of Mixed Monotone Operators

(1)

Volume 2011, Article ID 563136,8pages doi:10.1155/2011/563136

Research Article

On Fixed Point Theorems

of Mixed Monotone Operators

Xinsheng Du and Zengqin Zhao

School of Mathematics Sciences, Qufu Normal University, Qufu, Shandong 273165, China

Correspondence should be addressed to Xinsheng Du,[email protected] Received 11 November 2010; Accepted 4 January 2011

Academic Editor: T. Benavides

Copyrightq2011 X. Du and Z. Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We obtain some new existence and uniqueness theorems of positive fixed point of mixed monotone operators in Banach spaces partially ordered by a cone. Some results are new even for increasing or decreasing operators.

1. Introduction

Mixed monotone operators were introduced by Guo and Lakshmikantham in 1 in 1987.

Thereafter many authors have investigated these kinds of operators in Banach spaces and obtained a lot of interesting and important results. They are used extensively in nonlinear diﬀerential and integral equations. In this paper, we obtain some new existence and uniqueness theorems of positive fixed point of mixed monotone operators in Banach spaces partially ordered by a cone. Some results are new even for increasing or decreasing operators.

Let the real Banach spaceE be partially ordered by a coneP of E, that is, x ≤ y if and only if y − x ∈ P. A : P × P → P is said to be a mixed monotone operator if Ax, y is increasing in x and decreasing in y, that is, ui, vi i 1,2 ∈ P, u1 ≤ u2, v1 ≥ v2 implies Au1, v1 ≤ Au2, v2. Element x ∈ P is called a fixed point of A if Ax, x x.

Recall that cone P is said to be solid if the interiorP^◦ is nonempty, and we denote x0 ifx∈P.^◦ P is normal if there exists a positive constantNsuch that 0≤x≤yimplies x ≤Ny,Nis called the normal constant ofP.

For all x, y ∈ E, the notation x ∼ y means that there exist λ > 0 and μ > 0 such thatλx ≤ y ≤ μx. Clearly,∼is an equivalence relation. Givenh > θ i.e.,h ≥ θ andh /θ,

(2)

we denote byPh the setPh {x ∈ E | x ∼ h}. It is easy to see that Ph ⊂ P is convex and λP_hP_hfor allλ >0. IfP /^◦ ∅andh∈P^◦, it is clear thatP_hP^◦.

All the concepts discussed above can be found in2,3. For more facts about mixed monotone operators and other related concepts, the reader could refer to4–9and some of the references therein.

2. Main Results

In this section, we present our main results. To begin with, we give the definition ofτ-ϕ- concave-convex operators.

Definition 2.1. LetEbe a real Banach space andPa cone inE. We say an operatorA:P×P → P isτ-ϕ-concave-convex operator if there exist two positive-valued functionsτt, ϕt on intervala, bsuch that

H1τt:a, b → 0,1is a surjection, H2ϕt> τt, for allt∈a, b,

H3Aτtx,1/τty≥ϕtAx, y, for allt∈a, b,x, y∈P.

Theorem 2.2. LetP be normal cone ofE, and letA :P ×P → P be a mixed monotone andτ-ϕ- concave-convex operator. In addition, suppose that there existsh > θsuch thatAh, h∈Ph, thenA has exactly one fixed pointx^∗inPh. Moreover, constructing successively the sequence

xnA

xn−1, yn−1

, ynA

yn−1, xn−1

, n1,2, . . . , 2.1

for any initialx0, y0∈P_h, one has

x_n−x^∗ −→0, y_n−x^∗−→0, n−→ ∞. 2.2

Proof. We divide the proof into 3 steps.

Step 1. We prove thatAhas a fixed point inPh.

