Abstract
Full Text
MATHEMATICS
I. A. BAKHTIN
ON A CLASS OF EQUATIONS WITH POSITIVE OPERATORS
(Presented by Academician P. S. Aleksandrov, 27 IV 1957)
In the present article a new definition of concave operators is proposed. It turns out that for equations with such operators many theorems are valid which were previously established for more special classes of equations ((^{1,3-5,7})). We use the terminology of the theory of cones of M. G. Krein ((^{4,6})). We note that the main result of the article (Theorem 1) is obtained by topological methods.
- Below, (K) and (K_1) denote two cones in a real Banach space (E), and it is assumed that (K \subset K_1). The sign (\ll) denotes the semi-ordering generated by the “larger” cone (K_1): one writes (X \ll Y) if (y - x \in K_1). By (K_r) is denoted the intersection of the “smaller” cone (K) with the ball (|\varphi| \le r). It is assumed that (|x| \le |y|) if (\theta \ll x \ll y). By (A) is denoted an operator (in general, nonlinear) defined on (K_r). We shall say that the operator (A) is positive if (AK_r \subset K), and that the operator (A) is monotone if from (\varphi \ll \psi) ((\varphi,\psi \in K_r)) it follows that (A\varphi \ll A\psi). We emphasize that the concepts of positivity and monotonicity have been introduced with the aid of different cones. By (u_0) we denote some fixed nonzero element of (K).
Definition. A positive and monotone operator (A) will be called ({K_1,u_0})-concave if:
1) to each (\varphi \in K_r) ((\varphi \ne 0)) there correspond numbers (\alpha,\beta > 0) such that
[
\alpha u_0 \ll A\varphi \ll \beta u_0;
]
2) if (\varphi \in K_r) and (\varphi \gg \gamma u_0) ((\gamma > 0)), then
[
At\varphi \gg tA\varphi,\qquad At\varphi \ne tA\varphi \quad (0<t<1);
\tag{1}
]
3) for each pair of elements (\varphi_1,\varphi_2 \in K_r) ((\varphi_1,\varphi_2 \gg \varkappa u_0,\ \varkappa>0,\ \varphi_1-\varphi_2 \in K_1)), from (\varphi_1 \gg t\varphi_2) ((t>0,\ \varphi_1 \ne t\varphi_2)) it follows that
[
A\varphi_1 - tA\varphi_2 \gg \delta u_0 \qquad (\delta>0).
\tag{2}
]
- The equation
[
A\varphi=\lambda\varphi
\tag{3}
]
with ({K_1,u_0})-concave operators has important properties, analogous to those established for other equations in ((^{1,3-5,7})). We formulate the most important properties in the form of a theorem.
Nonzero solutions of equation (3) are called eigenvectors, and the corresponding values (\lambda) are called eigenvalues.
Theorem 1. Let the operator (A) ((A\theta=\theta)) be completely continuous and ({K_1,u_0})-concave.
Then the following assertions hold:
1) The eigenvectors of the operator (A) form in (K_r) a continuous branch* of length (r).
2) The corresponding eigenvalues completely fill a certain interval.
3) To each eigenvalue (\lambda) there corresponds a unique nonzero solution (\varphi(\lambda)) of equation (3) in (K_r).
4) From (\lambda_1 < \lambda_2) it follows that (\varphi(\lambda_2) \ll \varphi(\lambda_1)).
5) The function (\varphi(\lambda)) is strongly continuous in (\lambda).
The endpoints of the interval of eigenvalues can be found by the same method as in ((4)).
- Assertions analogous to Theorem 1 were first established by P. S. Uryson for an integral equation of the form
[
\varphi(x)=\lambda\int_0^1 \mathcal{K}(x,s,\varphi(s))\,ds .
