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ON A PERIODIC BOUNDARY VALUE PROBLEM FOR A LINEAR DIFFERENTIAL EQUATION OF THE FOURTH ORDER
A. M. KOLOBOV
In recent years, at the Izhevsk mathematical seminar the question of the behavior of the Green’s function for certain boundary value problems for differential and difference equations has been intensively developed. By establishing the character of the behavior of the Green’s function for the corresponding linear boundary value problem, one can prove theorems of the A. S. Chaplygin type on differential inequalities. Linear theorems, in turn, make it possible to obtain, for some nonlinear problems, existence theorems and estimates of solutions [1].
In connection with this there arises the problem of the existence and uniqueness of a solution of the nonlinear boundary value problem
\[ y^{(n)}+f(x,y)=0, \]
\[ y^{(i)}(\alpha)-y^{(i)}(\beta)=0,\quad (i=0,1,2,\ldots,n-1), \tag{1} \]
where the function \(f(x,y)\) is defined and continuous in some convex domain \(G\).
The central point of this problem is the behavior of the Green’s function of the corresponding linear boundary value problem for the equation
\[ y^{(n)}+p(x)y=f(x), \tag{2} \]
where \(p(x)\), \(f(x)\) are functions continuous on \([\alpha,\beta]\).
In the present note, theorems on the existence and uniqueness of the solution of the linear boundary value problem with periodic boundary conditions are established, and the question of the behavior of the Green’s function of this problem is also investigated.
It is known that the linear nonhomogeneous boundary value problem (2), (1) has a unique solution if for the homogeneous boundary value problem
\[ y^{(n)}+p(x)y=0, \tag{3} \]
\[ y^{(i)}(\alpha)-y^{(i)}(\beta)=0\quad (i=0,1,2,\ldots,n-1) \]
there exists only the unique trivial solution \(y(x)=0\).
In what follows we shall everywhere assume that \(p(x)\ne 0\).
Theorem 1. Let \(p(x)\) be a sign-constant continuous function on \([\alpha,\beta]\). Then, if \(n\) is an odd number, the boundary value problem (3), (1) has a unique solution. If \(n=2k\), then the boundary value problem has a unique solution for \(p(x)>0\) when \(k=2,4,\ldots,2l\), and for \(p(x)<0\) when \(k=1,3,\ldots,2l-1\).
Proof. 1) Let \(n\) be even. Put \(n=2k\). Consider the equation
\[ y^{(2k)}+p(x)y=0. \]
Suppose that, besides the trivial solution, the boundary value problem (3), (1) also has a solution \(y(x)\) different from it. We multiply the equation under consideration by this solution and integrate the result from \(\alpha\) to \(\beta\), transforming the first term by integration by parts:
\[ \int_{\alpha}^{\beta} y^{(2k)} y\,dx = \left. \left(y^{(2k-1)} y\right)\right|_{\alpha}^{\beta} - \int_{\alpha}^{\beta} y^{(2k-1)} y'\,dx = \]
\[ = \left[y^{(2k-1)}(\beta)y(\beta)-y^{(2k-1)}(\alpha)y(\alpha)\right] - \int_{\alpha}^{\beta} y^{(2k-1)} y'\,dx . \]
The expression in square brackets vanishes by virtue of the boundary conditions; we transform the integral once more by parts, repeating this process \(k-1\) times. After all the transformations we obtain
\[ \int_{\alpha}^{\beta} \left[(y^{(k)})^2+p(x)y^2\right]\,dx=0 \qquad (k=2,4,\ldots,2l), \]
\[ \int_{\alpha}^{\beta} \left[(y^{(k)})^2-p(x)y^2\right]\,dx=0 \qquad (k=1,3,5,\ldots,2l-1). \]
If \(p(x)\geq 0\), then the first equality can be fulfilled only in the case \(y(x)\equiv 0\). If \(p(x)\leq 0\), then the second equality is fulfilled when \(y(x)\equiv 0\).
