Fixed points and topological degree in nonlinear analysis

Fixed Points and Topological Degree in Nonlinear Analysis
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Fixed points and topological degree in nonlinear analysis We complete the proof by showing that 3. We prove 3. The proof of 3. Now suppose 3. Let fli f2 be simplicial mappings from K into an open set U C -Rn. But by Theorem 3. Let Na p be the a-neighborhood of a point p, i. Let the point p be in Bn - lJ.

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The exposition and mathematical content is improved throughout. By the quasimonotonicity of S , 2. In this chapter, I will present the Brouwer degree and, in the next chapter, I will demonstrate certain properties of it. Lagarias , Robert C. Dirichletsche Problem in Grossen fur nichtlineare elliptische Differentialgleichungen.

First by Corollaries 3. By Theorem 3. By Corollary 3. Hence by Theorem 2. Since a and b are similar, it is only necessary to prove a. But a follows from Theorem 2. The preceding Theorem 3. Let K be a complex consisting of oriented n — l -simplexes. Let p be a point such that p e Rn - LT. See Figure 7.

A homotopy is a continuous mapping from A' x [0, 1] into Rn. Let p t denote the image under a continuous mapping p from [0,1] into Rn. Let K denote the same type of complex as in Definition 3. Let tQ e [0, 1]. The proof is completed by applying the Heine- Borel Theorem.

Fixed Points and Topological Degree in Nonlinear Analysis - Jane Cronin - Google книги

By the remark following Definition 2. From one point of view, our definition of topological degree is now practically complete. V However, this places two undesirable limitations on the local degree. First we have not considered the degree as a sum of "indices" of points or as a "covering number. Our next job is to escape these limitations. If all the points are isolated and S is bounded, the set of g-points is finite. Let a:J. Using the p o i n t y as the center of projection, project from 6. A g-point p is regular if p is in the interior of an n-simplex in K and p is an isolated g-point.

Note that the index of a regular g-point is defined. In the following, let K be an n-dimensional simplicial complex in an oriented Rn with its?? The notions of algebraic number of g-points and local degree at q which we are about to define can be defined more generally by including a wider class of chains on A', but in applications the definitions we will give are the only ones used. The proof then follows from Theorem 2. This completes the proof of the lemma. Follows from Theorem 5. Now we show that there is a finite succession of simplicial sub- divisions of A such that each n-simplex of the resulting complex contains at most one g-point.

If xn contains more than one g-point, let o be the center of gravity of xn. Let 8 be less than the minimum of Si, 82, - - -, 8m. Continue to make subdivisions as described above until the diameters of the resulting simplexes are all less than 8. The diameters can be made less than S if we choose ex sufficiently small.

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Theorem 3. This completes the proof for the case in which the set of g-points is finite. Theorems 5. Follows from Definition 5. This is done by "'approximating" I by a complex K. There is an n-dimensional simplicial complex K such that: 6. First since I is open, the set D — D is closed. Subdivide B by hyperplanes parallel to the coordinate hyperplanes so that each rectangular parallelopiped in the subdivision has diameter less than 5, 2. Let K be the simplicial subdivision given by Theorem 3.

Analysis II Lecture 14 Part 3 examples of the index for vector fields

If K1, K2 are n-dimensional simplicial complexes satisfying condition 6. The intersection of two simplexes, one from Kx and one from K2, is a cell. Then for each t e [0. Theorem 0. Note, however, that Theorem 0. We will give illustrations of this in Chapter H. From Definitions 5. We will be able to make significant applications of this theorem to ordinary differential equations. The theorem is based on a theorem due to A. Sard [1] of which Lemma 7. At each point p0 e G, the Jacobian is denoted by Jf p0.

Suppose f is differentiable on the open set G C Rn. Let p0 be a fixed point in G. A proof of this theorem can be found in an advanced calculus text. For example, see Buck [1, pp. Let f be a continuous mapping from D into Rn where D is an open set such that f is differ entiable on D. Suppose q E J? We first show that p0 is an isolated -point. Using Theorem 7. By Theorem 7. Hence pQ is an isolated g-point.

