Existence of unique common solution to the system of non linear integral equations via fixed point results in incomplete metric spaces

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R E S E A R C H

Open Access

Existence of unique common solution to

the system of non-linear integral equations

via ﬁxed point results in incomplete metric

spaces

Mian Bahadur Zada

1

_{, Muhammad Sarwar}

1*

_{and Stojan Radenovi´c}

2

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

University of Malakand, Chakdara, Dir(L), Pakistan

Full list of author information is available at the end of the article

Abstract

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 Volterra-Hammerstein integral equations, respectively:

u(s) =

φ

i(s) + b

a

Ki

(

s,r,u(r)

)

dr,

wheres∈(a,b)⊆R;u,

φ

i∈C((a,b),Rn) andKi: (a,b)×(a,b)×Rn→Rn,i= 1, 2,. . ., 6 and

u(s) =pi(s) +

λ

t

0

m(s,r)gi

(

r,u(r)

)

dr+

μ

_∞

0

n(s,r)hi

(

r,u(r)

)

dr,

wheres∈(0,∞),

λ

,

μ

∈R,u,pi,m(s,r),n(s,r),gi(r,u(r)) andhi(r,u(r)),i= 1, 2,. . ., 6, are real-valued measurable functions both insandron (0,∞).

MSC: 47H10; 54H25

Keywords: common ﬁxed point; weakly compatible maps; common (CLR)-property;

common (E.A)-property; Urysohn integral equations; Volterra-Hammerstein integral equations

1 Introduction and preliminaries

Mathematical models are very powerful and important parts of the mathematical anal-ysis with numerous applications to real world problems. Several problems that appear in applied mathematics, physical sciences, geology, mechanics, engineering, economics, and biology generate mathematical models interpreted by functional equations, integral equations, matrix equations, and diﬀerential equationsetc.There are multifarious and ad-vanced methods, focusing on the existence of unique solutions to these models. To handle the existence of unique solution to such equations, one of these methods is the ﬁxed point

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method; for example, refer to [–]. In metric fixed point theory the first remarkable result was given by Banach, usually known as the Banach contraction principle. This principle is a prominent tool for solving problems in non-linear analysis. Several mathematicians improved and extended this principle by modifying the interpretation and pattern of the metric function for instance: cone metric spaces [],G-metric spaces [], partial metric spaces [] and fuzzy metric spaces []etc.After the proper introduction of cone metric space by Huang and Zhong [], there was a drawback that fixed point results under rational type contractions are unsubstantial in a cone metric space as it is a vector-valued metric. Azamet al.[] offered the conception of a complex-valued metric space for finding the fixed point results satisfying rational type contractive conditions.

Deﬁnition .([]) LetY be non-empty set andC+={c∈C:c}. Then the mapping

d:Y×Y→C+is a complex-valued metric if it satisﬁes the following axioms:

() d(c,c) = ⇔c=c;

() d(c,c) =d(c,c), for allc,c∈Y;

() d(c,c)d(c,c) +d(c,c), for allc,c,c∈Y.

The setYtogether withdis called a complex-valued metric space.

In this setting, Azamet al.[] generalized the Banach contraction principle for two self-maps under rational type contraction. Inspired by the impact of a complex-valued 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 non-linear integral equations, for example, refer to [, , , –]. Particularly, Sintu-navaratet 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, Pathaket al.[] and Rashwan and Saleh [] studied the existence of unique common solution to the system of Volterra-Hammerstein non-linear integral equations.

Throughout this manuscriptY represents a complex-valued metric space, unless oth-erwise speciﬁed. For two self-mapsf andf deﬁned on a non-empty setY,w∈Y is a

common ﬁxed point off andfiffw=fw=w. To study common ﬁxed points, Jungck [] initiated the concept of weak compatibility of maps thus:f andf onY are weakly compatible maps ifffw=ffwwheneverfw=fw, for somew∈Y. In the study of

com-mon 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. Liuet al.[] extended the (E.A)-property [] to the com-mon (E.A)-property and Imdadet al.[] extended the (CLR)-property [] to common (CLR)-property. Sarwar and Bahadur Zada [] deﬁned these views in the complex-valued metric space as follows.

Deﬁnition . Letf,f,f,f:Y→Ybe four maps. If there are two sequences{zn}and

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() the common(E.A)-property if

lim

n→∞fzn=n→∞lim fzn=n→∞limfwn=n→∞lim fwn=t∈Y;

() the common(CLRff)-property if

lim

n→∞fzn=n→∞lim fzn=n→∞limfwn=n→∞lim fwn=t∈f(Y)∩f(Y).

