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Mathematics
Corresponding Member of the Academy of Sciences of the USSR A. N. Tikhonov
ON THE SOLUTION OF NONLINEAR INTEGRAL EQUATIONS OF THE FIRST KIND
In the present article we shall consider a stable method for solving integral equations of the form
\[ K[x,z(s)] = \int_a^b K(x,s,z(s))\,ds = u(x), \qquad c \leq x \leq d. \tag{1} \]
The problem under consideration is an ill-posed problem, since a solution of this problem does not exist for every \(u(x)\), and if for some \(\bar u(x)\) solutions \(\bar z(s)\) do exist, then small perturbations of \(\bar u\) (in the \(L_2\) norm) may correspond to large perturbations of \(\bar z\) (in the \(C\) norm).
In \((^{1,2})\) regularization methods were developed for linear integral equations of the first kind, making it possible, from \(\tilde u(x)\), the perturbed values of \(u(x)\) in the \(L_2\) norm, and \(\delta\), the order of the perturbation, to find uniform approximations for \(\bar z(s)\in W_2^1\). The aim of the present article is to extend these methods to nonlinear equations (see also \((^{3,4})\)). We shall assume that:
\(1^\circ\). \(K[x,z_1(s)] \ne K[x,z_2(s)]\), if \(z_1(s) \ne z_2(s)\).
\(2^\circ\). The operator \(K[x,z(s)]\) is continuous from \(C\) into \(L_2\), i.e., if a sequence of functions \(z_n(s)\in C\) converges uniformly to \(z_0(s)\), then \(K[x,z_n(s)]_{L_2}\to K[x,z_0(s)]\) as \(n\to\infty\).
\(3^\circ\). \(K'_z(x,s,z)\) and \(K''_{zz}(x,s,z)\) are continuous in \(z\) in a neighborhood of \(\bar z(s)\).
\(4^\circ\). The linear equation of the first kind
\[ \int_a^b K'_z(x,s,\bar z(s))\,w(s)=0 \]
has in \(W_2^1\) only the trivial solution.
These conditions, in particular, are satisfied by the nonlinear integral equation of the first kind to which the inverse problem of gravimetry is reduced.
Here we restrict ourselves to the consideration of equations with one variable \(s\). The transition to equations with many variables proceeds as indicated in \((^1)\). Moreover, similarly to \((^2)\), regularization methods are constructed for uniform approximation together with \(n\) derivatives, if \(\bar z(s)\in W_2^{(n+1)}\).
- Consider the “smoothing” functional
\[ M^\alpha[z,\tilde u]=N[z(s),\tilde u(x)] + \alpha \Omega[z], \]
where
\[ N[z(s),\tilde u(x)] = \int_c^d |K[x,z(s)]-\tilde u(x)|^2\,dx \]
quadratic deviation of the operator \(K[x,z(s)]\) from the given function \(\tilde u(x)\), and
\[ \Omega[z]=\int_a^b\left[k_1(s)\left(\frac{dz}{ds}\right)^2+k_0(s)z^2(s)\right]\,ds,\qquad k_0(s),\,k_1(s)\geq x_0>0. \]
Theorem 1. Whatever the function \(\tilde u(x)\in L_2\) and \(\alpha>0\), there exists at least one function \(z^\alpha(s)\in W_2^1\) realizing the minimum of \(M^\alpha[z(s),\tilde u(x)]\).
Let \(M_0\) be the lower bound of \(M^\alpha[z,\tilde u(x)]\) for \(z(s)\in W_2^1\). This lower bound exists by virtue of the nonnegativity of \(M^\alpha\). Let, further, \(z_n(s)\) be a sequence of functions from \(W_2^1\) minimizing \(M^\alpha\), so that
\[ M_n=M^\alpha[z_n(s),\tilde u(x)]\to M_0 \quad\text{as } n\to\infty \qquad (M_n\leq M_{n-1}). \]
It is obvious that
\[ |z_n(s)|\leq C_1, \]
where \(C_1\) is some constant, since
\[ \Omega[z_n]\leq \frac{1}{\alpha}M_1. \]
Thus, from \(\{z_n\}\) one can choose a subsequence \(z_{n_k}(s)\) converging uniformly to \(z_0(s)\in W_2^1\), and \(\Omega[z_0]\leq \frac{1}{\alpha}M_1\). Hence it follows that
\[
N[z_{n_k}(s),\tilde u(x)]\to N[z_0(s),\tilde u(x)]
\quad\text{as } k\to\infty.
\]
Thus,
\[ M^\alpha[z_0(s),\tilde u(x)]=\lim_{k\to\infty} M_{n_k}=M_0, \]
which proves the existence of a function \(z^\alpha(s)\) \((z^\alpha(s)=z_0(s))\) realizing the minimum of \(M^\alpha[z(s),\tilde u(x)]\).
