UDC 517.946.9

ON THE CONSTRUCTION OF SMALL SOLUTIONS OF A CLASS OF NONLINEAR INTEGRAL EQUATIONS

P. I. GOLOVACHIK

The paper considers the problem of finding all continuous small solutions of nonlinear integral equations in a neighborhood of a branching point of solutions. The solution of this problem is closely connected with the solution of various physical problems. Results of the theory of branching of solutions of nonlinear integral equations are used in solving problems on equilibrium figures of a rotating fluid [1, 3], in the theory of motion of an incompressible heavy fluid [4, 5], in the theory of elasticity [7], and in the many-particle theory [8]. The solution of boundary-value problems for nonlinear differential equations [3, 6] is also reducible to the solution of nonlinear integral equations.

To find small solutions of nonlinear integral equations, two analytic methods are used: the Lyapunov–Schmidt method [1, 2] and the method of A. I. Nekrasov [4, 6].

In papers [10] and [11], the investigations of A. I. Nekrasov and N. N. Nazarov were continued and extended to nonlinear equations in Banach spaces. A further development of the Lyapunov–Schmidt method and its extension to nonlinear operator equations is given in papers [12, 13, 18, 19].

When the Lyapunov–Schmidt method is applied, the problem of finding all small solutions of nonlinear integral and operator equations is reduced to the problem of finding all small solutions of the corresponding branching equation. Some cases in which it is possible to find directly all small solutions of the one-dimensional Lyapunov–Schmidt branching equation were considered by L. Lichtenstein [3]. A complete solution of the problem of finding all small solutions of the one-dimensional Lyapunov–Schmidt branching equation is given in papers [13] and [17] with the aid of the Newton polygon [16]. In papers [9, 13] and [14], some methods were also indicated for finding small solutions of systems of analytic implicit functions. V. A. Trenogin, in paper [12], proposed a method making it possible to find all small solutions of nonlinear operator equations by applying the Newton polygon directly to the operator equation, without constructing the branching equation.

When the method of A. I. Nekrasov is used, solutions of nonlinear integral equations are sought in the form of series arranged in integral or fractional powers of a parameter. To find in this way all continuous small solutions, one must establish which fractional powers are possible for the given equation. In papers [13] and [17], this problem is solved by constructing the Lyapunov–Schmidt branching equation and studying it by the Newton polygon method.

In the present paper, the solution of the equation under consideration is sought in the form of a series arranged in arbitrary, not fixed, powers-

of a parameter, and for such a general series recurrent formulas are established that determine the desired solution. As a result of studying the formulas obtained, a simple algorithm is established that makes it possible to find all small solutions of the integral equation under consideration directly from A. I. Nekrasov’s method, without compiling and studying the Lyapunov—Schmidt branching equation.

Let us consider the nonlinear integral equation

\[ v(x)=\mu\int_0^1 A_{01}(x,y)\,dy+\int_0^1 A_{10}(x,y)v(y)\,dy+ \]

\[ +\int_0^1 \sum_{k+l\geq 2}\mu^l A_{kl}(x,y)v^k(y)\,dy, \tag{1} \]

where \(\mu\) is a parameter; \(v(x)\) is the unknown, and \(A_{kl}(x,y)\) are known functions continuous in the square \(0\leq x,\ y\leq 1\).

We shall seek a solution of equation (1) in the form of a series

\[ v(x)=\sum_{m=1}^{\infty} v_m(x)\mu^{\frac{m}{s}} =\sum_{m=1}^{\infty} v_m(x)\nu^m =\sum_{m=1}^{\infty} v_m(x,\nu), \tag{2} \]

where \(s\) is some natural number to be determined. Substituting this series into equation (1) and comparing terms of the same order in \(\mu\), we arrive at the recurrent system of linear integral equations

\[ v_m(x,\nu)=F_m(x,\nu)+\int_0^1 A_{10}(x,y)v_m(y,\nu)\,dy \qquad (m=1,\infty), \tag{3} \]

where

\[ F_m(x,\nu)=\nu^m F_m(x)= \int_0^1 \sum_{k+sl=2}^{m}\nu^{ls}A_{kl}(x,y)\times \]

