UDC 517.948:513.88

MATHEMATICS

Yu. G. BORISOVICH, Yu. I. SAPRONOV

ON THE TOPOLOGICAL THEORY OF CONDENSING OPERATORS

(Presented by Academician P. S. Aleksandrov on 11 III 1968)

In the note \((^1)\), B. N. Sadovskii drew attention to a new interesting class of nonlinear operators, which he called condensing, and proved for them an analogue of Schauder’s fixed-point theorem.

In the present note we develop a method that makes it possible to investigate more general problems for this class of operators. The basic idea is that a condensing operator can be narrowed down to a compact convex subspace, relative to which the rotation of the corresponding vector field, studied earlier by one of the authors \((^4,^5)\), is defined. This permits one to prove for condensing operators a number of assertions having analogues in the theory of completely continuous and weakly continuous vector fields \((^2,^3)\).

1. In this section we study an important property of condensing operators—the property of having invariant compact subspaces.

Let us recall the basic definition \((^1)\). The measure of noncompactness \(\chi(B)\) of a set \(B\) is the infimum of those \(\varepsilon\) for which the set \(B\) has a finite \(\varepsilon\)-net. An operator \(F:\Omega \to E\) is called condensing on the set \(\Omega\) if \(\chi(FB)<\chi(B)\) for every bounded noncompact subset \(B \subset \Omega\). The equality \(\chi(B)=\chi(\overline{\operatorname{co}}B)\) holds, where \(\overline{\operatorname{co}}B\) is the convex closure of the set \(B\).

Let now \(F:\overline U \to E\) be a condensing mapping defined on the closure of a bounded open set \(U \subset E\).

Theorem 1. For any compact set \(R\), \(R \cap U \ne \varnothing\), there exists a compact closed convex set \(A\) such that:
1) \(R \subset A\), 2) \(U_A=U\cap A\ne \varnothing\), 3) the restriction of the mapping \(F\) to \(\overline U_A\) acts in \(A\): \(F\overline U_A \subset A\).

Proof. Consider the class \(\{B\}\) of all bounded convex closed sets satisfying the conditions: a) \(R\subset B\), b) \(F\overline U_B\subset B\). We shall call the sets \(B\) admissible. The class of admissible sets is nonempty; for example, \(B=\overline{\operatorname{co}}(R\cup FU)\) is admissible. Let \(A=\bigcap B\). This is the minimal set in the class \(\{B\}\), since a) is fulfilled and \(F\overline U_A\subset F\overline U_B\subset B\) for every \(B\).

Consider the operator \(PB=\overline{\operatorname{co}}(R\cup FU_B)\), which assigns to sets from \(\{B\}\) the convex closed sets \(PB\). We have the inclusions \(PB\subset B\), \(PB\subset\{B\}\). The first follows from conditions a), b), and the second from the inclusions \(R\subset PB\), \(F\overline U_{PB}\subset F\overline U_B\subset PB\). In particular, \(PA\subset A\), which, by the minimality of \(A\), leads to \(PA=A\). The latter is possible only for compact \(A\), since otherwise, when \(\chi(A)>0\), we have the contradictory inequality

\[ \chi(A)=\chi(PA)=\chi(F\overline U_A)<\chi(A). \]

Thus theorem 1 is proved.

Let now \(K\subset E\) be a cone in the Banach space \(E\), and let \(U_K=U\cap K\) be a bounded open set relative to \(K\). Let \(F:\overline U_K\to K\) be a condensing positive mapping. By \(K_\theta=\theta[A]\) we shall denote the cone formed by all rays issuing from the point \(\theta\) and intersecting the convex set \(A\subset E\). It is obvious that if \(A\) is compact and separated from zero, then the cone \(\theta[A]\) is locally compact. We shall show that a positive condensing mapping, under the condition \(\theta\in \overline{FU_K}\)

induces, in a certain way, a mapping \(\widetilde F\) acting in a locally compact cone and closely connected with \(F\); we shall call \(\widetilde F\) the restriction of \(F\).

Theorem 2. For every compact set \(R\subset K\), \(\theta\in R\), \(R\cap U\ne\varnothing\), there exists a compact closed convex set \(A\subset K\), \(\theta\in A\), such that: 1) \(R\subset A\); 2) \(U\cap K_0\ne\varnothing\), where \(K_0=\theta[A]\); 3) the restriction of the mapping \(F\) to \(\overline U_{K_0}\) acts in \(K_0\): \(\widetilde F\overline U_{K_0}\subset K_0\).

