extremal solutions of φ laplacian diffusion scalar problems with nonlinear functional boundary conditions in a unified way abstract the aim of this paper is to ensure the existence of extremal solutions lying between a pair of lower and upper solutions for a second order differential equation that includes both the laplacian and some classes diffusion equations the results hold from a suitable change of variables an example is given to expose the applicability of the obtained results

existence of solutions for weighted p r laplacian system boundary value problems this paper investigates the existence of solutions for weighted p r laplacian ordinary system boundary value problems via leray schauder degree the sufficient conditions for the existence of solutions be given as an application we obtain the existence of the radial solutions for the p x laplacian partial differential system dirichlet problems

commutators and commutator subgroups in profinite groups abstract let g be a profinite group we prove that the commutator subgroup g is finite by procyclic if and only if the set of all commutators of g is contained in a union of countably many procyclic subgroups

implicit nonlinear discontinuous functional boundary value φ laplacian problems extremality results this paper is devoted to the study of the following implicit nonlinear discontinuous functional boundary value problem ip l 0u t f t u l 0u fora e t i a b l 1 u b 1 u l 1 u 0 l 2 u a u b wherel 0u t ddt f t u t u t g t u u t u t andl 1 u l 1 u a u b u a u b u supposing that there is a lower solution a and an upper solution b such that a b the existence of extremal solutions lying between both functions is proved

approximate solutions to a new class of nonlinear diffusion problems in this paper it is considered a new class of nonlinear differential equations that arises in the study of some diffusion processes the authors present an existence and uniqueness result by constructing a sequence of approximate solutions theoretical and numerical aspects are considered

commutators in finite p groups with 3 generator derived subgroup abstract it is well known that in general the set of commutators of a group g may not be a subgroup guralnick showed that if g is a finite p group with p 5 such that g is abelian and 3 generator then all the elements of the derived subgroup are commutators in this paper we extend guralnick s result by showing that the condition of g to be abelian is not needed in this way we complete the study of this property in finite p groups in terms of the number of generators of the derived subgroup we will also see that the same result is true when the action of g on g is uniserial modulo g p and g g p does not exceed p p 1 finally we will prove that analogous results are satisfied when working with pro p groups

a note on a class of problems for a higher order fully nonlinear equation under one sided nagumo type condition abstract the purpose of this work is to establish existence and location results for the higher order fully nonlinear differential equation u n t f t u t u t u n 1 t n 2 with the boundary conditions u i a a i for i 0 n 3 u n 1 a b u n 1 b c or u i a a i for i 0 n 3 c 1 u n 2 a c 2 u n 1 a b c 3 u n 2 b c 4 u n 1 b c with a i b c r for i 0 n 3 and c 1 c 2 c 3 c 4 real positive constants it is assumed that f a b r n 1 r is a continuous function satisfying one sided nagumo type conditions which allows an asymmetric unbounded behaviour on the nonlinearity the arguments are based on the leray schauder topological degree and lower and upper solutions method

nonlinear second order equations with functional implicit impulses and nonlinear functional boundary conditions abstract in this paper we present existence results for solutions of nonlinear second order boundary value problems with impulses our impulses are applied at p points in the interval and given implicitly by nonlinear functions of the solution moreover we allow functional dependence on the solution our existence results follow from the existence of a pair of well ordered lower and upper solutions we generalize earlier results of cabada and tomec ek allowing more general compatible boundary conditions impulses and laplacian equations

commutators and companion matrices over rings of stable rank 1 abstract we consider the group gl n a of all invertible n by n matrices over a ring a satisfying the first bass stable range condition we prove that every matrix is similar to the product of a lower and upper triangular matrix and that it is also the product of two matrices each similar to a companion matrix we use this to show that when n 3 and a is commutative every matrix in sl n a is the product of two commutators

an upper bound on reidemeister moves we provide an explicit upper bound on the number of reidemeister moves required to pass between two diagrams of the same link this leads to a conceptually simple solution to the equivalence problem for links

on singular functional nonsmooth and implicit phi laplacian initial and boundary value problems in this paper we apply fixed point results for mappings in partially ordered spaces to derive existence results for extremal solutions of phi laplacian initial and boundary value problems the considered problems can be singular functional discontinuous nonlocal and implicit concrete examples are also solved

positive solutions for a class of implicit and discontinuous second order functional differential equations with singularities we derive existence and approximation results for a type of implicit second order differential equations related with diffusion processes singularities discontinuities and functional dependence are allowed

extremality results for discontinuous explicit and implicit diffusion problems extremality and comparison results are derived for explicit and implicit diffusion problems differential equations of problems may involve discontinuous nonlinearities and functional terms

