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UDC 517.947.33
MATHEMATICS
P. P. ZABREIKO
ON THE DIFFERENTIABILITY OF NONLINEAR OPERATORS IN THE SPACES \(\mathscr L_p\)
(Presented by Academician I. G. Petrovsky, 29 V 1965)
Many methods for the study of nonlinear operator equations (the theory of implicit functions, approximate and topological methods, etc.) require smoothness of the nonlinear operators entering these equations. In connection with this, the problem arises of differentiating specific operators, for example integral operators, acting from one space \(E_1\) into another space \(E_2\). In the present note the question of differentiability is solved for the Uryson operator
\[ Ax(t)=\int_{\Omega_1} K[t,s,x(s)]\,ds, \tag{1} \]
acting from the space \(\mathscr L_p=\mathscr L_p(\Omega_1)\) \((1\le p\le \infty)\) into the space \(\mathscr L_q=\mathscr L_q(\Omega_2)\) \((0<q\le \infty)\); here, as usual, \(\Omega_1\) and \(\Omega_2\) are sets of finite Lebesgue measure. Some results on the differentiability of the operator (1) were obtained earlier in \((^{1-5})\).
Below we use certain special spaces of functions of two variables. Denote by \(N_p\) \((1\le p\le \infty)\) the Banach space of functions \(u(t,s)\) \((t\in\Omega_2,\ s\in\Omega_1)\) for which the norm is meaningful and finite
\[ \|u(t,s)\|_{N_p} = \left\|\operatorname*{vrai\,sup}_{t\in\Omega_2}|u(t,s)|\right\|_{\mathscr L_p(\Omega_1)} . \tag{2} \]
Next, denote by \(B_{p,q}\) \((B^0_{p,q})\) the space of functions \(k(t,s)\) \((t\in\Omega_2,\ s\in\Omega_1)\) for which the integral operators
\[ Kx(t)=\int_{\Omega_1} k(t,s)x(s)\,ds, \qquad |K|x(t)=\int_{\Omega_1} |k(t,s)|x(s)\,ds \tag{3} \]
act from \(\mathscr L_p\) to \(\mathscr L_q\) and are continuous (completely continuous). We introduce the norm (6) in the spaces \(B_{p,q}\) and \(B^0_{p,q}\) by the equality
\[ \|k(t,s)\|_{B_{p,q}} = \sup_{\|x\|_{\mathscr L_p}\le 1} \left\| \int_{\Omega_1} |k(t,s)x(s)|\,ds \right\|_{\mathscr L_q}. \tag{4} \]
- Suppose that the functions \(K(t,s,u)\) \((t\in\Omega_2,\ s\in\Omega_1,\ -\infty<u<\infty)\) are measurable on \(\Omega_2\times\Omega_1\) for all \(u\in(-\infty,\infty)\) and for almost all \(\{t,s\}\in\Omega_2\times\Omega_1\) have a derivative with respect to \(u\) that is continuous. Introduce the superposition operator
\[ \mathfrak G u(t,s)=K'_u[t,s,u(t,s)] . \tag{5} \]
Theorem 1. Let \(A\theta\in\mathscr L_q\) and let the operator (5) act from \(N_p\) into \(B_{p,q}\) and be continuous.
Then the nonlinear integral operator (1) acts from \(\mathscr L_p\) into \(\mathscr L_q\) and is continuously differentiable on \(\mathscr L_p\), moreover
\[ A'(x_0)h=\int_{\Omega_1} K'_u[t,s,x_0(s)]h(s)\,ds \qquad (x_0,h\in\mathscr L_p). \tag{6} \]
The main assumption of Theorem 1 is the continuity of the superposition operator (5). This assumption, for example, is satisfied if the inequality
\[ \left|K'_u(t,s,u_1)-K'_u(t,s,u_2)\right| \leq \sum_{i=1}^{n} R_i(t,s,u)\, |u_1-u_2|^{\delta_i} \quad (|u_1|,|u_2|\leq u), \tag{7} \]
holds, where \(0<\delta_1<\cdots<\delta_n<p-1\), and the functions \(R_i(t,s,u)\) \((i=1,\ldots,n)\) define operators acting from \(N_p\) into \(B_{\frac{p}{1+\delta_i},q}\) and bounded operators \(\mathfrak R_i u(t,s)=R_i[t,s,u(t,s)]\). More delicate criteria for the continuity of operator (5) are based on the following assertion.
