R E S E A R C H
Open Access
Fixed points for cyclic
R
-contractions and
solution of nonlinear Volterra
integro-differential equations
Mujahid Abbas
1,2, Abdul Latif
2and Yusuf I Suleiman
3**Correspondence:
3Department of Mathematics, Kano
University of Science and Technology, P.M.B. 3042, Wudil, Kano, Nigeria
Full list of author information is available at the end of the article
Abstract
In this paper, we introduce the notion of cyclicR-contraction mapping and then study the existence of fixed points for such mappings in the framework of metric spaces. Examples and application are presented to support the main result. Our result unify, complement, and generalize various comparable results in the existing literature.
MSC: 47H10; 47H09; 54H25
Keywords: fixed point; cyclic contractions; Meer-Keeler functions; simulation functions;R-contractions
1 Introduction and preliminaries
Let (X,d) be any metric space,Ya subset ofX, andf :X→Y. A pointxinXthat remains invariant underf is called a fixed point off. The set of all fixed points off is denoted byF(f). A sequence{xn} inX defined byxn+=f(xn) =fn(x),n= , , , . . . , is called a
sequence of successive approximations offstarting fromx∈X. If it converges to a unique
fixed point off, thenf is called a Picard operator.
Fixed point theory plays a vital role in the study of existence of solutions of nonlin-ear problems arising in physical, biological, and social sciences. Some fixed point results simply ensure the existence of a solution but provide no information about the unique-ness and determination of the solution. The distinguishing feature of Banach-Caccioppoli contraction principle is that it addresses three most important aspects known as exis-tence, uniqueness, and approximation or construction of a solution of linear and non-linear problems. The simplicity and usefulness of this principle has motivated many researchers to extend it further, and hence there are a number of generalizations and modifications of the principle. One way to extend the Banach theorem is to weaken the contractive condition by employing the concept of comparison functions. For a detailed survey of such extensions obtained in this direction, we refer to [, ] and references therein.
We denote byPcl(X),N,N,R, andR+the collection of nonempty closed subsets of a
metric space (X,d), the set of positive integers, the set of nonnegative integers, the set of real numbers, and the set of positive real numbers, respectively.
Let (X,d) be a metric space. A self mappingf onXis called aϕ-contractionif
d(fx,fy)≤ϕd(x,y)
for allx,yinX, whereϕis a suitable function on [,∞), called a comparison function. Definition . A mapϕ: [,∞)→[,∞) is said to be aBrowder functionifϕis right
continuous and monotone increasing.
Browder functionsare examples of comparison functions. A self-mappingfonXis called aBrowder contractionif
d(fx,fy)≤ϕ
d(x,y)
for allx,y∈X, whereϕis aBrowder function. EveryBrowder contractionon a complete
metric space is a Picard operator []. EveryBanach-contraction is aBrowder contraction ifϕ(t) =γtforγ ∈[, ).
Boyd and Wong [] introduced a class of comparison functions as follows.
Definition . A functionϕ: [,∞)→[,∞) is called aBoyd-Wong functionifϕ is
upper semicontinuous from the right andϕ(t) <tfor allt> .
A self-mappingf onXis called aBoyd-Wong contractionif for allx,y∈X, d(fx,fy)≤ϕ
d(x,y),
whereϕ is aBoyd-Wong function. EveryBoyd-Wong contractionon a complete metric
space is a Picard operator []. Note that Browder functions are Boyd-Wong functions. Matkowski [] initiated another class of comparison functions as follows.
Definition . A functionφ: [,∞)→[,∞) is called aMatkowski functionifφis in-creasing andlimn→∞φn(t) = for allt≥.
Every Matkowski function is a Boyd-Wond function ([]). Geraghty [] defined the following class of comparison functions.
Letbe the class of all mappingsβ: [,∞)→[, ) satisfying the condition:β(tn)→
impliestn→. Elements ofare calledGeraghty functions.
Note that=φ. For example, if a mappingβ: [,∞)→[, ) is defined byβ(x) =+x, x∈[,∞), thenβ∈.
Let (X,d) be a complete metric space, andf :X→X. If there exists aGeraghty function βsuch that for anyx,y∈X, we have
d(fx,fy)≤βd(x,y)d(x,y), thenf is a Picard operator.
