equilibrium points of a singular cooperative system with free boundary abstract in this paper we initiate the study of maps minimising the energy d u 2 2 u d x which due to lipschitz character of the integrand gives rise to the singular euler equations u u u u 0 u u 1 u m our primary goal in this paper is to set up a road map for future developments of the theory related to such energy minimising maps our results here concern regularity of the solution as well as that of the free boundary they are achieved by using monotonicity formulas and epiperimetric inequalities in combination with geometric analysis

multiple solutions to a three point boundary value problem for higher order ordinary differential equations in this paper we provide sufficient conditions for the existence of at least three solutions to a three point boundary value problem for higher order ordinary differential equations the nonlinear term f in the differential equation under consideration may depend on higher order derivatives of arbitrary order and this is where the main novelty of this work lies by applying the two pairs of upper and lower solutions method of henderson and thompson as well as degree theory the existence of at least three solutions of the problem is given

third order boundary value problems with nonlocal boundary conditions we treat the existence of multiple positive solutions for a third order boundary value problem with nonlocal boundary conditions in our work each of the three boundary conditions is a linear functional on c 0 1 that is given by a stieltjes integral but is not necessarily a positive functional

a theorem of real analysis and its application to free boundary problems in this note we show how comparison theorems for solutions to equations with non smooth coefficients yield in a very simple fashion some free boundary regularity theorems in cases where there is a priori geometrical information for instance when one knows that the solution u is monotone in some given direction

the two phase membrane problem an intersection comparison approach to the regularity at branch points for the two phase membrane problem delta u lambda 2 chi u 0 lambda 2 chi u 0 and lambda 0 we prove in two dimensions that the free boundary is in a neighborhood of each branch poin

variational problems with two phases and their free boundaries the problem of minimizing vu 2 q2 x 2 v dx in an appropriate class of functions v is considered here q x 0 and a2 t x2 if v 0 any minimizer u is harmonic in u 0 and vu 2 has a jump a2 x 22 across the free boundary u 0 regularity and various properties are established for the minimizer u and for the free boundary introduction in this paper we consider the problem of minimizing j v f vv 2 q2 x x2 v dx vgk where q2 x 0 a22 ifu 0 and x2 u is lower semicontinuous at v 0 it is assumed that a2 0 and a a2 a22 0 the class k consists of all functions v in l loc s2 with vu g l2 i2 such that v u on a given open subset s of 3i2 and s is a domain in r the analogous problem for functions in k v k v 0 a e was studied in 1 in that paper it was proved that any local minimizer of j v in k is lipschitz continuous and if n 2 the free boundary 8 w 0 is analytic if q x is analytic the present variational problem is motivated by applications to the now of two liquids in models of jets and cavities these applications will be studied in other forthcoming papers 5 6 the present work is aimed at extending results of 1 in particular we shall establish nondegeneracy theorems the lipschitz continuity of the solution and some properties of the free boundary for n 2 the free boundary is proved to be continuously differentiable a new and rather powerful tool introduced in this paper is the monotonicity formula lemma 5 1 asserting that for a minimizer u if u x0 0 then

positive solutions to differential inclusions with nonlocal nonlinear boundary conditions we demonstrate the existence of at least one positive solution to the differential inclusion u t f t u t email protected 0 1 equipped with the boundary conditions u 0 h f u and u 1 0 here f is linear functional realized as a lebesgue stieltjes integral and h is a nonlinear function consequently the boundary condition at t 0 can be both nonlocal and nonlinear by imposing an asymptotic condition on h we provide conditions under which the problem will have at least one positive solution

weak and strong convergence of the ishikawa iteration process with errors for two asymptotically nonexpansive mappings in this paper we prove the weak and strong convergence of the ishikawa iterative scheme with errors to a common fixed point for two asymptotically nonexpansive mappings in a uniformly convex banach space under a condition weaker than compactness our theorems improve and generalize recent known results in the literature

convergence of iterates with errors of asymptotically quasi nonexpansive mappings and applications in this paper we consider a generalized iterative process with errors to approximate the common fixed points of two asymptotically quasi nonexpansive mappings a convergence theorem has been obtained which generalizes a known result this theorem has then been used to prove another convergence theorem which in turn generalizes a number of results

