ON CALCULATION OF THE RELATIVE INDEX OF A FIXED POINT IN THE NONDEGENERATE CASE

The paper is devoted to the calculation of the index of a zero and the asymptotic index of a linear completely continuous nonnegative operator. Also the case of a nonlinear completely continuous operator A whose domain and image are situated in a closed convex set Q of a Banach space is considered. For this case, we formulate the rules for calculating the index of an arbitrary fixed point and the asymptotic index under the assumption that the corresponding linearizations exist and the operators of derivative do not have eigenvectors with eigenvalue 1 in some wedges.


Introduction
Let Q be a closed convex set in a Banach space and let A : Q → Q be a completely continuous operator.In [4] the calculation problem of a fixed point index of a vector field I − A was formulated.In the simplest case, when Q is a cone, this problem was investigated in the articles by Isaenko [3], Mukhamadiev and Sabirov [5], and Pokornyi [6] (see also references in [8]).Later, in [1,2] Dancer presented the general formula for the fixed point index of a completely continuous operator A with its domain and image in an arbitrary closed convex set.However, the case of an asymptotic index was not considered.Note that for the case of a cone this problem was earlier considered in the articles by Pokornyi [6] and Pokornyi and Astaf 'eva [7].The present paper concerns the cases of a wedge and an arbitrary closed convex set.In the latter case, the calculation of a fixed point index is reduced to the index calculation with respect to a specially constructed wedge.We also show that for the infinity singular point one needs to take the wedge (1.1)

