Abstract
This paper is concerned with the mixed boundary value problem of the second order singular ordinary differential equation[Φ(ρ(t)x'(t))]' + f(t, x(t), x'(t)) = 0, t ∈ R,limt→−∞ x(t) = ∫−∞+∞ g(s, x(s), x'(s)) ds,limt→+∞ ρ(t)x'(t) = ∫−∞+∞h(s, x(s), x' (s)) ds.Sufficient conditions to guarantee the existence of at least one positive solution are established. The emphasis is put on the one-dimensional p-Laplacian term [Φ(ρ(t)x'(t))]' involved with the nonnegative function ρ satisfying ∫−∞+∞1/ρ(s) ds = +∞.
Highlights
The study of multi-point boundary value problems for linear second order ordinary differential equations was initiated by Il’in and Moiseev [1], motivated by the work of Bitsadze and Samarskii on nonlocal linear elliptic boundary problems [2]
Ds; (ii) T : P → P is well defined; (iii) T : P → P is completely continuous; (iv) x ∈ P is a positive solution of boundary value problems (BVPs) (6) if and only if x is a fixed point of T in P
T : P → P is well defined. (iii) we prove that T is completely continuous
Summary
The study of multi-point boundary value problems for linear second order ordinary differential equations was initiated by Il’in and Moiseev [1], motivated by the work of Bitsadze and Samarskii on nonlocal linear elliptic boundary problems [2]. In [7], a class of boundary value problems for the second order nonlinear ordinary differential equations on the whole line is studied. To the author’s knowledge, there is no paper concerned with the existence of positive solutions of the boundary value problems to the second order differential equations ρ(t)Φ x (t) + f t, x(t), x (t) = 0, t ∈ (−∞, +∞) =: R. Limt→−∞ x (t) = limt→+∞ x (t), under adequate hypothesis and using the Bohnenblust–Karlin fixed point theorem, the existence of solutions of BVP (3) is established. We consider the mixed boundary value problem for second order singular differential equation on the whole line with quasi-Laplacian operator (6).
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