A self-mappingf onXis called aMeir-Keeler mappingif for any> , there existsδ>
such that for allx,y∈Xwith≤d(x,y) <+δ, we haved(fx,fy) <.
Definition . A mappingη: [,∞)→[,∞) is called aLim function or L-functionif η() = ,η(t) > for allt> and for any> , there existsδ> such thatη(t)≤for all
t∈[,+δ].
A self-mapf on a metric space (X,d) is aMeir-Keeler mapping iff there exists an L-functionηsuch thatd(fx,fy) <η(d(x,y)) for allx,y∈Xwithd(x,y) > .
The notion of simulation functions was introduced by Khojastehet al.[] and then mod-ified in [] and [].
Definition . A mappingζ : [,∞)×[,∞)→Ris called asimulation functionif the following conditions hold:
(ζ) ζ(t,s) <s–tfor allt,s> ;
(ζ) if{tn}and{sn}are sequences in(,∞)such thatlimn→∞tn=limn→∞sn∈(,∞)and
tn<snfor alln∈Nthenlim supn→∞ζ(tn,sn) < .
Note that Boyd-Wong functions are simulation functions.
Consistent with Rodan-Lopez-de-Hierro and Shahzad [], the following definitions, examples, and results will be needed in the sequel.
Definition . LetA⊂Rbe a nonempty set. A function:A×A→Ris called an R-functionif:
() for any sequence{an} ⊂(,∞)∩Awith(an+,an) > ∀n∈N, we havelimn→∞an=
;
() for any sequences{an},{bn}in(,∞)∩Asatisfying(an,bn) > ∀n∈N,limn→∞an= limn→∞bn=L≥andL<animply thatL= .
Example .([], Example ) Define: [,∞)×[,∞)→Rby
(t,s) = ⎧ ⎨ ⎩
s–t ift<s,
ift≥s.
Thenis anR-function that is not a simulation function.
Rodan-Lopez-de-Hierro and Shahzad [] also considered the following condition:
() If{an}and{bn}are sequences in(,∞)∩Asuch thatlimn→∞bn= and(an,bn) >
∀n∈N, thenlimn→∞an= .
Example .([], Lemma ) Every simulation function is anR-function that satisfies ().
Example . ([]) If φ : [,∞)→[, ) is a Geraghty function, then φ : [,∞)×
[,∞)→Rdefined by φ(t,s) =φ(s)s–t
Example .([]) Ifφ: [,∞)→[,∞) is anL-function, thenφ: [,∞)×[,∞)→R
defined byφ(t,s) =φ(s) –tis anR-function satisfying ().
Definition . Let (X,d) be a metric space. A self-mapfofXis called anR-contractionif there exists∈RAsuch thatran(d)⊆Aand(d(fx,fy),d(x,y)) > for allx,y∈Xwithx=y,
whereRAis the family of all functions:A×A→Rsatisfying the conditions () and
(), andran(d) is the range of the metricddefined byran(d) ={d(x,y) :x,y∈X} ⊆[,∞).
Definition . LetXbe a nonempty set,pa positive integer, andf a self-map onX. If
{Bi:i= , , . . . ,p}is a finite family of nonempty subsets ofXsuch thatf(B)⊂B,f(B)⊂
B, . . . ,f(Bp–)⊂Bp,f(Bp)⊂B. Then the setpi=Biis called acyclic representation of X
with respect to f.
Kirket al.[] introduced the notion of cyclicϕ-contraction mappings as follows.
Definition . Let (X,d) be a metric space, and{Bi:i= , , . . . ,p}be a finite family of
nonempty closed subsets ofX. An operatorf :pi=Bi→pi=Biis said to be acyclicϕ
-contractionifpi=Biis a cyclic representation ofXwith respect tof and
d(fx,fy)≤ϕd(x,y)
for allx∈Bi,y∈Bi+, ≤i≤p, whereBp+=B, andϕis a Boyd-Wong function.
Kirket al.[] established the following fixed point results for Geraghty, Boyd-Wong, and Caristi cyclicϕ-contractions.
