Fixed Point Theorems for Weakly Hardy-Rogers Type [620041]

Fixed Point Theorems for Weakly Hardy-Rogers Type
Operators on Generalized Metric Spaces
Liliana Guran, Khurram Shabbir
Abstract. The purpose of this paper is to present some xed point results in generalized metric
spaces in Perov's sense using a contractive condition of Hardy-Rogers type. The data dependence
of the xed point set, the well-posedness of the xed point problem, as well as, the Ulam-Hyres
stability are also studied.
1. Introduction and preliminaries
In 1964 the classical Banach contraction principle was extended for contraction mappings on
spaces endowed with vector-valued metrics by Perov. The concept of vector-valued metric was
introduced by Perov [ 29] as follows:
LetXbe a nonempty set. A mapping ed:XX!Rmwhereed=0
@d1(x;y)




dm(x;y)1
Afor every
m2Nis called vector-valued metric on Xif the following properties are satised:
(1)ed(x;y)0 for allx;y2X; anded(x;y) = 0 implies x=y;
(2)ed(x;y) =ed(y;x);
(3)ed(x;y)ed(x;z) +ed(z;y) for allx;y;z2X.
A setXequipped with a vector-valued metric edis called a generalized metric space in Perov'
sense and we will denote it by ( X;ed).
Throughout this paper we will denote by Mm;m(R+) the set of all mmmatrices with positive
elements, by  the zero mmmatrix and 0 1m=0
@0




01
A, byIthe identity mmmatrix and
I1m=0
@1




01
Aand byUthe unitymmmatrix. IfA2Mm;m(R+), then the symbol Astands
for the transpose matrix of A.
Recall that a matrix Ais said to be convergent to zero if and only if An! asn!1 .
For the proof of the main result we need the following theorem, see [ 36].
2010 Mathematics Subject Classication. 46T99; 47H10; 54H25.
Key words and phrases. xed point, coupled xed points, Perov space, generalized w-distance, Ulam-Hyers
stability, well-posedness, data dependence .
This paper was presented at "Conference on Ulam's Type Stability" held in Timi soara (Romania), October
8-13, 2018.
1

2 L. GURAN, K. SHABBIR
Theorem 1.1.LetA2Mm;m(R+). The following are equivalents:
(i)Ais a matrix convergent to zero;
(i)An!asn!1 ;
(ii)The eigen-values of Aare in the open unit disc, i.e. jj<1, for every 2Cwith
det(AI) = 0 ;
(iii) The matrix IAis non-singular and
(IA)1=I+A+:::+An+:::;
(iv) The matrix IAis non-singular and (IA)1has nonnenegative elements.
Let us note that Perov's metric is a very particular case of the so-called K-metric (see [ 42] and
the references therein), which in turn was rediscovered by Huang and Zhang [ 19] under the name
cone metric .
In 1973, Hardy and Rogers ([ 16]) gave a generalization of Reich xed point theorem. Since
then, many authors have been used dierent Hardy-Rogers contractive type conditions in order to
obtain xed point results.
Let (X;d) be a metric space. We will use the following notations:
P(X) -the set of all nonempty subsets of X ;
Pcl(X) -the set of all nonempty closed subsets of X ;
Pcp(X) -the set of all nonempty compact subsets of X ;
Fix(F) :=fx2Xjx2F(x)g-the set of the xed points of F;
SFix (F) :=fx2Xjfxg=F(x)g-the set of the strict xed points of F.
We also denote by Nthe set of all natural numbers and by N:=N[f0g.
Let (X;ed) be a generalized metric space in Perov' sense. Here, if v;r2Rm,v:= (v1;v2;


;vm)
andr:= (r1;r2;


;rm), then byvrwe meanviri, for eachi2f1;2;


;mg, whilev < r
vi<ri, for eachi2f1;2;


;mg. Also,jvj:= (jv1j;jv2j;


;jvmj) and, ifc2Rthenvcmeans
vic, for eachi2f1;2;


