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UDC 510.8 Вестник СПбГУ. Математика. Механика. Астрономия. 2021. Т. 8 (66). Вып. 3
MSC 47H10, 54H25
Perov multivalued contraction pair in rectangular cone metric spaces
M. Abbas1'2, V. Rakocevic3, Z. Noor1
1 Department of Mathematics, Government College University, Lahore, 54000, Pakistan
2 Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0002, South Africa
3 Faculty of Sciences and Mathematics, Department of Mathematics, University of Nis, Nis, 18000, Serbia
For citation: Abbas M., Rakocevic V., Noor Z. Perov multivalued contraction pair in rectangular cone metric spaces. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 2021, vol. 8(66), issue 3, pp. 484-501. https://doi.org/10.21638/spbu01.2021.310
Perov studied the Banach contraction principle in the framework of a generalized metric space and presented Perov contraction condition where the contractive constant is replaced by a matrix with nonnegative entries and spectral radius less than 1. Azam et al. presented the notion of rectangular cone metric space following the idea of Branciari, Huang and Zhang by replacing the triangular inequality in the cone metric space by rectangular inequality. Motivated by the work of Abbas and Vetro and Radenovic, the purpose of this paper is to introduce a new class of Perov type multivalued mappings and present a common fixed point result for such mappings on a complete rectangular cone metric space. Furthermore, an example is also presented to demonstrate the validity of our results. Our results extend, unify and generalize various comparable results in the existing literature.
Keywords: fixed point, cone metric space, rectangular metric space.
1. Introduction. Let (X,d) be a metric space. A mapping T : X ^ X is called a contraction if there exists a constant k € [0,1) such that for any x,y € X, we have d(Tx, Ty) < kd(x, y). The famous Banach contraction theorem [1] states that a contraction mapping on a complete metric space has a unique fixed point, that is, there exists a point x in X such that x = Tx. The set of all fixed point of T is denoted by Fix(T). Banach contraction principle and its variants have various applications in many areas of mathematics and related disciplines such as differential equations, optimization theory, computer science, economics and telecommunication (see, e.g., [2]). This intrigued several mathematicians to extend Banach contraction theorem to different directions. One such way is to extend the domain of a mapping.
Branciari [3] gave the notion of rectangular metric spaces by replacing the triangular inequality, that is, d(x, y) < d(x, z) + d(z, y) for x, y, z in X by a rectangular inequality, i. e., d(x, y) < d(x, u)+d(u, w)+d(w, y) for any x, y, z, w in X, u = v and u, v € X\{x, y}. Since then, many authors have established various fixed point theorems in rectangular metric spaces. For more results in this direction under different contractive conditions, we refer to [4-10]. In an ordinary metric space X, for any two points in an abstract set X, there is a positive real number that measures the distance between them. On the other hand, if X is an arbitrary set and the set R+ is replaced with a set E equipped with
© St. Petersburg State University, 2021
some order structure, whereas the distance mapping d : X x X ^ E satisfies properties analogous to the conditions of ordinary metric d : X x X ^ R+, then the notion of a metric can be extended in several ways. For example, if E = C the set of all complex numbers, then we have the concept of a complex valued metric space [11], if E = Rn, we have the notion of a generalized metric space. If E is a topological vector space, we obtain the vector-valued metric space [12]. If E is a C*-algebra, then we obtain the C*-algebra valued metric space [13]. Later, the concept of K-metric spaces was introduced by taking E as a real Banach space [14]. In 2007, Huang and Zhang [15] rediscovered the notion of K-metric space under the name cone metric space. The reader interested in fixed and common fixed point results in the setup of cone metric spaces is referred to [16-18].
In 1964, Perov [19] studied the Banach contraction principle in the framework of a generalized metric space and presented Perov contraction condition where the contractive constant is replaced by a matrix with nonnegative entries and spectral radius less than 1. He also obtained some fixed point theorems with various applications in coincidence problems, coupled fixed point problems and systems of semilinear differential inclusions [19, 20]. It must be noted that this generalized metric space is a special case of a normal cone metric space. Azam et al. [21] presented the notion of rectangular cone metric space following the idea of Branciari and Huang and Zhang by replacing the triangular inequality in the cone metric space by rectangular inequality. They also studied the fixed point results for both Banach and Perov type contractions in rectangular cone metric spaces. Shukla et al. [22] proved a generalized Banach fixed point theorem for the setting of cone rectangular Banach algebra valued metric spaces.
Recently, Radenovic and Vetro [23] introduced the notion of Sehgal — Guseman — Perov type mappings and established a result of existence and uniqueness of fixed points for this class of mappings. Vetro and Radenovic [24] studied fixed point results under various Perov type contractive conditions in rectangular cone metric spaces. Markin [25] initiated the study of fixed points for multivalued contractions and nonexpansive maps using the Hausdorff metric. He also developed an interesting and rich fixed point theory for multivalued maps having applications in control theory, convex optimization, differential inclusions and economics. The concept of multivalued contractions was initiated by Nadler [26]. He showed that a multivalued contraction possesses a fixed point in a complete metric space. Later several generalizations of Nadler's fixed point theorem were obtained (see, [27, 28]).
