critical point theory for nonsmooth functionals abstract in this paper we develop critical point theory for nonsmooth functional f h 0 1 r defined by f u 1 2 i j 1 a i j x u d i u d j u d x g x u d x the corresponding deformation lemmas are proved with the application to the bifurcation for quasilinear schrodinger equations in mind we extend the obtained results to that for a functional defined on a product space and prove a generalized saddle point theorem

uniform stability of switched linear systems extensions of lasalle s invariance principle this paper addresses the uniform stability of switched linear systems where uniformity refers to the convergence rate of the multiple solutions that one obtains as the switching signal ranges over a given set we provide a collection of results that can be viewed as extensions of lasalle s invariance principle to certain classes of switched linear systems using these results one can deduce asymptotic stability using multiple lyapunov functions whose lie derivatives are only negative semidefinite depending on the regularity assumptions placed on the switching signals one may be able to conclude just asymptotic stability or uniform exponential stability we show by counter example that the results obtained are tight

nonlinear systems of the burgers type equations lie and q conditional symmetries ansatze and solutions abstract a class of nonlinear diffusion convection systems containing two burgers type equations is considered a complete description of lie symmetries is obtained for these systems new results of finding q conditional symmetries are also presented moreover a variety of lie and non lie ansatze and exact solutions of a diffusion convection system invariant under the generalized galilei algebra are constructed

the monotone iterative technique for a parabolic boundary value problem with discontinuous nonlinearity let be a bounded lebesgue measurable domain of r n with a lipschitz boundary the following initial boundary value problem is considered u t lu f u h in q bu o on u 0 where q o t o t and 0 0 l is a uniformly elliptic operator b a boundary operator defined by bu u v bu in which v denotes the outer normal derivative on and f represents a discontinuous non linearity

ℋ stabilisation of switched linear stochastic systems under dwell time constraints based on the determination of a minimum dwell time this article addresses the problem of characterising a switching strategy for stabilisation of switched linear stochastic systems with adapted external inputs sufficient conditions that assure exponential mean square stability and an performance index are established by analysing the time evolution of the second order moment of the state and a recursive dynamic programming inequality respectively alternative conditions are derived for numerical implementations the proposed method is illustrated by numerical simulations

infinitely many solutions for a differential inclusion problem in r n abstract in this paper we consider the differential inclusion problem di p u u p 2 u x f u x x r n u w 1 p r n where 2 n p l 1 r n l r n is radially symmetric and f stands for the generalized gradient of a locally lipschitz function f r r under suitable oscillatory assumptions on the potential f at zero or at infinity we show the existence of infinitely many radially symmetric solutions of di no symmetry requirement on f is needed our approach is based on a non smooth ricceri type variational principle developed by marano and motreanu j differential equations 182 2002 108 120

a new class of multilevel decomposition algorithms for non monotone problems based on the quasidifferentiability concept abstract a convex multilevel decomposition approach is proposed for the solution of static analysis problems involving non monotone possibly multivalued laws the theory is developed here for model problems of structures having non monotone interface or boundary conditions first we decompose appropriately the non monotone laws writing them as a difference of monotone constituents in the general case this is related to the quasidifferentiability concept this permits us to obtain a system of convex variational inequalities the solution s of which describe the position s of static equilibrium of the considered structure then the problems are formulated as min min problems for appropriately defined lagrangian functions convex optimization algorithms of various complexity are used in a multilevel scheme for the numerical solution of the considered structural analysis problem numerical results concerning the calculation of an elastic contact problem and an elastic stamp problem illustrate the theory

nonlinear hemivariational inequalities at resonance in this paper we examine a semilinear hemivariational inequality at resonance in the first eigenvalue 1 of h01 z we prove two existence theorems for such problems our approach is variational and is based on the nonsmooth critical point theory of chang which uses the subdifferential calculus of clarke for locally lipschitz functions