;mg.
Notice that in a generalized metric space in Perov' sense the concepts of Cauchy sequence,
convergent sequence, completeness, open and closet subsets are similar dened as those in a metric
space.
On the other hand, in [ 21], Kada et al. have introduced the notion w-distance on a metric space
and improved several results replacing the involved metric by a generalized distance. Recently,
Suzuki and Takahashi [ 40] introduced notions of single-valued and multivalued weakly contractive
maps with respect to w-distance and improved Nadler's xed point result for such maps. Recent
xed point results concerning w-distance can be found in [ 15, 22, 23, 24, 26, 40 ].
Definition 1.1.A function w:XX![0;1)is aw-distance onXif it satises the
following conditions for any x;y;z2X:
(1)w(x;z)w(x;y) +w(y;z);
(2) the map w(x;:) :X![0;1)is lower semicontinuous;
(3) for any ">0;there exists >0such thatw(z;x)andw(z;y)implyd(x;y)":
For the following notations see I.A. Rus [ 37] and [ 38], I.A. Rus, A. Petru sel, A. S^ nt am arian
[35] and A. Petru sel [ 32].
Definition 1.2.Let (X,d) be a metric space and f:X!Xbe an operator. By denition, f
is weakly Picard operator (brie
y WPO) if the sequence (fn(x))n2Nof successive approximations
forfstarting from x2Xconverges, for all x2Xand its limit is a xed point for f.
Iffis WPO, then we consider the operator
f1:X!Xdened byf1(x) := lim
n!1fn(x):

FIXED POINT THEOREMS FOR WEAKLY HARDY-ROGERS TYPE OPERATORS ON GENERALIZED METRIC SPACES 3
Notice that f1(X) =Fix(f).
Definition 1.3.Let (X,d) be a metric space, f:X!Xbe a WPO and c > 0be a real
number. By denition, the operator f is c-weakly Picard operator (brie
y c-WPO) if and only if
d(x;f1(x))cd(x;f(x));for allx2X:
For the theory of weakly Picard operators for the singlevalued case see [ 37].
The Ulam stability of various functional equations have been investigated by many authors
(see [9], [10], [17], [18], [20], [31], [34], [38], [39]).
In [38] are given the denition of Ulam-Hyers stability as follows.
Definition 1.4.Let (X,d) be a metric space and f:X!Xbe an operator. By denition,
the xed point inclusion
(1) x=f(x)
is Ulam-Hyers stable if there exists a real number cf>0such that: for each " > 0and each
solutionyof the inequation
(2) d(y;f(y))"
there exists a solution xof the inclusion (1) such that
d(y;x)cf":
Remark 1.1.Iffis ac-weakly Picard operator, then the xed point inclusion (1) is Ulam-
Hyers stable.
The purpose of this paper is to present some xed point results in generalized metric spaces
in Perov's sense using a contractive condition of Hardy-Rogers type. The data dependence of the
xed point set, the well-posedness of the xed point problem, as well as, the Ulam-Hyers stability
are also studied.
2. Fixed Point Results
First, let us recall the notion of generalized w-distance in the setting of generalized metric
spaces dened in [ 13] by L. Guran.
Definition 2.1.Let(X;ed)be a generalized metric space. The mapping ew:XX!Rm
+is
called generalized w-distance on Xif it satises the following conditions:
(1)ew(x;y)ew(x;z) +ew(z;y), for every x;y;z2X;
(2)ewis lower semicontinuous in its second variable.;
(3) For any ":= ("1;"2;:::;"m)>0, there exists := (1;2;:::;m)>0such thatew(z;x)
andew(z;y)impliesed(x;y)".
Examples of generalized w-distance and some of its useful properties are also given [ 13]. Let
us recall the following useful result.
Lemma 2.1.Let(X;ed)be a generalized metric space, and let ew:XX!Rm
+be a generalized
w-distance on X. Let (xn)and(yn)be two sequences in X;letn= ((1)
n;(2)
n;:::;(m)
n)2Rm
+and
n= ((1)
n;(2)
n;:::;(m)
n)2Rm
+be two sequences such that (i)
nand(i)
nconverge to zero for each
i2f1;2;:::;mg. Letx;y;z2X:Then the following hold for every x;y;z2X:
(a) Ifew(xn;y)nandew(xn;z)nfor anyn2N;theny=z:
(b) Ifew(xn;yn)nandew(xn;z)nfor anyn2N;then (yn)converges to z:
(c) Ifew(xn;xm)nfor anyn;m2Nwithm>n; then (xn)is a Cauchy sequence.
(d) Ifew(y;xn)nfor anyn2N;then (xn)is a Cauchy sequence.

