Abstract
In this paper, we discuss third-order full nonlinear singularly perturbed vector boundary value problems. We first present the existence of solutions for the nonlinear vector boundary value problems without perturbation by using the upper and lower solutions method and topological degree theory. Then the existence, uniqueness and asymptotic estimates of solutions for the singularly perturbed vector boundary value problems are established by constructing appropriate a lower solution–upper solution pair, as well as analysis technique. Some known results are extended.
Highlights
In the past few decades, nonlinear boundary value problems (BVPs) and singularly perturbed boundary value problems (SPBVPs) have been studied widely [1,2,3,4,5,6,7,8,9,10,11]
The geometric singular perturbation theory has received a great deal of interests in studying the Burgers–KdV equation [13], the vector-disease model [14], the perturbed BBM equation [15], the perturbed Camassa–Holm equation [16] and the perturbed shallow water wave model [17] etc
The boundary value problems in the above-mentioned references are all scalar and little work has been published for vector systems [18,19,20]
Summary
In the past few decades, nonlinear boundary value problems (BVPs) and singularly perturbed boundary value problems (SPBVPs) have been studied widely [1,2,3,4,5,6,7,8,9,10,11]. Du et al [9] were concerned with a more generalized third-order singularly perturbed differential equations with multi-point boundary conditions and obtained the existence and uniqueness as well as the asymptotic estimates of solutions. X(t) = T1x has solutions x(t), it is clear that there exists some x(t) ∈ C3([0, 1], RN ) satisfying (3.10).
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