UDC 517.948.34

ON THE QUESTION OF THE EXISTENCE OF SOLUTIONS OF DIFFERENTIAL AND INTEGRO-DIFFERENTIAL EQUATIONS HAVING ASYMPTOTIC PARABOLAS

Yu. A. Ved

The definition of an asymptotic parabola of a plane curve is given in [1]. According to this definition, the parabola

\[ y=\sum_{k=0}^{m} A_k x^k \tag{Y} \]

is called an asymptotic parabola of degree \(m\) of the curve \(y=y(x)\), \(x\geq a>0\), \(m\) times continuously differentiable on the half-interval \(I=[a,\infty)\), if the relations

\[ \lim_{x\to\infty}\sum_{i=k}^{m}\frac{(-1)^{i-k}}{(i-k)!}\,x^{i-k}y^{(i)}(x)=k!A_k \quad (k=0,1,\ldots,m) \tag{Z} \]

hold.

In other words, the asymptotic parabola of degree \(m\) of the curve \(y=y(x)\), \(x\geq a>0\), \(m\) times continuously differentiable on the half-interval \(I\), is the parabola (Y) having contact of order \(m\) with the curve \(y=y(x)\) at infinity.

The question of the existence of solutions of linear second-order differential equations having asymptotic parabolas of degree 1 was considered in [2, 3, 7], where sufficient conditions were established for the presence of asymptotic parabolas not higher than degree 1 (in (Y), for \(m=1\), \(A_1=0\) or \(\ne 0\)) for all solutions of the equations under consideration, and also for the existence of solutions having asymptotic parabolas below degree 1 (in (Y), for \(m=1\), \(A_1=0\)); the results of [2, 7] on the indicated question are special cases of the results of [3].

Papers [9, 11] are devoted to the existence of asymptotic parabolas not higher and below degree 1 for solutions of nonlinear second-order integro-differential equations with weak nonlinearities, and also to the existence of solutions of the equations under consideration for which prescribed parabolas of degree 1 serve as asymptotic parabolas.

In [6] the question of the existence of asymptotic parabolas of degree \((n-1)\)

\[ y=\sum_{k=0}^{n-1} A_k x^k \tag{Y'} \]

in solutions of a linear nonhomogeneous differential equation of order \(n\); here sufficient conditions are established under which every solution of the equation under consideration has an asymptotic parabola of degree not higher than \(n-1\) (in \((Y')\): \(A_{n-1}=0\) or \(\ne 0\)). We note that by the method of [6] one cannot obtain an analogous result for a nonlinear differential equation.

In the present paper we study the question of the existence of solutions with asymptotic parabolas of degree \(n-1\) of the nonlinear differential equation

\[ y^{(n)}(x)+\sum_{j=1}^{n}p_j(x)y^{(n-j)}=f(x)+F(x,Y) \tag{1} \]

and of the integro-differential equation

\[ y^{(n)}(x)+\sum_{j=1}^{n}\left[p_j(x)y^{(n-j)}+\int_a^x K_j(x,\tau)y^{(n-j)}(\tau)\,d\tau\right] = \]

\[ = f(x)+F\left(x,Y,\int_a^x H(x,\tau,Y(\tau))\,d\tau\right), \tag{1'} \]

\[ n\geqslant 2,\qquad x\geqslant a>0,\qquad Y=(y,y',\ldots,y^{(n-1)}), \]

where the functions \(p_j(x)\), \(K_j(x,\tau)\) \((j=1,\ldots,n)\) and \(f(x)\) are defined and continuous in the domain \(D=\{a\leqslant \tau\leqslant x<\infty\}\), while the functions \(F(x,Y)\), \(F(x,Y,u)\), and \(H(x,\tau,Y)\) are defined and continuous in the domain \(D\times E_{n+1}\) (\(E_{n+1}\) is the \((n+1)\)-dimensional Euclidean space of the components of the vector \(Y\) and the variable \(u\)) and satisfy in this domain a Lipschitz condition with respect to \(Y\), \(u\), respectively, with “Lipschitz constants” \(g(x)\) and \(h_1(x,\tau)\), which are nonnegative continuous functions in the domain \(D\).

Every solution of equations (1) and \((1')\) is defined and \(n\) times continuously differentiable on the half-interval \(I\).

Sufficient conditions are established for the existence of solutions of equations (1) and \((1')\) possessing asymptotic parabolas exactly and below degree \(n-1\), asymptotic parabolas with nonzero coefficients of \(x^k\) \((k=0,1,\ldots,n-1)\), and also prescribed asymptotic parabolas of degree \(n-1\). The method is similar to that of [3]; at the same time it gives, to some extent, a constructive way of constructing solutions of equations (1) and \((1')\) with asymptotic parabolas of degree \(n-1\).

§ 1. AUXILIARY PROPOSITIONS

Lemma 1.1. A function \(y(x)\), defined and \(m\) times continuously differentiable on the half-interval \(I\), has an asymptotic parabola of degree \(m\) \((m\geqslant 1)\) if and only if: a) there exists on \(I\) a function \(\varphi(x)\), continuous for \(x\in I\) and tending to a finite limit as \(x\to\infty\): \(\lim_{x\to\infty}\varphi(x)=B_0<\infty\); b) for \(x\geqslant a>0\) the representation holds

\[ y(x)=\sum_{k=1}^{m}\frac{x^k}{(k-1)!}\left[(k-1)!B_k-\delta_m\frac{(-1)^{m-k}}{(m-k)!}\int_x^\infty \eta^{-1-k}\varphi(\eta)\,d\eta\right]= \]

\[ = \sum_{k=1}^{m} \frac{x^k}{(k-1)!} \left[ C_{k-1}+\delta_m \frac{(-1)^{m-k}}{(m-k)!} \int_a^x \eta^{-1-k}\varphi(\eta)\,d\eta \right], \tag{α} \]

where

\[ B_k=\frac{1}{(k-1)!} \left[ C_{k-1}+\delta_m \frac{(-1)^{m-k}}{(m-k)!} \int_a^\infty \eta^{-1-k}\varphi(\eta)\,d\eta \right] \qquad (k=1,\ldots,m) \]

are certain constants; \(\delta_m=(-1)^m m!\). Thus

\[ y=\sum_{k=0}^{m} B_k x^k \]

is the equation of the asymptotic parabola of the curve \(y=y(x)\).

Lemma 1.1 is borrowed, in a somewhat modified form, from [1] (here another representation (α) of the function \(y(x)\) is given, equivalent to its representation in [1]).

In what follows we introduce the following notation:

\[ L(y)\equiv \sum_{j=1}^{n} p_j(x)y^{(n-j)};\qquad h(x,\tau)\equiv g(x)h_1(x,\tau); \]

\[ F(x)\equiv \left|F\left(x,0,\int_a^x H(x,\tau,0)\,d\tau\right)\right|; \qquad P_k(x)\equiv \sum_{j=n-k}^{n} \frac{x^{j-1+k}}{(j-n+k)!}p_j(x); \]

\[ Q_k(x,\tau)\equiv x^{n-1}\sum_{j=n-k}^{n} \frac{\tau^{j-n+k}}{(j-n+k)!}K_j(x,\tau) \qquad (k=1,\ldots,n-1); \]

\(\Gamma\) is the class of functions continuous on the half-interval \(I\) and tending to finite limits as \(x\to\infty\); \(\Gamma_0\) is the class of functions continuous on \(I\) and tending to one and the same constant \(A_0\) as \(x\to\infty\); \(C^n\) is the class of functions defined and \(n\) times continuously differentiable on the half-interval \(I\).