\[ \times \sum_{k_1,\ldots,k_{m-1}}^{k,l} \frac{k!}{k_1!\cdots k_{m-1}!}\, v_1^{k_1}(y,\nu)\cdots v_{m-1}^{k_{m-1}}(y,\nu)\,dy \tag{4_m} \]

for \(m>1\), and

\[ F_1(x,\nu)= \begin{cases} \nu\displaystyle\int_0^1 A_{01}(x,y)\,dy, & \text{if } s=1,\\[1.2ex] 0, & \text{if } s>1, \end{cases} \tag{4_1} \]

and the summation is carried out over all such nonnegative integer values of the indices \(k_1,\ldots,k_{m-1}, k, l\) for which

\[ k_1+\cdots+k_{m-1}=k,\qquad k_1+\cdots+(m-1)k_{m-1}+sl=m. \tag{5} \]

If unity is an eigenvalue of the kernel \(A_{10}(x,y)\) of rank \(r=1\), and \(\varphi(x)\) and \(\psi(x)\) are eigenfunctions of the kernels \(A_{10}(x,y)\) and \(A_{10}(y,x)\), respectively, corresponding to this eigenvalue, then, under the condition

\[ \int_0^1 F_m(x)\psi(x)\,dx=0 \qquad (m=1,\infty) \tag{6} \]

the system (3) will have the solution

\[ v_m(x,\nu)=P_m(x,\nu)+\varphi(x)C_m(\nu) =\nu^m[P_m(x)+\varphi(x)C_m], \tag{7} \]

where

\[ P_m(x,\nu)=F_m(x,\nu)+\int_0^1 H(x,y)F_m(y,\nu)\,dy, \tag{8} \]

and \(H(x,y)\) is the generalized resolvent of the kernel \(A_{10}(x,y)\).

From \((4_m)\), (7), and (8) it follows that the functions \(F_m(x,\nu)\) contain products of powers of the quantities \(C_n(\nu)\) with indices \(n<m\). Grouping the terms of the functions \(F_m(x,\nu)\) according to these products and denoting by \(f_{pq}(x)\) the coefficients of the \(p\)-th powers of the quantities \(C_n(\nu)\) and the \(q\)-th powers of the parameter \(\mu\), we obtain

\[ F_m(x)= \sum_{p+qs=2}^{m} f_{pq}(x) \sum_{l_1,\ldots,l_{m-1}}^{p,q} \frac{p!}{l_1!,\ldots,l_{m-1}!} C_1^{l_1}\cdots C_{m-1}^{l_{m-1}} \tag{9_m} \]

for \(m>1\), and

\[ F_1(x)= \begin{cases} f_{01}(x) & \text{for } s=1,\\ 0 & \text{for } s>1. \end{cases} \tag{9_1} \]

The summation is carried out over all such nonnegative integer values of the quantities \(l_1,\ldots,l_{m-1},p,q\) for which

\[ l_1+\cdots+l_{m-1}=p,\qquad l_1+\cdots+(m-1)l_{m-1}+qs=m. \tag{10} \]

Using formulas (7), (8), and \((9_m)\), we find expressions for the functions \(v_m(x)\) in terms of the constants \(C_n\):

\[ v_m(x)=C_m\varphi(x)+ \sum_{p+qs=2}^{m} h_{pq}(x)\times \]

\[ \times \sum_{l_1,\ldots,l_{m-1}}^{p,q} \frac{p!}{l_1!,\ldots,l_{m-1}!} C_1^{l_1}\cdots C_{m-1}^{l_{m-1}} \qquad (m>1), \tag{11_m} \]

\[ v_1(x)= \begin{cases} C_1\varphi(x)+h_{01}(x) & \text{for } s=1,\\ C_1\varphi(x) & \text{for } s>1, \end{cases} \tag{11_1} \]

where

\[ h_{10}(x)=\varphi(x),\qquad h_{pq}(x)=f_{pq}(x)+\int_0^1 H(x,y)f_{pq}(y)\,dy. \tag{12} \]

And with the aid of formulas (6) and \((9_m)\) we obtain an analogous expression for the solvability condition of the recurrent system (3):

\[ \sum_{p+qs=2}^{m}\Psi_{p,q} \sum_{l_1,\ldots,l_{m-1}}^{p,q} \frac{p!}{l_1!\cdots l_{m-1}!} C_1^{l_1}\cdots C_{m-1}^{l_{m-1}}=0, \tag{13_m} \]

for all values of the quantity \(s\) when \(m>1\), and

\[ \Psi_{0,1}=0. \tag{13_1} \]

for \(m=1,\ s=1\). Here

\[ \Psi_{p,q}=\int_{0}^{1} f_{pq}(x)\psi(x)\,dx, \tag{14} \]

and the summation, as above, is carried out over all such integer nonnegative values of the quantities \(l_1,\ldots,l_{m-1},p,q\) that satisfy condition (10).

