We consider the problem of the existence of positive solutions u to the problem
u(n)(x)=g(u(x)), u(0)=u′(0)=...=u(n−1)(0)=0 (g ≥ 0,x > 0, n ≥ 2).
It is known that if g is nondecreasing then the Osgood condition
∫0δ​1/s[s/g(s)]1/nds<∞
is necessary and sufficient for the existence of nontrivial solutions to the above problem. We give a similar condition for other classes of functions g
For an initial value problem u'''(x) = g(u(x)), u(0) = u'(0) = u''(0) = 0, x > 0, some theorems on existence and uniqueness of solutions are established
We study the equation u = k∗g(u) with k such that ln k is convex or concave and g is monotonic. Some necessary and sufficient conditions for the existence of nontrivial continuous solutions u of this equation are given