stability and higher integrability of derivatives of solutions in double obstacle problems abstract higher integrability of the derivatives of solutions to double obstacle problems associated with the second order quasilinear elliptic differential equation a x u 0 is obtained under natural assumptions on obstacles this result is used to prove a stability result for solutions to double obstacle problems for varying equations

well posedness for nonlinear schrodinger equations with boundary forces in low dimensions by strichartz estimates abstract in this paper we study the well posedness of solutions for nonlinear schrodinger equations on one and two dimensional domains with boundary where the boundary is disturbed by an external inhomogeneous type of dirichlet or neumann force we first prove the local existence of solutions at the energy level for quadratic and superquadratic sources using the strichartz estimates on domains secondly we obtain conditional uniqueness and local stability then we prove the boundedness of solutions in the energy space to pass from the local theory to the global theory regarding subquadratic sources we appeal to classical methods and trudinger s inequality to prove the uniqueness which combined with the existence of weak energy solutions mass and energy inequalities eventually implies the continuity of solutions in time

nonlinear schrodinger equation with inhomogeneous dirichlet boundary data in this article we study the following nonlinear schrodinger equation iut u g u p 1u in a domain rn with initial condition u x 0 x and the dirichlet boundary condition u x t q x t on where q are given smooth functions the nonlinear term contributes a negative term to the energy i e g 0 we present the existence theorem for a global solution of finite energy when p 1 2 n

monotonicity method applied to the complex ginzburg landau and related equations abstract global existence of unique strong solutions is established for the complex ginzburg landau equation tu i u i u p 1u u 0 where 0 0 r p 1 and 1 2 p p 1 the key is a new inequality in monotonicity methods it is based on the sectorial estimates of in lp 1 and the nonlinear operator u u p 1u appearing in the equation the key inequality also yields the global existence of unique strong solutions of the nonlinear schrodinger type equation with monotone nonlinearity tu i u u p 1u 0 for all p 1

some parseval goldstein type identities involving the fs 2 transform the fc 2 transform and the p4 transform and their applications abstract in the present paper the authors consider several new integral transforms including the f s 2 transform the f c 2 transform and the p 4 transform as generalizations of the classical fourier sine transform the classical fourier cosine transform and the classical stieltjes transform respectively many identities involving these transforms are given by making use of these identities a number of new parseval goldstein type identities are obtained for these and many other well known integral transforms the identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions some examples are also given as illustrations of the results presented here

global existence and open loop exponential stabilization of weak solutions for nonlinear schrodinger equations with localized external neumann manipulation abstract in this paper we consider the nonlinear schrodinger equations nls with focusing defocusing interior source and possibly nonlinear damping on a bounded regular domain in the euclidean space moreover it is assumed that the solutions are subject to external neumann boundary manipulation on one portion of the boundary our aim is to obtain global existence of the weak solutions under various assumptions on the sign of the source and powers of the nonlinearities in the case of a linear damping we also prove that solutions decay exponentially under the assumption that the neumann type control at the boundary decays in a similar manner

well posedness and energy decay estimates in the cauchy problem for the damped defocusing schrodinger equation abstract in this paper we study the existence at the h 1 level as well as the stability for the damped defocusing schrodinger equation in r d the considered damping coefficient is time dependent and may vanish at infinity to prove the existence we employ the method devised by ozsari kalantarov and lasiecka 27 which is based on monotone operators theory in particular when d 1 or d 2 we obtain the uniqueness decay estimates for the l 2 level and h 1 l p 2 level energies are established with the help of direct multipliers method coupled with refined energy estimates and a lower semi continuity argument

fractional schrodinger equation solvability and connection with classical schrodinger equation abstract we consider the dirichlet boundary problem for semilinear fractional schrodinger equation with subcritical nonlinear term local and global in time solvability and regularity properties of solutions are discussed but our main task is to describe the connections of the fractional equation with the classical nonlinear schrodinger equation including convergence of the linear semigroups and continuity of the nonlinear semigroups when the fractional exponent approaches 1