If the values \((11_m)\) of the functions \(v_m(x)\) are substituted into the expressions \((4_m)\) for the functions \(\overline F_{rn}(x,v)\), taking, according to (2), \(v_m(x,v)=v^m v_m(x)\), and the resulting expressions are grouped according to powers of the quantities \(C_n(v)=v^n C_n\) and of the parameter \(\mu=v^s\), then one likewise obtains recurrent formulas for determining the quantities \(f_{pq}(x)\):

\[ f_{01}(x)=\int_{0}^{1} A_{01}(x,y)\,dy \]

and

\[ f_{pq}(x)= \int_{0}^{1} \sum_{k+l=2}^{p+q} A_{kl}(x,y) \sum_{\alpha_i,\beta_i,\gamma_i}^{k,l,p,q} \frac{k!}{\gamma_1!\cdots \gamma_n!} \times \]

\[ \times h_{\alpha_1\beta_1}^{\gamma_1}(y) h_{\alpha_2\beta_2}^{\gamma_2}(y)\cdots h_{\alpha_n\beta_n}^{\gamma_n}(y)\,dy, \tag{15_{pq}} \]

for all \(p\) and \(q\) satisfying the condition \(p+q\ge 2\). Here the summation is carried out over all such integer nonnegative values of the quantities \(k,l,\alpha_i,\beta_i,\gamma_i\) for which

\[ \gamma_1+\cdots+\gamma_n=k,\qquad \gamma_1\alpha_1+\cdots+\gamma_n\alpha_n=p, \]

\[ \gamma_1\beta_1+\cdots+\gamma_n\beta_n+l=q. \tag{16} \]

An important property of the quantities \(f_{pq}(x)\), \(h_{pq}(x)\), and \(\Psi_{p,q}\) entering the formulas \((9_m)\), \((11_m)\), and \((13_m)\) is that, as is seen from \((15_{pq})\), (12), and (14), they do not depend on the quantity \(s\) entering the series (2), so that when \(s\) is changed only the number of the functions \(F_m(x)\), or, respectively, \(v_m(x)\), or of the equations \((13_m)\), in which these quantities first appear, changes. Owing to this property, the quantities \(f_{pq}(x)\), \(h_{pq}(x)\), and \(\Psi_{p,q}\), for a known resolvent \(H(x,y)\) and eigenfunctions \(\varphi(x)\) and \(\psi(x)\) of the kernels \(A_{10}(x,y)\) and \(A_{10}(y,x)\), can be found in advance, without fixing the quantity \(s\) and without determining the constants \(C_n\) entering the formulas \((9_m)\), \((11_m)\), and \((13_m)\). The system \((13_m)\) of algebraic equations with respect to the constants \(C_n\) will be called the branching-equation system of the method of A. I. Nekrasov. In view of the fact that the functions \(v_m(x)\) are uniquely determined by the formula \((11_m)\), the form and number of solutions of the nonlinear integral equation (1) are completely determined by the properties of the system \((13_m)\).

Let us note some properties of this system.

Property 1. In passing from the \(m\)-th equation of the system \((13_m)\) to the \((m+1)\)-st, the index of one of the constants in the quantities \(\Psi_{p,q}\) entering the \(m\)-th equation of this system increases by one.

This property is a direct consequence of relation (10), which must be satisfied by the indices of the constants \(C_n\) entering the system \((13_m)\). From this property it follows:

Property 2. The constants \(C_n\) and their powers appear in each of the quantities \(\Psi_{p,q}\) in the natural order, i.e. each of these quantities first appears with the \(p\)-th power of the constant \(C_1\), and with each re-

at the repeated occurrence of this quantity in subsequent equations of the system \((13_m)\), the maximum value of the indices of the constants \(C_n\) entering the terms containing the given quantity \(\Psi_{p,q}\) increases by one. With the aid of this property one easily obtains

Property 3. For any \(s\), the quantities \(\Psi_{p,q}\) appear in the equations of the system \((13_m)\) in their natural order, i.e., the quantities \(\Psi_{p,q}\) with larger values of the indices \(p\) and \(q\) appear in equations with larger number \(m\).