Theorem 2. Let the function \(Q(t,s,u)\) \((t\in\Omega_2,\ s\in\Omega_1,\ -\infty<u<\infty)\) be measurable on \(\Omega_2\times\Omega_1\) for all \(u\in(-\infty,\infty)\) and continuous in \(u\) for almost all \(\{t,s\}\in\Omega_2\times\Omega_1\). Let the superposition operator \(\mathfrak Q u(t,s)=Q[t,s,u(t,s)]\) act from \(N_p\) into \(B_{p,q}\), where \(p>1,\ q<\infty\). Then the operator (5) acts from \(N_p\) into \(B^0_{p,q}\).
Then the superposition operator \(\mathfrak Q\) is continuous.
From Theorems 1 and 2 it follows:
Theorem 3. Let \(1<p\leq\infty,\ 0<q<\infty\). Let \(A\theta\in\mathscr L_q\), and let operator (5) act from \(N_p\) into \(B^0_{p,q}\).
Then the assertion of Theorem 1 is valid.
Applying various criteria for complete continuity of linear integral operators, from Theorem 3 one can obtain more particular criteria of differentiability. We give one example.
Theorem 4. Let \(1<p\leq\infty,\ 0<q<\infty\), and let \(A\theta\in\mathscr L_q\). Let the function \(K'_u(t,s,u)\) satisfy the inequality
\[ |K'_u(t,s,u)|\leq \sum_{j=1}^{m} S_j(t,s)g_j(s,u), \tag{8} \]
where \(g_j(s,u)\) \((j=1,\ldots,m)\) are nonnegative functions satisfying the Carathéodory conditions, and each operator \(g_j=g_j[s,x(s)]\) acts from \(\mathscr L_p\) into some \(\mathscr L_{r_i}\), \(1/r_i\leq 1-1/p\), while \(S_j(t,s)\in B_{\frac{r_i p}{r_i+p},q}\) for \(r_i<\infty\), or \(S_j(t,s)\in B^0_{p,q}\) for \(r_i=\infty\).
Then the assertion of Theorem 1 is valid.
- We give a theorem on the differentiability of operator (1) at a fixed point. Suppose that the function \(K(t,s,u)\) \((t\in\Omega_2,\ s\in\Omega_1,\ -\infty<u<\infty)\) is measurable on \(\Omega_2\times\Omega_1\) for all \(u\in(-\infty,\infty)\) and is continuous in \(u\) for almost all \(\{t,s\}\in\Omega_2\times\Omega_1\). Let \(x_0(s)\) be a fixed function, and suppose that there exists
\[ \lim_{u\to 0}\frac{1}{u}\{K[t,s,x_0(s)+u]-K[t,s,x_0(s)]\}=K_0(t,s), \tag{9} \]
Introduce the superposition operator
\[ \mathfrak G_0 u(t,s)=G_0[t,s,u(t,s)], \tag{10} \]
\[ G_0(t,s,0)=K_0(t,s),\qquad G_0(t,s,u)=\frac{1}{u}\{K[t,s,x_0(s)+u]-K[t,s,x_0(s)]\}. \]
Theorem 5. Let \(x_0\in\mathscr L_p,\ Ax_0\in\mathscr L_q\), and let operator (10) act from \(\mathscr L_p\) into \(B_{p,q}\) and be continuous at the point \(\theta\).
Then the nonlinear integral operator (1) acts from \(\mathscr L_p\) into \(\mathscr L_q\), is differentiable at the point \(x_0\), and
\[ A'(x_0)h=\int_{\Omega_1} K_0(t,s)h(s)\,ds. \tag{11} \]
Operator (10) acts from \(\mathscr L_p\) into \(B_{p,q}\) and is continuous if \(p>1,\ q<\infty\) and if it acts from \(N_p\) into \(B_{p,q}\). In particular, the latter is ful-
when the inequality holds
\[ |G_0(t,s,u)-K_0(t,s)|\leqslant \sum_{j=1}^{m} S_j(t,s)|u|^{\delta_j}, \tag{12} \]
where \(0=\delta_0<\delta_1<\cdots<\delta_m\leqslant p-1\), \(S_0(t,s)\in B^0_{p,q}\) and \(S_j(t,s)\in B_{\frac{p}{1+\delta_j},q}\) \((j=1,\ldots,m)\).
