1. WCNA 2008 1 New Perspectives of the Krasnoselskii Cone Fixed Point Theorem Dedicated 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 Please revisit later (possibly Sept, 2008). Will add explanation notes.

2. WCNA 2008 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, 2 nd order, 4 th order, superlinear, sublinear, f changes sign or not, 2-point BVP, m-point, (non-)homogeneous, ……

3. WCNA 2008 3 Linear BVP: well studied Eigenvalue method Green’s function For each v, there exists a solution u We have a mapping G : X → X, G(v) = u Completely continuous Maps +ve functions to +ve functions

4. WCNA 2008 4 Nonlinear BVP Solving FP problem T completely continuous Maps +ve functions to +ve functions Many Techniques Brouwer-Leary-Schauder FPT, degree theory, critical point (minimax, mountain pass lemma), monotone iterative, normed function spaces, …

5. WCNA 2008 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. WCNA 2008 6 Brouwer’s FPT T is continuous 0 1 1 T 0 1 1 Rothe’s Improvement T

7. WCNA 2008 7 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 1 1 1 1

8. WCNA 2008 8 (Simplified) Krasnoselskii’s FPT Cone convex subset of Banach space, closed under positive multiple ( ) and addition 1 1 1 1 0 0 S0S0 S0S0 S1S1 S1S1 Compresssive Form Expansive Form

9. WCNA 2008 9 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. FP2 FP1 FP3

10. WCNA 2008 10 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 0 a b c 2b

11. WCNA 2008 11 New Contributions 1.New proof: BFPT K 2.Free K from the “norm and sphere” setting: new form more flexible, wider applications 3.More general LS-type alternative for both B and K: unifies several well-known extensions of K: Leggett-Williams, 5- functional 4.(K 2007, also with James Wong 2007) For certain 2 nd 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. WCNA 2008 12 Proving B K 1.Topological Deformation 2.Compressive Form: Retract to tame image points that have wondered outside the domain 3.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 -1 u is a FP for T. Deform the cone (in K) into a cylinder. Need only prove K for a cylinder.

13. WCNA 2008 13 Proving B K (Compressive Form) Retract (used to prove the Compressive Form) R : X → X R 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 it has 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 T

14. WCNA 2008 14 Proving B K (Expansive Form) The Expansive Form Represent points by cylindrical coordinate u = (x, t ), Tu = (y, s ) Expansive means: t = 0 s ≤ 0 t = 1 s ≥ 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 u is a FP of T t = 0 t = 1

15. WCNA 2008 15 Another New Proof of Expansive K T Corollary of the following extension of the Rothe’s form of BFPT: B 1, B 2 are 2 (in general 2n) disjoint sub-balls. T(∂B)  B, T(∂B 1 )  B 1, T(∂B 2 )  B 2 FP in B – B 1 – B 2 Proof using degree and homotopy theory on the mapping x - Tx X FP X FP B1B1 B2B2

16. WCNA 2008 16 New Application of KFPT Bernstein (1912), Granas, Guenther, Lee (1978) There exists a solution y’’ = f(x,y,y’), y(0) = y(1) = 0 if | f(x,y,p) | 0 Petryshyn (1986) Periodic BVP New Result: improve conditions to if f(x,y,p) > - (Ap 2 ln( p) + B), p > 0 and f(x,y,p) < Ap 2 ln(-p) + B, p < 0 For Dir/Neumann problems, use classical shooting method. For periodic BVP, use KFPT

17. WCNA 2008 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. WCNA 2008 18 New Application of KFPT Bernstein (1912), Granas, Guenther, Lee (1978) There exists a solution y’’ = f(x,y,y’), y(0) = y(1) = 0 if | f(x,y,p) | 0 Petryshyn (1986) Periodic BVP New Result: improve conditions to if f(x,y,p) > - (Ap 2 ln( p) + B), p > 0 and f(x,y,p) < Ap 2 ln(-p) + B, p < 0 For Dir/Neumann problems, use classical shooting method. For periodic BVP, use KFPT ABC DEF α = M α = - M β = Pβ = - P B’ C’ F’ A’ D’ E’

19. WCNA 2008 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. WCNA 2008 20 Exercises 1.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. 2.T : C → C (3-dim cube): T(upper face) below upper F T(lower F) above lower F T(left F) left of left F T(right F) right of right F T(front F) front of front F T(back F) back of back F T has a FP. Generalize. u Tu T x T