positive periodic solutions of singular systems with a parameter abstract the existence and multiplicity of positive periodic solutions for second order non autonomous singular dynamical systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter our results provide a unified treatment for the problem and significantly improve several results in the literature the proof of our results is based on the krasnoselskii fixed point theorem in a cone

nonlinear periodic boundary value problems with sign changing green s function abstract we study the existence and nonexistence of positive solutions of nonlinear periodic boundary value problems u a t u b t f u a e t 0 t u 0 u t u 0 u t where b 0 a 0 and the green s function g t s of the linear problem u a t u 0 a e t 0 t u 0 u t u 0 u t may change sign on 0 t 0 t

multiple positive solutions to superlinear repulsive singular impulsive differential equations the existence of multiple positive solutions to superlinear repulsive singular impulsive differential equations is discussed in this paper the proof relies on a nonlinear alternative of leray schauder type and on krasnoselskii fixed point theorem on compression and expansion of cone

numerical algorithm for solving two point second order periodic boundary value problems for mixed integro differential equations in this study the numerical solution of fredholm volterra integro differential equations for two point second order periodic boundary value problems is discussed in a reproducing kernel hilbert space a reproducing kernel hilbert space is constructed in which the periodic boundary conditions of the problem are satisfied the exact solution u x is represented in the form of series in the space w 2 3 in the mean time the n term approximate solution u n x is obtained and is proved to converge to the exact solution u x furthermore we present an iterative method for obtaining the solution in the space w 2 3 some examples are displayed to demonstrate the validity and applicability of the proposed method the numerical result indicates that the proposed method is straightforward to implement efficient and accurate for solving linear and nonlinear equations

multiplicity of positive periodic solutions to superlinear repulsive singular equations abstract in this paper we study positive periodic solutions to the repulsive singular perturbations of the hill equations it is proved that such a perturbation problem has at least two positive periodic solutions when the anti maximum principle holds for the hill operator and the perturbation is superlinear at infinity the proof relies on a nonlinear alternative of leray schauder type and on krasnoselskii fixed point theorem on compression and expansion of cones

numerical investigations for systems of second order periodic boundary value problems using reproducing kernel method the reproducing kernel method is a numerical as well as analytical technique for solving a large variety of ordinary and partial differential equations associated to different kind of boundary conditions and usually provides the solutions in term of rapidly convergent series in the appropriate hilbert spaces with components that can be elegantly computed the aim of the present analysis is to implement a relatively recent computational method reproducing kernel hilbert space for obtaining the solutions for systems of second order differential equations with periodic boundary conditions a reproducing kernel space is constructed in which the periodic conditions of the systems are satisfied whilst the smooth kernel functions are used throughout the evolution of the method to obtain the required grid points an efficient construction is given to obtain the approximate solutions for the systems together with an existence proof of the exact solutions is proposed based upon the reproducing kernel theory convergence analysis and error behavior of the presented method are also discussed in this approach computational results of some numerical examples are presented to illustrate the viability simplicity and applicability of the algorithm developed

existence of one signed periodic solutions of some second order differential equations via a krasnoselskii fixed point theorem this paper is devoted to study the existence of periodic solutions of the second order equation x 00 f t x where f is a caratheodory function by combining some new properties of green s function together with krasnoselskii fixed point theorem on compression and expansion of cones as applications we get new existence results for equations with jumping nonlinearities as well as equations with a repulsive or attractive singularity in this latter case our results cover equations with weak singularities and are compared with some recent results by i rachunkova m tvrdyand i vrkoc r 2002 elsevier science usa all rights reserved

multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces abstract the existence of multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces is discussed in this paper our nonlinearity may be singular in its dependent variable and our analysis relies on a nonlinear alternative of leray schauder type and on a fixed point theorem in cones

positive solutions of neumann problems with singularities abstract in this paper we study the existence of positive solutions to second order singular equations with neumann boundary conditions the proof of the main result relies on a nonlinear alternative principle of leray schauder together with a truncation technique

an accelerated cyclic reduction based solvent method for solving quadratic eigenvalue problem of gyroscopic systems abstract the quadratic eigenvalue problem qep 2 m g k x 0 with m t m being positive definite k t k being negative definite and g t g is associated with gyroscopic systems in guo 2004 a cyclic reduction based solvent crs method was proposed to compute all eigenvalues of the above mentioned qep firstly the problem is converted to find a suitable solvent of the quadratic matrix equation qme m x 2 g x k 0 then using a cayley transformation and a proper substitution the qme is transformed into the nonlinear matrix equation nme z a t z 1 a q with a m k g and q 2 m k the problem finally can be solved by applying the cr method to obtain the maximal symmetric positive definite solution of the nme as long as the qep has no eigenvalues on the imaginary axis or for some cases where the qep has eigenvalues on the imaginary axis however when all eigenvalues of the qep are far away from or near the origin the cayley transformation seems not to be the best one and the convergence rate of the crs method proposed in guo 2004 might be further improved in this paper inspired by using a doubling algorithm to solve the qme we use a mobius transformation instead of the cayley transformation to present an accelerated crs acrs method for solving the qep of gyroscopic systems in addition we discuss the selection strategies of optimal parameter for the acrs method numerical results demonstrate the efficiency of our method