Latif and Beg [29] extended Kannan mappings to multivalued mappings and introduced the notion of a K-multivalued mapping. The term R-multivalued mapping as a generalization of K-multivalued mapping was presented by Rus [30]. Abbas and Rhoades [31] introduced the notion of a generalized R-multivalued mappings and established common fixed point results for such mappings. Recently, in 2020 Altun and Olgun [32] have introduced the concept of F-contraction on vector-valued metric space. Then they proved a fixed point result that includes the famous Perov fixed point theorem as properly. They provided a nontrivial and illustrative example showing this fact.
Motivated by the work of Abbas et al. [33] and Vetro and Radenovi c [24], the purpose of this paper is to introduce a new class of Perov type multivalued mappings and present a common fixed point result for such mappings on a complete rectangular cone metric space. Furthermore, an example is also presented to demonstrate the validity of our results. Our results extend, unify and generalize various comparable results in the existing literature [3-5, 34] and [6].
2. Preliminaries. Let E be a real Banach space. A subset P of E is called a cone if and only if:
1) P is nonempty, closed and P = {0} (where 0 is the zero element of E);
2) a, b € R, a, b > 0 and x, y € P implies that ax + by € P;
3) p n (-P) = {0}.
Partial ordering on E is defined with help of a cone P as follows: x ^ y if and only if y — x € P. We shall write x -< y to indicate that x ^ y but x = y and x C y stands for y — x € intP, where intP denotes the interior of P. A cone P is normal or semi monotone if
inf {||x + y|| : x,y € P and ||x|| = ||y|| =1} > 0
or equivalently, if there is a number K > 0 such that for all x,y € P, 0 < x ^ y implies that ||x|| < K ||y||. The least positive number satisfying the above inequality is called a normal constant of P. If x = (xi,x2 , ...,xn)T, y = (yi, y2,... yn)T € Rn, then a ^ b means that aj < bj, i = 1,...,n. In this case, the set P = < x = (x1,. .., xn )T € Rn : xj > 0 for i = 1, 2,..., n| is a normal cone with the normal constant K = 1. A cone P is called solid if it has a nonempty interior i.e. intP = 0.
Definition 2.1. Let X be a nonempty set. A mapping d : X x X ^ E is said to be a cone metric on X if for any x, y, z € X, the following conditions hold:
1) 0 ^ d(x, y) for all x, y € X and d(x, y) = 0 if and only if x = y;
2) d(x, y) = d(y, x);
3) d(x, y) ^ d(x, z) + d(z, y).
The pair (X, d) is called a cone metric space. If E = Rn, then a nonempty set X with a vector valued metric d is called a generalized metric. The concept of a cone metric space is more general than that of a metric space.
Lemma 2.2 [35]. Let (X, d) be a cone metric space over a cone P in E. Then one has the following:
(i) intP + intP C intP and yuintP C intP, ^ > 0;
(ii) for any given 0 and co 0, there exists no €= N such that — c;
no
(iii) if an, bn are sequences in E such that an ^ a, bn ^ b and an ^ bn for all n > 1, then a ^ b.
The following is a crucial result.
Lemma 2.3 [36]. Let (X, d) be a cone metric space. Then for each c € E with c ^ 0, there exists a > 0 such that (c — x) € IntP (i. e. x C c) whenever ||x|| < a, x € E.
Definition 2.4. Let X be a nonempty set. A mapping d : X x X ^ E is called a rectangular cone metric on X if it satisfies the following conditions:
(b1) d(u, v) ^ 0 for all u, v € X and d(u, v) = 0 if and only if u = v;
(b2) d(u, v) = d(v, u) for all u, v € X;
(b3) d(u,v) ^ d(u, z) + d(z,w) + d(w,v) for all u,v € X and for all distinct points w, z € X\ {u, v}.
Then the pair (X, d) is called a rectangular cone metric space over cone P.
Definition 2.5. Let (X, d) be a rectangular cone metric space over a solid cone P and {un} a sequence in X. We say that
(I) {un} is a Cauchy sequence if for every c € E with c ^ 0, there is an n(c) € N such that for all n, m > n(c), d(un, um) С c;
(II) {un} is a convergent sequence if for every c € E with c ^ 0, there is an n(c) € N such that for all n > n(c), d(un, u) С c for some u € X.
If the sequence {un} converges to u, we denote un ^u. A cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X.
If P is a normal solid cone, then un д u if and only if d(un,u) ^ 0 and {un} is a Cauchy sequence if and only if d(un, um) ^ 0 as n, m ^
Example 2.1 [24]. Let E = R2 and P = {v = (vi,v2) € E: Vj > 0 for j = 1, 2} . Clearly, P is a normal solid cone with normal constant K = 1. If X = N and d(v,v) = (0,0) for all v € X, d(2, 3) = d(3, 2) = (5,11) and d(v, u) = (2, 4) otherwise, then d is a rectangular cone metric on X. Clearly, (X, d) is not a cone metric space because it does not satisfy the triangular inequality. Indeed, we have
d(2, 3) ^ d(2, 5) + d(5, 3) gives that (5,11) ^ (2, 4) + (2, 4) which is not true.
