0Felhi and Aydi Fixed Point Theory and Applications (2016) 2016:22 DOI 10.1186/s13663-016-0510-y
0 Fixed Point Theory and Applications
a SpringerOpen Journal
RESEARCH Open Access
^ CrossMark
Best proximity points and stability results for controlled proximal contractive set valued mappings
Abdelbasset Felhi1 and Hassen Aydi2,3*
"Correspondence: hmaydi@uod.edu.sa 2Department of Mathematics, College of Education of Jubail, University of Dammam, P.O. Box 12020, Industrial Jubail, 31961, Saudi Arabia
3Department of MedicalResearch, China MedicalUniversity Hospital, China MedicalUniversity, Taichung, Taiwan
Fulllist of author information is available at the end of the article
Abstract
In this paper, we introduce first the concept of a Pompeiu-Hausdorff ¿-metric-like space. We also establish some best proximity points and stability results for controlled proximal contractive set valued mappings in the class of ¿-metric-like spaces and partial ¿-metric spaces. Moreover, we provide some examples and many nice consequences from our obtained results.
MSC: 47H10; 54H25
Keywords: Pompeiu-Hausdorff ¿»-metric-like; best proximity point; controlled proximal contraction; stability
ft Spri
ringer
1 Introduction and preliminaries
Markins [1] and Nadler [2] initiated the study of fixed point theorems for set valued operators. Since then, several other papers have been concerned with the study of multi-valued operators in variant (generalized) metric space. We cite for example, Ali et al. [3, 4], Aydi et al. [5, 6], Berinde and Berinde [7], Berinde and Pacurar [8], Boriceanu et al. [9], Bota [10], Ciric [11], Ciric and Ume [12,13], Czerwik [14], Daffer and Kaneko [15], Jleli et al. [16], Mizoguchi and Takahashi [17], etc. In this paper, we are interested first to initiate the concept of a Pompeiu-Hausdorff ¿-metric-like and to prove some best proximity points and stability results.
On the other hand, metric-like spaces were considered by Hitzler and Seda [18] under the name of dislocated metric spaces. In 2013, Alghamdi etal. [19] generalized the notion of a ¿-metric [14] by introducing the concept of a ¿-metric-like and proved some related fixed point results. After that, Hussain et al. [20] established some fixed point theorems in the setting of ¿-metric-like spaces.
Definition 1.1 Let X be a nonempty set and s > 1 bea given real. A function a : X x X ^ R+ is said to be a ¿-metric-like (or a dislocated ¿-metric) on X if for any x,y, z e X, the following conditions hold:
(bmi) a (x, y) = 0 ^ x = y;
(bm2) a(x,y)= a(y,x);
(bm3) a(x,z) < s(a(x,y) + a(y,z)).
The pair (X, a) is then called a ¿-metric-like space.
© 2016 Felhi and Aydi. This article is distributed under the terms of the Creative Commons Attribution 4.0 InternationalLicense (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use,distribution, and reproduction in any medium, provided you give appropriate credit to the originalauthor(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Let (X, a) be a ¿-metric-like space. An open a-ball [Ba (x, e):x e X, e > 0} is defined as Ba (x, e) = {y e X: |a(x,y) - a(x,x)| < e}, for all x e X and e >0. A sequence {xn} in X converges to x e X if and only if
lim a(xn,x) = a(x,x). (1.1)
Mention that the limit for a convergent sequence is not unique in general. {xn} is Cauchy if and only if limn,m—TO a(xn,xm) exists and is finite. We say that (X, a) is complete if and only if each Cauchy sequence in X is convergent.
Lemma 1.2 Let (X, a) be a b-metric-like space and {xn} be a sequence that converges to u with a(u, u) = 0. Then, for each y,z e X, one has
- a (u, z) < liminf a (xn, z) < limsup a (xn, z) < sa (u, z) and a (z, z) < 2sa (z, y).
s n—n—rn
In 2015, Aydi etal. [21] introduced the following concept.
Definition 1.3 Let (X, d) be a rectangular b-metric space. We say that (X, d) satisfies the property (SC) if for every sequence {xn} in X and all x, y e X, we have
lim d(xn,x) = 0 ^ lim d(xn,y) = d(x,y). We extend Definition 1.3 to the class of b-metric-like spaces.
Definition 1.4 Let (X, a)bea b-metric-like space. We say that (X, a) satisfies the property (GC) if for all sequences {xn}, {yn} in X and all x,y e X, we have
lim a(xn,x) = lim a (yn,y) = 0 ^ lim a (xn,yn) = a(x,y).
Remark 1.5
1. If (X, d) is a rectangular b-metric space satisfying the property (GC), then it also satisfies the property (SC). Indeed, let {xn} be a sequence in X and x,y e X such that limn—TO d(xn,x) = 0. Take {yn} in X such that yn = y for all n > 0. Then
d(yn,y) = d(y,y) = 0, and so limn—TO d(yn,y) = 0. Since (X, d) satisfies the property (GC), it follows that limn—TO d(xn,yn) = d(x,y), that is, limn—TO d(xn,y) = d(x,y), and so (X, d) satisfies the property (SC).
2. Let (X, a) be a b-metric-like space satisfying the property (GC). Take {xn} a sequence in X and x,y e X such that a(y,y) = 0 and limn—TO a(xn,x) = 0. Then
limn—TO a (xn, y) = a (x, y).
The following examples make effective use of the property (GC)•
Example 1.6 Let X = [0,1]. Consider the mapping a : X x X ^ [0, œ) defined by a (x, y) = (x + y - xy)2 for all x,y e X. Then (X, a) is a è-metric-like space with s = 2. Let {xn} and {yn} in X such that
lim a (xn, x) = lim a (yn, y) = 0.
n^œ n^œ
It follows that a(x,x) = a(y,y) = 0, and so x = y = 0. Then we get lim x2n = lim y2n = 0.
n—>to n—>TO
This leads to
lim xn = lim yn = 0.
n—TO n—TO
Hence,
lim a(xn,yn) = lim (xn + yn -xnyn)2 = 0 = a(0,0).
n—TO n—TO
Consequently, (X, a) satisfies the property (GC).
Example 1.7 Let X = {0,1,2}. Consider the mapping a : X x X — [0, to) defined by
a (0,0) = 0, a (1,1) = a (2,2) = 2, a (0,1)= a (1,0) =4, a (1,2) = a (2,1) = 2, a (0,2) = a (2,0) = 2.
Then (X, a) is a ¿-metric-like space with s = 2. Let {xn} and {yn} in X such that
lim a (xn, x) = lim a (yn, y) = 0.
n—TO n—TO
It follows that a(x,x) = a(y,y) = 0, and so x = y = 0. Moreover, there exists N e N, such that, for all n > N,
a(xn,0)< 2 and a0n,0)< 2.
