Three critical point theorem with comparative condition in strictly convex
Banach spaces
mgr inż. Piotr Kowalski
Theorem (Cabada,Iannizzotto - 2012)
Let (X , k·k) be a uniformly convex Banach space with strictly convex dual space, J ∈ C1 (X ) be a functional with compact derivative, x0, x1 ∈ X ,
p, r ∈ R be such that p > 1 and r > 0. Let the following conditions be satisfied:
J(x)
1. liminf kxk
p ≥ 0
kxk→+∞
2. inf J(x) <
x∈X
inf
kx−x0k≤r
J(x)
3. kx1 − x0k < r and J(x1) <
inf
kx−x0k=r
J(x).
Then there exists a nonempty open set A ⊆ (0, +∞) such that for all λ ∈ A the functional x →
kx−x0kp
p
+ λJ(x) has at least three critical points in X .
Theorem (Galewski,Wieteska - 2012)
Let (X , k·k) be a uniformly convex Banach space with strictly convex dual space, J ∈ C1 (X , R) be a functional with compact derivative,
µ ∈ C1 (X , R+) be a convex coercive functional such that its derivative is an operator µ0 : X → X ∗ admitting a continuous inverse, let e
x ∈ X and
r > 0 be fixed. Assume that the following conditions are satisfied:
J(x)
1. liminf µ(x)
≥0
kxk→∞
2. inf J(x) < inf J(x)
µx≤r
x∈X
3. µ (e
x ) < r and J(e
x ) < inf J(x).
µ(x)=r
Then there exists a nonempty open set A ⊆ (0, +∞) such that for all λ ∈ A the functional µ + λJ has at least three critical points in X .
Lemma
Let (X , k·k) be a reflexive Banach space, I ⊆ R+ be an interval, Φ ∈ C1 (X ) be a sequentially weakly l.s.c. functional whose derivative admits a
continuous inverse, J ∈ C1 (X ) be a functional with compact derivative. Moreover, assume that there exist x1, x2 ∈ X and σ ∈ R such that:
1. Φ(x1) < σ < Φ(x2)
1 )+(σ−Φ(x1 ))J(x2 )
2. inf J(x) > (Φ(x2)−σ)J(x
Φ(x2)−Φ(x1)
Φ(x)≤σ
3. lim [Φ(x) + λJ(x)] = +∞ for all λ ∈ I .
kxk→∞
Then there exists a nonempty open set A ⊆ I such that for all λ ∈ A the functional Φ + λJ has at least three critical points in X.
Theorem (Galewski,Kowalski - to appear)
Corollary (Comparative condition corollary)
Let (X , k·k) be a uniformly convex Banach space with strictly
convex dual space, J ∈ C1 (X , R) be a functional with compact
derivative. µ1 ∈ C1 (X , R) and µ2 ∈ C1 (X , R+) be a convex
coercive functional such that its derivative is an operator
µ02 : X → X ∗ admitting a continuous inverse, let y ∈ X and r > 0
be fixed. Assume the following conditions are satisfied:
1. liminf µJ(x)
≥
0
(x)
2
kxk→∞
2. inf J(x) < inf J(x)
x∈X
Let (X , k·k) be a uniformly convex Banach space with strictly
convex dual space, J ∈ C1 (X , R) be a functional with compact
derivative. µ1 ∈ C1 (X , R) is sequentially w.l.s.c and coercive,
µ2 ∈ C1 (X , R+) be a convex coercive functional. µ1 derivative is
an operator µ01 : X → X ∗ admitting a continuous inverse, let
y ∈ X and r > 0 be fixed. Assume the following conditions are
satisfied:
J(x)
1. liminf µ2(x) ≥ 0
kxk→∞
µ1x≤r
3. µ2 (e
x ) < r and J(e
x ) < inf J(x).
µ2(x)=r
4. For all x ∈ X if µ2(x) ≤ r then µ1(x) ≤ µ2(x).
Then there exists a non empty open set A ⊂ (0, +∞) such that
for all λ ∈ A the functional x → µ2(x) + λJ(x) has at least three
critical points in X .
Questions
Existence result for µ1 + λJ
I Implications on each assumption
I µ2 represent properties, J represents shapes. Can that properties
be mixed?
I µ2 is nonnegative. Is this property important?
I
2. inf J(x) < inf J(x)
x∈X
µ1(x)≤r
3. µ2 (y ) < r and J(y ) < inf J(x)
µ2(x)=r
4. ∀ µ2 (x) ≤ r ⇒ µ1 (x) ≤ µ2 (x) and µ1 (x) ≥ µ2 (x) for
x∈X
kxk ≥ M, where M > 0 is some constant.
5. J is convex on the convex hull of B := {x ∈ X : µ1(x) ≤ r }
Then there exists a non empty open set A ⊂ (0, +∞) such that
for all λ ∈ A the functional x → µ1 (x) + λJ(x) has at least three
critical points.
Conclusions
Three critical point Theorem version without nonnegative
assumption on µ1
I Improvement in the testability of conditions
I ”Shift” of the convexity condition from µ to J.
I
Reference
B. Ricceri, A three critical points theorem revisited, Nonlinear Anal. 70 (2009), 3084–3089.
A. Cabada, A. Iannizzotto, S. Tersian, Multiple solutions for discrete boundary value problems. J. Math. Anal. Appl. 356 (2009), no. 2, 418–428.
A. Cabada, A. Iannizzotto, A note on a question of Ricceri, Appl. Math. Lett. 25 (2012), 215-219.
M. Galewski, R. Wieteska, A note on the multiplicity of solutions to anisotropic discrete BVP’s, Appl. Math. Lett, DOI: 10.1016/j.aml.2012.11.002, (2012).