general variational inequalities and nonexpansive mappings in this paper we suggest and analyze some three step iterative schemes for finding the common elements of the set of the solutions of the noor variational inequalities involving two nonlinear operators and the set of the fixed points of nonexpansive mappings we also consider the convergence analysis of the suggested iterative schemes under some mild conditions since the noor variational inequalities include variational inequalities and complementarity problems as special cases results obtained in this paper continue to hold for these problems results obtained in this paper may be viewed as an refinement and improvement of the previously known results

stochastic nonlinear schrodinger equation on an upper right quarter plane with dirichlet random boundary in this paper we study the nonhomogeneous stochastic initial boundary value problem for the nonlinear schrodinger equation on an upper right quarter plane with random dirichlet boundary conditions the main novelty of this work is a convenient framework for the analysis of such equations excited by the wiener additive noise on the boundary our approach allows us to show the local existence and uniqueness of solutions in the space h2 the basic properties of the solutions such as the continuity and the boundary layer behavior are also studied using the ito calculus despite several technical difficulties we believe that the approach developed in this paper can be applied to the case of a large class of noise including fractional wiener space time white noise homogeneous noise and levy noise

implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings in this paper we propose a new implicit iteration scheme with perturbed mapping for approximation of common fixed points of a finite family of nonexpansive mappings we establish some convergence theorems for this implicit iteration scheme in particular necessary and sufficient conditions for strong convergence of this implicit iteration scheme were obtained

weakly damped focusing nonlinear schrodinger equations with dirichlet control abstract in this article we consider the weakly damped focusing nonlinear schrodinger equations on bounded domains at the natural h 1 energy level with dirichlet control acting on a portion of the boundary we introduce the dynamic extension method for homogenizing the inhomogeneous boundary input then we construct approximate solutions using monotone operator theory a hidden trace regularity is proved to control the norm of the solutions in a global sense this allows the use of compactness techniques by which we prove the existence of weak solutions finally using multiplier techniques we prove the exponential decay of solutions under the assumption that the boundary control also decays in a similar fashion

qualitative properties of solutions for nonlinear schrodinger equations with nonlinear boundary conditions on the half line in this paper we study the interaction between a nonlinear focusing robin type boundary source a nonlinear defocusing interior source and a weak damping term for nonlinear schrodinger equations posed on the infinite half line we construct solutions with negative initial energy satisfying a certain set of conditions which blow up in finite time in the h1 sense we obtain a sufficient condition relating the powers of nonlinearities present in the model which allows construction of blow up solutions in addition to the blow up property we also discuss the stabilization property and the critical exponent for this model

fixed point solutions of variational inequalities for a finite family of asymptotically nonexpansive mappings without common fixed point assumption let e be a real banach space with a uniformly gateaux differentiable norm and which possesses uniform normal structure k a nonempty bounded closed convex subset of e t i i 1 n a finite family of asymptotically nonexpansive self mappings on k with common sequence k n n 1 1 t n s n be two sequences in 0 1 such that s n t n 1 n 1 and f be a contraction on k under suitable conditions on the sequences s n t n we show the existence of a sequence x n satisfying the relation x n 1 1k n x n s nk nf x n t nk nt r n nx n where n l nn r n for some unique integers l n 0 and 1 r n n further we prove that x n converges strongly to a common fixed point of t i i 1 n which solves some variational inequality provided x n t ix n 0 as n for i 1 2 n as an application we prove that the iterative process defined by z 0 k z n 1 1 1k n z n s nk nf z n t nk nt r n nz n converges strongly to the same common fixed point of t i i 1 n

identities for the e2 1 transform and their applications in the present paper the authors introduce the e 2 1 transform with kernel the exponential integral function it is shown that the third iterate of the l 2 transform is the exponential integral transform and some identities involving the new transform the l 2 transform and the widder potential transform are given using the identities parseval goldstein type results involving these transforms are proved some illustrative examples are also given