existence theory for functional p laplacian equations with variable exponents in this paper we consider the solvability of equations of the form d dt t u u t u t f t u u t u t for a e t i a b subject to a general type of functional boundary conditions which cover dirichlet and periodic boundary data as particular cases our approach is that of upper and lower solutions together with growth restrictions of nagumo s type an example is provided where a p laplacian with variable p is shown to have a solution between given upper and lower solutions

a homotopic deformation along p of a leray schauder degree result and existence for u p 2u ƒ t u 0 u 0 u t 0 p 1 soit f 0 1 r r une fonction continue et soit o p r r definie par o p s s p 2 s pour p fixe 1 on considere le probleme de dirichlet suivant o p u f t u 0 u 0 u t 0 ou t est un nombre reel positif

mountain pass solutions to euler lagrange equations with general anisotropic operator abstract using the mountain pass theorem we show that the problem d d t l v t u t u t l x t u t u t for a e t a b u a u b 0 has a solution in anisotropic orlicz sobolev space we consider lagrangian l f t x v v t x f t x with growth conditions determined by anisotropic g function and some geometric conditions of ambrosetti rabinowitz type

a note on olssonʼs conjecture abstract for a p block b of a finite group g with defect group d olsson conjectured that k 0 b d d where k 0 b is the number of characters in b of height 0 and d denotes the commutator subgroup of d brauer deduced olsson s conjecture in the case where d is a dihedral 2 group using the fact that certain algebraically conjugate subsections are also conjugate in g we generalize brauer s argument for arbitrary primes p and arbitrary defect groups this extends two results by robinson for p 3 we show that olsson s conjecture is satisfied for defect groups of p rank 2 and for minimal non abelian defect groups

on the sobolev embedding theorem for variable exponent spaces in the critical range abstract in this paper we study the sobolev embedding theorem for variable exponent spaces with critical exponents we find conditions on the best constant in order to guaranty the existence of extremals the proof is based on a suitable refinement of the estimates in the concentration compactness theorem for variable exponents and an adaptation of a convexity argument due to p l lions f pacella and m tricarico

existence uniqueness and approximation of solutions to some nonlinear diffusion problems abstract we study the existence and approximation of a nontrivial positive solution for a nonlinear ordinary differential equation of second order to prove the uniqueness of positive solutions we use some estimates of the error between exact and approximate solutions the equation arises in the study of some nonlinear diffusion problems

existence of solutions for some nonlinear problems with p laplacian like operators abstract the purpose of this paper is to obtain some existence results of solutions for the nonlinear boundary value problems with p laplacian like operators

existence of periodic solutions to a p laplacian lienard differential equation with a deviating argument abstract by means of mawhin s continuation theorem a kind of p laplacian lienard differential equation with a deviating argument as follows p y t f y t y t h y t g y t t e t is studied a new result on the existence of periodic solutions is obtained the interest is that the relation between the existence of periodic solutions and the deviating argument t is investigated meanwhile the approaches used to estimate a priori bounds of periodic solutions are different from the corresponding ones in the known literature

existence of multiple solutions for weighted p r laplacian equation dirichlet problems abstract this paper deals with the existence of multiple solutions for weighted p r laplacian equation dirichlet problems several sub super solution type results have been given via leray schauder s degree method in particular we give a counterexample which means that 2 in paper j henderson h b thompson existence of multiple solutions for second order boundary value problems j differential equations 166 2000 443 454 is not an open subset of c 0 1

existence results and approximation methods for functional ordinary differential equations with singular diffusion type differential operators abstract we derive sufficient conditions for the existence of extremal solutions for a second order singular functional differential equation subject to initial data the type of equations that we study here can be regarded as stationary and one dimensional models for diffusion processes in which the diffusion coefficient is not a constant we have also tried to relax the regularity assumptions as far as possible in order to extend the applicability of our result we present an example in which the extremal solutions are approximated

cusp shapes under cone deformation a horospherical torus about a cusp of a hyperbolic manifold inherits a euclidean similarity structure called a cusp shape we bound the change in cusp shape when the hyperbolic structure of the manifold is deformed via cone deformation preserving the cusp the bounds are in terms of the change in structure in a neighborhood of the singular locus alone we then apply this result to provide information on the cusp shape of many hyperbolic knots given only a diagram of the knot specifically we show there is a universal constant c such that if a knot admits a prime twist reduced diagram with at least c crossings per twist region then the length of the second shortest curve on the cusp torus is bounded the bound is linear in the number of twist regions of the diagram