4 L. GURAN, K. SHABBIR
Next, let us give the denition of singlevalued weakly Hardy-Rogers type operators on gener-
alized metric space in Perov's sense.
Definition 2.2.Let(X;ed)be a generalized metric space in Perov's sense, ew:XX!Rm
+
be a generalized w-distance and f:X!Xbe a given singlevalued operator. We say that fis a
weakly Hardy-Rogers type operator if the following inequality is satisfyed:
ew(f(x);f(y))Aew(x;y) +B[ew(x;f(x)) +ew(y;f(y))] +C[ew(x;f(y)) +ew(y;f(x))];
for allx;y2RandA;B;C2Mm;m(R+).
The rst xed point result of this paper is the following.
Theorem 2.1.Let(X;ed)be a complete generalized metric space in Perov's sense, ew:XX!
Rm
+be a generalized w-distance and f:X!Xbe a singlevalued weakly Hardy-Rogers type
operator. Additionally we have the following conditions:
(a)fis continuous;
(b)there exists the matrices A;B;C2Mm;m(R+)such that:
(i)M= (I(B+C))1(A+B+C)converges to ;
(ii)I(B+C)is nonsingular and (I(B+C))12Mm;m(R+);
(iii)I(A+ 2B+ 2C)is nonsingular and [I(A+ 2B+ 2C)]12Mm;m(R+).
ThenFix(f)6=?. Moreover, if x=f(x), thenw(x;x) = 0 .
Proof. Fixx02X. Letx1=f(x0) andx2=f(x1). Then we have:
ew(x1;x2) =ew(f(x0);f(x1))Aew(x0;x1) +B[ew(x0;f(x0)) +ew(x1;f(x1))] +C[ew(x0;f(x1))+
+ew(x1;f(x0))] =Aew(x0;x1) +B[ew(x0;x1) +ew(x1;x2)] +C[ew(x0;x2) +ew(x1;x1)] =
= (A+B)ew(x0;x1)+B(ew(x1;x2))+C[ew(x0;x1)+ew(x1;x2)] = (A+B+C)ew(x0;x1)+(B+C)ew(x1;x2):
Then we have: [ I(B+C)]ew(x1;x2)(A+B+C)ew(x0;x1).
We get the inequality:
(3) ew(x1;x2)[I(B+C)]1(A+B+C)ew(x0;x1) =Mew(x0;x1):
For the next step we have:
ew(x2;x3) =ew(f(x1);f(x2))Aew(x1;x2) +B[ew(x1;f(x1)) +ew(x2;f(x2))] +C[ew(x1;f(x2))+
+ew(x2;f(x1))] =Aew(x1;x2) +B[ew(x1;x2) +ew(x2;x3)] +C[ew(x1;x3) +ew(x2;x2)] =
= (A+B)ew(x1;x2)+B(ew(x2;x3))+C[ew(x1;x2)+ew(x2;x3)] = (A+B+C)ew(x1;x2)+(B+C)ew(x2;x3):
Then we have: [ I(B+C)]ew(x2;x3)(A+B+C)ew(x1;x2).
Using (3) we obtain the inequality:
(4)ew(x2;x3)[I(B+C)]1(A+B+C)ew(x1;x2) =Mew(x1;x2)M2ew(x0;x1):
Continuing this process we shall obtain that there exists a sequence ( x)n2N2X, withxn=
f(xn1) such that
(5) ew(xn;xn+1)Mnew(x0;x1);
withM2Mm;m(R+) andn2N.
We will prove next that ( xn)n2Nis Cauchy, by estimating ew(xn;xm), for every m;n2Nwith
m>n .
ew(xn;xm)ew(xn;xn+1) +ew(xn+1;xn+2) +:::+ew(xm1;xm)
Mn(ew(x0;x1)) +Mn+1(ew(x0;x1)) +:::+Mm1(ew(x0;x1))
Mn(I+M+M2+:::+Mmn1)(ew(x0;x1))Mn(IM)1ew(x0;x1)):