Theorem . Let(X,d)be a complete metric space,and p a natural number.Suppose that a self-mapping f is a cyclicϕ-contraction onpi=Bi.Then there exists a unique element
z∈pi=Bisuch that f(z) =z.
Later, Pacurar and Rus [] introduced the notion ofweakly cyclicϕ-contraction. Kara-pinar [] improved the results in [] dropping the requirement of continuity. For more results in this direction, we refer to [–] and references therein.
We now introduce the following notion ofcyclic R-contraction mapping.
Definition . Let (X,d) be a metric space, and B,B, . . . ,Bp∈Pcl(X). A mappingf :
p i=Bi→
p
i=Biis said to be acyclic R-contractionif
(i) there exists∈RAwithran(d)⊆A;
(ii) pi=Biis a cyclic representation ofXwith respect tof, and
(iii) (d(fx,fy),d(x,y)) > for allx∈Bi,y∈Bi+,≤i≤p, whereBp+=B.
Meir-Keeler, Geraghty, and simulation contractions are typical examples ofR -contrac-tions that satisfy (). Consequently, the cyclic-R-contractions are a generalization of
2 Main results
We start with the following result.
Theorem . Let(X,d)be a complete metric space,and B,B, . . . ,Bp∈Pcl(X).Suppose
that a mapping f is a cyclic R-contraction onpi=Bi.Then there exists a unique element
z∈pi=Bisuch that f(z) =z.
Proof Letx be a given point in
p
i=Bi. Then there exists i in {, , . . . ,p} such that
x∈Bi. Sincef(Bi)⊂Bi+, we have thatf(x)∈Bi+. Thus, there existsx∈Bi+with f(x) =x. Similarly, there existsx ∈Bi+ withf(x) =x. Continuing in this way, we can construct a sequence inpi=Bibyxn=f(xn–) =fn(x)∈Bi+nfor alln∈N. Now, if xn+=xnfor somen∈N, then the result follows immediately. Suppose thatxn+=xnfor
alln∈N. Note that
d(fxn–,fxn),d(xn–,xn)
=d(xn,xn+),d(xn–,xn)
> for alln∈N. From property () of anR-function we have
lim
n→∞d(xn,xn+) = . (.)
We now show that{xn}is a Cauchy sequence. If not, then there existsL> such that for
anyk∈N, we can construct two subsequences{xmk}and{xnk}of{xn}withnk>mk≥k
satisfying
d(xmk,xnk) >L.
Without any loss of generality, we assume thatnkis the smallest integer greater thanmk
for which the last inequality holds. We can choosejk∈ {, , . . . ,p} such thatnk>mk>
mk–jkwithnkbelonging to the residue class ofmk–jk+ , and hencexmk–jkandxnklie in
different adjacently labeled setsBiandBi+for somei∈ {, , . . . ,p}. Thus,
d(xmk–jk,xnk) >L and d(xmk–jk,xnk–)≤L for allk∈N. (.)
By (.) we have
L< d(xmk–jk,xnk)
< d(xmk–jk,xnk–) +d(xnk–,xnk–) +d(xnk–,xnk)
≤L+d(xnk–,xnk–) +d(xnk–,xnk). (.)
Taking the limit ask→ ∞on both sides of this inequality, we have
lim
k→∞d(xmk–jk,xnk) =L. (.)
Similarly,
L<d(xmk–jk,xnk)
Also,
d(xmk–jk–,xnk–)≤d(xmk–jk–,xmk–jk) +d(xmk–jk,xnk) +d(xnk,xnk–). (.)
Taking the limit ask→ ∞on both sides of (.) and (.), we obtain that
lim
k→∞d(xmk–jk–,xnk–) =L. (.)