2) Let \(n\) be odd. Put \(n=2k+1\). Proceeding in exactly the same way as in the preceding case, after all transformations we arrive at an expression of the form
\[ \int_{\alpha}^{\beta} p(x)y^2\,dx=0, \]
which is possible only for \(y(x)\equiv 0\).
Corollary. If \(p(x)\) is a sign-definite function satisfying the conditions of Theorem 1, then the boundary value problem (2), (1) has one and only one solution.
Remark. Let us note that the assertion of Theorem 1, generally speaking, does not extend to the case when \(p(x)\leq 0\) and \(n=4\). Indeed, the periodic problem for the equation
\[ y^{\mathrm{IV}}-\left(\frac{2\pi}{\beta-\alpha}\right)^4 y=0, \]
besides the trivial solution, has a solution of the form
\[ y(x)=A\cos\frac{2\pi}{\beta-\alpha}x+B\sin\frac{2\pi}{\beta-\alpha}x . \]
It follows from this that the Green’s function of the problem under consideration does not exist.
It is known that a necessary and sufficient condition for the boundary value problem to have one and only one Green’s function is the existence of the unique solution \(y(x)\equiv 0\) of this problem.
Let us note that, generally speaking, for the fourth-order differential equation
\[ y^{\mathrm{IV}}+p(x)y=0, \]
the Green’s function of the periodic boundary-value problem for sign-changing \(p(x)\) does not exist. Indeed, it is easy to give an example of a homogeneous equation with periodic boundary conditions that has a nontrivial solution:
\[ y^{\mathrm{IV}}+\frac{\sin x}{2-\sin x}\,y=0. \]
Here, in addition to the trivial one, there also exists a solution different from it, \(y=2-\sin x\). It follows that for this boundary-value problem the Green’s function does not exist.
From the method of proof of Theorem 1 it follows that if \(n=2k\), where \(k=2,4,\ldots,2l\), and \(p(x)\le 0\), then the method set forth gives no result. Below we give the solution of this problem for \(n=4\).
Consider the periodic boundary-value problem
\[ L_4[y]\equiv y^{\mathrm{IV}}+p(x)y=0, \tag{4} \]
\[ y^{(i)}(\alpha)-y^{(i)}(\beta)=0\qquad (i=0,1,2,3), \]
where \(p(x)\) is a continuous periodic function with period \(T=\beta-\alpha\).
Theorem 2. If \(p(x)\le 0\) on \([\alpha,\beta]\), then the boundary-value problem (4), (1) has a unique solution on \([\alpha,\beta]\), belonging to an interval of nonoscillation \([1]\).
Proof. Suppose that, in addition to the trivial solution, the boundary-value problem (4), (1) has a solution \(y(x)\) different from it. Clearly, \(y(x)\) will be periodic with period \(T\). It is easy to see that, by virtue of the boundary conditions, the solution \(y(x)\) cannot be of constant sign. Without loss of generality, put \(y(\alpha)=0\). Using the boundary conditions, we obtain \(y(\beta)=0\). On the interval \([\alpha,\beta]\), by virtue of the condition \(y'(\alpha)=y'(\beta)\), there exists a point \(x=t\) at which \(y(x)\) vanishes, i.e. \(y(t)=0\). In passing through this point, \(y(x)\) changes sign, for from the definition of an interval of nonoscillation it follows that \(x=t\) cannot be a zero of order higher than the first. On \([\alpha,\beta]\) there will be found a value \(x=\lambda\) at which \(y(x)\) assumes its greatest value \(A>0\).