Lemma 1.2. If the functions \(x^{j-1}|p_j(x)|\) \((j=1,\ldots,n)\) \((A)\), \(f(x)\) \((B)\), \(x^{n-1}g(x)\) \((C)\), and \(|F(x,0)|\) \((D)\) are integrable on the half-interval \(I\), then for any solution \(y(x)\) of equation (1) there exist finite limits

\[ \lim_{x\to\infty} (i-1)!x^{1-i}y^{(n-i)}(x) \qquad (i=1,\ldots,n), \tag{II} \]

equal to one another. Moreover, the limits (II) are nonzero for those solutions whose initial values \(y^{(i)}(a)\) \((i=0,1,\ldots,n-1)\) satisfy the relation

\[ |R(\infty)|-S-[\sup_I |R(x)|+S]T\exp T>0, \tag{β} \]

where

\[ R(x)\equiv y^{(n-1)}(a) -\int_a^x \left[ \sum_{j=2}^{n}\sum_{k=0}^{j-2} \frac{1}{k!}y^{(n-j+k)}(a)(t-a)^k p_j(t)-f(t) \right]dt, \]

\[ R(\infty)=\lim_{x\to\infty}R(x); \]

\[ S=\int_a^\infty \left[g(t)\sum_{j=2}^n\left|\sum_{k=0}^{j-2}\frac{(t-a)^k}{k!}\,y^{(n-j+k)}(a)\right|+|F(t,0)|\right]dt; \]

\[ T=\int_a^\infty \left[|p_1(t)|+\int_a^t\left|\sum_{j=2}^n\frac{(t-s)^{j-2}}{(j-2)!}\,p_j(t)\right|ds +g(t)\sum_{j=1}^n\frac{(t-a)^{j-1}}{(j-1)!}\right]dt. \]

Proof. Let \(y(x)\) be an arbitrarily fixed solution of equation (1). As is known (see, for example, [4, 10]), in order to prove the first part of the lemma it suffices to show that there exists the finite limit \((\Pi)\) for \(i=1\). We have

\[ y^{(n-1)}(x)\equiv y^{(n-1)}(a)-\int_a^x[L(y(t))-f(t)-F(t,Y(t))]\,dt. \tag{1.1} \]

It is not difficult to see that, by virtue of the integrability on \(I\) of the functions \((A)\), \((B)\), \((C)\), and \((D)\), all the quantities entering into \((\beta)\) exist and are finite. Using, for each \(j=2,\ldots,n\), the identity

\[ y^{(n-j)}(x)\equiv \sum_{k=0}^{j-2}\frac{(x-a)^k}{k!}\,y^{(n-j+k)}(a) +\int_a^x\frac{(x-s)^{j-2}}{(j-2)!}\,y^{(n-1)}(s)\,ds, \tag{1.2} \]

from (1.1), for \(x\in I\), we obtain

\[ \begin{aligned} |y^{(n-1)}(x)|\le{}&\sup_I |R(x)|+S+\int_a^x\Biggl\{[|p_1(t)|+g(t)]|y^{(n-1)}(t)|\\ &+\int_a^t\left[\left|\sum_{j=2}^n\frac{(t-s)^{j-2}}{(j-2)!}\,p_j(t)\right| +g(t)\sum_{j=2}^n\frac{(t-s)^{j-2}}{(j-2)!}\right]|y^{(n-1)}(s)|\,ds\Biggr\}dt. \end{aligned} \]

Applying to the obtained inequality the basic lemma [8, p. 121], for \(x\in I\) we shall have

\[ |y^{(n-1)}(x)|\le [\sup_I |R(x)|+S]\exp T. \tag{1.3} \]

Consequently, the function \(y^{(n-1)}(x)\) is bounded on the half-interval \(I\). From the boundedness of the function \(y^{(n-1)}(x)\) and the integrability of the functions \((A)\), \((B)\), \((C)\), and \((D)\) on the half-interval \(I\), it follows that the right-hand side of the identity (1.1) belongs to the class \(\Gamma\). Consequently, \(y^{(n-1)}(x)\in\Gamma\). Further, from the identity (1.1), taking into account (1.2) and (1.3), we obtain

\[ \left|\lim_{x\to\infty} y^{(n-1)}(x)\right| \ge |R(\infty)|-S-[\sup_I |R(x)|+S]\,T\exp T, \]

which, according to \((\beta)\), completes the proof of the lemma.

In an analogous manner, applying instead of the basic lemma [8, p. 121] its generalization—Lemma 1 [10]—one proves

Lemma 1.2′. If the functions \((A)\), \((B)\), \((C)\), and

\[ \int_a^x \tau^{j-1}|K_j(x,\tau)|\,d\tau\quad (j=1,\ldots,n),\qquad \int_a^x \tau^{n-1}h(x,\tau)\,d\tau,\quad F(x) \]

integrable on the half-interval \(I\), then for every solution \(y(x)\) of equation \((1')\) there exist finite limits (II), equal to one another. Moreover, these limits (II) are nonzero for those solutions whose initial values \(y^{(i)}(a)\) \((i=0,1,\ldots,n-1)\) satisfy the relation

\[ |R_1(\infty)|-S' - [\sup_I |R_1(x)|+S']T'\exp T' >0, \tag{\(\beta'\)} \]

where

\[ R_1(x)\equiv R(x)- \int_a^x\int_a^t \sum_{j=2}^n \sum_{k=0}^{j-2}\frac{1}{k!}\, y^{(n-j+k)}(a)(\tau-a)^k K_j(t,\tau)\,d\tau\,dt; \]

\[ S'=\int_a^\infty \left\{ \sum_{j=2}^n \left[ g(t)\left|\sum_{k=0}^{j-2}\frac{(t-a)^k}{k!}\, y^{(n-j+k)}(a)\right| + \int_a^t h(t,\tau) \left|\sum_{k=0}^{j-2}\frac{(\tau-a)^k}{k!}\, y^{(n-j+k)}(a)\right|\,d\tau \right] +F(t) \right\}\,dt; \]

\[ T'=T+ \int_a^\infty\int_a^t \left[ |K_1(t,\tau)| + \int_a^\tau \left|\sum_{j=2}^n \frac{(\tau-s)^{j-2}}{(j-2)!}\,K_j(t,\tau)\right|\,ds + h(t,\tau)\sum_{j=1}^n\frac{(\tau-a)^{j-1}}{(j-1)!} \right]d\tau\,dt. \]

Remark 1.1. In the case where \(f(x)\equiv 0\) and the left-hand side of equation (1) (respectively \((1')\)) does not contain \(y, y',\ldots,y^{(n-2)}\), for condition \((\beta)\) (respectively \((\beta')\)) to hold it is necessary that the inequality \(T\exp T<1\) (\(T'\exp T'<1\), respectively) be satisfied.

From the first part of Lemma 1.2 (respectively \(1.2'\)), for \(F(x,Y)\equiv 0\) (respectively \(F(x,Y,u)\equiv 0\)) there follows the corresponding result of papers [4, 5] (respectively paper [10], with some strengthening).

§ 2. ON THE EXISTENCE OF SOLUTIONS OF EQUATIONS (1) AND \((1')\) HAVING ASYMPTOTIC PARABOLAS OF DEGREE \((n-1)\)

Lemma 2.1. For the existence of a solution of equation (1) having an asymptotic parabola of degree \((n-1)\), it is necessary and sufficient that the integral equation

\[ \varphi(x)+\int_a^x p_1(t)\varphi(t)\,dt +\frac{1}{\delta}\int_a^x \sum_{k=1}^{n-1} kP_k(t) \left[ C_{k-1} +\delta\frac{(-1)^{\,n-1-k}}{(n-1-k)!} \times \right. \]

\[ \left. \times\int_a^t \eta^{\,n-1-k}\varphi(\eta)\,d\eta \right]dt = C_{n-1} +\frac{1}{\delta}\int_a^x t^{\,n-1}f(t)\,dt + \]

\[ +\frac{1}{\delta}\int_a^x t^{\,n-1}F\bigl(t,\Phi^*(t;\varphi)\bigr)\,dt, \qquad \Phi^*\equiv(\Phi,\Phi',\ldots,\Phi^{(n-1)}), \tag{2.1} \]

where

\[ \delta=(-1)^{n-1}(n-1)!;\qquad \Phi(x;\varphi)\equiv\Phi(x;\varphi,C_0,C_1,\ldots,C_{n-2})\equiv \]

\[ \equiv \sum_{k=1}^{n-1} \frac{x^k}{(k-1)!} \left[ C_{k-1}+\delta \frac{(-1)^{\,n-1-k}}{(n-1-k)!} \int_a^x \eta^{-1-k}\varphi(\eta)\,d\eta \right], \]

for some values of the constants \(C_0, C_1,\ldots,C_{n-1}\), had a solution in the class \(\Gamma\).

