New Perspectives of the Krasnoselskii Cone Fixed Point Theorem Dedicated to Prof. Lakshmikantham

1. Please revisit later (possibly Sept, 2008). Will add explanation notes. New Perspectives of theKrasnoselskiiConeFixed Point TheoremDedicated to Prof. Lakshmikantham BVP & FPT New proof: Brouwer K New BVP results Man Kam Kwong The Hong Kong Polytechnic University myweb.polyu.edu.hk/~mankwong/wcna2008.ppt

2. Boundary Value Problems (BVP) Interested in the existence of (non-trivial) solutions Linear Dirichlet Nonlinear Dir-Neumann Nonlinear ground state Numerous Variations Ode, pde, difference eq, time scale, p-Laplacian, 2nd order, 4th order, superlinear, sublinear, f changes sign or not, 2-point BVP, m-point, (non-)homogeneous, ……

3. Linear BVP: well studied • Eigenvalue method • Green’s function • For each v , there exists a solution u We have a mappingG : X → X, G(v) = u • Completely continuous • Maps +ve functions to +ve functions

4. Nonlinear BVP • Solving • FP problem • T completely continuous • Maps+vefunctions to+vefunctions Many Techniques Brouwer-Leary-Schauder FPT, degree theory, critical point (minimax, mountain pass lemma), monotone iterative, normed function spaces, …

5. Periodic BVPs The integral equation approach no longer works. Usual techniques for tackling the problem: variational methods, upper and lower solutions, topological methods. Reformulate as a fixed point problem. Let be the solution of the initial value problem Define . Then a fixed point of T is a solution of the periodic BVP.

6. 1 T 1 0 T Brouwer’s FPTT is continuous Rothe’s Improvement 1 1 0

7. 1 1 1 1 Remarks on Brouwer’s FPT • ¥ dim extension: Leray-Schauder, T completely continuous • L-S alternative (extends Rothe): part of ∂B is mapped outside B • Usually not good enough for BVP. It only gives the trivial solution • Two variants of the 1-dim FPT: no apparent Brouwer analogs! • Krasnoselskii appears on stage

8. 1 0 1 1 0 1 (Simplified) Krasnoselskii’s FPT Cone convex subset of Banach space, closed under positive multiple ( ) and addition Compresssive Form S1 S0 Expansive Form S1 S0

9. FP3 FP2 FP1 Remarks on Krasnoselskii’s FPT • Cone ordering Original K uses conditions like: • Easy to deduce original theorem from simplified theorem • All known proofs start from first principle; most use degree theory • Krasnoselskii used it to study periodic solutions of systems of ode’s • Alternate compressive-expansive form • n x 100’s of papers written based on this idea.

10. c 2b 0 a b Typical BVP Results Recall BVP FP of If f(u)has an “oscillatory” behavior – small in some interval, large in the next, etc., the KFPT gives multiple solutions. Henderson Thompson 2000 Improved 2 positive solutions Ma 1998 Raffoul 2002:finite limits Improved: best possible constant

11. New Contributions • New proof:BFPT K • Free K from the “norm and sphere” setting:new form more flexible, wider applications • More general LS-type alternative for both B and K:unifies several well-known extensions of K:Leggett-Williams , 5-functional • (K 2007, also with James Wong 2007)For certain 2nd order BVPs (esp. m-point problems), the classical Shooting Method → better results than known topological methods; modified KFP approach, combined with super/sub solutions

12. Proving B K • Topological Deformation • Compressive Form: Retract to tame image points that have wondered outside the domain • Reduce Expansive form → Compressive form FP property is a topological invariant. BFPT holds not only for B (ball), but for a simplex, convex sets etc. – any set C = SB that is topologically equivalent to B. Given T: C → C. Then STS-1: B → B , so it has a FP u. It is easy to see that S-1u is a FP for T. Deform the cone (in K) into a cylinder.Need only prove K for a cylinder.

13. Proving B K (Compressive Form) Retract (used to prove the Compressive Form) R : X → X • R projects points above upper edge onto upper edge • points below lower edge onto l.e. • points inside rectangle B not changed • RT : B → B , so ithas a FP u = RT(u). • Claim: u is a FP for T • Case 1: u is an interior/side point of B • Case 2: u is on top/bottom of B R T

14. Proving B K (Expansive Form) The Expansive Form • Representpoints by cylindrical coordinate u = (x,t), Tu = (y,s) • Expansive means: t = 0 s ≤ 0 • t = 1s ≥ 1 • Define a new function T*u = (y,2t – s) • t = 0 2t – s ≥ 0 • t = 1 2t – s ≤ 1 • T *is compressive, so has FP • u = (x,t) = (y,2t – s) • x = y, t = s • uis a FP ofT t = 1 t = 0

15. Another New Proof of Expansive K Corollary of the following extension of the Rothe’s form of BFPT: B1, B2 are 2 (in general 2n) disjoint sub-balls. T(∂B) Ì B,T(∂B1) Ì B1, T(∂B2) Ì B2 FP in B – B1 – B2 Proof using degree and homotopy theory on the mapping x - Tx T B1 X FP X FP B2

16. New Application of KFPT • Bernstein (1912), Granas, Guenther, Lee (1978) There exists a solution • y’’ = f(x,y,y’), y(0) = y(1) = 0if | f(x,y,p) | < Ap2 + B, | x | < M y f(x,y,0) > 0 • Petryshyn (1986) Periodic BVP • New Result: improve conditions to • if f(x,y,p) > - (Ap2 ln( p) + B), p > 0and f(x,y,p) < Ap2 ln(-p) + B, p < 0For Dir/Neumann problems, use classical shooting method. For periodic BVP, use KFPT

17. Periodic BVPs The integral equation approach no longer works. Usual techniques for tackling the problem: variational methods, upper and lower solutions, topological methods. Reformulate as a fixed point problem. Let be the solution of the initial value problem Define . Then a fixed point of T is a solution of the periodic BVP.

18. New Application of KFPT F’ • Bernstein (1912), Granas, Guenther, Lee (1978) There exists a solution • y’’ = f(x,y,y’), y(0) = y(1) = 0if | f(x,y,p) | < Ap2 + B, | x | < M y f(x,y,0) > 0 • Petryshyn (1986) Periodic BVP • New Result: improve conditions to • if f(x,y,p) > - (Ap2 ln( p) + B), p > 0and f(x,y,p) < Ap2 ln(-p) + B, p < 0For Dir/Neumann problems, use classical shooting method. For periodic BVP, use KFPT B’ C’ A B C α = M α = - M D E F D’ A’ E’ β = - P β = P

19. Summary • New perspectives on KFPT • KFPT extended in several directions • Revisit and improve known results • Applications to PDE case scarce – opportunity for work (previous work by Anman, Nussbaum, Petryshyn, etc.) • Relationship with monotone operator (Browder, Brezis, …), and monotone iterative method? • Fertile area to spend time on. • myweb.polyu.edu.hk/~mankwong/wcna2008.ppt

20. u T T Tu x Exercises • T : B → B (2-dim circle), T(∂B)∩∂B = single pt u, Tu in B. Show T has a FP. False if T(∂B)∩∂B more pts. • T : C → C (3-dim cube): • T(upper face) below upper FT(lower F) above lower FT(left F) left of left FT(right F) right of right FT(front F) front of front FT(back F) back of back F • T has a FP. Generalize.