APPLICATION OF THE THEOREM ON DIFFERENTIAL INEQUALITIES TO THE GOURSAT PROBLEM FOR A LINEAR SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS

Yu. I. Kovach

Consider the system of differential equations

\[ \frac{\partial^{2n_i} U_i(x,y)}{\partial x^{n_i}\partial y^{n_i}} = \sum_{j=1}^{m} \left( a_{j00}^{(i)} U_j + a_{j10}^{(i)} U_{jx} +\cdots+ a_{jk_j\tau_j}^{(i)} \frac{\partial^{2n_j-1}U_j}{\partial x^{k_j}\partial y^{\tau_j}} \right) + \]

\[ + f_i(x,y) = L_i[U_1,U_2,\ldots,U_m]+f_i(x,y) \qquad (i=1,2,\ldots,m) \tag{1} \]

with data on the characteristics \(x=x_0,\ y=y_0\):

\[ \frac{\partial^{2k-2}U_i(x_0,y)}{\partial x^{k-1}\partial y^{k-1}} = \sigma_{ik}(y), \qquad \frac{\partial^{2k-2}U_i(x,y_0)}{\partial x^{k-1}\partial y^{k-1}} = \omega_{ik}(x), \]

\[ \sigma_{ik}(y_0)=\omega_{ik}(x_0) \qquad (k=1,2,\ldots,n_i). \tag{2} \]

\(L_i\) is a linear operator. We assume that the functions \(\sigma_{ik}(y)\) are continuously differentiable with respect to \(y\) \(n_i+1-k\) times, and \(\omega_{ik}(x)\) are continuously differentiable with respect to \(x\) \(n_i+1-k\) times.

Denote by \(\overline D\) a certain closed domain of the variables \(x,y,U_j,U_{jx},\ldots,\dfrac{\partial^{2n_j-1}U_j}{\partial x^{k_j}\partial y^{\tau_j}}\), whose projection onto the \(xoy\)-plane gives the domain \(\overline R\):

\[ \overline R=\{x_0\le x\le x_0+\alpha;\quad y_0\le y\le y_0+\beta;\quad \alpha>0,\ \beta>0\};\qquad \overline R\subset \overline D. \]

Let, in the domain \(\overline R\), the functions \(f_i(x,y)\), as well as the coefficients of system (1), \(a_{jk_j\tau_j}^{(i)}(x,y)\), be continuous functions, and in \(\overline R\) let the coefficients not take negative values, i.e.

\[ a_{jk_j\tau_j}^{(i)}(x,y)\ge 0. \tag{3} \]

We assume that in (3), in each equation of the right-hand side of system (1), a strict inequality is preserved everywhere in \(\overline R\) for at least one coefficient. Under these assumptions, the right-hand sides of the system of equations (1) will be known continuous functions in the prescribed closed domain \(\overline D\). Denote:

\[ M_i=\inf_{\overline D}\{L_i[U_1,U_2,\ldots,U_m]+f_i(x,y)\}-1, \]

\[ \widetilde M_i=\sup_D\{L_i[U_1,U_2,\ldots,U_m]+f_i(x,y)\}+1. \]

For the Goursat problem (1), (2) the following is valid.

Theorem 1. Suppose that for \((x,y)\in R\) systems of continuous functions \(V_i(x,y)\), \((Z_i(x,y))\) have been found such that:

a) they are continuous and have continuous partial derivatives of all those orders that the system (1) contains;

b) these functions satisfy the initial conditions (2);

c) the result of substituting them into the system (1) gives, in the closed domain \(\bar R\), residuals \(\gamma_i(x,y)<0\) \((\tilde\gamma_i(x,y)>0)\). Then for \((x,y)\in R\) the inequalities hold:
\[ V_i(x,y)<U_i(x,y)<Z_i(x,y),\ldots,\quad \frac{\partial^{r_i}V_i(x,y)}{\partial x^{s_i}\partial y^{m_i}} < \frac{\partial^{r_i}U_i(x,y)}{\partial x^{s_i}\partial y^{m_i}} < \]
\[ < \frac{\partial^{r_i}Z_i(x,y)}{\partial x^{s_i}\partial y^{m_i}} \quad (s_i+m_i=r_i\le 2n_i;\ s_i,m_i=0,1,2,\ldots,n_i). \tag{4} \]

