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On set-valued contractions of Nadler type in tus-G-cone metric spaces

Chi-Ming Chen

Correspondence: ming@mail.nhcue. edu.tw

Department of Applied Mathematics, NationalHsinchu University of Education, No. 521 Nanda Rd., Hsinchu City 300, Taiwan

Abstract

In this article, for a fus-G-cone metric space (X, G) and for the family A of subsets of X, we introduce a new notion of the tus - H - cone metric H with respect to G, and we get a fixed result for the stronger Meir-Keeler-G-cone-type function in a complete fus-G-cone metric space (A,H) Our result generalizes some recent results due to Dariusz Wardowski and Radonevic' et al. MSC: 47H10; 54C60; 54H25; 55M20.

Keywords: fixed point theorem, stronger Meir-Keeler-G-cone-type function, fus-G-cone metric space

1 Introduction and preliminaries

Recently, Huang and Zhang [1] introduced the concept of cone metric space by replacing the set of real numbers by an ordered Banach space, and they showed some fixed point theorems of contractive type mappings on cone metric spaces. The category of cone metric spaces is larger than metric spaces. Subsequently many authors like Abbas and Jungck [2] had generalized the results of Huang and Zhang [1] and studied the existence of common fixed points of a pair of self mappings satisfying a contractive type condition in the framework of normal cone metric spaces. However, authors like Jankovic' et al. [3], Rezapour and Hamlbarani [4] studied the existence of common fixed points of a pair of self and nonself mappings satisfying a contractive type condition in the situation in which the cone does not need to be normal. Many authors studied this subject and many results on fixed point theory are proved (see e.g., [4-15]).

Recently, Du [16] introduced the concept of tus-cone metric and tus-cone metric space to improve and extend the concept of cone metric space in the sense of Huang and Zhang [1]. Later, in the articles [16-19], the authors tried to generalize this approach by using cones in topological vector spaces tus instead of Banach spaces. However, it should be noted that an old result shows that if the underlying cone of an ordered tus is solid and normal, then such tus must be an ordered normed space. Thus, proper generalizations when passing from norm-valued cone metric spaces to tus-valued cone metric spaces can be obtained only in the case of nonnormal cones (for details, see [19]).

We recall some definitions and results of the tus-cone metric spaces that introduced in [19,20], which will be needed in the sequel.

ringer

© 2012 Chen; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let E be a real Hausdorff topological vector space (tvs for short) with the zero vector Q. A nonempty subset P of E is called a convex cone if P + P £ P and IP £ P for l > 0. A convex cone P is said to be pointed (or proper) if P n (-P) = {Q}; P is normal (or saturated) if E has a base of neighborhoods of zero consisting of order-convex subsets. For a given cone P £ E, we can define a partial ordering 4 with respect to P by x 4 y if and only if y - x e P; x < y will stand for x 4 y and x * y, while x « y will stand for y - x e intP, where intP denotes the interior of P. The cone P is said to be solid if it has a nonempty interior.

In the sequel, E will be a locally convex Hausdorff tvs with its zero vector Q, P a proper, closed, and convex pointed cone in E with int P * j and 4 a partial ordering with respect to P.

Definition 1 [16,18,19]Let X be a nonempty set and (E, P) an ordered tvs. A vector-valued function d: X x X ® E is said to be a tvs-cone metric, if the following conditions hold:

(Ci) vx,yex,x*y Q 4 d(x, y);

(C2) Vxyex d(x, y) = Q o x = y;

(C3) Vxyex d(x, y) = d(y, x);

(C4) Vx,y,zex d(x, z) 4 d(x, y) + d(y, z).

Then the pair (X, d) is called a tvs-cone metric space.

Definition 2 [16,18,19]Let (X, d) be a tvs-cone metric space, x e X and {xn} a sequence in X.

(1) {xn} tvs-cone converges to x whenever for every c e E with Q « c, there exists no e Nsuch that d(x№ x) « c for all n > no. We denote this by cone-lim

(2) {xn} is a tvs-cone Cauchy sequence whenever for every c e E with Q « c, there exists n0 e Nsuch that d(xn, xm) « c for all n, m > n0;

(3) (X, d) is tvs-cone complete if every tvs-cone Cauchy sequence in X is tvs-cone convergent in X.