Proof. Consider the class \(\{D\}\) of all sets satisfying the conditions\(*\): a) \(D=[\theta B]\); b) \(B\subset K\) is a convex closed bounded set; c) \(R\subset B\); d) \(F\overline U_D\subset B\); e) \(\theta\in B\).

As in the first theorem, it is easy to verify that the class \(\{D\}\) contains a minimal set \(D_0=\bigcap D\), with \(D_0=[\theta A]\), where \(A=\bigcap B\). It remains to show that \(A\) is compact. To this end consider the operator in the space of subsets
\[ PD=[\theta(R\cup F\overline U_D)]. \]
Analogously, we have \(PD\subset D\), \(PD\subset\{D\}\), whence it follows that \(PD_0=D_0\) and, consequently, \(\chi(F\overline U_{D_0})=\chi(A)\). The latter implies the compactness of \(A\), since for \(\chi(A)>0\) we have \(\chi(F\overline U_{D_0})<\chi(A)\). Extending \(F\) from \(\overline U_{D_0}\) to \(\overline U_{K_0}\) by constant values, we obtain \(\widetilde F\), which proves the theorem.

Suppose now that the condensing operator \(F_t\) depends on a parameter \(t\in[0,1]\) in such a way that the condensing property is preserved for every \(t\) and \(F_t x\) depends continuously on \(t\) uniformly in \(x\) on \(\overline U\) or \(\overline U_K\); let \(\theta\in F_t\overline U_K\), \(0\le t\le 1\). We shall call \(F_t\) a family of homotopic condensing operators.

Theorem 3. Let \(F_t\) be a family of homotopic condensing mappings \(\overline U\to E\) or \(\overline U_K\to K\). Then the assertions of Theorems 1 and 2 are valid with the modification that: 3) \(F_t\overline U_A\subset A\), \(\widetilde F_t\overline U_{K_0}\subset K_0\) for all \(t\in[0,1]\).

In the proof we use new properties of the function \(\chi(B)\). Let
\[ \theta(A,B)=\sup_{a\in A}\rho(a,B) \]
be the deviation of the set \(A\) from the set \(B\), and
\[ \rho(A,B)=\max\{\theta(A,B),\theta(B,A)\} \]
the distance between \(A\) and \(B\) in the sense of Hausdorff.

Lemma 1. \(\chi(B)=\inf \theta(B,\sigma)\), where \(\sigma\) is an arbitrary finite subset of \(E\).

Lemma 2. If \(G_t\) is a family of bounded sets, continuously dependent on \(t\in[0,1]\) in the Hausdorff metric, then \(\chi(G_t)\) is a continuous function of \(t\) and
\[ \chi\Bigl(\bigcup_t G_t\Bigr)=\sup_t \chi(G_t)=\chi(G_{t_0}) \]
for some \(t_0\in[0,1]\).

We now indicate what changes must be made in the proofs of Theorems 1, 2 in order to obtain Theorem 3. Suppose first that \(F_t:\overline U\to E\). Define the class of admissible sets \(\{B\}\) by the conditions: a) \(R\subset B\); b) \(F_t\overline U_B\subset B\) for every \(t\). This class includes, for example, the set \(\operatorname{co}(R\cup\bigcup_t F_t U)\). Consider the operator
\[ PB=\overline{\operatorname{co}}\,(R\cup\bigcup_t F_t U_B), \]
for which property 3) is satisfied.

The minimal set \(A\) in the class \(\{B\}\) satisfies the condition \(PA=A\), which is equivalent to the compactness of \(A\) and \(F_t\overline U_A\subset A\) for all \(t\in[0,1]\). Here we use the inequality
\[ \chi\Bigl(\bigcup_t F_t\overline U_A\Bigr)<\chi(A) \]
for \(\chi(A)>0\), proved with the aid of Lemma 2. The proof is carried out analogously in the case of positive mappings \(F_t\).

2. In this section we shall define the relative rotation of the vector field generated by a condensing operator.

Let \(F_t:\overline U\to E\) or \(\overline U_K\to K\) be a homotopic family of condensing operators, and suppose that on the boundary \(\dot U\) or \(\dot U_K\) (\(\dot U_K\) is understood relative to \(K\)) there are no fixed points of the operator \(F_t\), \(t\in[0,1]\). It is easy to see that the set \(G\) of all fixed points of the operators \(F_t\) is compact. Choose a compact set \(R\) so that \(R\supset G\), \(U\cap R\ne\varnothing\), and so that the relative boundary \(\dot U_R\) is nonempty. Then construct for the family

\(*\) We denote \(\overline{\operatorname{co}}(\theta\cup B)\) by \([\theta B]\).

