R E S E A R C H
Open Access
Existence of solutions of integral equations
via fixed point theorems
Selma Gülyaz
1and ˙Inci M Erhan
2**Correspondence: [email protected] 2Department of Mathematics, Atilim University, ˙Incek, Ankara 06836, Turkey
Full list of author information is available at the end of the article
Abstract
Existence and uniqueness of fixed points of a mapping defined on partially ordered G-metric spaces is discussed. The mapping satisfies contractive conditions based on certain classes of functions. The results are applied to the problems involving contractive conditions of integral type and to a particular type of initial value problems for the nonhomogeneous heat equation in one dimension. This work is a generalization of the results published recently in (Gordjiet al.in Fixed Point Theory Appl. 2012:74, 2012, doi:10.1186/1687-1812-2012-74) toG-metric space.
MSC: 47H10; 54H25
Keywords: fixed point theory;G-metric spaces; initial value problems
1 Introduction and preliminaries
One of the most common applications of the fixed point theory is the problem of existence and uniqueness of solutions of initial and boundary value problems for differential and integral equations. The number of studies dealing with such problems has increased con-siderably in the recent years. An important result in this direction has been reported by JJ Nieto and RR Lopez in []. They studied existence and uniqueness of fixed points on partially ordered metric spaces and applied their results to boundary value problems for ordinary differential equations. The research in this direction is advancing continuously and produces many interesting results; see [–].
In , Mustafa and Sims [] introduced the concept of aG-metric andG-metric space, which is a generalization of metric space. After this pioneering work,G-metric spaces and particularly fixed points of various maps onG-metric spaces have been intensively studied; see [–] and also [–].
In this work, we present some fixed point theorems onG-metric spaces and investigate the existence of solutions of an initial value problem for a partial differential equation, more precisely, a nonlinear one dimensional heat equation.
First, we briefly introduce some basic notions ofG-metric andG-metric space [].
Definition . LetXbe a nonempty set,G:X×X×X→[,∞) be a function satisfying the following conditions:
(G) G(x,y,z) = ifx=y=z,
(G) <G(x,x,y)for allx,y∈Xwithx=y,
(G) G(x,x,y)≤G(x,y,z)for allx,y,z∈Xwithz=y,
(G) G(x,y,z) =G(x,z,y) =G(y,z,x) =· · ·(symmetry in all variables),
(G) G(x,y,z)≤G(x,a,a) +G(a,y,z)for allx,y,z,a∈X(rectangle inequality).
Then the functionGis called aG-metric onXand the pair (X,G) is called aG-metric space.
Note that conditions (G) and (G) imply that
G(x,y,y) –G(x,z,z)≤G(y,z,z)
for allx,y,z∈X.
Definition .(see []) Let (X,G) be aG-metric space and let{xn}be a sequence inX.
. A pointx∈Xis said to be the limit of the sequence{xn}if
lim
n,m→∞G(x,xn,xm) =
and the sequence{xn}is said to beG-convergent tox.
. A sequence{xn}is called aG-Cauchy sequence if for everyε> , there is a positive
integerNsuch thatG(xn,xm,xl) <εfor alln,m,l≥N; that is, ifG(xn,xm,xl)→as n,m,l→ ∞.
. (X,G)is said to beG-complete (or a completeG-metric space) if everyG-Cauchy sequence in(X,G)isG-convergent inX.
Proposition .(see []) Let(X,G)be a G-metric space,{xn}be a sequence in X and x∈X. Then the following are equivalent:
. {xn}isG-convergent tox,
. G(xn,xn,x)→,asn→ ∞,
. G(xn,x,x)→,asn→ ∞,
. G(xn,xm,x)→,asn,m→ ∞.
Proposition .(see []) The following statements are equivalent on a G-metric space (X,G):
. The sequence{xn}isG-Cauchy.
. For everyε> ,there isN∈Nsuch thatG(xn,xm,xm) <ε,for alln,m≥N.
Definition .(see []) Let (X,G) and (X,G) beG-metric spaces. A functionf : (X,G)→ (X,G) is said to beG-continuous at a pointa∈Xif and only if for everyε> , there exists
δ> such thatx,y∈XandG(a,x,y) <δimpliesG(f(a),f(x),f(y)) <ε. A functionf is G-continuous onXif and only if it isG-continuous at all points inX.
Proposition . Let(X,G)be a G-metric space.Then the function G(x,y,z)is jointly con-tinuous in all of its three variables.
Definition . AG-metric space (X,G) is said to be symmetric if
G(x,x,y) =G(x,y,y)
It is obvious that for everyG-metric on the setX, the expression
dG(x,y) =G(x,x,y) +G(x,y,y)
is a standard metric onX.