SinceAh, h∈P_h, we can choose a suﬃciently small numbere0∈0,1such that

e0h≤Ah, h≤ 1

e0h. 2.3

It follows fromH1that there existst0∈a, bsuch thatτt0 e0, and hence

τt0h≤Ah, h≤ 1

τt0h. 2.4

(3)

ByH2, we know thatϕt0/τt0>1. So, we can take a positive integerksuch that ϕt0

τt0 k

≥ 1

τt0. 2.5

It is clear that

τt0 ϕt0

_k

≤τt0. 2.6

Letu0 τt0^kh, v0 1/τt0^kh. Evidently,u0, v0 ∈Phandu0 τt0^2kv0 < v0. By the mixed monotonicity ofA, we haveAu0, v0 ≤ Av0, u0. Further, combining the condition H3with2.4and2.6, we have

Au0, v0 A

τt0^kh, 1 τt0^kh

A

τt0τt0^k−1h, 1 τt0

1 τt0^k−1h

≥ϕt0A

τt0^k−1h, 1 τt0^k−1h

≥ · · ·

≥ ϕt0_k

Ah, h≥ ϕt0_k

τt0h

≥τt0^khu0.

2.7

Fort∈a, b, fromH3, we get

A x, y

A

τt 1 τtx, 1

τtτty

≥ϕtA 1

τtx, τty

, 2.8

and hence

A 1

τtx, τty

≤ 1 ϕtA

x, y

, ∀t∈a, b, x, y∈P. 2.9

(4)

Thus, we have

Av0, u0 A 1

τt0^kh,τt0^kh

A 1

τt0 1

τt0^k−1h, τt0τt0^k−1h

≤ 1 ϕt0A

1

τt0^k−1h,τt0^k−1h

≤ · · ·

≤ 1

ϕt0_kAh, h≤ 1

ϕt0_k · 1 τt0h

≤ 1

τt0^khv0.

2.10

Construct successively the sequences

u_nAu_n−1, v_n−1, v_nAv_n−1, u_n−1, n1,2, . . . . 2.11

It follows from2.7,2.10, and the mixed monotonicity ofAthat

u0≤u1≤ · · · ≤u_n≤ · · · ≤v_n≤ · · · ≤v1≤v0. 2.12

Note thatu0 τt0^2kv0, so we can getun ≥u0≥τt0^2kv0≥τt0^2kvn,n1,2, . . .. Let rnsup{r >0|un≥rvn}, n1,2, . . . . 2.13

Thus, we haveu_n≥r_nv_n, n1,2, . . ., and then

u_{n 1}≥u_n≥r_nv_n≥r_nv_{n 1}, n1,2, . . . . 2.14

Therefore, rn 1 ≥ rn, that is,{rn}is increasing with{rn} ⊂ 0,1. Suppose that rn → r^∗ as n → ∞, thenr^∗ 1. Indeed, suppose to the contrary that 0 < r^∗ < 1. ByH1, there exists t1∈a, bsuch thatτt1 r^∗. We distinguish two cases.

Case 1. There exists an integerNsuch thatRNr^∗. In this case, we know thatrnr^∗for all n≥N. So, forn≥N, we have

u_{n 1}Au_n, v_n≥A

r_nv_n, 1 rnu_n

A

τt1v_n, 1 τt1u_n

≥ϕt1v_{n 1}. 2.15

By the definition ofr_n, we getr_{n 1} r^∗≥ϕt1> τt1 r^∗, which is a contradiction.

(5)

Case 2. If for all integern,rn < r^∗, then 0< rn/r^∗ <1. ByH1, there existssn ∈a, bsuch thatτs_n r_n/r^∗. So, we have

u_{n 1}Au_n, v_n≥A

r_nv_n, 1 rnu_n

A

r_n

r^∗r^∗v_n,r^∗ rn

1 r^∗u_n

A

τs_nr^∗v_n, 1 τsn

1 r^∗u_n

≥ϕs_nA

r^∗v_n, 1 r^∗u_n

≥ϕs_nϕt1v_{n 1}.

2.16

By the definition ofr_n, we have

r_{n 1}≥ϕs_nϕt1≥τs_nϕt1 r_n

r^∗ϕt1. 2.17

Letn → ∞, we getr^∗≥ϕt1> τt1 r^∗, which is also a contradiction. Thus, lim_n_{→ ∞}r_n1.