Lemma 2.6 [24, 37]. Let E be a real Banach space and P С E a solid cone. Let v, w, z € E and {a„} С E. Then, we have the following properties:
(i) if z ^ w and w С v, then z С v;
(ii) if 0 ^ z С c for each c € intP then z = 0;
(iii) if c € intP and a„ ^ 0, then there exists n(c), such that for all n > n(c), we have a„ С c.
Definition 2.7 [38]. Let E be a topological vector space with a solid cone P and {un} a sequence in P. We say that {un} is a c-sequence if for every c € rntP, there exist n(c) € N such that un С c for all n > n(c).
Remark 2.8. Let (X, d) be a rectangular cone metric space over a solid cone, {un} С X and u € X:
(i) the sequence {un} converges to u if and only if {d(un, u)} is a c-sequence;
(ii) if there exist a c-sequence {vm} such that d(um, un) ^ vm for all m € N and n > m, then {un} is a Cauchy sequence.
The following proposition shows that the notion of c-sequence can be given by using -< or ^ instead of С .
Proposition 2.9 [39]. Let P be a solid cone in a topological vector space E and {un} a sequence in P. Then the following conditions are equivalent:
(i) {un} is a c-sequence;
(ii) for each c ^ 0, there exists n(c) G N such that un — c for n > n(c);
(iii) for each c ^ 0, there exists n(c) G N such that un ^ c for n > n(c);
(iv) there exists c ^ 0 such that for any A G (0,1), there is n(A) G N such that ^ Ac for all n > n(A);
(v) there exists a sequence {vn} such that ^ 0 for any n G N, ^ 0 and for any n G N, there exists n(c) G N such that ^ for each m > n(c).
u.
Lemma 2.10 [40]. Let E be a real Banach space, P C E a cone and A a linear operator on E. Then the following conditions are equivalent:
(i) A is nondecreasing, i. e., u ^ v implies that A(u) ^ A(v);
(ii) A is positive, i. e., A(P) C P.
We denote by Mn,n the set of all n x n matrices, and by Mn,n(R+), we mean the set of all n x n matrices with nonnegative elements. It is well known that if A € Mn n, then A(P) C P if and only if A € Mn,n(R+). We write © for the zero n x n matrix and /n for the identity n x n matrix. For the sake of simplicity we will identify row and column vector in Rn. A matrix A € Mn,n(R+) is said to be convergent to zero if An ^ © as n ^ TO.
Following is the extension of Lemma 2.3 in the setting of rectangular cone metric space which can be easily proved.
Lemma 2.11. Let (X, d) be a rectangular cone metric space. Then for each c ^ 0, c € E, there exists a > 0 such that (c — x) € intP (i. e. x C c) whenever ||x|| < a, x € E.
Theorem 2.12 [19, 20]. Let (X, d) be a complete generalized metric space, f : X i—>• X and A € Mn,n(R+) a matrix convergent to zero such that d(f (x),f (y)) ^ A(d(x,y)) holds for any x, y € X. Then:
(i) f has a unique fixed point x* € X;
(ii) the sequence of successive approximations xn = f (xn-1), n € N converges to x* for all xo € X;
(iii) d(xn, x*) ^ An(In — A)-1(d(x0,x1)), n € N;
(iv) if g : X ^ X satisfies the condition d(f (x),g(x)) ^ c for all x € X and some c € Rn, then by considering the sequence yn = gn(x0), n € N, one has
d(yn, **) d (In - A) (c) + An(/„ - A)-1 (d(xo, xi)), n G N.
Now we recall some results from Banach algebra theory (see, e.g., [2]). We write B(E) for the set of all bounded linear operators on E and L(E) for the set of all linear operators on E. Note that B(E) is a Banach algebra. If A € B(E), then
i i r(A) =lim||An||" =inf \\An\\~
is called the spectral radius of A. If r(A) < 1, then the series J2 +=o A® is absolutely convergent, consequently, ||An|| ^ 0 as n ^ Furthermore, I — A is invertible in B(E) and
= (1 - A)-1
n=0
Moreover, if ||A|| < 1, then r(A) < 1 and I — A is invertible and (I — A)
N-1
as well as r((/ — A) < . In addition, we have that (I — A) is nondecreasing
with respect to P.
Lemma 2.13 [24]. Let E be a real Banach space and P С E a solid cone. If A € L(E) is nondecreasing, then A is continuous.
Lemma 2.14 [24]. Let E be a real Banach space and P С E be a solid cone. Let A € L(E) be such that A(P) С P and r(A) < 1. Then the following properties hold:
(i) if a € P is such that a ^ A(a), then a = 0;
(ii) r(Am) < 1 for any fixed m € N.
Lemma 2.15 [24]. Let E be a real Banach space, P С E a solid cone, A € L(E) a nondecreasing operator and {un} С P a c-sequence. Then, A(un) is a c-sequence.
Lemma 2.16 [24]. Let (X, d) be a rectangular cone metric space and {un} a sequence in X such that
(i) {d(un,un+1)} is a c-sequence;
(ii) un = um whenever n = m;
(iii) u, v € {un : n € N} .
