Some fixed points theorems for multi-valued weakly uniform increasing operators

Some Fixed Point Theorems for Multi-Valued Weakly Uniform Increasing Operators.

ISHAKALTUN(*) - DURANTURKOGLU(**)

ABSTRACT- In the present work, we introduce the concept of weakly uniform in- creasing operators onanordered spaces. We also give some commonfixed point theorems for a pair of multivalued weakly uniform increasing operators in partially ordered metric space and in partially ordered Banach space.

1.Introduction.

The study on the existence of fixed points for single valued increasing operators is bounteous and successful, the results obtained are widely used to investigate the existence of solutions to the ordinary and partial dif- ferential equations (see [6], [7] and reference therein).

It is natural to extend this study to multi-valued case. Recently in [4], [7], [8], the authors have verified some fixed point theorems for multi-va- lued increasing operators.

Motivated by [3], [4], [9], [10] and [11] we prove some fixed point the- orems for multi-valued operators satisfying some weakly uniform in- creasing properties in metric space and Banach space, respectively.

2.Preliminaries

In this section some notations and preliminaries are given.

LetXbe a topological space andbe a partial order endowed onX.

(*) Indirizzo dell'A.: Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey.

E-mail: ialtun@kku.edu.tr, ishakaltun@yahoo.com

(**) Indirizzo dell'A.: Department of Mathematics, Faculty of Science and Arts, Gazi University, 06500 Teknikokullar, Ankara, Turkey.

E-mail: dturkoglu@gazi.edu.tr

In order to define the multi-valued weakly increasing and multi-valued weakly uniform increasing operators, we give the following relations be- tweentwo subsets ofX.

DEFINITION1 ([4]). Let A;B be two nonempty subsets of X, the rela- tions between A and B are defined as follows:

(1) If for every a2A, there exists b2B such that ab; then A₁B.

(2) If for every b2B, there exists a2A such that ab; then A2B.

(3) If A1B and A2B, then AB.

REMARK1. ₁ and₂ are different relations between A and B. For example, let XˆR, Aˆ[¹₂;1], Bˆ[0;1], be usual order on X, then A₁B but A6₂B;if Aˆ[0;1], Bˆ[0;¹₂], then A₂B while A6₁B.

REMARK 2. 1, 2 and are reflexive and transitive, but are not antisymmetric. For instance, let XˆR, Aˆ[0;3], Bˆ[0;1][ ‰2;3],be usual order on X, then AB and BA, but A6ˆB. Hence, they are not partial orders on2^X.

We canshow that is a partial order on CC(R), which denotes the family of closed and convex subsets ofR.

Let us recall some basic definitions and facts from multivalued analysis (see details [2]).

Let 2^X denote the family of all nonempty subset of X. A multivalued mapT:X!2^X is called upper semi continuous (u.s.c.) on X if for each x02X, the setTx0is a nonempty, closed subset ofX, and if for each open setVofXcontainingTx0, there exists anopenneighborhoodUofx0such thatT(U)V.

Tis said to be completely continuous ifT(B) is relatively compact for every bounded subsetBX.

If the multivalued map T is completely continuous with nonempty compact values, then T is u.s.c. if and only if T has a closed graph (i.e.

un!u0;vn!v0;vn2Tunimplyv02Tu0).

DEFINITION 2 ([4]). If fx_ng X satisfies x₁x₂ x_n or x₁x₂ x_n , thenfx_ngis called a monotone sequence.

DEFINITION 3 ([4]). A multivalued operator T:X!2^Xn; is called order closed if for monotone sequencesfung;fvng X;un !u0;vn!v0

and vn2Tunimply v₀2Tu₀.

DEFINITION4 ([4]). A function f :X!Ris called order upper (lower) semi continuous, if, for monotone sequencefung X;and u02X

un!u0ˆ)lim sup

n!1 f(un)f(u0); (f(u0)lim inff(un)):

REMARK 3. An operator with closed graph must be an order closed operator and an upper (lower) semi continuous function is an order upper (lower) semi continuous. There exist some examples in[4],showing that the converse is not true.

