Existence of unique common solution to the system of nonlinear integral equations via fixed point results in incomplete metric spaces
Bahadur Zada et al. Journal of Inequalities and Applications
Existence of unique common solution to the system of nonlinear integral equations via fixed point results in incomplete metric
Dir 0
Pakistan 0
Mian Bahadur Zada 0 1 3
Muhammad Sarwar 0 1 3
Stojan Radenovi c´ 0 2
0 University of Malakand , Chakdara
1 Department of Mathematics
2 Faculty of Mechanical Engineering, Universitry of Belgrade , Kraljice Marije 16, Beograd, 11 120 , Serbia
3 Department of Mathematics, University of Malakand , Chakdara, Dir(L) , Pakistan
In this article, we apply common fixed point results in incomplete metric spaces to examine the existence of a unique common solution for the following systems of Urysohn integral equations and VolterraHammerstein integral equations, respectively:
and; common fixed point; weakly compatible maps; common (CLR)property

→ Rn, i = 1, 2, . . . , 6
Ki(s, r, u(r)) dr,
n(s, r)hi(r, u(r)) dr,
MSC: 47H10; 54H25
1 Introduction and preliminaries
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indicate if changes were made.
In this setting, Azam et al. [] generalized the Banach contraction principle for two
selfmaps under rational type contraction. Inspired by the impact of a complexvalued metric
space, several authors [, –] proceeded with the investigation of common fixed point
results.
Many mathematicians applied fixed point methods to the existence of unique solutions
to nonlinear integral equations, for example, refer to [, , , –]. Particularly,
Sintunavarat et al. [] and Rashwan and Saleh [] established fixed point results to find the
existence of a unique common solution to a system of Urysohn integral equations. On the
other hand, Pathak et al. [] and Rashwan and Saleh [] studied the existence of unique
common solution to the system of VolterraHammerstein nonlinear integral equations.
Throughout this manuscript Y represents a complexvalued metric space, unless
otherwise specified. For two selfmaps f and f defined on a nonempty set Y , w ∈ Y is a
common fixed point of f and f if fw = fw = w. To study common fixed points, Jungck
[] initiated the concept of weak compatibility of maps thus: f and f on Y are weakly
compatible maps if ffw = ffw whenever fw = fw, for some w ∈ Y . In the study of
common fixed point results of weakly compatible mappings we often require the assumption
of the continuity of mappings or the completeness of the underlying space. Regarding this
Aamri and Moutawakil [] relaxed these conditions by introducing the notion of the
(E.A)property. In , the new notion of Common Limit in the Range property (for short
(CLR)property) was given by Sintunavarat and Kumam [], which does not enforce the
above mention conditions. Liu et al. [] extended the (E.A)property [] to the
common (E.A)property and Imdad et al. [] extended the (CLR)property [] to common
(CLR)property. Sarwar and Bahadur Zada [] defined these views in the complexvalued
metric space as follows.
Definition . Let f, f, f, f : Y → Y be four maps. If there are two sequences {zn} and
{wn} in Y . Then the pairs (f, f) and (f, f) satisfy
() the common (E.A)property if
() the common (CLRff )property if
lim fzn = lim fzn = lim fwn = lim fwn = t ∈ Y ;
n→∞ n→∞ n→∞ n→∞
lim fzn = lim fzn = lim fwn = lim fwn = t ∈ f(Y ) ∩ f(Y ).
n→∞ n→∞ n→∞ n→∞
Note that the (E.A)property tolerates the condition of closeness of the range of
subspaces of the involved mappings. However, the significance of the (CLR)property reveals
that closeness of the range of subspaces is not essential.
Sarwar and Bahadur Zada [] established the following common fixed point results.
Theorem . Let f, f, f, f, f, and f be six maps on Y such that
() f(Y ) ⊆ f(Y ), f(Y ) ⊆ f(Y ), f(Y ) ⊆ f(Y ) and f(Y ) ⊆ f(Y );
() for all u, v ∈ Y and < k < ,
d(fu, fv)
d(fu, fu)d(fu, fu)d(fu, fv)d(fu, fv)
+ d(fu, fv)d(fu, fv) + d(fv, fu)d(fv, fu)
d(fv, fv)d(fv, fv)d(fv, fu)d(fv, fu)
+ d(fu, fv)d(fu, fv) + d(fv, fu)d(fv, fu)
() the pairs (f, f), (f, f), (f, f), and (f, f) are weakly compatible;
() either both the pairs (f, f) and (f, f) satisfies common (CLRf )property or both the
pairs (f, f) and (f, f) satisfies common (CLRf )property.