Index of a linear operator
Let X be a Banach space, W a wedge in X (this means that W is a closed subset of X such that W + W ⊆ W and λW ⊂ W for λ ≥ 0), and A a linear completely continuous operator such that AW ⊆ W. (2.1) Let L = W ∩ (−W).Then L is the maximal subspace which is contained in W. From (2.1) and the linearity of A it follows that the inclusion holds.
Consider a quotient space X/L and a quotient mapping [•] : X → X/L.It is easy to check that the image W of W under the quotient mapping [•] is a cone in X/L.
From (2.2) it follows that the operator A induces a linear mapping A of the quotient space X/L into itself such that A[x] = [Ax].And by (2.1) we have that the cone W is invariant under the operator A, that is, A is nonnegative in the quotient space X/L.Theorem 2.1.Let A be a linear completely continuous operator, acting in a Banach space X, and let W be a wedge that is invariant under the operator A. Suppose that Ax where A| L is the restriction of the operator A to the space L, β(A| L ) is the sum of multiplicities of eigenvalues of A| L , greater than 1, and ρ( A) is the positive spectral radius of the operator A.
Proof.If Ax = x for x ∈ W, x = 0, then zero and infinity singular points of the vector field Φ = I − A are isolated in W. Hence the relative indices ind(0,I − A;W) and ind(∞,I − A;W) are well posed.In this case, by the definition of index at infinity (see, e.g., [4]), since the operator A has no more fixed points in W, it follows that ind(∞, To calculate the index ind(0, Φ;W) we will consider two possible cases: when the spectral radius ρ( A) of the operator A is less than 1 and when it is greater than 1.The case ρ( A) = 1 is impossible.Indeed, if this is not true, then there exists an element [x] * = 0 of the quotient space X/L such that A[x] * = [x] * .In other words, there exist u ∈ W, u ∈ L, and y ∈ L such that the equality u − Au = y holds.Let us explore the solvability in L of (I − A)w = y.If it had a solution w ∈ L, then the vector u − w would be an eigenvector of the operator A, corresponding to the eigenvalue 1 and would be in W, which would contradict our assumptions.If we supposed that the equation had no solutions in L, then the operator I − A would be invertible in L, which is impossible.
In the case ρ( A) < 1, let us show that the vector field Φx = x − Ax is linearly relatively homotopic on S W = {x ∈ W : x = 1} to the field Φ 1 x = x − AQ(x), where Q : X → L is A. V. Guminskaya and P. P. Zabreiko 3 a projection (in general, nonlinear) of X on L (see [4,Theorem 18.1]).To prove this fact assume the converse, that is, that has a solution for some x ∈ S W and λ ∈ [0,1].Now if we rewrite this equation for the operator A and recall (2.2), we get where Under our assumption on the spectral radius of the operator A this yields [x] = 0.In this case, we obtain x = Qx and then (2.4) implies x = Ax for x ∈ S W , which contradicts the assumption of our theorem.
Thus the relative index ind(0,I − A;W) is equal to the relative index ind(0,I − AQ;W), which actually is the Leray-Schauder zero fixed point index of the restriction A| L of the operator A to the subspace L and we can calculate it by the well-known formula (see, e.g., [4,Theorem 21 where β(A| L ) is the sum of multiplicities of eigenvalues of A| L , greater than 1.Now assume that ρ( A) > 1.In this case, there exists an element [x] * in the cone W such that A[x] * = ρ[x] * (ρ > 1).In other words, there exist an element x * ∈ L of the wedge W (we assume that x * = 1) and an element z of the subspace L such that Ax * = ρx * + z.
Show that the vector field Φx = x − Ax is linearly relatively homotopic on S W to the field Φ 2 x = x − cx * , where the constant c will be defined later.Let us show that For λ = 0, (2.7) coincides with x = Ax.The latter equation has no solutions for x ∈ S W .For 0 < λ ≤ 1 from (2.7), it follows that there exists a real t > 0 such that (2.8) We claim that there exists the maximal of such reals: ξ = max x∈SW ,x≥tx * t.We argue by contradiction.The inequality x * ≤ x/t implies that if t tends to infinity, then x * ≤ 0. On the other hand, x * is an element of the wedge W, thus x * ≥ 0. Hence we get that x * ∈ L, which contradicts its choice.Further, (2.7) and (2.8) imply (2.9) Choose c > ξ and observe that for this c the inequality (1 − λ)ξρ + λc > ξ holds.Indeed, for λ = 0 this follows from ρ > 1.For λ = 1 this follows from our assumption for c.For other λ ∈ (0,1) the inequality holds as the result of two previous cases.
Therefore for the chosen c, the real ξ is not maximal among t such that x ≥ tx * .This contradiction proves the nondegeneracy of the linear homotopy connecting vector fields Φ and Φ 2 .Consequently, ind(0,Φ;W) = ind(0,Φ 2 ;W).
To calculate ind(0,Φ 2 ;W) we will use the corollary of the Hahn-Banach theorem.According to it there exists a functional l ∈ X * such that l(W) ≥ 0 and l(x * ) = 1 (then l ≥ 1).Since (2.8), we obtain ξ ≤ l(x) ≤ l for x ∈ S W .And our assumption of c yields c > l ≥ 1.Now we can show that the vector field Φ (2.10) Hence, by the relative rotation property, ind(0, Φ 2 ;W) = 0.This completes the proof.
In applications there usually exist a complement X 1 of the linear hull Then X can be presented as the direct sum of subspaces (see Figure 2.1) (2.11) From (2.1) and the linearity of the operator A it follows that the inclusions hold.
Assume that an intersection W ∩ X 2 is not empty.Then it is easy to prove that this set is a cone K in X 2 .It generates the order relation in X 2 by the following rule: It can be proved that under such decomposition of the space X, the wedge W is invariant under a linear operator A if and only if A is determined by the matrix where ρ(a 22 ) is the spectral radius of the operator a 22 and β(a 33 ) is the sum of multiplicities of eigenvalues of the operator a 33 , greater than 1.