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Now let xn be an n-simplex such that. If the diameter of xn is sufficiently small, then by 7. Let f be continuous arid differentiate on an open set 0 C -ffn ichich contains D the closure of a bounded open set and suppose the range off is contained in a subset of Rn. Before proving the theorem, we point out the following corollary which is the form in which the theorem will be used most frequently. The proof follows from Theorem 7. Let f be a differentiate mapping of a bounded open set G Q Rn into Rn.

Let D be an open set such that I C G. Then the image under f of the set of critical points in D has n-measure zero. Since G is bounded, there is a rectangular parallelopiped R with sides parallel to the coordinate planes such that G Q R. Let rt be such a cube with side-length I. Now divide rx into mn cubes of side Ijm. Call these cubes r i ;. Take p2 fixed in rxj. Now suppose rxj contains a critical p o i n t y.

Hence if e is sufficiently small, the right side can be made as small as desired. This completes the proof of Lemma 7. To prove Theorem 7. Hence by Theorem 7. Also the index of each point of S is 4- 1 or — 1. By Defini- tions 5. The conclusion of the theorem then follows.

To facilitate the computation of the local degree of certain mappings, we obtain a theorem about the degree of a product mapping. Let f be a mapping, differentiate on a bounded open set G such that G 3 D the closure of an open set D in Rn, into Rn. Let q e Rn. Let N6 q be a neighborhood of q such that d[gf, D, q'] is constant for q' e N6 q.

By Corollary 7. The problem of computing the local degree is, to a considerable extent, unsolved except in the plane. First we point out tha t in the one-dimensional case, the local degree can be easily computed for polynomial mappings. Extensions of this result can be made using Theorems 6. First let M :. We assume that d[M, Slf 0] is defined, i. This means we assume that P and Q have no common real linear factors. Because of this assumption, the only 0-point of M is 0 itself.

Since P and Q are homogeneous, it follows that for every solid circle S with center 0, the degree d[M, S, 0] is defined and d[M, S, 0] has the same value for allfif. Now write P and Q in factored form, i. First we show that the following factors either do not change the signs of x" or y" or do not change them significantly, i. Then product y — atix y — ai2x is positive for all x and y. I t is fairly reasonable, intuitively, that we may disregard the factors listed above; nevertheless we give formal proofs which, incidentally, illustrate the use of the Invariance under Homotopy Theorem 6.

We describe the homotopies that are used. I t is easy to show that these homotopies satisfy the hypotheses of Theorem 6. If all the factors in n! For definiteness, take the first case. Start on the boundary of Sx at 0, — 1 and proceed in a counterclockwise direction. Proceeding in this manner around the boundary of Sx and observing the changes in sign and using Theorem 2. See Figure 8. Hence applying the Poincare-Bohl Theorem 6. Applying the Poincare-Bohl Theorem 6. So far we have computed the local degree by studying the behavior of the mapping on the boundary of the set, i. We consider mappings from B2n into B2n which can be described in terms of functions of n complex variables: for example, the mapping M :.

Let S be a circle of center 0 the origin and arbitrary fixed radius r. To compute d[M'. Theorem 6. Thus by the Cauchy-Riemann equations, Jacobian J is positive. So by the Definition 6. Hence the degree is kj. Then if plt p2e Rq — Vf, there is q a continuous path in Rq — Vf which has pl and p2 as its endpoints.

Hence by the Invariance under Homotopy Theorem 6. By using the same kind of arguments as in subsection 9. Then the Jacobian of J is always non-negative. Let J be the Jacobian of sJft, i. Now consider again the mapping. From Lemma 9. Thus d[J?

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Now let M be a mapping of Rn into P n such that M can be extended to a mapping 9J2 of the complex Euclidean n-space Sin into 3 n which satisfies the hypotheses of Lemma 9. But from Lemma 9. Let 2? Then at least one coordinate of qv has a nonzero imaginary part. But J p 0 by hypothesis. We complete the proof by showing that d[M 2 , AS 2 — F, 0] is an even number. Although we will not make use of it here, another possible method for computing the local degree is to use the Kronecker integral Alexandroff and Hopf [1, pp.

