A. Lomtatidze and H. Štěpánková

On Sign Constant and Monotone Solutions of Second Order Linear Functional Differential Equations

abstract:
In this paper, the question on the existence and uniqueness of a constant sign solution of the initial value problem
$$u''(t)=\ell(u)(t)+q(t),\quad u(a)=c_1,\quad u'(a)=c_2$$ is studied. More precisely, the nonimprovable effective sufficient conditions for a linear operator $\ell:C([a,b];\R)\rightarrow L([a,b];\R)$ are established guaranteeing that the considered problem with $q\in L([a,b];\R_+)$ and $c_1,c_2\in\R_+$ has a unique solution and this solution is nonnegative. The question on the existence and uniqueness of a monotone solution of the same problem is discussed, as well.

Mathematics Subject Classification: 34K06, 34K10

Key words and phrases: Second order linear functional differential equation, initial value problem, nonnegative solution, monotone solution