FIXED POINT THEOREMS FOR WEAKLY HARDY-ROGERS TYPE OPERATORS ON GENERALIZED METRIC SPACES 5
, Note that ( IM) is nonsingular since Mis convergent to zero. This implies:
lim
n!1w(xn;xm)lim
n!1Mn(IM)1ew(x0;x1))d!01m:
Then, by Lemma 2.1(3) result that the sequence ( xn)n2Nis Cauchy. Since ( X;ed) is complete
then there exists x2Xsuch thatew(xn;x)d!01m;asn!1 .
By (a) we have that ew(f(xn1);f(x))d!01m, asn! 1 . SinceXis complete, there
existsx2Xsuch that lim
n!1xnd!xasn!1 . From the continuity of f, it follows that
xn+1=f(xn)d!f(x) asn!1 . By the uniqueness of the limit, we get x=f(x), that is,x
is a xed point of f. ThenFix(f)6=?.
Letx2Xsuch thatx=f(x). We have that:
ew(x;x) =ew(f(x);f(x))Aew(x;x)+B[ew(x;f(x))+ew(x;f(x))]+C[ed(x;f(x))+ed(x;f(x))]
(6) = Aew(x;x) + 2Bew(x;x) + 2Cew(x;x):
This implies that [ I(A+ 2B+ 2C)]ew(x;x)01m. By hypothesis ( iii) result that
ew(x;x) = 0 1m. 
If we renounce to the continuity condition of the operator fwe get another xed point theorem.
Theorem 2.2.Let(X;ed)be a complete generalized metric space in Perov's sense, ew:XX!
Rm
+be a generalized w-distance and f:X!Xbe a singlevalued weakly Hardy-Rogers type
operator. Additionally we have the following conditions:
(a)inffew(x;y) +ew(x;f(x)) :x2Xg>0;
(c)there exists the matrices A;B;C2Mm;m(R+)such that:
(i)M= (I(B+C))1(A+B+C)converges to ;
(ii)I(B+C)is nonsingular and (I(B+C))12Mm;m(R+);
(iii)I(A+ 2B+ 2C)is nonsingular and [I(A+ 2B+ 2C)]12Mm;m(R+).
ThenFix(f)6=?. Moreover, if x=f(x), thenw(x;x) = 0 .
Proof. Following the same steps as in the previous theorem, Theorem 2.1 we have the esti-
mation:
(7) ew(xn;xm)Mn(IM)1ew(x0;x1):
withM2Mm;m(R+) andn2N.
By Lemma 2.1(3) we have that the sequence ( xn)n2Nis a Cauchy sequence. Since ( X;ed) is
complete we have that there exists x2Xsuch thatxnd!x. Letn2Nbe xed. Then, since
(xm)m2Nd!xandew(xn;
) is lower semicontinuous, we have:
(8) ew(xn;x)lim inf
m!1ew(xn;xm)Mn(IM)1ew(x0;x1):
Assume that x6=f(x). Then, for every x2X, by hypothesis ( a) we have:
0<inffew(x;x) +ew(x;f(x)) :x2Xginffew(xn;x) +ew(xn;xn+1) :n2Ng
inffMn(IM)1ew(x0;x1) +Mnew(x0;x1)g= 0
This is a contradiction. Therefore x=f(x), soFix(f)6=?. For the proof of the last part of
this theorem we use the same steps as is the previous theorem, Theorem 2.1. 
Further we give a more general xed point result concerning this new type of operators.