Then f, f, f, f, f, and f have a unique common fixed point in Y .
Theorem . Let f, f, f, f, f, and f be six maps on Y such that all the conditions of
Theorem . except condition () holds. In addition if either the pairs (f, f) and (f, f) or
the pairs (f, f) and (f, f) satisfy the common (E.A)property such that either f(Y ) and
f(Y ) or f(Y ) and f(Y ) are closed subspaces of Y , then f, f, f, f, f, and f have a unique
common fixed point in Y .
2 Existence of unique common solution to the systems of Urysohn integral equations
Ki s, r, u(r) dr,
Let us denote
u(s) =
Assume that the following conditions hold:
where i = , , . . . , .
u(s) + φ (s) + φ (s) –
i
u(s) + φ (s) + φ (s) = ,
i
u(s) + φ (s) + φ (s) –
j
u(s) + φ (s) + φ (s) = ,
j
φ (s) + φ (s) +
j
u(s) – φ (s) = u(s),
j
φ (s) + φ (s) +
i
u(s) – φ (s) = u(s).
i
Let Y = C((a, b),
), a > be an incomplete complexvalued metric space with metric
d(u, v) =
s∈(a,b)
u(s) – v(s)
for all u, v ∈ Y .
Define six operators f, f, f, f, f, f : Y →
Now, we are in a position to formulate the existence results.
() there exist two sequences {zn} and {wn} in Y such that
n→∞
n→∞
n→∞
n→∞
f w = z ∈ f(Y );
n
s∈(a,b) ϒ) · (max
s∈(a,b) ϒ) · (max
u(s) – v(s) + φ (s) – φ (s)
ϒ = u(s) – u(s) – u(s) – φ (s) – φ (s)
ϒ = u(s) – u(s) – u(s) – φ (s) – φ (s)
ϒ = u(s) – u(s) – v(s) – φ (s) – φ (s)
ϒ = u(s) – u(s) – v(s) – φ (s) – φ (s)
ϒ = v(s) – v(s) – v(s) – φ (s) – φ (s)
ϒ = v(s) – v(s) – v(s) – φ (s) – φ (s)
ϒ = v(s) – v(s) – u(s) – φ (s) – φ (s)
ϒ = v(s) – v(s) – u(s) – φ (s) – φ (s)
() f (Y) ⊆ f(Y), f (Y) ⊆ f(Y), f (Y) ⊆ f(Y), and f (Y) ⊆ f(Y) such that (f ,f ), (f ,f ),
(f ,f ), and (f ,f ) are weakly compatible.
s∈(a,b)
⎧ d(f u,f v) = max u(s) – v(s) + φ (s) – φ (s)
s∈(a,b)
⎪⎪ d(f v,f u) = max v(s) – v(s) – u(s) – φ (s) – φ (s)
⎩
s∈(a,b)
⎪
⎪
⎪
⎪
⎪
ιarctana ⎪
˙
+ a e
,⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
ιarctana ⎪
˙
+ a e
,⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
ιarctana ⎪
˙
+ a e
From condition () of Theorem ., we have
which implies that
sm∈(aa,xb)ϒ√ + aeι˙arctana
using (.), we obtain
λ (maxs∈(a,b) ϒ) · (maxs∈(a,b) ϒ) · (maxs∈(a,b) ϒ) · (maxs∈(a,b) ϒ)
+ (maxs∈(a,b) ϒ) · (maxs∈(a,b) ϒ) + (maxs∈(a,b) ϒ) · (maxs∈(a,b) ϒ)
d(fu,fu)d(fu,fu)d(fu,fv)d(fu,fv)
+ d(fu,fv)d(fu,fv) + d(fv,fu)d(fv,fu)
d(fv,fv)d(fv,fv)d(fv,fu)d(fv,fu)
+ + d(fu,fv)d(fu,fv) + d(fv,fu)d(fv,fu) .
Using (C), we get f(fu(s) + φ(s)) = fu(s), which implies that f(Y) ⊆ f(Y). Similarly, one
can prove that f(Y) ⊆ f(Y), f(Y) ⊆ f(Y) and f(Y) ⊆ f(Y).
