlinear and possibly degenerate parabolic equations under natural assumptions it is established that the initial value problem for a degenerate parabolic isaacs equation set in 0 t r n has a unique continuous viscosity solution with at most exponential growth at infinity

near maximum principle for general recursive utility optimal control problem in this study we introduce a general recursive utility optimal control problem driven by a fully nonlinear stochastic differential system in order to investigate pontryagin s stochastic maximum p

the fundamentals of non singular dislocations in the theory of gradient elasticity dislocation loops and straight dislocations abstract the fundamental problem of non singular dislocations in the framework of the theory of gradient elasticity is presented in this work gradient elasticity of helmholtz type and bi helmholtz type are used a general theory of non singular dislocations is developed for linearly elastic infinitely extended homogeneous and isotropic media dislocation loops and straight dislocations are investigated using the theory of gradient elasticity the non singular fields which are produced by arbitrary dislocation loops are given modified mura peach koehler and burgers formulae are presented in the framework of gradient elasticity theory these formulae are given in terms of an elementary function which regularizes the classical expressions obtained from the green tensor of the helmholtz navier equation and bi helmholtz navier equation using the mathematical method of green s functions and the fourier transform exact analytical and non singular solutions were found the obtained dislocation fields are non singular due to the regularization of the classical singular fields

bsdes with weak terminal condition we introduce a new class of backward stochastic differential equations in which the t terminal value y t of the solution y z is not fixed as a random variable but only satisfies a weak constraint of the form e psi y t ge m for some possibly random non decreasing map psi and some threshold m we name them textit bsdes with weak terminal condition and obtain a representation of the minimal time t values y t such that y z is a supersolution of the bsde with weak terminal condition it provides a non markovian bsde formulation of the pde characterization obtained for markovian stochastic target problems under controlled loss in bouchard elie and touzi we then study the main properties of this minimal value in particular we analyze its continuity and convexity with respect to the m parameter appearing in the weak terminal condition and show how it can be related to a dual optimal control problem in meyer form these last properties generalize to a non markovian framework previous results on quantile hedging and hedging under loss constraints obtained in follmer and leukert and in bouchard elie and touzi finally we observe a surprisingly strong connection between bsdes with weak terminal condition and 2nd order bsdes in the quasi linear case

dynamics of a quadratic map in two complex variables abstract we study the map c 2 c 2 defined by w z z z w2 and the associated collection of sequences zj satisfying the recurrence zj 2 zj 1 z2j iteration of is equivalent to the study of such sequences we analyze growth rates of the sequences positive sequences periodic and asymptotically periodic sequences and establish the existence of doubly infinite homoclinic sequences non zero sequences whose limits as j are 0 we investigate some associated functional equations f x 2 f x 1 f x 2 and l l x l x x2