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Series in Mathematical Analysis and Applications Edited by Ravi P. Agarwal and Donal O’Regan VOLUME 10 TOPOLOGICAL DEGREE THEORY AND APPLICATIONS Copyright 2006 by Taylor & Francis Group, LLC SERIES IN MATHEMATICAL ANALYSIS AND APPLICATIONS Series in Mathematical Analysis and Applications (SIMAA) is edited by Ravi P. Agarwal, Florida Institute of Technology, USA and Donal O’Regan, National University of Ireland, Galway, Ireland. The series is aimed at reporting on new developments in mathematical analysis and applications of a high standard and or current interest. Each volume in the series is devoted to a topic in analysis that has been applied, or is potentially applicable, to the solutions of scientific, engineering and social problems. Volume 1 Method of Variation of Parameters for Dynamic Systems V. Lakshmikantham and S.G. Deo Volume 2 Integral and Integrodifferential Equations: Theory, Methods and Applications Edited by Ravi P. Agarwal and Donal O’Regan Volume 3 Theorems of Leray-Schauder Type and Applications Donal O’Regan and Radu Precup Volume 4 Set Valued Mappings with Applications in Nonlinear Analysis Edited by Ravi P. Agarwal and Donal O’Regan Volume 5 Oscillation Theory for Second Order Dynamic Equations Ravi P. Agarwal, Said R. Grace, and Donal O’Regan Volume 6 Theory of Fuzzy Differential Equations and Inclusions V. Lakshmikantham and Ram N. Mohapatra Volume 7 Monotone Flows and Rapid Convergence for Nonlinear Partial Differential Equations V. Lakshmikantham, S. Koksal, and Raymond Bonnett Volume 8 Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems Leszek Gasin´ski and Nikolaos S. Papageorgiou Volume 9 Nonlinear Analysis Leszek Gasin´ski and Nikolaos S. Papageorgiou Volume 10 Topological Degree Theory and Applications Donal O’Regan, Yeol Je Cho, and Yu-Qing Chen Copyright 2006 by Taylor & Francis Group, LLC Series in Mathematical Analysis and Applications Edited by Ravi P. Agarwal and Donal O’Regan VOLUME 10 TOPOLOGICAL DEGREE THEORY AND APPLICATIONS Donal O’Regan Yeol Je Cho Yu-Qing Chen Boca Raton London New York Copyright 2006 by Taylor & Francis Group, LLC C648X_Discl.fm Page 1 Wednesday, February 22, 2006 9:37 AM Published in 2006 by Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 1-58488-648-X (Hardcover) International Standard Book Number-13: 978-1-58488-648-8 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. 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Library of Congress Cataloging-in-Publication Data Catalog record is available from the Library of Congress Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com Taylor & Francis Group and the CRC Press Web site at is the Academic Division of Informa plc. http://www.crcpress.com Copyright 2006 by Taylor & Francis Group, LLC Contents 1 BROUWER DEGREE THEORY 1 1.1 Continuous and Differentiable Functions . . . . . . . . . . . 2 1.2 Construction of Brouwer Degree . . . . . . . . . . . . . . . . 4 1.3 Degree Theory for Functions in VMO . . . . . . . . . . . . . 15 1.4 Applications to ODEs . . . . . . . . . . . . . . . . . . . . . . 19 1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2 LERAY SCHAUDER DEGREE THEORY 25 2.1 Compact Mappings . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Leray Schauder Degree . . . . . . . . . . . . . . . . . . . . . 