Proof of necessity. Let \(y(x)\) be a solution of equation (1) possessing an asymptotic parabola of degree \((n-1)\). Then, on the basis of Lemma 1.1, we conclude that for \(x\in I\) the representation

\[ y(x)=\sum_{k=1}^{n-1}\frac{x^k}{(k-1)!} \left[ C_{k-1}+\delta\frac{(-1)^{\,n-1-k}}{(n-1-k)!} \int_a^x \eta^{-1-k}\psi(\eta)\,d\eta \right], \tag{2.2} \]

holds, where \(\psi(x)\) is some function of the class \(\Gamma\); \(C_0,C_1,\ldots,C_{n-2}\) are some constants. From (2.2), for \(x\in I\), we find

\[ y^{(i)}(x)= \sum_{k=i}^{n-1}\frac{k}{(k-i)!}x^{k-i} \left[ C_{k-1}+\delta\frac{(-1)^{\,n-1-k}}{(n-1-k)!}\times \right. \]

\[ \left. \times \int_a^x \eta^{-1-k}\psi(\eta)\,d\eta \right] +\Delta_i(x) \quad (i=0,1,\ldots,n-1), \tag{\(2.2_i\)} \]

where

\[ \Delta_i(x)\equiv \begin{cases} 0, & \text{for } i=0,1,\ldots,n-2,\\ \delta x^{1-n}\psi(x), & \text{for } i=n-1 \end{cases} ; \]

\[ y^{(n)}(x)=\delta x^{1-n}\psi'(x). \tag{2.3} \]

Substituting the expressions for \(y^{(i)}(x)\) \((i=0,1,\ldots,n)\) from \((2.2_i)\) and (2.3) into equation (1), after elementary transformations we obtain

\[ \psi'(x)+p_1(x)\psi(x) +\frac{1}{\delta}\sum_{k=1}^{n-1}kP_k(x) \left[ C_{k-1}+\delta\frac{(-1)^{\,n-1-k}}{(n-1-k)!}\times \right. \]

\[ \left. \times\int_a^x \eta^{-1-k}\psi(\eta)\,d\eta \right] \equiv \frac{1}{\delta}x^{n-1}\bigl[f(x)+F(x,\Phi^*(x;\psi))\bigr]. \tag{2.4} \]

Integrating the identity (2.4) from \(a\) to \(x\) and setting \(\psi(a)=C_{n-1}\), we obtain that the function \(\psi(x)\) satisfies the integral equation (2.1).

Proof of sufficiency. Let the integral equation (2.1), for some values of \(C_0,C_1,\ldots,C_{n-1}\), have a solution \(\varphi_0(x)\in\Gamma\); \(\varphi_0(a)=C_{n-1}\). Then on \(I\) there exists a function \(y(x)\), continuously differentiable \((n-1)\) times, defined by relations (a) with \(m=n-1\), where \(\varphi(x)\equiv\varphi_0(x)\). Thus the conditions a) and b) of Lemma 1.1 are fulfilled. Therefore the function \(y(x)\) has an asymptotic parabola of degree \((n-1)\). It remains to prove that the function \(y(x)\) is a solution of equation (1). From the identity obtained after substituting \(\varphi_0(x)\) into the integral equation (2.1), it follows that the function \(\varphi_0(x)\) is continuously differentiable on \(I\). Differentiating (a) (for \(m=n-1\)) with \(\varphi(x)\equiv\varphi_0(x)\), for \(x\in I\) we obtain \((2.2_i)\) and (2.3), where \(\psi(x)\equiv\varphi_0(x)\). Substituting \((2.2_i)\) and (2.3) with \(\psi(x)\equiv\varphi_0(x)\) into equation (1), we obtain that the function \(y(x)\) satisfies equation (1).

Analogously to Lemma 2.1 one proves

Lemma 2.1′. For the existence of a solution of the integro-differential equation \((1')\) possessing an asymptotic parabola of degree \((n-1)\), it is necessary and sufficient that the integral equation

\[ \begin{aligned} \varphi(x)&+\int_a^x\left[p_1(t)\varphi(t)+t^{n-1}\int_a^t \tau^{1-n}K_1(t,\tau)\varphi(\tau)\,d\tau\right]dt+\\ &+\frac{1}{\delta}\int_a^x\sum_{k=1}^{n-1}k\left\{P_k(t)\left[C_{k-1}+\delta\frac{(-1)^{n-1-k}}{(n-1-k)!}\int_a^t\eta^{-1-k}\varphi(\eta)\,d\eta\right]+\right.\\ &\left.+\int_a^t Q_k(t,\tau)\left[C_{k-1}+\delta\frac{(-1)^{n-1-k}}{(n-1-k)!}\int_a^\tau\eta^{-1-k}\varphi(\eta)\,d\eta\right]d\tau\right\}dt=\\ &=C_{n-1}+\frac{1}{\delta}\int_a^x t^{n-1}f(t)\,dt+\frac{1}{\delta}\int_a^x t^{n-1}F\left(t,\Phi^*(t;\varphi),\right.\\ &\left.\int_a^t H\left(t,\tau,\Phi^*(\tau;\varphi)\right)d\tau\right)dt \end{aligned} \tag{2.1′} \]

for some values of the constants \(C_0, C_1,\ldots,C_{n-1}\) have a solution in the class \(\Gamma_0\).

Corollary 2.1. If \(\varphi(x)\) is a solution of the integral equation \((2.1)\) (respectively \((2.1')\)) in the class \(\Gamma_0\), then the function

\[ y(x)=\sum_{k=1}^{n-1}\frac{x^k}{(k-1)!}\left[C_{k-1}+\delta\frac{(-1)^{\,n-1-k}}{(n-1-k)!}\int_a^x\eta^{-1-k}\varphi(\eta)\,d\eta\right] \tag{2.5} \]

is a solution of equation \((1)\) (respectively \((1')\)), possessing the asymptotic parabola \((Y')\), where

\[ A_k=\frac{1}{(k-1)!}\left[C_{k-1}+\delta\frac{(-1)^{\,n-1-k}}{(n-1-k)!}\int_a^\infty\eta^{-1-k}\varphi(\eta)\,d\eta\right]\quad (k=1,\ldots,n-1). \]

If \(y(x)\) is a solution of equation \((1)\) (respectively \((1')\)) with the asymptotic parabola \((Y')\), then the function

\[ \varphi(x)=\sum_{i=0}^{n-1}\frac{(-1)^i}{i!}x^i y^{(i)}(x) \tag{2.6} \]

is a solution in the class \(\Gamma_0\) of the integral equation \((2.1)\) (respectively \((2.1')\)) for

\[ C_{k-1}=(k-1)!\,A_k-\delta\frac{(-1)^{\,n-1-k}}{(n-1-k)!}\int_a^\infty\eta^{-1-k}\varphi(\eta)\,d\eta \]

\[ (k=1,\ldots,n-1). \tag{2.6′} \]

Remark 2.1. Between the constants \(C_i\) \((i=0,1,\ldots,n-1)\), appearing in the integral equation \((2.1)\) (respectively \((2.1')\)), and the ...

initial values \(y^{(i)}(a)\) \((i=0,1,\ldots,n-1)\) of a solution with an asymptotic parabola of degree \(n-1\) of equation (1) (respectively (1′)) there exists a one-to-one correspondence, determined by the relations

\[ y^{(i)}(a)=\sum_{k=i}^{n-1}\frac{k}{(k-i)!}\,a^{k-i} C_{k-1} \quad (i=0,1,\ldots,n-2), \]

\[ y^{(n-1)}(a)=(n-1)C_{n-2}+\delta a^{1-n}C_{n-1}. \tag{2.7} \]

Theorem 2.1. If the functions \(|p_1(x)|\) \((A_1)\), \(x^{n-1}f(x)\) \((B_1)\), \(|P_k(x)|\) \((k=1,\ldots,n-1)\) \((A_2)\), \(x^{2n-2}g(x)\) \((C_1)\), and \(x^{n-1}|F(x,0)|\) \((D_1)\) are integrable on the half-interval \(I\), then every solution of equation (1) has an asymptotic parabola of degree not higher than \(n-1\). Moreover, solutions for whose initial values \(y^{(i)}(a)\) \((i=0,1,\ldots,n-1)\) relation (P) is satisfied possess asymptotic parabolas of exactly degree \(n-1\).