If {un} converges to both u and v, then u = v.
Remark 2.17 [33]. Let P С E be a cone in E, A : E ^ E a linear operator with r(A) < 1 and A(P) С P. Then
(a) (I — A/2)-1 € B(E). If B = (I — A/2)-1A/2, then B € B(E), r(B) < r(A) and ||B||<||A||;
(b) for any u, v in P, we have
. ( u + v\ 1 . , S 1,,/N u^A^—j =-A(u) + -A(v),
then u ^ B(v).
Proof of (b) From the assumption we have (I — A/2)u ^ A/2(v), and so u ^ B(v). We point out that part (b) is correction of the result Remark 1.16 (b) from [33].
Let (X, d) be a rectangular cone metric space. Denote by P(X) the family of all nonempty subsets of X, by Pcl(X) the family of all nonempty closed subsets of X .A point x in X is a fixed point of a multivalued mapping T : X ^ P(X) if x € T (x). The set of all fixed points of multivalued mapping T is denoted by Fix(T).
3. Common fixed points of multivalued mappings. We start this section with the following definition of Perov contraction pair.
Definition 3.18. Let T1,T2 : X ^ Pcl(X) be two multivalued mappings. A pair (T1,T2) is said to form a Perov contraction pair if there exist a linear bounded operator A : E ^ E with ||A|| < 1 and A(P) C P such that for any x,y € X with ux € T,(x), there exists uy € Tj (y) for i, j € {1, 2} with i = j such that
d(ux, uy) ^ A(M1(x, y, ux, uy)), (3.1)
holds, where
d(x,ux) + d(y,uy)
Mi(x, y, ux, Uy) € < d(x, y), d(x, Ux), Uy),
2
Lemma 3.19. Let (X, d) be a complete rectangular cone metric space over a solid cone P and T1 ,T2 : X ^ Pcl(X). If the pair (T1,T2) is a Perov contraction pair then Fix(T1) = ^ or Fix(T2) = ^ if and only if Fix(T1) = Fix(T2) =
Proof. Let x* € T1(x*). As, the pair (T1,T2) forms a Perov contraction pair, there exists x € T2(x*) we have
) ^ A(M1(x*,x*,x*,x))
where
d(x*, x*) + d(x, x
Mi(x*, x*, x*,x) G < d(x*, x*), d(x*, x*), d(x, x*),
2
d(x, ж*)
= < d(x*,x*), d(x,x*),
2
Now we have three possibilities: if M1(x*,x*,x*,x) = d(x*,x*) = 0, then we have
= x, if M1(x*, x*, x*, x) = d(x, x*), we have
d(x,x*) ^ A(d(x, x*)),
by the Lemma 2.14 (i), we have x* = x.
If Mx{x*,x*,x*,x) = then we obtain that x* = x. Hence x* G T2(x*) and
so Fix(T1) C Fix(T2). Similarly Fix(T2) C Fix(T1) and therefore Fix(T1) = Fix(T2). □
Theorem 3.20. Let (X, d) be a complete rectangular cone metric space over a solid cone P. If T ,T2 : X ^ Pcl(X) forms a Perov contraction pair. Then Fix(T1) = Fix(T2) =
Proof. Suppose that x0 is an arbitrary point of X. If x0 € T1(x0) or x0 € T2(x0), then by Lemma 3.19, the proof is complete. We now assume that x0 € Tj(x0) for i €
x*
{1,2}. Let i, j G {1, 2} with i = j, and xi G Ti(x0). As the pair (T,T2) is a Perov contraction pair, there exist x2 G Tj (x1) such that
d(xi, X2) ^ A(Mi(xo, xi, xi, X2)),
where
Mi(xo, xi, xi, x2) G |<i(xo, xi), (i(xo, xi), d(x\, x2), —^—— ^ 11
= < d(xo, xi), d(xi, x2),
2
d(xo, xi) + d(xi, x2) 2
Now, if Mi(xo,xi,xi,x2) = d(xo,xi), we have d(xi,x2) ^ A(d(xo,xi)). If Mi(x0, xi, xi, x2) = d(xi, x2) then d(xi, x2) ^ A(d(xi, x2)), which by Lemma 2.14 (i) implies that xi = x2, that is, xi G Tj(xi) and hence by the Lemma 3.19, proof is complete. If
d(xo ,xi)+ d(xi,x2)
M i(x0,xi,xi,x2) = ---,
then 1 1
d(xi,x2) ^ -A(ci(xo,xi)) + -A(ci(xi,x2)),
which by Remark 2.17 (b) gives that d(xi,x2) ^ B(d(x0,xi)). Let x2 G Tj (xi), there exist x3 e Tj (x2) such that
d(x2, x3) ^ A(Mi(xi, x2, x2, x3)),
where
Mi(xi, x2, x2, X3) G ici(xi, x2), d(x\, x2), <i(x2, X3), —^ ^ 2'
d(x1, X2), d(x2, X3),
2
d(x1, X2) + d(x2, X3) 2
Now, if Mi(xi,x2,x2,x3) = d(xi,x2), we have d(x2,x3) ^ A(d(xi,x2)). If Mi(x0,xi,xi,x2) = d(x2,x3) then d(x2,x3) ^ A(d(x2,x3)), which by Lemma 2.14 (i) implies that x2 = x3, that is, x2 G Tj(x2) and hence by Lemma 3.19, proof is complete. If
d(xi ,x2)+ d(x2,x3)
M i(x0,xi,xi,x2) = ---,
then 1 1
d(x2,x3) ^ -A{d{xi,x2)) + -A(d(x2,x3)),
which by Remark 2.17 (b) implies that d(x2,x3) ^ B(d(xi,x2)).