The existence of the functions \(V_i(x,y)\), \((Z_i(x,y))\) for \((x,y)\in R_1,\ R_1\Subset R\) follows from the system of equations
\[ \frac{\partial^{2n_i}V_i(x,y)}{\partial x^{n_i}\partial y^{n_i}} =M_i\left( \frac{\partial^{2n_i}Z_i(x,y)}{\partial x^{n_i}\partial y^{n_i}} =\widetilde M_i \right) \tag{5} \]
with the initial conditions (2), since
\[ \frac{\partial^{2n_i}V_i(x,y)}{\partial x^{n_i}\partial y^{n_i}} =L_i[V_1,\ldots,V_m]+f_i(x,y)+\gamma_i(x,y)=M_i, \]
\[ \gamma_i(x,y)=M_i-\{L_i[V_1,\ldots,V_m]+f_i(x,y)\}<0. \]

The solution of the system (5) with the initial conditions (2) is given in [1] by the following formulas:
\[ V_i(x,y)=\omega_{i1}(x)+\sigma_{i1}(y)-\sigma_{i1}(y_0) -\sum_{k=2}^{n_i} \frac{(x-x_0)^{k-1}}{(k-1)!}\, \frac{(y-y_0)^{k-1}}{(k-1)!}\, \sigma_{ik}(y_0)+ \]
\[ +\sum_{k=2}^{n_i}\left[ \frac{(y-y_0)^{k-1}}{(k-1)!} \int_{x_0}^{x}\frac{(x-t)^{k-2}}{(k-2)!}\,\omega_{ik}(t)\,dt +\right. \]
\[ \left. +\frac{(x-x_0)^{k-1}}{(k-1)!} \int_{y_0}^{y}\frac{(y-\tau)^{k-2}}{(k-2)!}\,\sigma_{ik}(\tau)\,d\tau \right]+ \tag{6} \]
\[ +\int_{x_0}^{x}dt\int_{y_0}^{y} \frac{(x-t)^{n_i-1}}{(n_i-1)!}\, \frac{(y-\tau)^{n_i-1}}{(n_i-1)!}\, M_i\,d\tau =F_i(x,y)+ \]
\[ +\int_{x_0}^{x}dt\int_{y_0}^{y}K_{n_i}(x,y,t,\tau)M_i\,d\tau, \]
where
\[ K_{n_i}(x,y,t,\tau)= \frac{(x-t)^{n_i-1}}{(n_i-1)!}\, \frac{(y-\tau)^{n_i-1}}{(n_i-1)!} \le \frac{(\alpha,\beta)^{\,n_i-1}}{[(n_i-1)!]^2}. \]

We divide the proof of the theorem into three parts.

1. We shall prove that one can construct sequences \(\{V_{ip}\}\), \(\{Z_{ip}\}\) such that everywhere in \(R\) one has \(V_{ip}<V_{ip+1}\) \((Z_{ip}>Z_{ip+1})\).

Denote by \(V_i(x,y)\), \((Z_i(x,y))\), the functions corresponding to the residuals \(\gamma_i(x,y)<0\) \((\tilde\gamma_i(x,y)>0)\), \(V_i=V_{i1}\) \((Z_i=Z_{i1})\), and define the sequences of functions \(\{V_{ip}\}\), \(\{Z_{ip}\}\) from the system

\[ \frac{\partial^{2n_i} V_{ip+1}}{\partial x^{n_i}\partial y^{n_i}} = L_i[V_{1p},\ldots,V_{mp}]+f_i(x,y)=\varphi_i(x,y), \tag{7} \]

\[ \frac{\partial^{2n_i} Z_{ip+1}}{\partial x^{n_i}\partial y^{n_i}} = Z_i[Z_{1p},\ldots,Z_{mp}]+f_i(x,y)=\psi_i(x,y) \]

under the initial conditions (2) on the characteristics. The solution of system (7) under the initial conditions (2) is given by the formulas [1]:

\[ V_{ip+1}=F_i(x,y)+\int_{x_0}^{x} dt \int_{y_0}^{y} K_{n_i}(x,y,t,\tau)\varphi_i(t,\tau)\,d\tau, \]

\[ Z_{ip+1}=F_i(x,y)+\int_{x_0}^{x} dt \int_{y_0}^{y} K_{n_i}(x,y,t,\tau)\psi_i(t,\tau)\,d\tau, \tag{8} \]

where \(\varphi_i(x,y)\), \(\psi_i(x,y)\) are known functions.