Proof. In view of Lemma 2.1, Remark 2.1, and the unique solvability of the Cauchy problem for equation (1), in particular of the Cauchy problem (2.7), to prove the first part of the theorem it suffices to show that, for arbitrarily fixed constants \(C_0,C_1,\ldots,C_{n-1}\), the integral equation (2.1) has a solution in the class \(\Gamma\). Fix arbitrarily the constants \(C_0,C_1,\ldots,C_{n-1}\) and, for the integral equation (2.1), construct successive approximations

\[ \varphi_0(x)\equiv T(x),\qquad \varphi_m(x)\equiv T(x)+U(x;\varphi_{m-1})\quad (m=1,2,\ldots), \tag{2.8} \]

where \(T(x)\) is the collection of terms of equation (2.1) that do not contain the unknown function \(\varphi(x)\), and \(U(x;\varphi)\) denotes the remaining terms of equation (2.1).

From (2.8) it is evident that each of the functions \(\varphi_m(x)\) \((m=0,1,\ldots)\) is defined and continuous on \(I\). From the integrability on \(I\) of the functions \((A_2)\) and \((B_1)\) it follows that the function \(T(x)\in\Gamma\). By the method of complete mathematical induction we verify that \(\varphi_m(x)\in\Gamma\) \((m=0,1,\ldots)\). Introduce on the half-interval \(I\) the function

\[ B(x)\equiv \int_a^x \left\{ |p_1(t)| +\sum_{k=1}^{n-1} a^{-k} \left[ |P_k(t)|+g(t)\sum_{i=0}^{k} t^{\,n-1+k-i} \right] \right\}\,dt. \]

According to its definition, the function \(B(x)\), for \(x\in I\), is nonnegative, continuously differentiable, monotonically nondecreasing, and, moreover, owing to the integrability on \(I\) of the functions \((A_1)\), \((A_2)\), and \((C_1)\), belongs to the class \(\Gamma\). Therefore \(\sup_I B(x)<\infty\). By the method of complete mathematical induction one can show that, for any natural number \(m\), the estimate

\[ |\psi_m(x)|\le N\frac{[B(x)]^{m-1}}{(m-1)!},\qquad x\in I, \]

is valid, where

\[ \psi_m(x)\equiv \varphi_m(x)-\varphi_{m-1}(x)\quad (m=1,2,\ldots); \qquad N=\sup_I|\psi_1(x)|<\infty. \]

Consequently, the series

\[ \varphi_0(x)+\sum_{m=1}^{\infty}\psi_m(x) \tag{2.9} \]

converges absolutely and uniformly on the half-interval \(I\). Since all terms of the series (2.9) belong to the class \(\Gamma\) and the series (2.9) converges uniformly on \(I\),

then the sum, say \(\varphi(x)\), of this series also belongs to \(\Gamma\). Passing in (2.8) to the limit as \(m \to \infty\), we obtain that the function \(\varphi(x)\) satisfies the integral equation (2.1) with arbitrarily fixed constants \(C_0, C_1, \ldots, C_{n-1}\). By contradiction one can show that in the class of functions continuous on the half-interval \(I\), the solution of the integral equation (2.1), for arbitrarily fixed values \(C_0, C_1, \ldots, C_{n-1}\), is unique.

Further, one can show that the integrability on \(I\) of the functions \((A_1)\), \((A_2)\), \((B_1)\), \((C_1)\), and \((D_1)\) guarantees the integrability on \(I\) of the functions \((A)\), \((B)\), \((C)\), and \((D)\). Therefore, since the function \(y(x)\) has an asymptotic parabola of exactly degree \((n-1)\) if and only if there exist finite limits \((Z)\) for \(m=n-1\), with
\[ \lim_{x\to\infty} y^{(n-1)}(x) \ne 0, \]
according to Lemma 1.2, the second part of the theorem is also valid.

Theorem 2.1′ is proved analogously to Theorem 2.1.

Theorem 2.1′. If the functions \((A_1)\), \((A_2)\), \((B_1)\), \((C_1)\), and
\[ \int_a^x x^{n-1} t^{1-n}\, |K_1(x,\tau)|\, d\tau \tag{\(A_1'\)} \]
\[ \int_a^x |Q_k(x,\tau)|\, d\tau \quad (k=1,\ldots,n-1) \tag{\(A_2'\)} \]
\[ \int_a^x x^{n-1}\tau^{\,n-1} h(x,\tau)\, d\tau \tag{\(C_1'\)} \qquad x^{n-1}F(x) \tag{\(D_1'\)} \]
are integrable on the half-interval \(I\), then every solution of equation \((1')\) has an asymptotic parabola of degree not higher than \((n-1)\). Moreover, those solutions for whose initial values \(y^{(i)}(a)\) \((i=0,1,\ldots,n-1)\) the relation \((\beta')\) is fulfilled have asymptotic parabolas of exactly degree \((n-1)\).

Corollary 2.2. If the conditions of the first part of Theorem 2.1 are fulfilled, then at least one solution of the differential equation (1) has an asymptotic parabola of exactly degree \((n-1)\).

Indeed, by virtue of the integrability on the half-interval \(I\) of the functions \((A_1)\), \((A_2)\), and \((C_1)\), there exists on \(I\) the function
\[ A(x) \equiv \int_x^\infty \sum_{j=1}^{n} t^{j-1}\,[|p_j(t)|+g(t)]\,dt, \]
which, for \(x\in I\), is nonnegative, continuous, and tends monotonically to zero as \(x\to\infty\). Consequently, there is a sufficiently large value \(x=x_0>a\) such that the inequality \(A(x_0)\exp A(x_0)<1\) is satisfied. Since the differential equation (1) does not contain the point \(a\), one may take \(x_0=a\), and hence, by virtue of \(T<A(a)\), the inequality \((T)\) is satisfied. Thus, by choosing a sufficiently large value of \(a\), in the case of the differential equation (1) the validity of the inequality \((T)\) is achieved. Let \(y(x)\) be a solution of equation (1) with initial data \(y^{(i)}(a)=0\) \((i=0,1,\ldots,n-2)\) and such \(y^{(n-1)}(a)\ne0\) that
\[ |y^{(n-1)}(a)|>(1-T\exp T)^{-1}\left[\left|\int_a^\infty f(t)\,dt\right|+T\exp T\times \]

\[ \times \sup_I \left|\int_a^x f(t)\,dt\right|+(1+T\exp T)\int_a^\infty |F(t,0)|\,dt \right]. \]

It is not difficult to verify that such initial data satisfy relation \((\beta)\). Therefore, on the basis of Theorem 2.1, the function \(y(x)\) has an asymptotic parabola of exactly degree \((n-1)\).

