Full Text
UDC 517.911
ON ONE APPROACH
TO NONLINEAR BOUNDARY VALUE PROBLEMS
FOR ORDINARY DIFFERENTIAL EQUATIONS
A. Ya. Lepin, A. D. Myshkis
We obtain here existence theorems for solutions of boundary value problems by means of the method of “a priori estimates,” which has been widely used in recent years, especially in the theory of partial differential equations. The general scheme of this method is as follows.
Suppose it is required to prove the existence of at least one solution of a certain problem, where the solution is chosen from among the elements of a topological space \(H\); in other words, it is required to prove the nonemptiness of some set \(G \subseteq H\) (the set of solutions of the problem). To this end one constructs a sequence of problems of analogous type, i.e., a sequence of sets \(G_p \subseteq H\) \((p = 1, 2, \ldots)\), approximating the original problem in the sense that from \(g_{p_k} \in G_{p_k}\), \((p_k \to \infty)\), \(g_{p_k} \to g\) it follows that \(g \in G\). Next one constructs a scalar function \(\omega(h) \geq 0\) \((h \in H)\), which realizes an a priori estimate; here from \(\sup_p \omega(g_p) < \infty\) \((g_p \in G_p;\ p = 1, 2, \ldots)\) there must follow the existence of a convergent subsequence \(g_{p_k} \to g\). Finally, it is proved that each of the approximating problems has at least one solution \(g_p\), and one such that \(\sup_p \omega(g_p) < \infty\). It is clear that from all this the nonemptiness of \(G\) follows.
Thus, after the construction of an approximating sequence of problems, the proof of the existence of a solution of the original problem is divided into two parts: the proof of the existence of a solution of the approximating problems (which may turn out to be simpler than the original one) and the construction of an a priori estimate.
Let us consider one of the many inequivalent refinements of this general method, which we need for application to boundary value problems.
Suppose that in a linear topological space \(H\) (over the field of real numbers) a set \(G \subseteq H\) and a mapping \(L\) of the space \(H\) into the \(n\)-dimensional Euclidean space \(E_n\) are given; it is required to prove the existence of a solution of the problem
\[ g \in G,\qquad Lg = a, \tag{1} \]
where the vector \(a \in E_n\) is given. (In applying the method to boundary value problems, \(G\) is the totality of all solutions of the given differential equation.) We shall assume that in \(H\) there is defined a scalar lower semicontinuous function \(\omega\), satisfying the inequalities \(0 \leq \omega(h) \leq \infty\), \(\omega(h_1 + h_2) \leq \omega(h_1) + \omega(h_2)\), \(\omega(\alpha h) \leq \omega(h)\) \((\forall \alpha,\ 0 < \alpha < 1)\).
Suppose the following conditions hold:
- The operator \(L\) can be represented in the form of a sum of continuous operators \(L^0 + L^1\).
- There exists a set \(G^0 \subseteq H\) such that the problem
\[ v \in G^0,\qquad L^0v=a \tag{2} \]
for every \(a \in E_n\) has a unique solution \(v_a\), depending continuously on \(a\).
- There exists a “comparison problem”
\[ w \in G^0,\qquad L^*w=b, \]
where \(L^*\) is a continuous operator mapping \(H\) into \(E_n\), whose solution \(w_b\) exists and is unique for every \(b \in E_n\); moreover, for every ball \(R \subset E_n\) one has \(\sup\limits_{b\in R}\omega(w_b)<\infty\).
The preceding conditions guarantee the existence of an operator \(A\), acting according to the rule \(Aa \underset{\mathrm{def}}{=} L^*v_a\) and effecting a continuous one-to-one mapping of \(E_n\) onto itself; as is known, such a mapping is a homeomorphism. From the definition of the operator \(A\) we have \(v_a=w_{Aa}\), whence \(w_b=v_{A^{-1}b}\) and \(L^0w_b=Bb\) \((B \underset{\mathrm{def}}{=} A^{-1})\).