iteration processes for nonlinear mappings in convex metric spaces on considere le schema iteratif d ishikawa pour construire des points fixes d applications quasi contractives quasi contractives generalisees et quasi non dilatante dans des espaces metriques convexes

approximating the common fixed points of two sequences of uniformly quasi lipschitzian mappings in convex metric spaces in this paper a kind of ishikawa type iterative scheme with errors for approximating a common fixed point of two sequences of uniformly quasi lipschitzian mappings is introduced and studied in convex metric spaces under some suitable conditions the convergence theorems concerned with the ishikawa type iterative scheme with errors to approximate a common fixed point of two sequences of uniformly quasi lipschitzian mappings were proved in convex metric spaces the results presented in the paper generalize and improve some recent results of wang and liu c wang l w liu convergence theorems for fixed points of uniformly quasi lipschitzian mappings in convex metric spaces nonlinear anal tma 70 2009 2067 2071

higher regularity of the free boundary in the obstacle problem for the fractional laplacian abstract we prove a higher regularity result for the free boundary in the obstacle problem for the fractional laplacian via a higher order boundary harnack estimate

new harnack inequalities and existence theorems for radially symmetric solutions of elliptic pdes with sign changing or vanishing green s function abstract we consider the perturbed hammerstein integral equation y t 1 t h 1 1 y 2 t h 2 2 y 0 1 g t s f s y s d s where 1 and 2 are linear functionals realized as stieltjes integrals with associated signed stieltjes measures we demonstrate that by utilizing a nonstandard order cone together with an associated nonstandard open set existence of a positive solution to the integral equation can be guaranteed under relatively mild hypotheses on the various constituent functions and functionals in the case in which the green s function t s g t s is allowed to both vanish and change sign as an example we apply our results to radially symmetric elliptic pdes of the form u h x g u x x r 1 r 2 r u x x b r 2 0 u r 1 x u r x x b 1 h u where 0 r 1 r r 2 is a linear functional and h is a continuous function as part of this analysis we demonstrate a new methodology for constructing a suitable harnack like inequality associated to the boundary value problem

the structure of the free boundary for lower dimensional obstacle problems we study the regularity of the free surface in boundary obstacle problems we show that near a non degenerate point the free boundary is a c 1 a n 2 dimensional surface in r n 1

existence of positive solutions to third order differential equations with advanced arguments and nonlocal boundary conditions abstract we use a fixed point theorem due to avery and peterson to establish the existence of at least three non negative solutions of some nonlocal boundary value problems to third order differential equations with advanced arguments an example is given to illustrate the main results

optimal regularity for the signorini problem and its free boundary we will show optimal regularity for minimizers of the signorini problem for the lame system in particular if mathbf u u 1 u 2 dots u n in w 1 2 b 1 mathbb r n minimizes begin aligned j mathbf u int b 1 nabla mathbf u nabla bot mathbf u 2 lambda hbox div mathbf u 2 end aligned in the convex set begin aligned k left mathbf u u 1 u 2 dots u n in w 1 2 b 1 mathbb r n u n ge 0 quad hbox on quad pi right left mathbf u f in c infty partial b 1 quad hbox on quad partial b 1 right end aligned where lambda ge 0 say then mathbf u in c 1 1 2 b 1 2 moreover the free boundary given by begin aligned gamma mathbf u partial x u n x 0 x n 0 cap b 1 end aligned will be a c 1 alpha graph close to points where mathbf u is not degenerate similar results have been know before for scalar partial differential equations see for instance 5 6 the novelty of this approach is that it does not rely on maximum principle methods and is therefore applicable to systems of equations

on the regularity of the free boundary in the parabolic obstacle problem application to american options this paper is devoted to local regularity results on the free boundary of the one dimensional parabolic obstacle problem with variable coefficients we give an energy criterion and a density criterion for characterising the subsets of the free boundary which are holder continuous in time with exponent 1 2 our results apply in the theory of american options as an illustration we apply these results to the generalised black scholes model of a complete market which rules out arbitrage if the volatility and the interest rate do not depend on time in this case we prove that the exercise boundary of the american put and call options are holder continuous with exponent 1 2 in time for every time