Remark 1 Clearly, a cone metric space in the sense of Huang and Zhang [1] is a special case of tvs-cone metric spaces when (X, d) is a tvs-cone metric space with respect to a normal cone P.

Remark 2 [19-21]Let (X, d) be a tvs-cone metric space with a solid cone P. The following properties are often used, particularly in the case when the underlying cone is nonnormal.

(p1) If u 4 v and v « w, then u « w;

(p2) If u « v and v 4 w, then u « w;

(p3) If u « v and v « w, then u « w;

(p4) If Q 4 u « c for each c e intP, then u = Q;

(p5) If a 4 b + c for each c e intP, then a 4 b;

(p6) IfE is tvs with a cone P, and if a 4 la where a e P and l e [0, 1), then a = Q;

(p7) If c e intP, an e E and an ® Q in locally convex tvs E, then there exists n0 e Nsuch that an « c for all n > n0.

Metric spaces are playing an important role in mathematics and the applied sciences. To overcome fundamental laws in Dhage's theory of generalized metric spaces [22], flaws that invalidate most of the results claimed for these spaces, Mustafa and Sims [23] introduced a more appropriate and robust notion of a generalized metric space as follows:

Definition 3 [23]Let X be a nonempty set, and let G : X x X x X ® [0, be a function satisfying the following axioms:

(G1) Vx,y,z^X G(x, y, z) = 0 o x = y = z;

(G2) Vx,y^X,x*y G(x, x, y) > 0;

(G3) Vx,y,z^X G(x, y, z) > G(x, x, y);

(G4) Vx,y,zeX G(x, y, z) = G(x, z, y) = G(z, y, x) = ... (symmetric in all three variables);

(G5) 'ix.y.z.w^X G(x, y, z) < G(x, w, w) + G(w, y, z).

Then the function G is called a generalized metric, or, more specifically a G-metric on X, and the pair (X, G)is called a G-metric space.

By using the notions of generalized metrics and tus-cone metrics, we introduced the below notion of tus-generalized-cone metrics.

Definition 4 Let X be a nonempty set and (E, P) an ordered tus, and let G : X x X x X ® E be a function satisfying the following axioms:

(G1) Vx,y,zeX G(x, y, z) = 0 if and only if x = y = z;

(G2) ^x,y^X,x*y 0 « G(x, x, y);

(G3) Vxy.z^X G(x, x, y) 4 G(x, y, z);

(G4) VxyzeX G(x, y, z) = G(x, z, y) = G(z, y, x) = ... (symmetric in all three variables);

(G5) Vx,yzwex G(x, y, z) 4 G(x, w, w) + G(w, y, z).

Then the function G is called a tus-generalized-cone metric, or, more specifically a tus-G-cone metric on X, and the pair (X, G) is called a tus-G-cone metric space.

Definition 5 Let (X, G) be a tus-G-cone metric space, x e X and {xn} a sequence in

(1) {xn} tus-G-cone converges to x whenever for every c e E with 0 « c, there exists n0 e Nsuch that G(xn, xm, x) « c for all m, n> n0. Here x is called the limit of the sequence {xn} and is denoted by G-cone-limn®roxn = x;

(2) {xn} is a tus-G-cone Cauchy sequence whenever for every c e E with 0 « c, there exists n0 e Nsuch that G(xn, xm, xl) « c for all n, m, l > n0;

(3) (X, G) is tus-G-cone complete if every tus-G-cone Cauchy sequence in X is tus-G-cone convergent in X.

Proposition 1 Let (X, G) be a tus-G-cone metric space, x e X and {xn} a sequence in X. The following are equivalent

(i) {xn} tus-G-cone converges to x;

(ii)G(xn, x№ x) ® 0 as n ®

(iii)G(xn, x, x) ® 0 as n ®

(iv)G(xn, xm x) ® 0 as n, m ®

We also recall the notion of Meir-Keeler type function (see [24]). A function $ : [0, ® [0, ro) is said to be a Meir-Keeler type function, if $ satisfies the following condition:

Vn> 0 35 > 0 Vt e [0, ra) (n < t <5 + n ^ p(t) < n).