- We give one theorem on the differentiability of the operator (1) on dense sets. Suppose, as in Sec. 1, that the function \(K(t,s,u)\), for almost all \(\{t,s\}\in\Omega_2\times\Omega_1\), has a derivative \(K'_u(t,s,u)\) continuous in \(u\).
Theorem 6. Let \(1<p<p_0\leqslant\infty\), \(0<q<\infty\). Let \(A\theta\in\mathscr L_q\) and let the superposition operator
\[ \mathfrak H(u,v)=H[t,s,u(t,s),v(t,s)], \tag{13} \]
where \(H(t,s,u,v)=\dfrac{1}{1+|v|}\{K(t,s,u+v)-K(t,s,u)\}\), act from \(N_{p_0}\times N_p\) into \(B^0_{p,q}\). Suppose the operator (5) acts from \(N_{p_0}\) into \(B^0_{p,q}\).
Then the nonlinear integral operator (1) acts from \(\mathscr L_p\) into \(\mathscr L_q\), is continuous, differentiable at every point of \(\mathscr L_{p_0}\), and
\[ A'(x_0)h=\int_{\Omega_1} K'_u[t,s,x_0(s)]\,h(s)\,ds \quad (x_0\in\mathscr L_{p_0},\ h\in\mathscr L_p). \tag{14} \]
Suppose, for example, that \(K(t,s,u)\) and \(K'_u(t,s,u)\) satisfy the inequalities
\[ |K(t,s,u)-K(t,s,0)|\leqslant \sum_{i=1}^{n} R_i(t,s)|u|^{k_i}, \tag{15} \]
\[ |K'_u(t,s,u)|\leqslant \sum_{j=1}^{m} S_j(t,s)|u|^{l_j}. \tag{16} \]
Then the conditions of Theorem 6 are satisfied if:
\(1^\circ.\) \(1<p<p_0\leqslant\infty\), \(0<q<\infty\), \(A\theta\in\mathscr L_q\).
\(2^\circ.\) For each \(i=1,\ldots,n\), one of the following relations holds:
a) \(p_0<\infty\), \(k_i=0\), and \(R_i(t,s)\in B^0_{p,q}\);
b) \(p_0<\infty\), \(0<k_i<\dfrac{p_0}{p_0-p}\), and
\[ R_i(t,s)\in B_{\frac{pp_0}{pk_i+p_0},q}; \]
c) \(p_0<\infty\), \(\dfrac{p_0}{p_0-p}\leqslant k_i\), and \(R_i(t,s)\in B_{\frac{p}{k_i},q}\);
d) \(p_0=\infty\), \(0\leqslant k_i\leqslant 1\), and \(R_i(t,s)\in B^0_{p,q}\);
e) \(p_0=\infty\), \(1<k_i\), and \(R_i(t,s)\in B_{\frac{p}{k_i},q}\).
\(3^\circ.\) For each \(j=1,\ldots,m\), one of the following relations holds:
a) \(p_0<\infty\), \(l_j=0\), and \(S_j(t,s)\in B^0_{p,q}\);
b) \(p_0<\infty\), \(l_j>0\), and \(S_j(t,s)\in B_{\frac{pp_0}{pl_j+p_0},q}\);
c) \(p_0=\infty\), and \(S_j(t,s)\in B^0_{p,q}\).
- For the nonlinear integral operator
\[ Ax(t)=\int_{\Omega_1} K(t,s)\,f[s,x(s)]\,ds \tag{17} \]
the assertions of Secs. 1–3 can be strengthened, since it can be represented in the form of a superposition of the nonlinear operator \(fx=f[s,x(s)]\) and a linear integral operator. The differentiability conditions for the superposition operator \(f\) have been studied in detail in (2). We give here two further assertions.