- The set \(G\) can be approximated by sets \(G_p\) \((p=1,2,\ldots)\) in such a way that from \(g_{p_k}\in G_{p_k}\), \((p_k\to\infty)\), and \(g_{p_k}\to g\) it follows that \(g\in G\), and each of the problems \(s\in G_p,\ L^*s=b\) \((p=1,2,\ldots)\) for every \(b\in E_n\) has a unique solution \(s_{pb}\), which depends continuously on \(b\), and
\[ \Omega \underset{\mathrm{def}}{=} \sup_{p,b}\omega(s_{pb}-w_b)<\infty . \]
-
From \(\sup\limits_p \omega(g_p)<\infty\) \((g_p\in G_p;\ p=1,2,\ldots)\) there follows the existence of a convergent subsequence \(g_{p_k}\to g\).
-
For every \(M>0\) and every \(a\in E_n\) there is a ball \(R\subset E_n\) such that, for all \(b\in R,\ s\in G,\ \omega(s-w_b)<M\), one has
\[ A(L^0w_b-Ls+a)\in R . \]
Theorem 1. If conditions 1–6 are fulfilled, then problem (1) has at least one solution.
Proof. For \(M=1+\Omega\) and a given \(a\), choose a ball \(R\) according to condition 6. Define mappings \(T_p\) \((p=1,2,\ldots)\) of the space \(E_n\) into itself by the formula
\[ T_pb \underset{\mathrm{def}}{=} A(L^0w_b-Ls_{pb}+a) \]
and denote \(R^p=T_pR\), while by \(R_\varepsilon\) \((\varepsilon>0)\) we denote the ball with the same center as \(R\) and with radius \(\varepsilon\) times larger. We shall show that for any \(\varepsilon>1\) there is \(p_\varepsilon\) such that \(R^p\subseteq R_\varepsilon\) for \(p>p_\varepsilon\). Suppose the contrary. Then there is a sequence \(b_k\in R\) \((k=1,2,\ldots)\) such that \(b_k\to b_0\) and \(d_k \underset{\mathrm{def}}{=} T_{p_k}b_k\in R_\varepsilon\) \((p_k\to\infty)\). Put \(s_k=s_{p_kb_k}\). Then
\[ \sup_k \omega(s_k)\le \sup_k\bigl(\omega(s_{p_kb_k}-w_{b_k})+\omega(w_{b_k})\bigr) \le \sup_{p,b}\omega(s_{pb}-w_b)+ \]
\[ +\sup_{b\in R}\omega(w_b)<\infty . \]
But then from conditions 4 and 5 there follows the existence of a subsequence \(s_{k_i}\to s\in G\). Hence
\[ d_{k_i}=A(L^0w_{b_{k_i}}-Ls_{p_{k_i}b_{k_i}}+a)\to A(L^0w_{b_0}-Ls+a)\in R, \]
since from the lower semicontinuity of \(\omega\) we have
\[ \omega(s-w_{b_0}) \leq \sup_i \omega(s_{p_{k_i}b_{k_i}}-w_{b_{k_i}})<M . \]
Thus the inclusion \(Rp\subseteq R_\xi\) for sufficiently large \(p\) has been proved. From the fixed-point theorem there follows the existence of a sequence of points \(b_p\in R\) for which \(T_p b_p-b_p=e_p\to0\). From the equality
\(A(L^0w_{b_p}-Ls_{pb_p}+a)=b_p+e_p\), we obtain
\(Ls_{pb_p}=a+L^0w_{b_p}-B(b_p+e_p)=a+Bb_p-B(b_p+e_p)\to a\).
Since \(\sup_p\omega(s_{pb_p})<\infty\), there is a subsequence \(s_{p_i b_{p_i}}\) for which
\(s_{p_i b_{p_i}}\to g\), \(Ls_{p_i b_{p_i}}\to a\), whence \(g\) is a solution of problem (1).
Remark 1. If there is a ball \(R\subset E_n\) such that, for sufficiently large \(p\), \(T_pR\subseteq R\), then condition 6 is not needed for the proof of Theorem 1.