Now, since
d(xmk–jk,xnk) >L for allk∈N, lim
k→∞d(xmk–jk–,xnk–) = lim
k→∞d(xmk–jk,xnk) =L, and
d(fxmk–jk–,fxnk–),d(xmk–jk–,xnk–)
=d(xmk–jk,xnk),d(xmk–jk–,xnk–)
> ,
then by property () of anR-function, we conclude that =L> , a contradiction. Hence,
{xn} is a Cauchy sequence in X. Since (X,d) is complete, there existsγ ∈X such that limn→∞xn=γ. Since
p
i=Biis a cyclic representation ofXwith respect tof, there
ex-ist subsequences{xnp},{xnp+},{xnp+}, . . . ,{xnp+p–},{xnp+p–}, and{xnp+p} of{xn} such that
{xnp} ⊂B,{xnp+} ⊂B,{xnp+} ⊂ B, . . . ,{xnp+p–} ⊂Bp–,{xnp+p–} ⊂ Bp, and {xnp+p} ⊂
Bp+=B. Since eachBi,i∈ {, , , . . . ,p}, is a closed subset ofXandlimn→∞xn=γ, we
deduce thatγ ∈pi=Bi.
Note that for eachn∈N, there existsin∈ {, , . . . ,p}such thatxn–∈Bin–,xn∈Bin, and
γ ∈Bin. Thus,
d(fγ,fxn–),d(γ,xn–)
=d(fγ,xn),d(γ,xn–)
> for alln∈N.
Using property () of anR-function, we obtain thatlimn→∞d(fγ,xn) =d(fγ,γ) = .
Therefore,γ is a fixed point off inpi=Bi.
Uniqueness: Suppose that there exists another fixed point x∗ off in pi=Bi, that is,
d(x∗,γ) > andd(fγ,fx∗) =d(γ,x∗). Sincef is a cyclicR-contraction, we have
dfγ,fx∗,dγ,x∗> .
By property () of anR-function we have <d(x∗,γ) =limn→∞d(x∗,γ) = , a
contradic-tion. This establishes the result.
Example . LetX=Rbe endowed with the Euclidean metric d(x,y) =|x–y| for all x,y∈X. Suppose thatB= [–, ],B= [, ], andA=ran(d)⊂[,∞). Definef :
i=Bi→
i=Biand:A×A→Ras
f(x) = –x
and (t,s) = ⎧ ⎨ ⎩
s–t ift<s,
ift≥s.
Note that (X,d) is a complete space andBandBare closed inX. Ifx∈B, that is, –≤
x≤, then ≤–x≤ implies thatf(x)∈B. Similarly, ifx∈B, that is, ≤x≤, then
Further,(d(fx,fy),d(x,y)) =d(x,y) –d(fx,fy) =|x–y|> for allx∈B,y∈B. Thus,
all conditions of Theorem . are satisfied. Moreover,z= ∈i=Biis a fixed point off.
Example . LetX=Randd(x,y) =|x–y|for allx,y∈X. Suppose thatB={n}n∈N∪{},
B={–n–}n∈N∪{}, andA=ran(d)⊂[,∞). Definef :
i=Bi→
i=Biand:A×A→
Ras
f(x) = ⎧ ⎨ ⎩
–x ifx∈B,
–x
ifx∈B,
and (t,s) = ⎧ ⎨ ⎩
s–t ift<s,
ift≥s.
It is clear thatB andB are closed subsets of a complete metric space (X,d) such that
f(B)⊂Bandf(B)⊂B. Note that
d(fx,fy),d(x,y)=
d(x,y) –d(fx,fy)
=
|x–y|– x–y
>
|x–y|– x– x
– y =
|x–y|>
for allx∈B,y∈B. Hence, all conditions of Theorem . are satisfied, andz= ∈
i=Bi
is a fixed point off.
Remark . In this example, the mapping is a cyclicR-contraction that is neither a Meir-Keeler cyclic contraction nor a simulative cyclic contraction and hence neither a Boyd-Wong nor a Geraghty cyclic contraction. Indeed, if we taket=s= , then (ζ) fails.
Corollary . Let(X,d)be a complete metric space,and B,B, . . . ,Bp∈Pcl(X).Suppose
that a mapping f is a manageable cyclic contraction,or a simulative cyclic contraction,or a Geraghty cyclic contraction,or a Boyd-Wong cyclic contraction,or a Meir-Keeler cyclic contraction onpi=Bi.Then there exists a unique element z∈
p
i=Bisuch that f(z) =z.
3 Application to nonlinear Volterra integral equations
Motivated by the work in [], we obtain the existence and uniqueness of solutions for nonlinear Volterra integral differential equations.