6 L. GURAN, K. SHABBIR
Theorem 2.3.Let(X;ed)be a complete generalized metric space in Perov's sense, ew:XX!
Rm
+be a generalized w-distance and f:X!Xbe a singlevalued weakly Hardy-Rogers type
operator. There exists the matrices A;B;C2Mm;m(R+)such that:
(i)M= (I(B+C))1(A+B+C)converges to ;
(ii)I(B+C)is nonsingular and (I(B+C))12Mm;m(R+);
(iii)I(A+ 2B+ 2C)is nonsingular and [I(A+ 2B+ 2C)]12Mm;m(R+).
ThenFix(f)6=?. Moreover, if x=f(x), thenw(x;x) = 0 .
Proof. Following the same steps as in Theorem 2.1 we get the estimation:
(9) ew(xn;xm)Mn(IM)1ew(x0;x1):
withM2Mm;m(R+) andn2N.
By Lemma 2.1(3) we have that the sequence ( xn)n2Nis a Cauchy sequence. Since ( X;ed) is
complete we have that there exists x2Xsuch thatxnd!x.
Letn2Nbe xed. Then, since ( xm)m2Nd!x,ew(xn;
) is lower semicontinuous and letting
n!1 we have:
(10) ew(xn;x)lim inf
m!1ew(xn;xm)Mn(IM)1ew(x0;x1)d!01m:
Letf(x)2X. By the triangle inequality and using (6) we obtain the following result.
ew(xn;f(x)) =ew(xn;x) +ew(x;f(x))ew(xn;x) +ew(f(x);f(x))
(11)Mn(IM)1ew(x0;x1) + [I(A+ 2B+ 2C)]ew(x;x)d!01m:
Using Lemma 2.1(1), by (10) and (11) we get that x=f(x), thenFix(f)6=?.
For the last part of the proof we use the same steps as in Theorem 2.1. 
Another xed point result concerning the singlevalued weakly Hardy-Rogers operators in gen-
eralized metric space is the following.
Theorem 2.4.Let(X;ed)be a complete generalized metric space in Perov' sense, ew:XX!
Rm
+be a generalized w-distance and f:X!Xbe a singlevalued Hardy-Rogers type operator.
Suppose that all the hypothesis of Theorem 2.1 are accomplished we have:
(1)Fix(f)6=?.
(2)There exists a sequence (xn)n2N2Xsuch thatxn+1=f(xn), for alln2Nand converge
to a xed point of f.
(3)One has the estimation ed(xn;x)Mned(x0;x1), wherex2Fix(f):
Example 2.1.LetX=R2be normed liniar space with the generalized norm eddened by
ed(x;y)(=
jjx1y1jj
jjx2y2jj
andewa generalized w-distance dened by ew(x;y)(=
jjy1jj
jjy2jj
, for each
x= (x1;x2);y= (y1;y2)2R2. Letf:R2!R2be an operator given by:
f(x;y) =4x
5+6y
51;6y
51;for(x;y)2R2;withx5;
x
5+y
31;y
5; for(x;y)2R2;withx>5:
We takef(x;y) = (f1(x;y);f2(x;y))wheref1(x;y) =4x
5+6y
51;for(x;y)2R2;withx5;
x
5+y
31;for(x;y)2R2;withx>5:
andf2(x;y) =6y
51;for(x;y)2R2;withx5;
y
5; for(x;y)2R2;withx>5:
Next we will prove that weakly Hardy-Rogers type condition it is true. Let A=4
56
5
06
5
.

FIXED POINT THEOREMS FOR WEAKLY HARDY-ROGERS TYPE OPERATORS ON GENERALIZED METRIC SPACES 7
Case 1. If 1x1;x2;y1;y25we have:
ew(f(x);f(y)) =
jjf1(y1;y2)jj
jjf2(y1;y2)jj
=
jj4
5y1+6
5y21jj
jj0
y1+6
5y21jj
4
5jjy1jj+6
5jjy2jj1
0
jjy1jj+6
5jjy2jj1

4
56
5
06
5
jjy1jj
jjy2jj
=Aew(x;y):
Case 2. Ifx1;x2;y1;y2>5we have:
ew(f(x);f(y)) =
jjf1(y1;y2)jj
jjf2(y1;y2)jj
=
jj1
5y1+1
3y21jj
jj0
y1+1
5y2jj
1
5jjy1jj+1
3jjy2jj1
0
jjy1jj+1
5jjy2jj

1
51
3
01
5
jjy1jj
jjy2jj
<4
56
5
06
5
jjy1jj
jjy2jj
=Aew(x;y):
Case 3. For other choices of x1;x2;y1;y2we have:
ew(f(x);f(y)) =
0
0
4
56
5
06
5
jjy1jj
jjy2jj
=Aew(x;y):
Thus, the weakly Hardy-Rogers type condition is accomplished for A=4
56
5
06
5
andB=C=
orB+C= .
Since all the hypothesis of the Theorem 2.2 hold we get that fhas a xed point and is easy to
check thatx=f(x) = (f1(x);f2(x)), wherex= (1;1).
Next, let us give some common xed point results.
Theorem 2.5.Let(X;ed)be a complete generalized metric space in Perov's sense, ew:XX!
Rm
+be a generalized w-distance and let f;g:X!Xbe two continuous singlevalued weakly Hardy-
Rogers type operators. There exists the matrices A;B;C2Mm;m(R+)such that:
(i)I(B+C)is nonsingular and (I(B+C))12Mm;m(R+);
(ii)M= (I(B+C))1(A+B+C)converges to .
Thenfandghave a common xed point x2X.
Proof. (1) Letx02X. We consider ( xn)n2Nthe sequence of successive approximations for f
andg, dened by:
x2n+1=f(x2n);n= 0;1;:::
x2n+2=g(x2n+1);n= 0;1;:::
Then we have:
ew(x2n;x2n+1) =ew(g(x2n1);f(x2n))Aew(x2n1;f(x2n)+
+B[ew(x2n;f(x2n)) +ew(x2n1;g(x2n1))] +C[ew(x2n;g(x2n1)) +ew(x2n1;f(x2n))] =
=Aew(x2n1;x2n) +B[ew(x2n;x2n+1) +ew(x2n1;x2n)] +Cew(x2n1;x2n+1)
Aew(x2n1;x2n) +B[ew(x2n;x2n+1) +ew(x2n1;x2n)] +C[ew(x2n1;x2n) +ew(x2n;x2n+1)]:
Then we have: ew(x2n;x2n+1)(I(B+C))1(A+B+C)ew(x2n1;x2n) =Mew(x2n1;x2n):
Continuing the process we get:
ew(x2n+1;x2n+2) =ew(f(x2n);g(x2n+1))Aed(x2n;f(x2n+1)+
+B[ew(x2n;f(x2n)) +ew(x2n+1;g(x2n+1))] +C[ew(x2n;g(x2n+1)) +ew(x2n+1;f(x2n))] =
=Aew(x2n;x2n+1) +B[ew(x2n;x2n+1) +ew(x2n+1;x2n+2)] +Cew(x2n;x2n+2)