Theorem 2. Suppose conditions 1—5 are satisfied, the set \(G^0\) is invariant under multiplication by any number \(\alpha>0\), the operators \(L^0\) and \(L^*\) are homogeneous: \(L^0\alpha g=\alpha^\mu L^0g\), \(L^*\alpha g=\alpha^\nu L^*g\) \((\mu,\nu>0)\), for all \(\alpha>0\), and suppose that for some norm \(\|\ \|\) in \(E_n\)
\[ \lim_{\alpha\to\infty}\ \sup_{\omega(g)\leq M,\ \alpha g\in G}\|\alpha^{-\mu}L^1\alpha g\|=0, \]
where \(M=\Omega+\sup_{\|b\|\leq1}\omega(w_b)\), and also
\[ \lim_{\alpha\to\infty}\ \overline{\lim}_{p\to\infty}\ \sup_{\|b\|\leq1} \left\|\alpha^{-\frac{\mu}{\nu}}\bigl(L^0w_{\alpha b}-L^0s_{p,\alpha b}\bigr)\right\|=0. \]
Then problem (1) has at least one solution.
Proof. For the proof we shall use Remark 1. By \(R\) denote the ball of unit radius with center at the origin, and show that, for each sufficiently large \(\alpha\) and for all \(p>p_\alpha\), one has \(T_pR_\alpha\subseteq R_\alpha\).
First we prove some relations. From \(L^0v_a=a\) it follows that
\(L^0\alpha^{\frac1\mu}v_a=\alpha a\), whence, by the conicity of \(G^0\), \(v_{\alpha a}=\alpha^{\frac1\mu}v_a\). Similarly
\(w_{\alpha b}=\alpha^{\frac1\nu}w_b\). Further,
\[ A\alpha a=L^*v_{\alpha a}=L^*\alpha^{\frac1\mu}v_a =\alpha^{\frac\nu\mu}L^*v_a=\alpha^{\frac\nu\mu}Aa . \]
We show that for any \(\varepsilon>0\) there exists \(\alpha_\varepsilon>0\) such that, for any \(\alpha>\alpha_\varepsilon\), there exists \(p_{\alpha\varepsilon}\) for which
\[ \sup_{\omega(g)\leq M,\ \alpha g\in G_p}\|\alpha^{-\mu}L^1\alpha g\|<\varepsilon \quad\text{for }p>p_{\alpha\varepsilon}. \tag{3} \]
Indeed, let \(\alpha_\varepsilon\) be such that
\[ \sup_{\omega(g)\leq M,\ \alpha g\in G_p}\|\alpha^{-\mu}L^1\alpha g\|<0.5\varepsilon \quad\text{for }\alpha>\alpha_\varepsilon . \]
Suppose that condition (3) is not satisfied for this \(\alpha_\varepsilon\). Then for some \(\alpha>\alpha_\varepsilon\) there is a sequence
\(\alpha g_k\in G_{p_k}\) \((k=1,2,\ldots;\ p_k\to\infty)\) such that
\(\sup_k\omega(g_k)\leq M\) and
\(\|\alpha^{-\mu}L^1\alpha g_k\|\geq0.5\varepsilon\).
But, by condition 5, we find a subsequence
\(\alpha g_{k_i}\to \alpha g\in G\), and from the semicontinuity ...
bounded below, \(\omega(g)\le M\). But then \(\|\alpha^{-\mu}L^1\alpha g\|<0.5\varepsilon\), while by the continuity of \(L^1\) we have
\[ \lim_{i\to\infty}\|\alpha^{-\mu}L^1\alpha g_{k_i}\| =\|\alpha^{-\mu}L^1\alpha g\|\ge 0.5\varepsilon, \]
which proves (3).
Since the operator \(A\) is continuous and \(A0=0\), there exists an \(\varepsilon>0\) such that \(Ac\in R\) for \(c\in R_\varepsilon\). We have
\[ T_p\alpha b=A(L^0w_{ab}-Ls_{p,ab}+a)= \]
\[ =\alpha A\left(\alpha^{-\frac{\mu}{\nu}}(L^0w_{ab}-L^0s_{p,ab}) -\alpha^{-\frac{\mu}{\nu}}L^1\alpha^{\frac{1}{\nu}} \left(w_b+\alpha^{-\frac{1}{\nu}}(s_{p,ab}-w_{ab})\right) +\alpha^{-\frac{\mu}{\nu}}a\right). \]
It is clear that if
\[ \left\|\alpha^{-\frac{\mu}{\nu}}(L^0w_{ab}-L^0s_{p,ab})\right\|<0.3\varepsilon \qquad (b\in R), \]
\[ \left\|\alpha^{-\frac{\mu}{\nu}}L^1\alpha^{\frac{1}{\nu}} \left(w_b+\alpha^{-\frac{1}{\nu}}(s_{p,ab}-w_{ab})\right)\right\|<0.3\varepsilon \qquad (b\in R) \]
and \(\|\alpha^{-\frac{\mu}{\nu}}a\|<0.3\varepsilon\), then \(T_pR_a\subseteq R_a\). But for every sufficiently large \(\alpha\), for all \(p>p_\alpha\), this is true. Thus Theorem 2 is proved.