We shall prove the validity of the inequalities \(V_{ip}<V_{ip+1}\) \((Z_{ip}>Z_{ip+1})\) only for \(p=1,2\).

We have

\[ \frac{\partial^{2n_i} V_{i1}(x,y)}{\partial x^{n_i}\partial y^{n_i}} = L_i[V_{11},\ldots,V_{m1}]+f_i(x,y)+\gamma_i(x,y), \tag{9} \]

where \(\gamma_i(x,y)<0\) in \(\overline{R}\). From system (9) and system (7) with \(p=1\), we obtain

\[ \frac{\partial^{2n_i}(V_{i2}-V_{i1})}{\partial x^{n_i}\partial y^{n_i}} = -\gamma_i(x,y)>0. \tag{10} \]

Multiplying the last inequality by \(dx>0\), we have

\[ \frac{\partial^{2n_i}(V_{i2}-V_{i1})}{\partial x^{n_i}\partial y^{n_i}}\,dx = \frac{\partial}{\partial x} \left[ \frac{\partial^{2n_i-1}(V_{i2}-V_{i1})}{\partial x^{n_i-1}\partial y^{n_i}} \right]dx = \]

\[ = d_x \left[ \frac{\partial^{2n_i-1}(V_{i2}-V_{i1})}{\partial x^{n_i-1}\partial y^{n_i}} \right]>0. \]

Integrating this inequality from \(x_0\) to \(x\), we obtain

\[ \frac{\partial^{2n_i-1}(V_{i2}-V_{i1})}{\partial x^{n_i-1}\partial y^{n_i}} - \left[ \frac{\partial^{2n_i-1}(V_{i2}-V_{i1})}{\partial x^{n_i-1}\partial y^{n_i}} \right]_{x=x_0} >0. \]

From the initial conditions (2) it follows that

\[ \left[ \frac{\partial^{2n_i-2}(V_{i2}-V_{i1})}{\partial x^{n_i-1}\partial y^{n_i-1}} \right]_{x=x_0} =0, \]

and therefore

\[ \left[ \frac{\partial^{2n_i-1}(V_{i2}-V_{i1})}{\partial x^{\,n_i-1}\partial y^{\,n_i}} \right]_{x=x_0}=0, \]

hence:

\[ \frac{\partial^{2n_i-1}V_{i2}}{\partial x^{\,n_i-1}\partial y^{\,n_i}} > \frac{\partial^{2n_i-1}V_{i1}}{\partial x^{\,n_i-1}\partial y^{\,n_i}} \tag{11} \]

for \(x>x_0,\ y>y_0\).

Similarly, from inequality (10) we obtain

\[ \frac{\partial^{2n_i-1}V_{i2}}{\partial x^{\,n_i}\partial y^{\,n_i-1}} > \frac{\partial^{2n_i-1}V_{i1}}{\partial x^{\,n_i}\partial y^{\,n_i-1}} \tag{12} \]

for \(x>x_0,\ y>y_0\). Starting from inequalities (11) and (12) and continuing this process further, we verify the validity of the inequalities:

\[ V_{i2}>V_{i1},\quad \frac{\partial V_{i2}}{\partial x}>\frac{\partial V_{i1}}{\partial x},\ \ldots,\ \frac{\partial^{r_i}V_{i2}}{\partial x^{s_i}\partial y^{m_i}} > \frac{\partial^{r_i}V_{i1}}{\partial x^{s_i}\partial y^{m_i}} \]

\[ (s_i+m_i=r_i\le 2n_i;\ s_i,\ m_i=0,\ 1,\ 2,\ldots,\ n_i;\ i=1,\ 2,\ldots,\ m) \]

for \(x>x_0,\ y>y_0\).