conservation laws and symmetries of radial generalized nonlinear p laplacian evolution equations abstract a class of generalized nonlinear p laplacian evolution equations is studied these equations model radial diffusion reaction processes in n 1 dimensions where the diffusivity depends on the gradient of the flow for this class all local conservation laws of low order and all lie symmetries are derived the physical meaning of the conservation laws is discussed and one of the conservation laws is used to show that the nonlinear equation can be mapped invertibly into a linear equation by a hodograph transformation in certain cases the symmetries are used to derive exact group invariant solutions from solvable three dimensional subgroups of the full symmetry group which yields a direct reduction of the nonlinear equation to a quadrature the physical and analytical properties of these exact solutions are explored some of which describe moving interfaces and green s functions

existence results for obstacle problems for nonlinear hemivariational inequality at resonance in this paper we are interested in studying the existence of solutions to obstacle problems for nonlinear hemivariational inequality at resonance driven by the p laplacian using a variational approach based on the nonsmooth critical point theory for nondifferentiable functionals we prove two existence theorems

exact solutions via equivalence transformations of variable coefficient fifth order kdv equations abstract in this paper a family of variable coefficient fifth order kdv equations has been considered by using an infinitesimal method based on the determination of the equivalence group differential invariants and invariant equations are obtained invariants provide an alternative way to find equations from the family which may be equivalent to a specific subclass of the same family and the invertible transformation which maps both equivalent equations here differential invariants are applied to obtain exact solutions

lie symmetries of nonlinear multidimensional reaction diffusion systems ii we present a complete description of the classical lie symmetries of a coupled system of partial differential equations comprising a pair of semilinear reaction diffusion equations with constant diffusivities and arbitrary nonlinearities in the reaction terms in any number of spatial dimensions part i cherniha r m and king j r 2000 j phys a math gen 33 267 82 j phys a math gen 33 7839 41 addressed the case of unequal diffusivities here we complete the analysis by treating the case of equal diffusivities in which the symmetry structure is richer still such models arise in the description of numerous physical chemical and biological systems and we also indicate the possible application in such contexts of some of the specific cases arising from the group classification specifically a variety of lie s ansatze and exact solutions of the so called reaction diffusion systems of a type that arises in mathematical biology are constructed

three dimensional adhesive contact laws with debonding a nonconvex energy bundle method abstract in the present paper we develop a new numerical approach to the problem of structures having two or more parts of them adhesively bonded together like e g in sandwich structures the adhesive material is idealized by a nonmonotone possibly multivalued stress strain law which is three dimensional 3d and introduces a nonconvex nonsmooth energy function in the problem the problem is formulated as a hemivariational inequality whose solution s must render the potential energy substationary we apply here the proximal bundle method and more specifically the optimization programme nsolib based on first order polyhedral approximations of the locally lipschitz continuous objective function this algorithm permits the determination of at least one substasionarity point e g of an equilibrium problem an example of a 3d finite element model illustrates the effectiveness of the proposed mehod

optimal control of discrete time bilinear systems with applications to switched linear stochastic systems abstract this paper aims at characterizing the most destabilizing switching law for discrete time switched systems governed by a set of bounded linear operators the switched system is embedded in a special class of discrete time bilinear control systems this allows us to apply the variational approach to the bilinear control system associated with a mayer type optimal control problem and a second order necessary optimality condition is derived optimal equivalence between the bilinear system and the switched system is analyzed which shows that any optimal control law can be equivalently expressed as a switching law this specific switching law is most unstable for the switched system and thus can be used to determine stability under arbitrary switching based on the second order moment of the state the proposed approach is applied to analyze uniform mean square stability of discrete time switched linear stochastic systems numerical simulations are presented to verify the usefulness of the theoretic results

elliptic problems with lack of compactness via a new fixed point theorem abstract this paper provides a new fixed point theorem for increasing self mappings g b b of a closed ball b x where x is a banach semilattice which is reflexive or has a weakly fully regular order cone x by means of this fixed point theorem we are able to establish existence results of elliptic problems with lack of compactness

exact polynomial solutions to some nonlinear diffusion equations abstract a class of non group invariant exact solutions to a number of nonlinear diffusion equations is constructed expressed in the appropriate variables these solutions are polynomials in the spatial variable the method is mainly applicable to equations which may be written in an appropriate quadratically nonlinear form a number of equivalence transformations between various nonlinear diffusion equations are also derived