Note that the (E.A)-property tolerates the condition of closeness of the range of sub-spaces 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)andf(Y)⊆f(Y); () for allu,v∈Y and <k< ,

d(fu,fv)k

d(fu,fu)d(fu,fu)d(fu,fv)d(fu,fv)

 +d(fu,fv)d(fu,fv) +d(fv,fu)d(fv,fu)

+ d(fv,fv)d(fv,fv)d(fv,fu)d(fv,fu)  +d(fu,fv)d(fu,fv) +d(fv,fu)d(fv,fu)

;

() 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.

The aim of this manuscript is to study the existence of unique common solution for the systems of:

• Urysohn integral equations in complex-valued metric spaces, • Volterra-Hammerstein integral equations in ordinary metric spaces.

2 Existence of unique common solution to the systems of Urysohn integral equations

Our plan is to apply Theorem . to the existence of a unique common solution to the following system:

u(s) =φi(s) +

b

a

Ki

s,r,u(r)dr, (.)

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Let us denote

i

u(s)=

b

a

Ki

s,r,u(r)dr,

wherei= , , . . . , .

Assume that the following conditions hold:

(C) fori= , ,

u(s) +φ(s) +φi(s) –i

u(s) +φ(s) +φi(s)

= ,

(C) forj= , ,

u(s) +φ(s) +φj(s) –j

u(s) +φ(s) +φj(s)

= ,

(C) forj= , ,

φ(s) + φj(s) + ju(s) +

u(s) +φ(s)

+j

u(s) –ju(s) –φj(s)

= u(s),

(C) fori= , ,

φ(s) + φi(s) + iu(s) +

u(s) +φ(s)

+i

u(s) –iu(s) –φi(s)

= u(s).

LetY=C((a,b),Rn_),_a_{>  be an incomplete complex-valued metric space with metric}

d(u,v) = max

s∈(a,b)

u(s) –v(s)_∞√ +a_e˙ιarctana_, _{for all}_u_,_v_∈_Y_.

Deﬁne six operatorsf,f,f,f,f,f:Y→Yby

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

fu(s) =u(s) +φ(s),

fu(s) =u(s) +φ(s),

fu(s) = u(s) –u(s) –φ(s),

fu(s) = u(s) –u(s) –φ(s),

fu(s) = u(s) –u(s) –φ(s),

fu(s) = u(s) –u(s) –φ(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}inY such that

lim

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() for eachu,v∈Yand <λ< ,

ϒ √

 +a_e˙ιarctana

λ

ϒ×ϒ×ϒ×ϒ+ϒ×ϒ×ϒ×ϒ

 + (max_s∈₍_a_,_b₎ϒ)·(maxs∈(a,b)ϒ) + (maxs∈(a,b)ϒ)·(maxs∈(a,b)ϒ)

,

where

ϒ=u(s) –v(s) +φ(s) –φ(s)_∞ √

 +a_e˙ιarctana_,

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)_∞ √

 +a_e˙ιarctana_;

() f(Y)⊆f(Y),f(Y)⊆f(Y),f(Y)⊆f(Y),andf(Y)⊆f(Y)such that(f,f),(f,f),

(f,f),and(f,f)are weakly compatible.

Then the system(.)of Urysohn integral equations has a unique common solution.

Proof Notice that the system (.) of Urysohn integral equations has a unique common solution if and only if the system (.) of operators has a unique common ﬁxed point.

Now, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

d(fu,fv) = max

s∈(a,b)

u(s) –v(s) +φ(s) –φ(s)_∞ √

d(fu,fu) = max

s∈(a,b)

u(s) –u(s) –u(s) –φ(s) –φ(s)_∞ √

d(fu,fu) = max

s∈(a,b)u(s) –u(s) –u(s) –φ(s) –φ(s)∞ √

d(fu,fv) = max

s∈(a,b)

u(s) –u(s) –v(s) –φ(s) –φ(s)_∞ √

d(fu,fv) =max

s∈(a,b)

u(s) –u(s) –v(s) –φ(s) –φ(s)_∞ √

d(fv,fv) = max

s∈(a,b)

v(s) –v(s) –v(s) –φ(s) –φ(s)_∞ √

d(fv,fv) =max

s∈(a,b)

v(s) –v(s) –v(s) –φ(s) –φ(s)_∞ √

d(fv,fu) =max

s∈(a,b)

v(s) –v(s) –u(s) –φ(s) –φ(s)_∞ √

d(fv,fu) = max

s∈(a,b)

v(s) –v(s) –u(s) –φ(s) –φ(s)_∞ √

 +a_e˙ιarctana_.