Next, we need to show the weak compatibility of the pair (f,f). For this, we have
with the help of (C), we get ffu(s) – ffu(s) = , which implies that ffu(s) = ffu(s),
whenever fu(s) = fu(s). Thus (f, f) is weakly compatible. In a similarly way one can easily
show the weakly compatibility of the pairs (f, f), (f, f) and (f, f). Also, from condition
() of Theorem ., the pairs (f, f) and (f, f) satisfy the common (CLRf )property.
Thus by Theorem . we can find a unique common fixed point of f, f, f, f, f, and f
in Y , that is, the system (.) of Urysohn integral equations has a unique common solution
in Y .
Theorem . Under the assumptions (C)(C) and the conditions (), () of Theorem .,
if there exist two sequences {zn} and {wn} in Y such that
lim fzn = lim fzn = lim fwn = lim fwn = z, for some z ∈ Y ,
n→∞ n→∞ n→∞ n→∞
and both f(Y ) and f(Y ) are closed subspaces of Y , then the system (.) of Urysohn integral
equations has a unique common solution.
3 Existence of unique common solution to the systems of
VolterraHammerstein integral equations
In this section, we present the realvalued metric version of Theorem . and Theorem .
and the proof can easily be obtained, so we omit its proof here.
d(fv, fv)d(fv, fv)d(fv, fu)d(fv, fu)
+ d(fu, fv)d(fu, fv) + d(fv, fu)d(fv, fu)
() the pairs (f, f), (f, f), (f, f) and (f, f) are weakly compatible;
() either both the pairs (f, f) and (f, f) satisfies common (CLRf )property or both the
pairs (f, f) and (f, f) satisfies common (CLRf )property.
Then f, f, f, f, f, and f have a unique common fixed point in Z.
Corollary . Let f, f, f, f, f, f be six maps on a metric space (Z, d) such that all the
conditions of corollary . except condition () holds. In addition if either the pairs (f, f)
and (f, f) or (f, f) and (f, f) satisfy the common (E.A)property such that either f(Z) and
f(Z) or f(Z) and f(Z) are closed subspaces of Z, then f, f, f, f, f, and f have a unique
common fixed point in Z.
We apply the above results to study the existence of unique common solution to the
following system (.) of nonlinear VolterraHammerstein integral equations.
n(s, r)hi r, u(r) dr,
where i = , , . . . , .
Assume that
(C∗) for i = , ,
iu(s) =
m(s, r)gi r, u(r) dr
∇iu(s) =
n(s, r)hi r, u(r) dr,
(C∗) for j = , ,
(C∗) for j = , ,
(C∗) for i = , ,
u(s) + ∇u(s) + p(s) + pi(s) –
u(s) + ∇u(s) + p(s) + pi(s)
– ∇i
u(s) + ∇u(s) + p(s) + pi(s) = ,
u(s) + ∇u(s) + p(s) + pj(s) –
u(s) + ∇u(s) + p(s) + pj(s)
– ∇j
u(s) + ∇u(s) + p(s) + pj(s) = ,
p(s) + pj(s) + ju(s) + ∇ju(s) +
u(s) + ∇u(s) + p(s)
j u(s) –
ju(s) – ∇ju(s) – pj(s) + ∇
u(s) + ∇u(s) + p(s)
+ ∇j u(s) –
ju(s) – ∇ju(s) – pj(s) = u(s),
p(s) + pi(s) + iu(s) + ∇iu(s) +
u(s) + ∇u(s) + p(s)
i u(s) –
iu(s) – ∇iu(s) – pi(s) + ∇
u(s) + ∇u(s) + p(s)
+ ∇i u(s) –
iu(s) – ∇iu(s) – pi(s) = u(s).
u(s) – v(s) ,
for all u, v ∈ Z.
Define the six operators f, f, f, f, f, and f on Z by
fu(s) = u(s) + ∇u(s) + p(s),
fu(s) = u(s) + ∇u(s) + p(s),
Now, we are in a position to formulate the existence results.