closed geodesics with lipschitz obstacle abstract geodesics on a general subsetmof r nare considered whenmis the closure of an open subset with lipschitz boundary and suitable topological conditions are fulfilled the existence of a nonconstant closed geodesic is proved

nonlocal ansatze reduction and some exact solutions for the system of the van der waals equations abstract in this paper two types of nonlocal equivalence transformations of the system of convection diffusion equations are found using these transformations we obtained two images of the van der waals system lie symmetries of these images are used to construct nonlocal ansatze and reduce that system owing to this it has become possible to have built classes of exact solutions of the van der waals system lie symmetry of this system is used to construct nonlocal ansatze and reduce its both images

group classification of a class of generalized nonlinear kolmogorov equations and exact solutions a class of generalized nonlinear kolmogorov equations is investigated we present the group classification of lie symmetries of the class with respect to the group of equivalence transformations we find a number of exact solutions of nonlinear kolmogorov equations which have the maximal symmetry properties a class of generalized nonlinear kolmogorov equations is investigated we present the group classification of lie symmetries of the class with respect to the group of equivalence transformations we find a number of exact solutions of nonlinear kolmogorov equations which have the maximal symmetry properties

group classification of a generalization of the heath equation abstract the complete group classification of a generalization of the heath model is carried out by connecting it to the heat equation with nonlinear source examples of invariant solutions are given under the terminal and the barrier option condition

coercive and semicoercive hemivariational inequalities variational inequalities have a concrete mechanical meaning they are expressions of the principle of virtual work or power in its inequality form in the case of lack of monotonicity of the underlying stress strain or reaction displacement conditions or equivalently if the corresponding energy functionals are nonconvex the variational expressions are no longer variational inequalities another type of inequality expression arises as variational formulation of the problem these expressions have been introduced in mechanics by the author and are called hemivariational inequalities the aim of this paper is the formulation of an abstract mathematical theory for coercive and semicoercive hemivariational inequalities

lie symmetries and form preserving transformations of reaction diffusion convection equations abstract new lie symmetry classification of the known class of reaction diffusion convection equations is presented the classification method is based on combining the standard group classification method and the form preserving transformation approach

root mean square gains of switched linear systems in this note we compute the root mean square rms gain of a switched linear system when the interval between consecutive switchings is large the algorithm proposed is based on the fact that a given constant spl gamma provides an upper bound on the rms gain whenever there is a separation between the stabilizing and the antistabilizing solutions to a set of spl gamma dependent algebraic riccati equations the motivation for this problem is the application of robust stability tools to the analysis of hybrid systems

global actions of lie symmetries for the nonlinear heat equation by restricting to a natural class of functions we show that the lie point symmetries of the nonlinear heat equation exponentiate to a global action of the corresponding lie group remarkably in most of the cases the action turns out to be linear

enhanced group analysis and conservation laws of variable coefficient reaction diffusion equations with power nonlinearities a class of variable coefficient 1 1 dimensional nonlinear reaction diffusion equations of the general form f x ut g x unux x h x um is investigated different kinds of equivalence groups are constructed including ones with transformations which are nonlocal with respect to arbitrary elements for the class under consideration the complete group classification is performed with respect to convenient equivalence groups generalized extended and conditional ones and with respect to the set of all local transformations usage of different equivalences and coefficient gauges plays the major role for simple and clear formulation of the final results the corresponding set of admissible transformations is described exhaustively then using the most direct method we classify local conservation laws some exact solutions are constructed by the classical lie method

on the existence of small energy solutions for a sublinear neumann problem abstract in this paper we are concerned with the sublinear problem 0 1 u u p 2 u in u 0 on where r n is a bounded domain and 1 p 2 for p 1 the nonlinearity u p 2 u will be identified by sgn u in contrast to previous work on the dirichlet problem some difficulties arise due to the fact that the associated energy functional is not bounded from below complementing recent work by parini and weth in 15 on least energy solutions we prove that 0 1 has infinitely many solutions with small negative energy

