Sturm-Liouville Problem on Unbounded Interval (joint work with Alois Kufner)
|
|
- Eleanor Stevens
- 5 years ago
- Views:
Transcription
1 (joint work with Alois Kufner) Pavel Drábek Department of Mathematics, Faculty of Applied Sciences University of West Bohemia, Pilsen Workshop on Differential Equations Hejnice, September 16-20, 2007 Pavel Drábek 19th September / 22
2 Functional setting Introduction Let p > 1 be a real number and let ϕ : R R be defined as ϕ(s) = s p 2 s for s 0, ϕ(0) = 0. Let r = r(t), c = c(t) be continuous and positive functions on [0, ). For x = x(t) defined on [0, ) denote x( ) := lim t x(t). We study the eigenvalue problem { (r(t)ϕ(x (t))) + λc(t)ϕ(x(t)) = 0, t 0, x (1) (0) = 0, x( ) = 0, where λ R is a spectral parameter. Pavel Drábek 19th September / 22
3 Functional setting Introduction Let p > 1 be a real number and let ϕ : R R be defined as ϕ(s) = s p 2 s for s 0, ϕ(0) = 0. Let r = r(t), c = c(t) be continuous and positive functions on [0, ). For x = x(t) defined on [0, ) denote x( ) := lim t x(t). We study the eigenvalue problem { (r(t)ϕ(x (t))) + λc(t)ϕ(x(t)) = 0, t 0, x (1) (0) = 0, x( ) = 0, where λ R is a spectral parameter. A function x W 1,p (r) (the weighted Sobolev space being defined later) is called a weak solution of (1) if the integral identity 0 r(t)ϕ(x (t))y (t)dt = λ holds for all y W 1,p (r) (with both integrals being finite). 0 c(t)ϕ(x(t))y(t)dt (2) Pavel Drábek 19th September / 22
4 Introduction Eigenvalues, eigenfunctions The parameter λ is called an eigenvalue of (1) if this problem has a nontrivial (i.e. nonzero) weak solution (called an eigenfunction of (1)). Pavel Drábek 19th September / 22
5 Introduction Eigenvalues, eigenfunctions The parameter λ is called an eigenvalue of (1) if this problem has a nontrivial (i.e. nonzero) weak solution (called an eigenfunction of (1)). (S.L.) Property for (1): "The set of all eigenvalues of (1) forms an increasing sequence {λ n } n=1 such that λ 1 > 0 and lim λ n =. n Every eigenvalue λ n, n = 1, 2,..., is simple in the sense that there exists a unique normalized eigenfunction x λn associated with λ n. Moreover, the eigenfunction x λn has precisely n 1 zeros in (0, ). In particular, x λ1 does not change sign in (0, ). For n 3, between two consecutive zeros of x λn 1 in (0, ) there is exactly one zero of x λn." Pavel Drábek 19th September / 22
6 Condition on weights Introduction Our main result depends on the following condition on the weight functions (coefficients in the equation) r and c: ( t ) 1/p ( ) 1/p lim c(τ)dτ r 1 p (τ)dτ t 0 t = 0. (3) Pavel Drábek 19th September / 22
7 Example Sturm-Liouville Problem on Unbounded Interval Introduction Let p = 2, r c 1. Then evidently there is no (weak) solution of for any λ R. x (t) + λx(t) = 0, x (0) = 0, x( ) = 0 Pavel Drábek 19th September / 22
8 Introduction Example Let p = 2, r c 1. Then evidently there is no (weak) solution of x (t) + λx(t) = 0, x (0) = 0, x( ) = 0 for any λ R. Note that in this case we have ( t ) 1/p ( c(τ)dτ 0 t ) 1/p r 1 p (τ)dτ, t (0, ). Pavel Drábek 19th September / 22
9 Example Sturm-Liouville Problem on Unbounded Interval Introduction Let p = 2, r(t) = (t + 1) 2, c(t) 1. Then we have for t. ( t ) 1/p ( c(τ)dτ 0 0 t t ) 1/p r 1 p (τ)dτ ( t ) 1/2 ( dτ ) 1/2 ( t ) 1/2 = dτ = 1 (1 + τ) t Pavel Drábek 19th September / 22
10 Example Sturm-Liouville Problem on Unbounded Interval Introduction Let p = 2, r(t) = (t + 1) 2, c(t) 1. Then we have for t. ( t ) 1/p ( c(τ)dτ 0 0 t t ) 1/p r 1 p (τ)dτ ( t ) 1/2 ( dτ ) 1/2 ( t ) 1/2 = dτ = 1 (1 + τ) t The boundary value problem has no eigenvalue. ((t + 1) 2 x (t)) + λx(t) = 0, x (0) = 0, x( ) = 0 (4) Pavel Drábek 19th September / 22
11 Remark Sturm-Liouville Problem on Unbounded Interval Introduction Pavel Drábek 19th September / 22
12 Remark Sturm-Liouville Problem on Unbounded Interval Introduction The foregoing examples indicate that if the condition (3) is violated, then the (S.L.) Property for (1) need not be satisfied. Pavel Drábek 19th September / 22
13 Weighted spaces Functional setting Let L p (c) denote the weighted Lebesgue space of all functions x = x(t) defined on (0, ), for which ( x p;c := 0 ) 1/p c(t) x(t) p dt is finite. Then L p (c) equipped with the norm p;c is a uniformly convex Banach space. Pavel Drábek 19th September / 22
14 Weighted spaces Functional setting Let L p (c) denote the weighted Lebesgue space of all functions x = x(t) defined on (0, ), for which ( x p;c := 0 ) 1/p c(t) x(t) p dt is finite. Then L p (c) equipped with the norm p;c is a uniformly convex Banach space. Let W 1,p (r) be the set of all absolutely continuous functions x = x(t) defined on [0, ) such that x( ) = 0 and ( x 1,p;r := 0 ) 1/p r(t) x (t) p dt is finite. Then W 1,p (r) equipped with the norm 1,p;r is a uniformly convex Banach space. Pavel Drábek 19th September / 22
15 Weak solution Functional setting A function x W 1,p (r) is called a weak solution of (1) if the integral identity 0 r(t)ϕ(x (t))y (t)dt = λ holds for all y W 1,p (r) (with both integrals being finite). 0 c(t)ϕ(x(t))y(t)dt (2) Pavel Drábek 19th September / 22