Index of a nonlinear operator
Let A be a nonlinear operator and let Q be a closed convex set of a Banach space X that is invariant under the operator A. In this part, we discuss the relative fixed point index calculations of A under the assumption that A has a Fréchet derivative at its fixed point.
Recall that an operator A is called differentiable at the point and differentiable at infinity with respect to where A (x 0 ) and A (∞) are linear operators.
Let Q be an arbitrary closed convex set of a Banach space X.Assume that Q is invariant under a completely continuous operator A. Let x 0 be a fixed point of the operator A and let A have a Fréchet derivative at the point x 0 with respect to Q.
The following lemmas show that in the case when the fixed point x 0 is not the infinity singular point some wedge W x0 comprising the set Q is invariant under the mapping of the derivative A (x 0 ), whereas in the case of the infinity singular point some wedge W ∞ lying in the set Q is invariant under the mapping of derivative A (∞). Lemma 3.1.Let A be a completely continuous operator, acting in a Banach space X, and let Q ⊆ X be a closed convex set that is invariant under A. Let x 0 ∈ Q be a fixed point of the operator A and let A have Fréchet derivative at x 0 with respect to Q. Then the wedge is invariant under the mapping of the derivative A (x 0 ).
Let h be an arbitrary nonzero element of W x0 .Then there exist x ∈ Q and t > 0 such that h = t(x − x 0 ).From the differentiability of the operator A at x 0 and the linearity of the operator A (x 0 ) it follows that Since A(Q) ⊆ Q, the element in the left-hand side of this equality is in W x0 .Taking the limit as t → ∞, by the closedness of W x0 , we get A (x 0 )h ∈ W x0 .This completes the proof.
Lemma 3.2.Let A be a completely continuous operator, acting in a Banach space X, and let Q ⊆ X be a closed convex unbounded set that is invariant under A. Let A have Fréchet derivative at infinity with respect to Q. Then the wedge

is invariant under the mapping of the derivative A (∞).
Proof.To prove that W ∞ is invariant under the operator A (∞), it suffices to show that there exists an element Then, by the convexity of Q, for any λ ≥ 0 we have Taking the limit as λ → ∞, by the closedness of Q, we get x * + tA (∞)h ∈ Q for any t ≥ 0, h ∈ W ∞ .This completes the proof.
The following theorem specifies the main result of Dancer [2, Theorem 1].
Theorem 3.3.Let A be a completely continuous operator, acting in a Banach space X, and let Q ⊆ X be a closed convex set that is invariant under A. Let x 0 ∈ ∂Q be a fixed point of the operator A and let A have Fréchet derivative at x 0 with respect to Q. Then the wedge is invariant under the mapping of the derivative A (x 0 ).If A (x 0 )x = x for x ∈ W x0 , x = 0, then the fixed point x 0 of the vector field (3.10) Proof.Without loss of generality it can be assumed that x 0 = 0 (in the opposite case, the whole argument needs to be made for the operator A(x 0 + x) − x 0 ).From the differentiability of the operator A at the point 0 it follows that there exists a linear operator B = A (0) such that Ax = Bx + w(x), (3.11)where the operator w meets the condition lim x∈Q, x→0 w(x) x = 0. (3.12) If Bx = x for x ∈ W 0 , x = 0, there exists a positive real c > 0 such that for any x ∈ W 0 the inequality holds.
Choose a real r > 0 such that inequalities hold for x ∈ Q, x ≤ r, where ρ r = sup x∈W0, x =r ρ(x,Q) and ρ(x,Q) denotes the distance from the point x to the set Q.
Show that on the intersection of the sphere S r = {x ∈ X : x = r} of radius r with the set W 0 the vector field Φ = I − B is linearly homotopic to the field Φ α x = x − AP α x where α meets the condition and P α is a projection (in general, nonlinear) on Q that has the following property: (the existence of such projection follows from [4, Theorem 18.1]).
To prove this, consider the linear deformation that connects vector fields Φ and Φ α .From the convexity of W 0 and the invariance of the sets W 0 and Q ⊆ W 0 under operators B and A, respectively, it follows that the element (3.18) Hence the vector fields Φ and Φ α are homotopic on S r ∩ W 0 .Thus, by the first property of the relative rotation, ind 0,I − B;W 0 = ind 0,I − AP α ;W 0 . ( By definition, the relative index ind(0,I − AP α ;W 0 ) is equal to the relative rotation γ(I − AP α ,B r ∩ W 0 ;W 0 ) of the vector field I − AP α on the boundary of an open set B r ∩ W 0 , where B r = {x ∈ X : x < r}.By the additivity property of rotation, From the fact that AP α has no fixed points beyond Q, it follows that γ(I − AP α ,B r ∩ (W 0 \ Q);W 0 ) = 0. On the other hand, the relative rotation γ(I − AP α ,B r ∩ Q;W 0 ) can be considered as the rotation γ(I − A,B r ∩ Q;Q) of the vector field I − A on the boundary of the open set B r ∩ Q with respect to Q.By the definition of relative index, this rotation coincides with ind(0,I − A;Q).This completes the proof.
As it appears, the analogous statement is true for the case of asymptotic index.
A. V. Guminskaya and P. P. Zabreiko 9 Theorem 3.4.Let A be a completely continuous operator, acting in a Banach space X, and let Q ⊆ X be a closed convex unbounded set that is invariant under A. Let A have Fréchet derivative at infinity with respect to Q. Then the wedge is invariant under the mapping of the derivative A (∞).
If W ∞ = {0} and A (∞)x = x for x ∈ W ∞ , x = 0, then the infinity singular point of the vector field Proof.From the differentiability of the operator A at infinity it follows that there exists linear operator B = A (∞) such that If Bx = x for x ∈ W ∞ , x = 0, then there exists a positive real c > 0 such that for all x ∈ W ∞ the inequality holds.
Choose a real R > 0 such that inequalities hold for all x ∈ Q, x ≥ R, where ρ R = sup x∈Q, x =R ρ(x,W ∞ ) and ρ(x,W ∞ ) denotes the distance from the point x to the set W ∞ .Show that on the intersection of the sphere S R = {x ∈ X : x = R} of radius R with the set Q the vector field Φ = I − A is linearly homotopic to the field Φ α x = x − BP α , where a real α meets the condition and P α is a projection (in general, nonlinear) on W ∞ that has the following property: Consider the linear deformation that connects vector fields Φ and Φ α .Since the convexity of Q and the invariance of the sets W ∞ ⊆ Q and Q under the operators B and A, respectively, it follows that the element