boundary controllability for the semilinear schrodinger equations on riemannian manifolds we study the boundary exact controllability for the semilinear schrodinger equation defined on an open bounded connected set of a complete n dimensional riemannian manifold m with metric g we prove the locally exact controllability around the equilibria under some checkable geometrical conditions our results show that exact controllability is geometrical characters of a riemannian metric given by the coefficients and equilibria of the semilinear schrodinger equation we then establish the globally exact controllability in such a way that the state of the semilinear schrodinger equation moves from an equilibrium in one location to an equilibrium in another location

convergence of hybrid steepest descent method for variational inequalities in banach spaces abstract let e be a q uniformly smooth real banach space with constant dq q 1 let ti e e i 1 2 r be a finite family of nonexpansive mappings with k i 1 r fix t i and k fix trtr 1 t1 fix t1tr t2 fix tr 1tr 2 tr let g e e be an strongly accretive map which is also lipschitzian a hybrid steepest descent method introduced by yamada 25 and studied by various authors is proved to converge strongly to the unique solution of the variational inequality problem vi g k in q uniformly smooth real banach space in particular in lp spaces 1

local exact controllability and stabilizability of the nonlinear schrodinger equation on a bounded interval this paper studies the exact controllability and the stabilization of the cubic schrodinger equation posed on a bounded interval both internal and boundary controls are considered and the results are given first in a periodic setting and next with dirichlet resp neumann boundary conditions it is shown that the systems with either an internal control or a boundary control are locally exactly controllable in the classical sobolev space h s for any s geq0 it is also shown that the systems with an internal stabilization are locally exponentially stabilizable in h s for any s geq0

identities and parseval type relations for the l2 transform abstract in the present paper the authors introduce identities for the l 2 transform which yield a parseval type relationship for the l 2 transform the parseval type relationship proven in this paper give rise to useful corollaries for evaluating indefinite integrals of special functions some examples are also given as illustrations of the results presented here

the generalized projection operator on reflexive banach spaces and its applications abstract in this paper we extend the definition of the generalized projection operator k b k where b is a reflexive banach space with dual space b and k is a nonempty closed and convex subset of b and we study its properties and applications to solving variational inequality

on the implicit iterative process for strictly pseudo contractive mappings in banach spaces in this paper we consider a composite implicit iterative process for an infinite family of strictly pseudo contractive mappings strong convergence theorems are established in an arbitrary real banach space the results presented in this paper improve and extend the recent ones announced by many others

an inhomogeneous boundary value problem for nonlinear schrodinger equations abstract we consider a nonlinear schrodinger equation in a domain r n with the inhomegeneous dirichlet boundary condition u q where q is a given smooth function the nonlinear term contributes a positive term to the energy we prove the existence of global solutions of finite energy

inhomogeneous dirichlet boundary value problem for nonlinear schrodinger equations with a power nonlinearity on the upper half plane abstract we consider the inhomogeneous dirichlet boundary value problem for nonlinear schrodinger equations with a power nonlinearity in the upper half plane we present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to the problem by using the classical energy method and factorization techniques

on reich s strong convergence theorems for resolvents of accretive operators let e be a real banach space with norm i then an operator a c e x e with domain d a and range r a is said to be accretive if xi xz 0 an accretive operator a c e x e is m accretive if r i ra e for all r 0 where z is the identity if a is accretive we can define for each positive r a singlevalued mapping j r z ra d a by j z ra it is called the resolvent of a in 6 reich proved the following theorem let e be a uniformly smooth banach space and let a c e x e be m accretive if 0 e r a then for each x in e the strong lim z x exists and belongs to a io he remarked also that the assumption that a is m accretive can be replaced with the assumption that cl a the closure of d a is convex and that a satisfies the range condition r z ra 1 cl d a for all r 0 in this paper we first prove a theorem that generalizes simultaneously the above results furthermore we generalize another reich strong convergence theorem asserting that the strong lim z x exists in uniformly convex and uniformly smooth banach spaces though our proofs are similar to those in 6 they are slightly simple on account of using banach limits