Theorem 7. Let \(1\leqslant r<p\leqslant\infty\), \(x_0\in\mathscr L_p\), \(fx_0\in\mathscr L_r\), and suppose there exists a function
\[ g(s)=\lim_{u\to 0}\frac{1}{u}\{f[s,x_0(s)+u]-f[s,x_0(s)]\},\quad g(s)\in\mathscr L_{\frac{pr}{p-r}}. \]
Suppose
\[ |f[s,x_0(s)+u]-f[s,x_0(s)]-g(s)u|\leqslant c(s)|u|^k\quad (|u|\leqslant u_0), \]
where \(1 \leq k < \infty,\ c(s)\in \mathscr L_\mu\) and
\[ |f[s,x_0(s)+u]-f[s,x_0(s)]| \leq \sum_{i=1}^{n} c_i(s)|u|^{\gamma_i}+b|u|^{p/r} \qquad (-\infty<u<\infty), \]
where \(0\leq \gamma_0<\gamma_1<\cdots<\gamma_n\leq 1,\ c_i(s)\in \mathscr L_{\mu_i}\). Suppose that for each \(i=1,\ldots,n\)
\[ \frac1r-\frac1p\geq \frac1{\mu_i},\qquad \frac1r-\frac1p\geq \frac{k-1}{k-\gamma_i}\frac1{\mu_i} +\frac{1-\gamma_i}{k-\gamma_i}\frac1\mu . \]
Then the operator \(fx=f[s,x(s)]\) maps \(\mathscr L_p\) into \(\mathscr L_r\), is differentiable at the point \(x_0\), and
\(f'(x_0)h=g(s)h(s)\) \((h\in \mathscr L_p)\).
Theorem 8. Let \(1\leq r<p<p_0\leq\infty\). Suppose that for almost all \(s\in\Omega_1\) the function \(f(s,u)\) has a derivative \(f'_u(s,u)\) continuous in \(u\), and that the superposition operator
\(\mathfrak G x=f'_u[s,x(s)]\) maps \(\mathscr L_{p_0}\) into \(\mathscr L_{\frac{pr}{p-r}}\).
Suppose that \(f\theta\in \mathscr L_q\) and that one of the following conditions is satisfied:
a) \(p_0<\infty\) and
\[ \frac{|f(s,u+v)-f(s,u)|}{1+|v|} \leq a(s)+b_1|u|^{p_0\frac{p-r}{pr}}+b_2|v|^{\frac{p-r}{r}}, \]
where \(a(s)\in \mathscr L_{\frac{pr}{p-r}}\), and \(b_1,b_2\) are constants;
b) \(p_0=\infty\) and for every \(R>0\)
\[ \frac{|f(s,u+v)-f(s,u)|}{1+|v|} \leq a_R(s)+b_R|v|^{\frac{p-r}{r}} \quad (|u|\leq R), \]
where \(a_R(s)\in \mathscr L_{\frac{pr}{p-r}}\) and \(b_R\) is a constant.
Then the operator \(fx=f[s,x(s)]\) maps \(\mathscr L_p\) into \(\mathscr L_r\), is continuous, and is differentiable at every point of the space \(\mathscr L_{p_0}\):
\[
f'(x_0)h=\mathfrak G x_0(s)\cdot h(s)
\]
\((x_0\in \mathscr L_{p_0},\ h\in \mathscr L_p)\).
Suppose, for example, that the function \(f(s,u)\) satisfies the inequality
\[ |f(s,u)-f(s,0)| \leq \sum_{i=1}^{n} c_i(s)|u|^{k_i}, \tag{18} \]
where \(0\leq k_0<k_1<\cdots<k_n,\ c_i(s)\in \mathscr L_{\mu_i}\). Then the conditions of Theorem 8 are fulfilled if
\(1\leq r<p<p_0\leq\infty,\ f(s,0)\in \mathscr L_r\), for each \(i=1,\ldots,n\),
\[ \frac1{\mu_i}+\frac{k_i}{p}\leq \frac1r,\qquad \frac1{\mu_i}+\frac{k_i}{p_0}\leq \frac1r-\frac1p \]
and if the operator
\(\mathfrak G x=f'_u[s,x(s)]\) maps \(\mathscr L_{p_0}\) into
\(\mathscr L_{\frac{pr}{p-r}}\).
- Theorems of §§ 1–4 are easily carried over to nonlinear integral operators acting in spaces of vector-functions.
Above, derivatives of nonlinear operators in the Fréchet sense were studied. Assertions analogous to Theorems 1–8 can also be proved for Gateaux derivatives.
The author expresses his gratitude to his supervisor M. A. Krasnosel’skii. Some results of this paper were obtained by the author jointly with M. A. Krasnosel’skii (7).
Voronezh State
University
Received
26 V 1965
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