Some new critical point theorems and application

Abstract

In this memory, we have studied new theorems : First, we proved the critical point theorem without satisfying the Palais-Smale condition, ensuring the existence of a critical point. Then, we demonstrated the existence of a critical point in Riesz-Banach space ordered by a cone k, followed by applying the abstract result to the following problem : ( −(p(t)u 0 (t))0 = f(t, u(t)), a.e.t ∈ [0, +∞) u(0) = u(+∞) = 0, (3) Where f : [0, +∞)R → R is a Caratheodory function, and may change sign, p : [0, +∞) → (0, +∞) satisfies 1 p ∈ L 1 [0, +∞), and Z +∞ 0 Z +∞ t 1 p(s) ds dt < +∞. In the second case, we established new theorems of fixed points in Hilbert spaces for potential α−positively homogeneous operators using the weak Ekeland principle, then applied our abstract result to the following problem : ( −u 00(t) = q(t)f(u(t))), t ∈ (0, 1), u(0) = u(1) = 0, (4) Where f : R → R is a continuous function, q ∈ L 2 (0, 1).