twin monotone positive solutions to a singular nonlinear third order differential equation abstract this paper discusses the existence of at least one or two nondecreasing positive solutions for the following singular nonlinear third order differential equation x t t f t x t 0 a t b x a x a x b 0 where 0 is a parameter c a b r f c a b 0 r t may be singular at t a b and f t s may be singular at s 0 green s function and the fixed point theorem of cone expansion and compression type are utilized to illustrate the results four nontrivial examples are included

positive solutions of nonlinear singular third order two point boundary value problem abstract in this paper we are concerned with the existence of single and multiple positive solutions to the nonlinear singular third order two point boundary value problem u t a t f u t 0 0 t 1 u 0 u 0 u 1 0 where is a positive parameter under various assumptions on a and f we establish intervals of the parameter which yield the existence of at least one at least two and infinitely many positive solutions of the boundary value problem by using krasnoselskii s fixed point theorem of cone expansion compression type

obstacle problems for integro differential operators higher regularity of free boundaries abstract we study the higher regularity of free boundaries in obstacle problems for integro differential operators our main result establishes that once free boundaries are c 1 then they are c this completes the study of regular points initiated in 5 in order to achieve this we need to establish optimal boundary regularity estimates for solutions to linear nonlocal equations in c k domains these new estimates are the core of our paper and extend previously known results by grubb for k and by the second author and serra for k 1

convergence theorems for fixed points of uniformly quasi lipschitzian mappings in convex metric spaces abstract in this paper we consider an ishikawa type iteration process with errors to approximate the fixed point of two uniformly quasi lipschitzian mappings in convex metric spaces some results have been obtained which generalize and unify many important known results in recent literature

optimal convergence for discrete variational inequalities modelling signorini contact in 2d and 3d without additional assumptions on the unknown contact set the basic h 1 finite element error estimate of order h with only h 2 regularity on the solution has not been yet established for the simplest two dimensional signorini problem approximated by a discrete variational inequality or the equivalent mixed method and linear finite elements to obtain an optimal error bound in this basic case and also when considering more general cases e g the three dimensional problem quadratic finite elements additional assumptions on the exact solution in particular on the unknown contact set see z belhachmi and f ben belgacem math comp 72 2003 pp 83 104 s hueber and b wohlmuth siam j numer anal 43 2005 pp 156 173 b wohlmuth a popp m gee and w wall comput mech 49 2012 pp 735 747 had to be used in this paper we consider finite element approximations of the two dimensional and three dimensional signorini problems with linear and quadratic finite elements in the analysis we remove all the additional assumptions and pr

higher regularity of the free boundary in the elliptic signorini problem abstract in this paper we study the higher regularity of the free boundary for the elliptic signorini problem by using a partial hodograph legendre transformation we show that the regular part of the free boundary is real analytic the first complication in the study is the invertibility of the hodograph transform which is only c 0 1 2 which can be overcome by studying the precise asymptotic behavior of the solutions near regular free boundary points the second and main complication in the study is that the equation satisfied by the legendre transform is degenerate however the equation has a subelliptic structure and can be viewed as a perturbation of the baouendi grushin operator by using the l p theory available for that operator we can bootstrap the regularity of the legendre transform up to real analyticity which implies the real analyticity of the free boundary

new monotonicity formulas and the optimal regularity in the signorini problem with variable coefficients abstract we study the interior signorini or lower dimensional obstacle problem for a uniformly elliptic divergence form operator l div a x with lipschitz continuous coefficients our main result states that similarly to what happens when l the variational solution has the optimal interior regularity c loc 1 1 2 m when m is a codimension one flat manifold which supports the obstacle we achieve this by proving some new monotonicity formulas for an appropriate generalization of the celebrated almgren s frequency functional

solvability of parabolic variational hemivariational inequalities involving space fractional laplacian abstract the paper is devoted to investigate a new kind of variational hemivariational inequality of parabolic type driven by a generalized space fractional laplace operator and two multivalued terms which are expressed by the clarke generalized gradient and convex subgradient respectively an existence theorem of weak solutions to the problem is established by applying a surjectivity theorem for the sum of operators combined with results from nonsmooth analysis

