s shaped bifurcation curve for a nonlocal boundary value problem abstract in this paper by using tools from bifurcation theory particularly the crandall rabinowitz bifurcation theorem the implicit function theorem and the persistence results of degenerate solutions due to e n dancer a result will be proved for a s shaped bifurcation curve of a nonlocal boundary value problem the main results of this paper partially extend the main results in 11 12 to the nonlocal boundary value problems

nodal solutions for second order m point boundary value problems with nonlinearities across several eigenvalues abstract we study the existence of nodal solutions of the m point boundary value problem u f u 0 0 t 1 u 0 0 u 1 i 1 m 2 i u i where i q i 1 2 m 2 with 0 1 2 m 2 1 and i r i 1 2 m 2 with i 0 and 0 i 1 m 2 i 1 we give conditions on the ratio f s s at infinity and zero that guarantee the existence of nodal solutions the proofs of our main results are based upon bifurcation techniques

braid groups and the co hopfian property abstract let b n be the braid group on n 4 strands we prove that b n modulo its center is co hopfian we then show that any injective endomorphism of b n is geometric in the sense that it is induced by a homeomorphism of a punctured disk we further prove that any injection from b n to b n 1 is geometric for n 7 additionally we obtain analogous results for mapping class groups of punctured spheres the methods use thurston s theory of surface homeomorphisms and build upon work of ivanov mccarthy

nodal solutions of multi point boundary value problems abstract we study the nonlinear boundary value problem consisting of the equation y w t f y 0 on a b and a multi point boundary condition by relating it to the eigenvalues of a linear sturm liouville problem with a two point separated boundary condition we obtain results on the existence and nonexistence of nodal solutions of this problem we also discuss the changes in the existence question for different types of nodal solutions as the problem changes

spectral properties of p laplacian problems with neumann and mixed type multi point boundary conditions abstract we consider the boundary value problem consisting of the p laplacian equation 1 p u p u on 1 1 where p 1 p s s p 1 sgn s for s r r together with the multi point boundary conditions 2 p u 1 i 1 m i p u i or 3 u 1 i 1 m i u i or a mixed pair of these conditions with one condition holding at each of x 1 and x 1 in 2 3 m 1 are integers i 1 1 1 i m and the coefficients i satisfy i 1 m i 1 we term the conditions 2 and 3 respectively neumann type and dirichlet type boundary conditions since they reduce to the standard neumann and dirichlet boundary conditions when 0 given a suitable pair of boundary conditions a number is an eigenvalue of the corresponding boundary value problem if there exists a non trivial solution u an eigenfunction the spectrum of the problem is the set of eigenvalues in this paper we obtain various spectral properties of these eigenvalue problems we then use these properties to prove rabinowitz type global bifurcation theorems for related bifurcation problems and to obtain nonresonance conditions in terms of the eigenvalues for the solvability of related inhomogeneous problems

linear and nonlinear second order problems with sturm liouville type multi point boundary conditions abstract we consider the nonlinear boundary value problem consisting of the equation u f u on 1 1 where f r r is continuous together with general sturm liouville type multi point boundary conditions at 1 we obtain rabinowitz type global bifurcation results and then use these to obtain nodal solutions of the problem we conclude with a nonresonance result for an inhomogeneous form of the problem these results rely on the spectral properties of the eigenvalue problem consisting of the equation u u on 1 1 together with the multi point boundary conditions in a previous paper it was shown that under certain optimal conditions the basic spectral properties of this eigenvalue problem are similar to those of the standard sturm liouville problem with single point boundary conditions in particular for each integer k 0 there exists a unique simple eigenvalue k whose eigenfunctions have oscillation count equal to k where the oscillation count was defined in terms of a complicated prufer angle construction unfortunately it seems to be difficult to apply the prufer angle construction to the nonlinear problem accordingly in this paper we use alternative non optimal oscillation counting methods to obtain the required spectral properties of the linear problem and these are then applied to the nonlinear problem to yield the results mentioned above

nodal solutions for nonlinear eigenvalue problems we are concerned with determining values of for which there exist nodal solutions of the boundary value problems u ra t f u 0 0 t 1 u o u 1 0 the proof of our main result is based upon bifurcation techniques

on injective homomorphisms between teichmuller modular groups i in this paper and its sequel we prove that injective homomorphisms between teichmuller modular groups of compact orientable surfaces are necessarily isomorphisms if an appropriately measured size of the surfaces in question differs by at most one in particular we establish the co hopfian property for modular groups of surfaces of positive genus

nodal solutions for m point boundary value problems using bifurcation methods abstract in this paper we first obtain some results on the structure of solution sets of some m point boundary value problems then using these results we obtain some multiplicity results for nodal solutions of m point boundary value problems the method to show our main results is based on the structure of solution sets due to p h rabinowitz