Therefore
a(xn,0) = 0 and a(yn,0) = 0, Vn >N.
Thus, for all n > N,we have xn = yn = 0. This yields a(xn,yn) = a(0,0) for all n > N, and so limn—TO a(xn,yn) = a(x,y). Hence, (X, a) satisfies the property (GC).
Lemma 1.8 Let (X, a) be a b-metric-like space. Let {xn} and {yn} be two sequences in X and x,y e X such that limn—TO a(xn,x) = limn—TO a(yn,y) = 0. Then one has
s~2a(x,y) < liminfa(xn,yn) < limsupa(xn,yn) < s2a(x,y).
n—TO n—TO
We also have the following useful lemma. Lemma 1.9 Any metric-like space satisfies the property (GC).
Proof It suffices to take s = 1 in Lemma 1.8. □
Recently, Aydi et al. [21, 22] introduced the concept of a Pompeiu-Hausdorff metric-like. The aim of the first part of paper is to extend this concept to the class of b-metric-like spaces and then to prove some results on best proximity points and stability for controlled proximal contractions, so generalizing the very recent paper of Kiran et al. [23]. In the second part of paper, the analogous of above results in the class of partial b-metric spaces is studied.
From now on, let (X, a) be a b-metric-like space. As in [21,22,24], let Cb(X) be the family of all nonempty, closed and bounded subsets of the b-metric-like space (X, a), induced by the b-metric-like a. For A,B e Cb(X) and x e X, define
a(x, A) = inf{a(x, a): a e A}, Sa (A, B) = sup{ a (a, B): a e A}, Sa (B, A) = sup{a(b, A): b e B}.
Hb (A, B) = max {5CT (A, B), (B, A)}. (.2)
The above is called a Pompeiu-Hausdorff b-metric-like. For A and B two nonempty subsets of a b-metric-like space (X, a), define
a (A, B) = infja (a, b): a e A, b e B},
A0 = {a e A : a (a, b) = a (A, B),for some b e B},
B0 = {b e B: a (a, b) = a (A, B),for some a e A}.
As in [25], the concept of a weak P-property is stated as follows.
Definition 1.10 Let A and B be nonempty subsets of a b-metric-like space (X, a) with A0 = 0. The pair (A, B) is said to have the weak P-property if and only if
fa (x1, yi)=a (A, B),
\ , , m ^ a (x^ x2) < a ^ У'l),
[a (x2, y2) = a (A, B)
where xi,x2 e A0 and y1,y2 e B0.
Example 1.11 Let X = {(1,2), (0,1), (1,3), (3,1)} be endowed with the b-metric-like a((x1, x2),(y1,y2)) = (x1 + x2 + y1 + y2)2 for all (x1,x2), (y1,y2) e X. Let A = {(1,2), (0,1)} and B = {(1,3), (3,1)}. Clearly,
a ((0,1), (1,3)) =25 = a (A, B) and a ((0,1), (3,1)) = a (A, B).
a((0,1), (0,1)) = 4 < 64 = a((1,3), (3,1)).
Moreover, A0 = 0. Hence, the pair (A,B) satisfies the weak P-property.
Example 1.12 Let A and B be nonempty subsets of a ¿-metric-like space (X, a) with A0 = 0 and a (A, B) = 0. Then the pair (A, B) satisfies the weak P-property.
On the other hand, the definition of a best proximity point is as follows.
Definition 1.13 Let (X, a) be a ¿-metric-like space. Consider A and B two nonempty subsets of X. An element a e X is said to be a best proximity point of T: A ^ B if
a (a, Ta) = a (A, B).
It is clear that a fixed point coincides with a best proximity point if a (A, B) = 0. For more results on best proximity points, see for example [26-31].
In this paper, we give first some properties of H%. Second, we establish some existence results on best proximity points and some stability results for controlled proximal set valued contractive mappings in the setting of two (generalized) metric spaces. We will support the obtained theorems by some concrete examples. We also provide many interesting consequences and corollaries.
2 Properties and preliminaries
First, we present some useful properties of the Pompeiu-Hausdorff ¿-metric- like Hba.
Lemma 2.1 [21, 22] Let (X, a) ¿e a ¿-metric-like space and A any nonempty set in (X, a), then
if a (a, A) = 0, then a e A. (.1)
Lemma 2.2 Let (X, a) ¿e a ¿-metric-like space. For x e X and A, B, C e C(X), we have
(i) Hi (A, A) = Sa (A, A) = sup {a (a, A):a e A};
(ii) H(A,B)=H(B,A);
(iii) Hi (A, B) = 0 implies that A = B;
(iv) Ha (A, B) < s(H (A, C) + Ha (C,B));
(v) a(x, A) < s(a (x,y) + a (y, A)).
Proof (i)-(iii) are clear. (iv) Let a e A, ¿e B, and c e C. By a triangular inequality
a (a, ¿) < s(a (a, c) + a (c, ¿)).
The points and c are arbitrary, so
a (a, B) < s(a (a, c) + a (c, B)) < s(a (a, c) + Sa (C, B)) < s(a (a, C) + Sa (C, B)).
Again, a is arbitrary, so
5a (A,B) < s(5, (A, C) + 5, (C,B)) < sHba (A, C) + sHba (C,B).
Similarly, by symmetry of , we have
5a(B,A) < s(Hba(A, C) +Hba(C,B)).
Combining the two above inequalities, we get (iv).
(v) For a e A and x,y e X, we have a(x, A) < a(x, a) < s(a(x,y) + a(y, a)). Again, a is arbitrary, then
a(x,A) < s(a(x,y) + a(y,A)). □
The following two lemmas are very essential for best proximity points and stability results stated in the next section. The proofs are very classical.
Lemma 2.3 Let (X, a) be a b-metric-like space. Let A, B e Cb(X) and h >1. For any x e A, there exists y = y(a) e B such that
a(x,y) < hHb(A,B). (2.2)
Lemma 2.4 Let (X, a) be a b-metric-like space. Let A, B e Cb(X) and a e A. Then, for all e >0, there exists a pointy e B such that a (a, y) < (A, B) + e.
3 Best proximity points and stability results on the class of b-metric-like spaces 3.1 Best proximity points
First, we need the following definition.
Definition 3.1 Let A and B be nonempty subsets of a b-metric-like space (X, a) such that A0 = 0. Let xo e A0 and r > 0. A mapping T: A — Cb(B) is called a proximal contraction on Ba (x0, r), if there exists a e (0,1) such that
Hba (Tx, Ty) < aa(x,y), (3.1)
for all x,y e Ba (x0, r) n A.
Our first main result is the following theorem.