Remark 2. If the operator \(L^0\) is linear, the set \(G^0\) is linear and \(n\)-dimensional, and \(v^1,v^2,\ldots,v^n\) is some basis of \(G^0\), then the unique solvability of problem (2) and the continuous dependence of the solution on \(a\) are equivalent to the nonvanishing of the determinant
\[ \Delta=\operatorname*{det}_{\mathrm{def}}(L^0v^1,\ldots,L^0v^n). \]
Let us apply the results obtained to the boundary-value problem (in scalar notation)
\[ x_i^{(n_i)}-f_i(t,x_1,\ldots,x_m^{(n_m-1)})=0 \quad (i=1,\ldots,m;\ t_1\le t\le t_2), \tag{4'} \]
\[ L_k(x_1,\ldots,x_m)=a_k \quad (k=1,\ldots,n=n_1+\cdots+n_m), \]
where the functions \(f_i\) are defined for \(t_1\le t\le t_2\), \(-\infty<x_1,\ldots,x_m^{(n_m-1)}<\infty\), all \(n_i>0\), and the functionals \(L_k\) are defined on the totality of functions \(x(t)=\{x_i(t)\}\) \((t_1\le t\le t_2)\) for which all derivatives \(x_i^{(n_i-1)}\) are absolutely continuous. With respect to \(f_i\) we shall assume that the Carathéodory conditions are satisfied: all \(f_i\) are measurable in \(t\) for each fixed set of values \(x_1,\ldots,x_m^{(n_m-1)}\), continuous in the totality of these values for almost every fixed \(t\); for any \(M>0\) there exists a function \(f_M\), summable on \((t_1,t_2)\), such that all \(|f_i|\le f_M(t)\) for \(t_1\le t\le t_2\), \(\|x\|<M\). We shall write this problem more briefly in vector form as
\[ Dx=0,\qquad Lx=a. \tag{4} \]
Here \(H\) is the space of functions \(x(t)=\{x_i(t)\}\) \((t_1\le t\le t_2)\), for which all derivatives \(x_i^{(n_i-1)}\) are absolutely continuous; the topology in \(H\) is defined by the norm
\[ \|x\|=\max_{[t_1,t_2]}\sum_{i=1}^{m}\sum_{j=0}^{n_i-1}|x_i^{(j)}(t)|. \]
\(G\) is the set of solutions of the equation \(Dx=0\),
\[ \omega(x)=\|x\|_1=\sum_{i=1}^{m}\left(\sum_{j=0}^{n_i-1}|x_i^{(j)}(t_1)|+\int_{t_1}^{t_2}|x_i^{(n_i)}(\tau)|\,d\tau\right). \]
The operator \(D^0\) is obtained from \(D\) by setting all \(f_i \equiv 0\); then \(G^0\) is the set of all solutions of the equation \(D^0x=0\).
If, as the comparison problem, one takes the Cauchy problem
\[ D^0w=0;\quad w_1(t_1)=b_1\ldots,\quad w_m^{(n_m-1)}(t_1)=b_n\quad (L^*w=b), \]
then we easily obtain the following theorems.
Theorem 3. Suppose the following conditions are satisfied:
-
There exists a function \(f\in \mathcal L_1(t_1,t_2)\) such that all \(|f_i|\le f(t)\) for \(t_1\le t\le t_2\) and arbitrary \(x\).
-
The operator \(L\) can be represented in the form \(L=L^0+L^1\), where the operators \(L^0\) and \(L^1\) are continuous in \(H\).
-
The boundary-value problem \(D^0v=0,\ L^0v=a\), for any \(a\in E_n\), has a unique solution depending continuously on \(a\).
-
For any \(M>0\) and any \(a\in E_n\) there is a ball \(R\subset E_n\) such that for any \(b\in R,\ s\in G,\ \omega(s-w_b)<M\) one has \(A(L^0w_b-Ls+a)\in R\) (the meaning of the notation was explained before Theorem 1).