boundary hemivariational inequality problems with doubly nonlinear operators in this paper we investigate a class of nonlinear boundary hemivariational inequality problems under suitable hypotheses the existence of solutions is established via rewriting these problems into a class of evolution inclusions and using the discretization method and the theory of pseudomonotone operators moreover the continuous dependence result of the solutions to the initial data is given

stability and stabilization of discrete time switched systems this paper addresses two strategies for stabilization of discrete time linear switched systems the first one is of open loop nature trajectory independent and is based on the determination of an upper bound of the minimum dwell time by means of a family of quadratic lyapunov functions the relevant point on dwell time calculation is that the proposed stability condition does not require the lyapunov function be uniformly decreasing at every switching time the second one is of closed loop nature trajectory dependent and is designed from the solution of what we call lyapunov metzler inequalities from which the stability condition is expressed being non convex a more conservative but simpler to solve version of the lyapunov metzler inequalities is provided the theoretical results are illustrated by means of examples

optimal control of switched systems based on parameterization of the switching instants this paper presents a new approach for solving optimal control problems for switched systems we focus on problems in which a prespecified sequence of active subsystems is given for such problems we need to seek both the optimal switching instants and the optimal continuous inputs in order to search for the optimal switching instants the derivatives of the optimal cost with respect to the switching instants need to be known the most important contribution of the paper is a method which first transcribes an optimal control problem into an equivalent problem parameterized by the switching instants and then obtains the values of the derivatives based on the solution of a two point boundary value differential algebraic equation formed by the state costate stationarity equations the boundary and continuity conditions along with their differentiations this method is applied to general switched linear quadratic problems and an efficient method based on the solution of an initial value ordinary differential equation is developed an extension of the method is also applied to problems with internally forced switching examples are shown to illustrate the results in the paper

brief paper markov jump linear systems with switching transition rates mean square stability with dwell time the stability of a class of markov jump linear systems characterized by piecewise constant transition rates and system dynamics is investigated for these switching markov jump linear systems mean square stability is analyzed through the time evolution of the second order moment of the state the main result is a sufficient condition that guarantees mean square stability under constraints on the dwell time between switching instants an alternative condition based on kronecker calculus is worked out it is shown that both the stability criteria admit an lmi implementation

existence of solutions for elliptic hemivariational inequalities abstract this paper is devoted to the existence of solutions concerning the dirichlet problem for quasilinear elliptic hemivariational inequalities at the first eigenvalue using the notion of the generalized gradient of clarke and the property of the first eigenfunction some existence results of solutions have been proved

on the approximation of nonmonotone multivalued problems by monotone subproblems abstract structures involving zig zag type reaction displacement or stress strain laws i e nonmonotone possibly multivalued laws cannot be effectively treated by the classical numerical methods for nonlinear laws in order to calculate the arising free boundaries accurately we propose a new approximation method of the nonmonotone problem by monotone ones the proposed method finds its justification in the approximation of a hemivariational inequality by a sequence of variational inequalities also the case of combining this procedure with fixed point algorithms to treat more complicated problems is considered the numerical methods proposed are illustrated mainly by numerical examples which are described in detail

variational methods for non differentiable functionals and their applications to partial differential equations the interest in extending the theory of nonlinear partial differential equations pde to pde with discontinuous nonlinearities dnde has been increasing daily the reasons are twofold 1 many free boundary problems may be reduced to boundary value problems of dnde cf chang 16 71 chang and jiang 9 and fleishman and mahar 121 2 sometimes in dealing with a pde problem it is beneficial to put the original pde into a large category for instance dnde cf chang 7 simply speaking a dnde is as follows

existence for minimization with nonconvex constraints abstract in this paper we present an extension of the abstract existence result by v barbu and t i seidman existence for minimization in banach spaces with some applications j math anal appl to appear to the case of a minimization problem with nonconvex constraints also one obtains a geometric version extending the palais smale existence theorem an application of the main result deals with a distributed parameter optimal control problem