Theorem 3.2 Let A and B be nonempty closed subsets of a complete b-metric-like space (X, a) and r >0. Let T: A — Cb(B) be a multi-valued mapping. Suppose that
(i) A0= 0;
(ii) for each x e A0, we have Tx c B0;
(iii) the pair (A, B) satisfies the weak P-property;
(iv) there exists x0 e A0 such that T is a proximal contraction on Ba (x0, r) and 5a(Tx0, {x0})+a(A,B) < (1 - \/as)r;
(v) (X, a) satisfies the property (GC).
Then T has a best proximity point in Ba (x0, r) n A. We also have a (x*, x*) = 0.
Proof By assumption (iv), there exists x0 e A0 such that T is a proximal contraction on Ba (x0, r) and
5a (Tx0, {x0}) + a (A, B) < (1 - VOs)r.
Let yo e Tx0. By condition (ii), we have Tx0 c B0. Then there exists xi e A0 such that
a(x1, y0)=a(A, B). (3.2)
We have
a(x0,x1) < s[a(x0,y0) + a(y0,x1)]
< s[5ff (Tx0, {x0}) + a (A,B)]
< 2sTZ~s (1- -a)r. (3.3) On the other hand, we have
a(x0,x0) - a(x0,x1) < (2s - 1)a(x0,x1).
a(x0,x1) -a(x0,x0) < a(x0,x1) < (2s - 1)a(x0,x1). Then
|a(x0,x1)-a(x0,x0)| < (2s-1)a(x0,x1) (2s -1), < 2^ = s-1(1 - sfas)r < r.
Thus, x1 e Ba (x0, r) n A0. By Lemma 2.3, there exists y1 e Tx1 such that
a(y0,y1) < -=Hb(Tx0, Tx1). (3.4)
So, by (3.1), we get
a(yo,yi) <ysa(xo,xi). (3.5)
Since yi e Txi ç Bo, there exists x2 e Ao such that
a(x2, yi) = a (A, B). (3.6)
From condition (iii), (3.2), and (3.6)
a(xi,X2) < a(yo,yi). (3.7)
Therefore,
a(xi, x2) < J aa(xo, xi). (3.8)
We have
\a(x0,x2)-a(x0,x0)| < (25 -l)a(x0,x2)
< 5(25 - l)[a(x0,xl) + a(xl)x2)]
< 5(25 -l)[a(x0,xl) + 5a(xl,x2)]
< 5(25 -1)
l + 5,/ ^
a(x0, xl)
< 5(25- l)[l + Vos]—-(l - \fas)r
252 - 5
= (l - a5)r < r.
Then x2 e Ba (x0, r) n A0. Again, by Lemma 2.3, there exists y2 e Tx2 such that
a(ylty2) Hb(Txi, Tx2).
So, by (3.l), we get
a(yl,y2) <J 5a(xl,x2).
Since y2 e Tx2 ç B0, then there exists X3 e A0 such that
c(x3, y2) =a(A, B). By condition (iii), (3.8), and (3.10)
a(x2,x3) < a(yl,y2) <J^a(xl,x2) <IJ^ ) a(x0,xi).
We have
\a(x0,x3)-a(x0,x0)\ < (25 -l)a(x0,x3)
< (25 -1) [5a(x0,xi) +
(XI, x2) +
(x2, x3)]
< (25 - l) [5a (x0, xl) +
(xl, x2) +
(x2, x3)]
< 5(25 -1)
l+ 5 + 5V 5
(x0, xl)
(3.l0)
(3.ll)
(3.l2)
< s(2s - 1)[1 + fas + (fas)2] —-(1 - fas)r
2s2 - s
= (1 - (Jasf) r < r.
Then x3 e Ba (x0, r) n A0.
Continuing this process, we complete two sequences {xn} c Ba (x0, r) n A0 and {yn} c B0 such that
a(x„, y„_l) = a (A, B),
a(x„,x„+l) < a(y„-l,y„) < (^ffa(x0,xi), yn e Txn, for all n = l, 2,____
For m > n,we have
m-1 m-1
a(xn, xm) < ^ ska (xk, xk+1) < ^ (*fsa)ka (x0, x1)
k=n k=n
< y^(*fsa)ka(xo,x1) — 0 as n — ro.
We supposed that 0 < as < 1, so limn,m—ro a(xn,xm) = 0. Hence, {xn} is a Cauchy sequence in Ba(x0,r) n A. A similar reasoning shows that limn,m—ro a(yn,ym) = 0 and so {yn} is a Cauchy sequence in B. Since Ba (x0, r) n A and B are closed subsets of the complete ¿-metric-like space (X, a), there exist x* e Ba (x0, r) n A and y* e B such that
lim a (xn,= a (x*,x*) = lim a(xn,xm) = 0 and
i^œ v / v 7 n,m^œ
lim a (yn,y*) = a (y*,y*) = lim aftn,ym) = 0.
1_l(Vl V ' V ' TA VM_llVl
Since, for all n > 1, we have a(xn, yn-1) = a (A, B) and by condition (v), (X,a) satisfies the property (GC), by letting n — ro, we conclude that
a(x*,y*) = a (A, B).
On the other hand, since yn e Txn, we have, for all n > 1,
a(y*, Tx*) < sa(y*,yn) + sa(yn, Tx*) < saiy*,yn) + sH^(Txn, Tx*) < sa(y*, y^ + saa (xn, x*).
Letting n —^ ro, we obtain
a(y*, Tx*) < 0,
and so a(y*, Tx*) = 0. By Lemma 2.1, we have y* e Tx* = Tx*. Also, we have
a (A, B) < a(x*, Tx*) < a(x*, y*) = a (A, B). Thus, x* is a best proximity point of T. Moreover, we have a (x*,x*) = 0. □
The following example illustrates Theorem 3.2.
Example 3.3 Let X = [0, ro) x [0, ro). Consider the mapping a : X x X — [0, ro)as follows:
I, w ^ f(|x1-y1| + |x2-y21)2 if(x1,x2),(y1,y2) e [0,10]2, a((x1, x2), (Уl, y2))=\, ,2 ., ^
[ (x1 + x2 + y1 + y2)2 if not.
It is easy to see that (X, a) a complete ¿-metric-like space with s = 2.
Take A = (ljx [0,10] and B = |0}x [0,10]. Define the mapping T: A ^ Cb(B) by T (1, x) =
{(0,0), (0,x )} if 0 < x < 8,
{0} x [0,1] if 8 < x < 10.
Note that for all (1,x) e A, we have T(1,x) is closed and is bounded in (X, a). Remark that a (A, B) = 1, A0 = A and B0 = B. So, for each (1, x) e A0, we have T(1, x) c B0. Moreover, A and B are closed subsets of X. Consider the ball Ba (x0, r) with x0 = (1,0) and r = 82. Now, let (1,x1), (1,x2) e A and (0,y1), (0,y2) e B such that
Ja((1, x1),(0, y1))=a(A, B) = 1, [ a((1,x2),(0,y2))=a(A,B) = 1.