regularity estimates for the solution and the free boundary of the obstacle problem for the fractional laplacian we use a characterization of the fractional laplacian as a dirichlet to neumann operator for an appropriate differential equation to study its obstacle problem we write an equivalent characterization as a thin obstacle problem in this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary

the viscosity approximation method for asymptotically nonexpansive mappings in banach spaces abstract a recent trend in the iterative methods for constructing fixed points of nonlinear mappings is to use the viscosity approximation technique the advantage of this technique is that one can find a particular solution to the associated problems and in most cases this particular solution solves some variational inequality in this paper we try to extend this technique to find a particular common fixed point of a finite family of asymptotically nonexpansive mappings in a banach space which is reflexive and has a weakly continuous duality map both implicit and explicit viscosity approximation schemes are proposed and their strong convergence to a solution to a variational inequality is proved

a homogeneity improvement approach to the obstacle problem this paper contains a new approach to regularity and singularity of the obstacle problem by means of an epiperimetric inequality and a monotonicity formula an energy decay estimate is derived as a consequence the increase in the solution s homogeneity when passing to a smaller ball can be estimated this leads to uniqueness of blow up limits and regularity in that order

the structure of the free boundary in the fully nonlinear thin obstacle problem abstract we study the regularity of the free boundary in the fully nonlinear thin obstacle problem our main result establishes that the free boundary is c 1 near any regular point this extends to the fully nonlinear setting the celebrated result of athanasopoulos caffarelli salsa 1 the proofs we present here are completely independent from those in 1 and do not rely on any monotonicity formula furthermore an interesting and novel feature of our proofs is that we establish the regularity of the free boundary without classifying blow ups a priori they could be non homogeneous and or non unique we do not classify blow ups but only prove that they are 1d on x n 0

an elementary partial regularity proof for vector valued obstacle problems soit r n et m r n des ensembles ouverts bornes on suppose n 3 n 2 et on requiert que la frontiere de m est une hyper surface lisse de classe c 2 l espace de fonction h 1 2 m des fonctions de comparaison consiste en toutes les fonctions de sobolev v h 1 2 r n avec la propriete v x m pour presque tous les points x dans si u h 1 2 m minimise localement l energie alors il y a un sous ensemble ouvert 0 de tel que u c 1 0 r n pour tout 0 1 h n 2 0 0 et x 0 si et seulement si lim 1 0 r 2 n br x du 2 dz 0

convergence of an ishikawa type iterative scheme for asymptotically quasi nonexpansive mappings in convex metric spaces the ishikawa iteration process and the ishikawa iteration process with errors is defined for asymptotically quasi nonexpansive mappings it is proved some sufficient and necessary conditions for the iterative scheme converges to the fixed point of the asymptotically quasi nonexpansive mappings these results generalize and unify many important known results in recent literature

an accurate local average contact method for nonmatching meshes the present paper deals with linear and quadratic finite element approximations of the two and three dimensional unilateral contact problems between two elastic bodies with nonmatching meshes we propose a simple noninterpenetration condition on the displacements which is local as the well known node to segment and node to face conditions and accurate like the mortar approach this condition consists of averaging locally on a few elements the noninterpenetration we prove optimal convergence rates in 2d and 3d using various linear and quadratic elements the taylor patch test and the hertzian contact test illustrate the theoretical results and show the capabilities of the method

convergence theorems for common fixed points of asymptotically quasi nonexpansive mappings in convex metric spaces abstract in this paper we consider an implicit iteration process to approximate the common fixed points of two finite families of asymptotically quasi nonexpansive mappings in convex metric spaces as a consequence of our result we obtain some related convergence theorems our results generalize some recent results of khan and ahmed 4 khan et al 6 sun 12 wittmann 14 and xu and ori 15

