Phon-onetal. Fixed Point Theory and Applications (2015) 2015:57 DOI 10.1186/s13663-015-0307-4

0 Fixed Point Theory and Applications

a SpringerOpen Journal

RESEARCH

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Reich type weak contractions on metric spaces endowed with a graph

Aniruth Phon-on, Areeyuth Sama-Ae*, Nifatamah Makaje and Pakwan Riyapan

CrossMark

"Correspondence: sassaare@gmail.com Department of Mathematics and Computer Science, Faculty of Science and Technology, Prince of Songkla University, Pattani Campus, Pattani, 94000, Thailand

Abstract

In this paper, we define a new class of Reich type multi-valued contractions on a complete metric space satisfying the g-graph preserving condition and we study the fixed point theorem for such mappings. In addition, we present the existence and uniqueness of the fixed point for at least one of two multi-valued mappings. The results of this paper extend and generalize several well-known results. Some examples illustrate the usability of our results.

MSC: 47H04;47H10

Keywords: fixed point theorem; multi-valued mapping; graph preserving; Reich type contraction

ft Spri

ringer

1 Introduction

The classical contraction mapping principle of Banach states that if (X, d) is a complete metric space and f : X ^ X such that d(f (x),f (y)) < ad(x,y) for all x,y e X, where a e [0,1), thenf has a unique fixed point. The Banach fixed point theorem plays an important role in studying the existence of solutions of nonlinear integral equations, system of linear equations, nonlinear differential equations, and proving the convergence of algorithms in computational mathematics. The Banach fixed point theorem has been extended in many directions; see [1-17]. Fixed point theory of multi-valued mappings plays a central role in control theory, optimization, partial differential equations, and economics. For a metric space (X, d), we let CB(X) be the set of all nonempty, closed, and bounded subsets of X. A point x e X is a fixed point of a multi-valued mapping T: X ^ 2X if x e Tx. Nadler [18] has proved a multi-valued version of the Banach contraction principle which we state as the following theorem.

Theoreml.l Let (X, d) be a complete metric space and T: X ^ CB(X). Assume that there exists k e [0,1) such that H(Tx, Ty) < kd(x,y) for all x,y e X. Then there exists z e X such that z e Tz.

Reich [19] generalized the Banach fixed point theorem for single-valued maps and multivalued maps as the two following theorems.

Theorem 1.2 Let (X, d) be a complete metric space and letf: X ^ Xbea Reich type single-valued (a, b, c)-contraction, that is, there exist nonnegative numbers a, b, c with a + b + c <1

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such that

d(f (x),f (y)) < ad(x,y) + bd(x,f (x)) + cd{y,f (y)) for each x, y e X. Thenf has a unique fixed point.

Theorem 1.3 Let (X, d) be a complete metric space and let a mapping T : X ^ Pcl(X), where Pcl(X) is the set of all nonempty closed subsets of X, be a Reich type multi-valued (a, b, c)-contraction, that is, there exist nonnegative numbers a, b, c with a + b + c <1 such that

H(Tx, Ty) < ad(x,y) + bD(x, Tx) + cD(y, Ty) for each x, y e X. Then there exists z e X such that z e Tz.

In 2008, Jachymski [20] introduced the concept of a contraction concerning a graph, called a G-contraction, and proved some fixed point results of the G-contraction in a complete metric space endowed with a graph and he showed that the results of many authors can be derived by his results.

Definition 1.4 Let (X, d) be a metric space and G = (V(G),E(G)) a directed graph such that V(G) = X and E(G) contains all loops, i.e., A = {(x,x) | x e Xjc E(G). We say that a mappingf: X ^ X is a G-contraction iff preserves edges of G, i.e., for every x,y e X,

(x, y) e E(G) ^ f (x),f (y)) e E(G)

and there exists a e (0,1) such that, for x, y e X,

(x,y) e E(G) ^ df (x),f(y)) < ad(x,y).

Jachymski showed in [20] that assuming some properties for X, a G-contractionf: X ^ X has a fixed point if and only if there exists x e X such that (x,f (x)) e E(G). The results of Jachymski were generalized by several authors (see, for example, Bojor [3]; Chifu and Petrusel [6]; Samreen and Kamran [13]; Asl et al. [2]; Abbas and Nazir [1]). Recently, Tiammee and Suantai [21] introduced the concept ofg-graph preserving for multi-valued mappings and proved their fixed point theorem in a complete metric space endowed with a graph.