Indeed, from (10) and Property 2 there follows the inequality

\[ p+qs\leqslant m, \tag{17} \]

so that the smallest number \(m_0\) of an equation containing the quantity \(\Psi_{p,q}\) will be \(m_0=p+qs\). In view of this, for any fixed \(s\), when \(p\) is increased by one the quantity \(m_0\) increases by one, and when \(q\) is increased by one, \(m_0\) increases by \(s\) units.

A direct consequence of the noted properties of the system \((13_m)\) is

Theorem 1. The constants \(C_n\) appear in the equations of the system \((13_m)\) in their natural order, so that, when these constants are successively determined from the system \((13_m)\), the first nonzero constant \(C_{n_k}\) is found from an equation containing, in fact, only different powers of this constant.

The equation of the system \((13_m)\) from which it is possible to find the first nonzero constant \(C_{n_k}\) will be called a nontrivial determining equation, and the constant \(C_{n_k}\) entering it a nontrivial determining constant. Equations of the system \((13_m)\) that make it possible to find only zero values of the constants \(C_n\) entering them \((n<n_k)\) will be called trivial determining equations.

Theorem 2. The nontrivial determining equation has the form

\[ \Psi_{p_k,q_k} C_{n_k}^{p_k} + \Psi_{p_k-1,\,q_k+\frac{n_k}{s_k}} C_{n_k}^{p_k-1} +\cdots+ \]

\[ + \Psi_{p_{k+1}+1,\,q_{k+1}-\frac{n_k}{s_k}} C_{n_k}^{p_{k+1}+1} + \Psi_{p_{k+1},q_{k+1}} C_{n_k}^{p_{k+1}} =0, \tag{18} \]

where

\[ \frac{n_k}{s_k}=\frac{q_{k+1}-q_k}{p_k-p_{k+1}} \tag{19} \]

and at least two quantities \(\Psi_{p_k,q_k}\) and \(\Psi_{p_{k+1},q_{k+1}}\) entering this equation are different from zero.

Proof. As already noted, in fact only the sought constant \(C_{n_k}\) enters the determining equation; therefore it has the form

\[ a_{p_k}\Psi_{p_k,q_k} C_{n_k}^{p_k} +\cdots+ a_p\Psi_{p,q} C_{n_k}^{p} +\cdots+ a_{p_{k+1}}\Psi_{p_{k+1},q_{k+1}} C_{n_k}^{p_{k+1}} =0. \tag{20} \]

In this case, according to (10), in order to determine the indices of the quantities \(\Psi_{p,q}\) entering this equation, one may use the formula

\[ n_k p+s_k q=m. \tag{21} \]

In order that the quantities \(\Psi_{p_k,q_k}\) and \(\Psi_{p_{k+1},q_{k+1}}\) enter the \(m\)-th equation of the form (20), it is necessary and sufficient that

\[ n_k p_k+s_k q_k=m \tag{22} \]

and

\[ n_k p_{k+1}+s_k q_{k+1}=m. \tag{23} \]

From this system we obtain (19), and with the help of formula (19) we find the values, indicated in (18), of the second indices of the quantities \(\Psi_{p,q}\) entering equation (20). The quantities \(a_j\), according to formula \((13_m)\), will be equal to one. The values of the quantities \(p_k, q_k, p_{k+1}, q_{k+1}\), and \(s_k\) are determined in the manner indicated below. We note that, in order to determine which of the quantities \(\Psi_{p,q}\), besides \(\Psi_{p_k,q_k}\) and \(\Psi_{p_{k+1},q_{k+1}}\), enter the nontrivial determining equation (18), it is necessary to find only the integer values of the second indices of these quantities, since by definition these values cannot be fractional.