Theorem 2. Let the function \(\bar u(x)\in L_2\) correspond to a solution of equation (1), equal to \(\bar z(s)\in W_2^1\). For any \(\varepsilon>0\) there exists such a \(\delta_0(\varepsilon,\bar z)\) that, if \(\|\tilde u(x)-\bar u(x)\|_{L_2}\leq\delta\) and \(q_1\delta^2\leq \alpha(\delta)\leq q_2\delta^2\) \((q_1>0)\), then \(|z^{\alpha(\delta)}(s)-\bar z(s)|\leq\varepsilon\) for \(\delta\leq\delta_0\), whatever the function \(z^{\alpha(\delta)}(s)\) realizing the minimum of \(M^{\alpha(\delta)}[z(s),\tilde u(x)]\).
Indeed, since \(M^\alpha[z^\alpha(s),\tilde u(x)]\leq M^\alpha[\bar z(s),\tilde u(x)]\), we have:
1) \(\alpha(\delta)\Omega[z^{\alpha(\delta)}]\leq \delta^2+\alpha\Omega[\bar z]\leq \alpha\left[\dfrac{1}{q_1}+\Omega[\bar z]\right]\), or \(\Omega[z^{\alpha(\delta)}]\leq C_0=\dfrac{1}{q_1}+\Omega[\bar z]\), i.e., all \(z^{\alpha(\delta)}(s)\), for arbitrary \(\delta\), belong to the set \(Z_0(\bar z)\), compact in the sense of \(C\), of functions defined by the condition \(\Omega[z]\leq C_0\).
2) Let \(U_0\) be the set of functions \(u(x)=K[x,z]\) \((z\in Z_0)\). By the continuity of the mapping \(Z_0\to U_0\) and the uniqueness of the solution, the inverse mapping \(U_0\to Z_0\) is defined and continuous, i.e., for any \(\varepsilon\) there exists such an \(\eta_0(\varepsilon,\bar z)\), depending on \(Z_0(\bar z)\), that if, for \(\tilde u_1(x),\,u_2(x)\in U_0\), the norm \(\|\tilde u_1-u_2\|_2\leq \eta_0(\varepsilon,\bar z)\), then \(|z_1(s)-z_2(s)|<\varepsilon\). Further,
\[ N[z^{\alpha(\delta)}(s),\tilde u(x)] =\int_a^b [u^{\alpha(\delta)}-\tilde u(x)]^2\,dx= \]
\[ =\|u^{\alpha(\delta)}(x)-\tilde u(x)\|_2^2 \leq \delta^2\bigl(1+q_2\Omega[\bar z]\bigr), \]
where \(u^{\alpha(\delta)}(x)=K[x,z^{\alpha(\delta)}(s)]\). Thus, since \(\bar z(s), z^{\alpha(\delta)}(s)\in Z_0(\bar z)\) and
\[ \|u^{\alpha(\delta)}(x)-\bar u(x)\|_2 \le \|u^{\alpha(\delta)}-\tilde u\|_2+\|\tilde u-\bar u\|_2 \le \delta C_2 \]
\[ \left(C_2=1+\sqrt{1+q_2\Omega[\bar z]}\right), \]
we find that
\[ |z^{\alpha(\delta)}(s)-\bar z(s)|_C<\varepsilon \quad \text{for } \delta \le \frac{1}{C_2}\eta_0(\varepsilon,\bar z)=\delta_0(\varepsilon,\bar z), \]
which proves Theorem 2.
Consider the Euler equation corresponding to the functional \(M^\alpha[z(s),\tilde u(x)]\). Let \(z^\alpha(s)\) be some function from \(W_2^1\) realizing the minimum of \(M^\alpha[z,\tilde u]\); putting \(z=z^\alpha(s)+\eta \zeta(s)\) \((\zeta(s)\in W_2^1)\), we obtain
\[ \frac{d}{d\eta}\left[M^\alpha\right] = \int_c^d F(\xi) \left[ \int_a^b K'_z(\xi,s,z^\alpha(s))\,\zeta(s)\,ds \right]d\xi + \alpha\int_a^b\left[k_1(z^\alpha)'_s\zeta'+k_0z^\alpha\zeta\right]ds=0, \]
where
\[ F(\xi)=\left.[K[\xi,z(s)]-\tilde u(\xi)]\right|_{z=z^\alpha(s)} = [u^\alpha(\xi)-\tilde u(\xi)]. \]
If \(k_1=\mathrm{const}\), then, choosing \(\zeta(s)\) to be a piecewise-linear function equal to
\[ \zeta(s_0)=\frac{1}{h},\qquad \zeta(s_0-h)=\zeta(s_0+h)=0, \]
and identically equal to zero outside the interval \((s_0-h,s_0+h)\), we obtain (for \(k_1=\mathrm{const}\))
\[ -\alpha k_1 \frac{z^\alpha(s_0+h)+z^\alpha(s_0-h)-2z^\alpha(s_0)}{h^2} + \]
\[ + \left\{ \int_{s_0-h}^{s_0+h} \zeta(s) \left[ \int_c^d F(\xi)K'_z(\xi,s,z^\alpha(s))\,d\xi + \alpha k_0 z^\alpha \right]ds \right\}, \]
and as \(h\to 0\),
\[ -\alpha\bigl(k_1(z^\alpha)''-k_0z^\alpha\bigr) + \int_c^d F(\xi)K'_z(\xi,s,z^\alpha(s))\,d\xi =0. \]
It is also not difficult to verify that
\[ (z^\alpha)'_{s=a}=(z^\alpha)'_{s=b}=0. \]
If \(k_1\) depends on \(s\), then analogously
\[ -\alpha\left[ \frac{d}{ds}\left(k_1\frac{dz^\alpha}{ds}\right)-k_0z^\alpha \right] + \int_c^d F(\xi)K'_z(\xi,s,z^\alpha(s))\,d\xi =0, \qquad (z^\alpha)'_a=(z^\alpha)'_b=0. \]
Until now we have left aside the question of the uniqueness of \(z^\alpha(s)\).