Definition 1.5 [21] Let X be a nonempty set and G = (V(G),E(G)) be a graph such that V(G) = X, and let T: X ^ CB(X). T is said to be graph preserving if it satisfies the following:

if (x, y) e E(G), then (u, v) e E(G) for all u e Tx and v e Ty.

Definition 1.6 [21] Let X be a nonempty set and G = (V(G),E(G)) be a graph such that V (G) = X, and let T: X ^ CB(X), g: X ^ X. T is said to be g -graph preserving if it satisfies

the following: for each x, y e X,

if (g(x),g(y)) e E(G), then (u, v) e E(G) for all u e Tx, v e Ty.

By using the concept of 'g-graph preserving' introduced by Tiammee and Suantai [21] and the concept of a Reich type multi-valued contraction defined by Reich [19], we define a new class of Reich type multi-valued contraction on a complete metric space satisfying theg-graph preserving condition and then we shall study the fixed point theorem for such mappings. Moreover, we establish some results on common fixed points for two multivalued mappings. The results of this research extend and generalize several well-known results from previous work.

2 Main results

Let (X, d) be a metric space. Denote CB(X) the set of all nonempty closed and bounded subsets of X. For a e X and A,B e CB(X), define

d(x,A) = inf{d(x,y) | y e A}, D(A,B) = inf{d(x,B) | x e A}.

Also, define

H(A, B) = max , A), sup d(x, B) [.

The mapping H is said to be a Hausdorff metric induced by d. The next lemma will play central roles in our main results.

Lemma 2.1 Let (X, d) be a metric space. If A, B e CB(X) and x e A, then for each e > 0, there is b e B such that

d(a, b)<H(A, B) + e.

We start with the new class of Reich type multi-valued (а, в, Y)-contraction on a complete metric space.

Definition2.2 Let(X, d)beametricspace, G = (V(G), E(G)) be a directed graph such that V(G) = X, g: X ^ X, and T: X ^ CB(X). T is said to be a Reich type weak G-contraction with respect to g or a (g, а, в, Y)-G-contraction provided that

(1) T is g-graph preserving;

(2) there exist nonnegative numbers а, в, Y with а + в + Y <1 and

xgB xeA

H(Tx, Ty) < ad(g(x),g(y)) + ßD(g(x), Tx) + yD{g(y), Ty)

for all x,y e X such that (g(x),g(y)) e E(G).

Example 2.3 Let N be a metric space with the usual metric. Consider the directed graph defined by V(G) = X and E(G) = {(2n -1,2n + 1): n e N}U {(2n, 2n + 2): n e N - {1}} U

{(2n, 2n + 4): n e N - {1}} U {(2n, 2n): n e N - {1}} U {(1,1), (6,4)}. Let T: X ^ CB(X) be defined by

{2k, 2k +2} if n = 2k - 1,k e N, {1} if n = 2k,k e N,

and g : N ^ N be defined by

g(n) =

2k if n = 2k + 2,k e N, 2k-1 if n = 2k + 1,k e N, 2 if n = 1,2.

We will show that T isa(g, a, ff, y )-G-contraction with a = 0,f = y = |.Let(g(x), g(y)) e E(G). If (g(x),g(y)) = (2k -1,2k +1) for k e N, then (x,y) = (2k +1,2k + 3), Tx = {2k + 2,2k + 4}, and Ty = {2k + 4,2k + 6}.We obtain (2k + 2,2k + 4), (2k + 2,2k + 6), (2k + 4,2k + 4), (2k + 4,2k + 6) e E(G). Also, £>(g(x), Tx) = 3, £(g(y), Ty) = 3, and

H (Tx, Ty) = max] sup d(a, Tx), sup d(b, Ty) \

'■aeTy beTx '

= max J sup d(a, {2k + 2,2k + 4}), sup d(b, {2k + 4,2k + 6})}

UcTii fccTV J

lae2y beTx

= max {2,2} = 2

< 0d(g(x),g(y)) + 3(3)+ 3(3)

< 0d(g(x),g(y)) + 3D(g(x), Tx) + 3D(g(y), Ty).