Continuing this way, for x2n G Tj(x2n-i), there exist x2n+i G Tj(x2n) such that the following holds:
d(x2„,x2„+i) ^ A(Mi(x2n-i,x2„,x2„,x2„+i)),
where
Mi(x2n-i,x2„,x2„,x2„+i) G
^ i,/ N N N d(x2n-i,x2n) + d(x2n ,x2n+i)l G < a{X2„-i, X2„), a{X2„-i, X2„), a{X2„, X2„+i),---> =
,, ,, \ d(x2n-i ,x2n)+ d(x2n ,x2n+i)l a[X2n-l,X2n), d(X2n, x2n+l), --- f •
If Mi(x2n-i,x2n,x2n,x2n+i) = d(x2n-i,x2n), then
d(x2„,x2„+i) ^ A(d(x2n-i,x2n)).
For Mi(x2n-i,x2n,x2n,x2n+i) = d(x2„, x2n+i), we obtain d(x2„,x2„+i) ^ A(d(x2n,x2n+i)), which by Lemma 2.14 (i) gives x2n = x2n+i. Finally, for
,, , s d(x2n-i,x2n) + d(x2„,x2„+i) Mi(x2n_i,x2n,x2n,x2n+ij = ---,
we obtain that
d(x2„,x2„+1) ^ ^A(d(x2„-i,x2„) + d(x2„,x2„+i)) ^
^ ^A(d(x2„-i,x2n)) + ^A(d(x2„,x2„+1))
and hence by Remark 2.17 (b) we have
d(x2„,x2n+i) ^ B(d(x2n-i,x2n)).
Similarly, for x2n+i G Tj (x2n), there exist x2n+2 G Tj(x2n+i) such that
d(x2n+i,x2n+2) ^ A(d(x2n,x2n+i)) or d(x2„+i,x2n+2) ^ B(d(x2„,x2n+i)).
Therefore, we obtain a sequence {xn} in X such that for xn G Tj(xn-i), there exist xn+i G Tj(x„) and it satisfies
d(x„,x„+i) ^ A(d(x„-i,x„)) for all n G N or d(x„,x„+i) ^ B(d(x„-i, x„)). (3.2)
First, we prove that xn = xm for all n, m G N with n = m. Assume on contrary that there exist n,p G N such that xn = xn+p with p > 2. Then for xn+i = xn+p+i, because A and B commute it follows from (3.2) that
d(x„,x„+i) = d(x„+p,x„+p+i) < ApiBp-pi(d(x„,x„+i)),
where p G {0,1,2,... ,p}. Which by the Lemma 2.14 gives that d(xn,xn+1) = в and xn = xn+1. Hence, xn G Tj(xn) for i G {1,2}, a contradiction. Thus xn = xm for all n, m G N with n = m. Therefore
d(xn,xn+i) d AniBn-n (d(xo,xi)), where n G {0,1,2,..., n}. For xm-2 G Tj(xm-3), there exist xm G Tj(xm-1 ) such that
d(xm-2 , xm ) d A(Mi (xm-3,xm-1 ,xm-2,xm)), 492 Вестник СПбГУ. Математика. Механика. Астрономия. 2021. Т. 8(66). Вып. 3
(xm-3xm-1 xm-2xm) €
J 7/ \ 7/ \ 7/ \ d(xm —3? xm—2) d(xm—1? xm)
t \ a{Xm-3, xm — 1J d(Xm-31 xm—2 J 1, xm J, —
If M1(xm_3,xm_1,xm_2,xm) = d(xm_3,xm_1), then we have
d(xm_2,xm) ^ A(d(xm_3,xm_1)) ^ AaiBm_2_a(d(x0,x2)), (3.3)
where a € {1, 2,..., m — 2}. If M1(xm_3, xm_1, xm_2, xm) = d(xm_3, xm_2), then we obtain that
d(xm_2,xm) ^ A(d(xm_3,xm_2)) ^ Bm_2_ft(d(x0,x1)), (3.4) where $ € {1, 2,..., m — 2}. If M1(xm_3, xm_1, xm_2, xm) = d(xm_1, xm), then
d(xm_2,xm) ^ A(d(xm_1 ,xm)) ^ AYiBm_Yi(d(x0,x1)), (3.5) where 7j € {1, 2,..., m — 2}. Finally, the case
_ d(xm_3, xm_2) + d(xm_1, xm)
m—3 ■¿■m— 1 ^m- ™
^^1(xm-3 , xm-1, xm-2 , xm
2
implies that
7/ ч . л d(xm—3? xm—2) d(xm— 1 xm)\ , d{xm-2 , xm) li -Л I --- I ^
^ ^ (A(d(xm-3,xm-2) + A(d(xm_i,xm))) ^
^ I(^Bm-2-^(d(xo,xi)) + ^Bm-%(ci(xo,xi))), (3.6)
where ¿j € {1, 2,..., m — 2} and n € {1, 2,..., m}. Now for m, n € N with m > n, we consider the following cases.
