Abstract
Full Text
UDC 517.948.33
MATHEMATICS
P. P. ZABREIKO, A. I. POVOLOTSKII
BIFURCATION POINTS OF THE HAMMERSTEIN EQUATION
(Presented by Academician I. N. Vekua on March 10, 1970)
The paper considers the problem of bifurcation points of the equation
\[ \int_{\Omega} k(t,s) f[s,x(s)]\,ds=\lambda x(t). \tag{1} \]
Here \(\Omega\) is a bounded closed subset of a finite-dimensional space. In what follows it is assumed that the function \(f(s,u)\), defined on \(\Omega \times R^n\) with values in \(R^n\), satisfies the Carathéodory conditions and is potential \((f(s,u)=\operatorname{grad} G(s,u);\ G(s,0)=0)\); the matrix \(k(t,s)\) \((t,s\in\Omega)\) is symmetric, measurable in the aggregate of the variables, and the linear operator defined by it
\[ Kx(t)=\int_{\Omega} k(t,s)x(s)\,ds \tag{2} \]
in the Hilbert space of vector functions \(H=\mathcal L_2\) is self-adjoint and has no more than a finite number of negative eigenvalues (each of finite multiplicity). The paper is a natural continuation of work \((^1)\) on eigenvectors of the Hammerstein operator.
Equation (1) has the form \(Ax=\lambda x\). Suppose that the Hammerstein operator \(A\) acts in some real Banach space \(E\). If the condition \(f(s,0)=0\) is fulfilled, then equation (1) has the zero solution for all real \(\lambda\). A number \(\lambda_0\) is called a bifurcation point of equation (1) if for any \(\varepsilon,\delta>0\) there exists a \(\lambda\), \(|\lambda-\lambda_0|<\varepsilon\), to which there corresponds a nonzero solution \(x(t)\) of equation (1), with \(\|x\|_E<\delta\). The paper also considers asymptotic bifurcation points (in this case fulfillment of the condition \(f(s,0)=0\) is not assumed). A number \(\lambda_0\) is called an asymptotic bifurcation point of equation (1) if for any \(\varepsilon,\rho>0\) there exists a \(\lambda\), \(|\lambda-\lambda_0|<\varepsilon\), to which there corresponds a solution \(x(t)\) of equation (1), with \(\|x\|_E>\rho\).
The problem of bifurcation points of equation (1), in the case where the role of \(E\) is played by various \(\mathcal L_p\), was considered in a number of works by M. A. Krasnosel’skii \((^{2,3})\) and others, M. M. Vainberg \((^4)\) and others. A detailed analysis of the properties of operators in general Banach spaces of measurable functions (the so-called ideal spaces \((^{5,6})\)) made it possible to obtain essentially new results for this problem; these results are presented in the present note.
1. Let \(E\) be an ideal space of measurable vector functions finite almost everywhere on \(\Omega\) (see \((^{1,5})\)). By \(E'\) we denote the ideal space dual to \(E\); by \(E^0\), the set of vector functions from \(E\) with absolutely continuous norm; by \(E'/E\), the space of multipliers from \(E\) into \(E'\). The spaces \(E^0\) and \(E'/E\) are themselves ideal. Examples of ideal spaces are the space \(M_{u_0}\) (\(u_0\) a nonnegative measurable function) of vector functions for which the norm \(\|x\|_{M_{u_0}}=\inf\{\lambda:\ |x|\leq \lambda u_0\}\) is meaningful, the spaces \(\mathcal L_p\), and Orlicz spaces.
A set in \(E\) is called \(w\)-bounded if for every \(\varepsilon>0\) this set has an \(\varepsilon\)-net \(T\) such that \(|x|\leq u_0\) for all \(x\in T\) and for some \(u_0\in E\). An operator acting from one ideal space \(E_1\) into another \(E_2\) is called \(w\)-bounded if it maps every norm-bounded set into a \(w\)-bounded set.
Let \(H_1\) be the linear span of the eigenvectors of the operator (2) in \(H\) corresponding to negative eigenvalues. Denote \(J=-P_1+P_2\), where \(P_1\) and \(P_2\) are the projection operators, respectively, onto \(H_1\) and \(H_2=H\ominus H_1\).