Substituting \(p=1,\ p=2\) into system (7), we obtain

\[ \frac{\partial^{2n_i}(V_{i3}-V_{i2})}{\partial x^{n_i}\partial y^{n_i}} = L_i[V_{12},\ldots,V_{m2}]-L_i[V_{11},\ldots,V_{m1}] = \]

\[ = \sum_{j=1}^{m} \left\{ a_{j00}^{(i)}(V_{j2}-V_{j1}) + a_{j10}^{(i)} \left( \frac{\partial V_{j2}}{\partial x} - \frac{\partial V_{j1}}{\partial x} \right) +\ldots+ \tag{13} \]

\[ + a_{jk_j\tau_j}^{(i)} \left( \frac{\partial^{2n_j-1}V_{j2}}{\partial x^{k_j}\partial y^{\tau_j}} - \frac{\partial^{2n_j-1}V_{j1}}{\partial x^{k_j}\partial y^{\tau_j}} \right) \right\}>0. \]

Arguing in an analogous manner, from inequality (13) we obtain the inequalities:

\[ V_{i3}>V_{i2},\quad \frac{\partial V_{i3}}{\partial x}>\frac{\partial V_{i2}}{\partial x},\ \ldots,\ \frac{\partial^{r_i}V_{i3}}{\partial x^{s_i}\partial y^{m_i}} > \frac{\partial^{r_i}V_{i2}}{\partial x^{s_i}\partial y^{m_i}} \]

\[ (s_i+m_i=r_i\le 2n_i;\ s_i,\ m_i=0,\ 1,\ 2,\ldots,\ n_i;\ i=1,\ 2,\ldots,\ m). \]

1. It remains to prove that the mentioned sequences of functions
   \(\{V_{ip}\}, \{V_{ipx}\}, \{V_{ipy}\}, \ldots\) tend uniformly, as \(p\) increases, to the corresponding functions
   \(U_i,\ U_{ix},\ U_{iy},\ldots\).

For this purpose we make the following estimates. From expression (10) we obtain

\[ \frac{\partial^{2n_i}(V_{i2}-V_{i1})}{\partial x^{n_i}\partial y^{n_i}} = -\gamma_i(x,y)\le P\quad (P>0). \tag{14} \]

Integrating inequality (14) and taking into account the initial data on the characteristics, we obtain the estimates

\[ \frac{\partial^{2n_i-1}(V_{i2}-V_{i1})}{\partial x^{n_i-1}\partial y^{n_i}} \leq P(x-x_0+y-y_0); \]

\[ \frac{\partial^{2n_i-1}(V_{i2}-V_{i1})}{\partial x^{n_i}\partial y^{n_i-1}} \leq P(x-x_0+y-y_0), \ldots, \]

\[ \frac{\partial^{r_i}(V_{i2}-V_{i1})}{\partial x^{k_i}\partial y^{\tau_i}} \leq P\,\frac{(x-x_0+y-y_0)^{\,2n_i-k_i-\tau_i}}{(2n_i-k_i-\tau_i)!}, \ldots, \]

\[ V_{i2}-V_{i1} \leq P\,\frac{(x-x_0+y-y_0)^{2n_i}}{(2n_i)!}. \]

Indeed, multiplying inequality (14) by \(dx>0\), we obtain

\[ \frac{\partial^{2n_i}(V_{i2}-V_{i1})}{\partial x^{n_i}\partial y^{n_i}}\,dx = \frac{\partial}{\partial x} \left[ \frac{\partial^{2n_i-1}(V_{i2}-V_{i1})}{\partial x^{n_i-1}\partial y^{n_i}} \right]dx = \]

\[ = d_x \left[ \frac{\partial^{2n_i-1}(V_{i2}-V_{i1})}{\partial x^{n_i-1}\partial y^{n_i}} \right] \leq Pdx. \]

Whence

\[ \frac{\partial^{2n_i-1}(V_{i2}-V_{i1})}{\partial x^{n_i-1}\partial y^{n_i}} \leq P(x-x_0)\leq P(x-x_0+y-y_0). \]

Similarly

\[ d_x \left[ \frac{\partial^{2n_i-2}(V_{i2}-V_{i1})}{\partial x^{n_i-2}\partial y^{n_i}} \right] \leq P(x-x_0+y-y_0)\,dx = \]

\[ = P(x-x_0+y-y_0)\,d_x(x-x_0+y-y_0). \]

