# On the Equivalence between Two Fixed Point Theorems for Concave-Type Operators

Authors: Mohamed Jleli; Bessem Samet
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this note, we prove the equivalence between a fixed point theorem forφ–h,e–concave operators and a previous one for generalized concave operators.


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## Body

## 1. Introduction

In this note, we show that the fixed point result established by Zhai and Wang [1] for φ–h,e–concave operators is equivalent to a previous result due to Zhai and Wang [2] for generalized concave operators. We first recall some notions regarding Banach spaces partially ordered by a cone. For the convenience of readers, we refer them to [3, 4] for more details.LetE,· be a real Banach space. A nonempty closed convex subset P⊂E is said to be a cone in E, if the following two conditions are satisfied:(a)
λ≥0,x∈Pλx∈P(b)
x∈P,−x∈Px=0E, where 0E is the zero vector of EIfP is a cone in E, then we can define a partial order ≤P in E as follows: for all x,y∈E,(1)x≤Py⇔y−x∈P.A coneP in E is said to be normal, if there exists K>0 such that for all x,y∈E,(2)0E≤Px≤py⇒x≤Ky.LetP be a cone in E. We define the equivalence relation ~P in E as follows: for all x,y∈E,(3)x~Py⇔thereexistλ,μ>0suchthatλx≤Py≤Pμy.Givenh∈P with h≠0E, we denote by Ph the equivalence class of h with respect to the equivalence relation ~P, i.e.,(4)Ph=x∈E∣x~Ph=x∈E∣∃λ,μ>0:λh≤Px≤Pμh.It can be easily seen thatPh⊂P.Definition 1.
LetP be a cone in E and A:D⟶E be a given operator, where D is a nonempty subset of E. We say that A is increasing if(5)x,y∈D,x≤Py⇒Ax≤PAy.


## 2. A Fixed Point Theorem forφ–h,e–Concave Operators

In this section, we recall the fixed point theorem established in [1] for φ–h,e–concave operators.LetE,· be a real Banach space and P be a cone in E. Given e,h∈P with 0E≤Pe≤Ph, we define the set(6)Ph,e=x∈E∣x+e∈Ph,that is,(7)Ph,e=x∈E∣∃λ,μ>0:λh≤Px+e≤Pμh.It can be easily seen thatPh⊂Ph,e. In particular, for e=0E, we have(8)Ph,0E=Ph.Zhai and Wang [1] introduced the notion of φ–h,e–concave operators as follows.Definition 2 (φ−h,e–concave operators).
LetA:Ph,e⟶E be a given operator. Suppose that for any x∈Ph,e and λ∈0,1, there exists φλ>λ such that(9)Aλx+λ−1e≥PφλAx+φλ−1e.
ThenA is called a φ–h,e–concave operator.Notice that for allλ>0 and x∈Ph,e, we have λx+λ−1e∈Ph,e. In the special case e=0E, the above notion reduces to the generalized concavity concept.Definition 3 (generalized concavity).
LetA:Ph⟶E be a given operator. We say that A is a generalized concave operator, if for any x∈Ph and λ∈0,1, there exists φλ∈λ,1 such that(10)Aλx≥PφλAx.The main result obtained in [1] is the following fixed point theorem.Theorem 4.
LetP be a normal cone in E and A:Ph,e⟶E be an increasing φ–h,e–concave operator with Ah∈Ph,e. Then A admits a unique fixed point x∗inPh,e. Moreover, for any x0∈Ph,e, the sequence xn defined by(11)xn=Axn−1,n=1,2,⋯,converges to x∗, that is, limn⟶∞∥xn−x∗∥=0.In the special casee=0E, we deduce from Theorem 4 the following fixed point theorem which was established by Zhai and Wang [2].Theorem 5 (the casee=0E).
LetP be a normal cone in E and A:Ph⟶E be an increasing φ–h,0E–concave operator (i.e., generalized concave operator) with Ah∈Ph. Then A admits a unique fixed point x∗ in Ph. Moreover, for any x0∈Ph, the sequence xn defined by(12)xn=Axn−1,n=1,2,converges to x∗, that is, limn⟶∞∥xn−x∗∥=0.In the next section, we will show that Theorem5⇒ Theorem 4. So, both theorems are equivalent.