\(F_t\) has a common invariant subspace \(A\) or \(K_0\), as in Theorem 3, and we narrow the family \(F_t\) to this subspace: \(F_t:\bar U_A\to A\), \(F_t:\bar U_{K_0}\to K_0\). On the relative boundaries \(\dot U_A,\dot U_{K_0}\) the operators \(F_t\) have no fixed points. For the corresponding vector fields \(\Phi_t x=F_t x-x\), relative to \(A\) or \(K_0\), on the boundaries \(\dot U_A,\dot U_{K_0}\) there is defined the rotation \(\gamma(\dot U_A,\Phi_t)\), \(\gamma(\dot U_{K_0},\Phi_t)\), studied earlier by one of the authors \((^4)\). It is constant under changes of \(t\) and has the usual properties.

If one analyzes a number of topological theorems \((^{2-4})\), one observes that all constructions can be carried out in an invariant convex compact subset. This makes it possible, with the aid of the relative rotation \(\gamma(\dot U_A,\Phi_t)\), \(\gamma(\dot U_{K_0},\Phi_t)\), to obtain analogous propositions for condensing operators.

1. We formulate a number of such assertions. In notes \((^{4,5})\) one of the authors considered various strengthenings of Brouwer’s theorem \((^6)\). We shall formulate their analogues and a number of other propositions for condensing mappings.

Theorem 4. Let \(F:\bar U\to E\) be a condensing mapping, \(D\subset U\) some convex closed set, \(F^jD\subset U\) for all \(j=1,2,\ldots,m-1\), and let \(F^m\) map its natural domain of definition into \(D\). Then the relative rotation of the vector field \(\Phi=F-I\) is equal to 1; consequently, the mapping \(F\) has a fixed point in \(D\).

Theorem 5. Let \(U\) be an open convex set. Let \(F,F^1,\ldots,F^{2m-1}\) be condensing mappings \(\bar U\to E\), with \(F^k(\bar U)\subset U\), \(k=m,\ldots,2m-1\). Then the operator \(F\) has a fixed point.

We note that in this case the relative rotation is also equal to 1.

Theorem 6. Let \(F_t,\ 0\le t\le 1:\bar U\to E\) be a family of homotopic condensing mappings, where \(F_0\) satisfies the conditions of Theorem 4 or 5, and on the boundary \(\dot U\) there are no fixed points of any of the operators \(F_t\). Then the operator \(F_1\) has a fixed point, and the rotation of the vector field \(\Phi_1=F_1-I\) is equal to \(+1\).

Corollary. If \(F_0\) is a condensing mapping \(\bar U\to E\) satisfying the conditions of Theorems 4 or 5, and if the operator \(F_0+\Omega:\bar U\to E\) is condensing, then there exists \(\varepsilon>0\) such that the mapping \(F_0+\mu\Omega\), for all \(\mu,\ 0\le\mu\le\varepsilon\), has a fixed point.

Theorem 7. If \(F\) is a condensing mapping \(\bar U\to E\) in a Hilbert space, \(\theta\in U\), and \((Fx,x)\le (x,x)\), \(x\in\dot U\), then there exists a fixed point.

Theorem 8. If \(F\) is a condensing mapping, defined in a neighborhood of the point \(\theta\), having a Fréchet derivative \(F'_\theta\) at the point \(\theta\), which is also a condensing operator, then every eigenvalue \(\lambda\), \(|\lambda|>1\), of the Fréchet derivative of odd multiplicity is a bifurcation point of small solutions of the equation \(Fx=\lambda x\).

Now let \(F:\bar U_K\to K\) be a positive condensing operator. We note the simple proposition:

If \(U\ni\theta\) and on the boundary \(\dot U_K\) the inequality \(Fx\ge x\) holds (“not a single element goes forward” \((^3)\)), then \(\gamma(\Phi,\dot U_{K_0})=1\).

The results of the article are valid for countably normed spaces, if one sets \(\chi=\sup\chi_\alpha\), where \(\chi_\alpha\) is the measure of noncompactness constructed from the seminorm \(\|x\|_\alpha\).

Voronezh State
University

Received
21 II 1968

REFERENCES

1. B. N. Sadovskii, Functional Analysis and Its Applications, 1, no. 2, 1967, p. 74.
2. M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations, 1956.
3. M. A. Krasnosel’skii, Positive Solutions of Operator Equations, 1962.
4. Yu. G. Borisovich, DAN, 153, no. 1, 12 (1963).
5. Yu. G. Borisovich, Transactions of the Seminar on Functional Analysis, Voronezh State University, issue 7, 17 (1963).
6. F. E. Browder, Duke Math. J., 26, no. 2 (1959).