Integrating this inequality from \(x\) to \(x_0\) \((x>x_0,\ y>y_0)\) and taking into account the initial data on the characteristics, we obtain

\[ \frac{\partial^{2n_i-2}(V_{i2}-V_{i1})}{\partial x^{n_i-2}\partial y^{n_i}} \leq P\frac{(x-x_0+y-y_0)^2}{2!} - \]

\[ - P\frac{(y-y_0)^2}{2!} \leq P\frac{(x-x_0+y-y_0)^2}{2!}. \]

Continuing this process further, we obtain the estimates indicated above. Denote

\[ \max\left[1,\ \max_R (x-x_0+y-y_0)^r\right] = E \quad (r=1,2,\ldots,2n_i-1). \]

Then the preceding inequalities take the form:

\[ V_{i2}-V_{i1}<PE(x-x_0+y-y_0),\quad \frac{\partial V_{i2}}{\partial x}-\frac{\partial V_{i1}}{\partial x} <PE(x-x_0+y-y_0), \]

\[ \frac{\partial V_{i2}}{\partial y}-\frac{\partial V_{i1}}{\partial y} <PE(x-x_0+y-y_0),\ \ldots \]

\[ \ldots,\ \frac{\partial^{r_i}V_{i2}}{\partial x^{s_i}\partial y^{m_i}} - \frac{\partial^{r_i}V_{i1}}{\partial x^{s_i}\partial y^{m_i}} <PE(x-x_0+y-y_0) \]

\[ (s_i+m_i=r_i\le 2n_i;\quad s_i,\ m_i=0,\ 1,\ 2,\ \ldots,\ n_i;\ i=1,\ 2,\ \ldots,\ m). \]

Consider inequality (13). Denote

\[ T=\max\left\{\sup_R a^{(i)}_{j k_i \tau_j}\right\}\quad (T>0). \]

Then from inequality (13) we obtain the inequalities

\[ \frac{\partial^{2n_i}(V_{i3}-V_{i2})}{\partial x^{n_i}\partial y^{n_i}} <CPET(x-x_0+y-y_0), \tag{15} \]

where \(C\) is the number of coefficients \(a^{(i)}_{j k_i \tau_j}\) in the right-hand side of the system of differential equations (1).

In an analogous way, from (15) we obtain

\[ V_{i3}-V_{i2}<CPE^2T\,\frac{(x-x_0+y-y_0)^2}{2!},\ \ldots \]

\[ \ldots,\ \frac{\partial^{r_i}V_{i3}}{\partial x^{s_i}\partial y^{m_i}} - \frac{\partial^{r_i}V_{i2}}{\partial x^{s_i}\partial y^{m_i}} <CPE^2T\,\frac{(x-x_0+y-y_0)^2}{2!}. \]

From system (7), for \(p=2,\ p=3\), we obtain

\[ \frac{\partial^{2n_i}(V_{i4}-V_{i3})}{\partial x^{n_i}\partial y^{n_i}} <PN^2\,\frac{(x-x_0+y-y_0)^2}{2!},\quad N=CET. \tag{16} \]

Continuing this method further and applying the method of mathematical induction, we are convinced of the validity of the inequalities

\[ V_{ip+1}-V_{ip}<\frac{A^p}{p!},\quad \frac{\partial V_{ip+1}}{\partial x}-\frac{\partial V_{ip}}{\partial x}<\frac{A^p}{p!},\ \ldots \]

\[ \ldots,\ \frac{\partial^{r_i}V_{ip+1}}{\partial x^{s_i}\partial y^{m_i}} - \frac{\partial^{r_i}V_{ip}}{\partial x^{s_i}\partial y^{m_i}} <\frac{A^p}{p!}, \tag{17} \]

where \(A\) is some positive constant.