Then the boundary-value problem (4) has at least one solution.
Proof. In order that Theorem 3 follow directly from Theorem 1, we must verify the validity of conditions 4 and 5 (before Theorem 1). But the \(f_i\) can always be approximated in a suitable way by means of \(f_{ip}\), for which the Lipschitz condition is satisfied and there is a uniform estimate of the form \(|f_{ip}|\le f_0(t)\in \mathcal L_1(t_1,t_2)\). (This can be achieved by passing to bounded right-hand sides of system (4), triangulating the space \((x_1,\ldots,x_m^{(n_m-1)})\), and applying piecewise-linear approximation.) Hence both conditions 4 and 5 follow. Thus the theorem is proved.
Theorems 4 and 5 are proved analogously.
Theorem 4. Suppose conditions 1–3 of Theorem 3 are satisfied. If \(L^0\) is homogeneous \((L^0\alpha x=\alpha^\mu Lx,\ \alpha>0,\ \mu=\mathrm{const}>0)\) and for any \(M>0\)
\[ \lim_{\alpha\to\infty}\sup_{\omega(x)\le M}\|\alpha^{-\mu}L^1\alpha x\|=0, \]
then the boundary-value problem (4) has at least one solution.
Theorem 5. Suppose conditions 1 and 2 of Theorem 3 are satisfied. If the operator \(L^0\) is linear, \(\Delta\ne 0\), and for any \(M>0\)
\[ \lim_{\alpha\to\infty}\sup_{\omega(x)\le M}\|\alpha^{-1}L^1\alpha x\|=0, \]
then the boundary-value problem (4) has at least one solution.
Applying this method, one can obtain a strengthening of the corresponding results of papers [1, 2], and possibly also [3].
Condition 1 of Theorem 3 is, of course, very restrictive in applications. However, in many cases it turns out to be possible to use this condition only for the approximating equations. Namely, if for the solution of boundary-value problem (4) there is an a priori estimate, then one may try to replace \(f\) in problem (4) by a function \(f^*\) so that solvability of the modified problem would follow on the basis of the general theorems stated above, while the solution of the modified problem, on the basis of the a priori estimates, would at the same time serve as a solution of the original problem. It may also be possible to construct a sequence of functions \(f_p^*\) and apply the general theorems to it.
If instead of \(\sup_p \omega(x_p)<\infty\) one can obtain the estimate \(\sup_p \omega_1(x_p)<\infty\), where
\[ \omega_1(x)=\max_{[t_1,t_2]}\sum_{i=1}^{m}\sum_{j=0}^{k_i}\left|x_i^{(j)}(t)\right| \quad (0\leq k_i\leq n_i-1), \]
then, using analogues of the Kolmogorov–Gorny inequality for derivatives, one can easily indicate an estimate of the form \(|f|<\varphi\) (an analogue of Bernstein’s inequality), under whose fulfillment \(\sup_p \omega_1(x_p)<\infty\) implies \(\sup_p \omega(x_p)<\infty\) (see [4]).
Let us illustrate these considerations by simple examples. Consider the boundary-value problem
\[ x_1'=f_1(t,x_1,x_2),\quad x_2'=f_2(t,x_1,x_2)\quad (t_1\leq t\leq t_2), \]
\[ x_1(t_1)=a_1,\quad x_2(t_2)=a_2. \tag{5} \]
Let \(f_1\) and \(f_2\) satisfy the Carathéodory conditions, and suppose there exist two measurable functions \(\psi_{1,2}(x)>1\) \((-\infty<x<\infty)\) and a constant \(M>0\) such that
\[ \int_0^\infty \psi_{1,2}^{-1}(x)\,dx=\infty,\quad \int_{-\infty}^0 \psi_{1,2}^{-1}(x)\,dx=\infty \]
and, for any \(x_1,x_2\) and \(t\in[t_1,t_2]\),
\[ f_1(t,x_1,x_2)\operatorname{sign}x_1\leq \psi_1(x_1)\quad \text{for } |x_1|>M, \]
\[ -f_2(t,x_1,x_2)\operatorname{sign}x_2\leq \psi_2(x_2)\quad \text{for } |x_2|>M. \]
Then there exists a solution of the boundary-value problem (5).