Let X be a real Banach space with a normk k _X:A non-empty closed subsetPofXis said to be anorder cone inXif

(i) P‡PP;

(ii) lPP, forl0 an d

(iii) P\ Pˆ fug, whereudenotes the zero element ofX.

Thenthe relationxy if and only if y x2P defines the partial ordering in X: P is called regular, if every nondecreasing and bounded above inorder sequence inX has a limit. i.e. if fx_ng X satisfies x1x2 xn y( y2X ), thenthere exists x2X such that

xn x

k k_X!0 asn! 1. We do not require any property of the cones in the present discussion, however, the details of order cones and their properties may be found in Guo and Lakshmikantham [5].

Now we introduce the following definition.

DEFINITION 5. Let X be a topological space, be a partial order endowed on X and S;T:X!2^X be two maps. The pair(S;T)is said to be weakly uniform increasing with respect to₁if for any x2X we have Sx₁Sy;Sx₁Sz for all y2Sx;z2Tx and Tx₁Ty;Tx₁Tz for all y2Tx;z2Sx. Similarly the pair (S;T) is said to be weakly uniform increasing with respect to2if for any x2X we have Sy2Sx;Sz2Sx for all y2Sx;z2Tx and Ty2Tx;Tz2Tx for all y2Tx;z2Sx.

Now we give some examples.

EXAMPLE1. LetRdenote the real line with the usual order relation and let Xˆ‰0;1Š R. Consider two mappings S;T:X!2^X defined

by

Sxˆ (0;1]; if xˆ0 f1g; if x6ˆ0 8<

: and Txˆ

n1 3;1

2

o; if xˆ0 n2

3

o; if x6ˆ0 8>

><

>>

:

for x2‰0;1Š. Then the pair of mappings (S;T) is weakly uniform in- creasing with respect to1but not weakly uniform increasing with respect to2on‰0;1Š:Indeed, since

Sx ˆ (0;1]; if xˆ0 f1g; if x6ˆ0 (

1f1g

ˆSy; for all y2Sx

;

Sx ˆ (0;1]; if xˆ0 f1g; if x6ˆ0 (

1f1g

ˆSz; for all z2Tx;

and

Tx ˆ 1

3;1 2

; if xˆ0

2 3

; if x6ˆ0

8>

>>

<

>>

>: ₁

2 3

ˆTy; for all y2Tx

;

Tx ˆ 1

3;1 2

; if xˆ0

2 3

; if x6ˆ0

8>

>>

<

>>

>: ₁

2 3

ˆTz; for all z2Sx so(S;T)is weakly uniform increasing with respect to₁. Note that the pair (S;T)is not weakly uniform increasing with respect to₂since S¹₂ˆ f1g 6₂(0;1]ˆS0for¹₂2S0.

EXAMPLE2. Let Xˆ[1;1)andbe usual order on X. Consider two mappings S;T:X!2^X defined by Sxˆ[1;x²] and Txˆ[1;2x] for all x2X. Then the pair of mappings (S;T) is weakly uniform increasing with respect to₂but not₁. Indeed, since

Syˆ[1;y²]₂[1;x²]ˆSx for all y2Sx;

Szˆ[1;z²]2[1;x²]ˆSx for all z2Tx;

Tyˆ[1;2y]₂[1;2x]ˆTx for all y2Tx;

Tzˆ[1;2z]₂ [1;2x]ˆTx for all z2Sx;

then the pair(S;T)is weakly uniform increasing with respect to2. But S2ˆ[1;4]6₁f1g ˆS1for12S2;so the pair(S;T)is not weakly uniform increasing with respect to₁:

3.Fixed point theorems.

Inthis section, we prove some fixed point theorems for multivalued weakly uniform increasing operators with respect to₁and₂ inpartial order metric space and partial order Banach space.

We will use the following lemma in our theorems.

LEMMA1 ([1]). Let(X;d)be a metric space,W:X!R. Define the re- lationon X as follows:

xy()d(x;y)W(x) W(y):

Thenis a partial order on X and X is called a partial ordered metric space.

Now we give a fixed point theorem.