If (g(x),g(y)) = (2k, 2k + 2) or (2k, 2k + 4) or (2k, 2k) for k e N - {1}, then Tx = Ty = {1} and (1,1) e E(G). It follows that

H(Tx, Ty) = max] sup d(a, Tx), sup d(b, Ty) \

'■aeTy beTx '

< 0d(g(x),g(y)) + 3D(g(x), Tx) + 3D(g(y), Ty).

If (g(x),g(y)) = (1,1), then x = y = 3, Tx = Ty = {4,6}, and (4,4), (4,6), (6,4), (6,6) e E(G). It follows that

H(Tx, Ty) = max] sup d(a, Tx), sup d(b, Ty) \

'■aeTy beTx '

< 0d(g(x),g(y)) + 3D(g(x), Tx) + 3D(g(y), Ty).

If (g(x),g(y)) = (6,4), then x = 8, y = 6, and Tx = Ty = {1} and (1,1) e E(G). Note that d(g(x),g(y)) = 2, D(g(x), T8) = 5, D(g(y), T6) = 3, and so

H(Tx, Ty) = max] sup d(a, Tx), sup d(b, Ty) |

UeTy beTx '

< 0d(g(x),g(y)) + 3D(g(x), Tx) + 3D(g(y), Ty). It follows that T is a (g, 0,1, |)-G-contraction.

Property A For any sequence {xn}neN in X, if xn ^ x and (xn, xn+i) e E(G) for n e N, then there is a subsequence {xnk }nkeN such that (xnk, x) e E(G) for nk e N.

Lemma 2.4 Let (X, d) be a metric space with the directed graph G, g1, g2: X ^ Xbe surjec-tive maps and, for i = 1,2, Ti: X ^ CB(X) be g-graph preserving satisfying the following: there exist nonnegative numbers a, ¡i, y with a + ¡i + y <1 such that, for all x,y e X, if (g1(x),g2(y)) e E(G), then

H(T1 x, T2y) < ad(g1(x),g2(y)) + iD(g1(x), T1x) + yD(g2(y), Tjy),

and if (g2(x),g1(y)) e E(G), then

H(T2x, T1y) < ad(g2(x),g1(y)) + iD(g2(x), T2x) + yD(g1(y), T^).

(A) there emte x0 e X sMch that (g1(x0), u) e E(G) for some u e T1x0, and

(B) if (g1(x),g2(y)) e E(G), then (z, w) e E(G)for a// z e T1x, w e T2y, and f (g2(x),g1(y)) e E(G), then (b, r) e E(G) for all b e T2x, r e T1y,

then there exists a sequence {xk}keNU{0} in X such that, for each k e N,

g1(x2k) e T2x2k-1, g2(x2k-1) e T1x2k-2, (g1(x2k-2),g2(x2k-1^, (g2(x2k-1),g1(x2k)) e E(G),

and {S(xk)} is a Cauchy sequence in X where

S(xk ) =

gi{xk) ifk is even, g2(xk ) ifk is odd.

Proof Sinceg2 is surjective, there exists x\ e X such thatg2(xi) e Tixo and (gi(xo),g2(xi)) e E(G). By Lemma 2.1, there exists x2 e X such that g1(x2) e T2x1 and d(g2(x1),g1(x2)) < H(T1 xo, T2x1) + (a + ß). By (B), it follows that (g2(x1),g1(x2)) e E(G). By assumption,

^g2(xi), gi(x2)) < H (Tixo, T2xi) + (a + ß )

< a^gi(xo),g2(xi^ + ß^gi(xo), Tix^ + y^g2(xi), T2X1)

+ (a + ff)

< ad(g1(x0),g2(x0) + f d(g1(x0),g2(x0) + Yd(g2(x0,g^)) + (a + f).

Hence,

(1 - Y)d(g2(x0,g^)) < (a + f )4g1(x0),g2(x0) + (a + f). It follows that

d(g2(x1),g1(x2)) < (a- f)d(g1(x0),g2(xi)) + ^

= n^g1(x0), g2(x0) + n, (1)

where n = t-t < 1. Next, by Lemma 2.1, we can choose x3 e X such that g2(x3) e Tix2 and

d(g1(x2),g2(x3^ < H(T2x1, T1x2) + (a + f )n.