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

on the existence of multiple solutions of the boundary value problem for nonlinear second order differential equations abstract we consider the boundary value problem for nonlinear second order differential equations of the form u a x f u 0 0 u 0 u 1 0 we establish the precise condition concerning the behavior of the ratio f s s at infinity and zero for the existence of solutions with prescribed nodal properties then we derive the existence and the multiplicity of nodal solutions to the problem our argument is based on the shooting method together with the strum s comparison theorem the results obtained here can be applied to the study of radially symmetric solutions of the dirichlet problem for semilinear elliptic equations in annular domains

global bifurcation for 2mth order boundary value problems and infinitely many solutions of superlinear problems we consider the boundary value problem lu x p x u x g x u 0 x u 2m 1 x u x x 0 where i l is a 2mth order self adjoint disconjugate ordinary differential operator on 0 together with separated boundary conditions at 0 and ii p is continuous and p 0 on 0 while p 0 on any interval in 0 iii g 0 r2m r is continuous and there exist increasing functions u l 0 0 such that u 1 g x l 1 0 lim 0g x 0 with limt l t the non linear term in is superlinear as u x we obtain a global bifurcation result for a related bifurcation problem we then use this to obtain infinitely many solutions of having specified nodal properties

second order multi point problems with variable coefficients abstract in this paper we consider the eigenvalue problem consisting of the equation u r u on 1 1 where r c 1 1 1 r 0 and r together with the multi point boundary conditions u 1 i 1 m i u i where m 1 are integers and for i 1 m i r i 1 1 with i 1 i 1 we show that if the coefficients i r are sufficiently small depending on r then the spectral properties of this problem are similar to those of the usual separated problem but if the coefficients i are not sufficiently small then these standard spectral properties need not hold the spectral properties of such multi point problems have been obtained before for the constant coefficient case r 1 but the variable coefficient case has not been considered previously apart from the existence of principal eigenvalues some nonlinear multi point problems are also considered we obtain a partial rabinowitz type result on global bifurcation from the eigenvalues and various nonresonance conditions for the existence of general solutions and also of nodal solutions these results rely on the spectral properties of the linear problem

spectral properties of second order multi point p laplacian boundary value problems abstract we consider the multi point boundary value problem p u p u on 1 1 u 1 i 1 m i u i where p 1 p s s p 1 sgn s for s r r m 1 are integers i 1 1 1 i m and the coefficients i satisfy i 1 m i 1 a number r is said to be an eigenvalue of the above problem if there exists a non trivial solution u the spectrum is the set of eigenvalues in this paper we obtain some basic spectral and degree theoretic properties of this eigenvalue problem these results have numerous applications to more general problems as an example a rabinowitz type global bifurcation theorem is briefly described

nodal solutions of boundary value problems with boundary conditions involving riemann stieltjes integrals abstract we study the nonlinear boundary value problem consisting of the equation y i 1 m w i t f i y and a boundary condition involving a riemann stieltjes integral by relating it to the eigenvalues of the corresponding linear sturm liouville problem with a two point separated boundary condition we obtain results on the existence and nonexistence of nodal solutions of this problem the shooting method and an energy function are used to prove the main results

second order three point boundary value problems with jumping non linearities abstract we consider the non linear three point boundary value problem consisting of the equation 1 u f u h a e on 0 1 where h l 1 0 1 together with the boundary conditions 2 u 0 0 u 1 u where 0 1 the function f r r is continuous and we assume that the following limits exist and are finite f lim s f s s we allow f f such a non linearity f is said to be jumping related to 1 is the equation 3 u a u b u u on 0 1 where a b r and u x max u x 0 for x 0 1 the problem 2 and 3 is positively homogeneous and jumping values of for which 2 and 3 has a non trivial solution u will be called half eigenvalues while the corresponding solutions u will be called half eigenfunctions we show that a sequence of half eigenvalues exists the corresponding half eigenfunctions having specified nodal properties and we obtain certain spectral and degree theoretic properties of the set of half eigenvalues these properties lead to solvability and non solvability results for the problem 1 and 2 the set of half eigenvalues is closely related to the fucik spectrum of the problem and equivalent solvability and non solvability results for 1 and 2 are obtained from either the half eigenvalue or the fucik spectrum approach

the automorphism groups of the braid groups in the first of two papers published in the annals in 1947 3 emil artin mentioned the problem of determining all automorphisms of the braid groups of the euclidean plane and in the second 4 took a first step towards a solution the main result of this paper is a complete determination of these automorphism groups the outer automorphism group is of order two generated by the automorphism class containing the element which inverts each of the standard braid generators visually a plane projection of the automorphic image of a braid m is obtained from one for f by replacing each over crossing by an under crossing this result establishes a recent conjecture of pietrowski and solitar to whom we are indebted for bringing the problem to our attention they obtained the first determination of the automorphism group of the fourstring braid group using techniques for computing automorphism groups of amalgamated free products 13 the result for braids on at most three strings is easy and seems part of the folklore the argument we employ is algebraic and is patterned after that

on matrix inverses modulo a subspace given a pair of matrices a b rn n rn m with coefficients in a commutative ring we study the problem of finding a matrix f such that a bf becomes invertible we point out some relations between the problem of finding feedback inverses of a modulo b and the pole shifting problem for the pair a b in fact we give feedback invertibility results over a well known large class of rings related with the pole shifting the class of pa rings on the other hand we also give a pointwise global characterization the ring r of rational integers and the coordinate ring of the real unit circle r x y x2 y2 1 are studied in some detail finally the problem of derivative feedback standardization of generalized linear systems is reviewed as a particular case

spectral properties and nodal solutions for second order m point boundary value problems abstract we consider the m point boundary value problem consisting of the equation u f u on 0 1 where f r r is c 1 with f 0 0 together with the boundary conditions u 0 0 u 1 i 1 m 2 i u i where m 3 i 0 1 and i 0 for i 1 m 2 with i 1 m 2 i 1 we first show that the spectral properties of the linearisation of this problem are similar to the well known properties of the standard sturm liouville problem with separated boundary conditions with a minor modification to deal with the multi point boundary condition these spectral properties are then used to prove a rabinowitz type global bifurcation theorem for a bifurcation problem related to the above problem finally we use the global bifurcation theorem to obtain nodal solutions that is sign changing solutions with a specified number of zeros of the above problem under various conditions on the asymptotic behaviour of f