16 Weak solution Functional setting A function x W 1,p (r) is called a weak solution of (1) if the integral identity 0 r(t)ϕ(x (t))y (t)dt = λ holds for all y W 1,p (r) (with both integrals being finite). 0 c(t)ϕ(x(t))y(t)dt (2) Let us note that if x W 1,p (r) is a weak solution of (1), then rϕ(x ) C 1 [0, ), the equation in (1) is satisfied at every point and x (0) = x( ) = 0. Pavel Drábek 19th September / 22
17 Weak solution Functional setting A function x W 1,p (r) is called a weak solution of (1) if the integral identity 0 r(t)ϕ(x (t))y (t)dt = λ holds for all y W 1,p (r) (with both integrals being finite). 0 c(t)ϕ(x(t))y(t)dt (2) Let us note that if x W 1,p (r) is a weak solution of (1), then rϕ(x ) C 1 [0, ), the equation in (1) is satisfied at every point and x (0) = x( ) = 0. The reason for considering weak solutions consists in reformulating (1) as an abstract nonlinear eigenvalue problem and finding the eigenvalues in a "constructive way" employing the tools of nonlinear functional analysis. Pavel Drábek 19th September / 22
18 Compact embedding Functional setting In order to use compactness arguments (associated with the Palais-Smale condition) we need the compact embedding: W 1,p (r) L p (c). (5) Pavel Drábek 19th September / 22
19 Compact embedding Functional setting In order to use compactness arguments (associated with the Palais-Smale condition) we need the compact embedding: W 1,p (r) L p (c). (5) This compactness result holds if and only if condition (3) is satisfied. It follows e.g. from the results collected in the book [1] B. Opic and A. Kufner Hardy-Type Inequalities, Pitman Research Notes in Mathematics Series 279, Longman Scientific and Technical, Harlow Pavel Drábek 19th September / 22
20 Theorem Sturm-Liouville Problem on Unbounded Interval Main result The condition (3) is satisfied if and only if the (S.L.) Property for (1) holds. Pavel Drábek 19th September / 22
21 Tools Sturm-Liouville Problem on Unbounded Interval Main result Pavel Drábek 19th September / 22
22 Tools Sturm-Liouville Problem on Unbounded Interval Main result Variational characterization of eigenvalues. Pavel Drábek 19th September / 22
23 Tools Sturm-Liouville Problem on Unbounded Interval Main result Variational characterization of eigenvalues. Oscillatory criteria for half-linear equations. Pavel Drábek 19th September / 22
24 Tools Sturm-Liouville Problem on Unbounded Interval Main result Variational characterization of eigenvalues. Oscillatory criteria for half-linear equations. Properties of the initial-value problem for half-linear equation. Pavel Drábek 19th September / 22
25 Principal eigenvalue Variational approach Pavel Drábek 19th September / 22
26 Principal eigenvalue Variational approach Having the compact embedding (5) available, we can apply the Lagrange multiplier method to get the following assertion: Pavel Drábek 19th September / 22
27 Principal eigenvalue Variational approach Having the compact embedding (5) available, we can apply the Lagrange multiplier method to get the following assertion: Let us assume (3). Then (1) has the least eigenvalue λ 1 > 0 which can be characterized as follows: 0 r(t) x (t) p dt λ 1 = min 0 c(t) x(t) p dt, (7) where the miminum is taken over all x W 1,p (r), x 0. Pavel Drábek 19th September / 22
28 Higher eigenvalues Variational approach Pavel Drábek 19th September / 22
29 Higher eigenvalues Variational approach Let S := {x W 1,p (r) : x p;c = 1}. Let S k be the unit sphere in R k. For k N define a family of sets F k := {A S : A = A, A = h(s k ) with h a continuous odd mapping from R k into S}. Pavel Drábek 19th September / 22
30 Higher eigenvalues Variational approach Let S := {x W 1,p (r) : x p;c = 1}. Let S k be the unit sphere in R k. For k N define a family of sets F k := {A S : A = A, A = h(s k ) with h a continuous odd mapping from R k into S}. Let us consider the functional I(x) := x p 1,p;r. The compact embedding (5) allows to prove that for any k N, λ k = min max A F k x A I(x) (8) are eigenvalues of (1). The interested listener is referred to [2] P. Drábek and S.B. Robinson Resonance problems for the p-laplacian, Journal of Functional Analysis 169 (1999), for the proof which can be literally adapted in our situation. Pavel Drábek 19th September / 22
31 Proposition Sturm-Liouville Problem on Unbounded Interval Variational approach Pavel Drábek 19th September / 22
32 Proposition Sturm-Liouville Problem on Unbounded Interval Variational approach Let (3) be satisfied. The sequence {λ k } k=1 defined by (8) forms the sequence of eigenvalues of (1) and lim λ n =. n Pavel Drábek 19th September / 22
33 Proposition Sturm-Liouville Problem on Unbounded Interval Variational approach Let (3) be satisfied. The sequence {λ k } k=1 defined by (8) forms the sequence of eigenvalues of (1) and lim λ n =. n Warning! We do not know that this is the entire set of all eigenvalues yet!!! It remains to show that the sequence {λ k } k=1 exhausts the set of all eigenvalues of (1). Pavel Drábek 19th September / 22