Necessarily, (x1 = y1 e [0,10]) and (x2 = y2 e [0,10]). In this case,
a((1, x1),(1, *2)) = a((0, y1),(0, y0),
that is, the pair (A,B) has the weak P-property. Now, we shall show that T is a proximal contraction on Ba (x0, r) with a = It is easy to see that B„ (x0, r) n A = {1}x [0, V82 - 2].
Let (1,x) and (1,y) e Ba (x0, r) n A. Then x,y e [0, v/82 - 2] c [0,8]. In this case, we have T(1,x) = {(0,0), ^0, 0 }, T(1,y) = {(0,0), (0, 2 Then
(T(1,x), T(1,y))
= maxL ((0,0), {(0,0), (0, y) 1),a ((0, ^ J(0,0)^0, |
{x2 (x -y)2\ (x - y)2
—,-i <-.
4 4 J " 4
Similarly, we have
5a (T(1,y), T(1,x)) < ^^ This yields
Hba (T(1,x), T(1,y)) = max{5a (T(1,x), T(1,y)), 5a (T(1,y), T(1,x))} < ((x—y-= aa((1,x), (1,y)).
We also have 5a (Tx0, {xo}) + a (A, B) = 2 < (1 - fas)r. Furthermore, (X,a) satisfies the (GC) property. In fact, let {(x„,y„)}, {(zn, tn)} in X and (x,y), (z, t) e X such that
lim a t(x„,y„),(x,y)) = lim a t(z„, t„),(z, t)) = 0.
Then cr((x,y), (x,y)) = cr((z, t), (z, t)) = 0. It follows that (x,y), (z, t) e [0,10]2. There also exists N e N such that (xn,yn), (zn, tn) c [0,10]2 for all n > N. This yields, for all n > N,
c((x„,yn),(x,y)) = (|Xn -x| + |yn -y\)2 and cr((zn, tn), (z, t)) = ( | zn - z | + | tn - t|)2.
lim \xn - x\ = lim \yn -y\ = lim \z„ - z\ = lim \tn - t\ = 0.
n^œ n^œ n^œ n^œ
lim a((x„,yn),(zn, tn)) = lim (\x„ - z„ \ + y - tn\)2
n^œ v 7 n^œ '
= (\x - z\ + \y -1\)2 = a((x, y),(z, t^.
Therefore, all conditions of Theorem 3.2 are verified. So, T has a best proximity point, which is x* = (1,0). It also verifies cr(x*,x*) = 0.
As consequences of our first result, we give the following immediate corollaries.
Corollary 3.4 Let A and B be nonempty closed subsets of a complete metric-like space (X, c) and r >0. Let T: A ^ Cb(B) be a multi-valued mapping. Suppose that
(i) A0= 0;
(ii) for each x e A0, we have Tx c B0;
(iii) the pair (A, B) satisfies the weak P-property;
(iv) there exists x0 e A0 such that T is a proximal contraction on Bc (x0, r) and
(Tx0, {x0}) + c(A,B) < (1 - vO)r. Then T has a best proximity point in Bc (x0, r) n A. We also have c (x*, x*) = 0.
Proof It suffices to take s = 1 in Theorem 3.2. By Lemma 1.9, (X,c) satisfies the property (Gc ). □
Corollary 3.5 Let A and B be nonempty closed subsets of a complete metric-like space (X, c) and r >0. Let T: A ^ B be a given mapping. Suppose that
(i) A0= 0;
(ii) for each x e A0, we have Tx e B0;
(iii) the pair (A, B) satisfies the weak P-property;
(iv) there exists x0 e A0 such that T is a proximal contraction on Bc (x0, r) and c(x0, Txq) + a (A, B) < (1 - v/os)r;
(v) (X, a) satisfies the property (GC).
Then T has a best proximity point in Ba (x0, r) n A. We also have a (x*, x*) = 0.
Proof It suffices to take s = 1 and T as a single-valued mapping in Theorem 3.2. □
Corollary 3.6 Let A and B be nonempty closed subsets of a complete metric space (X, d) and r >0. Let T: A ^ Cb(B) be a multi-valued mapping. Suppose that
(i) A0= 0;
(ii) for each x e A0, we have Tx c B0;
(iii) the pair (A, B) satisfies the weak P-property;
(iv) there exists x0 e A0 such that T is a proximal contraction on Bd(x0, r) and 5d(Tx0, {x0}) + d(A,B) < (1 - Va>.
Then T has a best proximity point in Bd(x0, r) n A.
If we choose A = B = X, then we have the following fixed point theorem.
Corollary 3.7 Let (X,a) be a complete b-metric-like space, r >0, and T: X ^ Cb(X) be a multi-valued mapping. Suppose there exist x0 e X and a e (0,1) such that
H(Tx, Ty) < aa(x,y),
for all x, y e Ba (x0, r) and 5a (Tx0, {x0}) < ^ 2 (1 - fas)r. Then T has a fixed point.
Proof Following the proof of Theorem 3.2, we construct two sequences {xn} c Ba (x0, r) and {yn} c X such that
a(xn, yn-1) = a(X, X), a(xn, xn+1) < a (yn-1, yn) < (ff)na (x0, x1), yn e Txn, for all n = 1,2,____
Moreover, there exist x* e Ba (x0, r) and y* e X such that
lim a (xn,x*) = a (x*,x^ = lim a(xn,xm) = 0 and
n^TO v / v / n,m^TO
lim a (yn, y*) = a (y*,y^ = lim a(ym ym) = 0.
n^TO v 7 v 7 n,m^TO
We have, for all n > 1,
a(x*,y^ < sa(x*,xn) + sa(xn,y*) < sa(x*,x^ + s2a(xn,yn-1) + s2a(yn-1,y*) = sa (x*, x^ + s2a(A, B) +s2a (yn-1, y*).
Letting n ^ to, we obtain
a(x*,y*) < sa(x*,x*) + s2a(X,X) + s2a(y*,y*) = s2a(X,X). (3.13)
Also, for all n > 1,
a(X,X) = a(xn,yn-1) < sa(xn,x*) + s2a(x*,y^ + s2a(y*,yn-1).
We pass to the limit n ^to,
a(X,X) < s2a(x*,y*). (.14)
Combining (3.13) and (3.14), we get
s-2a(X,X) < a(x*,y*) < s2a(X,X). (3.15)
On the other hand, since yn e Txn, we have, for all n > 1,
a(y*, Tx*) < sa(y*,yn) + sa(yn, Tx*) < saiy*,yn) + sHba{Txn, Tx*) < sa{y*,yn) + saa(xn,x*).