Perform the change of coordinates according to the rule
\[ y=Y+A,\qquad x=x. \]
Then the equation transformed to the new coordinates will have the form
\[ Y^{\mathrm{IV}}+p(x)Y=-Ap(x). \]
The solution of this equation will be nonpositive:
\[ Y(x)\le 0\quad \text{on }[\alpha,\beta]. \tag{5} \]
We represent the solution of the boundary-value problem
\[ Y^{\mathrm{IV}}+p(x)Y=-Ap(x), \]
\[ Y(\lambda)=Y'(\lambda)=Y(\lambda+T)=Y'(\lambda+T)=0 \]
by Green’s formula
\[ Y(x)=\int_{\lambda}^{\lambda+T}\Gamma(x,s)[-Ap(s)]\,ds, \]
where \(\Gamma(x,s)\) is the Green’s function of the problem under consideration. Since \(\Gamma(x,s)>0\) for \(x,s\in(\alpha,\beta)\) [3], it follows that \(Y(x)>0\), which contradicts (5). The contradiction obtained proves Theorem 2.
It is known that the boundary value problem for the equation
\[ L_4[y]\equiv y^{\mathrm{IV}}+p(x)y=f(x), \]
where \(p(x)\) is a continuous periodic function of period \(T=\beta-\alpha\), and \([\alpha,\alpha+T=\beta]\) belongs to an interval of nonoscillation of the equation \(L_4[y]=0\), has a unique periodic solution \(y(x)\) possessing the following property:
\[ y(x)>0\;(<0),\quad \text{if}\quad p(x)\cdot f(x)\geqslant 0\;(\leqslant 0). \]
The Green’s function of the corresponding boundary value problem, in turn, satisfies the condition \(\Gamma(x,s)\geqslant 0\;(\leqslant 0)\) for \(x,s\in(\alpha,\beta)\), when \(p(x)\geqslant 0\) \((\leqslant 0)\). This result makes it possible to establish a theorem on the existence and uniqueness of the solution of the boundary value problem (4), (1), when \(p(x)\) is a continuous sign-definite function on the interval \([\alpha,\beta]\), belonging to an interval of nonoscillation of the equation \(L[y]=0\). At the same time one can establish the character of the behavior of the Green’s function.
To prove the main theorem, consider the boundary value problems
\[ L[u]\equiv u^{\mathrm{IV}}+p_1(x)u=f(x), \tag{6} \]
\[ u^{(i)}(\alpha)-u^{(i)}(\beta)=0\quad (i=0,1,2,3), \tag{7} \]
\[ L[v]\equiv v^{\mathrm{IV}}+p_2(x)v=f(x), \tag{8} \]
\[ v^{(i)}(\alpha)-v^{(i)}(\beta)=0\quad (i=0,1,2,3), \tag{9} \]
where \(p_1(x)\), \(p_2(x)\), \(f(x)\) are continuous periodic functions of period \(T=\beta-\alpha\); \([\alpha,\beta]\) belongs to an interval of nonoscillation of the equations \(L[u]=0\), \(L[v]=0\). Then the following assertion, analogous to Mikusinski’s theorem, is valid.
Lemma. If \(p_1(x)\), \(p_2(x)\), \(f(x)\) are continuous sign-definite functions, with \(p_1(x)\) and \(p_2(x)\) periodic functions of period \(T=\beta-\alpha\); \(u(x)\) and \(v(x)\) are the solutions, respectively, of the boundary value problems (6), (7) and (8), (9), then
\[ 1)\quad v(x)<u(x),\quad \text{if}\quad p_1(x)\leqslant p_2(x),\quad f(x)\geqslant 0, \]
\[ 2)\quad v(x)>u(x),\quad \text{if}\quad p_1(x)\leqslant p_2(x),\quad f(x)\leqslant 0. \]
From this lemma there follows the obvious corollary.
Corollary. If \(p_1(x)\) and \(p_2(x)\) are sign-definite periodic continuous functions of period \(T\), and \(\Gamma_u(x,s)\) and \(\Gamma_v(x,s)\) are the Green’s functions of the homogeneous boundary value problems corresponding to problems (6), (7) and (8), (9), then from \(p_1(x)\leqslant p_2(x)\) it follows that \(\Gamma_u(x,s)\geqslant \Gamma_v(x,s)\) for \(x,s\in(\alpha,\beta)\).
