UDC 519.34

MATHEMATICS

M. M. VAINBERG

NONLINEAR EQUATIONS WITH POTENTIAL AND MONOTONE OPERATORS

(Presented by Academician G. I. Petrov on 11 IV 1968)

Nonlinear equations with potential and monotone operators were considered by us in a number of works (for example, in \((^{1-3})\)). Later such equations were studied in the works of J. L. Lions \((^{4})\), F. Browder (see, for example, \((^{5})\), where a bibliography is given), G. Minty \((^{6})\), and in the works of other authors. In the present paper we study such equations in separable locally convex spaces and establish new propositions for them.

1. Let \(E\) be a real reflexive locally convex space contained in some real Hilbert space \(H\). Everywhere we shall assume that condition \((\alpha)\) of \((^{7})\) is satisfied, i.e. \(E \subset H\), \(H\) is contained in the strong dual space \(E'\) and is dense in it, the topologies of the spaces \(E\) and \(H\), and also of \(H\) and \(E'\), respectively, are compatible, and the bilinear functional \(\langle x,y\rangle\), where \(x \in E\), \(y \in E'\), coincides for \(y \in H\) with the scalar product \((x,y)\) in \(H\). Examples of such spaces \(E\), \(H\), and \(E'\) are indicated in \((^{7})\). They include, in particular, the Lebesgue spaces \(E=L^p(D)\), \(H=L^2(D)\), where \(D\) is a set of finite measure in \(s\)-dimensional Euclidean space and \(p>2\). Other important examples of such spaces are rigged Hilbert spaces \((^{8})\).

2. Consider the equation

\[ x = BF(x), \qquad x \in E, \tag{1} \]

where \(F\) is a nonlinear operator from \(E\) into \(E'\) and \(B\) is a linear bounded operator from \(E'\) into \(E\).

Let \(B_H\) be the restriction of the operator \(B\) to \(H\). Suppose that \(B_H\) is a self-adjoint positive operator in \(H\). Then \((^{7})\) the positive square root \(A=B_H^{1/2}\) is bounded from \(H\) into \(E\), and the adjoint operator \(A'\) acts and is bounded from \(E'\) into \(H\), with \(B=AA'\).

In this case consider the equation

\[ y = A'F(Ay), \qquad y \in H. \tag{2} \]

Equations (1) and (2) are called equivalent if the sets of their solutions are equivalent.

Lemma 1. Equations (1) and (2) are equivalent; moreover, every solution \(y\) of equation (2) leads to a solution \(x=Ay\) of equation (1), and every solution \(x\) of equation (1) leads to a solution \(y=A'F(x)\) of equation (2).

1. Let \(B_H\) be an indefinite operator in \(H\),

\[ B_H^+ = \tfrac12(B_H+|B_H|), \qquad B_H^- = \tfrac12(B_H-|B_H|), \]

\[ A=(B_H^+)^{1/2}-|B_H^-|^{1/2}, \qquad T=(B_H^+)^{1/2}+|B_H^-|^{1/2}. \]

Then \((^{7})\) the operator \(A\) is bounded from \(H\) into \(E\), the operator \(T'\) is bounded from \(E'\) into \(H\), and \(B=AT'\),

Let us consider in this case the equation

\[ y = T'F(Ay), \qquad y \in H . \tag{3} \]

Lemma 2. Equations (1) and (3) are equivalent; moreover, every solution \(y\) of equation (3) leads to a solution \(x = Ay\) of equation (1), and every solution \(x\) of equation (1) leads to a solution \(y = T'F(x)\) of equation (3).

1. Let \(B_H\) be a self-adjoint positive operator in \(H\), and let \(x_1, x_2\) be distinct solutions of equation (1). They correspond to distinct solutions \(y_1\) and \(y_2\) of equation (2), so that

\[ 0 < (y_1 - y_2, y_1 - y_2) = (y_1 - y_2, A'[F(Ay_1) - F(Ay_2)]) = \]

\[ = \langle x_1 - x_2, F(x_1) - F(x_2)\rangle . \]

An operator \(F\) from \(E\) into \(E'\) (not necessarily linear) satisfying the condition

\[ \langle x_1 - x_2, F(x_1) - F(x_2)\rangle \geq 0 \tag{4} \]

for arbitrary \(x_1, x_2 \in E\) was called in \((^3)\) positive definite, and in \((^9)\) monotone. Thus, on the set of solutions of equation (1), the operator \(F\) is strictly monotone, i.e. in (4) equality is excluded when \(x_1 \ne x_2\). Hence the following assertion follows. If the operator \((-1)F\) is monotone, i.e.

\[ \langle x_1 - x_2, F(x_1) - F(x_2)\rangle \leq 0, \tag{5} \]

then equation (1) cannot have more than one solution.

Theorem 1. Suppose the following conditions are satisfied: 1) a continuous operator \(F(x)\), acting from \(E\) into \(E'\), satisfies inequality (5); 2) \(B_H\) is a self-adjoint positive operator in \(H\). Then equation (1) has a unique solution, and it belongs to the space \(E\).

Let us note that this theorem is an analogue of Theorem 2.1 from \((^2)\), established by us for Lebesgue spaces, and the conditions of Theorem 2.1 from \((^2)\) ensure the uniqueness of the solution established there.

1. Here we shall assume that \(B_H\) is a quasi-positive operator \((^1)\), \(F(x)\) is a potential operator \((^{1,10})\), continuous along every segment of the space \(E\) (hemicontinuous \((^{11})\)), and the potential of the operator \(F(x)\), i.e. \(f(x)\), is continuous on every bounded set \(C \subset E\) endowed with the topology \(\tau\) induced in \(C\) by the space \(E\) \((^4)\).