Examples. 2.1. For the equation

\[ y^{IV}(x)-x^{-4}(8x-1)^{-1}\bigl(x^3y'''-3x^2y''+6xy'-6y\bigr)=0,\qquad x\geq 2^{-1}, \tag{2.10} \]

the integral equation (2.1), for arbitrarily fixed constants \(C_0, C_1, C_2, C_3\), has the solution
\(\varphi(x)\equiv [4/3-(6x)^{-1}]C_3\). Substituting this function into formula (2.5) with \(n=4\) and \(a=2^{-1}\), we obtain all solutions of equation (2.10):
\(y(x)\equiv d_1x+d_2x^2+d_3x^3+d_4(32-x^{-1})\)
(\(d_1,d_2,d_3,d_4\) are constants); they have asymptotic parabolas of degree not higher than the 3rd:
\(y=32d_4+d_1x+d_2x^2+d_3x^3\). The initial values
\(y(2^{-1}), y'(2^{-1}), y''(2^{-1})\), connected by the relation

\[ 8y(2^{-1})-4y'(2^{-1})+y''(2^{-1})=0, \tag{a} \]

and arbitrary \(y'''(2^{-1})\ne 0\) satisfy condition \((\beta)\), and the corresponding solutions
\(y(x)\equiv b_1x^3+b_2x^2+b_3x+b_4(32-x^{-1})\), where
\(b_1=13y'''(2^{-1})/72\),
\(b_2=2^{-1}y'''(2^{-1})-5y''(2^{-1})/18\),
\(b_3=7y''(2^{-1})/48-2^{-1}y''(2^{-1})+y'(2^{-1})\),
\(b_4=-1152^{-1}y'''(2^{-1})\), have asymptotic parabolas of exactly the 3rd degree:
\(y=32b_4+b_3x+b_2x^2+b_1x^3\). Condition \((\beta)\) is essential. Indeed, the initial values
\(y(2^{-1}), y'(2^{-1}), y''(2^{-1})\), connected by relation (a), and \(y'''(2^{-1})=0\), do not satisfy condition \((\beta)\), and the corresponding solutions
\(y(x)\equiv 2^{-1}y''(2^{-1})x^2+[y'(2^{-1})-2^{-1}y''(2^{-1})]x\)
(parabolas of the 2nd degree) have asymptotic parabolas below the 3rd degree.

2.2. For the equation

\[ y'''(x)+x^{-3}(x+1)^{-1}\bigl(x^2y''-2xy'+2y\bigr)- \]

\[ -\int_{2^{-1}}^x x^{-2}(x+1)^{-2}\bigl[\tau y''(\tau)-2y'(\tau)+2\tau^{-1}y(\tau)\bigr]\,d\tau =x^{-4},\qquad x\geq 2^{-1}, \tag{2.11} \]

the integral equation \((2.1')\), for arbitrarily fixed constants \(C_0, C_1, C_2\), has the solution
\(\varphi(x)\equiv C_2+1-(C_2+3/2)x^{-1}\ln[2(x+1)/3]-(2x)^{-1}(1-\ln 2x)\).
Substituting this function into formula (2.5) with \(n=3\) and \(a=2^{-1}\), we obtain all solutions of equation (2.11):
\(y(x)\equiv d_1x+d_2x^2+\)
\(+d_3\{1+(2x^2+3x-x^{-1})\ln[2(x+1)/3]-(2x^2+3x)\ln 2x\}+20x^2/9-x-\)
\(-(36x)^{-1}+[x^2+3x/2-(2x)^{-1}]\ln[2(x+1)/3]-[x^2+3x/2-(6x)^{-1}]\ln 2x\)
(\(d_1,d_2,d_3\) are constants); they have asymptotic parabolas of degree not higher than the 2nd.

2.3. Condition \((\beta')\) is essential, which is confirmed, for example, by the integro-differential equation

\[ y'''(x)+3x^{-5}y'+\int_1^x 12x^{-4}\tau^{-2}y''(\tau)\,d\tau=0,\qquad x\geq 1, \tag{2.12} \]

for which condition \((\beta')\) is not fulfilled, since \(T'=7/4\) (hence the inequality \((T')\) is violated), and all its solutions
\(y(x)\equiv d_1+d_2x+d_3x^{-1}\)
have asymptotic parabolas below the 2nd degree.

Remark 2.2. Example 2.3 shows that if only the conditions of the first part of Theorem 2.1′ are fulfilled, the integro-differential equation (1′) may have no solutions possessing asymptotic parabolas of exactly degree \((n-1)\).

Theorem 2.2. Suppose that 1) the functions \((A_1)\), \((B_1)\), \((C_1)\), \((D_1)\), and \(|P_k(x)|\) \((k=1,\ldots,n-2)\), \((A_3)\) are integrable on the half-interval \(I\); 2) the function \(P_{n-1}(x)\) does not change sign for sufficiently large values of \(x\) and is not integrable on the half-interval \(I\). Then the solutions of equation (1) cannot possess asymptotic parabolas of exactly degree \((n-1)\).

Proof. Suppose the contrary: let \(y(x)\) be a solution of equation (1) with asymptotic parabola \((Y')\), where \(A_{n-1}\ne 0\). Then, by Corollary 2.1, the function \(\varphi(x)\), defined by formula (2.6), is, in the class \(\Gamma_0\), a solution of the integral equation (2.1) for the values \(C_{k-1}\) \((k=1,\ldots,n-1)\), defined by the equalities (2.6′). The functions \(\Phi_k(x)\) \((k=1,\ldots,n-1)\), which are the corresponding expressions in square brackets in formula (2.5), belong to the class \(\Gamma\). Let \(\Phi\) be the limit of the function \(\Phi_{n-1}(x)\) as \(x\to\infty\). We have \(\Phi=(n-2)!\,A_{n-1}\). Since, by assumption, \(A_{n-1}\ne0\), it follows that \(\Phi\ne0\). On the other hand, from the identity

\[ \varphi(x)+\int_a^x p_1(t)\varphi(t)\,dt+\frac{1}{\delta}\int_a^x \sum_{k=1}^{n-2} kP_k(t)\Phi_k(t)\,dt-C_{n-1}- \]

\[ -\frac{1}{\delta}\int_a^x t^{\,n-1}\bigl[f(t)+F(t,\Phi^*(t;\varphi))\bigr]\,dt \equiv \]

\[ \equiv -\frac{n-1}{\delta}\int_a^x P_{n-1}(t)\Phi_{n-1}(t)\,dt \]

by virtue of condition 1) it follows that the function \(P_{n-1}(x)\Phi_{n-1}(x)\) is integrable on the half-interval \(I\). Hence, from condition 2) it follows that \(\Phi=0\). The contradiction obtained completes the proof of the theorem.

Analogously to Theorem 2.2 one proves

Theorem 2.2′. Suppose that the functions \((A_1)\), \((B_1)\), \((C_1)\), \((A_1')\), \((A_2')\), \((C_1')\), \((D_1')\), and \((A_3)\) are integrable on the half-interval \(I\), and condition 2) of Theorem 2.2 is fulfilled. Then the solutions of equation (1′) cannot possess asymptotic parabolas of exactly degree \((n-1)\).

Example 2.4. The equation \(y''-y=0\) satisfies the conditions of Theorem 2.2, and all its solutions do not possess asymptotic parabolas of exactly degree 1 (it has a one-parameter family of solutions with asymptote \(y=0\)).

Theorem 2.3. If the conditions of Theorem 2.2 are fulfilled and, in addition, the function \(x^{1-n}P_{n-1}(x)\) \((A_4)\) is integrable on the half-interval \(I\), then for arbitrarily fixed values of the \((n-1)\) parameters

\[ \sum_{r=s}^{n-2}\sum_{i=0}^{r}\frac{(-1)^{i+s}r!}{(r-s)!\,i!}\,a^{i-s-1}y^{(i)}(a) \qquad (s=0,1,\ldots,n-3), \]

\[ \sum_{i=0}^{n-1}\frac{(-1)^i}{i!}a^i y^{(i)}(a) \tag{*} \]

there exists a unique solution of equation (1) possessing an asymptotic parabola of degree lower than \((n-1)\).

Proof. On the basis of Theorem 2.2, for every solution \(\varphi(x)\in\Gamma\) of the integral equation (2.1) the relation

\[ C_{n-2}+\delta\int_a^\infty \eta^{-n}\varphi(\eta)\,d\eta=0 \tag{2.13} \]

holds.