Theorem 3. If \(p(x)\) is a sign-definite continuous function, then the boundary value problem (4), (1) on the interval \([\alpha,\beta]\), belonging to an interval of nonoscillation of the equation \(L_4[y]=0\), has a unique solution.
Proof. For definiteness put \(p(x)\leqslant 0\). Let \(a(x)\) be a continuous periodic function of period \(T\), such that \(p(x)\geqslant a(x)\) for \(x\in[\alpha,\beta]\). We shall also assume that the length of the maximal interval of nonoscillation of the equation
\[ y^{\mathrm{IV}}+a(x)y=0 \]
is not less than \(T\).
Choose an ordered finite sequence of functions \(b_1(x), b_2(x), \ldots, b_n(x)\) such that
\(0 \geq p(x)=b_n(x) \geq b_{n-1}(x) \geq b_{n-2}(x) \geq \cdots \geq b_2(x) \geq b_1(x) \geq b_0(x)=a(x)\), and in such a way that
\(\max_{x\in[\alpha,\beta]} |b_{k+1}(x)-b_k(x)|\) \((k=0,1,2,\ldots,n-1)\) is sufficiently small.
Consider the boundary-value problem
\[ y^{\mathrm{IV}}+b_1(x)y=0, \tag{10} \]
\[ y^{(i)}(\alpha)-y^{(i)}(\beta)=0 \quad (i=0,1,2,3). \]
The solution of this problem \(y_1(x)\) must, obviously, satisfy the equation
\[ y^{\mathrm{IV}}+a(x)y=[a(x)-b_1(x)]y. \]
It is known [4] that the boundary-value problem
\[ y^{\mathrm{IV}}+a(x)y=[a(x)-b_1(x)]y, \]
\[ y^{(i)}(\alpha)-y^{(i)}(\beta)=0 \quad (i=0,1,2,3) \]
is equivalent to a homogeneous integral equation of the form
\[ y(x)=\int_{\alpha}^{\beta}\Gamma_0(x,s)[a(s)-b_1(s)]y(s)\,ds, \]
where \(\Gamma_0(x,s)\) is the Green’s function of the corresponding homogeneous boundary-value problem. The kernel of the written integral equation,
\[ \Gamma_0(x,s)[a(s)-b_1(s)], \]
is, by assumption, sufficiently small. And such a homogeneous integral equation, as is known, has a unique, i.e. trivial, solution [5]. Hence follows the uniqueness of the solution of problem (10), (1), as well as the existence of the Green’s function \(\Gamma_1(x,s)\) of this problem. Using the consequence of the lemma, from the condition \(0 \geq b_1(x) \geq a(x)\) we obtain \(0 \geq \Gamma_0(x,s) \geq \Gamma_1(x,s)\) for \(x,s\in(\alpha,\beta)\).
Suppose further that the problem
\[ y^{\mathrm{IV}}+b_{n-1}(x)y=0, \]
\[ y^{(i)}(\alpha)-y^{(i)}(\beta)=0 \quad (i=0,1,2,3) \]
has the unique solution \(y_{n-1}(x)\equiv0\) and a Green’s function satisfying the inequality \(0 \geq \Gamma_{n-2}(x,s) \geq \Gamma_{n-1}(x,s)\) for \(x,s\in(\alpha,\beta)\). We shall then show that the boundary-value problem
\[ y^{\mathrm{IV}}+b_n(x)y=0, \]
\[ y^{(i)}(\alpha)-y^{(i)}(\beta)=0 \quad (i=0,1,2,3) \]
has only the trivial solution \(y_n(x)\equiv0\). For the proof, as before, consider the boundary-value problem
\[ y^{\mathrm{IV}}+b_{n-1}(x)y=[b_{n-1}(x)-b_n(x)]y, \]
\[ y^{(i)}(\alpha)-y^{(i)}(\beta)=0 \quad (i=0,1,2,3), \]
equivalent to the integral equation
\[ y(x)=\int_{\alpha}^{\beta}\Gamma_{n-1}(x,s)[b_{n-1}(s)-b_n(s)]y(s)\,ds \]
with sufficiently small kernel \(\Gamma_{n-1}(x,s)[b_{n-1}(s)-b_n(s)]\). And this homogeneous integral equation has the unique solution \(y_n(x)=0\). Since, by assumption, \(p(x)\equiv b_n(x)\), it follows that the boundary value problem (4), (1) has a unique solution, which proves the theorem.