By (B) again, it follows that (g1(x2),g2(x3)) e E(G). By assumption again, we have

^g1(x2),g2(x3^ < H(T2x1, T1x2) + (a + f )n

< ad^fo),g1(x2)) + f Dg2(x1), T2x^ + YDg1(x2), T1 x2) + (a + f)n

< ad(g2(x0,g^)) + f d(jg2(x\),g1(x2)) + Yd^fe),g2(x3))

+ (a + f)n.

Hence,

(1 - Y)d(g1(x2),g2(x3^ < (a + f )4g2(x1),g^)) + (a + f )n. It follows that

d(g1(x2),g2(x3^ < n4g2(x1),g^)) + n2. By the inequality of (1), we have

d(g1(x2),g2(x3)) < n24g1(x0),g2(x0) + 2n2. Continuing in this fashion, we obtain sequences {xk} and {S(xk)} with the property that

S(xk ) =

g1(xk) if k is even, g2(x/) if k is odd,

and for each k e N,

g1(x2k) e T2x2k-1, g2(x2k-1) e T^2k-2, (g1(x2k-2),g2(x2k-1^, (g2(x2k-1),g1(x2k)) e E(G)

d(S(xk),S(xk+1^ < nkd(g1(x0),g2(x1)) + knk. Since n < 1, we have

to TO TO

^2(d(S(xk),S(xk+1))) < 4g1(xo),g2(x0) ^ nk + knk < to.

k=0 k=0 k=0

It is straightforward to check that {S(xk)} is a Cauchy sequence in X. □

Theorem 2.5 Let (X, d) be a complete metric space with the directed graph G, g1, g2: X ^ X be surjective maps and, for i = 1,2, Ti: X ^ CB(X) be g-graph preserving satisfying the following: there exist nonnegative numbers a, ff, y with a + ff + y <1 such that, for all x,y e X, if (g1(x),g2(y)) e E(G), then

H(T1 x, T2y) < ad(g1(x),g2(y)) + f D(g1(x), T1x) + YD(g2(y), Tjy),

and if (g2(x),g1(y)) e E(G), then

H(T2x, T1y) < ad(g2(x),g1(y)) + fD(g2(x), T2x) + yD(g1(y), T^).

If the following hold:

(1) there exists x0 e X s^ch that (g1(x0), u) e E(G)for some u e T1x0;

(2) if (g1(x),g2(y)) e E(G), then (z, w) e E(G) for all z e T1x, w e T2y and if (g2(x),g1(y)) e E(G), then (b, r) e E(G) for all b e T2x, r e T1y;

(3) X has Property A,

then there exist u, v e X such thatg1(u) e T1u org2(v) e T2v.

Proof By (1), let xo e X be such that (gi(X0),g2(Xi)) e E(G) for some g2(xi) e Tixo. By Lemma 2.4, there exists a sequence {xk}keNU{0} in X such that, for each k e N,

gi(x2k) e T2X2k-1, g2(X2k-l) e TiX2k-2, (gl(X2k-2),g2(X2k-i^, (g2(X2k-i),gi(X2k)) e E(G),

and {5(Xk)} is a Cauchy sequence in X where

S(x/) =

g1(xk) if k is even, g2(xk) if k is odd.

Since X is complete, the sequence {S(xk)} converges to a point w for some w e X. Let u, v e X be such that g1(u) = w = g2(v). By Property A in (3), there is a subsequence {S(xkn)} such that (S(xkn),g1(u)) e E(G) for any n e N. We claim thatg1(u) e T1u org2(v) e T2v. Let A = {kn | kn is even} and B = {kn | kn is odd}. Since A U B is infinite, at least A or B must be infinite. If A is infinite, for each g2(xkn+1), where kn e A, we have

D(g2(v), T2v) < d(g2(v),g2(xk„+1)) + Dg2(xkH+1), T2v)

< 4g2(v),g2(xkn+1)) + H(Txkn, T2v)

< d(g2(v),g2(xkH+1^ + ad(g1(xkn),g2(v^ + iD(g1(xk„), Tx^) + Y Dg2(v), T2 v)

< 4g2(v),g2(xkn+1)) + ad(g1(xkn),g2(v^ + i4g1(xk„),g2(xkH+1))

+ YD(g2(v), T2^.