Now consider the following series:

\[ \tilde U_i=V_{i1}+(V_{i2}-V_{i1})+\ldots+(V_{ip}-V_{ip-1})+\ldots, \tag{18_1} \]

\[ \tilde U_{ix}=V_{i1x}+(V_{i2x}-V_{i1x})+\ldots+(V_{ipx}-V_{ip-1x})+\ldots \tag{18_2} \]

Denote the sums of the series \((18_{r_i})\) respectively by \(\tilde U_i\), \(\tilde U_{ix}\), \(\tilde U_{iy}\), \(\tilde U_{ixx},\ldots\), and consider the series \((18_{r_i})\) \((r_i \le C+1)\). The terms of the series \((18_{r_i})\) for \((x,y)\in R\) are continuous functions. From inequalities (17) it follows that the series \((18_{r_i})\) converges absolutely and uniformly in the domain \(R\). Hence there follows the existence and continuity of the sum of the series \((18_{r_i})\), and also the fact that this sum satisfies the initial conditions (2). It remains to prove that the functions \(\tilde U_i\) obtained in this way satisfy the system of differential equations (1). From system (7), with initial conditions (2), according to (8), we obtain

\[ V_{ip+1}=F_i(x,y)+\int_{x_0}^{x}dt\int_{y_0}^{y}K_{n_i}(x,y,t,\tau)\{L_i[V_{1p},\ldots,V_{mp}]+ \tag{19} \]

\[ +f_i(t,\tau)\}\,d\tau . \]

Using inequalities (17), we pass in the preceding formula to the limit as \(p\to\infty\). The left-hand side in formula (19) tends uniformly to \(\tilde U_i\), and

\[ \iint_R K_{n_i}(x,y,t,\tau)L_i[V_{1p},\ldots,V_{mp}]\,dt\,d\tau \]

tends uniformly to

\[ \iint_R K_{n_i}(x,y,t,\tau)L_i[\tilde U_1,\ldots,\tilde U_m]\,dt\,d\tau . \]

Thus, passing to the limit in (19), we obtain

\[ \tilde U_i(x,y)=F_i(x,y)+\int_{x_0}^{x}dt\int_{y_0}^{y}K_{n_i}(x,y,t,\tau)\{L_i[\tilde U_1,\ldots,\tilde U_m]+ \]

\[ +f_i(t,\tau)\}\,d\tau . \tag{20} \]

Differentiating the left- and right-hand sides of formula (20), we obtain

\[ \frac{\partial^{2n_i}\tilde U_i}{\partial x^{n_i}\partial y^{n_i}} = L_i[\tilde U_1,\ldots,\tilde U_m]+f_i(x,y), \]

therefore: \(\tilde U_i=U\).

1. We shall prove that \(V_{ip}\to U_i\), while always remaining smaller, i.e. everywhere in \(R\), \(U_i-V_{ip}>0\).

Suppose the contrary. Let for some number \(p\), \(V_{ip}>U_i\). Then as \(k\to\infty\) for the sequence \(V_{ip+k}\) we shall have \(V_{ip+k}>U_i\). In item 1 it was proved that \(V_{ip+1}>V_{ip}\), and therefore from the inequalities \(V_{ip+k}>U_i\) it follows that the series \((18_{r_i})\) at any point of \(R\) does not converge to \(U_i\), and this contradicts what was proved in item 2.

Similarly, if one assumes that \(V_{ip}\) intersect with \(U_i\), i.e. at some points of the domain \(R\) we have \(V_{ip}<U_i\), and at others \(V_{ip}>U_i\), then at those points of \(R\) at which \(V_{ip}>U_i\), in view of the inequalities \(V_{ip+1}>V_{ip}\), we obtain that \(V_{ip}\) at these points will not converge to the solution of system (1).

Consequently, if the conditions of the theorem are fulfilled, then everywhere for \((x,y)\in R\), \(V_{i1}<U_i\) for negative residual \(\gamma_i(x,y)\).

The theorem formulated is also valid for a system of linear differential equations of the form (1), when the number of independent variables in the functions \(U_i(x,y)\) is greater than two.

The method set forth makes it possible to enclose the solution of the Goursat problem (1), (2) in a “fork,” and then to narrow this “fork,” which may be used in engineering calculations.