It is easy to see that the Green’s function of this problem satisfies the inequality
\[ \Gamma(x,s)\leq 0 \quad \text{for } x,\ s\in(\alpha,\beta). \]
The case \(p(x)\geq 0\) is proved analogously.
Remark. An essential point in the proof of Theorem 3 is the possibility of choosing an ordered finite sequence of continuous functions
\[ b_1(x),\ b_2(x),\ \ldots,\ b_{n-1}(x),\quad b_n(x)\equiv p(x), \]
having the property that the expression
\[ \Gamma_k(x,s)[b_k(s)-b_{k+1}(s)] \quad (k=0,\ 1,\ 2,\ \ldots,\ n-1) \]
is sufficiently small, or, in other words, the essential uniform boundedness of the functions \(\Gamma_k(x,s)\) \((k=0,\ 1,\ 2,\ \ldots,\ n-1)\).
But the uniform boundedness of \(\Gamma_k(x,s)\) is easy to verify if one notes the following. For a given function \(p(x)\leq 0\) one can specify such a \(T\)-periodic function \(c(x)\leq 0\) that \(p(x)\leq c(x)\). Then the Green’s function \(\Gamma_c(x,s)\) of the periodic boundary value problem for the equation
\[ y^{IV}+c(x)y=0 \]
exists and, according to the corollary to the lemma of the present paper,
\[ \Gamma_c(x,s)\leq \Gamma_k(x,s)\leq \Gamma_0(x,s) \quad (k=0,\ 1,\ 2,\ \ldots,\ n-1). \]
Corollary 1. If \(p(x)\) is a sign-definite function on \([\alpha,\beta]\) belonging to a nonoscillation interval, then the Green’s function of the boundary value problem (4), (1) is sign-definite; moreover,
\[ \Gamma(x,s)\geq 0 \quad \text{for } x,\ s\in[\alpha,\beta],\quad \text{when } p(x)\geq 0, \]
\[ \Gamma(x,s)\leq 0 \quad \text{for } x,\ s\in[\alpha,\beta],\quad \text{when } p(x)\leq 0. \]
Corollary 2. If \(p(x)\) and \(f(x)\) are sign-definite continuous functions, then for the solution \(y(x)\) of the problem \(L_4[y]=f(x)\), \(y^{(i)}(\alpha)-y^{(i)}(\beta)=0\) the inequalities
\[ y(x)>0,\quad \text{if } p(x)f(x)\geq 0, \]
\[ y(x)<0,\quad \text{if } p(x)f(x)\leq 0 \quad (i=0,\ 1,\ 2,\ 3) \]
hold.
References
- N. V. Azbelev, A. Ya. Khokhryakov, Z. B. Tsalyuk, Matem. sb., 59, 1962, pp. 137–143.
- A. Ya. Khokhryakov, Matem. sb., 63/4, 1964, pp. 639–645.
- R. G. Aliev, Dissertation, Kazan, 1963.
- M. A. Naimark, Linear Differential Operators. Moscow, 1954.
- S. G. Mikhlin, Lectures on Linear Integral Equations. Moscow, 1959.
Received by the editors
December 21, 1964
Minsk Radio Engineering
Institute