tailieunhanh - Solvability of boundary value problems for coupled impulsive differential equations with one-dimensional p-Laplacians

This paper is concerned with a boundary value problem of impulsive differential systems on the whole line with one-dimensional p-Laplacians. By constructing a weighted Banach space and defining a nonlinear operator, together with Schauder’s fixed point theorem, sufficient conditions to guarantee the existence of at least one solution are established (Theorems –). | Turk J Math (2017) 41: 896 – 917 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Solvability of boundary value problems for coupled impulsive differential equations with one-dimensional p-Laplacians Yuji LIU∗ Department of Mathematics, Guangdong University of Business Studies, Guangzhou, . China Received: • Accepted/Published Online: • Final Version: Abstract: This paper is concerned with a boundary value problem of impulsive differential systems on the whole line with one-dimensional p-Laplacians. By constructing a weighted Banach space and defining a nonlinear operator, together with Schauder’s fixed point theorem, sufficient conditions to guarantee the existence of at least one solution are established (Theorems –). Two examples are given to illustrate the main results. Key words: Impulsive differential system on whole line, boundary value problem, increasing odd homeomorphisms, sub-Carath´eodory function, discrete Carath´eodory function, fixed point theorem 1. Introduction Boundary-value problems for linear second order ordinary differential equations were initiated by Il’in and Moiseev [17] and studied by many authors; see the textbooks [1, 16], the papers [9, 26, 27], and the references therein. In recent years, many authors have studied the existence of positive radial solutions for elliptic systems in annular/exterior domains, which is equivalent to that of positive solutions for the corresponding systems of ordinary differential equations (see [12–15, 19, 20] and the references therein). The usual method used is the fixed point theorems of cone expansion/compression type, the upper and lower solutions method, and the fixed point index theory in cones. In [10, 23], the following system and its special case were discussed: [ϕp (u′ (t))]′ + λh1 (t)f (t, u(t), v(t)) = 0, t ∈ (0, 1), [ϕp (v ′ (t))]′ + µh2 (t)g(t, u(t), v(t)) = 0, t ∈ (0,