Fixed points and topological degree in nonlinear analysis

The order v[f, K, p] can be shown to be equal to an integral over A', the Kronecker integral. Indeed, historically the Kronecker integral precedes the definition of local degree we have developed here. However, computing the integral is generally difficult. We add two theorems which will be useful in applications. The first theorem is due to Leray and Schauder [1]. The proof is completed by showing that Hence Let Fu y be a mapping from Rn x [ — E, e] into Rn which is differentiable as a function of y and continuous in fj..

Also these solutions do not have any common values, i. Let xQ be such a point which is not the image of a critical point. A proof of the fixed point theorem. I t is easy to show that the fixed point theorem follows from the corresponding statement for a solid unit sphere S with center 0 in En. So we prove this latter statement.

Let yeS'. Thus by the Invariance under Homotopy Theorem 6. Hence by the Existence Theorem 6. A special case of the fixed point theorem which follows at once from Theorem If K is a closed convex set and f is a continuous mapping of K into itself, then there is a point x e K such that f x — x. I t is convenient in some applications to work with the concept of the index of a vector field rather than the local degree.

A continuous mapping V from Bn into the space of n-vectors is a vector field on Bn. If V x0 is the zero vector for some x0 e Bn, then x0 is a critical point of the vector field V. If for each x e Bn, the vector V x is a vector 0a, i. See Figure 9. If the vector field n V has no critical points on Bn ', the boundary of Bn, then d[Mv, Bn, 0] is defined. A rotation of the vector in the counterclockwise [clockwise] direction is counted as a positive [negative] rotation. Now define the continuous family of vector fields for t e [0, 1]: 1 - t W x 4- tV x.

Then is a continuous mapping from a11 x [0, 1] into Rn — 0.

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Applying the Invariance under Homotopy Theorem 6. Now IMV is the mapping corresponding to a vector field all of whose vectors on a n ' are directed outward. Hence by the Product Theorem 8. The theory of fixed points and local degree described in this chapter is only for mappings whose domains and ranges are subsets of Euclidean space. In Chapter I I I, the theory will be extended to certain mappings of which the domains and ranges are subsets of linear normed spaces.

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This theory will be sufficient for the applications to analysis, but there are far wider generalizations. The most important is the Lefschetz fixed point theory see Alexandroff and Hopf [1], and Lefschetz [2; 3]. The fixed point theorems in Chapters I and I I I may be obtained as very special cases of this general theorem. A fixed point index for mappings in abstract spaces has been defined by Browder [1] and Bourgin [1].

A generalization in another direction has been carried out by Kakutani [2] who studied mappings from points to sets. Since our object is to provide only an introduction to this kind of study, we give a fairly self-contained account of some of the simplest applications to quasilinear systems, but a much briefer description of the more difficult applications to systems with large nonlin- earities. Also, the discussion is not complete. For example, we omit the work of Cesari [3] on quasilinear systems; in Chapter IV, however, we describe the extended form of Cesari's method to systems with large non- linearities.

Topological Methods in Nonlinear Analysis

11 J. Cronin, Fixed points and topological degree in nonlinear analysis, 10 R. Ayoub, An introduction to the analytic theory of numbers, 9 Arthur. Jan 5, The topological methods based on fixed-point theory and on local topological degree which have been developed by Leray, Schauder.

Throughout this chapter, we will use vector notation and apply certain existence theorems, and we begin with a discussion of these. Such a system is called an w-dimensional first-order system. An wth-order differential equation ,. Then equation 1. The subject of existence theorems for solutions of 1. We will prove the simplest version of the basic existence theorem and a theorem on continuation of solutions. We will merely state and explain the significance of the other theorems we need.

A domain is a connected open set in Rn. A convex set E in Rn is a set such that if p, q are points in E, then the line segment joinings and q is contained in E. Let f x; t be a real n-vector function i. Let a be the minimum of ajM and b. Suppose f x; t is continuous on a domain D such that D D R and suppose f satisfies a Lipschitz condition with respect to x in D. Then the n-dimensional system 1. Weaker hypoth- eses than the Lipschitz condition imply uniqueness of solution, but the Lipschitz condition, because of its simplicity, is very frequently used. A striking feature of the basic existence theorem above which holds generally in existence theorems for ordinary differential equations is that the solution x t, x 0 , t0 is only defined locally.