30 2.3 Leray Schauder Degree for Multi-Valued Mappings . . . . . . 38 2.4 Applications to Bifurcations . . . . . . . . . . . . . . . . . . 43 2.5 Applications to ODEs and PDEs . . . . . . . . . . . . . . . . 46 2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3 DEGREE THEORY FOR SET CONTRACTIVE MAPS 55 3.1 Measure of Noncompactness and Set Contractions . . . . . . 55 3.2 Degree Theory for Countably Condensing Mappings . . . . . 65 3.3 Applications to ODEs in Banach Spaces . . . . . . . . . . . . 68 3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4 GENERALIZEDDEGREETHEORYFORA-PROPERMAPS 75 4.1 A-Proper Mappings . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Generalized Degree for A-Proper Mappings . . . . . . . . . . 80 4.3 Equations with Fredholm Mappings of Index Zero . . . . . . 82 4.4 Equations with Fredholm Mappings of Index Zero Type . . . 87 4.5 Applications of the Generalized Degree . . . . . . . . . . . . 95 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5 COINCIDENCE DEGREE THEORY 105 5.1 Fredholm Mappings . . . . . . . . . . . . . . . . . . . . . . . 105 5.2 Coincidence Degree for L-Compact Mappings . . . . . . . . . 110 5.3 Existence Theorems for Operator Equations . . . . . . . . . 116 5.4 Applications to ODEs . . . . . . . . . . . . . . . . . . . . . . 119 5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 1 Copyright 2006 by Taylor & Francis Group, LLC 0 6 DEGREE THEORY FOR MONOTONE-TYPE MAPS 127 6.1 Monotone Type-Mappings in Reflexive Banach Spaces . . . . 128 6.2 Degree Theory for Mappings of Class (S ) . . . . . . . . . . 142 + 6.3 Degree for Perturbations of Monotone-Type Mappings . . . . 145 6.4 Degree Theory for Mappings of Class (S ) . . . . . . . . . 149 + L 6.5 Coincidence Degree for Mappings of Class L-(S ) . . . . . . 152 + 6.6 Computation of Topological Degree . . . . . . . . . . . . . . 156 6.7 Applications to PDEs and Evolution Equations . . . . . . . 159 6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7 FIXED POINT INDEX THEORY 169 7.1 Cone in Normed Spaces . . . . . . . . . . . . . . . . . . . . . 169 7.2 Fixed Point Index Theory . . . . . . . . . . . . . . . . . . . . 176 7.3 Fixed Point Theorems in Cones . . . . . . . . . . . . . . . . 179 7.4 Perturbations of Condensing Mappings . . . . . . . . . . . . 186 7.5 Index Theory for Nonself Mappings . . . . . . . . . . . . . . 189 7.6 Applications to Integral and Differential Equations . . . . . . 191 7.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 8 REFERENCES 195 Copyright 2006 by Taylor & Francis Group, LLC Chapter 1 BROUWER DEGREE THEORY Let R be the real numbers, Rn = {x = (x ,x ,··· ,x ) : x ∈ R for i = 1 2 n i 1,2,··· ,n} with |x| = (Pni=1x2i)21 and let Ω ⊂ Rn, and let f : Ω → Rn be a continuous function. A basic mathematical problem is: Does f(x) = 0 have a solution in Ω? It is also of interest to know how many solutions are distributed in Ω. In this chapter, we will present a number, the topological degree of f with respect to Ω and 0, which is very useful in answering these questions. To motivate the process, let us first recall the winding number of plane curves, a basic topic in an elementary course in complex analysis. Let C be the set of complex numbers, Γ ⊂ C an oriented closed C1 curve and a∈C\Γ. Then the integer 1 Z 1 w(Γ,a)= dz (1) 