Abstract
In this paper we consider a coupled system of second-order boundary value problems with nonlocal, nonlinear boundary conditions. By imposing only a condition of asymptotic sublinear growth on the nonlinear boundary functions, we are able to achieve generalizations over existing works and, in particular, we allow for the nonlocal terms to be able to be represented as Lebesgue-Stieltjes integrals possessing signed Borel measures. Because we only suppose the sublinearity of the nonlinear boundary functions at positive infinity, we also remove many of the restrictive growth assumptions found in other recent works on closely related problems. We conclude with a numerical example to explicate the consequences of our main result.
Highlights
In this paper we consider a system of nonlocal boundary value problems with nonlinear boundary conditions
The nonlocal terms here are very general, being as they are realized as Lebesgue-Stieltjes integrals – that is, (1.2)
Note that condition (1.3) implies that Hi may grow either sub- or superlinearly at z = 0. These two relatively simple modifications allow for considerably weaker conditions on problem (1.1), for we may assume that each of the measures μα1 and μα2 is signed and that neither H1 nor H2 is sublinear at z = 0, assumptions that seem to be made in most problems related to (1.1) as we indicate in the sequel
Summary
In this paper we consider a system of nonlocal boundary value problems with nonlinear boundary conditions. Select numbers ε12, ε22 > 0 sufficiently small such that each of the following inequalities holds. These inequalities may be satisfied because condition (H7) holds.
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