If we prove that, for arbitrarily fixed values of the constants \(C_0, C_1,\ldots,C_{n-3}, C_{n-1}\), there exists in the class \(\Gamma\) a unique solution of the integral equation which is obtained from equation (2.1) with equality (2.13) taken into account (we denote this integral equation by \((2.1_0)\)), then, on the basis of Lemma 2.1, Corollary 2.1, Remark 2.1, and the one-to-one solvability of the Cauchy problem (2.7) for equation (1) (from (2.7) we find that the constants \(C_s\) \((s=0,1,\ldots,n-3)\) are equal, respectively, to the first \((n-2)\) parameters, and \(C_{n-1}\) to the last parameter from (*)), the validity of the assertion of the theorem will thereby be established. We fix arbitrarily the constants \(C_0, C_1,\ldots,C_{n-3}, C_{n-1}\) and, for the integral equation \((2.1_0)\), construct successive approximations

\[ \varphi_0(x)\equiv T_1(x),\qquad \varphi_m(x)\equiv T_1(x)+U_1(x;\varphi_{m-1}) \quad (m=1,2,\ldots), \tag{2.14} \]

where \(T_1(x)\) is the collection of terms of equation \((2.1_0)\) not containing the unknown function \(\varphi(x)\), and \(U_1(x;\varphi)\) is the remaining terms of equation \((2.1_0)\) (\(U_1(x;\varphi)\) contains, in particular, improper integrals). By the method of complete mathematical induction one can verify that the improper integrals occurring in (2.14) converge absolutely and uniformly on \(I\), and \(\varphi_m(x)\in\Gamma\) \((m=0,1,\ldots)\). The integrability on the half-interval \(I\) of the functions \((A_1)\), \((A_3)\), \((A_4)\), and \((C_1)\) ensures the existence on \(I\) of the function

\[ q(x)\equiv \int_x^\infty \Biggl\{ |p_1(t)|+t^{1-n}|P_{n-1}(t)|+ g(t)\left[1+\sum_{i=0}^{n-1}\frac{t^{\,n-1-i}}{(n-1-i)!}\right]+ \]

\[ +\sum_{k=1}^{n-2}\frac{x^{-k}-t^{-k}}{(n-1-k)!} \left[|P_k(t)|+g(t)\sum_{i=0}^{k}\frac{t^{\,n-1+k-i}}{(k-i)!}\right] \Biggr\}\,dt, \]

which for \(x\in I\) is nonnegative, continuous, and, as \(x\to\infty\), decreases monotonically to zero. Therefore there is a sufficiently large value \(x=x_0\ge a\) such that \(q(x_0)<1\). Without loss of generality one may assume that the inequality \(q(a)<1\) holds. From (2.14) we obtain \(M_m\le M_1 q^{m-1}(a)\) \((m=2,\ldots)\), where \(M_m=\sup_I|\varphi_m(x)-\varphi_{m-1}(x)|<\infty\) \((m=1,2,\ldots)\). Consequently, the sequence of successive approximations (2.14) converges uniformly on the half-interval \(I\) to a certain function \(\varphi(x)\in\Gamma\). Passing in (2.14) to the limit as \(m\to\infty\), we obtain that the function \(\varphi(x)\) satisfies the integral equation \((2.1_0)\). By contradiction one can prove the uniqueness of the solution of the integral equation \((2.1_0)\) (for arbitrarily fixed values of \(C_0, C_1,\ldots,C_{n-3}, C_{n-1}\)) in the class \(\Gamma\).

Analogously to Theorem 2.3 one proves

Theorem 2.3′. If 1) the conditions of Theorem 2.2′ are satisfied and, moreover, the function \((A_4)\) is integrable on the half-interval \(I\);

\[ 2)\quad q(a)+\int_a^\infty\int_a^t \left\{t^{\,n-1}\tau^{\,1-n} \left[|K_1(t,\tau)|+t^{\,1-n}|Q_{n-1}(t,\tau)|+\right.\right. \]

\[ {}+ h(t,\tau)\left(1+\sum_{i=0}^{n-1}\frac{\tau^{\,n-1-i}}{(n-1-i)!}\right)\Bigg] +\sum_{k=1}^{n-2}\frac{a^{-k}-\tau^{-k}}{(n-1-k)!}\times \]

\[ \times\left[|Q_k(t,\tau)|+t^{\,n-1}h(t,\tau)\sum_{i=0}^{k}\frac{\tau^{\,k-i}}{(k-i)!}\right\}d\tau\,dt<1, \]

then for equation \((1')\) the assertion of Theorem 2.3 holds.

Examples. 2.5. The equation

\[ y'''(x)+(2x^4)^{-1}(x^2y''-xy'+y)=0,\quad x\geqslant 1, \tag{2.15} \]

satisfies the conditions of Theorem 2.3, and all its solutions

\[ y(x)\equiv d_1x+d_2(2x^2+x\ln x)+ \]

\[ +d_3\bigl[(2x^2+x\ln x)\exp(2x)^{-1}+2^{-1}xE(x)\bigr], \]

where \(d_1,d_2,d_3\) are constants;

\[ E(x)\equiv\int_1^x t^{-2}\ln t\,\exp(2t)^{-1}\,dt, \]

do not possess asymptotic parabolas of exact degree 2. For arbitrarily fixed values \(C_0=2y(1)-y'(1)\) and \(C_2=y(1)-y'(1)+2^{-1}y''(1)\), equation (2.15) has the unique solution

\[ y(x)\equiv C_0x+2(\exp 2^{-1}-1)^{-1}C_2[x+2x^2+x\ln x- \]

\[ -(2x^2+x\ln x)\exp(2x)^{-1}-2^{-1}xE(x)] \]

with an asymptotic parabola of degree lower than 2:

\[ y=2^{-1}(\exp 2^{-1}-1)^{-1}C_2+[C_0-C_2(\exp 2^{-1}-1)^{-1}E(\infty)]x. \]

2.6. The equation

\[ y'''(x)+(6x^5)^{-1}\left\{2x^3y''+x(1-2x)y'+2(x-1)y-\right. \]

\[ \left.-\int_1^x x\tau^{-2}[\tau^2y''(\tau)-2\tau y'(\tau)+2y(\tau)]\,d\tau\right\}=0,\quad x\geqslant 1, \tag{2.16} \]

satisfies the conditions of Theorem \(2.3'\). For arbitrarily fixed values \(C_0\) and \(C_2\) (see Example 2.5), equation (2.16) has the unique solution

\[ y(x)\equiv(\exp 3^{-1}-1)^{-1}\{C_0[2^{-1}(\exp 3^{-1}-1)-x-3x^2-x\ln x+ \]

\[ +(3x^2+x\ln x)\exp(3x)^{-1}+3^{-1}xE_1(x)]+C_2[2x+6x^2+2x\ln x- \]

\[ -(6x^2+2x\ln x)\exp(3x)^{-1}-2xE_1(x)/3]\}, \]

where

\[ E_1(x)\equiv\int_1^x t^{-2}\ln t\,\exp(3t)^{-1}\,dt, \]

with an asymptotic parabola of degree lower than 2.

From the first part of Theorem 2.1, for \(F(x,Y)\equiv0\), the corresponding result of [6] follows. For \(n=2\) and \(F(x,Y)\equiv0\) (respectively

\(F(x,Y,u)\) is a linear function with respect to \(u\)) from the first part of Theorem 2.1 (respectively, \(2.1'\)) and from Theorems 2.2 and 2.3 (respectively, \(2.2'\) and \(2.3'\)) there follow the corresponding results of [3] (respectively, [9]).