If m — n is odd, then by the rectangular inequality, we have
d(x„, xm) - d(x„, x„+i) + d(x„+i, x„+2) + ... + d(xm_i, xm) -
— [AP"+ B1-Zl AP"+ • • • + + AZl+-Cm-1 Bm-1-Zi--Cm-1 Ap"Bn-P"](d(xo,x1)) - AP"Bn-P"W(d(xo,x1)).
where pn € {0,1,.. .,n}, Z € {0,1},i = 1, 2,.. .,m — 1, W = (I + ^ +=1 Afffc Bfc-fffc) € B(E), afc = £k=1 Ci. Let c > 0. Choose 6 > 0 such that c + N(0) С P, where N(0) = {x € E : || x ||< 6}. Also, choose N1 € N such that AP"Bn-p"W(d(x0,x1)) € N(0) for all n > N1. Thus {xn} is a Cauchy sequence, i.e.,
d(xn,xm) - AP"B"-p"W(d(x0,x1)) < c.
If m — n is even, we consider following four cases.
Case 1. If Mi(xm-3, xm-i, xm-2, xm) = d(xm-3,xm-i), then by (3.3), we have d(xm-2,xm) ^ AaBm-2-a(d(xo,x2)), where a® G {1, 2,..., m — 2}. Also,
d(x„, xm) ^ d(x„, x„+i) + ... + d(xm-3, xm-2) + d(xm-2, xm) ^
^ AP"Bn-p"W(d(xo,xi)) + AMiBm-2-^i(d(xo,x2)),
where G {1, 2,..., m — 2}.
Case 2. If Mi(xm-3, xm-i, xm-2, xm) = d(xm-3,xm-2), then by (3.4) we obtain
that
d(xm-2,xm) ^ AAiBm-2-Ai(d(xo,xi)), where A® G {1, 2,..., m — 2}, which implies that
d(x„, xm) ^ d(x„, x„+i) + ... + d(xm-3, xm-2) + d(xm-2, xm) ^
^ Ap"Bn-p"W(d(xo,xi)) + AAiBm-2-Ai(d(xo,xi)).
Case 3. If Mi(xm-3, xm-i, xm-2, xm) = d(xm-i,xm), then it follows from (3.5)
that
d(xm-2,xm) ^ AYiBm-Yi(d(xo,xi)), where y® G {1, 2,..., m — 2}. Which gives
d(x„, xm) ^ d(x„, x„+i) + ... + d(xm-3, xm-2) + d(xm-2, xm) ^
^ Ap"Bn-p"W(d(xo,xi)) + AYBm-Y(d(xo,xi)),
where y® G {1, 2,..., m — 2}. Case 4. If
j, ^ / \ 3, Xm—2 ) d(xm— \ , Xm)
1 vi\\xm—3:xm—\:xm—2:xm} — ,
then by (3.6) we have
d(xm-2,xm) ^ 1/2(A*Bm-2-5i(d(xo,xi))+ AnBm-n(d(xo,xi))), where J® G {1, 2,..., m — 2} and n G {1, 2,..., m}. And
d(x„, xm) ^ d(x„, x„+i) + ... + d(xm-3, xm-2) + d(xm-2, xm) ^
r< Ap™Bn-p™W(d(x0, xi)) + xi)) + AViBm~Vi(d(xo, xi))).
Thus in all the cases we obtain that {xn} is a Cauchy sequence as n, m ^ to. By completeness of X, there exist an element x* G X such that xn ^ x* as n ^ to.