periodic solutions of lienard equations with singular forces of repulsive type in this paper we will use the coincidence degree to give an existence result of periodic solutions for the scalar lienard equations with singular forces of repulsive type the main result of this paper clearly describes what balance conditions between the singular force at the singularity and at the infinity should be imposed if the equation has a periodic solution as an application we will improve the results of ding ye and wang for the brillouin electron beam focusing system

positive periodic solutions of singular systems of first order ordinary differential equations the existence and multiplicity of positive periodic solutions for first non autonomous singular systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter the proof of our results is based on the krasnoselskii fixed point theorem in a cone

generalized quasilinearization method for mixed boundary value problems a generalized quasilinearization method for a nonlinear mixed bvp is developed and a sequence of approximate solutions converging monotonically and quadratically to a solution of the given problem is presented

on existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations the existence and multiplicity of positive solutions are established to periodic boundary value problems for singular nonlinear second order ordinary differential equations the arguments are based only upon the positivity of the green s functions and the krasnosel skii fixed point theorem as an example a periodic boundary value problem is also considered which comes from the theory of nonlinear elasticity

type i operators and the approximation of singular two point boundary value problems a class of self adjoint operators associated with second order singular ordinary differential expressions arises naturally when the problem is weakly formulated and integration by parts is performed we call this class type i operators it turns out that this class can be successfully used to tackle numerical approximations of singular two point boundary value problems they can also be approximated by regular differential operators in a straightforward manner without having to bring the delicate structure of singular differential operators to the forefront of the investigation

necessary conditions for constant solutions of coupled riccati equations in nash games abstract the asymptotic behaviour of solutions as t for coupled matrix riccati equations occurring in open loop linear quadratic nash games is studied in this paper a general formula representing all possible solutions is given necessary conditions for constant real solutions are derived and an estimate for the rate of convergence is obtained two examples illustrate these results

existence multiplicity and infinite solvability of positive solutions to a nonlinear fourth order periodic boundary value problem abstract the existence of n and infinitely many positive solutions is proved for the nonlinear fourth order periodic boundary value problem u 4 t u t u t f t u t 0 t 1 u i 0 u i 1 i 0 1 2 3 where n is an arbitrary natural number and 2 2 0 1 2 2 2 2 4 2 1 0 this kind of fourth order boundary value problems usually describes the equilibrium state of an elastic beam with periodic boundary condition the main results show that the problem may have n or infinitely many positive solutions provided the growth rates of nonlinear term f t l are appropriate on some bounded subsets of its domain

a relationship between the periodic and the dirichlet bvps of singular differential equations in this paper a relationship between the periodic and the dirichlet boundary value problems for second order ordinary differential equations with singularities is established this relationship may be useful in explaining the difference between the nonresonance of singular and nonsingular differential equations using this relationship we give in this paper an existence result of positive periodic solutions to singular differential equations when the singular forces satisfy some strong force condition at the singularity 0 and some linear growth condition at infinity

solving singular nonlinear second order periodic boundary value problems in the reproducing kernel space abstract in this paper we present a new method for solving a singular nonlinear second order periodic boundary value problem its analytical solution is represented in the form of series in the reproducing kernel space in the mean time the n term approximation u n x to the analytical solution u x is obtained and is proved to converge to the analytical solution some numerical examples are studied to demonstrate the accuracy of the present method results obtained by the method indicate the method is simple and effective

periodic solutions for second order differential equations with a singular nonlinearity abstract this paper deals with the existence of positive t periodic solutions for the following differential equation x a t x f t x c t where a c l 1 0 t and f car 0 t r r the existence results are obtained by using a fixed point theorem in cones

existence of multiple solutions for some nonlinear boundary value problems abstract by imposing one sided conditions on the nonlinearity where neither regularity nor uniformity is required we prove the existence of either a nonnegative or a nonpositive solution for first and second order ordinary differential equations with periodic neumann or dirichlet boundary conditions positone and nonpositone problems are considered some nonexistence results are also obtained when using generalized ambrosetti prodi type conditions we get the existence of nonnegative and nonpositive solutions for first and second order periodic or neumann boundary value problems our method of proof makes use of topological degree arguments in cones