We now define a new notion of stronger Meir-Keeler type function, as follows:

Definition 6 We call $ : [0, ® [0, 1) a stronger Meir-Keeler type function if the function $ satisfies the following condition:

Vn> 0 35 > 0 3yn e [0,1) Vt e [0, <x>) (n < t <5 + n ^ v(t) < Yn).

And, we introduce the below concept of the stronger Meir-Keeler tus-G-cone-type function in a tus-G-cone metric space.

Definition 7 Let (X, G) be a tus-G-cone metric space with a solid cone P. We call $ : P ® [0, 1) a stronger Meir-Keeler tus-G-cone-type function in X if the function $ satisfies the following condition:

Vn » e 35 » e 3yn e [0,1) Vx, y,z e X (n ^ G(x, y,z) « S+n ^ y(G(x, y,z)) < yn).

The Nadler's results [25] concerning set-valued contractive mappings in metric spaces became the inspiration for many authors in the metric fixed point theory (see for example [26-28]). Particularly Wardowski [29] established a new cone metric H : A x A ^ E for a cone metric space (X, d) and for the family A of subsets of X, and introduced the concept of set-valued contraction of Nadler type and prove a fixed point theorem. Later, in [21], the concept of set-valued contraction of Nadler type in the setting of tus-cone spaces was introduced and a fixed point theorem in the setting of tus-cone spaces with respect to a solid cone was proved.

In this article, for a tus-G-cone metric space (X, G) and for the family A of subsets of X, we introduce a new notion of the tus - H - cone metric H with respect to G, and we get a fixed result for the stronger Meir-Keeler type function in a complete tus-general-ized-cone metric space (A, H . Our result generalizes some recent results due to Rado-nevic' et al. [21] and Dariusz Wardowski [29].

2 Main results

Let E be a locally convex Hausdorff tus with its zero vector Q, P a proper, closed, and convex pointed cone in E with intP * j and 4 a partial ordering with respect to P. We introduce the below notion of the tus - H - cone metric H with respect to tus-G-cone metric G.

Definition 8 Let (X, G) be a tus-G-cone metric space with a solid cone P and let Abe a collection of nonempty subsets of X. A map H : A x A x A ^ E is called a tus — H — cone metric with respect to G if for any A1, A2, A3 e Athe following conditions hold:

(H1) H(Ai, A2, A3) = e ^ Ai= A2 = A3;

(H2) H(A1r A2, A3) = H(A1r A2, A3) = H(A1r A2, A3) = ••• (symmetry in all variables;

(H3) H(A1, A2, A3) ^ H(A1, A2, A3);

(H4) V'see,e ^s V'xea1,yea2 3z€ a3G(x, y,z) ^ H(A1, A2, A3) + s

(H5) one of the following is satisfied:

(i) VSGE,e«s 3XeAl VyeA2,zA H(Ai, A2, A3) ^ G(x, y, z) + S;

(ii) VseE,e«s 3xA VxeAi,zeA3 H(Ai,A2, A3) 4 G(x, y, z) + S;

(iii) VseE,e«s ^zGA3 ^yeA^zeAi H(Ai, A2, A3) 4 G(x, y, z) + S.

Lemma 1 Let (X, G) be a tus-G-cone metric space with a solid cone P and let A be a collection of nonempty subsets of X. A = </>. If H : A x A x A ^ Eis a tus — H — cone metric with respect to G, then pair (A, H) is a tus-G-cone metric space.

Proof Let {en} c E be a sequence such that Q « en for all n e N and G-cone-limn®„ en = Q. Take any A1f A2, A3 e A and x e A1, y e A2. From (H4), for each n e N, there exists zn e A3 such that

G(x, y,Zn) 4 H(A1,A2,As) + En.

Therefore, H(A1, A2, A3) + En e P for each n e N. By the closedness of P, we conclude that 0 4 H(A1, A2, A3 .

Assume that A1 = A2 = A3. From H5, we obtain H(A1, A2, A3) 4 En for any n e N. So H(A1, A2, A3) = 6.