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: .30) Hence the vector fields Φ and Φ α are homotopic on S R ∩ Q.Thus, by the first property of the relative rotation, ind(∞,I − A;Q) = ind ∞, I − BP α ;Q .(3.31)By definition, the relative index ind(∞,I − BP α ;Q) is equal to the relative rotation γ(I − BP α ,B R ∩ Q;Q) of the vector field I − BP α on the boundary of an open set B R ∩ Q, where B R = {x ∈ X : x < R}.By the additivity property of rotation, .32) From the fact that the operator BP α has no fixed points beyond W ∞ , it follows that the relative rotation γ(I − BP α ,B R ∩ (Q \ W ∞ );Q) is equal to zero.Finally, the relative rotation γ(I − BP α ,B R ∩ W ∞ ;Q) can be considered as the rotation γ(I − B,B R ∩ W ∞ ;W ∞ ) of the vector field I − B on the boundary of open set B R ∩ W ∞ with respect to W ∞ .By the definition of relative index and since the operator B has no nonzero fixed points in W ∞ , the latter rotation coincides with ind(0, I − B;W ∞ ).This completes the proof.

First
Round of ReviewsMay 1, 2009 22 ≥ 0. Furthermore, one can show that zero and infinity singular points of such operator A are isolated in W if and only if 1 is not an eigenvalue of the operators a 22 and a 33 .In this case, Theorem 2.1 can be formulated in the following way.Theorem 2.2.Let A be a linear completely continuous operator, acting in a Banach space X, and let W be a wedge that is invariant under the operator A. Then A can be defined by matrix(2.13).If 1 is not an eigenvalue of the operators a 22 and a 33 , then zero and infinity singular points of the operator A are isolated in W and ind(0,I −