Some new critical point theorems and application
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Date
2024-06-11
Journal Title
Journal ISSN
Volume Title
Publisher
Mohamed Boudiaf University of M’sila, Faculty of Mathematics and Informatics, Department of Mathematics
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).
Description
Keywords
Critical point, Fixed point, Ekeland principle, Weak solution, Boundary problem