Consider the following problem:
u(x,y) =f(x,y) + x
gx,y,ξ,u(ξ,y)dξ+ x
y
hx,y,σ,τ,u(σ,τ)dτdσ, (.)
where f ∈C(R+×R+,R),g∈C(E
×R+,R),h∈C(E×R+,R),E={f(x,y,s) :s≤x∈
[,∞),y∈[,∞)}, andE={f(x,y,s,t) :s≤x∈[,∞),t≤y∈[,∞)}.
LetXbe the space of functionsz∈C(R+×R+,R) satisfying|z(x,t)|=O(eλ(x+y)), where
λis a positive constant, that is,|z(x,y)| ≤Meλ(x+y)for some constantM> .
Note that (X, · X) is a Banach space. Define the mappingT:X→Xby
Tu(x,y)=f(x,y) + x
gx,y,ξ,u(ξ,y)dξ+ x
y
hx,y,σ,τ,u(σ,τ)dτdσ
for everyu∈X. It is easy to see thatu∗∈Xis a solution of problem (.) ifT(u∗) =u∗. Theorem . Suppose that problem(.)satisfies the following conditions:
(I)
g(x,y,ξ,u) –g(x,y,ξ,u¯) ≤h(x,y,ξ)|u–u¯|
and
h(x,y,σ,τ,u) –h(x,y,σ,τ,u¯) ≤h(x,y,σ,τ)|u–u¯|,
whereh∈C(E, [,∞))andh∈C(E, [,∞));
(II) There existα,βinXandα,βinRwithα≤α(x,t)≤β(x,t)≤β(x,t)such that
α(x,t)≤f(x,t) + x
gt,s,ξ,β(ξ,s)dξ+ x
y
ht,s,σ,τ,β(σ,τ)dτdσ
and
β(x,t)≥f(x,t) + x
gt,s,ξ,α(ξ,s)dξ+ x
y
ht,s,σ,τ,α(σ,τ)dτdσ
for allx,t∈[,∞); (III)
x
h(x,y,ξ)eλ(x+y)dξ+
x
y
h(x,y,σ,τ)eλ(σ+τ)dτdσ≤δeλ(x+y)
and
f(x,t) +
x
g(x,y,ξ, )dξ+ x
y
h(x,y,σ,τ, )dτdσ ≤δeλ(x+y)
for some nonnegative constantsδ,δ< ;
(IV) There existα,βinXsuch thatα(t)≤β(t),T(α(x,t))≤β(x,t),and
T(β(x,t))≥α(x,t).Then the integral Eq. (.)has a unique solutionu∗in
={u∈X:α(x,y)≤u(x,y)≤β(x,y)}.
Proof LetB={u∈X:u(x,t)≤β(x,t)}andB={u∈X:u(x,t)≥α(x,t)}. ThenBandB
are closed subsets of the complete metric spaceX. Ifu∈B, then by conditions (I), (II),
and (IV) we conclude thatT(u(x,t))≥α(x,t). Hence,Tu∈B. Similarly,u∈B implies
thatTu∈B, and henceT(B)⊂BandT(B)⊂B.
Ifu∈Bandv∈B, thenu(x,t)≤β(x,t)≤βandv(x,t)≥α(x,t)≥α. From conditions
(I) and (III) we obtain that
Thus,
Tu–TvX≤ςu–vXeλ(x+y), whereς=δ< .
Taking(t,s) =ςs–t, we have
Tu–TvX,u–vX
=ςu–vX–Tu–TvX> , u=v.
Consequently,Tis a cyclicR-contraction oni=Bi. By Theorem .,Thas a unique fixed
pointu∗ini=Bi∈, which is the solution of the integral-differential Eq. (.).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. The authors declare that they have no competing interests. All authors read and approved the final manuscript.
Author details
1Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, South Africa.2Department of
Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.3Department of Mathematics, Kano
University of Science and Technology, P.M.B. 3042, Wudil, Kano, Nigeria.
Acknowledgements
The authors are very indebted to both reviewers for a number of useful suggestions to improve this paper.
Received: 16 December 2015 Accepted: 4 May 2016
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