]
The investigations of P. S. Uryson served as the starting point for the creation of the theory of (u_0)-concave operators ((^{4,5})). Subsequently, assertions analogous to Theorem 1 were established ((^1)) for the nonlinear integral equation
[
\lambda\varphi(x)=\rho(x)\int_0^1 G(x,s)\varphi(s)\,ds \times
\left[1-\left(\int_0^1 \frac{\partial G(x,s)}{\partial x}\varphi(s)\,ds\right)^2\right]^{1/2},
]
arising in the study of the longitudinal bending of rods of variable stiffness. This equation gave occasion for singling out the class of (u_0)-monotone operators ((^4)), for which assertions of the type of Theorem 1 are also valid. The classes of (u_0)-concave and (u_0)-monotone operators (for definitions see ((^4))) turned out not to contain one another. In connection with this there arose the problem of finding broader classes of equations for which Theorem 1 is valid. In a certain sense this problem is solved, since the following theorem holds.
Theorem 2. Every (u_0)-concave and every (u_0)-monotone operator is a ({K_1,u_0})-concave operator.
- The application of Theorem 1 to the study of concrete equations requires verification of the conditions of ({K_1,u_0})-concavity. By Theorem 2, examples of ({K_1,u_0})-concave operators will be the operators studied in ((^{1,3-5,7})).
As a new example, consider the operator (A), defined in the space (C_n) of vector-functions continuous on the interval ([0,1]), (\vec{\varphi}(x)={\varphi_1(x),\varphi_2(x),\ldots,\varphi_n(x)}), by the formula
[
A\vec{\varphi}(x)={A_1\vec{\varphi}(x),A_2\vec{\varphi}(x),\ldots,A_n\vec{\varphi}(x)},
\tag{4}
]
where
[
A_i\vec{\varphi}(x)=\int_0^1 \mathcal{K}_i(x,s,\varphi_1(s),\varphi_2(s),\ldots,\varphi_n(s))\,ds .
\tag{5}
]
To transfer the theory of P. S. Uryson to systems
[
A\vec{\varphi}(x)=\lambda\vec{\varphi}(x)
\tag{6}
]
one may use Theorem 1. For this—
* By a continuous branch, following M. A. Krasnosel’skii ((^4)), we mean such a set (\mathfrak{A}) of eigenvectors that (\mathfrak{A}\cap\Gamma\ne 0) for the boundary (\Gamma) of any domain of the space (E) containing zero (\theta) and contained in the ball (|x|\le r).
one must find conditions for ({K_1,u_0})-concavity of the operator (A\vec\varphi(x)) with respect to certain cones (K) and (K_1) and to a certain element (u_0).
As (K_1) we shall consider the set of nonnegative vector-functions of the space (C_n). As the cone (K) we consider the set (K=K(u_0,c)) of vector-functions (\vec\psi(x)={\psi_1(x),\psi_2(x),\ldots,\psi_n(x)}), whose components satisfy the conditions
[
\alpha u_i^{(0)}(x)\leqslant \psi_i(x)\leqslant \beta u_i^{(0)}(x)
\quad (i=1,2,\ldots,n),
\tag{7}
]
where (\alpha,\beta>0); (\beta\leqslant c_i\alpha); (u_0(x)={u_1^{(0)}(x),u_2^{(0)}(x),\ldots,u_n^{(0)}(x)}) is a fixed element of (K_1) with components not identically equal to zero; (c={c_1,c_2,\ldots,c_n}) ((c_i\geqslant 1)) is a fixed vector, and the element (\vec\theta(x)={0,0,\ldots,0}).
Assume that the nonnegative functions (\mathcal K_i(x,s,u_1,u_2,\ldots,u_n))
((0\leqslant x,\ s\leqslant 1,\ 0\leqslant u_1,\ u_2,\ldots,u_n<\infty)) ((i=1,2,\ldots,n)) are continuous and satisfy the conditions:
1) (\mathcal K_i(x,s,u_1,u_2,\ldots,u_n)) are nondecreasing, and
[
\frac{\mathcal K_i(x,s,u_1,u_2,\ldots,u_n)}
{\sqrt{u_1^2+u_2^2+\cdots+u_n^2}}
]
are nonincreasing functions of the variables (u_1,u_2,\ldots,u_n).