Indeed, if \(f_{1,2}^*=f_{1,2}\) for \(\|x\|<N\) and \(f_{1,2}^*=0\) for \(\|x\|>N\), where \(N\) is a sufficiently large number, then an a priori estimate in the norm \(C[t_1,t_2]\) is obtained elementarily by means of the functions \(\psi_{1,2}\).
This example illustrates the case of splitting: the condition on \(f_1\), together with the boundary condition \(x_1(t_1)=a_1\), independently of the conditions on \(f_2\) and the second boundary condition, guarantees the presence of an a priori estimate for \(x_1\).
Consider also the boundary-value problem
\[ x_1''=f_1(t,x_1,\ldots,x_2'),\quad x_2''=f_2(t,x_1,\ldots,x_2')\quad (t_1\leq t\leq t_2), \]
\[ x_1'(t_1)=a_1,\quad x_1'(t_2)=a_2,\quad x_1(t_1)=a_3, \tag{6} \]
\[ x_2'(t_1)=a_4,\quad x_2'(t_2)=a_5,\quad x_2(t_2)=a_6. \]
Let \(f_1\) and \(f_2\) be continuous, and suppose there exist a twice continuously differentiable function \(\sigma(x_1',x_2')\) \((-\infty<x_1',x_2'<\infty)\) and constants \(M>0\) and \(\varepsilon\in(0,1)\) such that \(\sigma(x_1',x_2')\to\infty\) as \(|x_1'|+|x_2'|\to\infty\),
\[ \frac{\partial^2\sigma}{\partial x_1'^2}x_1''^2 +2\frac{\partial^2\sigma}{\partial x_1'\partial x_2'}x_1''x_2'' +\frac{\partial^2\sigma}{\partial x_2'^2}x_2''^2 +\alpha\left(\frac{\partial\sigma}{\partial x_1'}f_1+\frac{\partial\sigma}{\partial x_2'}f_2\right)>0 \]
when
\[ \frac{\partial\sigma}{\partial x_1'}x_1''+\frac{\partial\sigma}{\partial x_2'}x_2''=0,\quad |x_1'|+|x_2'|>M \]
and for every \(\alpha\in(0,1]\); and suppose, moreover, that for all values of the arguments an analogue of Bernstein’s inequality holds:
\[ |f_1|+|f_2|<A+B\left(|x_1''|^{2-\varepsilon}+|x_2''|^{2-\varepsilon}\right). \]
Then there exists a solution of the boundary-value problem (6).
Indeed, set
\[ f^{*}_{1,2}(t, x_1, \ldots, x''_2) = \left(1+\chi\left(N^{-1}(|x_1|+\ldots+|x''_2|)\right)(|f_1|+|f_2|)\right)^{-1} f_{1,2}(t, x_1, \ldots, x''_2), \]
where \(\chi(x)=0\) for \(0\le x\le 1\), \(\chi(x)=1\) for \(x\ge 2\), and is linear for \(1\le x\le 2\), while \(N\) is a sufficiently large number. Then the a priori estimate for
\[ \max_{[t_1,t_2]} \left(|x'_1(t)|+|x'_2(t)|\right) \]
is obtained through \(\sigma\), since from the boundary conditions there follows the boundedness of the function \(\sigma(t)=\sigma(x'_1(t),x'_2(t))\) at the endpoints of the interval, and it cannot have a sufficiently large maximum by virtue of the imposed conditions.
This example illustrates the possibility of passing to functions of one variable with the aid of the function \(\sigma\). The boundary conditions and the conditions on the right-hand sides ensure the boundedness of \(\sigma\), which corresponds to the boundedness of a certain norm.
References
- Santagati G. Ann. mat. pura ed appl., 62, 335—370, 1963.
- Conti R. Ann. mat. pura ed appl., 57, 49—61, 1962.
- Lasota A. Bull. Acad. polon, sci Sér. sci. math. astron. et phys., 10, No. 11, 565—570, 1962.
- Lepin A. Ya., Myshkis A. D. Differential Equations, 1, No. 9, 1260—1263, 1965.
Received by the editors
May 24, 1966
Latvian State University
named after P. Stuchka,
Physico-Technical Institute of Low Temperatures
Academy of Sciences of the Ukrainian SSR