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From condition () of Theorem ., we have

ϒ

√

λ

ϒ×ϒ×ϒ×ϒ+ϒ×ϒ×ϒ×ϒ

,

which implies that

max

s∈(a,b)ϒ √

λ

(max_s∈(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)  + (maxs∈(a,b)ϒ)·(maxs∈(a,b)ϒ) + (maxs∈(a,b)ϒ)·(maxs∈(a,b)ϒ)

,

using (.), we obtain

d(fu,fv)λ

d(fu,fu)d(fu,fu)d(fu,fv)d(fu,fv)  +d(fu,fv)d(fu,fv) +d(fv,fu)d(fv,fu)

+ d(fv,fv)d(fv,fv)d(fv,fu)d(fv,fu)  +d(fu,fv)d(fu,fv) +d(fv,fu)d(fv,fu)

.

Now, to show thatf(Y)⊆f(Y), we have

ffu(s) +φ(s)

= fu(s) +φ(s)

–

fu(s) +φ(s)

–φ(s)

=fu(s) +fu(s) +φ(s) –

fu(s) +φ(s)

=fu(s) +u(s) +φ(s) +φ(s) –

u(s) +φ(s) +φ(s)

.

Using (C), we getf(fu(s) +φ(s)) =fu(s), which implies thatf(Y)⊆f(Y). Similarly, one

can prove thatf(Y)⊆f(Y),f(Y)⊆f(Y) andf(Y)⊆f(Y).

Next, we need to show the weak compatibility of the pair (f,f). For this, we have

ffu(s) –ffu(s)=f

u(s) +φ(s)

–f

u(s) –u(s) –φ(s)

=u(s) +φ(s)

–

u(s) +φ(s)

–φ(s)

–

u(s) –u(s) –φ(s)

–φ(s). (.)

Iffu(s) =fu(s), foru(s)∈Y. Thenu(s) +φ(s) = u(s) –u(s) –φ(s), thus (.)

be-comes

ffu(s) –ffu(s)=u(s) –u(s) –φ(s)

–

u(s) –u(s) –φ(s)

–φ(s) –

u(s) +φ(s)

–φ(s)

=u(s) – u(s) – φ(s) –

u(s) –u(s) –φ(s)

–

u(s) +φ(s)

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with the help of (C), we get ffu(s) –ffu(s) = , which implies thatffu(s) =ffu(s),

wheneverfu(s) =fu(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 off,f,f,f,f, andf

inY, that is, the system (.) of Urysohn integral equations has a unique common solution

inY.

In the next result we use the common (E.A)-property and the proof is simple, so we omit it.

Theorem . Under the assumptions(C)-(C)and the conditions(), ()of Theorem., if there exist two sequences{zn}and{wn}in Y such that

lim

n→∞fzn=n→∞limfzn=n→∞limfwn=n→∞lim fwn=z, for some z∈Y, (.)

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 Volterra-Hammerstein integral equations

In this section, we present the real-valued metric version of Theorem . and Theorem . and the proof can easily be obtained, so we omit its proof here.

Corollary . Let f,f,f,f,f,fbe six maps on a metric space(Z,d)such that

() f(Z)⊆f(Z),f(Z)⊆f(Z),f(Z)⊆f(Z),andf(Z)⊆f(Z);

() for allu,v∈Zand <k< ,

d(fu,fv)≤λ

d(fu,fu)d(fu,fu)d(fu,fv)d(fu,fv)  +d(fu,fv)d(fu,fv) +d(fv,fu)d(fv,fu)

+ d(fv,fv)d(fv,fv)d(fv,fu)d(fv,fu)  +d(fu,fv)d(fu,fv) +d(fv,fu)d(fv,fu)

;

() 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.

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LetZ= (L(,∞),R) be the space of real-valued measurable functions on (,∞):

u(s) =pi(s) +λ

t



m(s,r)gi

r,u(r)dr+μ

∞



n(s,r)hi

r,u(r)dr, (.)

for alls∈(,∞), whereλ,μ∈R,u,pi,m(s,r),n(s,r),gi(r,u(r)) andhi(r,u(r)),i= , , . . . , , are real-valued measurable functions insandron (,∞).