THEOREM 1. Let (X;d) be a complete metric space, W:X!Rbe a bounded below function, be a partial order induced by W and S;T:X!2^Xn;be two order closed with respect toand weakly uniform increasing operator with respect to₁. Then there exist z₁;z₂2X such that z₁2Sz₂and z₂2Tz₁. Further if Sz₂₁fz₂gand Tz₁₁fz₁g, then S and T have a common fixed point (z₁ˆz₂) in X.

PROOF. Let x0 2X be arbitrary point. SinceSx06ˆ ;, we canchoose x₁2Sx₀. Now since (S;T) is weakly uniform increasing with respect to₁; we havex₁2Sx₀₁Syfor ally2Sx₀and sox₁2Sx₀₁Sx₁. Again, since (S;T) is weakly uniform increasing with respect to₁we haveSx₁₁Sz for all z2Tx1. Now we choose x22Tx1, then Sx11Sx2 and so x12Sx01Sx11Sx2. Now using the definition of 1, we canchoose x₃2Sx₂ such that x₁x₃. Similarly, since (S;T) is weakly uniform in- creasing with respect to₁, we havex₂2Tx₁₁Tyfor ally2Tx₁and so x₂2Tx₁₁Tx₂. Againsince (S;T) is weakly uniform increasing with re- spect to1, we haveTx21Tzfor allz2Sx2. Now sincex32Sx2, we have Tx21Tx3 and so x22Tx11Tx21Tx3. Now using the definition of

1we canchoosex42Tx3such thatx2x4. Continue this process, we will get a sequencefxng¹_nˆ1which its subsequencesfx2n‡1g¹_nˆ0andfx2n‡2g¹_nˆ0 are increasing and satisfiesx_2n‡12Sx_2n;x_2n‡22Tx_2n‡1;nˆ0;1;2; :::.

By the definitionwe have

W(x₁)W(x₃) W(x_2n‡1) and

W(x2)W(x4) W(x2n‡2) :

SinceWis bounded from below, thenfW(x_2n‡1)g¹_nˆ0 andfW(x_2n‡2)g¹_nˆ0 are convergent. Thus fore>0, there existsN, ifn>m>Nthen

d(x2m‡1;x2n‡1)W(x2m‡1) W(x2n‡1)<e and

d(x_2m‡2;x_2n‡2)W(x_2m‡2) W(x_2n‡2)<e

which indicatesfx2n‡1g¹_nˆ0andfx2n‡2g¹_nˆ0are Cauchy sequence inX. Due to completeness of X, we know there exist z1;z22X, x2n‡1!z1 and x2n‡2!z2. Since S and T are order closed, fx2n‡1g¹_nˆ0 and fx2n‡2g¹_nˆ0 monotone and x_2n‡12Sx_2n;x_2n‡22Tx_2n‡1, we deduce that z₁2Sz₂ and z₂2Tz₁. Now ifSz₂₁fz₂g, thenz₁z₂ and ifTz₁₁fz₁g, thenz₂z₁ and soz1ˆz2is a commonfixed point ofSandT. p The proof of the following theorem carries over in the same manner.

THEOREM 2. Let (X;d) be a complete metric space, W:X!R be a bounded above function, be a partial order induced by W and S;T:X!2^Xn;be two order closed with respect toand weakly uniform increasing operator with respect to2. Then there exist z1;z2 2X such that z₁2Sz₂and z₂2Tz₁. Further iffz₂g ₂Sz₂andfz₁g ₂Tz₁, then S and T have a common fixed point (z₁ˆz₂) in X.

PROOF. Letx02X be arbitrary point. SinceSx06ˆ ;, we canchoose x12Sx0. Now since (S;T) is weakly uniform increasing with respect to2; we haveSx₁₁Sx₀ forx₁2Sx₀. Again, since (S;T) is weakly uniform in- creasing with respect to ₂ we have Sz₂Sx₁ for all z2Tx₁. Now we choosex₂ 2Tx₁, thenSx_{2 2}Sx₁and soSx₂₂Sx₁₂Sx₀. Now using the definition of2, we canchoosex32Sx2such thatx3x1. Similarly, since (S;T) is weakly uniform increasing with respect to2, we haveTx21Tx1

