On new critical point theorems without the Palais–Smale condition

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On new critical point theorems without the Palais–Smale condition

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Mabrouk Briki a, Toufik Moussaoui a,*, Donal O’Regan b,c a

Laboratory of Fixed Point Theory and Applications, École Normale Supérieure, Kouba, Algiers, Algeria School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland c NAAM Research Group, King Abdulaziz University, Jeddah, Saudi Arabia

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A R T I C L E

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I N F O

A B S T R A C T

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Article history:

In this paper we prove new theorems on critical point theory based on the weak Ekeland’s

Received 20 May 2015

variational principle.

Received in revised form 4

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September 2015

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© 2015 Production and hosting by Elsevier B.V. on behalf of Mansoura University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/

Accepted 4 September 2015

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licenses/by-nc-nd/4.0/).

Available online

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Keywords:

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Weak Ekeland’s variational

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principle

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Almost critical point

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Critical point

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1.

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Introduction

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The weak Ekeland variational principle is an important tool in critical point theory and nonlinear analysis, and in this paper we will use this principle to establish some new results in critical point theory.

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Lemma 1 (Weak Ekeland variational principle). Let (E, d) be a complete metric space and let ϕ : E →  be a lower semicontinuous functional, bounded from below. Then for every ε > 0, there exists a point u* ∈ E such that

ϕ ( u* ) < ϕ ( v ) + ε d ( u*, v ) , ∀v ∈ E such that

v ≠ u*.

54 Definition 1. We say that a functional ϕ ∈ C1 ( E, R ) has a sequence of almost critical points if there exists a sequence ( vn )n in E such that φ′(vn) → 0 in E* as n → ∞.

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2.

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Preliminaries

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We need the following weak Ekeland variational principle which can be found for example in Ref. 1.

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Lemma 2 (Minimization principle). ( [2]) Let E be a Banach space and ϕ : E →  a functional, bounded from below and Gâteaux differentiable. Then, there exists a minimizing sequence ( vn )n of almost critical points of φ in the sense that

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* Corresponding author. Tel.: +213560313516. E-mail address: [email protected] (T. Moussaoui). http://dx.doi.org/10.1016/j.ejbas.2015.09.003 2314-808X/© 2015 Production and hosting by Elsevier B.V. on behalf of Mansoura University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Mabrouk Briki, Toufik Moussaoui, Donal O’Regan, On new critical point theorems without the Palais–Smale condition, Egyptian Journal of Basic and Applied Sciences (2015), doi: 10.1016/j.ejbas.2015.09.003

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egyptian journal of basic and applied sciences ■■ (2015) ■■–■■

limϕ ( vn ) = infϕ ( v )

n →+∞

2

v∈E

and

limϕ ′ ( vn ) = 0.

n →+∞

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A slight modification of Theorem 1.26 in Ref. 3 (see also Corollary 4 in this paper) gives the following lemma with a dilatation type condition.

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Lemma 3. Let ( E, . 1 ) be a Banach space and ( F, . 2 ) be a normed space. If A is a closed set in E, f : A → F is continuous, and

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∃k > 0 : ∃θ > 0,

θ

f (x) − f ( y) 2 ≥ k x − y 1

∀x, y ∈ A,

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Proof. We consider the complete metric space J′(E) of E* and define the functional φ on J′(E) by

ϕ : J′ ( E ) →  J ′ ( u ) → ϕ ( J ′ ( u ) ) = J ′ ( u ) E *.

then f(A) is closed.

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3.

Main results

J ′ ( u* ) E * ≤ J ′ ( v ) E * + ε J ′ ( u* ) − J ′ ( v ) E * ,

∀v ∈ E.

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Theorem 1. Let E be a reflexive Banach space, Ω be a bounded and weakly closed set of E with J ∈ C1 ( E,  ), and let J′ be strongly continuous on Ω. Suppose also that J satisfies J′ (ϕ ( u ) ) E * ≤ k J′ ( u ) E * for all u ∈ Ω, where φ : Ω → Ω is a function such that φ(u) ≠ u for all u ∈ Ω, and 0 < k < 1 is a constant . Then J has at least one critical point in Ω.

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Proof. We first show J′(Ω) is closed. Let g ∈ J′ ( Ω ) . There exist a sequence gn ∈ J′ ( Ω ) , such that limn →+∞ gn = g , and so there exist (un) ⊂ Ω with limn →+∞ J′ ( un ) = g . Since Ω is bounded and E is reflexive, there exist ( unk ) ⊂ ( un ) such that unk  u ∈ Ω . Since J′ is strongly continuous then

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g = lim J′ ( unk ) = J′ ( u ) ∈ J′ ( Ω ) . n →+∞

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Then φ is lower semicontinuous and bounded from below on J′(E). Let ε ∈ (0, 1). From the weak Ekeland variational principle, there exists u* in E such that

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Theorem 2. Let E be a Banach space and let the functional J ∈ C1 ( E,  ) with J′(E) a closed set in E* . Suppose also that J admits a sequence of almost critical points. Then J has at least one critical point in E.