Let A1, A2, A3, A4 e A. Assume that Ai, A2, A3 satisfy the condition (H5)(i). Then for each n e N, there exists xn e A1 such that H(A1, A2, A3) 4 G(xn, y,z) + En for all y e A2 and z e A3. From (H4), there exists a sequence {wn} c A4 satisfying G(xn, wn, wn) 4 H(A1, A4, A4) + En for every n e N. Obviously for any y e A2 and any z e A3 and n e N, we have

H(A1,A2,A3) 4 G(xn,y,z) + En

4 G(xn,Wn, Wn) + G(wn,y, z) + En.

Now for each n e N, there exists yn e A2, zn e A3 such that G(wn,yn,zn) 4 H(A4, A2, A3) + En. Consequently, we obtain that for each n e N

HA1A2A3) 4 H(AAA4,) + H(A4,A2A3i) + 3En.

Therefore,

H(AlA2A3) 4 H(AhA4,A4,) + H(A4A2A%).

In the case when (H5)(ii) or (H 5)(iii) hold, we use the analog method. □

Our main result is the following.

Theorem 1 Let (X, G) be a tus-G-cone complete metric space with a solid cone P and let Abe a collection of nonempty closed subsets of X, A = <, and let H : A x A x A ^ Ebe a tus — H — conemetric with respect to G. If the mapping T : X ^ Asatisfies the condition that exists a stronger Meir-Keeler tus-G-cone-type function $ : P ® [0, 1) such that for all x, y, z e X holds

H(Tx, Ty, Tz) 4 p(G(x, y,z)) • G(x, y,z), (1)

then T has a fixed point in X.

Proof. Let us choose x0 e X arbitrarily and x1 e Tx0. If G(x0, x0, x1) = 0, then x0 = x1 e T(x0), and we are done. Assume that G(x0, x0, x1) « 0. Put G(x0, x0, x1) = h0, h0 » 0. By the definition of the stronger Meir-Keeler tus-G-cone-type function $ : P ® [0, 1), corresponding to h0 use, there exist S0 » 0 and Yn0 e (0,1) with h0 4 G(x0, x0, x1) < h0 + 80 such that p(G(x0, x0, x1)) < Yn0. Let s e intP and s1 e E such that 0 « e1 and E1 4 Yn0 • E. Taking into account (1) and (H4), there exists x2 e Tx1 such that

G (x1, x1, x2) 4 H (Tx0, Tx0, Tx1) + e1

4 p (G (x0, x0, x1)) • G (x0, x0, x1) + E1 (2)

4 Yn0 • G (x0, x0, x1) + E'1.

Now, put G(x1, x1, x2) = h1, h1 » 0. By the definition of the stronger Meir-Keeler tus-G-cone-type function $ : P - [0, 1), corresponding to h1 use, there exist 8 1 » 0 and Yn1 e (0,1) with h 1 4 G(x1, x1, x2) < h 1 + 8 1 such that p(G(x1, x1, x2)) < Yn1. Put a0 = Ync and a1 = max{Yn0, Ym}. Then a0, a 1 e (0, 1) and

p (G (x0,x0,x1)) < Yn0 < a < 1andp(G (x1,x1,x2)) < Yn1 < a1 < 1.

Let e2 e E such that 9 « e2 and £2 4 y2x ' e• Then

si 4 ai • £ and £2 4 a2 • £.

Taking into account (1), (2), and (H4), there exists x3 e Tx2 such that

G(x2, x2, x3) 4 H (Txi, Txi, Tx2) + s2

4 p (G (xi,xi,x2)) • G (xi,xi,x2) + s2 4 ai • G(xi,xi,x2) + s2 4 ai(ai • G(x0,x0,xi) + si) + s2 4 a2 • G (x0, x0, xi) + ai • si+ s2 4 a\ • G (x0, x0, xi) + 2a\ • s.

We continue in this manner. In general, for xn, n e N, xn+1 is chosen such that xn+1 e Txn• Put G(xn, xn, xn+1) = hn, hn » 9. By the definition of the stronger Meir-Keeler tus-G-cone-type function ^ : P ® [0, 1), corresponding to hn use, there exist Sn » 9 and Ynn e (0,1) with hn < G(xm xn, xn+i) < hn + ¿n such that p (G (xn, xn, xn+i)) < yVn. Put an = max {yno, yni,..., yn„}, n e N. Then an e (0, 1) and

p (G (xi , xi, xi+i)%) < yn — an < ~1, for all i e {0, l,t2, . .. , n} . (4)

On the other hand, for each n e N, corresponding to ynn use, we choose en+1 e E such that 9 « en+1 and £n+i 4 y• £• Then

£n+i 4 an+1 • s. (5)

From above argument, we can construct a sequence {xn} in X, a non-decreasing sequence {an} and a sequence {en} recursively as follow:

xn+1 e Txn ,

an = max {yno, ym,..., ynn} < 1,

£n+i 4 y^1 • £ 4 al+i • s, for all n e N U {0}.