x22Tx1. Againsince (S;T) is weakly uniform increasing with respect to2,

we haveTz2Tx2for allz2Sx2. Now sincex32Sx2, we haveTx32Tx2

and so Tx32Tx22Tx1. Now using the definition of2we canchoose x₄2Tx₃ such thatx₄x₂. Continue this process, we will get a sequence fx_ng¹_nˆ1which its subsequencesfx_2n‡1g¹_nˆ0andfx_2n‡2g¹_nˆ0are decreasing and satisfiesx_2n‡12Sx_2n;x_2n‡22Tx_2n‡1;nˆ0;1;2; :::.

By the definitionwe have

W(x1)W(x3) W(x2n‡1) and

W(x₂)W(x₄) W(x_2n‡2) :

SinceWis bounded from above, thenfW(x2n‡1)g¹_nˆ0 andfW(x2n‡2)g¹_nˆ0 are convergent. Thus fore>0, there existsN, ifn>m>N then

d(x_2m‡1;x_2n‡1)W(x_2n‡1) W(x_2m‡1)<e and

d(x2m‡2;x2n‡2)W(x2n‡2) W(x2m‡2)<e

which indicatesfx2n‡1g¹_nˆ0andfx2n‡2g¹_nˆ0are Cauchy sequence inX. Due to completeness of X, we know there exist z₁;z₂2X, x_2n‡1!z₁ and x_2n‡2!z₂. Since S and T are order closed, fx_2n‡1g¹_nˆ0 and fx_2n‡2g¹_nˆ0 monotone and x2n‡12Sx2n;x2n‡22Tx2n‡1, we deduce that z12Sz2 and z22Tz1. Now iffz2g 2Sz2, then z2z1 and iffz1g 2Tz1, thenz1z2

and soz1ˆz2is a commonfixed point ofSandT. p Now we give some fixed point theorem in ordered Banach space.

THEOREM3. Let X be a Banach space, PX be a regular cone,be a partial order induced by P and S;T:X!2^Xn;be two order closed with respect toand weakly uniform increasing operator with respect to₁. If the following condition hold:

(i) there exists some u2X such that Sx1fugand Tx1fugfor all x2X.

Then there exist z₁;z₂2X such that z₁2Sz₂ and z₂2Tz₁. Further if Sz₂₁fz₂g and Tz₁₁fz₁g, then S and T have a common fixed point (z₁ˆz₂) in X.

PROOF. We can construct a sequencefxng¹_nˆ1 inX as inthe proof of Theorem 1. By (i), and the definition of 1, we have xnu;nˆ1;2; :::.

SincePis regular,fx2n‡1g¹_nˆ0andfx2n‡2g¹_nˆ0are nondecreasing and boun- ded above inorder, there exist z1;z22X such that x2n‡1!z1 and x_2n‡2!z₂. Now the proof cancomplete as inthe proof of Theorem 1.u p

The following theorem can be similarly verified.

THEOREM4. Let X be a Banach space, PX be a regular cone,be a partial order induced by P and S;T:X!2^Xn;be two order closed with respect toand weakly uniform increasing operator with respect to₂. If the following condition hold:

(ii) there exists some u2X such thatfug ₂Sx andfug ₂Tx for all x2X.

Then there exist z1;z22X such that z12Sz2and z22Tz1. Further if fz2g 2Sz2 andfz1g 2Tz1, then S and T have a common fixed point (z₁ˆz₂) in X.

REMARK4. By Theorems1, 2, 3 and4,we give some improved ver- sions of Theorems3.2, 3.3, 4.2and4.3of [4],respectively.

Now we introduce the following condition.

CONDITION1. Let X be a Banach space. Two maps S;T:X!2^X are said to satisfy Condition1if for any countable subset A of X and for any fixed a2X the condition

A fag [S(A)[T(A)implies A is compact;

here T(A)ˆ [_x2ATx.

Using the above condition, we can give the following theorems.