local c1 ω minimizers versus local w1 p ω minimizers of nonsmooth functionals abstract we study not necessarily differentiable functionals of the form j u 1 p u p d x 1 p u p d x j 1 x u d x j 2 x u d with 1 p involving locally lipschitz functions j 1 r r as well as j 2 r r we prove that local c 1 minimizers of j must be local w 1 p minimizers of j

dynamic contact problem for viscoelastic piezoelectric materials with normal damped response and friction in this paper we deal with a class of inequality problems for dynamic frictional contact between a piezoelectric body and a foundation the model consists of a system of the hemivariational inequality of hyperbolic type for the displacement the time dependent elliptic equation for the electric potential the contact is modeled by a general normal damped response condition and a friction law which are nonmonotone possibly multivalued and have the subdifferential form the existence of a weak solution to the model is proved by embedding the problem into a class of second order evolution inclusions and by applying a surjectivity result for multivalued operators

on the eigenvalue problem for hemivariational inequalities existence and multiplicity of solutions in this article the eigenvalue problem for hemivariational inequalities is studied first some existence results are obtained by applying a critical point method suitable for nonconvex nonsmooth energy functions further some results concerning the multiplicity of solutions are proved applications from mechanics illustrate the theory

existence for minimization in banach space with some applications abstract existence of a minimum is shown for a coercive isc differentiable function restricted to a finite intersection of half spaces under a structural assumption on the g teaux derivative this abstract existence result is illustrated by two examples from optimal control theory for distributed systems

a unified approach for stochastic and mean square stability of continuous time linear systems with markovian jumping parameters and additive disturbances necessary and sufficient conditions for stochastic stability ss and mean square stability mss of continuous time linear systems subject to markovian jumps in the parameters and additive disturbances are established we consider two scenarios regarding the additive disturbances one in which the system is driven by a wiener process and one characterized by functions in l 2 m omega cal f mathbb p which is the usual scenario for the h infty approach the markov process is assumed to take values in an infinite countable set mathcal s it is shown that ss is equivalent to the spectrum of an augmented matrix lying in the open left half plane to the existence of a solution for a certain lyapunov equation and implies is equivalent for mathcal s finite asymptotic wide sense stationarity awss it is also shown that ss is equivalent to the state x t belonging to l 2 n omega cal f mathbb p whenever the disturbances are in l 2 m omega cal f mathbb p for the case in which mathcal s is finite ss and mss are equivalent and the lyapunov equation can be written down in two equivalent forms with each one providing an easier to check sufficient condition

on stabilizability and exact observability of stochastic systems with their applications this paper mainly studies the stabilizability and exact observability of stochastic linear controlled systems and their applications with the aid of the operator spectrum a necessary and sufficient condition is given for the stabilizability of stochastic systems some new concepts such as unremovable spectrum and strong solution are introduced an unremovable spectral theorem and a stochastic popov belevith hautus criterion for exact observability are presented as applications a comparison theorem for stochastic algebraic riccati equations and a result on lyapunov type equations are obtained

a converse lyapunov theorem for linear parameter varying and linear switching systems we study families of linear time varying systems where time variations have to satisfy restrictions on the dwell time that is on the minimum distance between discontinuities as well as on the derivative in between discontinuities such classes of systems may be formulated as linear flows on vector bundles the main objective of this paper is to construct parameter dependent lyapunov functions which characterize the exponential growth rate this is possible in the generic irreducible case as an application the gelfand formula is generalized to the class of systems studied here in other words the maximal exponential growth rate may be approximated by only considering the periodic systems in the family of time varying systems a perspective on the question of continuous dependence of the exponential growth rate on the data is given

existence of solutions for wave type hemivariational inequalities with noncoercive viscosity damping abstract in this paper we prove the existence of solutions for a hyperbolic hemivariational inequality of the form u au bu j u f where b is a linear elliptic operator and a is linear and nonnegative not necessarily coercive

periodic problems with asymmetric nonlinearities and nonsmooth potentials abstract we consider a nonlinear periodic boundary value problem driven by the ordinary scalar p laplacian and with a nonsmooth locally lipschitz potential which exhibits an asymmetric behaviour at and at using the fucik spectrum and a variational approach based on the nonsmooth critical point theory we obtain a solution for the problem using a general minimax principle concerning linking sets