Analogous arguments can be made for the interval Step 1. Jto Jto Now by induction, we obtain an estimate for dn t. Step 3. Hence for all t e [tQ, tQ -f a], the function f[x t ] t] is defined and 1. J t0 We obtain the desired conclusion. Step 4. Suppose there exist continuous solutions x t and xa t of 1. Solution xll t is called a continua- tion of solution x t.

Let D be a bounded domain in x, t -space such that f x, t is continuous in D and f x, t satisfies a Lipschitz condition with respect to x in some neighborhood of each point in D. Suppose that the solution x t of 1. Let 8 0 be the distance from the point t, x t to the boundary of D, i. Since x t is continuous, the gib is a minimum. Let S be the open set in D which consists of those points of D which have distance greater than e1 from D' where e1 is a fixed positive number.

Since S is bounded. Then a solution of 1. As the ex used at the beginning of the proof to define S is arbitrary, this completes the proof of the first statement in the theorem. For the proof of the second statement, suppose that conclusion i does not hold. Then we may apply the first state- ment of the theorem and obtain conclusion ii. Now we state two extensions of the basic existence theorem: for proofs see Coddington and Levinson [1] ; Hurewicz [1] or Lefschetz [4].

A question about solutions which frequently occurs is: if f x, t has certain differentiability properties, does the solution x t have the same differentiability properties? The specific answrers to this question which we need are given by the following theorem: 1. For our study of quasilinear systems, we will need some facts concerning linear systems. For proofs, see Coddington and Levinson [1]. Let 2. System 2. An n x n matrix whose columns are n linearly independent solutions of 2. If M t is a fundamental matrix, then each solution of 2.

A fundamental matrix its elements are functions of t is nonsingular for all values of t. The solution x t of 2. For linear inhomogeneous systems, we have the useful variation of constants formula. Suppose M t is a fundamental matrix of 2. Jt0 3. We study the existence and stability of periodic and almost periodic solutions of the? Because LL is small, equation 3. We consider first the problem of periodic solutions of 3. The hypotheses to be imposed on 3.

The problem we study is: does equation 3. First a word about the significance of this problem. The fact that LL is kept small might seem to imply that this i. That is. For certain circumstances called the nonresonance case this is a rough description of what actually occurs. Starting with Poincare [1] a great deal of work has been done on various aspects of this problem. A detailed discussion of much of this work, including applications, is given by Malkin [1]. We impose one further condition on 3. We require that git 0 0 or that fix, t. Equation 3. If the right-hand side of 3. Indeed we may say that A.

In this case periodic solutions may be studied but the problem considered is quite different from the problem stated above. Now we proceed to a detailed study of 3. Applying the Existence Theorem for a System with a Parameter 1. We want to determine conditions under which this solution. Suppose 3. Then 3. But y t, JJL, c is a solution of 3. Hence by the uniqueness con- dition of the Basic Existence Theorem 1.

Now we use the variation of constants formula 2. If we can solve 3. Thus the problem of finding solutions of 3. Case I. This is called the non- resonance case or the nondegenerate case. That [M T — M Q ] is non- singular implies that the implicit function theorem can be applied to solve 3.

This is because the determinant of [M T — M 0 ] is exactly the Jacobian which appears in the hypothesis of the implicit function theorem. Thus the existence of a unique periodic solution of 3. This solution of the nonresonance case is due originally to Poincare. The drawback to this result is that many of the most important physical problems described by systems of the form 3. See Malkin [1] and Minorsky [2]. The number r is called the degree of degeneracy of the system. This is the more complicated case and its treatment requires the use of some linear algebra. See MacDuffee [1, pj.

We rewrite the branching equations 3. Then multiplying 3. A necessary condition that 3. Multiply 3. Solving 3. The expressions on the left in 3.