And, we have that for each n e N U {0}

G (xn+i, xn+i, xn+2) 4 H (Txn, Txn, Txn+i ) + £n+i.

Taking into account (4), (5), and (H4), there exists xn+2 e Txn+1 such that

G (xn+i,xn+i, xn+2) 4 H (Txn, Txn, Txn+i ) + £n+i

4 p ( G ( xn , xn, xn+i )) • G (xn, xn, xn+i ) + £n+i 4 an • G (xn, xn, xn+i) + an • £

4 an [H (Txn-i, Txn-i, Txn) + sj + a^+l • s

an p ( G ( xn -1, xn-i, xn)) • G (xn, xn, xn+i ) + sn\ + an • s ir\

n+i (6)

4 an [an • G (xn, xn, xn+i ) + sn\ + an • s

4 an • G (xn, xn, xn+i ) + an • sn + an • s

4 an • G (xn/ xn, xn+i ) +2an • s

4......

4 a"+i • G (x0, x0, xi) + (n +1) a"+i • s.

Let m, n e N be such that m > n. From (6) we conclude that

G(Xn,XnXm) ' ^ ; G(xj,Xj,Xj+i) j=n

4 [a— ' G(xo,Xo,xi) + ja— • s],

From above argument and the inequality (7), we put a = max{«n-i, an, &n+i, ..., an 2}. Then, we get a = am-2 <1 and

G(xn, xn, xm) 4 [a' • ^xi) + ja' • e]

4 -G(.Xo,.Xo,.Xi) + ViV • s

l — a

an , n n+a

4 -G(.Xo,.Xo,.Xi) + a

l - a v (l - a)

n+a (I - i/y

a l* n Since lim- = 0 and a ~ = ^ we obtain that

11111 .. ^ n_i/Vi f 1 ~.\2

a ,, n n + a

-G{xo,xo,xi) + a -j • e 0

1 - a (1 - a)2

in locally convex space E as ®

Apply Remark 2, we conclude that for every t e E with 0 « t there exists n0 e N such that G(xn, xn, xm) « t for all m, n > n0. So {xn} is a tus-G-cone Cauchy sequence. Since (X, G) is a tus-G-cone complete metric space, {xn} is tus-G-cone convergent in X and G-cone-limn®„ xn = x. Thus, for every t e intP and sufficiently large n, we have

H(Txn Txn Tx) 4 a ■ G(xn xn x) ■ a— = —.

Since for n e N U {0}, xn+1 e Txn, by (H4), we obtain that for all n e N there exist yn e Tx such that

G(xn+1, Xn+1, Yn+1) 4 H(Txn,Txn,Tx) + en+1 4 a • G(xn,xn,x) + an+1 e.

Then for sufficiently large n, we obtain that

G(yn+ix,x) 4 G(yn+i,.rn+i,.rn+i) + G(.rn+i «; — + - = x,

which implies G-cone-limn®„ yn = x. Since Tx is closed, we obtain that x e Tx. □

Follows Theorem 1, we immediate get the following corollary.

Corollary 1 Let (X, G) be a tus-G-cone complete metric space with a solid cone P and let Abe a collection of nonempty closed subsets of X, A = <, and let H : A x A x A ^ Ebe a tus — H — conemetric with respect to G. If the mapping T : X ^ Asatisfies the condition that exists a e (0, 1) such that for all x, y, z e X holds

H(Tx, Ty, Tz) 4 a ■ G(x, y, z) then T has a fixed point in X.

Acknowledgements

The authors would like to thank referee(s) for many usefulcomments and suggestions for the improvement of the

article.

Competing interests

The authors declare that they have no competing interests.

Received: 31 October 2011 Accepted: 26 March 2012 Published: 26 March 2012

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doi:10.1186/1687-1812-2012-52

Cite this article as: Chen: On set-valued contractions of Nadler type in tus-G-cone metric spaces. Fixed Point Theory and Applications 2012 2012:52.