global regularity for the free boundary in the obstacle problem for the fractional laplacian we study the regularity of the free boundary in the obstacle problem for the fractional laplacian under the assumption that the obstacle varphi satisfies delta varphi leq 0 near the contact region our main result establishes that the free boundary consists of a set of regular points which is known to be a n 1 dimensional c 1 alpha manifold by the results obtained by caffarelli salsa and silvestre it invent math 2008 and a set of singular points which we prove to be contained in a union of k dimensional c 1 submanifold k 0 ldots n 1 such a complete result on the structure of the free boundary proved by l a caffarelli it acta math 1977 and it j fourier anal appl 1998 was known only in the case of the classical laplacian and it is new even for the signorini problem which corresponds to the particular case of the 1 over 2 fractional laplacian a key ingredient behind our results is the validity of a new non degeneracy condition sup b r x 0 u varphi geq c r 2 valid at all free boundary points x 0

regularity for the nonlinear signorini problem abstract we study the regularity of the solution to a fully nonlinear version of the thin obstacle problem in particular we prove that the solution is c 1 for some small 0 this extends a result of luis caffarelli of 1979 our proof relies on new estimates up to the boundary for fully nonlinear equations with neumann boundary data developed recently by the authors

an iterative scheme for equilibrium problems and fixed point problems of strict pseudo contraction mappings in this paper we propose an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudo contraction mapping in the setting of real hilbert spaces we establish some weak and strong convergence theorems of the sequences generated by our proposed scheme our results combine the ideas of marino and xu s result g marino h k xu weak and strong convergence theorems for strict pseudo contractions in hilbert spaces j math anal appl 329 2007 336 346 and takahashi and takahashi s result s takahashi w takahashi viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 331 2007 506 515 in particular necessary and sufficient conditions for strong convergence of our iterative scheme are obtained

optimal regularity and structure of the free boundary for minimizers in cohesive zone models we study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics under smoothness assumptions on the boundary conditions and on the fracture energy density we show that minimizers are c 1 1 2 and that near non degenerate points the fracture set is c 1 alpha for some alpha in 0 1

regularity of a free boundary with application to the pompeiu problem in the unit ball b 0 1 let u and a domain in n solve the following overdetermined problem u in b 0 1 02 u jruj 0 inb 0 1 n where denotes the characteristic function and the equation is satisfled in the sense of distributions if the complement of does not develop cusp singularities at the origin then we prove is analytic in some small neighborhood of the origin the result can be modifled to yield for more general divergence form operators as an application of this then we obtain the regularity of the boundary of a domain without the pompeiu property provided its complement has no cusp singularities

three solutions of an nth order three point focal type boundary value problem abstract we consider the following differential equation u n t f t u t t 0 1 together with the three point focal type boundary conditions u j 0 0 0 j n 3 u n 2 p 0 u n 1 1 0 where 1 2 p 1 by using two different fixed point theorems we offer criteria for the existence of three positive solutions of this problem examples are also included to illustrate the results obtained

on the fine structure of the free boundary for the classical obstacle problem in the classical obstacle problem the free boundary can be decomposed into regular and singular points as shown by caffarelli in his seminal papers caffarelli in acta math 139 155 184 1977 j fourier anal appl 4 383 402 1998 regular points consist of smooth hypersurfaces while singular points are contained in a stratified union of c 1 manifolds of varying dimension in two dimensions this c 1 result has been improved to c 1 alpha by weiss invent math 138 23 50 1999 in this paper we prove that for n 2 singular points are locally contained in a c 2 curve in higher dimension n ge 3 we show that the same result holds with c 1 1 manifolds or with countably many c 2 manifolds up to the presence of some anomalous points of higher codimension in addition we prove that the higher dimensional stratum is always contained in a c 1 alpha manifold thus extending to every dimension the result in weiss 1999 we note that in terms of density decay estimates for the contact set our result is optimal in addition for n ge 3 we construct examples of very symmetric solutions exhibiting linear spaces of anomalous points proving that our bound on their hausdorff dimension is sharp

a parabolic free boundary problem with double pinning abstract we study the cauchy problem for the equation tue ue e ue in 0 rn as e 0 where the nonlinearity e is assumed to converge to a measure concentrated at 0 in this paper we allow for sign changes of e and ue the solutions are uniformly lipschitz continuous in space and holder continuous in time we show that each limit of ue is a solution of the free boundary problem tu u 0 in u 0 0 rn u 2 u 2 g on u 0 u e the function g in the condition on the free boundary need not be a constant

