UPPER AND LOWER SOLUTIONS FOR Φ-LAPLACIAN THIRD-ORDER BVPs ON THE HALF LINE

Abstract

In this paper, we investigate the existence of positive solution for a class of singular third-order boundary value problem associated with a φ-Laplacian operator and posed on the positive half-line:where µ ≥ 0. By using the upper and lower solution approach and the fixed point theory, the existence of positive solutions is proved under a monotonic condition on f. The nonlinearity f may be singular at x = 0. An example of application is included to illustrate the main existence result. RESUMENEn este artículo investigamos la existencia de una solución positiva de una clase de problema singular de valores de frontera de tercer-orden asociado con el operador φ-Laplaciano y colocado sobre la semirecta real positiva:donde µ ≥ 0. Usando la técnica de sub y súper soluciones y la teoría del punto fijo, se prueba la existencia de soluciones positivas bajo una condición de monotonicidad sobre f. La nolinealidad f puede ser singular en x = 0. Se incluye un ejemplo de aplicación para ilustrar el resultado principal de existencia.

Positive solution for singular third-order BVPs on the half line with first-order derivative dependence

“…Naturally, in such boundary value problems, the nonlinearity may have a singular dependence on time or on the space variable. This was the case in the papers [3,6,7,8,20,21,27,28,29], which motivated this work.…”

“…Study of existence of positive solutions for third-order bvps has received a great deal of attention and was the subject of many articles, see, for instance, [10,11,12,13,14,21,25,27,28,29,30,31], for the case of finite intervals and [1,2,3,4,6,7,8,9,16,19,20,24,26] for the case posed on the halfline. Naturally, in such boundary value problems, the nonlinearity may have a singular dependence on time or on the space variable.…”

“…Step 1. Existence in the case where (7) holds Let > 0 be such that (f 0 + ) < Γ . For such a , there exists R 1 > 0 such that f(t, e kt w, e kt z) ≤ (f 0 + )(w + z) for all w, z with |(w, z)| ≤ R 1 and let…”

Singular $\phi $-Laplacian third-order BVPs with derivative dependance

“…(b) Problem (1.1) is considered in [11] when f does not depend on the first derivative. When f depends as well on the first derivative, it has also been studied in [10] via a topological method; the hypotheses on the nonlinearity rather involve the growth of the function f (t, (1+t)x, y).…”

“…However problems with higher-order differential equations on [0, +∞) have not been so extensively investigated; we can only cite [13], [14], [18], and [19]. When f does not depend on the first derivative, problem (1.1) in investigated in [11].…”