34 Proposition Sturm-Liouville Problem on Unbounded Interval Variational approach Let (3) be satisfied. The sequence {λ k } k=1 defined by (8) forms the sequence of eigenvalues of (1) and lim λ n =. n Warning! We do not know that this is the entire set of all eigenvalues yet!!! It remains to show that the sequence {λ k } k=1 exhausts the set of all eigenvalues of (1). To prove this fact is made possible by using purely ODE techniques. Pavel Drábek 19th September / 22
35 Existence and uniqueness The initial value problem Pavel Drábek 19th September / 22
36 The initial value problem Existence and uniqueness Let t 0 [0, ), A, B R. Consider the initial value problem { (r(t)ϕ(x (t))) + λc(t)ϕ(x(t)) = 0, t > t 0, x(t 0 ) = A, x (t 0 ) = B. (9) Pavel Drábek 19th September / 22
37 The initial value problem Existence and uniqueness Let t 0 [0, ), A, B R. Consider the initial value problem { (r(t)ϕ(x (t))) + λc(t)ϕ(x(t)) = 0, t > t 0, x(t 0 ) = A, x (t 0 ) = B. (9) Existence, uniqueness, global extensibility of the solution up to follows from [3] A. Elbert A half-linear second order differential equation, Colloq. Math. Soc. János Bolyai 30 (1979), Pavel Drábek 19th September / 22
38 The initial value problem Disconjugacy and comparison of zeros Pavel Drábek 19th September / 22
39 The initial value problem Disconjugacy and comparison of zeros In order to compare zeros of different eigenfunctions we use the disconjugacy criteria for half-linear equations: [4] O. Došlý Oscillation criteria for half-linear second order differential equation, Hiroshima J. Math. 28 (1998), Pavel Drábek 19th September / 22
40 Oscillatory criteria The initial value problem Pavel Drábek 19th September / 22
41 Oscillatory criteria The initial value problem Došlý [4]: Let lim sup t ( ) p 1 ( t r 1 p (τ)dτ t Then the equation in (9) is non oscillatory. 0 ) (p 1)p 1 c(τ)dτ < λp p. Pavel Drábek 19th September / 22
42 The initial value problem Oscillatory criteria Došlý [4]: Let lim sup t ( ) p 1 ( t r 1 p (τ)dτ t Then the equation in (9) is non oscillatory. 0 ) (p 1)p 1 c(τ)dτ < λp p. Let lim inf t ( ) p 1 ( t r 1 p (τ)dτ t Then the equation in (9) is oscillatory. 0 ) (p 1)p 1 c(τ)dτ > λp p. Pavel Drábek 19th September / 22
43 Corollary Sturm-Liouville Problem on Unbounded Interval The initial value problem Pavel Drábek 19th September / 22
44 Corollary Sturm-Liouville Problem on Unbounded Interval The initial value problem Let us assume (3). Then the equation in (9) is nonoscillatory for any λ R. Let us assume that (3) does not hold, i.e., there is a sequence t n (as n ) such that ( t n lim inf c(τ)dτ t n 0 ) 1/p ( t n ) 1/p r 1 p (τ)dτ > 0. Then there exists λ 0 > 0 such that the equation in (9) is oscillatory provided λ λ 0. Pavel Drábek 19th September / 22
45 Necessity of (3) Main result Pavel Drábek 19th September / 22
46 Necessity of (3) Main result Assume the (S.L.) Property for (1). Then the condition (3) must hold. Pavel Drábek 19th September / 22
47 Necessity of (3) Main result Assume the (S.L.) Property for (1). Then the condition (3) must hold. Idea of the proof: Assume that (3) is violated. Then according to the assertion above there exists λ 0 > 0 such that the equation in (9) is oscillatory provided λ λ 0. But then taking the eigenvalue λ n λ 0 the corresponding eigenfunction does not vanish on [a, b] with a and b large enough. This would imply that (9) is disconjugate on [a, b] which contradicts its oscillatory behavior. Pavel Drábek 19th September / 22
48 Sufficiency of (3) Main result Pavel Drábek 19th September / 22
49 Sufficiency of (3) Main result Assume (3). Then the (S.L.) Property for (1) must hold. Pavel Drábek 19th September / 22
50 Sufficiency of (3) Main result Assume (3). Then the (S.L.) Property for (1) must hold. Idea of the proof: With the sequence of "variational eigenvalues" {λ n } n=1 and corresponding eigenfunctions x λn in hands we employ comparison, disconjugacy and oscillatory criteria and proceed to show that besides of x λn there are no other eigenfunctions of (1). The sequence {λ n } n=1 thus exhausts the set of all eigenvalues of (1). Pavel Drábek 19th September / 22
51 END Thank you very much for your attention Pavel Drábek 19th September / 22
52 END Thank you very much for your attention Pavel Drábek 19th September / 22
Existence of Multiple Positive Solutions of Quasilinear Elliptic Problems in R N
Advances in Dynamical Systems and Applications. ISSN 0973-5321 Volume 2 Number 1 (2007), pp. 1 11 c Research India Publications http://www.ripublication.com/adsa.htm Existence of Multiple Positive Solutions
More informationVariational eigenvalues of degenerate eigenvalue problems for the weighted p-laplacian
Variational eigenvalues of degenerate eigenvalue problems for the weighted p-laplacian An Lê Mathematics Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720 e-mail: anle@msri.org Klaus