8 L. GURAN, K. SHABBIR
Aew(x2n;x2n+1) +B[ew(x2n;x2n+1) +ew(x2n+1;x2n+2)] +C[ew(x2n;x2n+1) +ew(x2n+1;x2n+2)]:
Then we have: ew(x2n+1;x2n+2)(I(B+C))1(A+B+C)ew(x2n;x2n+1) =Mew(x2n;x2n+1):
Further we obtain that ew(xn;xn+1)Mnew(x0;x1) for eachn2N.
Following the same steps like in the proof of the Theorem 2.1 we estimate ew(xn;xm), for every
m;n2Nwithm>n .
ew(xn;xm)ew(xn;xn+1) +ew(xn+1;xn+2) +:::+ew(xm1;xm)
Mn(ew(x0;x1)) +Mn+1(ew(x0;x1)) +:::+Mm1(ew(x0;x1))
Mn(I+M+M2+:::+Mmn1)(ew(x0;x1))Mn(IM)1ew(x0;x1)):
Note that ( IM) is nonsingular since Mis convergent to . Using Lemma 2.1(3) result that
the sequence ( xn)n2Nis a Cauchy sequence.
Using the lower semicontinuity property of the generalized w-distance, by relation (8) we have
thatew(xn;x)d!01masn!1 . Then we have that ew(x2n;x)d!01masn!1 . By the
continuity of fit follows that x2n+1=f(x2n)d!f(x) asn!1 . By the uniqueness of the limit
we get that x=f(x).
In the same time, by ew(xn;x)d!01masn!1 we have that ew(x2n+1;x)d!01mas
n!1 . By the continuity of git follows that x2n+2=g(x2n+1)d!g(x) asn!1 . By the
uniqueness of the limit we get that x=g(x).
Then we get that xis a common xed point for fandg. 
If we renonce of the continuity condition of the maps fandgwe get the following result.
Theorem 2.6.Let(X;ed)be a complete generalized metric space in Perov's sense, ew:XX!
Rm
+be a generalized w-distance and let f;g:X!Xbe two singlevalued Hardy-Rogers type
operators. There exists the matrices A;B;C2Mm;m(R+)such that:
(i)I(B+C)is nonsingular and (I(B+C))12Mm;m(R+);
(ii)I(A+ 2B+ 2C)is nonsingular and [I(A+ 2B+ 2C)]12Mm;m(R+);
(iii)M= (I(B+C))1(A+B+C)converges to .
Thenfandghave a common xed point x2X.
Proof. (1) As in the proof of the previous theorem, Theorem 2.5, for x02Xwe consider
(xn)n2Nthe sequence of successive approximations for fandg, dened by:
x2n+1=f(x2n);n= 0;1;:::
x2n+2=g(x2n+1);n= 0;1;:::
In the same way we dene the sequence ( xn)nN2Xsuch that
ew(x2n+1;x2n+2)(I(B+C))1(A+B+C)ew(x2n;x2n+1) =Mew(x2n;x2n+1):
Further we obtain that ew(xn;xn+1)Mned(x0;x1) for eachn2N.
Following the same steps like in the proof of the Theorem 2.5 we estimate ew(xn;xm), for every
m;n2Nwithm>n and we getew(xn;xm)Mn(IM)1ew(x0;x1)):
Note that ( IM) is nonsingular since Mis convergent to . Using Lemma 2.1(3) we get
that the sequence ( xn)n2Nis a Cauchy sequence. Using the lower semicontinuity property of the
generalized w-distance, by relation (8) we have that ew(xn;x)d!01m;asn!1 . By (11) we
have thatew(xn;f(x))d!01m;asn!1 . Then, using Lemma 2.1(2) we get that x=f().
Let us show that g(x) =x. Then, by the denition of Hardy-Rogers type operators we have:
ew(x;g(x)) =ed(f(x);g(x))