Theorem . Under the assumptions (C∗)(C∗), if
() there exist two sequences {zn} and {wn} in Z such that
lim fzn = lim fzn = lim fwn = lim fwn = z ∈ f(Z);
n→∞ n→∞ n→∞ n→∞
() for each u, v ∈ Z and < λ < ,
() f(Z) ⊆ f(Z), f(Z) ⊆ f(Z), f(Z) ⊆ f(Z) and f(Z) ⊆ f(Z) such that the pairs (f, f),
(f, f), (f, f) and (f, f) are weakly compatible,
then the system (.) of VolterraHammerstein equations has a unique common solution.
Proof Notice that the system of VolterraHammerstein nonlinear integral equations (.)
has a unique common solution if and only if the system of operators (.) has a unique
common fixed point.
From condition () of Theorem ., we have
ϒ × ϒ × ϒ × ϒ + ϒ × ϒ × ϒ × ϒ
+ (max
s∈(a,b) ϒ) · (max
s∈(a,b) ϒ) · (max
+ (max
s∈(,∞) ϒ) · (max
s∈(,∞) ϒ) · (max
+ (max
s∈(,∞) ϒ) · (max
s∈(,∞) ϒ) · (max
which implies that
s∈(,∞)
using (.), we get
d(fu, fu)d(fu, fu)d(fu, fv)d(fu, fv)
+ d(fu, fv)d(fu, fv) + d(fv, fu)d(fv, fu)
d(fv, fv)d(fv, fv)d(fv, fu)d(fv, fu)
+ d(fu, fv)d(fu, fv) + d(fv, fu)d(fv, fu)
Now, to show that f (Z) ⊆ f(Z), we have
f f u(s) + p (s)
= f u(s) + p (s) –
= f u(s) + f u(s) + p (s) –
f u(s) + p (s) – ∇ fu(s) + p (s)
– ∇
u(s) + ∇u(s) + p(s) + p(s)
u(s) + ∇u(s) + p(s) + p(s) .
Using (C∗), we get f(fu(s) + p(s)) = fu(s), which implies that f(Z) ⊆ f(Z). Similarly, one
can prove that f(Z) ⊆ f(Z), f(Z) ⊆ f(Z) and f(Z) ⊆ f(Z).
Next, we need to show the weak compatibility of the pair (f, f). For this purpose,
ffu(s) – ffu(s)
= f
=
ffu(s) – ffu(s)
u(s) + ∇u(s) + p(s) –
u(s) + ∇u(s) + p(s)
– ∇
If fu(s) = fu(s), for u(s) ∈ Z. Then u(s) + ∇u(s) + p(s) = u(s) – u(s) – ∇u(s) – p(s),
thus the above equation becomes
u(s) + ∇u(s) + p(s)
– ∇
u(s) + ∇u(s) + p(s) – p(s)
u(s) + ∇u(s) + p(s) – ∇
u(s) + ∇u(s) + p(s)
– u(s) – u(s) – ∇u(s) – p(s)
– ∇ u(s) – u(s) – ∇u(s) – p(s) ,
with the help of (C∗), we get ffz(s) – ffz(s) = , which implies that ffz(s) = ffz(s),
whenever fz(s) = fz(s). Thus the pair (f, f) is weakly compatible. In a similar way one can
easily show the weakly compatibility of the pairs (f, f), (f, f), and (f, f). Also, from
condition () of Theorem . the pairs (f, f) and (f, f) satisfy the common (CLRf )property.
Thus by Corollary ., we can find a unique common fixed point of f, f, f, f, f, and f
in Z, that is, the system (.) of VolterraHammerstein nonlinear integral equations has
a unique common solution in Z.
In the next theorem we use the common (E.A)property.
Theorem . Under the assumptions (C∗)(C∗) and the conditions (), () of Theorem .,
if there exist two sequences {zn} and {wn} in Z such that
lim fzn = lim fzn = lim fwn = lim fwn = z, for some z ∈ Z,
n→∞ n→∞ n→∞ n→∞
and both f(Z) and f(Z) are closed subspaces of Z, then the system (.) of
VolterraHammerstein equations has a unique common solution.
Competing interests
The authors declare that they have no competing interests regarding this manuscript.
Authors’ contributions
All authors contributed equally to the writing of this manuscript. All authors read and approved the final version.
Acknowledgements
The authors wish to thank the editor and anonymous referees for their comments and suggestions, which helped to
improve this paper. The authors are also grateful to Springer International Publishing for granting full fee waiver.
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