Let c ^ d be given. Choose a natural number N such that d(xm, x*) C c for all m > N. As {x2n} converges to x* as n ^ to, for x2n G Tj(x2n-i) there exist un G T®(x*) such that
d(x2n,Un) ^ A(Mi(x2„-i, x* ,x2n,Un))
M1(x2n-1 ,x*,x2n,Un) €
€ <j d(x2n-1 ,x*), d(x2n-1,x2n),d(x,U")
Note that
d(U",x*) - d(U", x2n) + d(x2n, x2n+1) + d(x2n+1 ,x*) -
* , d(x2n-1,x2n) + d(x*,U")
- A(M1(x2n-1, x*, x2n, U")) + d(x2n, x2n+1) + d(x2n+1, x*). Now, if M1 (x2n-1, x*, x2", U") = d(x2n-1, x*), then
d(U", x*) - A(d(x2n-1, x*)) + d(x2n, x2"+1) + d(x2"+1, x*) < A(c) + c + c. As c ^ 0 is arbitrary, for m > 1, we have
1, *n . i c \ c c A(c) c c
d(un, x*) z<A( —)+ — + — = + — +--> 0
vm/ mm m m m
as m —у то. If M1(x2n-1, x*, x2n, Un) = d(x2n-1, x2n), then from Lemmas 2.15 and 2.16, we obtain that
d(U", x*) - A(d(x2"-1 ,x2n)) + d(x2", x2"+1) + d(x2"+1, x*) < c + c + c where c ^ 0 is arbitrary. For m > 1, we have
.... c c c 3c _ d(un, x ) <--1---1--=--> 0
m m m m
as m — то. In case M1(x2n-1, x*, x2n, Un) = d(x*, Un), we have
d(U",x*) - A(d(x*,U")) + d(x2",x2"+1) + d(x2"+1,x*)
which gives
(I — A)(d(U", x*)) - d(x2n, x2n+1) + d(x2n+1, x*) - 2c, where c ^ 0 is arbitrary. For m > 1
d(un, x*) r< (I-A)-1 (—) ^ —(I - A)-1^) ^ 0 mm
as то ^ 00. Finally, if Mi(x2n_i, ж*, ж2„, u„) = ^ wg haye
d{un, x j^AI --- I + d(X2n, X2n+1) + d(x2„+i,x
^ ^A(<i(x2n_i,x2n)) + + ci(x2n,x2n+i) + <i(x2n+i,x*).
2
Therefore,
I - —j d(un, x*) < -c + c + c
d(un, x*) A/2)-1 + - + -
V2m m m
-)• 0
as m ^ to. Thus un ^ x* as n ^ to. Since T®(x*) is closed, x* G Fix(Tj) = Fix(Tj).
Remark 3.21. We notice that the concept of a rectangular cone metric space is more general than one of the rectangular metric space. The results presented in this research article generalized some results of Branciari [3] and Ahmad et al. [4]. □
Example 3.1. Let E = R2 , P = {(x,y) G R2 : x, y > 0}, and ||x|| =max{|xi |, |x2|}, where x = (xi,x2) G E. Suppose that X = {(x, 0) G R2 : x > 0} U {(0,x) G R2 : x > o}. Define d : X x X ^ E by:
(0,0), if (x, 0) = (y, 0),
d((x 0) (y 0)) = ^ (3ft 3)if x and yare in {1,2}, x = y,
(( , )l ( , )) i (g, 1), if x and y are not in {1,2} simultaneously, x = y,
(| | x — y I, | x — y I), otherwise,
d((0,x), (0,y)) =
(0,0), if (x, 0) = (y, 0), (3g, 3), if x and y are in {1, 2} , x = y, (g, 1), if x and y are not in {1,2} simultaneously, x = y,
otherwise,
and
(3g, 3), if x and y are in {1, 2} , x = y, d((x, 0), (0, y)) = d((0, y), (x, 0)) = { (e, 1), if x and y are not both in {1, 2}, x = y,
(|x + y, x + |y), otherwise,
where e > 0 is a constant. Note that (X, d) is a complete rectangular cone metric space. For
(x,y) G {((0,0), (0,0))} U j((0, 1), (0,0)) J U |(0, I), (0, ±)j U
u|(i,o),(o,o)|u|(i,o),(i,o)|u{(i,o),(o,2)},
define a mapping Ti,T2 : X ^ Pcl(X) by
and
Ti(x,y) =
T2(x,y) =
{(0, x)}, {(§,0) :x >0},
{(0, x)}, {(f,0) >0},
if y = 0, if y = 0
if y = 0, if У = 0.
First, we show that for x,y G X with G T1(x), there exists G T2(y) such that (3.1) is satisfied. We consider the following cases.
(i) If x = y = (0,0), then (3.1) is satisfied obviously if we take = = (0,0).
(ii) For x = (0, i), у = (0,0) and ux = (0,0) G Ti(x), take uy = (0,0) G T2(y).
(iii) When X = (0, i), y = (0, ±) and ux = (0,0) G Ti(x), take uy = (0,0) G T2(y).
(iv) In case x = (±,0), y = (0,0) and ux = (0, ±) G Ti(x), take uy = (0,0) G T2(y).
Note that
and
Now
d(ux,uy) = d[ (0,i ) ,(0,0)
1 1 2' 3
d(x,y) = d[ ( ± oY(0,0)) =
X ""У
) =
1 t ■ 3 0 2
2 4 3
1 0 2 1
3 3 . 2
= A(d(x,y)),
where d(x, G Mi(x,y, wx,wy).
(v) For x = (i,0), y = (i,0) and
have
d(ux,uy) = d ( ( 0, -
and
Now
<i(x, мж) = d ( ( —, 0
(0, i) G Ti(x), for
0.1
1 1
4' 6
7 5 6' 6
(0,i) &T2(y), we
Г 1 1 t Г 7 1 t ■ 3 0 2 Г 7 1 t
d(ux uy) = 4 1 < 8 5 = 4 0 6 5 = A(d(x,ux )),
6 9 3 . 6
where d(x,ux) G Mi(x,y, wx,wy).
(vi) In case x = (1,0), y = (0,2) and = (0,1) G Ti(x), take Uy = (0,0) G T2(y). Note that
and
Now
d(ux, uy) = d((0,1), (0, 0)) = (q, 1), q > 0 d(x,y) = d((1,0), (0, 2)) = (3q, 3), q > 0.