Theorem 2. Suppose the following conditions are satisfied: 1) \(B\) is a linear bounded operator from \(E'\) into \(E\), whose restriction \(B_H\) is a quasi-positive operator in \(H\); 2) \(F(x)\) is a potential monotone and hemicontinuous operator in \(E\), whose potential \(f(x)\) is continuous on every bounded set in \(E\) with respect to the topology \(\tau\) and satisfies the inequality

\[ f(x) \geq \lambda_1 (x,x) + b(x,x)^{\alpha/2} + c, \qquad x \in E, \]

where \(\lambda_1\) is the largest (positive) characteristic number of the operator \(B_H\), \(0 \leq \alpha < 2\), and \(b\) and \(c\) are arbitrary negative numbers. Then equation (1) has a solution belonging to the space \(E\).

The proof is carried out according to the scheme indicated in \((^{1,2})\), and uses the following proposition.

Lemma 3. In order that a real functional \(f(x)\), defined on a locally convex space, be sequentially weakly lower semicontinuous, it is necessary and sufficient that the following condition be satisfied: for any real number \(c\), the set

\[ E_c = \{x: f(x) \leq c\} \]

is sequentially weakly closed.

The property is also used that if a functional defined in a vector space (not necessarily a Banach space) is Gateaux differentiable, then for its convexity (strict convexity) it is sufficient that its gradient be a monotone (strictly monotone) operator.

If the potentiality of the operator \(F\) is abandoned, we arrive at the following proposition.

Theorem 3. Suppose that the following conditions are satisfied: 1) \(B_H\) is a quasi-positive operator in \(H\); 2) the continuous operator \(F\) from \(E\) into \(E'\) satisfies the condition

\[ \langle h, F(x+h)-F(x)\rangle \geqslant 2\lambda_1(h,h), \qquad h\in E, \]

where \(\lambda_1\) is the largest (positive) characteristic number of the operator \(B_H\). Then equation (1) has a unique solution in \(H\).

1. Here we shall consider the operator \(\Gamma=BF\) under the assumption that \(B_H\) is a quasi-positive operator in \(H\) and \(F\) is a potential operator satisfying the conditions: it is strictly positive, i.e. \(\langle x,F(x)\rangle>0\) for \(x\ne0\), and \(F(0)=0\). In this case various propositions concerning the eigenvectors of the operator \(\Gamma\) are valid. We give one such proposition.

Theorem 4. Let \(F\) be a strictly positive operator, \(F(0)=0\), and let \(B\) be a bounded operator from \(E'\) into \(E\) such that \(B_H\) is a quasi-positive operator in \(H\). Then there exists an \(r>0\) such that, whatever hyperboloid in \(H\), \(((u))=c<r\), generated by the operator \(B_H\), is taken, there exist in \(E\) two eigenvectors of the operator \(\Gamma\), and they are representable in the form

\[ x_c^{(1)}=Au_c^{(1)}, \qquad x_c^{(2)}=Tu_c^{(2)}, \qquad ((u_c^{(1)}))=((u_c^{(2)}))=c. \]

These eigenvectors correspond respectively to the positive eigenvalues

\[ \mu_c^{(i)}=c^{-2}\langle x_c^{(i)},F(x_c^{(i)})\rangle, \qquad i=1,2. \]

Among the eigenvectors \(x_c^{(1)}\) (and \(x_c^{(2)}\)) there is a continuum of such vectors whose norms in \(H\) are less than an arbitrary positive number.

Let us also note that if, in the hypotheses of Theorem 4, one additionally requires that the potential \(f(x)\) of the operator \(F\) satisfy, for vectors \(v\in V\subset H\) (where \(V\) is the cone \((^1)\) generated by the operator \(B_H\)), the conditions

\[ \lim_{\|v\|\to\infty} f(Av)=+\infty, \qquad \lim_{\|v\|\to+\infty} f(Tv)=+\infty, \]

then the assertion of Theorem 4 is valid for every hyperboloid; moreover, among the eigenvectors of the operator \(\Gamma\) there is also a continuum of such vectors whose norms in \(H\) are greater than any positive number.

Moscow Regional Pedagogical Institute
named after N. K. Krupskaya

Received
11 IV 1968

REFERENCES

\(^{1}\) M. M. Vainberg, Variational Methods for the Investigation of Nonlinear Operators, 1956.
\(^{2}\) M. M. Vainberg, Uch. zap. Mosk. obl. ped. inst. im. N. K. Krupskoi, 77, 131 (1959).
\(^{3}\) M. M. Vainberg, UMN, 15, no. 1, 243 (1960).
\(^{4}\) Ya. L. Engelson, Nauchn. dokl. vyssh. shkoly, 4, 75 (1958).
\(^{5}\) F. E. Browder, Arch. Rational Mech. and Analysis, 26, no. 1, 33 (1967).
\(^{6}\) G. J. Minty, Duke Math. J., 29, 341 (1962).
\(^{7}\) M. M. Vainberg, Ya. L. Engelson, DAN, 112, no. 5, 755 (1958).
\(^{8}\) I. M. Gel'fand, N. Ya. Vilenkin, Generalized Functions, 4, 1961.
\(^{9}\) N. V. Kirpotina, V Vsesoyuzn. konf. po funktsional’nomu analizu, Baku, 1961, p. 113.
\(^{10}\) Ya. L. Engelson, Uch. zap. Latv. gos. univ., 20, no. 3, 27 (1958).
\(^{11}\) F. E. Browder, Ann. Math., 80, no. 3, 485 (1964).

* Note added in proof. The assertion of the theorem remains valid if \(2\lambda_1\) is replaced in the inequality by \((1+\alpha)/m\), \(\alpha>0\), if \((0,m)\) contains no spectral points.