For any natural numberp, we have

θ≤u_{n p}−u_n≤v_n−u_n≤1−r_nv0,

θ≤v_{n p}−v_n≤v_n−u_n≤1−r_nv0. 2.18

SinceP is normal, we have

un p−un≤N1−rnv0 asn → ∞,

vn p−vn≤N1−rnv0 asn → ∞. 2.19 Here,Nis the normality constant.

So,{u_n}and{v_n}are Cauchy sequences. BecauseEis complete, there existu^∗, v^∗such thatu_n → u^∗, v_n → v^∗n → ∞. By2.12, we know thatu_n≤u^∗≤v^∗≤v_nand

θ≤v^∗−u^∗≤1−r_nv0. 2.20

Further,

v^∗−u^∗ ≤N1−rnv0 n → ∞, 2.21

and thusu^∗v^∗. Letx^∗:u^∗v^∗, we obtain

un 1Aun, vn≤Ax^∗, x^∗≤Avn, un vn 1. 2.22

Letn → ∞, we getx^∗Ax^∗, x^∗. That is,x^∗is a fixed point ofAinP_h.

(6)

Step 2. We prove thatx^∗is the unique fixed point ofAinPh.

In fact, suppose thatxis a fixed point ofAinP_h. Sincex^∗, x ∈P_h, there exist positive numbersβ > α > 0 such that αx ≤ x^∗ ≤ βx. Let e1 sup{e > 0 | 1/ex ≥ x^∗ ≥ ex}.

Evidently,e1 ∈ 0,1. We now prove thate1 1. If otherwise, 0< e1 < 1. FromH1, there existst2∈a, bsuch thatτt2 e1. Then,

x^∗Ax^∗, x^∗≥A

e1x, 1 e1x

A

τt2x, 1 τt2x

≥ϕt2Ax, x ϕt2x.

x^∗Ax^∗, x^∗≤A 1

e1x, e1x

A 1

τt2x, τt2x

≤ 1

ϕt2Ax, x 1 ϕt2x.

2.23

Sinceϕt2 > τt2 e1, this contradicts the definition of e1. Hence,e1 1, thus,x^∗ x.

Therefore,Ahas a unique fixed pointx^∗inPh. Step 3. We prove2.2.

For anyx0, y0∈P_h, we can choose a small numbere2∈0,1such that e2h≤x0≤ 1

e2h, e2h≤y0≤ 1

e2h. 2.24

Also fromH1, there ist3∈a, bsuch thatτt3 e2, and hence τt3h≤x0≤ 1

τt3h, τt3h≤y0≤ 1

τt3h. 2.25

We can choose a suﬃciently large integerm, such that ϕt3

τt3 m

≥ 1

τt3. 2.26

Let u0 τt3^mh, v0 1/τt3^mh. It is easy to see that u0, v0 ∈ P_h and u0 ≤ x0 ≤ v0, u0 ≤ y0 ≤ v0. Putu_n Au_n−1, v_n−1,v_n Av_n−1, u_n−1,x_n Ax_n−1, y_n−1, yn Ayn−1, xn−1,n 1,2, . . .. Similarly toStep 1, it follows that there existsy^∗ ∈ Ph such that Ay^∗, y^∗ y^∗, lim_n_{→ ∞}u_n lim_n_{→ ∞}v_n y^∗. By the uniqueness of fixed points of operatorAinP_h, we gety^∗ x^∗. And by induction,u_n ≤x_n ≤v_n, u_n ≤y_n ≤ v_n, n1,2. . ..

SinceP is normal, we have lim_n_{→ ∞}xnx^∗, lim_n_{→ ∞}ynx^∗.

3. Concerned Remarks and Corollaries

If we suppose that the operatorA:Ph×Ph → PhorA:P^◦ ×P^◦ → P^◦ withP is a solid cone, thenAh, h∈P_his automatically satisfied. This proves the following corollaries.

