| Abstract: | In this paper, the existence and uniqueness of solution of the limit boundary value problem
$\\ddot x = f(t,x)g(\dot x)\]$(F)
$\a\dot x(0) + bx(0) = c\]$(A)
$\x( + \infty ) = 0\]$(B)
is considered, where $\f(t,x),g(\dot x)\]$ are continuous functions on $\\{ t \ge 0, - \infty < x,\dot x < + \infty \} \]$ such that the uniqueness of solution together with thier continuous dependence on initial value are ensured, and assume: 1)$\f(t,0) \equiv 0,f(t,x)/x > 0(x \ne 0);\]$; 2) f(t,x)/x is nondecreasing in x>0 for fixed t and non-increasing in x<0 for fixed t, 3)$\g(\dot x) > 0\]$,
In theorem 1, farther assume: 4) $\\int\limits_0^{ \pm \infty } {dy/g(y) = \pm \infty } \]$
Condition (A) may be discussed in the following three cases
$x(0)=p(p \neq 0)$(A_1)
$\x(0) = q(q \ne 0)\]$(A_2)
$\x(0) = kx(0) + r{\rm{ }}(k > 0,r \ne 0)\]$(A_3)
The notation $\f(t,x) \in {I_\infty }\]$ will refer to the function f(t,x) satisfying $\\int_0^{ + \infty } {\alpha tf(t,\alpha )dt = + \infty } \]$ for each $\alpha \neq 0$,
Theorem. 1. For each $p \neq 0$, the boundary value problem (F), (A_1), (B) has a solution if and only if $f(t,x) \in I_{\infty}$
Theorem 2. For each$q \neq 0$, the boundary value problem (F), (A_2), (B) has a solution if and only if $f(t, x) \in I_{\infty}$.
Theorem 3. For each k>0 and $r \neq 0$, the boundary value problem (F), (A_3), (B) has a solution if and only if f(t, x) \in I_{\infty},
Theorem 4. The boundary value problem (F), (A_j), (B) has at most one solution for j=l, 2, 3. . |