Letting n ^ro, we obtain
a(y*, Tx*) < 0,
and so a(y*, Tx*) = 0. By Lemma 2.1, we have y* e Tx* = Tx*. Again
a(X,X) < a(x*, Tx*) < a(x*,y*) < s2a(X,X).
We also have a(x*,x*) = 0. Thus, a(X,X) < a(x*,x*) = 0, and so a(X,X) = 0. It follows that a(x*, Tx*) = 0. By Lemma 2.1, we get x* e Tx* = Tx*. Here, we do not need the conditions (i), (ii), (iii) and (v) of Theorem 3.2. □
3.2 Stability results
In this paragraph, we extend and generalize the stability results due to Kiran et al. [23] to b-metric-like spaces.
Let A and B be nonempty subsets of a b-metric-like space (X, a) and T: A ^ Cb(B) be a multi-valued mapping. Take the set B(T) = {a e A : a(A,B) = a(a, Ta)}. It corresponds to the set of best proximity points of T.
Theorem 3.8 Let A and B be nonempty closed subsets of a complete b-metric-like space (X, a) and ri, r2 > 0. Let Ti: A ^ Cb(B), i = 1,2, be two multi-valued mappings. Suppose that
(i) Aq= 0;
(ii) for each x e A0, we have Tix c B0, i = 1,2;
(iii) the pair (A, B) satisfies the weak P-property;
(iv) (X, a) satisfies the property (GC);
(v) for each i = 1,2, there exists ai e A0 such that Ti is a proximal contraction on B a (ai, r) n A with the same Lipschitz constant a e (0,1), that is,
Hb(Tix, Tiy) < aa(x,y), (3.16)
for all x, y e Ba (ai, r) n A and Sa (Tiai, {ai}) + a (A, B) < 2s31s2 (1 -^/as)ri.
s4 r 1
Hba(B(T1),B(T2)) < --= supHba(Tx, T2x) +1 + s~1)a(A,B) . (3.17)
1 - sfas
■xeA
Proof Let e > 0 and x0 e B(T1), then there exists z0 e Tixo such that
a(x0, z0) < a(x0, Tix0) + e = a (A, B) + e.
By Lemma 2.4, there exists y0 e T2x0 such that
a(zo,y0) < H(Tix0, T2x0) + e.
Then, from (3.18) and (3.19), we get
a(x0,y0) < s[a(x0,Z0) + a(z0,y0)]
< s[Hba (Tix0, T2x0) + a (A, B) + 2e].
Since y0 e T2x0 c B0, there exists x1 e A0 such that
a(xi, y0) = a (A, B).
By Lemma 2.3, there exists y1 e T2x1 such that
a(y0,y1) < —=H(T2x0, T2x1).
Without loss generality, we take a2 = x0 and r2 = r such that
Sa Tx0,1x0}) + a (A, B) < 2s31- s2 (1- —as)r. As (3.3), we have
|a(x0,x1)-a(x0,x0)| < (2s-1)a(x0,x1)
< —S—)(1 - —as)r = s-1(1 - —as)r < r. 2s2 - s
Thus, x1 e Ba (x0, r) n A0. By Lemma 2.3, there exists y1 e T2x1 such that
a(y0,y1) < —= H(T2x0, T2x1). as
So, we get
a(y0,yi) <Ja(x0,xi).
Again, yi e T2x\ ç hence there exists x2 e such that
c(x2, yi) = a(A, B). By condition (iii), it follows that c(xi,X2) < o(yo,yi).
(3.2i)
(3.22)
(3.23)
(3.24)
(3.25)
(3.26)
Applying (3.24),
a(xi,x2) <ysa(x°,Xi). (3.27)
Repeating the same process and similar to the proof of Theorem 3.2, we construct two sequences {xn} c Ba (xo, r) n A0 and {yn} c B0 such that
o(Xn, yn-i) = a (A, B),
a(Xn,Xn+i) < o(yn-i,yn) < (^)no(xo,Xi),
yn e T2xn, for all n = i, 2,____
It follows that limn,m—TO a(xn,xm) = 0. Thus, {xn} is a Cauchy sequence in Ba (xo, r) n A. A similar reasoning shows that limn,m—TO o(yn,ym) = 0 and so {yn} is a Cauchy sequence in B. SinceBa (x0, r) n A and B are closed subsets of a complete ¿-metric-like space (X,a), there exist u e Ba (x0, r) n A and v e B such that
lim a(xn, u) = a(u, u) = lim a(xn, xm) = 0 and
n—n,m—TO
lim a(yn, v) = a(v,v)= lim a(yn,ym) = 0.
Similarly, we have u e T2u and a (A, B) = a(u, T2u). Thus, u e B(T2). On the other hand, for all n > i
a(x0, u) < sa(x0,xn) + sa(xn, u) < s2a(x0,xi) +s2a(xi,xn) + sa (xn, u)
< 5 a(xo,+ s a(x^x2) + ••• + sn+ a(xn-l,xn) + sa (xn, u)
= s2 ^ sV (xk, x^+l) + sa (xn, u)
< s2 ^ '(Vsa)ka(xo,xi) + sa (xn, u).
Letting n — to, we obtain
a(x0, u) < s2Y(^)ka(x0,xl) =-= a(x0,xl).
Thus, from (3.20),
a(x0,u) < --— [a(x0,y0) + a(y0,Xi)]
i — sa
l - Jsa
l - Jsa
(s[Hba (Tlxo, T2xo) + a (A, B) + 2e] + a (A, B))
Hb(Tlxo, T2xo) + (l + s-l)a(A,B) + 2e].
Similarly, if y0 e B(T2), then there exists u' e B(Ti) such that s4
a(y0, u') < --= H (Tyo, T2y0) +1 + s—1)a(A,B) + 2e].
l — sa
Consequently, we obtain
H(B(Ti),B(T2)) < --= supH(Ti*, T2*) +1 + s—1)a(A,B) + 2e
1 — fsal xgA
The real e > 0 is arbitrary, so the proof is completed, that is, (3.17) is satisfied. □
We provide the following example.
Example 3.9 Let X = [0, to) x [0, to) be endowed with the b-metric-like a : X x X ^ [0, to) defined by
I, w v, f(i*i — y1l + 1*2— y21)2 if (*1,*2),(y1,y2) e [0,10]2, a((*l, *2),(Уl, y2))=\i .2 +
[ (*1 + *2 + y1 + y2)2 if not. Take A = {1}x [0,10] and B = {0}x [0,10]. Define the mapping Ti, T2: A ^ Cb(B) by
Ta^i{(0,0),(0,2)} if0<*<8,
l(,*) { {0} x [0,1] if 8 < * < 10
T a^i {(0,0), (0, ¥)} if 0 < * < 8,
2(,*) { {0} x [0,5] if 8 <* < 10.