2πi z−a Γ is called the winding number of Γ with respect to a∈C\Γ. Now, let G⊂C be a simply connected region and f :G→C be analytic and Γ⊂G a closed C1 curve such that f(z)6=0 on Γ. Then we have 1 Z 1 1 Z f0(z) X w(f(Γ),0)= dz = dz = w(Γ,z )α , (2) 2πi z 2πi f(z) i i f(Γ) Γ i where z are the zeros of f in the region enclosed by Γ and α are the corre- i i spondingmultiplicites. IfweassumeinadditionthatΓhaspositiveorientation and no intersection points, then we know from Jordan’s Theorem, which will be proved later in this chapter, that w(Γ,z )=1 for all z . Thus (2) becomes i i X w(f(Γ),0)= α . (3) i i Sowemaysaythatf hasatleast|w(f(Γ),0)|zerosinG. Thewindingnumber is a very old concept which goes back to Cauchy and Gauss. Kronecker, Hadamard, Poincare, and others extended formula (1). In 1912, Brouwer [32] introducedtheso-calledBrouwerdegreeinRn (seeBrowder[35],Sieberg[277] forhistoricaldevelopments). Inthischapter,weintroducetheBrouwerdegree theoryanditsgeneralizationtofunctionsinVMO. Thischapterisorganized as follows: InSection1.1weintroducethenotionofacriticalpointforadifferentiable functionf. WethenproveSard’sLemma,whichstatesthatthesetofcritical 1 Copyright 2006 by Taylor & Francis Group, LLC 2 Topological Degree Theory and Applications points of a C1 function is “small”. Our final result in this section shows how a continuous function can be approximated by a C∞ function. In Section 1.2 we begin by defining the degree of a C1 function using the Jacobian. Also we present an integral representation which we use to define the degree of a continuous function. Also in this section we present some properties of our degree (see theorems 1.2.6, 1.2.12, and 1.2.13) and some useful consequences. For example, we prove Brouwer’s and Borsuk’s fixed pointtheorem,Jordan’sseparationtheoremandanopenmappingtheorem. In addition we discuss the relation between the winding number and the degree. In Section 1.3 we discuss some properties of the average value function and then we introduce the degree for functions in VMO. In Section 1.4 we use the degree theory in Section 1.2 to present some exis- tenceresultsfortheperiodicandanti-periodicfirstorderordinarydifferential equations. 1.1 Continuous and Differentiable Functions We begin with the following Bolzano’s intermediate value theorem: Theorem 1.1.1. Let f : [a,b] → R be a continuous function, then, for m between f(a) and f(b), there exists x ∈[a,b] such that f(x )=m. 0 0 Corollary 1.1.2. Let f : [a,b] → R be a continuous function such that f(a)f(b)<0. Then there exists x ∈(a,b) such that f(x )=0. 0 0 Corollary 1.1.3. Let f : [a,b] → [a,b] be a continuous function. Then there exists x ∈[a,b] such that f(x )=x . 0 0 0 Let Ω ⊂ Rn be an open subset. We recall that a function f : Ω → Rn is differentiable at x ∈ Ω if there is a matrix f0(x ) such that f(x +h) = 0 0 0 f(x )+f0(x )h+o(h), where x +h∈Ω and |o(h)| tends to zero as |h|→0. 