More informationLeighton Coles Wintner Type Oscillation Criteria for Half-Linear Impulsive Differential Equations
Advances in Dynamical Systems and Applications ISSN 973-5321, Volume 5, Number 2, pp. 25 214 (21) http://campus.mst.edu/adsa Leighton Coles Wintner Type Oscillation Criteria for Half-Linear Impulsive Differential
More informationThe Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:
Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply
More informationLayer structures for the solutions to the perturbed simple pendulum problems
Layer structures for the solutions to the perturbed simple pendulum problems Tetsutaro Shibata Applied Mathematics Research Group, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima,
More informationResearch Article On the Variational Eigenvalues Which Are Not of Ljusternik-Schnirelmann Type
Abstract and Applied Analysis Volume 2012, Article ID 434631, 9 pages doi:10.1155/2012/434631 Research Article On the Variational Eigenvalues Which Are Not of Ljusternik-Schnirelmann Type Pavel Drábek
More informationEXISTENCE THEORY FOR NONLINEAR STURM-LIOUVILLE PROBLEMS WITH NON-LOCAL BOUNDARY CONDITIONS. 1. Introduction
EXISTENCE THEORY FOR NONLINEAR STURM-LIOUVILLE PROBLEMS WITH NON-LOCAL BOUNDARY CONDITIONS DANIEL MARONCELLI AND JESÚS RODRÍGUEZ Abstract. In this work we provide conditions for the existence of solutions
More informationOscillation Criteria for Certain nth Order Differential Equations with Deviating Arguments
Journal of Mathematical Analysis Applications 6, 601 6 001) doi:10.1006/jmaa.001.7571, available online at http://www.idealibrary.com on Oscillation Criteria for Certain nth Order Differential Equations
More informationNontrivial Solutions for Boundary Value Problems of Nonlinear Differential Equation
Advances in Dynamical Systems and Applications ISSN 973-532, Volume 6, Number 2, pp. 24 254 (2 http://campus.mst.edu/adsa Nontrivial Solutions for Boundary Value Problems of Nonlinear Differential Equation
More informationEXISTENCE RESULTS FOR QUASILINEAR HEMIVARIATIONAL INEQUALITIES AT RESONANCE. Leszek Gasiński
DISCRETE AND CONTINUOUS Website: www.aimsciences.org DYNAMICAL SYSTEMS SUPPLEMENT 2007 pp. 409 418 EXISTENCE RESULTS FOR QUASILINEAR HEMIVARIATIONAL INEQUALITIES AT RESONANCE Leszek Gasiński Jagiellonian
More informationOn the discrete boundary value problem for anisotropic equation
On the discrete boundary value problem for anisotropic equation Marek Galewski, Szymon G l ab August 4, 0 Abstract In this paper we consider the discrete anisotropic boundary value problem using critical
More informationPOTENTIAL LANDESMAN-LAZER TYPE CONDITIONS AND. 1. Introduction We investigate the existence of solutions for the nonlinear boundary-value problem
Electronic Journal of Differential Equations, Vol. 25(25), No. 94, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) POTENTIAL
More informationOn the generalized Emden-Fowler differential equations
On the generalized Emden-Fowler differential equations Joint research with Mauro Marini Topological and Variational Methods for ODEs Dedicated to Massimo Furi Professor Emeritus at the University of Florence,
More informationNonvariational problems with critical growth
Nonlinear Analysis ( ) www.elsevier.com/locate/na Nonvariational problems with critical growth Maya Chhetri a, Pavel Drábek b, Sarah Raynor c,, Stephen Robinson c a University of North Carolina, Greensboro,
More informationBifurcation from the rst eigenvalue of some nonlinear elliptic operators in Banach spaces
Nonlinear Analysis 42 (2000) 561 572 www.elsevier.nl/locate/na Bifurcation from the rst eigenvalue of some nonlinear elliptic operators in Banach spaces Pavel Drabek a;, Nikos M. Stavrakakis b a Department
More informationOscillation Theorems for Second-Order Nonlinear Dynamic Equation on Time Scales
Appl. Math. Inf. Sci. 7, No. 6, 289-293 (203) 289 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/0.2785/amis/070608 Oscillation heorems for Second-Order Nonlinear
More informationAsymptotic theory of second order nonlinear differential equations: quadro completo
Asymptotic theory of second order nonlinear differential equations: quadro completo Zuzana Došlá Joint research with Mauro Marini Convegno dedicato a Mauro Marini, Firenze, Decembre 2-3, 2016 Table of
More informationBASIS PROPERTIES OF EIGENFUNCTIONS OF THE p-laplacian
BASIS PROPERTIES OF EIGENFUNCTIONS OF THE p-laplacian Paul Binding 1, Lyonell Boulton 2 Department of Mathematics and Statistics, University of Calgary Calgary, Alberta, Canada T2N 1N4 Jan Čepička3, Pavel
More informationSPECTRAL PROPERTIES AND NODAL SOLUTIONS FOR SECOND-ORDER, m-point, BOUNDARY VALUE PROBLEMS
SPECTRAL PROPERTIES AND NODAL SOLUTIONS FOR SECOND-ORDER, m-point, BOUNDARY VALUE PROBLEMS BRYAN P. RYNNE Abstract. We consider the m-point boundary value problem consisting of the equation u = f(u), on
More informationRNDr. Petr Tomiczek CSc.