We obtain

D(g2(v), T2v) < 1-Y [d(g2(v),g2(xkH+1)) + ad(g1(xkn),g2(v)) + id(g1(xkn),g2(xkH+1))].

Since {g1(xk^n)} and {g2(xkn+1)} are subsequences of S(xm), they converge tog2(v) as n ^ro, and hence D(g2(v), T2v) = 0. Since T2v is closed, we conclude thatg2(v) e T2v. Similarly, if B is infinite, we can show that g1(u) e T1u, completing the proof. Note that if A and B in Theorem 2.5 are both infinite theng2(v) e T2v andg1(u) e T1u. □

The following example illustrates Theorem 2.5.

Example 2.6 Let (X, d) be a metric space where X = [0,1] and d is a usual metric on R. Consider the directed graph G = (V(G),E(G)) defined by V(G) = X and

E(G) = (0,0),(i,i), 0,

i,oUi,i

2 / V2 2

Let T, S : X ^ CB(X) and ghg2 : X ^ X be defined by

{2} if x = i,

{0,2} if x e (0, i) },

if x = 02, --f,

if x = 0,i,i,i,

{0,|} if x e (0, i) }, gi(x) = x2 and g2(x) = x for all x e X.

It is clear that S, T aregi,g2-graph preserving, respectively. It is straightforward to check that the conditions (i), (2), (3) of Theorem 2.5 are satisfied. Next, we will show that, for all

x, y e X, if (g1(x),g2 (y)) e E(G), then

H(Tx,Sy) < ad(g1(x),g2(y)) + fD(g1(x), Tx) + yDg(y),Sy),

and if (g2(x),g1(y)) e E(G), then

H(Sx, Ty) < ad(g2(x),g1(y)) + fD(g2(x),Sx) + yD(g1(y), Ty),

where a = 0, f = y = ).

If (g1(x),g2(y)), (g2(x),g1(y)) e E(G) \ {(1,1)}, thenH(Tx,Sy) = 0 = H(Sx, Ty). So the above inequalities are satisfied.

If (g1(x),g2(y)) = (1,1), then g1(x) = 1, g2(y) = 1, which implies that x = 1 and y = 1. Thus we have Tx =2 and Sy = 4 and hence

h (Tx, Sy)=K{

< 0(0)+K2)+K4

= ad(g1(x),g2(y)) + fD(gi(x), Tx) + YD(g2(y),Sy).

If (g2(x),g1(y)) = (1,1), then g2(x) = 1 and g1(y) = 1, which yields x = 1 and y = 1. Thus we have Sx =4 and Ty = 2 and hence

h (Sx, Ty )=K{ 4112

< 0(0) + )(4) + )(2

= ad(g2(x),g^)) + £D(g2(x),Sx + YDg1(y), Ty).

By Theorem 2.5, there are u, v e X such that g1(u) e Tu or g2(v) e Sv. In this example, choose u =4. Sinceg2(u) = u, we have u e Su = {u}.

From Theorem 2.5 one deduces the next two corollaries.

Corollary 2.7 Let (X, d) be a complete metric space with the directed graph G, g: X ^ X be a surjective map, and T1, T2: X ^ CB(X) be g-graph preserving satisfying

H(T1 x, T2y) < ad(g(x),g(y)) + f D(g(x), T1x) + yD(g(y), T2y)

for allx, y e X with (g(x), g(y)) e E(G). If the following hold:

(1) there exists x0 e X s^ch that (g(x0), u) e E(G)for some u e T1x0;

(2) X has Property A,

then there exists u e X such thatg(u) e T1u org(u) e T2u.

Proof Set g1 = g2 = g. Then this corollary follows immediately from Theorem 2.5. □

Corollary 2.8 Let (X, d) be a complete metric space, G = (V(G), E(G)) be a directed graph such that V(G) = X, and let g: X ^ X be a surjective map. IfT : X ^ CB(X) is a multivalued mapping satisfying the following properties:

(1) T is a (g, a, i, y)-G-contraction;

(2) the set XT = {x e X | (g(x),y) e E(G) for some y e Tx} = 0;

(3) X has Property A,

then there exists u e X such thatg(u) e Tu.

Proof Set T1 = T2 = T and g1 = g2 = g. Then the result follows directly from Theorem 2.5.