dependence of the unique solution of a periodic boundary value problem on the parameter abstract in this paper we study the dependence of the unique positive solution of the periodic boundary value problem u 2 u w t u s 0 s 1 and 0 u 0 u 1 u 0 u 1 on the parameter as 0 we show that this solution u t is infinitely differentiable in and has singularity in the order of with 2 1 s as 0 furthermore the graph of u t has fluctuation in t only of the order o 2 this will help us to determine the location of the solution in numerical computations to study the behavior of the solution when is sufficiently close to 0

bifurcation from interval and positive solutions for second order periodic boundary value problems we give a global description of the branches of positive solutions of second order periodic boundary value problemsu q t u lf t u 0 0

existence of nontrivial periodic solutions for a nonlinear second order periodic boundary value problem abstract in this paper we study the existence of nontrivial periodic solutions to the following nonlinear differential equation u t a t u t f t u t t r u 0 u u 0 u where a r r is an periodic continuous function with a t 0 f r r r is continuous may take negative values and can be sign changing without making any nonnegative assumption on nonlinearity by using the first eigenvalue corresponding to the relevant linear operator and the topological degree the existence of nontrivial periodic solutions to the above periodic boundary value problem is established finally three examples are given to demonstrate the validity of our main results

a monotone iterative technique for stationary and time dependent problems in banach spaces an abstract monotone iterative method is developed for operators between partially ordered banach spaces for the nonlinear problem lu nu and the nonlinear time dependent problem u l n u under appropriate assumptions on l and n we obtain maximal and minimal solutions as limits of monotone sequences of solutions of linear problems the results are illustrated by means of concrete examples

existence and quasilinearization methods in hilbert spaces abstract in this paper we discuss some existence results and the application of quasilinearization methods developed so far for differential equations to the solution of the abstract problem l u f u in a hilbert space h under fairly general assumptions on l f and h we show that this problem has a solution that can be obtained as the limit of a quadratically convergent nondecreasing sequence of approximate solutions if the assumptions are strengthened we show that the abstract problem has a solution which is quadratically bracketed between two monotone sequences of approximate solutions of certain related linear equations

global structure of positive solutions for superlinear second order periodic boundary value problems in this paper we consider the nonlinear eigenvalue problems gather u lambda h t f u 0 quad 0 0 for i 1 ldots m 2 with sum i 1 m 2 alpha i eta i 0 for t 0 in 0 1 f in c 0 infty 0 infty and f s 0 for s 0 and f 0 infty where f 0 lim s rightarrow 0 f s s we investigate the global structure of positive solutions by using the nonlinear krein rutman theorem

necessary and sufficient condition for two player stackelberg strategy this technical note revisits the open loop stackelberg strategy for a two player game by introducing a new costate which captures the future information of the control input we present a necessary and sufficient condition for the existence and uniqueness of the two player game the optimal strategy is designed in terms of three decoupled and symmetric riccati equations which improves the existing results greatly on computation

positive solutions of even order system periodic boundary value problems we study a class of even order system boundary value problems with periodic boundary conditions a series of criteria are obtained for the existence of one two any arbitrary number and even a countably infinite number of positive solutions criteria for the nonexistence of positive solutions are also derived as for the second order case our results extend improve and supplement those in the literature for scalar and system boundary value problems several examples are given to demonstrate the applications moreover we obtain conditions for system periodic boundary value problems of a different form to have nontrivial solutions by transforming our main results to such problems

existence multiplicity and dependence on a parameter for a periodic boundary value problem the authors consider the boundary value problem y 2 y g t f y 0 0 t 2 y 0 y 2 y 0 y 2

existence of nontrivial solutions for a nonlinear sturm liouville problem with integral boundary conditions abstract this paper deals with the existence of nontrivial solutions for a nonlinear singular sturm liouville problem with integral boundary conditions

an open loop stackelberg strategy for the linear quadratic mean field stochastic differential game this paper is concerned with the open loop linear quadratic lq stackelberg game of the mean field stochastic systems in finite horizon by means of two generalized differential riccati equations the follower first solves a mean field stochastic lq optimal control problem then the leader turns to solve an optimization problem for a linear mean field forward backward stochastic differential equation by introducing new state and costate variables we present a sufficient condition for the existence and uniqueness of the stackelberg strategy in terms of the solvability of some riccati equations and a convexity condition furthermore it is shown that the open loop stackelberg equilibrium admits a feedback representation involving the new state and its mean finally two examples are given to show the effectiveness of the proposed results