THEOREM 5. Let X be a Banach space, PX be a cone (not ne- cessary regular),be a partial order induced by P and S;T:X!2^Xn;

be two order closed with respect to and weakly uniform increasing operator with respect to ₁. Suppose that S and T are satisfy the Condition 1,then there exist z₁;z₂2X such that z₁2Sz₂ and z₂2Tz₁. Further if Sz₂₁fz₂g and Tz₁₁fz₁g, then S and T have a common fixed point (z1ˆz2) in X.

PROOF. We can construct a sequencefxng¹_nˆ1 inXas inthe proof of Theorem 1.

Now letAˆ fx0;x1; :::g. SinceAis countable, Aˆ fx₀g [ fx₁;x₃; :::g [ fx₂;x₄; :::g

fx0g [S(A)[T(A);

andSandTare satisfy the Condition 1, thenAis compact. Thusfx2n‡1g¹_nˆ0 and fx2n‡2g¹_nˆ0 have convergent subsequences which converges to say z1;z22X, respectively. However fx2n‡1g¹_nˆ0 andfx2n‡2g¹_nˆ0 are increas- ing, sox_2n‡1!z₁andx_2n‡2!z₂forn! 1. Now the proof cancomplete

as inthe proof of Theorem 1. p

The following theorem can be similarly verified.

THEOREM6. Let X be a Banach space, PX be a cone (not necessary regular),be a partial order induced by P and S;T:X!2^Xn;be two order closed with respect to and weakly uniform increasing operator with respect to₂. Suppose that S and T are satisfy the Condition1,then there exist z₁;z₂2X such that z₁2Sz₂and z₂2Tz₁. Further iffz₂g ₂Sz₂ andfz1g 2Tz1, then S and T have a common fixed point (z1ˆz2) in X.

REMARK5. By Theorems5and6,we give some improved versions of Theorem3.1 of[3]and Theorem4of [12].

Acknowledgments. The authors are grateful to the referees for their valuable comments in modifying the first version of this paper.

REFERENCES

[1] A. BR6ONDSTED,On a lemma of Bishop and Phelps, Pacific J. Math.,55(1974), pp. 335-341.

[2] K. DEIMLING,Multivalued Differential Equations, De Gruyter, Berlin, 1992.

[3] B. C. DHAGE- D. O'REGAN- R. P. AGARWAL,Common fixed point theorems for a pair of countably condensing mappings in ordered Banach spaces, J. Appl.

Math. Stochastic Anal,16, 3 (2003), pp. 243-248.

[4] Y. FENG- S. LIU,Fixed point theorems for multi-valued increasing operators in partial ordered spaces, Soochow J. Math.,30, 4 (2004), pp. 461-469.

[5] D. GUO - V. LAKSHMIKANTHAM, Nonlinear problems in abstract cones, Academic Press, New York, 1988.

[6] S. HEIKKILA - V. LAKSHMIKANTHAM, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Dekker, New York, 1994.

[7] N. B. HUY, Fixed points of increasing multivalued operators and an application to the discontinuous elliptic equations, Nonlinear Anal. 51, 4 (2002), pp. 673-678.

[8] N. B. HUY- N. H. KHANH,Fixed point for multivalued increasing operators, J. Math. Anal. Appl.,250, 1 (2000), pp. 368-371.

[9] J. JACHYMSKI, Converses to fixed point theorems of Zermelo and Caristi, Nonlinear Anal.,52, 5 (2003), pp. 1455-1463.

[10] J. JACHYMSKI, Caristi's fixed point theorem and selections of set-valued contractions, J. Math. Anal. Appl.,227, 1 (1998), pp. 55-67.

[11] W. A. KIRK - L. M. SALIGA, The BreÂzis-Browder order principle and extensions of Caristi's theorem, Proceedings of the Third World Congress of Nonlinear Analysts, Part 4 (Catania, 2000), Nonlinear Anal.,47, 4 (2001), pp.

2765-2778.

[12] D. TURKOGLU- R. P. AGARWAL- D. O'REGAN - I. ALTUN,Some fixed point theorems for single-valued and multi-valued countably condensing weakly uniform isotone mappings, PanAmer. Math. J.,1(1) (2005), pp. 57-68.

Manoscritto pervenuto in redazione il 21 maggio 2007.