Theorem 2.4. If the functions \((B_1)\), \((C_1)\), \((D_1)\) and \(x^{j+n-2}|p_j(x)|\) \((j=1,\ldots,n)\) \((A_5)\) are integrable on the half-interval \(I\), then all solutions of equation (1) possess asymptotic parabolas of degree not higher than \((n-1)\). Moreover, if the initial data \(y^{(i)}(a)\) \((i=0,1,\ldots,n-1)\) satisfy the relation

\[ \left| \sum_{i=k}^{n-1}\frac{(-1)^{i-k}}{(i-k)!}a^{i-k}y^{(i)}(a) -\frac{(-1)^{n-1-k}}{(n-1-k)!}\int_a^\infty t^{\,n-1-k}U(t)\,dt \right| - \]

\[ -\int_a^\infty \frac{t^{\,n-1-k}}{(n-1-k)!}V(t)\,dt -\left[\sup_I |R(x)|\right] + \]

\[ +S|e^T|\int_a^\infty \frac{t^{\,n-1-k}}{(n-1-k)!}W(t)\,dt>0 \qquad (k=0,1,\ldots,n-1), \tag{\(\gamma\)} \]

where \(U(t)\) is the integrand in \(R(x)\), \(V(t)\) in \(S\), \(W(t)\) in \(T\) (see relation \((\beta)\)), then the solution determined by these data possesses an asymptotic parabola with a coefficient different from zero at \(x^k\) \((k=0,1,\ldots,n-1)\).

Proof. The validity of the first part and of the second part for \(k=n-1\) of this theorem follows from Theorem 2.1. We shall give the proof independently of Theorem 2.1. For an arbitrarily fixed solution \(y(x)\) of equation (1), we shall have the identities

\[ A_k(x)=\sum_{i=k}^{n-1}\frac{(-1)^{i-k}}{(i-k)!}a^{i-k}y^{(i)}(a) +\frac{(-1)^{n-k}}{(n-1-k)!}\times \]

\[ \times\int_a^x t^{\,n-1-k}\,[L(y(t))-f(t)-F(t,Y(t))]\,dt \qquad (k=0,1,\ldots,n-1), \tag{2.17} \]

where \(A_k(x)\) is the expression under the limit sign in \((Z)\) for \(m=n-1\). By the integrability on \(I\) of the functions \((A_5)\), \((B_1)\), \((C_1)\), \((D_1)\) and Lemma 1.2, we obtain that the right-hand sides of the identities (2.17), as \(x\to\infty\), tend to finite limits. Consequently, the function \(y(x)\) possesses an asymptotic parabola of degree not higher than \((n-1)\). Further, as in Lemma 1.2, we establish the validity of the second part of the present theorem.

In an analogous manner one proves

Theorem \(2.4'\). If the functions \((A_5)\), \((B_1)\), \((C_1)\), \((C'_1)\), \((D'_1)\) and

\[ \int_a^x x^{n-1}\times \tau^{j-1}|K_j(x,\tau)|\,d\tau \qquad (j=1,\ldots,n) \]

are integrable on the half-interval \(I\), then for equation \((1')\) the assertions of Theorem 2.4 hold, where for the given case in \((\gamma)\), instead of \(R(x)\), \(S\), and \(T\), there stand respectively \(R_1(x)\), \(S'\), and \(T'\); \(U(t)\) is the integrand in \(R_1(x)\), \(V(t)\) in \(S'\), \(W(t)\) in \(T'\) (see relation \((\beta')\)).

Example 2.7. For the equation

\[ y'''(x)+6[x^3(x+1)(x+2)(x+3)]^{-1}\times \]

\[ \times (x^2y''-2xy'+2y)=0,\qquad x\geqslant 1, \tag{2.18} \]

the conditions of the first part of Theorem 2.4 are fulfilled, and all its solutions
\(y(x) \equiv d_1x + d_2x^2 + d_3\{10 + 2x^{-1} + 2(x+2)^{-1} + 3(x^2+2x)\ln[(x+2)/x]\}\)
have asymptotic parabolas of degree not higher than the 2nd:
\(y = 16d_3 + (d_1 + 6d_3)x + d_2x^2\). The initial data \(y^{(i)}(1)\) \((i = 0, 1, 2)\), connected by the relations \(y(1) = y'(1) = 2y''(1) \ne 0\), satisfy condition \((\gamma)\) for \(k = 1\), and the solutions determined by these data,
\(y(x) \equiv 2048^{-1}\{(1142 - 162\ln 3)x + (564 - 81\ln 3)x^2 + 270 + 54x^{-1} + 54(x+2)^{-1} + 81(x^2+2x)\ln[(x+2)/x]\}y(1)\), have asymptotic parabolas with nonzero coefficients at \(x\):
\(y = 2048^{-1}[432 + (1304 - 162\ln 3)x + (564 - 81\ln 3)x^2]y(1)\). Condition \((\gamma)\) is essential. Indeed, the initial data \(y^{(i)}(1)\) \((i = 0, 1, 2)\), connected by the relations
\(y(1) = y'(1) = (372 + 81\ln 3)y''(1)/512 \ne 0\), do not satisfy condition \((\gamma)\) for \(k = 1\), and the solutions determined by such data
\(y(x) \equiv 1024^{-1}\{(564 - 81\ln 3)x^2 - 162x + 270 + 54x^{-1} + 54(x+2)^{-1} + 81(x^2+2x)\ln[(x+2)/x]\}y''(1)\)
have asymptotic parabolas not containing \(x\):
\(y = 3[144 + (188 - 27\ln 3)x^2]y''(1)/1024\).

§ 3. ON THE EXISTENCE OF SOLUTIONS OF EQUATIONS (1) AND (1′) WITH GIVEN ASYMPTOTIC PARABOLAS OF DEGREE \((n-1)\)

Let a parabola \((Y')\) be given. We shall establish the existence, in the class \(C^n\), of a solution \(y(x)\) of equations (1) and (1′) for which the parabola \((Y')\) is an asymptotic parabola, i.e., which satisfies the limiting conditions

\[ \lim_{x\to\infty} \frac{1}{k!}\sum_{i=k}^{n-1} \frac{(-1)^{i-k}}{(i-k)!}\,x^{i-k}y^{(i)}(x) = A_k \qquad (k=0,1,\ldots,n-1). \tag{2} \]

Analogously to Lemma 2.1, one proves

Lemma 3.1. For the solvability of problem (1), (2) in the class \(C^n\), it is necessary and sufficient that the integral equation

\[ \begin{aligned} \varphi(x) = A_0 + \int_x^\infty \Bigg\{& p_1(t)\varphi(t) + \frac{1}{\delta}\sum_{k=1}^{n-1} kP_k(t) \left[(k-1)!A_k -\delta\frac{(-1)^{\,n-1-k}}{(n-1-k)!} \int_t^\infty \eta^{\,n-1-k}\varphi(\eta)\,d\eta\right] \\ &\left. -\frac{1}{\delta}t^{\,n-1}\bigl[f(t)+F(t,\Psi^*(t;\varphi))\bigr]\right\}\,dt, \end{aligned} \tag{3.1} \]

\[ \Psi^* = \left(\Psi,\ \Psi',\ \ldots,\ \Psi^{(n-1)}\right), \]

where

\[ \Psi(x;\varphi)\equiv \sum_{k=1}^{n-1}\frac{x^k}{(k-1)!} \left[ (k-1)!A_k -\delta\frac{(-1)^{\,n-1-k}}{(n-1-k)!} \int_x^\infty \eta^{\,n-1-k}\varphi(\eta)\,d\eta \right], \]

have a solution in the class \(\Gamma_0\).

An analogous lemma holds for problem (1′), (2) (we shall denote the corresponding integral equation by (3.1′)).

Corollary 3.1. If \(\varphi(x)\) is a solution of the integral equation (3.1) (respectively (3.1′)) in the class \(\Gamma_0\), then the function

\[ y(x)=\sum_{k=1}^{n-1}\frac{x^k}{(k-1)!} \left[ (k-1)!A_k -\delta\frac{(-1)^{\,n-1-k}}{(n-1-k)!} \int_x^\infty \eta^{\,n-1-k}\varphi(\eta)\,d\eta \right] \tag{3.2} \]

is a solution of problem (1), (2) (respectively \((1'), (2)\)) in the class \(C^n\). If \(y(x)\) is a solution of problem (1), (2) (respectively \((1'), (2)\)) in the class \(C^n\), then the function (2.6) is a solution of the integral equation (3.1) (respectively \((3.1')\)) in the class \(\Gamma_0\). The number of distinct solutions of problem (1), (2) (respectively \((1'), (2)\)) in the class \(C^n\) is equal to the number of distinct solutions of the integral equation (3.1) (respectively \((3.1')\)) in the class \(\Gamma_0\).