We shall state the conditions that will be assumed to hold in what follows. Let the superposition operator \(fx(s)=f[s,x(s)]\) act from \(E\) into \(E'\), and let the linear operator \(K\) (and consequently also \(\widetilde K=JK\)) act from \(E'\) into \(E\), with \(E\subseteq H\). In this case in (1) the operator \(C=\widetilde K^{1/2}\) acts from \(H\) into \(E\), while the adjoint operator \(C'\) acts from \(E'\) into \(H\). The operator \(G[s,x(s)]\) acts from \(E\) into \(\mathscr L_1\), and therefore on \(H\) the Golomb functional is defined by
\[ \Phi(y)=\int_{\Omega}G[s,Cy(s)]\,ds. \tag{3} \]
This functional turns out to be differentiable on \(H\), and its gradient is the Krasnosel’skii operator \(\Gamma=C'fC\).
If the operator \(K\) is positive definite in \(H\), then \(\widetilde K=K\) and \(\widetilde K^{1/2}=K^{1/2}\). In this case the problem of bifurcation points of equation (1) reduces to the problem of bifurcation points of the equation \(\Gamma y=\lambda y\) with the Krasnosel’skii operator \(\Gamma\), acting in \(H\). If, however, \(K\) has a finite number of negative eigenvalues, then the problem reduces to the problem of bifurcation points of the equation \(J\Gamma y=\lambda y\). This is also true for asymptotic bifurcation points.
- Suppose that
\[ f(s,u)=a_0(s)u+h(s,u)u, \tag{4} \]
where \(a_0(s)\in E'/E\), and \(h(s,u)\) \((h(s,0)=0)\) is a function satisfying the Carathéodory conditions and defining the superposition operator \(hx=h[s,x(s)]\), acting from \(E\) into \(E'/E\). By \(K_0\) denote the linear integral operator with kernel \(k(t,s)a_0(s)\).
Assume that one of the following conditions is fulfilled:
a) the operator \(K\) is completely continuous;
b) the operator \(f\) is \(w\)-bounded (or the operator \(G\) is \(w\)-bounded, and the operator \(f\) is uniformly continuous), \(a_0(s)\in (E'/E)^0\), and the operator \(h\) is continuous at the zero point.
c) \(E=M_{u_0}\).
If any of these conditions is fulfilled, the Krasnosel’skii operator \(\Gamma=C'fC\) has as its potential the weakly continuous and smooth (uniformly differentiable on each pair) Golomb functional \(\Phi\). Moreover, the operator \(\Gamma\) has at the zero point the completely continuous Fréchet derivative \(B_0=C'A_0C\), where \(A_0\) is the operator of multiplication by the function \(a_0(s)\). Therefore, using the theorem of M. A. Krasnosel’skii \((^2)\) and the theorem from \((^3)\), we obtain the following statements.
Theorem 1. Let the operator \(K\) be positive definite and \(a_0(s)\ne 0\). Then every eigenvalue of the operator \(K_0\) is a bifurcation point for equation (1).
Theorem 2. Let the operator \(K\) have a finite number of negative eigenvalues and \(a_0(s)=c\ne 0\). Then for \(c<0\) every positive, and for \(c>0\) every negative, eigenvalue of the operator \(K_0\) is a bifurcation point for equation (1).
- Suppose now that
\[ f(s,u)=a_\infty(s)u+\omega(s,u), \tag{5} \]
where \(a_\infty(s)\in E'/E\), and \(\omega(s,u)\) is a function satisfying the Carathéodory conditions, with
\[ \lim_{u\to\infty}\operatorname*{as}\frac{\omega(s,u)}{u}=0, \tag{6} \]
\[ |\omega(s,u)|\leq a(s)+b(s)|u|, \tag{7} \]
where \(a(s)\in E'\), \(b(s)\in E'/E\) (under these assumptions, obviously, the superposition operator \(\omega x=\omega[s,x(s)]\) acts from \(E\) into \(E'\)). Denote by \(K_\infty\) the linear integral operator with kernel \(k(t,s)a_\infty(s)\).
Assume that one of the following conditions is satisfied:
a) the operator \(K\) is completely continuous;
b) the operator \(f\) is \(w\)-bounded (or the operator \(G\) is \(w\)-bounded, and the operator \(f\) is uniformly continuous); \(a_\infty(s)\in (E'/E)^0\); \(b(s)\in (E'/E)^0\);
c) \(E=M_{u_0}\).
