Abstract
In this article, we prove existence of solutions for a nonlocal boundary value problem with nonlinearity in a nonlocal condition. Our method is based upon Mawhin's coincidence theory.
Highlights
In this paper we consider the following ordinary differential equation x = f (t, x) (1.1)with the nonlocal condition h x(s) dg(s) = 0, (1.2)where f : [0, 1] × Rk → Rk is continuous, g = (g1, . . . , gk) : [0, 1] → Rk has bounded variation, h : Rk → Rk is continuous and x(s) dg(s) =x1(s) dg1(s), . . . , xk(s) dgk(s) .The subject of nonlocal boundary conditions for ordinary differential equations has been a topic of various studies in mathematical articles for many years
The multi-point conditions were studied at first and this kind of conditions has been initiated in [15], the significantly nonlocal conditions with the values of the unknown function occurring over the entire domain became the subject of interest
The methods are typical: searching for the fixed point of integral operator using contraction principle, Schauder’s fixedpoint theorem, topological-order methods, e.g. basing on the cone expansion and compression theorem, or the Leray–Schauder degree of compact mappings or the Mawhin degree of coincidence. In this paper both differential equations and boundary conditions are nonlinear which somehow forces to the use of the degree of coincidence – the linear part x has the nontrivial kernel
Summary
The methods are typical: searching for the fixed point of integral operator using contraction principle, Schauder’s fixedpoint theorem, topological-order methods, e.g. basing on the cone expansion and compression theorem, or the Leray–Schauder degree of compact mappings or the Mawhin degree of coincidence (for the multi-point boundary value problem in [16]). In this paper both differential equations and boundary conditions are nonlinear which somehow forces to the use of the degree of coincidence – the linear part x has the nontrivial kernel. In [21], the authors considered another kind of boundary conditions, namely where a, b ≥ 0
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