We deduce that u* is a critical point of J. If this is not true then J′ ( u* ) ≠ 0 . Let (vn) the sequence of almost critical point of J. Then we obtain that

J′ ( u* ) E* ≤ J′ ( vn ) E* + ε J′ ( u* ) − J′ ( vn ) E*,

∀n ∈ N.

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Because J′(vn) → 0 as n → ∞, by passing to the limit, we obtain that

J ′ ( u* ) E * ≤ ε J ′ ( u* ) E * ,

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which is a contradiction. ■ As a consequence of the last theorem, we obtain the following corollary.

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We consider the complete metric space J′(Ω), and define the functional ψ on J′(Ω) by

Remark 1. The two geometric conditions in the Mountain pass theorem suffice to get a sequence of almost critical points (see Ref. 2).

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ψ : J′ ( Ω ) →  J ′ ( u ) → ψ ( J ′ ( u ) ) = J ′ ( u ) E *.

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Then ψ is lower semicontinuous and bounded from below 1−k on J′(Ω). Let ε = ∈ ( 0, 1 ) . From the weak Ekeland varia1+k tional principle, there exists u* in Ω such that

Corollary 1. Let E be a Banach space, and let J ∈ C1 ( E,  ) satisfy J(0) = 0. Assume that J′(E) is a closed set in E* and there exist positive numbers ρ and α such that

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J(u) ≥ α if u = ρ , there exists e ∈ E such that e > ρ and J(e) < α.

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J ′ ( u* ) E * < J ′ ( v ) E * + ε J ′ ( u* ) − J ′ ( v ) E * ,

∀v ∈ Ω such that

v ≠ u*.

Then J admits at least one critical point u. It is characterized by

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We claim that u* is a critical point of J. If this is not true then J′ ( u* ) ≠ 0 . Now

J ( u ) = infmax J ( γ ( t ) ) γ ∈Γ t∈[0,1]

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where

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J′ ( u ) = 0,

J ′ ( u* ) E *

1+ε < J′ ( v ) E*, 1−ε

∀v ∈ Ω, such that

v ≠ u*,

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Γ = {γ ∈ C ([ 0, 1] , E ) γ ( 0 ) = 0, γ ( 1 ) = e} .

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and in particular for v = ϕ ( u* ), we have

Corollary 2. Let E be a Banach space and let the functional J ∈ C1 ( E,  ) satisfy

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J ′ ( u* ) E * <

1+ε J′ (ϕ ( u* ) ) E*, 1−ε

∃k > 0 : ∃θ > 0, J′ ( u ) − J′ ( v ) E * ≥ k u − v

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∀u, v ∈ E.

(1)

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i.e.,

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Suppose also that J admits a sequence of almost critical points. Then J has at least one critical point in E.

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θ E

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J′ (ϕ ( u* ) ) E* > k J′ ( u* ) E*.

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This contradicts the hypothesis J′ (ϕ ( u ) )

E*

≤ k J′ ( u ) E * . ■

Proof. This is a direct consequence of Theorem 2 using Lemma 3. ■

Please cite this article in press as: Mabrouk Briki, Toufik Moussaoui, Donal O’Regan, On new critical point theorems without the Palais–Smale condition, Egyptian Journal of Basic and Applied Sciences (2015), doi: 10.1016/j.ejbas.2015.09.003

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Corollary 3. Let E be a Banach space, and let J ∈ C1 ( E,  ) with J′(E) a closed set in E* . Suppose that J is bounded from below. Then J has at least one critical point.

J′ ( un ) − J′ ( u ) E = sup J′ ( un ) h − J′ ( u ) h

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h H1 ≤ 1

= sup h H1 ≤ 1

∫

1

0

( f ( t, un ( t ) ) − f ( t, u ( t ) ) ) h ( t ) dt

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Proof. The minimization principle ensures the existence of almost critical points. The conclusion follows from Theorem 2. ■

≤ sup

h H1 ≤ 1

(∫

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Corollary 4. Let E be a reflexive Banach space, let Ω a bounded and weakly closed subset of E with J ∈ C1 ( E, R ), and let J′ be strongly continuous on Ω. Suppose also that J admits a sequence of almost critical point in Ω. Then J has at least one critical point in Ω.