Let us denote

iu(s) =

t



m(s,r)gi

r,u(r)dr

and

∇iu(s) =

∞



n(s,r)hi

r,u(r)dr,

wherei= , , . . . , . Assume that

(C∗_) fori= , ,

u(s) +∇u(s) +p(s) +pi(s) –i

u(s) +∇u(s) +p(s) +pi(s)

–∇i

u(s) +∇u(s) +p(s) +pi(s)

= ,

(C∗_) forj= , ,

u(s) +∇u(s) +p(s) +pj(s) –ju(s) +∇u(s) +p(s) +pj(s)

–∇j

u(s) +∇u(s) +p(s) +pj(s)= ,

(C∗_) forj= , ,

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),

(C∗_) fori= , ,

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).

LetZ= (L(,∞),R) be an incomplete metric space with metric

d(u,v) = max

s∈(,∞)

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Deﬁne the six operatorsf,f,f,f,f, andfonZby

fu(s) =u(s) +∇u(s) +p(s),

fu(s) =u(s) +∇u(s) +p(s),

fu(s) = u(s) –u(s) –∇u(s) –p(s),

fu(s) = u(s) –u(s) –∇u(s) –p(s),

fu(s) = u(s) –u(s) –∇u(s) –p(s),

fu(s) = u(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}inZsuch that

lim

n→∞fzn=n→∞lim fzn=n→∞lim fwn=n→∞lim fwn=z∈f(Z); (.)

() for eachu,v∈Zand <λ< ,

u(s) +∇u(s) +p(s) –u(s) –∇u(s) –p(s)

≤λ

ϒ×ϒ×ϒ×ϒ+ϒ×ϒ×ϒ×ϒ

 + (max_s∈(a,b)ϒ)·(maxs∈(a,b)ϒ) + (maxs∈(a,b)ϒ)·(maxs∈(a,b)ϒ)

,

where

ϒ=u(s) –u(s) –∇u(s) –p(s) –u(s) –∇u(s) –p(s),

ϒ=u(s) –u(s) –∇u(s) –p(s) –u(s) –∇u(s) –p(s),

ϒ=u(s) –u(s) –∇u(s) –p(s) –v(s) –∇v(s) –p(s),

ϒ=u(s) –u(s) –∇u(s) –p(s) –v(s) –∇v(s) –p(s),

ϒ=v(s) –v(s) –∇v(s) –p(s) –v(s) –∇v(s) –p(s),

ϒ=v(s) –v(s) –∇v(s) –p(s) –v(s) –∇v(s) –p(s),

ϒ=v(s) –v(s) –∇v(s) –p(s) –u(s) –∇u(s) –p(s),

ϒ=v(s) –v(s) –∇v(s) –p(s) –u(s) –∇u(s) –p(s);

() f(Z)⊆f(Z),f(Z)⊆f(Z),f(Z)⊆f(Z)andf(Z)⊆f(Z)such that the pairs(f,f),

(f,f),(f,f)and(f,f)are weakly compatible,

then the system(.)of Volterra-Hammerstein equations has a unique common solution.

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Now, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

d(fu,fv) = max s∈(,∞)

u(s) +∇u(s) +p(s) –u(s) –∇u(s) –p(s),

d(fu,fu) = max s∈(,∞)

u(s) –u(s) –∇u(s) –p(s) –u(s) –∇u(s) –p(s),

d(fu,fu) = max s∈(,∞)

u(s) –u(s) –∇u(s) –p(s) –u(s) –∇u(s) –p(s),

d(fu,fv) = max s∈(,∞)

u(s) –u(s) –∇u(s) –p(s) –v(s) –∇v(s) –p(s),

d(fu,fv) = max s∈(,∞)

u(s) –u(s) –∇u(s) –p(s) –v(s) –∇v(s) –p(s),

d(fv,fv) = max s∈(,∞)

v(s) –v(s) –∇v(s) –p(s) –v(s) –∇v(s) –p(s),

d(fv,fv) = max s∈(,∞)

v(s) –v(s) –∇v(s) –p(s) –v(s) –∇v(s) –p(s),

d(fv,fu) = max s∈(,∞)

v(s) –v(s) –∇v(s) –p(s) –u(s) –∇u(s) –p(s),

d(fv,fu) = max s∈(,∞)

v(s) –v(s) –∇v(s) –p(s) –u(s) –∇u(s) –p(s).

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (.)