Each of these conditions guarantees that the Krasnosel’skii operator \(\Gamma\) has a weakly continuous and smooth potential—the Golomb functional \(\Phi\). Moreover, under any of these conditions the operator \(\Gamma\) has the completely continuous asymptotic derivative \(B_\infty=C'A_\infty C\), where \(A_\infty\) is the operator of multiplication by the function \(a_\infty(s)\). Therefore, using somewhat modified results from (3), we obtain the following assertions.
Theorem 3. Let the operator \(K\) be positive definite and \(a_\infty(s)\ne 0\). Then every eigenvalue of the operator \(K_\infty\) is a bifurcation point for equation (1).
Theorem 4. Let the operator \(K\) have a finite number of negative eigenvalues and let \(a_\infty(s)=c\ne 0\). Then for \(c<0\) every positive eigenvalue, and for \(c>0\) every negative eigenvalue of the operator \(K_\infty\), is a bifurcation point for equation (1).
- Let the superposition operator \(fx=f[s,x(s)]\) act from one ideal space \(E_1\) into another \(E_2\). In the case of a regular (see (5)) space \(E_2\), the requirement that it be \(w\)-bounded is equivalent to the relation
\[ \lim_{\operatorname{mes}D\to 0}\ \sup_{\|x\|\leq r}\|P_Dfx\|=0 \tag{8} \]
(\(P_D\) is the operator of multiplication by the characteristic function of the set \(D\subset\Omega\)), and the requirement of uniform continuity is equivalent to the relation
\[ \lim_{\operatorname{mes}D,\rho\to 0}\ \sup_{\|x\|,\|y\|\leq r;\ \|x-y\|\leq \rho}\|P_Dfx-P_Dfy\|=0. \tag{9} \]
Therefore, in the case under consideration, uniform continuity is a weaker restriction. If the space \(E_2\) does not possess the regularity property, the requirements of \(w\)-boundedness and uniform continuity are independent.
Consider, in particular, the case when \(E_1\) is the Orlicz space \(\mathcal L_M\), generated by the \(N\)-function \(m(s,u)\), and \(E_2\) is the Orlicz space \(\mathcal L_N\), generated by the \(N\)-function \(n(s,u)\).
Lemma 1. The superposition operator \(fx=f[s,x(s)]\) acting from \(\mathcal L_M\) into \(\mathcal L_N\) is \(w\)-bounded if there exists a function \(\Psi(u)\) satisfying the condition
\[ \lim_{u\to\infty}\frac{\Psi(u)}{u}=\infty, \tag{10} \]
such that for every \(r>0\) one can specify \(a_r\in\mathcal L_1\), \(b_r,\lambda_r>0\), such that
\[ \Psi\{n[s,\lambda_r f(s,u)]\}\leq a_r(s)+b_r m(s,u/r). \tag{11} \]
Lemma 2. The superposition operator acting from \(\mathscr L_m\) to \(\mathscr L_N\),
\(fx=f[s,x(s)]\), is uniformly continuous if and only if, for some \(c>0\) and arbitrary \(\varepsilon,r>0\), one can specify \(a_{\varepsilon,r}\in \mathscr L_1\), \(b_{\varepsilon,r}>0\) such that
\[ n\left[s,\frac{f(s,u)-f(s,v)}{\varepsilon}\right] \le a_{\varepsilon,r}(s)+m\left[s,b_{\varepsilon,r}(u-v)\right]+ c\left[m\left(s,\frac{u}{r}\right)+m\left(s,\frac{v}{r}\right)\right]. \tag{12} \]
Lemmas 1 and 2 also cover the cases when the functions \(n(s,u)\) do not satisfy the \(\Delta_2\)-condition.
- In conclusion, we note that the results of the paper carry over to equations with a Lebesgue integral with respect to an arbitrary measure, in particular to infinite systems; in this case, instead of Banach spaces, one may consider locally convex spaces.
Leningrad State Pedagogical Institute
named after A. I. Herzen
Received
28 IV 1969
CITED LITERATURE
- P. P. Zabreiko, A. I. Povolotskii, DAN, 183, No. 4 (1968).
- M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations, 1956.
- M. A. Krasnosel’skii, A. I. Povolotskii, DAN, 91, No. 1, 19 (1953).
- M. M. Vainberg, Variational Methods for the Study of Nonlinear Operators, 1956.
- P. P. Zabreiko, Tr. seminara po funktsional’n. analizu, vol. 8 (1966).
- P. P. Zabreiko, Studies in the Theory of Integral Operators in Ideal Spaces of Functions, Doctoral dissertation, Voronezh, 1968.