FIXED POINT THEOREMS FOR WEAKLY HARDY-ROGERS TYPE OPERATORS ON GENERALIZED METRIC SPACES 9
Aew(x;x) +B[ew(x;f(x)) +ew(x;g(x)] +C[ew(x;g(x)) +ew(x;f(x))]:
Then result that:
(12) ew(x;g(x))(I(B+C))1(A+B+C)ew(x;x):
By (6) we get that ew(x;g(x)) = 0 1m:
Letg(x)2X. By the triangle inequality and using (12) we obtain:
(13)ew(xn;g(x)) =ew(xn;x) +ew(x;g(x))Mn(IM)1ew(x0;x1) + 0 1md!01m:
Using (8) and (13), by Lemma 2.1(2), we obtain x=g(x). Thenxis a common xed point
forfandg. 
3. Ulam Hyers stability, well posedness and data dependence of xed point problems
First, let us present the extension of Ulam-Hyers stability for xed point inclusions for the
case of singlevalued operators on generalized metric space in Perov's sense. Then, let us recall the
denition for weakly Ulam-Hyers stability as follows.
Definition 3.1.Let(X;ed)be a metric space, ew:XX!Rm
+be a generalized w-distance
andf:X!Xbe an operator. By denition, the xed point inclusion
(14) x=f(x)
is weakly Ulam-Hyers stable if there exists a real positive matrix N2Mm;m(R+)such that, for
each">0and each solution yof the inecuation
(15) ew(y;f(y))"I1m
there exists a solution xof the inclusion (14) such that
ed(y;x)N"I 1m:
Theorem 3.1.Let(X;ed)be a generalized metric space in Perov's sense, ew:XX!Rm
+be
a generalized w-distance and f:X!Xbe a singlevalued Hardy-Rogers type operator dened in
(2.2). There exists the matrices A;B;C2Mm;m(R+)such that:
(i)N=Mn(IM)1is nonsingular and N=Mn(IM)12Mm;m(R+), whereM=
(I(B+C))1(A+B+C)converges to ;
(ii)I(A+ 2B+ 2C)is nonsingular and [I(A+ 2B+ 2C)]12Mm;m(R+);
(iii)IP2is nonsingular and IP22Mm;m(R+)whereP= [I(A+C)]1C2Mm;m(R+).
Then, the xed point inclusion (14) is weakly Ulam-Hyers stable.
Proof. LetI1m>01msuch thatew(x0;x1)I1m, for everyx0;x12Xwithx1=f(x0).
LetFix(f) =fxgandu2Xbe a solution of inclusion (14). Then ew(u;f(u))"I1m. By
the denition of the weakly Hardy-Rogres operators we obtain:
ew(x;u)ew(f(x);f(u))Aew(x;u) +B[ew(x;f(x)) +ew(u;f(u))] +C[ew(x;f(u)+
(16) +ew(u;f(x))] =Aew(x;u) +B[ew(x;x) +ew(u;u)] +C[ew(x;u) +ew(u;x)] =
= (A+C)ew(x;u) +B[ew(x;x) +ew(u;u)] +Cew(u;x)
By (6) we get that:
(17) ew(x;x) =ew(f(x);f(x))(A+ 2B+ 2C)ew(x;x) and
ew(u;u) =ew(f(u);f(u))(A+ 2B+ 2C)ew(u;u):
Using hypothesis ( ii) we get that ew(x;x) =ew(u;u) = 0 1m.