Note that A0 = A and B0 = B. So, for each * e A0, we have T* c B0. Moreover, A and B are closed subsets of X. Consider the balls Ba (aL, rL), Ba (a2, r2) with aL = (1,0), a2 = (1,0.2) and rL = 82, r2 = 84. We know that the pair (A, B) has the weak P-property. Moreover, it is easy to prove that Ti is a proximal contraction on Ba (a;, ri) for i = 1,2 with the same constant a = 4. We also have Sa (Tai, {a;}) + a (A, B) < ^¡3—2 (1 — Vas)ri, i = 1,2. Furthermore, (X,a) satisfies the (GC) property. Therefore, all conditions of Theorem 3.8 are verified. So, we have
h(b(ti),B(T2)) <
sup H ( Tx, T2X) + -
.xeA 2.
V2 —1
We derive the following interesting consequences from Theorem 3.8.
Corollary 3.10 Let A and B be nonempty closed subsets of a complete metric-like space (X, a) and rL, r2 > 0. Let Ti: A ^ Cb(B), i = 1,2, be two multi-valued mappings. Suppose that
(i) A0= 0;
(ii) for each * e A0, we have Ti* c B0, i = 1,2;
(iii) the pair (A, B) satisfies the weak P-property;
(iv) for each i = i, 2, there exists ai e A0 such that Ti is a proximal contraction on Ba (at, r) n A with the same Lipschitz constant a e (0, i), that is,
Ha (Ttx, Ty) < aa(x,y), (3.28)
for all x, y e Ba (at, r) n A and Sa (Tiai, {at}) + a (A, B) < (i — -/a)ri.
1 r 1 (3.29)
Ha (B(Tl),B(T2)) < —^ [supHa (Tlx, T2x) + 2a(A,B)
i — \/a L xeA
Proof It suffices to consider s = i in Theorem 3.8. □
Corollary 3.11 Let (X,a) be a complete b-metric-like space, ri, r2 > 0, and let Ti: X — Cb(X), i = i, 2, be two multi-valued mappings. Suppose there exist a e (0, s—i) and ai e X such that, for each i = i, 2, we have
Hb(Tix, Ty) < aa(x,y), (3.30)
forallx,y e Ba(ai,r) and Sa(Tiai, {ai}) < 31 2 (i — \Zas)ri. Then
Hb(F(Ti), F (T2)) < --= sup Ha (Tix, T2X), (3.3i)
i — Vsa xeA
where F(Ti) is the set of fixed points ofTi, i = i, 2.
Proof It suffices to consider A = B = X in Theorem 3.8. Here, we do not need the conditions (i), (ii), and (iii) of Theorem 3.8. □
Corollary 3.12 Let A and B be nonempty closed subsets of a complete metric space (X, d) and ri, r2 > 0. Let Ti: A — Cb(B), i = i, 2, be two multi-valued mappings. Suppose that
(i) A0= 0;
(ii) for each x e A0, we have Tix c B0, i = i, 2;
(iii) the pair (A, B) satisfies the weak P-property;
(iv) for each i = i, 2, there exists ai e A0 such that Ti is a proximal contraction on Bd(ai, r) n A with the same Lipschitz constant a e (0, i), that is,
H(Tix, Tiy) < ad(x,y), (3.32)
for all x,y e Bd(ai, r) n A and Sd(Tiai, {ai}) + d(A,B) < (i — v/a)ri.
H (B(Tl), B(T2)) < —[sup H (Tlx, T2x) + 2d(A, B)
(3.33)
i — V«
Proof It suffices to consider a as a metric in Corollary 3.i0. □
4 Best proximity points and stability results on the class of partial b-metric spaces
In 2014, Shukla [32] introduced a generalized metric space called a partial ¿-metric space and established the Banach contraction principle as well as the Kannan type fixed point theorem in partial ¿-metric spaces.
Definition 4.1 [32] Let X be a nonempty set and s > 1 be a given real number. A function b: X x X ^ R+ is called a partial ¿-metric on X if for all x, y, z e X, the following conditions are satisfied:
(Pbl) b(x, x) = b(x, y) = b(y, y), then x = y; (Pb2) b(x,x) < b(x,y); (Pb3) b(x, y) = b(y, x); (Pb4) b(x,z) + b(y,y) < s[b(x,y) + b(y,z)]. The pair (X, b) is then called a partial b-metric space.
Remark 4.2 Each partial b-metric space is a b-metric-like space, but the converse is not true.
Example 4.3 Let X = [0, то). Consider the mapping a : X x X ^ [0, то) defined by a(x,y) = (x + y)2 for all x,y e X. Then (X,a) is a b-metric-like space with s = 2, but it is not a partial b-metric space since a(x, x) > a(x, y) for all x > y.
Lemma 4.4 Let (X, b) be a partial b-metric space. We have
(1) if b(x,y) = 0, then x = y,
(2) ifx = y, then b(x,y) > 0.
Remark 4.5 If b is a partial b-metric, then Bb(x, e) = {y e X: b(x,y) - b(x,x) < ej.
Very recently, Felhi [33] introduced the concept of a partial Pompeiu-Hausdorff b-metric and he obtained some fixed point results.
Remark 4.6 If b is a partial b-metric, for simplicity we denote Hb = HI (defined as in (1.2)). b
Following [33], we have the following lemmas.
Lemma 4.7 [33] Let (X, b) be a partial b-metric space with coefficient s > 1. For A e Cb(X) (Cb(X) is the set of bounded and closed subsets in the partial b-metric space) and x e X, we have
b(x, A) = b(x, x) if and only if x e A = A, (.1)
where A is the closure of A.
Lemma 4.8 [33] Let (X, b) be a partial b-metric space with coefficient s > 1. For A, B, C e Cb(X), we have
(i) Hh(A, A) < Hh(A,B);
(ii) Hb(A,B)=Hb(B,A);
(iii) Hb(A,B) < s[Hb(A, C) + Hb(C,B)] - infceC b(c, c).
4.1 Best proximity results
The main result of this paragraph is the analogous of Theorem 3.2 on the class of partial b-metric spaces. It is stated as follows.
Theorem 4.9 Let A andB be nonempty closed subsets of a complete partial b-metric space (X, b) and r >0. Let T: A ^ Cb(B) be a multi-valued mapping. Suppose that
(i) A0 = 0;
(ii) for each * e A0, we have T* c B0;
(iii) the pair (A, B) satisfies the weak P-property;
(iv) there exists *0 e A0 such that T is a proximal contraction on Bb(*0, r) and Sb(T*0, {*0}) + b(A,B) < s—2(1 — Vas)r;
(v) (X, b) satisfies the property (GC).
Then T has a bestpro*imitypoint in Bb(*0, r) n A. We also have b(**,**) = 0.