equivalence and finite time blow up of solutions and interfaces for two nonlinear diffusion equations abstract in this work we construct a transformation between the solutions to the following reaction convection diffusion equation t u u m x x a x u m x b x u m posed for x r t 0 and m 1 where a b are two continuous real functions and the solutions to the nonhomogeneous diffusion equation of porous medium type f y m y y posed in the half line y 0 with 0 m 1 and suitable density functions f y we apply this correspondence to the case of constant coefficients a x 1 and b x k 0 for this case we prove that compactly supported solutions to the first equation blow up in finite time together with their interfaces as x we then establish the large time behavior of solutions to a homogeneous dirichlet problem associated to the first equation on a bounded interval we also prove a finite time blow up of the interfaces for compactly supported solutions to the second equation when f y y with 2

stability and stabilization of continuous time switched linear systems this paper addresses two strategies for the stabilization of continuous time switched linear systems the first one is of open loop nature trajectory independent and is based on the determination of a minimum dwell time by means of a family of quadratic lyapunov functions the relevant point on dwell time calculation is that the proposed stability condition does not require the lyapunov function to be uniformly decreasing at every switching time the second one is of closed loop nature trajectory dependent and is designed from the solution of what we call lyapunov metzler inequalities from which the stability condition including chattering is expressed being nonconvex a more conservative but simpler to solve version of the lyapunov metzler inequalities is provided the theoretical results are illustrated by means of examples

stability of switched systems a lie algebraic condition we present a sucient condition for asymptotic stability of a switched linear system in terms of the lie algebra generated by the individual matrices namely if this lie algebra is solvable then the switched system is exponentially stable for arbitrary switching in fact we show that any family of linear systems satisfying this condition possesses a quadratic common lyapunov function we also discuss the implications of this result for switched nonlinear systems c 1999 elsevier science b v all rights reserved

state feedback h_ infty control for a class of nonlinear stochastic systems this paper discusses the h infty control problem for a class of nonlinear stochastic systems with both state and disturbance dependent noise by means of hamilton jacobi equations both infinite and finite horizon nonlinear stochastic h infty control designs are developed some results on nonlinear h infty control of deterministic systems are generalized to a stochastic setting we introduce some useful concepts such as zero state observability and zero state detectability which together with the stochastic lasalle invariance principle yield some valuable consequences in infinite horizon nonlinear stochastic h infty control

brief paper a unified design for state and output feedback h control of nonlinear stochastic markovian jump systems with state and disturbance dependent noise for a class of continuous time nonlinear stochastic markovian jump systems with state and disturbance dependent noise both the state and output feedback h control problems are studied based on the solvability of a set of coupled hamilton jacobi inequalities hjis we give a unified design for both state and dynamic output feedback h control via a common function u r t i e u x r t is a state feedback h control while u email protected r t is the desired output feedback h control

on some elliptic hemivariational and variational hemivaritional inequalities abstract the existence of solutions to an elliptic hemivariational inequality involving the p laplacian is first established the variational hemivariational case which includes the above one is then examined when p 2 both of them are chiefly treated through mountain pass like critical point results for non differentiable functions

cal h _ infty and dwell time specifications of continuous time switched linear systems this technical note deals with continuous time switched linear systems and addresses the problem of characterizing open loop switching strategies based on dwell time specifications assuring a pre specified root mean square gain rms as a natural consequence of treating general systems of this particular class in terms of order and number of subsystems only sufficient conditions are worked out however as positive features they are expressed through linear matrix inequalities lmi being thus numerically solvable in polynomial time and allow the treatment of stable switched linear systems which do not admit a common lyapunov function the conservativeness of the proposed design method is evaluated by comparison with other results available in the literature by means of two simple illustrative examples