existence of nonnegative and nonpositive solutions for second order periodic boundary value problems we are interested in nonnegative and nonpositive solutions of the boundary value problem u f t u u 0 u 1 u 0 u 1 where f fulfils the caratheodory conditions on 0 1 r we generalize the results reached by m n nkashama j santanilla and l sanchez and present estimates for solutions in addition we apply our existence theorems to periodic boundary value problems for nonlinear duffing equations whose right hand sides have a repulsive or attractive singularity at the origin we extend or generalize existence results by a c lazer and s solimini and other authors moreover we get some multiplicity results and in the case of a repulsive singularity we also admit a weak singularity in constrast to the previous papers on this subject our proofs are based on the method of lower and upper functions and topological degree arguments and the results are tested on examples

discrete time robust hierarchical linear quadratic dynamic games in this paper the theory of robust min max control is extended to hierarchical and multiplayer dynamic games for linear quadratic discrete time systems the proposed game model consists of one leader and many followers while the performance of all players is affected by disturbance the stackelberg nash saddle equilibrium point of the game is derived and a necessary and sufficient condition for the existence and uniqueness of such a solution is obtained in the infinite time horizon it is shown that the solution of the riccati equation is upper bounded under a condition that is called individual controllability by assuming such a condition and using a time varying lyapunov function the input to state stability of the hierarchical dynamic game is achieved considering the optimal feedback strategies of the players and an arbitrary disturbance as the input

infinite horizon linear quadratic stackelberg games for discrete time stochastic systems abstract in this paper we consider infinite horizon linear quadratic stackelberg games for a discrete time stochastic system with multiple decision makers necessary conditions for the existence of the stackelberg strategy set are derived in terms of the solvability of cross coupled stochastic algebraic equations csaes as an important application the hierarchical h constraint control problem for the infinite horizon discrete time stochastic system with multiple channel inputs is solved using the stackelberg game approach computational methods for solving the csaes are also discussed a numerical example is provided to demonstrate the usefulness of the proposed algorithms

positive solutions for singular non linear third order periodic boundary value problems abstract in this paper we are concerned with the problem of the existence of positive solutions for non linear third order periodic boundary value problem u 3 u f t u 0 t 2 u i 0 u i 2 i 0 1 2 here 0 1 3 is a positive constant and our non linearity f t u may be singular at u 0 the proof relies on a non linear alternative of leray schauder type and on krasnoselskii fixed point theorem on compression and expansion of cones

on the existence of periodic solutions for certain differential equations here we are concerned with the problem of the existence of periodic solution for certain second and third order nonlinear differential equations our method here is to consider the problem as an eigenvalue problem and treat it by the topological degree theory in particular we establish the conditions of the existence of periodic solution first for a simpler system which is homotopic to the original system and then generalize the obtained results for the focal system the method employed here is applicable also for a system of nonlinear differential equations just with simple modifications finally we present some specific examples numerically to show that the results are valid and realizable

h sup spl infin control and estimation with preview part ii fixed size are solutions in discrete time h sup spl infin preview control and fixed lag smoothing problems are solved in general discrete time linear systems via a reduction to equivalent open loop differential games to prevent high order riccati equations found in some solutions the state of the hamilton jacobi system resides in a quotient space of an auxiliary extended state space system the dimension of that auxiliary space is equal to the state space dimension of the original system ignoring the delay

on linear quadratic stackelberg games with time preference rates this note deals with linear quadratic stackelberg differential games including time preference rates with an open loop information structure the properties of the characteristic matrix associated with the necessary conditions for a stackelberg strategy are pointed out it is shown that such a matrix exhibits a special symmetry property of its eigenvalues sufficient conditions to guarantee a predefined degree of stability are given based on the distribution of the eigenvalues in the complex plane

multiple closed orbits of fixed energy forn body type problems with gravitational potentials abstract using the equivariant ljusternik schnirelmann theory and the estimates of the upper bound and lower bound of the critical values for the variational functional we obtain results in general concerning multiple geometrically distinct periodic solutions of fixed energy for a class of planarn body type problems

periodic solutions of second order non autonomous singular dynamical systems abstract in this paper we establish two different existence results of positive periodic solutions for second order non autonomous singular dynamical systems the first one is based on a nonlinear alternative principle of leray schauder and the result is applicable to the case of a strong singularity as well as the case of a weak singularity the second one is based on schauder s fixed point theorem and the result sheds some new light on problems with weak singularities and proves that in some situations weak singularities may help create periodic solutions recent results in the literature are generalized and significantly improved

periodic solutions to some problems of n body type we prove the existence of at least one t periodic solution to a dynamical system of the type m i ddot u i sum limits j 1 j ne i n triangledown v ij u i u j text t 1 where the potentials v ij are t periodic in t and singular at the origin u i e r k i 1 n and k 3 we also provide estimates on the h 1 norm of this solution the proofs are based on a variant of the ljusternik schnirelman method the results here generalize to the n body problem some results obtained by bahri rabinowitz on the 3 body problem in 6