Theorem 3.1. If the functions \((A_1)\), \(P_k(x)\) \((A_6)\), \(x^{-k}|P_k(x)|\) \((k=1,\ldots,n-1)\), \((A_7)\), \((B_1)\), \((C_1)\), and \((D_1)\) are integrable on the half-interval \(I\), then problem (1), (2), with arbitrary fixed values \(A_0,A_1,\ldots,A_{n-1}\), has a unique solution in the class \(C^n\).

Proof. On the basis of Lemma 3.1 and Corollary 3.1 it is enough to prove that the integral equation (3.1), for arbitrary fixed constants \(A_0,A_1,\ldots,A_{n-1}\), has a unique solution in the class \(\Gamma_0\). The latter assertion is proved analogously to the proof of the existence of a unique solution in the class \(\Gamma\) of the integral equation (2.1), for arbitrary fixed constants \(C_0,C_1,\ldots,C_{n-1}\).

For problem \((1'), (2)\) the following holds.

Theorem \(3.1'\). If the functions \((A_1)\), \((A_7)\), \((B_1)\), \((C_1)\), \((A_1')\), \((C_1')\), \((D_1')\), and

\[ P_k(x)+\int_a^x Q_k(x,\tau)\,d\tau\quad (A_6'),\qquad \int_a^x \tau^{-k}|Q_k(x,\tau)|\,d\tau\quad (k=1,\ldots,n-1)\quad (A_7') \]

are integrable on the half-interval \(I\) and, moreover,

\[ \begin{aligned} q_1={}&\int_a^\infty \Biggl\{|p_1(t)|+g(t)+t^{\,n-1}\int_a^t \tau^{1-n}\bigl[|K_1(t,\tau)|+h(t,\tau)\bigr]\,d\tau \\ &\quad+\sum_{k=1}^{n-1}\frac{1}{(n-1-k)!}\Biggl[t^{-k}|P_k(t)|+\int_a^t \tau^{-k}|Q_k(t,\tau)|\,d\tau+t^{\,n-1}\\ &\qquad\qquad\times\sum_{i=0}^k \frac{1}{(k-i)!}\left(t^{-i}g(t)+\int_a^t \tau^{-i}h(t,\tau)\,d\tau\right)\Biggr]\Biggr\}\,dt<1, \end{aligned} \tag{q_1} \]

then problem \((1'), (2)\), with arbitrary fixed values \(A_0,A_1,\ldots,A_{n-1}\), has a unique solution in the class \(C^n\).

Remark 3.1. In the case where, in the prescribed parabola \((Y')\) of degree \(n-1\), the coefficients \(A_k=0\) \((k=s,\ldots,n-1;\ 0\le s\le n-1)\), for the unique solvability of problem (1), (2) (respectively \((1'), (2)\)) in the class \(C^n\), instead of the integrability on \(I\) of all the functions \((A_6)\) (respectively \((A_6')\)), it is enough to assume their integrability on \(I\) for \(k=1,\ldots,s-1\) [for \(s=0,1\) there is in general no need to assume integrability on \(I\) of the functions \((A_6)\) (respectively \((A_6')\))], and instead of the integrability on \(I\) of the function \((C_1)\) (respectively the functions \((C_1)\) and \((C_1')\))—the integrability on \(I\) of the functions \(x^{n-2+s}g(x)\) \((1\le s\le n-1)\) (respectively

\[ x^{n-1}\left[x^{s-1}g(x)+\int_a^x \tau^{s-1}h(x,\tau)\,d\tau\right]\quad (1\le s\le n-1)). \]

From Theorems 2.1 and 3.1 (respectively \(2.1'\) and \(3.1'\)) it follows:

Corollary 3.2. If the functions \((A_1)\), \((A_2)\), \((B_1)\), \((C_1)\), and \((D_1)\) are integrable on the half-interval \(I\) (respectively \((A_1)\), \((A_2)\), \((B_1)\), \((C_1)\), \((A_1')\), \((A_2')\),

\((C_1')\) and \((D_1')\), and the condition \((q_1)\) is satisfied, then between the integral curves of equation (1) (respectively, \((1')\)) in the class \(C^n\) and the parabolas of degree \((n-1)\) of the form \((Y')\) there exists a one-to-one correspondence under which, for each integral curve, there is a unique parabola as its asymptotic parabola of degree \((n-1)\), and each parabola serves as the asymptotic parabola of degree \((n-1)\) of a unique integral curve.

Examples. 3.1. Find a solution of the equation

\[ y^{IV}(x)+2x^{-5}\left(x^2y''-2xy'+2y\right)=x^{-6}(2x+1),\quad x\geq a>0, \tag{3.3} \]

with asymptotic parabola of the 3rd degree \(y=b_1x+b_2x^2+b_3x^3\), where \(b_3=0\), and \(b_1\) and \(b_2\) are arbitrary prescribed constants. The corresponding integral equation (3.1) has the solution \(\varphi(x)\equiv(3x)^{-1}\). Substituting this function into formula (3.2) for \(n=4\), we obtain the desired solution of equation (3.3):

\[ y(x)=b_1x+b_2x^2+(12x)^{-1}. \]

3.2. Find a solution of the equation

\[ y'''(x)-x^{-3}(x+1)^{-1}\left(x^2y''-2xy'+2y\right)+ \int_2^x x^{-4}(\tau+1)^{-1}\times \]

\[ \times\left[\tau^2y''(\tau)-2\tau y'(\tau)+2y(\tau)\right]\,d\tau =x^{-4},\quad x\geq2, \tag{3.4} \]

with asymptotic parabola \(y=x^2\). The corresponding integral equation \((3.1')\) has the solution \(\varphi(x)\equiv -3/4(x+1)\). Substituting this function into (3.2) for \(n=3\), we obtain the desired solution of equation (3.4):

\[ y(x)=x^2-3x/2-3/4+3x(x+1)\ln[(x+1)/x]/2. \]

3.3. Let us give an example showing the essential nature of condition \((q_1)\). Suppose a parabola \(y=A_0+A_1x+A_2x^2+A_3x^3\) \((A_3\ne0)\) is given. The equation

\[ y^{IV}(x)+5x^{-6}y'''+\int_1^x25x^{-6}\tau^{-1}y'''(\tau)\,d\tau=0,\quad x\geq1, \tag{3.5} \]

satisfies the conditions of Theorem \(3.1'\), with the exception of condition \((q_1)\), since here \(q_1=7\), and for none of its solutions
\(y(x)=d_1+d_2x+d_3x^2+d_4x^{-2}\)
(\(d_1,d_2,d_3,d_4\) are constants) is the given parabola an asymptotic parabola.

From Theorem \(3.1'\), for \(n=2\) and \(F(x,Y,u)\) a linear function with respect to \(u\), there follows the corresponding result of [11].

References

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4. Haupt O. Math. Zeit., 48, 289—292, 1942.
5. Wilkins J. E. Bull. Amer. Math. Soc., 50, No. 6, 388—394, 1944.
6. Соболь И. М. ДАН СССР, 61, No. 2, 219—222, 1948.
7. Hartman Ph., Wintner A. Amer. J. Math., 75, No. 4, 717—730, 1953.
8. Быков Я. В. On some problems in the theory of integro-differential equations. Frunze, 1957, pp. 1—328.
9. Ведь Ю. А. Collection “Investigations on integro-differential equations in Kirghizia,” vol. I. Publishing House of the Academy of Sciences of the Kirghiz SSR, Frunze, 1961, pp. 77—102.
10. Ведь Ю. А. Ibid., pp. 103—110.
11. Ведь Ю. А. Collection “Investigations on integro-differential equations in Kirghizia,” vol. II. Publishing House of the Academy of Sciences of the Kirghiz SSR, Frunze, 1962, pp. 151—165.

Received by the editors
30 July 1965

Institute of Physics and Mathematics,
Academy of Sciences of the Kirghiz SSR