positive solutions to boundary value problems with nonlinear boundary conditions abstract in this paper we consider the boundary value problem y t f t y t subject to the boundary conditions y a y and y 2 b 0 in this setting c rd a 2 b t r r is a continuous functional which represents a nonlinear nonlocal boundary condition by imposing sufficient structure on and the nonlinearity f we deduce the existence of at least one positive solution to this problem the novelty in our setting lies in the fact that may be strictly nonpositive for some y 0 our results are achieved by appealing to the krasnosel ski fixed point theorem we conclude with several examples illustrating our results and the generalizations that they afford

a note on multi point boundary value problems abstract multi point boundary value problems for a second order ordinary differential equation are considered in this note an existence result is obtained with the help of coincidence degree theory

regularity results of the thin obstacle problem for the p x laplacian abstract we study thin obstacle problems involving the energy functional with p x growth we prove higher integrability and holder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable exponent p x is holder continuous

a free boundary problem for semilinear elliptic equations etude de la nature de la frontiere libre u 0 pour un probleme de dirichlet semilineaire

viscosity approximation methods for asymptotically nonexpansive mappings abstract let e be a uniformly convex banach space with a uniformly g teaux differentiable norm k a nonempty closed convex subset of e t k k an asymptotically nonexpansive mapping with sequence k n 1 lim n k n 1 let n 0 1 be such that lim n n 0 lim n k n 1 n 0 and f be a contraction on k under suitable conditions we show the existence of a sequence z n satisfying the relation z n n f z n 1 n t n z n and prove that z n converges strongly to the fixed point of t which solves some variational inequality provided t is asymptotically regular as an application we prove that the iterative process defined by x 0 k x n 1 n f x n n x n n t n x n converges strongly to the same fixed point of t

the variable coefficient thin obstacle problem carleman inequalities abstract in this article we present a new strategy of addressing the variable coefficient thin obstacle problem our approach is based on a variable coefficient carleman estimate this yields semi continuity of the vanishing order lower and uniform upper growth bounds of solutions and sufficient compactness properties in order to carry out a blow up procedure moreover the carleman estimate implies the existence of homogeneous blow up limits along certain sequences and ultimately leads to an almost optimal regularity statement as it is a very robust tool it allows us to consider the problem in the setting of sobolev metrics i e working on the upper half ball b 1 r n 1 the coefficients are only w 1 p regular for some p n 1 these results provide the basis for our further analysis of the free boundary the optimal regularity of solutions and a first order asymptotic expansion of solutions at the regular free boundary which is carried out in a follow up article 21 in the framework of w 1 p p 2 n 1 regular coefficients and w 2 p p 2 n 1 regular non zero obstacles

on nonlinear boundary conditions satisfying certain asymptotic behavior abstract in this paper we consider a second order boundary value problem of the form y t a t g y t y 0 y y 1 0 we demonstrate that if the nonlinear functional y satisfies certain asymptotic behavior then the problem will possess at least one positive solution these results generalize and improve on some recent results in the literature on boundary value problems with nonlocal nonlinear boundary conditions and we provide some examples which illustrate these generalizations and improvements

nondegenerate motion of singular points in obstacle problems with varying data abstract recent work by serfaty and serra give a formula for the velocity of the free boundary of the obstacle problem at regular points 19 and much older work by king lacey and vazquez gives an example of a singular free boundary point in the hele shaw flow that remains stationary for a positive amount of time 13 the authors show how singular free boundaries in the obstacle problem in some settings move immediately in response to varying data three applications of this result are given and in particular the authors show a uniqueness result for sufficiently smooth elliptic divergence form operators on domains in r n and for the laplace beltrami operator on a smooth manifold the boundaries of distinct mean value sets of the type found in 7 and 5 which are centered at the same point do not intersect

implicit iterative algorithms for asymptotically nonexpansive mappings in the intermediate sense and lipschitz continuous monotone mappings in this paper we introduce some implicit iterative algorithms for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone lipschitz continuous mapping these implicit iterative algorithms are based on two well known methods extragradient and approximate proximal methods we obtain some weak convergence theorems for these implicit iterative algorithms based on these theorems we also construct some implicit iterative processes for finding a common fixed point of two mappings such that one of these two mappings is taken from the more general class of lipschitz pseudocontractive mappings and the other mapping is asymptotically nonexpansive