A partial order is a binary relation < over the set X which satisfies the following conditions:

(1) x < x (reflexivity);

(2) if x < y and y < x, then x = y (antisymmetry);

(3) if x < y and y < z, then x < z (transitivity),

for all x, y e X .A set with a partial order < is called a partially ordered set. We write x < y if x < y and x = y.

Definition 2.9 Let (X, <) be a partially ordered set. For each A, B c X, A < B if a < b for any a e A , b e B.

Definition 2.10 [21] Let (X, d) be a metric space endowed with a partial order <, g: X ^ X a surjective map, and T: X ^ CB(X). T is said to be g-increasing if for any x,y e X,

g(x)<g(y) ^ Tx < Ty.

In the case g is the identity map, the mapping T is called an increasing mapping.

Theorem 2.11 Let (X, d) be a metric space endowed with a partial order <, g: X ^ X be a surjective map and T: X ^ CB(X) be a multi-valued mapping. Suppose that

(1) T is g-increasing;

(2) there exist x0 e X and u e Tx0 such that g(x0) < u;

(3) for each sequence {xk} such that g(xk) <g(xk+1) for all k e N and g(xk) converges to g(x) for some x e X, then g(xk) < g(x) for all k e N;

(4) there exist nonnegative numbers a, i, y with a + i + y <1 such that

H(Tx, Ty) < ad(g(x),g(y)) + iD(g(x), Tx) + yD(g(y), Ty)

for all x, y e X such that g(x) < g(y);

(5) the metric d is complete.

Then there exists u e X such thatg(u) e Tu. Ifg is injective, then there is a unique u e X such thatg(u) e Tu.

Proof Define G = (V(G),E(G)), where V(G) = X and E(G) = {(x,y) | x < y}. Let x,y e X be such that (g(x),g(y)) e E(G). Then g(x) <g(y) so by (1) it implies that Tx < Ty. For each

u e Tx, v e Ty, we have u < v, thus (u, v) e E(G). That is, T is g-graph preserving. By assumption (2), there exist x0 e X and u e Tx0 such that g(x0) < u. So (g(x0), u) e E(G) and hence the property (1) in Corollary 2.7 is satisfied. Moreover, we obtain the property (2) of Corollary 2.7 from the assumption ()). Set T1 = T2 = T, then the T1, T2 are g-graph preserving mappings satisfying

H(Tx, T2y) < ad(g(x),g(y)) + pD(g(x), T x) + yD(g(y), T2y)

for all x,y e X with (g(x),g(y)) e E(G). Therefore, from the result of this theorem follows Corollary 2.7.

Assume that g is injective. Let u, v e X be such that g(u) e Tu and g(v) e Tv. Suppose that g(u) = g(v). Without loss of generality, assume that g(u) < g(v). Since g(u) e Tu and g(v) e Tv, it follows that D(g(u), Tu) = D(g(v), Tv) = 0 and hence

d(g(u),g(v)) < H(Tu, Tv)

< ad(g(u),g(v)) + f D(g(u), Tu) + yD(g(v), Tv)

< ^g(u),g(v^.

This leads to a contradiction. Thus g(u) = g(v). Since g is injective, we have u = v. □

We obtain the following result by considering g(x) = x for all x e X.

Corollary 2.12 Let (X, d) be a metric space endowed with a partial order < and T: X ^ CB(X) be a multi-valued mapping. Suppose that

(1) T is increasing;

(2) there exist x0 e X and u e Tx0 swch that x0 < u;

(3) for each sequence {xk} such that xk < xk+1 for all k e N and xk converges to x for some x e X, then xk < x for all k e N;

(4) there exist nonnegative numbers a, ff, y with a + ff + y <1 such that

H(Tx, Ty) < ad(x,y) + ffD(x, Tx) + yD(y, Ty)

for all x, y e X such that x < y;

(5) the metric d is complete.

Then there is a unique u e X such that u e Tu.

Competing interests

The authors declare that they have no competing interests. Authors' contributions

All authors read and approved the final manuscript. Acknowledgements

This paper was supported by the Faculty of Science and Technology, Prince of Songkla University, Pattani Campus. Received: 16 December 2014 Accepted: 6 April 2015 Published online: 22 April 2015

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