Remark 1. In the course of the proof of Theorem 1, an assertion was proved which can be formulated as the following theorem:

Theorem 2. If the functions \(V_i=V_{i1}\), \(Z_i=Z_{i1}\) satisfy the conditions of Theorem 1, then the sequences of functions \(\{V_{ik}\}\), \(\{Z_{ik}\}\), which are defined from system (7) under the initial conditions (2), for \((x,y)\in R\), satisfy the inequalities:

\[ V_{i1}<V_{i2}<V_{i3}<\cdots<U_i<\cdots<Z_{i3}<Z_{i2}<Z_{i1}, \]

\[ \frac{\partial V_{i1}}{\partial x}<\frac{\partial V_{i2}}{\partial x}<\frac{\partial V_{i3}}{\partial x}<\cdots< \frac{\partial U_i}{\partial x}<\cdots< \frac{\partial Z_{i3}}{\partial x}<\frac{\partial Z_{i2}}{\partial x}<\frac{\partial Z_{i1}}{\partial x}, \]

\[ \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot \]

\[ \frac{\partial^{r_i}V_{i1}}{\partial x^{s_i}\partial y^{m_i}} < \frac{\partial^{r_i}V_{i2}}{\partial x^{s_i}\partial y^{m_i}} < \cdots < \frac{\partial^{r_i}U_i}{\partial x^{s_i}\partial y^{m_i}} < \cdots < \frac{\partial^{r_i}Z_{i2}}{\partial x^{s_i}\partial y^{m_i}} < \frac{\partial^{r_i}Z_{i1}}{\partial x^{s_i}\partial y^{m_i}}. \]

\[ (s_i+m_i=r_i\leq 2n_i;\quad s_i,\ m_i=0,\ 1,\ 2,\ldots,n_i). \]

The theorem on differential inequalities proved above also has theoretical applications. As an example, let us consider the Goursat problem for the equation

\[ F[U]\equiv F\left(x,\ y,\ U,\ U_x,\ U_y,\ \ldots,\ \frac{\partial^{r}U}{\partial x^{s}\partial y^{m}},\ \ldots,\ \frac{\partial^{2n}U}{\partial x^{n}\partial y^{n}}\right)=0 \tag{21} \]

\[ (s+m=r<2n;\ s,\ m=0,\ 1,\ 2,\ldots,n) \]

with data on the characteristics \(x=x_0,\ y=y_0\):

\[ \frac{\partial^{2k-2}U(x_0,y)}{\partial x^{k-1}\partial y^{k-1}}=\sigma_k(y),\qquad \frac{\partial^{2k-2}U(x,y_0)}{\partial x^{k-1}\partial y^{k-1}}=\omega_k(x), \tag{22} \]

\[ \sigma_k(y_0)=\omega_k(x_0)\quad (k=1,\ 2,\ldots,n). \]

Denote

\[ p_{nn}=\frac{\partial^{2n}U}{\partial x^{n}\partial y^{n}}. \]

We assume that, in some domain \(D\), the continuous derivatives of the function \(F\) satisfy the conditions:

\[ \text{a)}\quad \frac{\partial F}{\partial p_{nn}}>0, \]

\[ \text{b)}\quad \frac{\partial F}{\partial p_{sm}}\leq 0 \quad (s+m<2n;\ s,\ m=0,\ 1,\ 2,\ldots,n), \]

where in condition b) at least one strict inequality holds everywhere in \(D\).

We shall prove that, under these conditions, a theorem on differential inequalities is valid for the function \(F\), which in a certain sense generalizes the results of [2].

We have \(F[V]=\gamma(x,y)\), where \(\gamma(x,y)<0\) in \(\overline R\). Expanding \(F[U]-F[V]\) in a Taylor series, we obtain

\[ \begin{aligned} F[U]-F[V] &=\left.\frac{\partial F}{\partial U}\right|_{H}(U-V) +\left.\frac{\partial F}{\partial U_x}\right|_{H} \left(\frac{\partial U}{\partial x}-\frac{\partial V}{\partial x}\right)+\ldots+ \\ &\quad+\left.\frac{\partial F}{\partial p_{nn}}\right|_{H} \left(\frac{\partial^{2n}U}{\partial x^n\partial y^n} -\frac{\partial^{2n}V}{\partial x^n\partial y^n}\right) \\ &=P_{00}(U-V)+P_{10} \left(\frac{\partial U}{\partial x}-\frac{\partial V}{\partial x}\right) +P_{01}\left(\frac{\partial U}{\partial y}-\frac{\partial V}{\partial y}\right)+\ldots \\ &\quad+P_{nn}\left(\frac{\partial^{2n}U}{\partial x^n\partial y^n} -\frac{\partial^{2n}V}{\partial x^n\partial y^n}\right) =-\gamma(x,y). \end{aligned} \tag{23} \]