HABILITAČNÍ PRÁCE RNDr. Petr Tomiczek CSc. Plzeň 26 Nonlinear differential equation of second order RNDr. Petr Tomiczek CSc. Department of Mathematics, University of West Bohemia 5 Contents 1 Introduction
More informationAsymptotic properties of solutions of ordinary differential equations
Masaryk University Brno Zuzana Došlá Asymptotic properties of solutions of ordinary differential equations Syllabus of the dissertation Brno, February 24 1 1 Subject, methods and results of the dissertation
More informationPseudo-monotonicity and degenerate elliptic operators of second order
2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 9 24. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu
More informationExistence of Solutions for a Class of p(x)-biharmonic Problems without (A-R) Type Conditions
International Journal of Mathematical Analysis Vol. 2, 208, no., 505-55 HIKARI Ltd, www.m-hikari.com https://doi.org/0.2988/ijma.208.886 Existence of Solutions for a Class of p(x)-biharmonic Problems without
More informationLayer structures for the solutions to the perturbed simple pendulum problems
J. Math. Anal. Appl. 35 (26) 725 739 www.elsevier.com/locate/jmaa Layer structures for the solutions to the perturbed simple pendulum problems Tetsutaro Shibata Applied Mathematics Research Group, Graduate
More informationEIGENVALUE QUESTIONS ON SOME QUASILINEAR ELLIPTIC PROBLEMS. lim u(x) = 0, (1.2)
Proceedings of Equadiff-11 2005, pp. 455 458 Home Page Page 1 of 7 EIGENVALUE QUESTIONS ON SOME QUASILINEAR ELLIPTIC PROBLEMS M. N. POULOU AND N. M. STAVRAKAKIS Abstract. We present resent results on some
More informationEXISTENCE OF SOLUTIONS FOR A RESONANT PROBLEM UNDER LANDESMAN-LAZER CONDITIONS
Electronic Journal of Differential Equations, Vol. 2008(2008), No. 98, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXISTENCE
More informationGeneralized trigonometric functions
7/211, Santiago de Compostela (Etwas allgemein machen heisst, es denken. G.W.F. Hegel) Based on: * David E. Edmunds, Petr Gurka, J.L., Properties of generalized Trigonometric functions, Preprint * David
More informationNONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 210, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian
More informationEXISTENCE RESULTS FOR AN ELLIPTIC DIRICHLET PROBLEM
LE MATEMATICHE Vol. LXVI (2) Fasc. I pp. 33 4 doi:.448/2.66..3 EXISTENCE RESULTS FOR AN ELLIPTIC DIRICHLET PROBLEM GIUSEPPINA D AGUÌ - GIOVANNI MOLICA BISCI The main purpose of this paper is to present
More informationEXISTENCE OF NONTRIVIAL SOLUTIONS FOR A QUASILINEAR SCHRÖDINGER EQUATIONS WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 05, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF NONTRIVIAL
More informationAN EXAMPLE OF FUNCTIONAL WHICH IS WEAKLY LOWER SEMICONTINUOUS ON W 1,p FOR EVERY p > 2 BUT NOT ON H0
AN EXAMPLE OF FUNCTIONAL WHICH IS WEAKLY LOWER SEMICONTINUOUS ON W,p FOR EVERY p > BUT NOT ON H FERNANDO FARRONI, RAFFAELLA GIOVA AND FRANÇOIS MURAT Abstract. In this note we give an example of functional
More informationPositive decaying solutions to BVPs with mean curvature operator
Rend. Istit. Mat. Univ. Trieste Volume 49 207), 47 64 DOI: 0.337/2464-8728/620 Positive decaying solutions to BVPs with mean curvature operator Zuzana Došlá, Mauro Marini and Serena Matucci Dedicated to
More informationDisconjugate operators and related differential equations
Disconjugate operators and related differential equations Mariella Cecchi, Zuzana Došlá and Mauro Marini Dedicated to J. Vosmanský on occasion of his 65 th birthday Abstract: There are studied asymptotic
More informationSpectrum of one dimensional p-laplacian Operator with indefinite weight
Spectrum of one dimensional p-laplacian Operator with indefinite weight A. Anane, O. Chakrone and M. Moussa 2 Département de mathématiques, Faculté des Sciences, Université Mohamed I er, Oujda. Maroc.
More informationEXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM
EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM JENICĂ CRÎNGANU We derive existence results for operator equations having the form J ϕu = N f u, by using
More informationPERIODIC SOLUTIONS OF THE FORCED PENDULUM : CLASSICAL VS RELATIVISTIC
LE MATEMATICHE Vol. LXV 21) Fasc. II, pp. 97 17 doi: 1.4418/21.65.2.11 PERIODIC SOLUTIONS OF THE FORCED PENDULUM : CLASSICAL VS RELATIVISTIC JEAN MAWHIN The paper surveys and compares some results on the
More informationNONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS. Shouchuan Hu Nikolas S. Papageorgiou. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 34, 29, 327 338 NONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS Shouchuan Hu Nikolas S. Papageorgiou
More informationA NECESSARY AND SUFFICIENT CONDITION FOR THE GLOBAL ASYMPTOTIC STABILITY OF DAMPED HALF-LINEAR OSCILLATORS
Acta Math. Hungar., 138 (1-2 (213, 156 172. DOI: 1.17/s1474-12-259-7 First published online September 5, 212 A NECESSARY AND SUFFICIEN CONDIION FOR HE GLOBAL ASYMPOIC SABILIY OF DAMPED HALF-LINEAR OSCILLAORS
More informationMAIN ARTICLES. In present paper we consider the Neumann problem for the operator equation
Volume 14, 2010 1 MAIN ARTICLES THE NEUMANN PROBLEM FOR A DEGENERATE DIFFERENTIAL OPERATOR EQUATION Liparit Tepoyan Yerevan State University, Faculty of mathematics and mechanics Abstract. We consider
More informationMultiple positive solutions for a class of quasilinear elliptic boundary-value problems
Electronic Journal of Differential Equations, Vol. 20032003), No. 07, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu login: ftp) Multiple positive
More informationp-laplacian problems with critical Sobolev exponents
Nonlinear Analysis 66 (2007) 454 459 www.elsevier.com/locate/na p-laplacian problems with critical Sobolev exponents Kanishka Perera a,, Elves A.B. Silva b a Department of Mathematical Sciences, Florida
More informationSHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction
SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms
More informationA NOTE ON THE EXISTENCE OF TWO NONTRIVIAL SOLUTIONS OF A RESONANCE PROBLEM
PORTUGALIAE MATHEMATICA Vol. 51 Fasc. 4 1994 A NOTE ON THE EXISTENCE OF TWO NONTRIVIAL SOLUTIONS OF A RESONANCE PROBLEM To Fu Ma* Abstract: We study the existence of two nontrivial solutions for an elliptic
More informationOSCILLATION AND SPECTRAL THEORY FOR LINEAR HAMILTONIAN SYSTEMS WITH NONLINEAR DEPENDENCE ON THE SPECTRAL PARAMETER
OSCILLATION AND SPECTRAL THEORY FOR LINEAR HAMILTONIAN SYSTEMS WITH NONLINEAR DEPENDENCE ON THE SPECTRAL PARAMETER Martin Bohner, Werner Kratz und Roman Simon Hilscher Preprint Series: 2011-09 Fakultät
More informationXiyou Cheng Zhitao Zhang. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 34, 2009, 267 277 EXISTENCE OF POSITIVE SOLUTIONS TO SYSTEMS OF NONLINEAR INTEGRAL OR DIFFERENTIAL EQUATIONS Xiyou
More informationON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM
ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM OLEG ZUBELEVICH DEPARTMENT OF MATHEMATICS THE BUDGET AND TREASURY ACADEMY OF THE MINISTRY OF FINANCE OF THE RUSSIAN FEDERATION 7, ZLATOUSTINSKY MALIY PER.,
More informationRolle s Theorem for Polynomials of Degree Four in a Hilbert Space 1
Journal of Mathematical Analysis and Applications 265, 322 33 (2002) doi:0.006/jmaa.200.7708, available online at http://www.idealibrary.com on Rolle s Theorem for Polynomials of Degree Four in a Hilbert
More informationSufficient conditions for functions to form Riesz bases in L 2 and applications to nonlinear boundary-value problems
Electronic Journal of Differential Equations, Vol. 200(200), No. 74, pp. 0. ISSN: 072-669. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Sufficient conditions
More informationCUTOFF RESOLVENT ESTIMATES AND THE SEMILINEAR SCHRÖDINGER EQUATION
CUTOFF RESOLVENT ESTIMATES AND THE SEMILINEAR SCHRÖDINGER EQUATION HANS CHRISTIANSON Abstract. This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schrödinger equation.
More informationA discrete analogue of Lyapunov-type inequalities for nonlinear systems
Computers Mathematics with Applications 55 2008 2631 2642 www.elsevier.com/locate/camwa A discrete analogue of Lyapunov-type inequalities for nonlinear systems Mehmet Ünal a,, Devrim Çakmak b, Aydın Tiryaki
More informationNonlinear elliptic systems with exponential nonlinearities
22-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 9, 22, pp 139 147. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu
More informationIntroduction to Optimization Techniques. Nonlinear Optimization in Function Spaces
Introduction to Optimization Techniques Nonlinear Optimization in Function Spaces X : T : Gateaux and Fréchet Differentials Gateaux and Fréchet Differentials a vector space, Y : a normed space transformation
More informationg 2 (x) (1/3)M 1 = (1/3)(2/3)M.
COMPACTNESS If C R n is closed and bounded, then by B-W it is sequentially compact: any sequence of points in C has a subsequence converging to a point in C Conversely, any sequentially compact C R n is
More informationEXISTENCE OF WEAK SOLUTIONS FOR A NONUNIFORMLY ELLIPTIC NONLINEAR SYSTEM IN R N. 1. Introduction We study the nonuniformly elliptic, nonlinear system
Electronic Journal of Differential Equations, Vol. 20082008), No. 119, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu login: ftp) EXISTENCE
More informationPERIODIC PROBLEMS WITH φ-laplacian INVOLVING NON-ORDERED LOWER AND UPPER FUNCTIONS
Fixed Point Theory, Volume 6, No. 1, 25, 99-112 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.htm PERIODIC PROBLEMS WITH φ-laplacian INVOLVING NON-ORDERED LOWER AND UPPER FUNCTIONS IRENA RACHŮNKOVÁ1 AND MILAN
More informationINEQUALITIES FOR SUMS OF INDEPENDENT RANDOM VARIABLES IN LORENTZ SPACES
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 45, Number 5, 2015 INEQUALITIES FOR SUMS OF INDEPENDENT RANDOM VARIABLES IN LORENTZ SPACES GHADIR SADEGHI ABSTRACT. By using interpolation with a function parameter,
More informationMath 104: Homework 7 solutions
Math 04: Homework 7 solutions. (a) The derivative of f () = is f () = 2 which is unbounded as 0. Since f () is continuous on [0, ], it is uniformly continous on this interval by Theorem 9.2. Hence for
More informationASYMMETRIC SUPERLINEAR PROBLEMS UNDER STRONG RESONANCE CONDITIONS
Electronic Journal of Differential Equations, Vol. 07 (07), No. 49, pp. 7. ISSN: 07-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ASYMMETRIC SUPERLINEAR PROBLEMS UNDER STRONG RESONANCE
More informationANOTHER LOOK AT CHEBYSHEV SYSTEMS
1 ANOTHER LOOK AT CHEBYSHEV SYSTEMS RICHARD A. ZALIK In memory of Samuel Karlin Abstract. We study Chebyshev systems defined on an interval, whose constituent functions are either complex or real valued,
More informationRemarks on the Spectrum of Bounded and Normal Operators on Hilbert Spaces
An. Şt. Univ. Ovidius Constanţa Vol. 16(2), 2008, 7 14 Remarks on the Spectrum of Bounded and Normal Operators on Hilbert Spaces M. AKKOUCHI Abstract Let H be a complex Hilbert space H. Let T be a bounded
More informationExistence of Positive Solutions to a Nonlinear Biharmonic Equation
International Mathematical Forum, 3, 2008, no. 40, 1959-1964 Existence of Positive Solutions to a Nonlinear Biharmonic Equation S. H. Al Hashimi Department of Chemical Engineering The Petroleum Institute,
More informationON NONHOMOGENEOUS BIHARMONIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT
PORTUGALIAE MATHEMATICA Vol. 56 Fasc. 3 1999 ON NONHOMOGENEOUS BIHARMONIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT M. Guedda Abstract: In this paper we consider the problem u = λ u u + f in, u = u
More informationCOMPARISON THEOREMS FOR THE SPECTRAL GAP OF DIFFUSIONS PROCESSES AND SCHRÖDINGER OPERATORS ON AN INTERVAL. Ross G. Pinsky
COMPARISON THEOREMS FOR THE SPECTRAL GAP OF DIFFUSIONS PROCESSES AND SCHRÖDINGER OPERATORS ON AN INTERVAL Ross G. Pinsky Department of Mathematics Technion-Israel Institute of Technology Haifa, 32000 Israel
More informationCRITICAL POINT METHODS IN DEGENERATE ANISOTROPIC PROBLEMS WITH VARIABLE EXPONENT. We are interested in discussing the problem:
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LV, Number 4, December 2010 CRITICAL POINT METHODS IN DEGENERATE ANISOTROPIC PROBLEMS WITH VARIABLE EXPONENT MARIA-MAGDALENA BOUREANU Abstract. We work on
More informationBounded and homoclinic-like solutions of a second-order singular differential equation
Bull. London Math. Soc. 44 (2012) 47 54 C 2011 London Mathematical Society doi:10.1112/blms/bdr060 Bounded and homoclinic-like solutions of a second-order singular differential equation Denis Bonheure
More informationBIHARMONIC WAVE MAPS INTO SPHERES
BIHARMONIC WAVE MAPS INTO SPHERES SEBASTIAN HERR, TOBIAS LAMM, AND ROLAND SCHNAUBELT Abstract. A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed.
More informationOn the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem
On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem Koichiro TAKAOKA Dept of Applied Physics, Tokyo Institute of Technology Abstract M Yor constructed a family
More informationSingular boundary value problems
Department of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic, e-mail: irena.rachunkova@upol.cz Malá Morávka, May 2012 Overview of our common research
More informationBehaviour of the approximation numbers of a Sobolev embedding in the one-dimensional case.
Behaviour of the approximation numbers of a Sobolev embedding in the one-dimensional case. D.E.Edmunds a, J.Lang b, a Centre for Mathematical Analysis and its Applications, University of Sussex, Falmer,
More informationMEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW
MEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW GREGORY DRUGAN AND XUAN HIEN NGUYEN Abstract. We present two initial graphs over the entire R n, n 2 for which the mean curvature flow
More informationMULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN
Electronic Journal of Differential Equations, Vol. 016 (016), No. 97, pp. 1 11. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIONS
More informationDynamic Systems and Applications xx (200x) xx-xx ON TWO POINT BOUNDARY VALUE PROBLEMS FOR SECOND ORDER DIFFERENTIAL INCLUSIONS
Dynamic Systems and Applications xx (2x) xx-xx ON WO POIN BOUNDARY VALUE PROBLEMS FOR SECOND ORDER DIFFERENIAL INCLUSIONS LYNN ERBE 1, RUYUN MA 2, AND CHRISOPHER C. ISDELL 3 1 Department of Mathematics,
More informationBoundary Value Problems For A Differential Equation On A Measure Chain
Boundary Value Problems For A Differential Equation On A Measure Chain Elvan Akin Department of Mathematics and Statistics, University of Nebraska-Lincoln Lincoln, NE 68588-0323 eakin@math.unl.edu Abstract
More informationEVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS
EVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS Cezar Avramescu Abstract The problem of existence of the solutions for ordinary differential equations vanishing at ± is considered. AMS
More informationHOMOLOGICAL LOCAL LINKING
HOMOLOGICAL LOCAL LINKING KANISHKA PERERA Abstract. We generalize the notion of local linking to include certain cases where the functional does not have a local splitting near the origin. Applications
More informationA Concise Course on Stochastic Partial Differential Equations
A Concise Course on Stochastic Partial Differential Equations Michael Röckner Reference: C. Prevot, M. Röckner: Springer LN in Math. 1905, Berlin (2007) And see the references therein for the original
More informationALUR DUAL RENORMINGS OF BANACH SPACES SEBASTIÁN LAJARA
ALUR DUAL RENORMINGS OF BANACH SPACES SEBASTIÁN LAJARA ABSTRACT. We give a covering type characterization for the class of dual Banach spaces with an equivalent ALUR dual norm. Let K be a closed convex
More informationOllivier Ricci curvature for general graph Laplacians
for general graph Laplacians York College and the Graduate Center City University of New York 6th Cornell Conference on Analysis, Probability and Mathematical Physics on Fractals Cornell University June
More information1 A complete Fourier series solution
Math 128 Notes 13 In this last set of notes I will try to tie up some loose ends. 1 A complete Fourier series solution First here is an example of the full solution of a pde by Fourier series. Consider
More informationThe complex periodic problem for a Riccati equation
The complex periodic problem for a Riccati equation Rafael Ortega Departamento de Matemática Aplicada Facultad de Ciencias Universidad de Granada, 1871 Granada, Spain rortega@ugr.es Dedicated to Jean Mawhin,
More informationASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES ABSTRACT
ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI ABSTRACT In this paper, we prove that if ρ is a convex, σ-finite modular function satisfying
More informationEXISTENCE AND MULTIPLICITY OF PERIODIC SOLUTIONS GENERATED BY IMPULSES FOR SECOND-ORDER HAMILTONIAN SYSTEM
Electronic Journal of Differential Equations, Vol. 14 (14), No. 11, pp. 1 1. ISSN: 17-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND MULTIPLICITY
More informationSome Properties of the Augmented Lagrangian in Cone Constrained Optimization
MATHEMATICS OF OPERATIONS RESEARCH Vol. 29, No. 3, August 2004, pp. 479 491 issn 0364-765X eissn 1526-5471 04 2903 0479 informs doi 10.1287/moor.1040.0103 2004 INFORMS Some Properties of the Augmented
More informationOn the spectrum of a nontypical eigenvalue problem
Electronic Journal of Qualitative Theory of Differential Equations 18, No. 87, 1 1; https://doi.org/1.143/ejqtde.18.1.87 www.math.u-szeged.hu/ejqtde/ On the spectrum of a nontypical eigenvalue problem
More informationOn Systems of Boundary Value Problems for Differential Inclusions
On Systems of Boundary Value Problems for Differential Inclusions Lynn Erbe 1, Christopher C. Tisdell 2 and Patricia J. Y. Wong 3 1 Department of Mathematics The University of Nebraska - Lincoln Lincoln,
More information("-1/' .. f/ L) I LOCAL BOUNDEDNESS OF NONLINEAR, MONOTONE OPERA TORS. R. T. Rockafellar. MICHIGAN MATHEMATICAL vol. 16 (1969) pp.