positive solutions and multiple solutions at non resonance resonance and near resonance for hemivariational inequalities with p laplacian in this paper we study eigenvalue problems for hemivariational inequalities driven by the p laplacian differential operator we prove the existence of positive smooth solutions for both non resonant and resonant problems at the principal eigenvalue of the negative p laplacian with homogeneous dirichlet boundary condition we also examine problems which are near resonance both from the left and from the right of the principal eigenvalue for nearly resonant from the right problems we also prove a multiplicity result

convex multilevel decomposition algorithms for non monotone problems a convex multilevel decomposition algorithm is proposed in this paper for the solution of static analysis problems involving non monotone possibly multivalued laws the theory is developed here for a model structure with non monotone interface or boundary conditions first the non monotone laws are written in the form of a difference of two monotone functions under this decomposition the non linear elastostatic analysis problem is equivalent to a system of convex variational inequalities and to non convex min min problems for appropriately defined lagrangian functions the solution s of each one of the aforementioned problems describe the position s of static equilibrium of the considered structure in this paper a multilevel optimization scheme due to auchmuty 1 is used for the numerical solution of the problem the most interesting feature of this method from the computational mechanics standpoint is the fact that each one of the subproblems involved in the multilevel algorithm is a convex optimization problem or in terms of mechanics an appropriately modified monotone unilateral problem thus existing algorithms and software can be used for the numerical solution with minor modifications numerical results concerning the calculation of elastic and rigid stamp problems and of material inclusion problems with delamination and non monotone stick slip frictional effects illustrate the theory

nonlinear algebraic systems with discontinuous terms abstract using a multiple critical points theorem for locally lipschitz continuous functionals we establish the existence of at least three distinct solutions for a parametric discrete differential inclusion problem involving a real symmetric and positive definite matrix applications to tridiagonal fourth order and partial difference inclusions are presented

a variational approach to discontinuous problems with critical sobolev exponents abstract we employ variational techniques to study the existence and multiplicity of positive solutions of semilinear equations of the form u h x h u a uq u2 1 in rn where a 0 are parameters h x is both nonnegative and integrable on rn h is the heaviside function 2 is the critical sobolev exponent and 0 q

lie symmetries and reductions of multi dimensional boundary value problems of the stefan type a new definition of lie invariance for nonlinear multi dimensional boundary value problems bvps is proposed by the generalization of known definitions to much wider classes of bvps the class of 1 3 dimensional nonlinear bvps of the stefan type modeling the process of melting and evaporation of metals is studied in detail using the definition proposed the group classification problem for this class of bvps is solved and some reductions with physical meaning to bvps of lower dimensionality are made examples of how to construct exact solutions of the 1 3 dimensional nonlinear bvp with the correctly specified coefficients are presented

systems of reaction convection diffusion equations invariant under galilean algebras abstract systems which are invariant under galilean algebras with and without mass operator and their main extensions by operators of scale and projective transformations are selected from a class of systems of reaction convection diffusion equations for two dimensional vector field with one space variable

non linear reaction diffusion systems with variable diffusivities lie symmetries ansatze and exact solutions abstract this work first considers the classical lie symmetry analysis of a class of systems of two quasilinear reaction diffusion equations having variable diffusivities subsequently non lie reductions to systems of first order ordinary differential equations are obtained for a subclass of these systems in particular families of exact solutions of a diffusive lotka volterra type system are constructed

nonsmooth critical point theory on closed convex sets and nonlinear hemivariational inequalities abstract in this paper we develop a critical point theory for nonsmooth locally lipschitz functionals defined on a closed convex set extending this way the work of struwe variational methods springer berlin 1990 through a deformation result we obtain minimax principles producing critical points then we use the theory to obtain positive and negative solutions of nonlinear and semilinear hemivariational inequalities in this context we improve a result on positive solutions for semilinear elliptic problems due to nirenberg variational methods in nonlinear problems in topics in calculus of variations lecture notes in mathematics vol 1365 springer berlin 1987

boundary value problems for discontinuous perturbations of singular ϕ laplacian operator abstract systems of differential inclusions of the form u f t u t 0 t where with strictly convex is a homeomorphism of the ball b a r n onto r n are considered under dirichlet periodic and neumann boundary conditions here f t x stands for the generalized clarke gradient of f t at x r n using nonsmooth critical point theory we obtain existence results under some appropriate conditions on the potential f examples of applications concerning filippov type solutions are also provided