Theorem 3. Let \(\bar u(x)\in L_2\) correspond to a function \(\bar z(s)\in W_2^1\) which is a solution of equation (1). There exists a \(\delta_0(\bar z)\) such that for \(\delta<\delta_0(\bar z)\) there exists only one function \(z^{\alpha(\delta)}(s)\) realizing the minimum of the functional \(M^{\alpha(\delta)}[z,\tilde u(x)]\).
This theorem will be proved if we prove the uniqueness of the solution of the Euler equation with the corresponding boundary conditions. Let
\(z_1^{\alpha(\delta)}(s)\) and \(z_2^{\alpha(\delta)}(s)\) are two different solutions of the Euler equation and \(w(s)=z_1^{\alpha(\delta)}-z_2^{\alpha(\delta)}\). Subtracting the corresponding equations, we find
\[ -\alpha\left[\frac{d}{ds}\left(k_1\frac{dw}{ds}\right)-k_0w\right] +\int_c^d [F_1(\xi)-F_2(\xi)]K_z'(\xi,s,z_1^{\alpha(\delta)}(s))\,d\xi+ \]
\[ +\int_c^d F_2(\xi)\left[K_z'(\xi,s,z_1^{\alpha(\delta)}(s))-K_z'(\xi,s,z_2^{\alpha(\delta)}(s))\right]\,d\xi=0,\quad w'(a)=w'(b)=0, \]
\[ F_i(\xi)=\left[K[\xi,z_i^{\alpha(\delta)}(s)]-\tilde u(\xi)\right]\quad (i=1,2). \]
Represent the integral term in the form
\[ \int_a^b\left\{\int_c^d K_z'(\xi,s,\bar z(s))K_z'(\xi,s_1,\bar z(s_1))\,d\xi\right\}w(s_1)\,ds_1+ \]
\[ +\int_a^b\int_c^d \eta(\xi,s,s_1,\delta)w(s_1)\,ds_1\,d\xi+w(s)\eta(s,\delta), \]
where \(\eta(\xi,s,s_1,\delta)\) and \(\eta(s,\delta)\) are uniformly small as \(\delta\to0\), which follows from Theorem 2 and the continuity of \(K_z'(x,s,\bar z(s))\) and \(K_{zz}''(x,s,\bar z(s))\) in a neighborhood of \(\bar z(s)\). Multiplying the equation by \(w(s)\), integrating and transforming, we obtain
\[ \alpha\int_a^b [k_1(w')^2+k_0w^2]\,ds+ \int_c^d\left[\int_a^b K_z'(\xi,s,\bar z(s))w(s)\,ds\right]^2\,d\xi+S=0, \]
where
\[ |S|\leq \eta(\delta)\int_a^b w^2(s)\,ds \]
and \(\eta(\delta)\to0\) as \(\delta\to0\).
We shall show that this equality is impossible for sufficiently small \(\delta\leq\delta_0\). Indeed,
\[ \int_c^d\left[\int_a^b K_z'(\xi,\bar z(s))w(s)\,ds\right]^2\,d\xi\geq \gamma_0\Omega[w], \]
where \(\gamma_0\) is the lower bound of the left-hand side of the inequality on the sphere \(\Omega[w]=1\). Here \(\gamma_0>0\), since
\[ \int_a^b K_z'(\xi,s,\bar z(s))w(s)\,ds\ne0 \]
and the sphere \(\Omega[w]=1\) is compact, which proves the impossibility of the last equality for \(w(s)\ne0\) when \(\delta<\delta_0\) such that \(\eta(\delta)<\gamma_0\).
The numerical algorithms following from the method presented show their high efficiency.
Received
29 III 1964
REFERENCES
- A. N. Tikhonov, DAN, 151, No. 3 (1963).
- A. N. Tikhonov, DAN, 153, No. 1 (1963).
- M. M. Lavrent’ev, On Certain Ill-Posed Problems of Mathematical Physics, Novosibirsk, 1962.
- M. M. Lavrent’ev, On One Class of Nonlinear Integral Equations, Materials for the Joint Soviet-American Symposium on Partial Differential Equations, Novosibirsk, 1963.