From condition () of Theorem ., we have

maxu(s) +∇u(s) +p(s) –u(s) –∇u(s) –p(s)

≤λmax

ϒ×ϒ×ϒ×ϒ+ϒ×ϒ×ϒ×ϒ

 + (max_s∈(a,b)ϒ)·(maxs∈(a,b)ϒ) + (maxs∈(a,b)ϒ)·(maxs∈(a,b)ϒ)

,

which implies that

max

s∈(,∞)

u(s) +∇u(s) +p(s) –u(s) –∇u(s) –p(s)

≤λ

(max_s∈(,∞)ϒ)·(maxs∈(,∞)ϒ)·(maxs∈(,∞)ϒ)·(maxs∈(,∞)ϒ)

 + (maxs∈(,∞)ϒ)·(maxs∈(,∞)ϒ) + (maxs∈(,∞)ϒ)·(maxs∈(,∞)ϒ)

+ (maxs∈(,∞)ϒ)·(maxs∈(,∞)ϒ)·(maxs∈(,∞)ϒ)·(maxs∈(,∞)ϒ)  + (max_s∈(,∞)ϒ)·(maxs∈(,∞)ϒ) + (maxs∈(,∞)ϒ)·(maxs∈(,∞)ϒ)

,

using (.), we get

d(fu,fv)≤λ

d(fu,fu)d(fu,fu)d(fu,fv)d(fu,fv)

 +d(fu,fv)d(fu,fv) +d(fv,fu)d(fv,fu)

+ d(fv,fv)d(fv,fv)d(fv,fu)d(fv,fu)  +d(fu,fv)d(fu,fv) +d(fv,fu)d(fv,fu)

.

Now, to show thatf(Z)⊆f(Z), we have

ffu(s) +p(s)

= fu(s) +p(s)–

fu(s) +p(s)–∇

fu(s) +p(s)–p(s)

=fu(s) +fu(s) +p(s) –

fu(s) +p(s)

–∇

fu(s) +p(s)

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–

u(s) +∇u(s) +p(s) +p(s)

–∇

u(s) +∇u(s) +p(s) +p(s).

Using (C∗_), we getf(fu(s) +p(s)) =fu(s), which implies thatf(Z)⊆f(Z). Similarly, one can prove thatf(Z)⊆f(Z),f(Z)⊆f(Z) andf(Z)⊆f(Z).

Next, we need to show the weak compatibility of the pair (f,f). For this purpose,

ffu(s) –ffu(s)

=fu(s) +∇u(s) +p(s)–fu(s) –u(s) –∇u(s) –p(s)

=u(s) +∇u(s) +p(s)–

u(s) +∇u(s) +p(s)

–∇

u(s) +∇u(s) +p(s)–p(s) –

u(s) –u(s) –∇u(s) –p(s)

–∇

u(s) –u(s) –∇u(s) –p(s)–p(s).

Iffu(s) =fu(s), foru(s)∈Z. Thenu(s) +∇u(s) +p(s) = u(s) –u(s) –∇u(s) –p(s),

thus the above equation becomes

ffu(s) –ffu(s)

=u(s) –u(s) –∇u(s) –p(s)

–

u(s) –u(s) –∇u(s) –p(s)

–∇

u(s) –u(s) –∇u(s) –p(s)

–p(s) –

u(s) +∇u(s) +p(s)

–∇

u(s) +∇u(s) +p(s)

–p(s)

=u(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 ffz(s) –ffz(s) = , which implies thatffz(s) =ffz(s), wheneverfz(s) =fz(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 con-dition () 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 off,f,f,f,f, andf

inZ, that is, the system (.) of Volterra-Hammerstein non-linear integral equations has

a unique common solution inZ.

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

n→∞fzn=n→∞limfzn=n→∞limfwn=n→∞lim fwn=z, for some z∈Z, (.)

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4 Conclusions

In the current work, we studied the existence of unique common solution for the systems of Urysohn and Volterra-Hammerstein integral equations in incomplete spaces. Several problems that appear in applied mathematics, physical sciences, geology, mechanics, en-gineering, economics, and biology generate mathematical models described by integral equations.

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.

Author details

1_{Department of Mathematics, University of Malakand, Chakdara, Dir(L), Pakistan.}2_{Faculty of Mechanical Engineering,}

Universitry of Belgrade, Kraljice Marije 16, Beograd, 11 120, Serbia.

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.

Received: 15 August 2016 Accepted: 22 December 2016

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