exact boundary controllability of the nonlinear schrodinger equation abstract this paper studies the exact boundary controllability of the semi linear schrodinger equation posed on a bounded domain r n with either the dirichlet boundary conditions or the neumann boundary conditions it is shown that if s n 2 or 0 s n 2 with 1 n 2 2 s or s 0 1 with n 2 then the systems are locally exactly controllable in the classical sobolev space h s around any smooth solution of the cubic schrodinger equation

a new method of solving hermite s differential equation using the l2 transform in the present paper a new method of solving hermite s differential equation is given using the l 2 transform

uniform decay rate estimates for schrodinger and plate equations with nonlinear locally distributed damping on a compact n dimensional riemannian manifold m g we establish uniform decay rate estimates for the linear schrodinger and plate equations subject to an internal nonlinear damping locally distributed on the manifold our approach can be also employed for other equations provided that inverse inequality for the linear model occurs in the particular case of the wave equation where the well known geometric control condition gcc is equivalent to the observability inequality our method generalizes the results due to cavalcanti et al 2010 2009 9 10 regarding the optimal choice of dissipative regions

uniform decay rates for the energy of weakly damped defocusing semilinear schrodinger equations with inhomogeneous dirichlet boundary control abstract in this paper we study the open loop stabilization as well as the existence and regularity of solutions of the weakly damped defocusing semilinear schrodinger equation with an inhomogeneous dirichlet boundary control first of all we prove the global existence of weak solutions at the h 1 energy level together with the stabilization in the same sense it is then deduced that the decay rate of the boundary data controls the decay rate of the solutions up to an exponential rate secondly we prove some regularity and stabilization results for the strong solutions in h 2 sense the proof uses the direct multiplier method combined with monotonicity and compactness techniques the result for weak solutions is strong in the sense that it is independent of the dimension of the domain the power of the nonlinearity and the smallness of the initial data however the regularity and stabilization of strong solutions are obtained only in low dimensions with small initial and boundary data

inequalities in banach spaces with applications well known that many problems in a hilbert space h are solved by applying the following identity ilax 1 a yi12 a412 1 llyl12 a 1 mx yl12 1 1 for all x y in h and 0 5 i 5 1 therefore one natural method to solve problems in a banach space xis to establish equalities and usually inequalities in xanalogous to 1 1 this is in fact shown by some authors recent work see for example reich l poffald and reich 2 lim 3 lim et af 4 prus and smarzewski 5 smarzewski 6 and reich 21 the purpose of this paper is to continue carrying out investigations in this line more precisely in section 3 we establish some inequalities analogous to 1 l in either uniformly convex or uniformly smooth banach spaces as corollaries we obtain new characteristics of p uniform convexity and q uniform smoothness of a banach space x via the functionals i lip and ii llq respectively we give in section 4 applications of the inequalities obtained in section 3 to fixed points of uniformly lipschitzian mappings and to best approximation theory

on global solutions to the initial boundary value problem for the damped nonlinear schrodinger equations where a 0 and q is a bounded domain in rn with smooth boundary x when a 0 undamped case the global existence of smooth solutions to the initial boundary value problem l 3 has been proved only when n 2 and f lu12 w lulp with p 2 brezis and gallouet cl with 2 i p i3 m tsutsumi 9 higher dimensional cases are very difficult pribl s and remain unproved as far as the author knows if we add a linear damping term iau a 0 then the mass ilull iztr1 and the energy ivu x t l dx j f lu x t l dx decay exponentially as t cc as is seen below which may assert the global existence of small smooth solutions even if the nonlinear term f lul u does not have a proper sign it is our purpose to establish the global existence and decay estimates of smooth solutions of the problem l 3 with small data there are a number of works on the nonlinear evolution equations see 3 5 71 and their references in the following spirit energy and decay estimates give global existence of solutions if the damping term is not added there is no hope that the decay estimates of solutions in some f q p 2 2 can be obtained since