2) To every set of numbers (u_1,u_2,\ldots,u_n\geqslant 0), for which the functions (\mathcal K_i(x,s,u_1,u_2,\ldots,u_n)\not\equiv 0), and to every (t\in(0,1)), there correspond such continuous nonnegative functions (\gamma_i(x)) that
[
\frac{\mathcal K_i(x,s,tu_1,tu_2,\ldots,tu_n)}
{t\sqrt{u_1^2+u_2^2+\cdots+u_n^2}}
-
\frac{\mathcal K_i(x,s,u_1,u_2,\ldots,u_n)}
{\sqrt{u_1^2+u_2^2+\cdots+u_n^2}}
\geqslant
\gamma_i(s)\,u_i^{(0)}(x),
]
where (\gamma_i(x)) is positive almost at all those points where (u_i^{(0)}(x)>0).
3) There exist such nonnegative continuous functions (\alpha(s)) and (\beta(s)) that for every set (u_1,u_2,\ldots,u_n\geqslant 0), for which (\mathcal K_i(x,s,u_1,u_2,\ldots,u_n)\not\equiv 0), the inequalities
[
\alpha(s)u_i^{(0)}(x)\leqslant
\frac{\mathcal K_i(x,s,u_1,u_2,\ldots,u_n)}
{\sqrt{u_1^2+u_2^2+\cdots+u_n^2}}
\leqslant
\beta(s)u_i^{(0)}(x)
\quad (i=1,2,\ldots,n),
]
hold, where (\alpha(x)u_i^{(0)}(x)\not\equiv 0) almost at all points where (u_i^{(0)}(x)\not\equiv 0), and (\beta(s)\leqslant c_i\alpha(s)) ((i=1,2,\ldots,n)).
4) For each (i) ((i=1,2,\ldots,n)) there exists its own point (s_0\in[0,1]), at which
[
\mathcal K_i\bigl(x,s_0,u_1^{(0)}(s_0),u_2^{(0)}(s_0),\ldots,u_n^{(0)}(s_0)\bigr)\not\equiv 0,
\quad
u_i^{(0)}(s_0)\not\equiv 0.
]
Under these assumptions the operator (6) is ({K_1,u_0})-concave.
The theory of (u_0)-concave operators makes use, in particular, of many properties of special classes of linear positive operators. These properties for linear integral operators were already found by Jentzsch ((^2)). The corresponding theorems were extended to the case of abstract positive linear operators by M. G. Krein and M. A. Rutman ((^6)), and to the special case by M. A. Krasnosel’skii and L. A. Ladyzhenskii ((^5)).
Our constructions required the study of a broader class of linear positive operators.
We shall say that a linear operator (A), leaving invariant the “smaller” cone (K), is (u_0)-strongly positive with respect to the “larger” cone (K_1) ((K_1 \supset K)), if there is an element (u_0 \in K) ((u_0 \ne \theta)) such that for every (\varphi \in K_1) ((\varphi \ne \theta)) there exist a natural number (n) and numbers (\alpha,\beta > 0) such that
[
\alpha u_0 \ll A^n \varphi \ll \beta u_0 .
]
Theorem 3. Let (A) be a monotone linear completely continuous operator that is (u_0)-strongly positive with respect to (K_1). Suppose that for every (\varphi \in E) one can indicate a natural number (p > 0) and a number (l > 0) such that (lA^p\varphi \ll u_0). Then the operator (A) has a simple positive eigenvalue, which is greater than the absolute value of all other eigenvalues and to which there corresponds an eigenvector (\varphi \in K).
This theorem is a direct generalization of known propositions on linear integral operators with positive kernels. It contains, as various special cases, Jentzsch’s theorem and some theorems from ((5,6)).
The author takes the opportunity to express his gratitude to M. A. Krasnosel’skii for his attention and advice.
Voronezh
State University
Received
20 IV 1957
REFERENCES CITED
¹ I. A. Bakhtin, M. A. Krasnosel’skii, DAN, 105, No. 4 (1955).
² R. Jentzsch, J. f. reine u. angew. Math., 141 (1912).
³ M. A. Krasnosel’skii, DAN, 77, No. 2 (1951).
⁴ M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations, 1956.
⁵ M. A. Krasnosel’skii, L. A. Ladyzhenskii, Tr. Moskovsk. matem. obshch., 3, 321 (1954).
⁶ M. G. Krein, M. A. Rutman, Uspekhi matem. nauk, 3, 1 (23) (1948).
⁷ P. S. Uryson, Works on Topology and Other Fields of Mathematics, 1, 1951, p. 45.