I l ("-1/'.. f/ L) I LOCAL BOUNDEDNESS OF NONLINEAR, MONOTONE OPERA TORS R. T. Rockafellar from the MICHIGAN MATHEMATICAL vol. 16 (1969) pp. 397-407 JOURNAL LOCAL BOUNDEDNESS OF NONLINEAR, MONOTONE OPERATORS
More informationSEQUENCES OF SMALL HOMOCLINIC SOLUTIONS FOR DIFFERENCE EQUATIONS ON INTEGERS
Electronic Journal of Differential Equations, Vol. 2017 (2017, No. 228, pp. 1 12. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu SEQUENCES OF SMALL HOMOCLINIC SOLUTIONS
More informationA Perron-type theorem on the principal eigenvalue of nonsymmetric elliptic operators
A Perron-type theorem on the principal eigenvalue of nonsymmetric elliptic operators Lei Ni And I cherish more than anything else the Analogies, my most trustworthy masters. They know all the secrets of
More informationIntroduction to Spectral Theory
P.D. Hislop I.M. Sigal Introduction to Spectral Theory With Applications to Schrodinger Operators Springer Introduction and Overview 1 1 The Spectrum of Linear Operators and Hilbert Spaces 9 1.1 TheSpectrum
More informationCzechoslovak Mathematical Journal
Czechoslovak Mathematical Journal Oktay Duman; Cihan Orhan µ-statistically convergent function sequences Czechoslovak Mathematical Journal, Vol. 54 (2004), No. 2, 413 422 Persistent URL: http://dml.cz/dmlcz/127899
More informationExistence and Multiplicity of Solutions for a Class of Semilinear Elliptic Equations 1
Journal of Mathematical Analysis and Applications 257, 321 331 (2001) doi:10.1006/jmaa.2000.7347, available online at http://www.idealibrary.com on Existence and Multiplicity of Solutions for a Class of
More informationLERAY LIONS DEGENERATED PROBLEM WITH GENERAL GROWTH CONDITION
2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 73 81. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
More informationl(y j ) = 0 for all y j (1)
Problem 1. The closed linear span of a subset {y j } of a normed vector space is defined as the intersection of all closed subspaces containing all y j and thus the smallest such subspace. 1 Show that
More informationTravelling fronts for the thermodiffusive system with arbitrary Lewis numbers
Travelling fronts for the thermodiffusive system with arbitrary Lewis numbers François Hamel Lenya Ryzhik Abstract We consider KPP-type systems in a cylinder with an arbitrary Lewis number (the ratio of
More informationON CONTINUITY OF MEASURABLE COCYCLES
Journal of Applied Analysis Vol. 6, No. 2 (2000), pp. 295 302 ON CONTINUITY OF MEASURABLE COCYCLES G. GUZIK Received January 18, 2000 and, in revised form, July 27, 2000 Abstract. It is proved that every
More informationConvergence Rates in a Weighted Fu cik Problem
Advanced Nonlinear Studies 14 (214), 385 42 Convergence Rates in a Weighted Fu cik Problem Ariel Martin Salort Departamento de Matemáticas FCEN - Universidad de Buenos Aires Ciudad Universitaria, Pabellón
More informationSOME STABILITY RESULTS FOR THE SEMI-AFFINE VARIATIONAL INEQUALITY PROBLEM. 1. Introduction
ACTA MATHEMATICA VIETNAMICA 271 Volume 29, Number 3, 2004, pp. 271-280 SOME STABILITY RESULTS FOR THE SEMI-AFFINE VARIATIONAL INEQUALITY PROBLEM NGUYEN NANG TAM Abstract. This paper establishes two theorems
More informationThe Contraction Mapping Theorem and the Implicit and Inverse Function Theorems
The Contraction Mapping Theorem and the Implicit and Inverse Function Theorems The Contraction Mapping Theorem Theorem The Contraction Mapping Theorem) Let B a = { x IR d x < a } denote the open ball of
More informationCONNECTIONS BETWEEN A CONJECTURE OF SCHIFFER S AND INCOMPRESSIBLE FLUID MECHANICS
CONNECTIONS BETWEEN A CONJECTURE OF SCHIFFER S AND INCOMPRESSIBLE FLUID MECHANICS JAMES P. KELLIHER Abstract. We demonstrate connections that exists between a conjecture of Schiffer s (which is equivalent
More informationApplications of generalized trigonometric functions with two parameters II
arxiv:194.1827v1 [math.ca] 3 Apr 219 Applications of generalized trigonometric functions with two parameters II Shingo Takeuchi Department of Mathematical Sciences Shibaura Institute of Technology Abstract
More information