10 L. GURAN, K. SHABBIR
Replacing in (16) we obtain:
(18) ew(x;u)[I(A+C)]1Cew(u;x):
By the denition of weakly Hardy-Rogers operators we get that:
ew(u;x)[I(A+C)]1Cew(x;u)
and replacing in (18) we obtain:
(19) ew(x;u)([I(A+C)]1C)2ew(x;u) =P2ew(x;u):
Then (IP2)ew(x;u)01m. By hypothesis ( iii) result thatew(x;u) = 0 1m.
Letxn2Xsuch that by (8) and (19) we have:
(20) ew(xn;x)Mn(IM)1ew(x0;x1)NI 1mand
ew(xn;u)ew(xn;x) +ew(x;u)Mn(IM)1ew(x0;x1) + 0 1mNI 1m:
Then, using the denition of the generalized w-distance, there exists "I1m>01msuch that:
ed(x;u)"I1mN"I 1m:
Then the xed point inclusion (14) is weakly Ulam-Hyers stable.

Let us give the following results which assure the well-posedness with respect to the generalized
w-distanceew.
Theorem 3.2.Let(X;ed)be a generalized metric space in Perov's sense, ew:XX!Rm
+be
a generalized w-distance and f:X!Xbe a singlevalued Hardy-Rogers type operator dened in
(2.2). If there are accomplished all the hypothesis of Theorem 2.1(respectively 2.2, 2.3) the xed
point inclusion (14) is well-posed; i.e. if Fix(f) =fxgandxn2N, withn2N, such that
ew(xn;f(xn))!01masn!1 , thenxn!xasn!1 . Then the xed point (14) is weakly
Ulam-Hyers stable.
Proof. Letx2Fix(f) and let (x)n2N2Xsuch thatew(xn;f(xn))d!01masn!1 . That
meansew(xn1;xn)d!01masn!1 .
By the lower semicontinuity of the generalized w-distance, using (8) we have that:
ew(xn1;x)lim inf
m!1ew(xn;xm)Mn(IM)1ew(x0;x1)d!01m:
Then, using Lemma 2.1(2) we get that xnd!xasn!1 . 
Next, let us give a data dependence result.
Theorem 3.3.Let(X;ed)be a generalized metric space in Perov's sense, ew:XX!Rm
+be
a generalized w-distance and f1;f2:X!Xbe a singlevalued operators which satisfy the following
conditions:
(i)forA;B;C;M2Mm;m(R+)withM= [I(B+C)]1(A+B+C)a matrix convergent
tosuch that, for every x;y2Xandi2f1;2g, we have:
ew(fi(x);fi(y))Aew(x;y) +B[ew(x;fi(x)) +ew(y;fi(y))] +C[ew(x;fi(y)) +ew(y;fi(x))];
(ii)there exists >0such thatew(f1(x);f2(x))I, for allx2X.
Then forx
1=f1(x
1)there existx
2=f2(x
2)such thated(x
1;x
2)(IM)1I1m; (respectively
forx
2=f2(x
2)there existx
1=f1(x
1)such thatew(x
2;x
1)(IM)1I1m).

FIXED POINT THEOREMS FOR WEAKLY HARDY-ROGERS TYPE OPERATORS ON GENERALIZED METRIC SPACES 11
Proof. As in the proof of Theorem 2.1 (respectively Theorem 2.2) we construct a sequence
of succesive approximations ( xn)n2N2Xoff2withx0:=x
1andx1=f2(x
1) having property
ew(xn;xn+1)Mnew(x0;x1), whereM= [I(B+C)]1(A+B+C).
If we consider that the sequence ( xn)n2N2Xconverges to x
2we have that x
2=f(x
2). More-
over, for each n;p2Nwe haveew(xn;xn+p)Mn(IM)1ew(x0;x1).
Lettingp!0 we get that ew(xn;x
2)I(IM)1ew(x0;x1).
Choosingn= 0 we get that ew(x0;x
2)I(IM)1ew(x0;x1) and using above notations we
get the conclusion: ew(x
1;x
2)(IM)1I1m. 
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Liliana Guran,
Department of Pharmaceutical Sciences, "Vasile Goldis " Western University of Arad, L. Rebre-
anu Street, no. 86, 310414, Arad, Romania,
Email address :lguran@uvvg.ro, gliliana.math@gmail.com.
Khurram Shabbir,
Department of Mathematics, GC University Lahore, Pakistan,
Email address :dr.khurramshabbir@gcu.edu.pk.