UDC 517.948:513.88:519.3

MATHEMATICS

A. M. RUBINOV

NECESSARY CONDITIONS FOR AN EXTREMUM AND THEIR APPLICATION TO THE STUDY OF CERTAIN EQUATIONS

(Presented by Academician L. V. Kantorovich, 11 XI 1965)

Let \(X\) be a Banach space, \(\Omega \subset X\). An element \(u \in X\) will be called an admissible direction for the set \(\Omega\) at the point \(x \in \overline{\Omega}\) if there exist a sequence \(u_s \in X\) and a numerical sequence \(\alpha_s\) such that 1) \(x+\alpha_s u_s \in \Omega\), 2) \(u_s \to u\), 3) \(\alpha_s > 0\), \(\alpha_s \to 0\). The admissible directions for the set \(\Omega\) at the point \(x\) form a closed cone, which we shall denote by \(M_x(\Omega)\). Note that always \(0 \in M_x(\Omega)\).

Let us describe the cone \(M_x(\Omega)\) in several important special cases.

1. Let \(\varphi\) be a functional defined and continuous in some neighborhood \(V(x_0)\) of the point \(x_0 \in X\), and let \(\Omega=\{x \in V(x_0)\mid \varphi(x)=\varphi(x_0)\}\). Then
   \[ M_{x_0}(\Omega)=\{u\in X\mid \operatorname{grad}\varphi(x_0)(u)=0\}. \]

2. Let \(\varphi\) be the functional considered above,
   \[ \Omega=\{x\in V(x_0)\mid \varphi(x)\leq \varphi(x_0)\}. \]
   In this case
   \[ M_{x_0}(\Omega)=\{u\in X\mid \operatorname{grad}\varphi(x_0)(u)\leq 0\}. \]

3. Suppose that \(\Omega\) is a convex set, \(x\in \overline{\Omega}\). In this case \(M_x(\Omega)\) is the closed conical hull of the set \(\Omega-x\).

4. Let \(X\) be a finite-dimensional Banach space, \(\Omega'\) a convex solid polyhedron in \(X\), \(\Omega\) the boundary of \(\Omega'\), and \(x_0\) a vertex of \(\Omega'\). It is easy to see that \(M_{x_0}(\Omega)\) is the boundary of the closed conical hull of the set \(\Omega'-x_0\). Note that in this case \(M_{x_0}(\Omega)\) is a nonconvex cone.

Consider a functional \(f\) defined on the space \(X\). If \(f\) is differentiable at some point \(x\in X\), we put \(Fx=\operatorname{grad} f(x)\).

We indicate necessary conditions for a minimum of the functional \(f\) on the set \(\Omega\). Here by a minimum we shall everywhere mean a local minimum.

Theorem 1. Let the functional \(f\) be defined on the set \(\Omega\), attain a minimum there at the point \(y\), and be Fréchet differentiable at this point. Then
\[ \min_{u\in M_y(\Omega)} Fy(u)=0. \]

We shall call the set \(\Omega\) pseudoconvex if, for any \(x\in\Omega\), the following conditions are satisfied: a) if \(h\in X^*\) is such that \(h(M_x(\Omega))\geq 0\) and for some \(u\in M_x(\Omega)\) \(h(u)>0\), then \(h(\Omega-x)\geq 0\); b) if \(h\in X^*\) is such that \(h(M_x(\Omega))=0\), then either \(h(\Omega-x)\geq 0\), or \(h(\Omega-x)\leq 0\).

It is clear that every convex set is pseudoconvex. An example of a pseudoconvex but nonconvex set is the boundary of a convex solid set.

Theorem 2. If the functional \(f\), defined on a pseudoconvex set \(\Omega\), attains a minimum there at the point \(y\) and is Fréchet differentiable at this point, then either
\[ \min_{x\in\Omega} Fy(x-y)=0, \]
or
\[ \max_{x\in\Omega} Fy(x-y)=0. \]

Theorem \(2'\) (see \((^1)\)). If the functional \(f\), defined on a convex set \(\Omega\), attains a minimum there at the point \(y\) and is Gâteaux differentiable at this point, then
\[ \min_{x\in\Omega} Fy(x-y)=0. \]

Let \(\Gamma \subset X^*\). In what follows we shall consider sets \(\Omega\) satisfying the following condition:

\((*)\) If \(h \in \Gamma\), then there exist unique elements \(y_h\) and \(z_h\) such that

\[ h(y_h)=\min_{x\in\Omega} h(x), \tag{1} \]

\[ h(z_h)=\max_{x\in\Omega} h(x). \tag{2} \]

Let \(\Omega\) satisfy condition \((*)\) with respect to \(\Gamma\). Consider the operators \(G_\Omega\) and \(H_\Omega\), acting from \(\Gamma\) into \(\Omega\), as follows: \(G_\Omega h=y_h\), \(H_\Omega h=z_h\) (here \(y_h\) and \(z_h\) are defined, respectively, by formulas (1) and (2)).

We give an example of the operators \(G_\Omega\) and \(H_\Omega\). Let \(H\) be a Hilbert space; \(X\) a Banach space; \(B\) a linear bounded operator acting from \(H\) into \(X\), whose range is dense in \(X\); \(\Gamma=X^*\setminus\{0\}\), \(\Omega=\{x\in X\mid x=Bz,\ \|z\|=1\}\). In this case, for \(h\in\Gamma\),
\(G_\Omega h=-BB^*h/\|B^*h\|\), \(H_\Omega h=BB^*h/\|B^*h\|\).