## 3. The Equivalence between Theorems4 and 5

We shall prove Theorem4 using Theorem 5.Proof of Theorem4.
LetP be a normal cone in E and A:Ph,e⟶E be an increasing φ–h,e–concave operator with Ah∈Ph,e. Let(13)h∗=h+e.
We claim that(14)Ph∗=Ph.
Lety∈Ph∗. Then there exist λ,μ>0 such that(15)λh∗=λh+e≤Py≤Pμh∗=μh+e.
Sincee≤ph, we have(16)μh+e≤P2μh.
Sinceh+e≥Ph, we have(17)λh+e≥Pλh.
It follows from (15), (16), and (17) that(18)λh≤Py≤P2μh,which shows that y∈Ph. This proves that Ph∗⊂Ph. Let z∈Ph. Then there exist δ,σ>0 such that(19)δh≤Pz≤Pσh.
Sinceh≤Ph∗, we have(20)σh≤Pσh∗.
Sincee≤ph, we have(21)δh=δ2h+δ2h≥Pδ2h+e=δ2h∗.
It follows from (19), (20), and (21) that(22)δ2h∗≤Pz≤Pσh∗,which shows that z∈Ph∗. This proves that Ph⊂Ph∗. Therefore, (14) is proved.
Now, we introduce the operator(23)B:Ph∗⟶Edefined by(24)By=Ay−e+e,y∈Ph∗.
Notice that for ally∈Ph∗=Ph (by (14)), we have y−e+e=y∈Ph, so y−e∈Ph,e. Hence, the operator B is well-defined.
SinceA is increasing, for all y1,y2∈Ph∗ with y1≤Py2, we have y1−e≤Py2−e, which implies that(25)By1−e=Ay1−e≤PAy2−e=By2−e,and then By1≤PBy2. This shows that B is an increasing operator.
Next, we show that(26)Bh∗∈Ph∗.
We have(27)Bh∗=Ah∗−e+e=Ah+e.
SinceAh∈Ph,e, then Bh∗=Ah+e∈Ph=Ph∗, which proves (26).
Now, we show thatB is a generalized concave operator. Let y∈Ph∗=Ph and λ∈0,1. Since x=y−e∈Ph,e and A is φ–h,e–concave, then there exists φλ>λ such that(28)Aλx+λ−1e≥PφλAx+φλ−1e,that is,(29)Aλy−e≥PφλAy−e+φλ−1e.
Hence, by the definition of the operatorB, we obtain(30)Bλy−e≥PφλBy−e+φλ−1e=φλBy−e,that is,(31)Bλy≥PφλBy.
This shows thatB is a generalized concave operator.
Next, by Theorem5, we deduce that B has a unique fixed point y∗ in Ph∗. Moreover, for any y0∈Ph∗, the sequence yn defined by(32)yn=Byn−1,n=1,2⋯,converges to y∗, that is, limn⟶∞∥yn−y∗∥=0. On the other hand, since y∗∈Ph∗=Ph is a fixed point of B, then(33)y∗=By∗=Ay∗−e+e,that is,(34)x∗=y∗−e=Ay∗−e=Ax∗.
This shows thatx∗∈Ph,e is a fixed point of A. Suppose now that x∗∗∈Ph,e is another fixed point of A. Then we have y∗∗=x∗∗+e∈Ph=Ph∗ and(35)By∗∗=Ay∗∗−e+e=Ax∗∗+e=x∗∗+e=y∗∗.
This shows thaty∗∗ is another fixed point of B in Ph∗, which contradicts the uniqueness of fixed points of B in Ph∗. Hence, we deduce that x∗ is the unique fixed point of A in Ph,e. Finally, given x0∈Ph,e, we define the sequence xn by(36)xn=Axn−1,n=1,2,⋯.
Takingy0=x0+e∈Ph=Ph∗ and define the sequence yn by (32), we obtain(37)limn⟶∞∥yn−y∗∥=0=limn⟶∞∥xn+e−x∗+e∥=limn⟶∞∥xn−x∗∥.
This concludes the proof of Theorem4.



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