'X ""У
)=
<
9g 1 l 4 2
I S 0 -u 3
3q 3
= A(d(x,y)),
where d(x,y) G Mi(x,y,uI,uy).
Now we show that for x, y G X with G T2(x), there exists uy G T1(y) such that (3.1) is satisfied. We consider the following cases.
(i) If x = y = (0,0), then (3.1) is satisfied obviously as = u
(0,0).
and
(ii) For x = (0, ¿), y = (0,0) and ux = (0,0) G T2(x), take uy = (0,0) G Ti(y).
(iii) When x = (0, i), y = (0, ±) and ux = (0,0) G T2(x), take uy = (0,0) G Tx{y).
(iv) In case x = (±,0), y = (0,0) and ux = (0, ±) G T2(x), take uy = (0,0) G Tx{y),
d(ux,uy)=d[ (0,^ ) ,(0,0)
1 1
2' 3
t
t
d(x, ux) = d[ ( -, 0
7 5 6' 6
Now
Г 1 1 t Г 7 1 t ■ 3 О 2 Г 7 1 t
d(ux, uy) = 2 1 < 8 5 = 4 О 6 5 = A(d(x,uœ )),
3 9 3 . 6
where d(x, G Mi(x, y, ).
(v) For x = (i,0), y = (i,0) and ux Note that
(0, i) G T2(x), take uy
(0,i)GTi(y).
and
Also,
d(ux, uy) = d d(x, ux) = d
' 2
0,1
1 1
4' 6
7 5 6' 6
i t 7 t ■ 3 О 2 r t
d(ux, uy) = 4 1 < 8 5 = 4 О 6 5 = A(d(x,uœ )),
6 9 3 . 6
where d(x,ux) G Mi(x, y, ).
Thus the pair (Ti.T2) is forms a Perov contraction with operator A
0 2
u 3 J
Indeed A" ->•
0 0 0 0
and ||A|| < 1. So all the conditions of theorem is satisfied.
Moreover (0,0) is the fixed point of mappings T1 and T2.
Our results extend and unify various comparable results in [24, 33] and [3]. The next result is a corollary of Theorem 3.20.
Corollary 3.22. Let (X, d) be a complete metric space, and Ti,T2 : X ^ CB (X) forms a generalized R-mulivalued pair, that is, if for each x, y G X, G T¿x, there exists a G Tjy for each i, j G {1, 2} with i = j such that
d (ux, uy) < h ma^ d (x, y), d (x, ux), d (y, uy),
d (x,
2
where 0 < h < 1.
Then Fix(Ti) = Fix(T2) = 0. Moreover, Fix(ï\) = Fix(T2) is closed.
References
l
О
2
l
l
О
О
2
2
3
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Received: August 27, 2020 Revised: February 26, 2021 Accepted: March 19, 2021
Authors' information:
Mujahid Abbas — Professor, PhD; abbas.mujahid@gmail.com Vladimir Rakocevic — Professor, PhD; vrakoc@sbb.rs Zahra Noor — znoor2175@gmail.com
Многозначная сжимающая пара Перова
в метрических пространствах прямоугольного конуса
М. Аббас1'2, В. Ракочевич3, З. Нур1
1 Государственный колледж Университета Лахора, Лахор, 54000, Пакистан
2 Университет Претории, Претория, 0002, Южная Африка
3 Университет Ниша, Ниш, 18000, Сербия
Для цитирования: Abbas M., Rakocevic V., Noor Z. Perov multivalued contraction pair in
rectangular cone metric spaces // Вестник Санкт-Петербургского университета. Математика.
Механика. Астрономия. 2021. Т. 8(66). Вып. 3. С. 484-501.
https://doi.org/10.21638/spbu01.2021.310
Перов изучил принцип Банахова сжатия в рамках обобщенного метрического пространства и представил условие сжатия, при котором сжимающая постоянная заменяется матрицей с неотрицательными входами и спектральным радиусом менее 1. Азам и др. представили понятие прямоугольного конусного метрического пространства, следуя идее Бранчиари, Хуана и Чжана, заменив треугольное неравенство в конусном метрическом пространстве прямоугольным неравенством. Мотивированная работой Аббаса, Ветро и Раденовича цель настоящей работы состоит в том, чтобы ввести новый класс многозначных отображений типа Перова и представить общий результат фиксированной точки для таких отображений на полном метрическом пространстве прямоугольного конуса. В работе приведен пример, демонстрирующий справедливость
полученных результатов. Наши выводы расширяют, объединяют и обобщают различные сопоставимые результаты в существующей литературе.
Ключевые слова: неподвижная точка, коническое метрическое пространство, прямоугольное метрическое пространство.
Статья поступила в редакцию 27 августа 2020 г.;
после доработки 26 февраля 2021 г.; рекомендована в печать 19 марта 2021 г.
Контактная информация:
Аббас Моджахед — abbas.mujahid@gmail.com Ракочевич Владимир — vrakoc@sbb.rs Нур Захра — znoor2175@gmail.com