(7)

Corollary 3.1. LetP be a normal cone ofE, and letA : Ph×Ph → Ph be a mixed monotone and τ-ϕ-concave-convex operator, thenAhas exactly one fixed pointx^∗ in P_h. Moreover, constructing successively the sequence

x_nA

x_n−1, y_n−1

, y_nA

y_n−1, x_n−1

, n1,2, . . . , 3.1

for any initialx0, y0∈P_h, one has

xn−x^∗ −→0, yn−x^∗−→0, n−→ ∞. 3.2

Corollary 3.2. LetP be a normal solid cone ofE, and letA : P^◦ ×P^◦ → P^◦ be a mixed monotone andτ-ϕ-concave-convex operator, thenAhas exactly one fixed pointx^∗inP. Moreover, constructing^◦ successively the sequence

x_nA

x_n−1, y_n−1

, y_nA

y_n−1, x_n−1

, n1,2, . . . , 3.3

for any initialx0, y0∈P, one has

x_n−x^∗ −→0, y_n−x^∗−→0, n−→ ∞. 3.4 Whenτt t,ϕt t^αt, 0 < αt < 1, 0 < t < 1, the conditions H1and H2are automatically satisfied. So, one has

Corollary 3.3. LetPbe a normal cone of a real Banach spaceE, h > θ. A:Ph×Ph → Phis a mixed monotone operator. In addition, suppose that for all 0< t <1, there exists 0< αt<1 such that

A

tx,1 ty

≥t^αtA x, y

, ∀x, y∈Ph, 0< t <1, 3.5

thenAhas exactly one fixed pointx^∗inPh. Moreover, constructing successively the sequence x_nA

x_n−1, y_n−1

, y_nA

y_n−1, x_n−1

, n1,2, . . . , 3.6

for any initialx0, y0∈Ph, one has

x_n−x^∗ −→0, y_n−x^∗−→0, n−→ ∞. 3.7

Remark 3.4. Corollary 3.3is the main result in5. So, our results generalized the result in5.

Whenτt t, ϕt t^β,0 < t < 1, the conditionsH1and H2are automatically satisfied. So, we have the following.

(8)

Corollary 3.5. LetPbe normal solid cone ofE, and letA:P^◦×P^◦ → P^◦ be a mixed monotone operator.

In addition, suppose that there exists 0< β <1 such that A

tx,1

ty

≥t^βA x, y

, ∀x, y∈P,˙ 0< t <1, 3.8

thenAhas exactly one fixed pointx^∗inP. Moreover, constructing successively the sequence^◦ xnA

xn−1, yn−1

, ynA

yn−1, xn−1

, n1,2, . . . , 3.9

for any initialx0, y0∈P, one has^◦

xn−x^∗ −→0, yn−x^∗−→0, n−→ ∞. 3.10 Remark 3.6. Corollary 3.5is the main result in4. So, our results generalized the result in4.

Acknowledgments

This research was supported by the National Natural Science Foundation of China nos.

10871116, 11001151, the Natural Science Foundation of Shandong Provinceno. Q2008A03, the Doctoral Program Foundation of Education Ministry of Chinano. 20103705120002, and the Youth Foundation of Qufu Normal Universitynos. XJ200910, XJ03003.

References

1 D. J. Guo and V. Lakshmikantham, “Coupled fixed points of nonlinear operators with applications,”

Nonlinear Analysis: Theory, Methods & Applications, vol. 11, no. 5, pp. 623–632, 1987.

2 D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988.

3 D. Guo, Nonlinear Functional Analysis, Shandong Scientific Technical, Jinan, China, 2000.

4 D. J. Guo, “Fixed points of mixed monotone operators with applications,” Applicable Analysis, vol. 31, no. 3, pp. 215–224, 1988.

5 Y. Wu and Z. Liang, “Existence and uniqueness of fixed points for mixed monotone operators with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 10, pp. 1913–1924, 2006.

6 Z. Zhao, “Existence and uniqueness of fixed points for some mixed monotone operators,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 6, pp. 1481–1490, 2010.

7 Z. Zhang and K. Wang, “On fixed point theorems of mixed monotone operators and applications,”

Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3279–3284, 2009.

8 Y. Wu, “New fixed point theorems and applications of mixed monotone operator,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 883–893, 2008.

9 K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.