Denote \(U-V=W\). Since, by assumption,
\[ \left.\frac{\partial F}{\partial p_{nn}}\right|_{H}=P_{nn}>0, \]
from equation (23) we obtain

\[ \frac{\partial^{2n}W}{\partial x^n\partial y^n} +B_{00}W+B_{10}W_x+\ldots+B_{sm} \frac{\partial^r W}{\partial x^s\partial y^m} +\frac{\gamma(x,y)}{P_{nn}(x,y)}=0, \tag{24} \]

\[ B_{sm}=\frac{P_{sm}(x,y)}{P_{nn}(x,y)}\leq 0. \]

For equation (24) the theorem on the differential inequality has been proved; the initial conditions (22) for the function \(W\) are zero. Substitution of zero in place of the function \(W\) gives the residual \(\dfrac{\gamma}{P_{nn}}<0\). Consequently, the function identically equal to zero will be a lower function in comparison with the function \(W\), and we obtain \(W=U-V>0,\ U>V\).

It follows from theorem (2) that in \(D\) the inequalities hold:

\[ V(x,y)<U(x,y),\quad \frac{\partial V}{\partial x}<\frac{\partial U}{\partial x},\ldots,\quad \frac{\partial^r V}{\partial x^s\partial y^m} < \frac{\partial^r U}{\partial x^s\partial y^m}. \]

Denote by \(Z\) the function that satisfies the initial conditions (2) and corresponds to the residual \(\widetilde\gamma(x,y)>0\). It is proved analogously that the function \(Z(x,y)\) will be an upper function in comparison with the function \(U(x,y)\). Thus, the pair of functions \([V,Z]\) contains within it the solution of the Goursat problem (22), (23), and in the domain \(D\) we shall have

\[ V(x,y)<U(x,y)<Z(x,y),\ldots,\quad \frac{\partial^r V}{\partial x^s\partial y^m} < \frac{\partial^r U}{\partial x^s\partial y^m} < \frac{\partial^r Z}{\partial x^s\partial y^m} \]

\[ (s+m=r\leq 2n;\ s,m=0,1,2,\ldots,n). \]

Remark 2. In the works of Kim Menyam [3, 4], approximating sequences of functions for the Goursat problem (1), (2) in the case of a linear system of second-order differential equations are constructed by the formulas:

\[ Z_{in+1}(x,y)=Z_{in}(x,y)-\sigma_{in}(x,y), \]

where \(\sigma_{in}\) is a solution of the equation
\[ \sigma_{inxy}-m_i\sigma_{inx}-k_i\sigma_{iny}-r_i\sigma_{in}=a_{in} \]
with zero initial conditions on the characteristics \(x=x_0,\ y=y_0\), where

\[ \begin{aligned} a_{in} =\min\bigg[& Z_{inxy}-a_{ii}Z_{inx}-b_{ii}Z_{iny}-c_{ii}Z_{in} -\left(\sum_{j=1}^{i-1}+\sum_{j=i+1}^{n}\right) (a_{ij}p_j+b_{ij}q_j+ \\ &\quad+c_{ij}U_j)-f_i(x,y)\bigg]. \end{aligned} \]

However, the practical application of the Kim–Menham method is associated with great difficulties, especially when \(2n_i > 2\). In this sense, the proposed method for solving the Goursat problem when \(2n_i > 2\), by means of approximating functions according to formula (8), is much simpler and more convenient.

References

1. Kovach Yu. I. Reports and Communications of Uzhgorod State University, ser. phys.-math. and historical sciences, No. 5, 1962, pp. 98–101.
2. Kovach Yu. I. Abstracts of Reports and Communications of Uzhgorod State University, ser. phys.-math. sciences, 1964.
3. Kim Menham. Scientific Notes of Kabardino-Balkarian State University, ser. phys.-math., issue 17. Nalchik, 1963.
4. Kim Menham. UMN 15, issue 1, 1960, p. 239.

Received by the editors
October 19, 1964

Uzhgorod State University