Proof By assumption (iv), there exists *0 e A0 such that T is a proximal contraction on Bb(*0,r) and Sb(T*0, {*0}) + b(A,B) < s—2(1 — fas)r. Let y0 e T*0. By condition (ii), we have T*0 c B0. Then there exists *l e A0 such that
b(xi, yo) = b(A, B).
We have
b(x0,Xi) - b(x0,x0) < b(x0,Xi) < s[b(xo,y0) + b(yo,xi)] - b(yo,y0)
< 5[5b(Txo, (xol) + b(A,B)]
< s[s-2(i ^VÖi)^ = s-1(l - sfas)r < r.
Then x\ e Bb(xo, r) n A0. By Lemma 2.3, there exists y\ e Txi such that
b(yo,yi) < Hb(Txo, Txi).
So, by (3.i), we get
Since yL e T*l c B0, there exists *2 e A0 such that
b(x2, yi) = b(A, B).
By condition (iii), (4.2), and (4.6)
b(xi,x2) < b(yo,yi).
The above inequality together with (4.7) implies that b(*1,*2) -b(*0,*l).
Using (4.3), we have
b(xo,x2) - b(xo,xo) < b{xo,x2) < sb{xo,xi) + sb(xi,x2) - b(xi,xi)
< sb(xo,xi) +s2b(xi,x2) < s
b(xo, x1)
< s[1 + *Jas]s - sfas)r =(1 - as)r < r. Then x2 e Bb(x0, r) n A0. Again, by Lemma 2.3, thereexists y2 e Tx2 such that
b(yi,y2) < —=Hh(Txi, Txi). J as
So, by (3.1), we get
b(yi,y2) <J ^b(xi,x2).
Again, y2 e Tx2 ç B0, so there exists x3 e A0 such that
b(x3, yi) = b(A, B). From condition (iii), (4.10), and (4.8)
b(x2,x3) < b(yi,y2) < J^b(xi,x2) <
b(xo, xi).
(4.io)
(4.ii)
(4.i2)
We have
b(xo,x3) - b(xo,xo) < b(xo,x3) < sb(xo,xi) + s2b(xi,x2) + s2b(x2,xi)
< sb(xo,xi) + s2b(xi,x2) + s3b(x2,xi) 2-1
i + s J — + s
21 „-i
b(xo, xi)
< 5[1 + *JaS + (Vos)2]s -*fas)r = (l - (Vas)3)r < r.
Then x3 e Bb(x0, r) n A0.
Continuing this process, we construct two sequences {xn} c Bb(xo, r) n A0 and {yn} c B0 such that
b(xn, yn-i) = b(A, 5),
b(xn,xn+i) < b(yn-i,yn) < ( Ja)nb(xo,xi), yn e Txn, for all n = i, 2,____
As in the proof of Theorem 3.2, there exist x* e Bb(x0, r) n A and y* e B such that lim b(xn,x^ = b(x*,x^ = lim b(xn,xm) = 0 and
n^TO v / v 7 n,m^TO
lim b(yn,y^ = b{y*,y*) = lim b(yn,ym) = 0. By the same strategy, we see that x* is a best proximity point of T and b(x*, x*) = 0. □
As consequences, we may provide the following corollaries.
Corollary 4.10 Let A and B be nonempty closed subsets of a complete partial b-metric space (X, b) and r >0. Let T: A ^ B be a given single-valued mapping. Suppose that
(i) A0 = 0;
(ii) for each * e A0, we have T* e B0;
(iii) the pair (A, B) satisfies the weak P-property;
(iv) there exists *0 e A0 such that T is a proximal contraction on Bb(*0, r) and b(*0, T*0) + b(A,B) < s—2(1 — v—s)r;
(v) (X, b) satisfies the property (GC).
Then T has a bestpro*imitypoint in Bb(*0, r) n A. We also have b(**,**) = 0.
In the setting of b-metric spaces, we have the following.
Corollary 4.11 Let A andB be nonempty closed subsets of a complete b-metric space (X, d), r >0, and T: A ^ Cb(B) be a multi-valued mapping. Suppose that
(i) A0 = 0;
(ii) for each * e A0, we have T* c B0;
(iii) the pair (A, B) satisfies the weak P-property;
(iv) there exists *0 e A0 such that T is a proximal contraction on Bd(*0, r) and Sd(T*0, {*0}) + d(A, B) < s—2(1 — V—s)r;
(v) (X, d) satisfies the property (GC).
Then T has a bestpro*imitypoint in Bd(*0, r) n A.
Corollary 4.12 Let (X, d) be a complete b-metric space and T : X ^ Cb(X) be a multivalued contractive non-self-mapping, that is,
H(T*, Ty) < ad(*,y),
for some a e (0,1) and for all*, y e Bd (*0, r) and Sd (T* 0, {*0}) < s—2(1 — fas)r. Then T has a fi*ed point.
Corollary 4.13 ([2], Theorem 1) Let (X, d) be a complete metric space and T: X ^ Cb(X) be such that
H(T*, Ty) < ad(*,y),
for some a e (0,1) and for all *, y e X. Then T has a fi*ed point.
Corollary 4.14 ([26], Theorem 2.1) Let (A, B) be a pair of nonempty closed subsets of a complete metric space (X, d) such that A0 = 0 and (A,B) satisfies the P-property. Let T : A ^ 2B be a multi-valued contraction non-self-mapping, that is,
H(T*, Ty) < ad(*,y),
for some a e (0,1) and for all *, y e A. If T (*) is bounded and is closed in Bfor all * e A, and T(*0) c B0 for each *0 e A, then T has a bestpro*imity point in A.
4.2 Stability results
As Theorem 3.8, we state the following stability result.
Theorem 4.15 Let A and B be nonempty closed subsets of a complete partial b-metric space (X, b) and r1, r2 > 0. Let Ti: A ^ Cb(B) with i = 1,2, be two multi-valued mappings. Suppose that
(i) A0= 0;
(ii) for each x e A0, we have Tix c B0, i = 1,2;
(iii) the pair (A, B) satisfies the weak P-property;
(iv) (X, b) satisfies the property (GC);
(v) for each i = 1,2, there exists at e A0 such that Ti is a proximal contraction on Bb(ai, r) n A with the same Lipschitz constant a e (0,1), that is,
Hb(Tix, Tiy) < ab(x,y), (4.13)
for all x,y e Bb(ai, r) n A and 8b(Tiai, {a,-}) + b(A, B) < s-2(1 - ^/Os)ri.