It is also easily obtained

Theorem 3. For all values of the nontrivial determining constants \(C_{n_k}\) for which the quantity

\[ \begin{aligned} b_k(C_{n_k})={}& p_k\Psi_{p_k,q_k} C_{n_k}^{p_k-1} +(p_k-1)\Psi_{p_k-1,\,q_k+\frac{n_k}{s_k}} C_{n_k}^{p_k-2} +\cdots+{}\\ &+(p_{k+1}+1)\Psi_{p_{k+1}+1,\,q_{k+1}-\frac{n_k}{s_k}} C_{n_k}^{p_{k+1}} +p_{k+1}\Psi_{p_{k+1},q_{k+1}} C_{n_k}^{p_{k+1}} \end{aligned} \tag{24} \]

is nonzero, the process of finding the constants \(C_n\) by means of the recurrent system \((13_m)\) stabilizes, i.e. all equations of this system following the nontrivial determining equation (18) will be linear with respect to the sought constants \(C_n\). The coefficient \(b_k(C_{n_k})\) for all these constants is determined by formula (24).

Indeed, from property 2 and formula \((13_m)\) it follows that, in order to obtain those terms of the equation following the determining one which contain the quantities \(\Psi_{p,q}\) entering the determining equation, it is sufficient, in each term entering equation (18), to replace one of the constants \(C_{n_k}\) by \(C_{n_k+1}\) and to multiply the result by the exponent of the constant \(C_{n_k}\).

The coefficient thus obtained at \(C_{n_k+1}\) will be determined by formula (24). The same coefficient will also occur at all subsequent constants with the greatest index, since, according to property 2 and formula \((13_m)\), the terms of the \((m+j)\)-th equation containing constants \(C_{n_k+i}\) with the greatest index \(n_k+i\) are obtained from the terms of the \((m+j-1)\)-th equation containing constants \(C_{n_k+i-1}\) by replacing these constants by \(C_{n_k+i}\).

Theorems analogous to 1 and 3 were proved in [23] and [22], but there only the case \(s=1\) was considered. The determining equation, and consequently also the coefficient \(b_k(C_{n_k})\) at the constants with the greatest index, according to formulas (18) and (24), depend on \(s\). We note that the expression for the coefficient \(b_\mu\) indicated in [22] differs somewhat from expression (24), and in that work the values of the coefficients at the constants \(C_n\) are not indicated.

Let us now suppose that at least one of the quantities \(\Psi_{p,q}\), determined by formulas (14), \((15_{pq})\), and (12), is different from zero, and first consider the case when the first terms of the solutions of equation (1) are distinct. Taking into account the above-noted properties of the system \((13_m)\), we indicate the following method for finding all continuous solutions of equation (1) representable in the form (2).

Using formulas (14), \((15_{pq})\), and (12), we examine the sequences of quantities \(\Psi_{p,q}\) with fixed second index and establish the least value of this index \(q_1\) for which the indicated sequences contain at least one nonzero quantity \(\Psi_{p,q_1}\).

Then we establish the least value of the first index \(p_1\) of the quantities \(\Psi_{p,q_1}\) for which \(\Psi_{p_1,q_1}\ne 0\).

We shall call the quantity \(\Psi_{p_1,q_1}\) the first determining quantity. After this we look for a nonzero quantity \(\Psi_{p_2,q_2}\), with the smallest index \(p_2<p_1\), such that all \(\Psi_{p,q}\) whose indices satisfy the conditions

\[ p_2<p<p_1,\qquad qs_1+pn_1<q_1s_1+p_1n_1, \tag{25} \]

where \(\dfrac{n_1}{s_1}\) is the irreducible fraction defined by the formula

\[ \frac{n_1}{s_1}=\frac{q_2-q_1}{p_1-p_2}, \tag{26} \]

are equal to zero. We shall call this quantity the second determining quantity.

If there exists a second determining quantity \(\Psi_{p_2,q_2}\), then equation (1) has \(p_1-p_2\) nontrivial complex solutions of the form (2) with the least exponent of the parameter \(\mu\) equal to \(\dfrac{n_1}{s_1}\).

