Abstract
We establish sufficient conditions for the existence of extremal solutions for a second-order problem on the half-line with discontinuous right-hand side and nonlinear boundary conditions. Our results are new even in the classical case of continuous nonlinearities. We illustrate their applicability with an example.
Highlights
We study the existence of solutions of the nonlinear equation on the half-line x (t) = f (t, x, x ), t ∈ [0, ∞), (1.1)coupled with the functional boundary conditionsL(x(0), x (0), x) = 0, x (+∞) := lim x (t) = B, (1.2) t→+∞where B ∈ R and L : C([0, ∞)) × R2 → R is a continuous function and it is nonincreasing in the second and third variables.Boundary value problems on unbounded intervals are becoming popular in the literature because of their applications to model real world problems in engineering or chemistry, see [1]
We shall assume the existence of well ordered upper and lower solutions on unbounded domains and a Nagumo condition to control the first derivative in order to obtain existence results for (1.1)–(1.2)
We prove the existence of extremal solutions for (1.1)–(1.2) by adapting the arguments in [5] to unbounded domains
Summary
We shall assume the existence of well ordered upper and lower solutions on unbounded domains and a Nagumo condition to control the first derivative in order to obtain existence results for (1.1)–(1.2) Another interesting point in this paper is that we are able to relax the usual definition of lower and upper solutions, cf [2,8,11,12,13], and our main existence result is new even in the classical case of continuous right-hand sides in (1.1). We prove the existence of extremal solutions for (1.1)–(1.2) by adapting the arguments in [5] to unbounded domains This is a new result even for a continuous function f (t, x, x ).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