Hb(B(T0,B(T2)) < 1-s^_ [supHb(T1 x, T2x) +1 + s-1)b(A,B)]. (4.14)
Proof The proof is similar to that of Theorem 3.8. □
Corollary 4.16 Let (X, d) be a complete b-metric space. Take r1, r2 >0. Let Tt: X ^ Cb(X), i = 1,2, be two multi-valued mappings. Suppose there exist a e (0, s-1) and at e Xsuch that, for each i = 1,2,
Hb(T-x, Tiy) < ad(x,y), (4.15)
for all x,y e Bd(at, r) and 8d(Tiai, {at}) < s-2(1 - ^/as)ri. Then
Hb{F (T1), F (T2) < sup Hb (T1 x, T2 x). (4.16)
1 a/sa
Competing interests
The authors declare that they have no competing interests. Authors' contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript. Author details
1 Department of Mathematics, College of Sciences, KFU, Al-Hasa, Saudi Arabia. 2Department of Mathematics, College of Education of Jubail, University of Dammam, P.O. Box 12020, Industrial Jubail, 31961, Saudi Arabia. 3Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan.
Received: 16 December 2015 Accepted: 28 February 2016 Published online: 05 March 2016
References
1. Matthews, SG: Partial metric topology. Research report 212, Department of Computer Science, University of Warwick (1992)
2. Nadler, SB: Multi-valued contraction mappings. Pac. J. Math. 30, 282-291 (1969)
3. Ali, MU, Kamran,T, Karapinar, E: A new approach to (a, ^)-contractive nonself multivalued mappings. J. Inequal. Appl. 2014,71 (2014)
4. Ali, MU, Kiran, Q, Shahzad, N: Fixed point theorems for multi-valued mappings involving «-function. Abstr. Appl. Anal. 2014, Article ID 409467 (2014)
5. Aydi, H, Abbas, M, Vetro, C: PartialHausdorff metric and Nadler's fixed point theorem on partialmetric spaces. Topol. Appl. 159, 3234-3242(2012)
6. Aydi, H, Abbas, M, Vetro, C: Common fixed points for multivalued generalized contractions on partialmetric spaces. Rev. R. Acad. Cienc. Exactas Fis. Nat., Ser. A Mat. 108,483-501 (2014)
7. Berinde, M, Berinde, V: On a generalclass of multi-valued weakly Picard mappings. J. Math. Anal. Appl. 326, 772-782 (2007)
8. Berinde, V, Pacurar, M: The role of the Pompeiu-Hausdorff metric in fixed point theory. Creative Math. Inform. 22(2), 143-150 (2013)
9. Boriceanu, M, Petrusel, A, Rus, IA: Fixed point theorems for some multivalued generalized contractions in b-metric spaces. Int. J. Math. Stat. 6,65-76 (2010)
10. Bota, M: DynamicalAspects in the Theory of Multivalued Operators. Cluj University Press, Cluj-Napoca (2010)
11. Ciric, LB: Fixed Point Theory, Contraction Mapping Principle. FME Press, Beograd (2003)
12. (Ciric, LB, Ume, JS: Common fixed point theorems for multi-valued non-self mappings. Publ. Math. (Debr.) 60(3-4), 359-371 (2002)
13. Ciric, LB, Ume, JS: On the convergence of Ishikawa iterates to a common fixed point of multi-valued mappings. Demonstr. Math. 36(4), 951-956 (2003)
14. Czerwik, S: Nonlinear set valued contraction mappings in b-metric spaces. Atti Semin. Mat. Fis. Univ. Modena 46(2), 263-276(1998)
15. Daffer, PZ, Kaneko, H: Fixed points of generalized contractive multi-valued mappings. J. Math. Anal. Appl. 192, 655-666(1995)
16. Jleli, M, Samet, B, Vetro, C, Vetro, F: Fixed points for multivalued mappings in b-metric spaces. Abstr. Appl. Anal. 2015, Article ID 718074 (2015)
17. Mizoguchi, N, Takahashi, W: Fixed point theorems for multi-valued mappings on complete metric spaces. J. Math. Anal. Appl. 141, 177-188(1989)
18. Hitzler, P, Seda, AK: Dislocated topologies. J. Electr. Eng. 51 (12/s), 3-7 (2000)
19. Alghamdi, MA, Hussain, N, Salimi, P: Fixed point and coupled fixed point theorems on b-metric-like spaces. J. Inequal. Appl. 2013, 402 (2013)
20. Hussain, N, Roshan, JR, Parvaneh, V, Kadelburg, Z: Fixed points of contractive mappings in b-metric-like spaces. Sci. World J. 2014, Article ID471827 (2014)
21. Aydi, H, Felhi, A, Sahmim, S: Common fixed points in b-rectangular metric spaces using (E.A) property. J. Adv. Math. Stud. 8(2), 159-169 (2015)
22. Aydi, H, Felhi, A, Sahmim, S: Fixed points of multivalued nonself almost contractions in metric-like spaces. Math. Sci. 9, 103-108 (2015)
23. Kiran, Q, Ali, MU, Kamran, T, Karapinar, E: Existence of best proximity points for controlled proximalcontraction. Fixed Point Theory Appl. 2015, 207 (2015)
24. Aydi, H, Felhi, A, Karapinar, E, Sahmim, S: A Nadler-type fixed point theorem in metric-like spaces and applications. Miskolc Math. Notes (2015, accepted)
25. Zhang, J, Su, Y, Cheng, Q: A note on 'Best proximity point theorem for Geraghty-contractions'. Fixed Point Theory Appl. 2013, 99 (2013)
26. Abkar, A, Gabeleh, M: The existence of best proximity points for multivalued non-self-mappings. Rev. R. Acad. Cienc. Exactas Fis. Nat., Ser. A Mat. 107, 319-325 (2013)
27. Ali, MU, Kamran, T, Shahzad, N: Best proximity point for a-ty-proximalcontractive multimaps. Abstr. Appl. Anal. 2014, Article ID 181598 (2014)
28. Jleli, M, Karapinar, E, Samet, B: On best proximity points under the P-property on partially ordered metric spaces. Abstr. Appl. Anal. 2013, Article ID 150970 (2013)
29. Karpagam, S, Agrawal, S: Best proximity points theorems for cyclic Meir-Keeler contraction maps. Nonlinear Anal. 74, 1040-1046 (2011)
30. Latif, A, Hezarjaribi, M, Salimi, P, Hussain, N: Best proximity point theorems for a-ty-proximalcontractions in intuitionistic fuzzy metric spaces. J. Inequal. Appl. 2014, 352 (2014)
31. Sankar Raj, V, Veeramani, P: A best proximity theorems for weakly contractive non-self mappings. Nonlinear Anal. 74, 4804-4808 (2011)
32. Shukla, S: Partialb-metric spaces and fixed point theorems. Mediterr. J. Math. 11, 703-711 (2014)
33. Felhi, A: Some fixed point results for multi-valued contractive mappings in partialb-metric spaces. J. Adv. Math. Stud. (2015, accepted)