Indeed, relation (26) ensures the appearance in some equation (18) of the system \((13_m)\) of at least two terms of different dimensions \(p_1\) and \(p_2\) with respect to the sought constant \(C_{n_1}\), and the indicated choice of the determining quantities \(\Psi_{p_1,q_1}\) and \(\Psi_{p_2,q_2}\) ensures the trivial solvability of all equations preceding this equation, because the values of the first nonzero constant \(C_{n_1}\) are found from equation (18) for \(k=1\). The values of all subsequent constants \(C_n\) \((n>n_1)\), according to Theorem 3, will be found from the equations of the system \((13_m)\), linear with respect to \(C_n\), and the coefficient \(b_1(C_{n_1})\) at all these constants will be determined by formula (24). This coefficient will be nonzero, since, according to formulas (18) and (24), it is equal to the derivative of the left-hand side of equation (18) with respect to the quantity \(C_{n_1}\). In view of this, all equations of the system \((13_m)\) following the nontrivial determining equation (18) will be uniquely solvable, and the number of nontrivial solutions of equation (1) will coincide with the number of nontrivial solutions of the determining equation (18). After the constants \(C_n\) have been determined, the values of the functions \(v_m(x)\) can be found by formula \((11_m)\).

The remaining solutions of equation (1) are found in the same way, only the role of the first determining quantity \(\Psi_{p_1,q_1}\) will be played by the second \(\Psi_{p_2,q_2}\), and the role of the second \(\Psi_{p_2,q_2}\) by the third \(\Psi_{p_3,q_3}\). Since the first indices \(p_1,\ldots,p_k,\ldots\) of the quantities \(\Psi_{p_1,q_1}, \Psi_{p_2,q_2}, \ldots, \Psi_{p_k,q_k}, \ldots\) satisfy the condition

\[ p_1>p_2>\cdots>p_k>\cdots \tag{27} \]

and cannot be fractional or negative, the number of all pairs of successive determining quantities \(\Psi_{p_i,q_i}\) and \(\Psi_{p_{i+1},q_{i+1}}\) \((i=1,2,\ldots,k,\ldots)\), and consequently also the number of all values of the quantity \(s\) found by means of the indicated algorithm, will be finite and will not exceed the value of the first index of the first determining quantity \(\Psi_{p_1,q_1}\). It also follows from the indicated method of constructing the determining quantities \(\Psi_{p_k,q_k}\) that the first index of the determining quantity \(\Psi_{p_0,q_0}\) with the smallest first index can turn out to be nonzero only in the case when there do not exist nonzero quantities \(\Psi_{p,q}\) with index \(p<p_0\).

Taking the above into account, one may state the following assertion.

Theorem 4. If the rank of the characteristic number of the kernel \(A_{10}(x,y)\) is equal to one and there exists at least one nonzero quantity \(\Psi_{p,q}\), then, under the condition

\[ b_k(C_{n_k}) \ne 0 \quad (k=1,2,\ldots,0) \tag{28} \]

the number of formal nontrivial complex solutions of equation (1), representable in the form (2), is finite and is equal to the difference of the first indices of the first \(\Psi_{p_1,q_1}\) and the last \(\Psi_{p_0,q_0}\) determining quantities. In particular, for \(p=0\) equation (1) will have no solutions of the form (2), while for \(p_1=1\), \(p_0=0\) such a solution will be unique. All these solutions can be found by the method indicated above.

Indeed, if there exists at least one nonzero quantity \(\Psi_{p,q}\), then there also exists the first determining quantity \(\Psi_{p_1,q_1}\). In this case, for \(p_0=0\) the method indicated above makes it possible to find \(p_1\) nontrivial complex solutions of equation (1), representable in the form of the series (2).

There can be no more such solutions, since, according to property 3, the quantities \(\Psi_{p,q}\) with indices \(p>p_1\), \(q>q_1\) appear in the equations of the system \((13_m)\) later than the first nonzero determining quantity \(\Psi_{p_1,q_1}\) and cannot enter into the determining equations, while all possible cases of formation of nontrivial determining equations by quantities \(\Psi_{p,q}\) with indices \(p<p_1\) are exhausted by the method indicated above.