Let \(\Omega\) be some set on which a differentiable functional \(f\) is defined. Put \(\Gamma_f=\{h\in X^*\mid h=F'x,\ x\in\Omega\}\).

Theorem 3. Let the functional \(f\) be defined and differentiable in the Fréchet sense on a pseudoconvex set \(\Omega\), which has property \((*)\) with respect to \(\Gamma_f\), and attain a minimum on \(\Omega\) at a point \(y\). Then \(y\) satisfies one of the two equations \(x=H_\Omega F'x\) or \(x=G_\Omega F'x\).

Theorem \(3'\). Let the functional \(f\) be defined and differentiable in the Gâteaux sense on a convex set \(\Omega\), which has property \((*)\) with respect to \(\Gamma_f\), and attain a minimum on \(\Omega\) at a point \(y\). Then \(y\) satisfies the equation \(x=G_\Omega F'x\).

We note that all the theorems formulated above carry over, with obvious modifications, to the case when a maximum is considered instead of a minimum.

Theorem 4. Let a strongly potential operator \(F\)* (the gradient of a functional \(f\)) be defined on a pseudoconvex set \(\Omega\), which has property \((*)\) with respect to \(\Gamma_f\). Suppose further that one of the following conditions is satisfied: a) \(\Omega\) is compact; b) \(f\) is weakly lower or upper semicontinuous, \(\Omega\) is weakly compact. Then one of the equations \(x=G_\Omega Fx\) or \(x=H_\Omega Fx\) has a solution.

Theorem \(4'\). Let a potential operator \(F\) (the gradient of a functional \(f\)) be given on a convex set \(\Omega\), which has property \((*)\) with respect to \(\Gamma_f\). Suppose further that one of the following conditions is satisfied: a) \(f\) is a continuous functional, \(\Omega\) is compact; b) \(f\) is a weakly continuous functional, \(\Omega\) is weakly compact. Then both equations \(x=G_\Omega Fx\) and \(x=H_\Omega Fx\) have solutions.

If in condition b) one requires only weak lower (upper) semicontinuity of \(f\), then one can guarantee only the existence of solutions of the equation \(x=G_\Omega Fx\) (respectively, \(x=H_\Omega Fx\)).

We give one consequence of Theorems 4 and \(4'\).

Theorem 5. Let \(H\) be a Hilbert space; \(X\) a Banach space; \(B\) a linear bounded operator acting from \(H\) into \(X\), whose range is dense in \(X\). Suppose further that \(F\) is a strongly potential operator defined on the set \(\Omega=\{x\in X\mid x=Bz,\ \|z\|=R\}\), and \(Fx\ne 0\) \((x\in\Omega)\). Assume that one of the following conditions is satisfied: a) \(B\) is a completely continuous operator; b) \(F\) is the gradient of a weakly lower or upper semicontinuous functional. Then there exists a number \(\lambda_0\) such that the equation \(x=\lambda_0BB^*Fx\) has at least

* An operator \(F\) is called potential (strongly potential) if it is the Gâteaux (Fréchet) derivative of some functional \(f\) (see (2)).

one solution of the form $x_0 = Bz_0$ ($\|z_0\| = R$). In this case either $\lambda_0 = R/\|B^*Fx_0\|$, or $\lambda_0 = -R/\|B^*Fx_0\|$.

Theorem 5′. Let $H$ and $X$ be the same spaces, and $B$ the same operator, as in Theorem 5; let $F$ be a potential operator defined on the set
$\Omega = \{x \in X \mid x = Bz,\ \|z\| \le R\}$, with $Fx \ne 0$ ($x \in \Omega$). Suppose that one of the following conditions is satisfied: a) $F$ is the gradient of a continuous functional, and $B$ is a completely continuous operator; b) $F$ is the gradient of a weakly continuous functional. Then for any $0 < r \le R$ there exist numbers $\lambda_1$ and $\lambda_2$ such that, for $i = 1, 2$, the equations $x = \lambda_i BB^*Fx$ have at least one solution of the form $x_i = Bz_i$ ($\|z_i\| = r$). In this case
$\lambda_1 = -r/\|B^*Fx_1\|$, $\lambda_2 = r/\|B^*Fx_2\|$. If in condition b) $F$ is the gradient of a functional weakly lower (upper) semicontinuous, then there exists $\lambda_0$ such that the equation $x = \lambda_0 BB^*Fx$ has at least one solution of the form $x_0 = Bz_0$ ($\|z_0\| = r$). In this case, from weak lower (upper) semicontinuity it follows that
$\lambda_0 = -r/\|B^*Fx_0\|$ ($\lambda_0 = r/\|B^*Fx_0\|$).

If $F$ is the gradient of a convex (concave) functional, then there exists a unique negative (positive) number $\lambda_0$ such that the equation $x = \lambda_0 BB^*Fx$ has a solution of the form $x_0 = Bz_0$ ($\|z_0\| = r$), and this solution is unique.

Theorems 5 and 5′ are a generalization of Theorems 15.1–15.4 in ($^2$).

Let us note in conclusion that, for solving the equations $x = G_\Omega Fx$ and $x = H_\Omega Fx$ in the case where $\Omega$ is a convex set, the method of successive approximations described in ($^1$) may be applied.

REFERENCES

$^1$ V. F. Demyanov, A. M. Rubinov, DAN, 160, 15 (1965).
$^2$ M. M. Vainberg, Variational Methods for the Study of Nonlinear Operators, 1956.