0 0 0 |h| We use Ck(Ω) to denote the space of k-times continuously differentiable functions. Iff isdifferentiableat x , wecallJ (x )=detf0(x )the Jacobian 0 f 0 0 off atx . IfJ (x )=0, thenx issaidtobeacriticalpointoff andweuse 0 f 0 0 S (Ω)={x∈Ω:J (x)=0} to denote the set of critical points of f, in Ω. If f f f−1(y)∩S (Ω)=∅, then y is said to be a regular value of f. Otherwise, y is f said to be a singular value of f. Lemma 1.1.4. (Sard’s Lemma) Let Ω ⊂ Rn be open and f ∈ C1(Ω). Then µ (f(S (Ω))=0, where µ is the n-dimensional Lebesgue measure. n f n Proof. Since Ω is open, Ω = ∪∞ Q , where Q is a cube for i = 1,2,···. i=1 i i We only need to show that µ (f(S (Q))) = 0 for a cube Q ⊂ Ω. In fact, let n f Copyright 2006 by Taylor & Francis Group, LLC BROUWER DEGREE THEORY 3 l be the lateral length of Q. By the uniform continuity of f0 on Q, for any given (cid:15)>0, there exists an integer m>0 such that |f0(x)−f0(y)|≤(cid:15) √ for all x,y ∈Q with |x−y|≤ nl. Therefore, we have m Z 1 |f(x)−f(y)−f0(y)(x−y)|≤ |f0(y+t(x−y))−f0(y)||x−y|dt 0 ≤(cid:15)|x−y| √ for all x,y ∈ Q with |x−y| ≤ nl. We decompose Q into r cubes, Qi, of √ m diameter nl, i = 1,2,··· ,r. Since l is the lateral length of Qi, we have m m r =mn. Now,supposethatQi∩S (Ω)6=∅. Choosingy ∈Qi∩S (Ω),wehave f f √ f(y+x)−f(y)=f0(y)x+R(y,x)forallx∈Qi−y,where|R(y,x+y)|≤(cid:15) nl. m Therefore, we have f(Qi)=f(y)+f0(y)(Qi−y)+R(y,Qi). Butf0(y)=0,sof0(y)(Qi−y)iscontainedinan(n−1)-dimensionalsubspace of Rn. Thus, µ (f0(y)(Qi−y))=0, so we have n √ nl µ (f(Qi))≤2n(cid:15)n( )n. n m Obviously, f(S (Q))⊂∪r f(Qi), so we have f i=1 √ nl √ µ (f(S (Q))≤r2n(cid:15)n( )n =2n(cid:15)n( nl)n. n f m By letting (cid:15)→0+, we obtain µ (f(S (Q)))=0. Therefore, µ (f(S (Ω)))= n f n f 0. This completes the proof. Proposition 1.1.5. LetK ⊂Rn beaboundedclosedsubset,andf :K → Rn continuous. Then there exists a continuous function f˜: Rn → convf(K) such that f˜(x) = f(x) for all x ∈ K, where convf(K) is the convex hull of f(K). Proof. Since K is bounded closed subset, there exists at most countable {k :i=1,2,···}⊂K such that {k :i=1,2,···}=K. Put i i |x−k | d(x,K)= inf |x−y|, α (x)=max{2− i ,0} y∈K i d(x,A) for any x∈/ K and ( f(x), x∈K, f˜(x)= Pi≥12−iαi(x)f(ki), x∈/ K. Pi≥12−iαi(x) Copyright 2006 by Taylor & Francis Group, LLC 4 Topological Degree Theory and Applications Then f˜is the desired function. Proposition 1.1.6. Let K ⊂ Rn be a bounded closed subset and f : K → Rn continuous. Then there exists a function g ∈ C∞(Rn) such that |f(x)−g(x)|<(cid:15). Proof. By Proposition 1.1.5, there exists a continuous extension fõf f to Rn. Define the following function ( ce−1−1|x|, |x|<1, φ(x)= (1.1) 0, |x|≥1, where c satisfies R φ(x)dx=1. Set φ (x)=λ−nφ(x) for all x∈Rn and Rn λ λ Z f (x)= f˜(y)φ (y−x)dx for all x∈Rn, λ>0. λ λ Rn It is obvious that suppf = {x∈Rn :f (x)6=0} = {x : |x| ≤ λ} for all λ λ λ > 0. Consequently, we have f ∈ C∞ and f (x) → f(x) uniformly on K λ λ as λ → 0+. Taking g as f for sufficiently small λ, g is the desired function. λ This completes the proof. 1.2 Construction of Brouwer Degree Now, we give the construction of Brouwer degree in this section as follows: Definition 1.2.1. Let Ω ⊂ RN be open and bounded and f ∈ C1(Ω). If p∈/ f(∂Ω) and J (p)6=0, then we define f X deg(f,Ω,p)= sgnJ (x), f x∈f−1(p) where deg(f,Ω,p)=0 if f−1(p)=∅. The next result gives another equivalent form of Definition 1.2.1. Proposition 1.2.2. Let Ω, f and p be as in Definition 1.2.1 and let (c(cid:15)−ne−1−|(cid:15)−11x|2, |x|<1, φ (x)= (1.2) (cid:15) 0, otherwise, R where c is a constant such that φ(x)=1. Then there exists (cid:15) =(cid:15) (p,f) Rn 0 0 such that Z deg(f,Ω,p)= φ (f(x)−p)J (x)dx for all (cid:15)∈(0,(cid:15) ). (cid:15) f 0 Ω Copyright 2006 by Taylor & Francis Group, LLC