Indeed, by means of the method described we obtain all possible nontrivial determining equations formed by adjacent determining quantities \(\Psi_{p_k,q_k}\) and \(\Psi_{p_{k+1},q_{k+1}}\). Each nontrivial determining equation obtained in this way gives \(p_k-p_{k+1}\) nontrivial complex solutions of the form (2) with the least exponent of the parameter \(\mu\) equal to

\[ \frac{n_k}{s_k}. \]

By means of the quantities \(\Psi_{p,q}\) for which \(p_{k+1}<p<p_k\), it is impossible to obtain any new nontrivial determining equation and to find a solution of the form (2) of equation (1) with the least exponent of the parameter

\[ \frac{n}{s} \ne \frac{n_k}{s_k}. \]

For

\[ \frac{n_{k-1}}{s_{k-1}} < \frac{n}{s} < \frac{n_k}{s_k}, \]

according to formula (18), the quantity \(\Psi_{p_k,q_k}\) will appear in the determining equation (18) earlier than the nonzero quantities \(\Psi_{p,q}\) with index \(p\) lying between \(p_1\) and \(p_0\). This equation will be trivial and will be satisfied only by the zero value of the constant entering it. For

\[ \frac{n_k}{s_k} < \frac{n}{s} < \frac{n_{k+1}}{s_{k+1}}, \]

the determining quantity \(\Psi_{p_{k+1},q_{k+1}}\) will appear earlier than all these quantities, so that in both cases the first nonzero constant, and consequently also the nonzero value of the least exponent of the degree of the parameter \(\mu\),

\[ \frac{n_k}{s_k}, \]

can be obtained only with the aid of nontrivial determining equations containing quantities \(\Psi_{p,q}\) with index \(p\) not belonging to the interval \((p_{k+1},p_k)\).

If \(p_0\) turns out to be greater than zero, then all possible determining equations (18) for which the senior determining quantity \(\Psi_{p_k,q_k}\) is the quantity \(\Psi_{p_0,q_0}\) will be trivial. Since in this case, for \(p_0\) values of the determining constant \(C_{n_0}\), no nonzero values can be found—

for the values, then the number of nontrivial solutions of equation (1) will be smaller by \(p_0\) units than when \(p_0=0\).

Let us now consider the case in which condition (28) is not fulfilled. Since the quantity \(b_k(C_{n_k})\) is a polynomial with respect to the constant \(C_{n_k}\), it can vanish either at a finite number of points, when not all quantities \(\Psi_{p,q}\) entering into the determining equation (18) are equal to zero, or identically, when all these quantities in fact turn out to be equal to zero.

The latter case can occur only when, for the given equation, there exist no nonzero quantities \(\Psi_{p,q}\). But if, for the given equation, there exist no nonzero quantities \(\Psi_{p,q}\), then all equations of the system \((13_m)\) will be satisfied by arbitrary values of the constants \(C_n\) \((n=1,\infty)\) entering into them; and for those values of the constants \(C_n\) for which the series (2) converges, the solution of equation (1) will have the form

\[ v(x)=P(x,\nu)+C(\nu)\varphi(x). \tag{29} \]

In this case, the solution of the nonlinear integral equation (1) will depend on one arbitrary quantity \(C(\nu)\).

The case in which \(b_k(w)\ne 0\), but vanishes for a value of \(w\) equal to the determining constant \(C_{n_k}\), reduces to the case considered earlier. Indeed, the first terms of the solutions (2) can be found by the method set forth above. If some of these terms have turned out to coincide and, in view of this, the quantity \(b_k(C_{n_k})\) has vanished, then to determine the next \((n_k+1)\) terms of the series (2) it suffices to set

\[ v(x)=v_{n_k}(x,\nu)+z(x). \tag{30} \]

The expansion into a series of the function \(z(x)\) will begin with the \((n_k+1)\)-st terms of the series (2), and to find the function \(z(x)\) one may use the method described above.

Since, according to [13], all formal solutions (2) of equation (1) obtained in the first and the last cases converge, the formal solutions of equation (1) constructed above will also be genuine solutions of this equation; moreover, according to the results of [13] and [20], the solutions found above exhaust all continuous small solutions of equation (1).

We note that, in the case \(q_0=0\), the number of analytic solutions of equations of the form (1) was established in [17] and [21]. For nonlinear operator equations this result was indicated in [13]. In [13] and [17], by means of the Newton diagram, some cases were also investigated in which the form of the solutions and the equations for determining the first terms of the solutions of the Lyapunov—Schmidt branching equations were specified.