quasilinear hemivariational inequalities with strong resonance at infinity we study quasilinear driven by p laplacian elliptic equations with strong resonance at infinity and with a nonsmooth potential using nonsmooth critical point theory we obtain existence and multiplicity results

generalized gradients of lipschitz functionals abstract the purpose of this article is threefold i to present in a unified fashion the theory of generalized gradients whose elements are at present scattered in various sources ii to give an account of the ways in which the theory has been applied iii to prove new results concerning generalized gradients of summation functionals pointwise maxima and integral functionals on subspaces of l these last mentioned formulas are obtained with an eye to future applications in the calculus of variations and optimal control their proofs can be regarded as applications of the existing theory of subgradients of convex functionals as developed by rockafellar ioffe and levin valadier and others

stochastic h infty we consider stochastic linear plants which are controlled by dynamic output feedback and subjected to both deterministic and stochastic perturbations our objective is to develop an h infty type theory for such systems we prove a bounded real lemma for stochastic systems with deterministic and stochastic perturbations this enables us to obtain necessary and sufficient conditions for the existence of a stabilizing compensator which keeps the effect of the perturbations on the to be controlled output below a given threshhold gamma 0 in the deterministic case the analogous conditions involve two uncoupled linear matrix inequalities but in the stochastic setting we obtain coupled nonlinear matrix inequalities instead the connection between h infty theory and stability radii is discussed and leads to a lower bound for the radii which is shown to be tight in some special cases

nonsmooth version of fountain theorem and its application to a dirichlet type differential inclusion problem abstract in this paper we establish a nonsmooth version of fountain theorem by adopting the framework of nonsmooth analysis theory which is a generalization of theorem 3 6 of willem 1996 2 then we present an application of this theorem to a dirichlet type differential inclusion problem

on form preserving point transformations of partial differential equations new identities are presented relating arbitrary order partial derivatives of and for the general point transformation these identities are used to study the nature of those point transformations which preserve the general form of a wide class of 1 1 partial differential equations these results facilitate the search for point symmetries both discrete and continuous lie and assist the search for point transformations which reduce equations to canonical but similar form a simple test for the existence of hodograph type transformations between equations of similar form is given

new q conditional symmetries and exact solutions of some reaction diffusion convection equations arising in mathematical biology abstract a theorem giving a complete description of q conditional symmetries of a class of nonlinear reaction diffusion convection equations is proved furthermore the q conditional symmetries obtained and the method of additional generating conditions are applied for finding exact solutions of the generalized fisher fitzhugh nagumo and kolmogorov petrovskii piskunov equations the symmetries and solutions constructed are compared with those obtained by other authors in particular it was established that the known travelling wave solutions of these equations are particular cases of more general non lie solutions the relation between q conditional symmetries and generalized conditional symmetries is also shown

surjection and inversion for locally lipschitz maps between banach spaces abstract we study the global invertibility of non smooth locally lipschitz maps between infinite dimensional banach spaces using a kind of palais smale condition to this end we consider the chang version of the weighted palais smale condition for locally lipschitz functionals in terms of the clarke subdifferential as well as the notion of pseudo jacobians in the infinite dimensional setting which are the analog of the pseudo jacobian matrices defined by jeyakumar and luc using these tools we obtain the existence and uniqueness of solution for certain nonlinear equations defined by locally lipschitz mappings along the way we also obtain a global surjection theorem for locally lipschitz maps in terms of pseudo jacobians

new results on group classification of nonlinear diffusion convection equations using a new method and additional conditional and partial equivalence transformations we performed group classification in a class of variable coefficient 1 1 dimensional nonlinear diffusion convection equations of the general form f x ut d u ux x k u ux we obtain new interesting cases of such equations with the density f localized in space which have non trivial invariance algebra exact solutions of these equations are constructed we also consider the problem of investigation of the possible local transformations for an arbitrary pair of equations from the class under consideration i e of describing all the possible partial equivalence transformations in this class