1

0

1 2 2

( ∫ ( f (t, u (t )) − f (t, u (t )) dt ) ) 1

n

0

h2 ( t ) dt

= sup h H1 ≤ 1

)

1 2 1 2 2

( ∫ ( f (t, u (t )) − f (t, u (t )) dt ) )

≤ C sup

h H1 ≤ 1

1

n

0

1 2 2

( ∫ ( f (t, u (t )) − f (t, u (t )) dt ) ) 1

n

0

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Proof. We show J′(Ω) is closed. Indeed let g ∈ J′ ( Ω ) . There exists ( gn ) ⊂ J′ ( Ω ) such that limn →+∞ gn = g , and there exists (un) ⊂ Ω with limn →+∞ J′ ( un ) = g . Since Ω is bounded and E is reflexive, there exists ( unk ) ⊂ ( un ) such that unk  u ∈ Ω . Since J′ is strongly continuous then

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g = lim J′ ( unk ) = J′ ( u ) ∈ J′ ( Ω )

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L2 ( 0,1 )

h

H1

1 2 2

( ∫ ( f (t, u (t )) − f (t, u (t )) dt ) ) . 1

≤C

n

0

Let K be the constant of the continuous embedding of H1(0, 1) in C[0, 1], and note that

f ( t, un ( t ) ) − f ( t, u ( t ) ) ≤ 2

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h

n →+∞

sup

( t , y )∈[0,1]×[ − Kρ , Kρ ]

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f ( t, y ) ,

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Following the same steps in the proof of Theorem 2, we obtain the result. ■

lim f ( t, un ( t ) ) = f ( t, u ( t ) ) .

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n →+∞

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From the Lebesgue dominated convergence theorem, we obtain

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4.

Application

1 2 2

1

lim ∫0 ( f ( t, un ( t ) ) − f ( t, u ( t ) ) dt )

n →+∞

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We consider the functional J defined on E = H ( 0, 1 ) by

)

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= 0, 69

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1

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and so

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J (u) = ∫

1

0

(∫

u( t )

0

)

f ( t, ξ ) dξ dt,

lim J′ ( un ) − J′ ( u ) E = 0.

n →+∞

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where f ∈ C ([ 0, 1] × ,  ) is a continuous function. Suppose that there exist a function φ : R → R and k ∈ ]0, 1[ such that

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1

Finally we show J′ satisfies J′ (ϕ ( u ) ) E * ≤ k J′ ( u ) E * for all u ∈ Ω, where φ is the Nemytskii’s operator associated with ϕ. Now from (f1) we have

1

( f1 ) ∫0 f ( t, φ ( u ( t ) ) ) h ( t ) dt ≤ k ∫0 f ( t, u ( t ) ) h ( t ) dt ,

J′ (ϕ ( u ) ) E = sup

for all u, h ∈ H1 ( 0, 1 ) .

h H1 ≤1

f ( t, u ) = q ( t ) ( u + k ) , φ ( s ) = ks + k2 − k, 2

h H1 ≤1

t ∈ [ 0, 1] , k ∈ ]0, 1[

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1

0

f ( t, u ( t ) ) h ( t ) dt

Theorem 3. Suppose that f satisfies (f1). Then J has at least one critical point.

Uncited references

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[4]

85 REFERENCES

1

J′ ( u ) .h = ∫ f ( t, u ( t ) ) h ( t ) dt, for all We now show Ω = B ( 0, ρ ) ⊂ H1 ( 0, 1 ).

J′

is

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u, h ∈ H1 ( 0, 1 ) .

strongly

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Proof. Note that J is well defined and J ∈ C1 ( H1 ( 0, 1 ) , R ) with

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∫

From Theorem 1, J has at least one critical point in Ω.

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f ( t, φ ( u ( t ) ) ) h ( t ) dt

0

where q is a positive function defined on [0, 1].

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continuous

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= k J ′ ( u ) E .

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∫

≤ k sup

One may take as examples of f and ϕ,

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on

Let (un) a sequence with (un) ⊂ Ω and un  u (Ω is bounded in H1(0, 1)) and note it converges uniformly to u on [0, 1]. Since Ω is weakly closed, u ∈ Ω. Let C be the constant of the continuous embedding of H1(0, 1) in L2(0, 1). We have

[1] Cellina A. Methods of Nonconvex Analysis, Lecture notes in mathematics, 1446; 1989. [2] Jabri Y. The mountain pass theorem, variants, generalizations and some applications. Cambridge University Press; 2003. [3] Rudin W. Functional analysis. McGraw-Hill, Inc; 1991. [4] Kavian O. Introduction à la Thèorie des Points Critiques et Application aux Problèmes Elliptique. Springer-Verlag; 1993.

Please cite this article in press as: Mabrouk Briki, Toufik Moussaoui, Donal O’Regan, On new critical point theorems without the Palais–Smale condition, Egyptian Journal of Basic and Applied Sciences (2015), doi: 10.1016/j.ejbas.2015.09.003

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