In [13], by the method of the Newton diagram, the boundedness of the number of all values of the quantity \(s\) for which nonlinear operator equations can have nontrivial solutions representable as series in integral or fractional powers of the parameter was proved, as well as the boundedness of the number of all these solutions, when unity is a simple eigenvalue of the corresponding linearized equation and the Lyapunov—Schmidt branching equation does not vanish identically. In [15] the case was also considered in which the multiplicity of the eigenvalue of the corresponding linearized equation is greater than one. Some conditions ensuring boundedness of the number of small solutions of one class of nonlinear operator equations were established in [24].

We note that the problem of constructing small solutions of equation (1) of the form (2) is also easily reduced, by the well-known method of A. I. Nekrasov, to the problem of investigating branch points of solutions of nonlinear integral equations of the form

\[ u(x)=\int_0^1 F[x,y,\lambda,u(y)]\,dy, \tag{31} \]

where \(\lambda\) is a parameter; \(u(x)\) is the unknown, and \(F[x,y,\lambda,u]\) is a known function continuous in \(x,y\) and analytic in \(\lambda,u\). Therefore, from the results of the present work there follow, as special cases, many theorems concerning the branching of solutions of nonlinear integral equations. Thus, for example, from Theorem 4 of the present work, for \(F[x,y,\lambda,u]=\lambda\Gamma[x,y,u]\) and under the corresponding assumptions concerning the defining quantities \(\Psi_{p_k,q_k}\) \((k=0,1,2)\), we obtain all the theorems of [17].

References

1. Lyapunov A. M. Zap. Akad. nauk, SPb., 1906, pp. 1–225.
2. Schmidt E. Math. Ann., 65, 1908.
3. Lichtenstein L. Vorlesungen über einige klassen nichtlinearen Integralgleichungen und Integro-Differentialgleichungen nebst Anwendungen. Berlin, 1931.
4. Nekrasov A. I. Exact theory of waves of steady form on the surface of a heavy fluid. Publishing House of the Academy of Sciences of the USSR, 1951.
5. Sekerzh-Zen’kovich Ya. I. Tr. TsAGI, vol. 299, 1937, pp. 1–47.
6. Nazarov N. N. Nonlinear integral equations of Hammerstein type. Tr. SAGU, 5, Math., 1941, p. 33.
7. Nazarov N. N. Tr. Instituta matematiki AN UzSSR, vol. 4, 1948.
8. Vlasov A. A. Theory of many particles. Moscow–Leningrad, Gostekhizdat, 1950.
9. Erugin N. P. Nonlinear functions. Publishing House of Leningrad State University, 1956.
10. Akhmedov K. T. UMN, 12, vol. 4, 135–153, 1957.
11. Trenogin V. A. UMN, 13, vol. 4, 197–203, 1958.
12. Trenogin V. A. DAN SSSR, 131, No. 5, 1032–1035, 1960.
13. Vainberg M. M., Trenogin V. A. UMN, 17, vol. 2, 13–72, 1962.
14. Vainberg M. M., Trenogin V. A. UMN, 17, vol. 5, 185–196, 1962.
15. Vainberg M. M., Trenogin V. A. UMN, 18, vol. 5, 223–224, 1963.
16. Chebotarev N. G. Theory of algebraic functions. Moscow–Leningrad, Gostekhizdat, 1948.
17. Stapan A. E. Uchen. zap. Rizhsk. ped. in-ta, 4, 1957, 31–43.
18. Bartle R. G. Trans. Amer. Math. Soc., 75, No. 2, 366–384, 1953.
19. Graves L. M. Trans. Amer. Math. Soc., 79, No. 1, 150–157, 1955.
20. Pokornyi V. V. Tr. seminar. po funkts. analizu, Voronezh, vol. 2, 1956, pp. 39–45.
21. Pokornyi V. V. Tr. seminar. po funkts. analizu, Voronezh, vol. 5, 1957, 15–20.
22. Pokornyi V. V., Rybin P. P. UMN, 15, vol. 4, 169–172, 1960.
23. Rybin P. P. Izv. vuzov, No. 6, 131–137, 1959.
24. Melamed V. B. DAN SSSR, 145, No. 3, 531–533, 1962.

Received by the editors
September 13, 1965

Irkutsk State University
named after A. A. Zhdanov