ishikawa and mann iterative processes with errors for nonlinear strongly accretive operator equations abstract the purpose of this paper is to revise the definitions of ishikawa and mann iterative processes with errors by using a new inequality to study the unique solution of the nonlinear strongly accretive operator equationtx fand the convergence problem of ishikawa and mann iterative sequences for strongly pseudo contractive mappings without the lipschitz condition the results presented in this paper improve and extend the corresponding results in 4 5 7 10 12 15 16 in the more general setting in particular the open problem mentioned by chidume in 5 has been given an affirmative answer

a minimization problem with free boundary related to a cooperative system we study the minimum problem for the functional u 2 q2 u 0 dx with the constraint ui 0 for i 1 m where rn is a bounded domain and u u1 um h1 rm first we derive the euler equation satisfied by each component then we show that the noncoincidence set u 0 is locally nontangentially accessible having this we are able to establish sufficient regularity of the force term appearing in the euler equations and derive the regularity of the free boundary u 0

some new monotonicity formulas and the singular set in the lower dimensional obstacle problem we construct two new one parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem the first one is a family of weiss type formulas geared for points of any given homogeneity and the second one is a family of monneau type formulas suited for the study of singular points we show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity this allows to prove a structural theorem for the singular set our approach works both for zero and smooth non zero lower dimensional obstacles the study in the latter case is based on a generalization of almgren s frequency formula first established by caffarelli salsa and silvestre

nonnegative solutions to nonlocal boundary value problems for systems of second order differential equations dependent on the first order derivatives abstract in this paper we consider nonlocal boundary value problems for systems of second order differential equations with dependence on the first order derivatives and deviating arguments by using a fixed point theorem due to avery and peterson sufficient conditions are obtained that guarantee the existence of at least three nonnegative solutions to such problems we investigate our problem both for delayed and advanced arguments i i and also for the case when i t i t t t 0 1 in all cases arguments i i can change the character on 0 1 so in some subinterval i of 0 1 they can be delayed in i and advanced in 0 1 i some remarks concern also the case when differential equations do not depend on the first order derivatives examples illustrate some results

strong convergence theorems with a noor type iterative scheme in convex metric spaces the paper introduces some new mappings in convex metric spaces and then it considers a noor type iterative scheme to approximate common fixed points of an infinite family of uniformly quasi sup f n lipschitzian mappings and an infinite family of g n expansive mappings in convex metric spaces its results generalize improve and unify some results in chang et al 2010 13 fukhar ud din and khan 2007 14 liu et al 2010 17 tian 2005 7 wang and liu 2009 19 and wang et al 2009 20 under some appropriate conditions

on the regularity of the non dynamic parabolic fractional obstacle problem abstract in the class of the non dynamic fractional obstacle problems of parabolic type it is shown existence uniqueness apriori bounds of the solution and optimal regularity of the space derivatives of the solution furthermore at free boundary points of positive parabolic density it is proven that the time derivative of the solution is holder continuous at such free boundary points space time regularity of the corresponding free boundary is obtained for any fraction s 0 1

conditions for the existence of positive solutions covering a class of boundary value problems in a uniform way abstract in this paper we will give conditions which will guarantee the existence of positive solutions for a variety of second order boundary value problems using the well known krasnoselskii fixed point theorem we will deal with a specific differential equation meeting a specific initial condition and use a general boundary condition involving a not necessarily linear functional our purpose is to pose conditions on that functional which will guarantee that the krasnoselskii fixed point theorem can be applied it is important to notice that only the values of this functional on two specific functions are involved in the conditions we pose this paper unifies the way we deal with a wide variety of boundary value problems and provides results which to the best of our knowledge are new

asymptotically strict pseudocontractive mappings in the intermediate sense abstract it is proved that the modified mann iteration process x n 1 1 n x n n t n x n n n where n is a sequence in 0 1 with n 1 for some 0 1 converges weakly to a fixed point of an asymptotically strict pseudocontractive mapping t in the intermediate sense which is not necessarily lipschitzian we also develop cq method for this modified mann iteration process which generates a strongly convergent sequence