Some applications of the topological degree to stability theory

We can then apply the Existence Theorem 6. This in turn yields periodic solutions of 3. This is sometimes called the totally degenerate case. We have proved: 3. Again we apply the Existence Theorem 6. For example, from Theorem 6. At this stage, Theorem 3. Of course, Theorems 6. For such cases, the local degree theory is too coarse. Before applying the technique described above to some concrete problems in nonlinear oscillations, we show that if 3. In applications, it is essential to know if the solution equilibrium point, periodic solution or whatever has some kind of stability.

If the solution is unstable, it will not correspond to an observed phenomenon in the physical system described by the equation. So from the point of view of applica- tions, stability is just as important as existence. Indeed in some applications such as certain problems in the differential equations of control systems, interest is centered mainly on the stability question. The stability problem is hard to solve because of the difficulty of obtaining a satisfactory criterion for stability which is practical enough to be applied in specific problems.

The treatment here will be based upon the version of the Lyapunov theory in Lefschetz [1] and in Malkin [2]. The Lyapunov theory is widely used and has several advantages. However, it is not a final solution to the stability problem. For example, systems can be exhibited which are stable in the sense of Lyapunov and which are unstable from the practical point of view.

I t is convenient to start by studying the stability of critical points. Assume that the hypotheses of the Basic Existence Theorem 1. The critical point 0 is stable with respect to 4. If critical point 0 is not stable, then it ip said to be unstable. Lefschetz [4] makes a finer analysis of stability by introducing the notion of conditional stability. Some of the results obtained here can be extended by using this notion.

If solution x 0 t is not stable, then it is said to be unstable. A weaker but often useful kind of stability is orbital stability. The other types of orbital stability, i. Stability implies orbital stability, but there are examples of solutions which are orbitally stable but not stable. See Lefschetz [4, pp. Now we state several versions of a classical stability criterion due to Lyapunov. In the n-di- mensiorial system 4. If the characteristic roots of A all have negative real parts, the origin 0 is asymptotically stable with respect to 4.

A detailed discussion of this and other stability theorems can be found in Malkin [2] or Cesari [2J. Now suppose that x t is a periodic solution of period T of the? Since x t is a solution of 4. S , we obtain 4. If B is a constant matrix and if the characteristic roots of B all have negative real parts, then solution x t of 4. If there is a characteristic root of B with a positive real part, solution x t is unstable. We also need a stability theorem for periodic solutions which is applicable if matrix B is not constant.

For this purpose, we introduce the characteristic exponents of a linear system with periodic coefficients. Let 4. Let M t be a fundamental matrix of 4. Since B t has period T. Hence there is a constant nonsingular matrix C such that 4. If N t is another fundamental matrix of 4. Substituting from 4. Thus the matrices C and DCD'1 have the same characteristic roots. This justifies the following definition. Now the following results can be obtained.

The origin 0 is a critical point of systems 4. In the n-dimensional system 4. If there is at least one fih with absolute value greater than one, then the origin is unstable. If there is a characteristic exponent of B t with absolute value greater than 1, then x t is unstable. Now we are ready to study the stability of the periodic solutions of 3. The existence of the periodic solutions is established by applying to the branching equations 3.

The result of this section is, essentially, to show that the sign of the local degree yields information about the stability of the periodic solutions. This result is a rigorization and extension of the Andronov-Witt stability method in the nonautonomous case. Assume throughout our discussion of stability that 3. Fixed points and topological degree in nonlinear analysis Share this page. Advanced search. Author s Product display : J Cronin. Abstract: The topological methods based on fixed-point theory and on local topological degree which have been developed by Leray, Schauder, Nirenberg, Cesari and others for the study of nonlinear differential equations are here described in detail, beginning with elementary considerations.

Volume: Publication Month and Year: Copyright Year: Page Count: Cover Type: Softcover. Print ISBN Online ISBN Print ISSN: Online ISSN: Annalen, , Krasnoselski, Topological methods in the theory of nonlinear integral equations, Moscow, "Trans onto English, Pergamon Press, ". Math 6 Join as an Editor-in-Chief. Open Access. Download Certificates.