Note that on a symmetricG-metric spacedG(x,y) = G(x,y,y), but on an asymmetric G-metric space, the inequality
G(x,y,y)≤dG(x,y)≤G(x,y,y)
holds for allx,y∈X.
Some examples ofG-metric spaces are presented below.
Example .
() Let(X,d)be a metric space. DefineGsby
Gs(x,y,z) =d(x,y) +d(y,z) +d(x,z)
for allx,y,z∈X. Then clearly,(X,Gs)is a symmetricG-metric space. Note that if
X=Randdis the Euclidean metric onX, thenGsmay be interpreted as the perimeter of the triangle with verticesx,y,z.
() LetX={a,b}. Define
G(a,a,a) =G(b,b,b) = ,
G(a,a,b) = , G(a,b,b) = ,
and extendGtoX×X×Xby using the symmetry in the variables. Then(X,G)is an asymmetricG-metric space.
2 The main results
The attempts to generalize the contractive conditions on the maps resulted in defini-tions of various classes of funcdefini-tions. Altering distance funcdefini-tions defined in [], weak
ψ-contraction presented in [] are some of these classes. In this study we employ con-tractive conditions based on the following classes of functions.
Let denote the class of the functionsψ: [, +∞)→[, +∞) satisfying the following conditions:
. ψis nondecreasing,
. ψis sub-additive, that is,ψ(s+t)≤ψ(s) +ψ(t), . ψis continuous,
. ψ(t) = ⇐⇒t= .
LetSdenote the class of the functionsβ: [, +∞)→[, ] such that for any bounded sequence{tn}of positive real numbers,β(tn)→⇒tn→.
Lemma . Let(X,G)be a G-metric space and let{xn}be a sequence in X such that the sequence{G(xn+,xn+,xn)}of nonnegative real numbers is decreasing and
lim
n→∞G(xn+,xn+,xn) = . ()
When the subsequence{xn}is not G-Cauchy,then there existε> and two sequences{mk} and{nk}of positive integers such that the sequences
G(xmk,xnk,xmk+), G(xmk,xnk+,xmk+),
G(xmk–,xnk,xmk), G(xmk–,xnk+,xmk)
()
converge toεas k→ ∞.
Proof From the Proposition ., if{xn}is notG-Cauchy, then there existε> and two sequences{mk}and{nk}ofNsatisfyingnk>mk>kfor which
G(xmk,xnk,xmk+)≥ε, ()
wherenkis chosen as the smallest integer satisfying (). In other words,
G(xmk,xnk–,xmk+) <ε. ()
By (), (), and using the symmetry (G) and the rectangle inequality (G), we easily derive
ε≤G(xmk,xnk,xmk+) =G(xmk,xmk+,xnk)
≤G(xmk,xmk+,xnk–) +G(xnk–,xnk–,xnk–)
+G(xnk–,xnk–,xnk)
<ε+G(xnk–,xnk–,xnk–) +G(xnk–,xnk–,xnk). ()
Taking the limitk→ ∞in () and using (), we obtain
lim
n→∞G(xmk,xnk,xmk+) =ε. ()
In addition, from the inequalities
G(xmk,xnk,xmk+)≤G(xmk,xmk+,xnk+) +G(xnk+,xnk+,xnk)
and
G(xmk,xmk+,xnk+)≤G(xmk,xmk+,xnk) +G(xnk,xnk,xnk+)
we deduce
lim
upon taking the limitk→ ∞and using () and (). In a similar way it can be shown that
the remaining two sequences in () also tend toε.
We state next our first main theorem about the existence of fixed points on partially orderedG-metric spaces.
Theorem .(Existence theorem) Let(X,)be a partially ordered set, (X,G)be a G-complete metric space and f :X→X be a nondecreasing function.Suppose that there exist functionsβ∈S andψ∈such that
ψGfx,fy,fx≤βψG(x,y,fx)ψG(x,y,fx) ()
for all x,y∈X with xy.Assume also that for any increasing sequence{xn}in X converging to x,
xnx for each n≥. ()
If there exists x∈X such that xfx,then f has a fixed point.
Proof By the assumption, there existsx∈Xsuch thatxfx. We construct a sequence {xn}in the following way:
fxn=xn+ for alln∈N∪ {}. ()
Sincef is nondecreasing, we havefxnfxn+for eachn∈N∪ {}. Hence,{xn}is a non-decreasing sequence. If xn =xn+ for some n∈N∪ {}, then xn is the fixed point
of f. Assume thatxn=xn+ for all n∈N∪ {}. Then, by the definition ofψ, we have ψ(G(fxn,fxn+,fxn)) > for alln∈N∪ {}. Takingx=xnandy=xn+in () we get
ψG(xn+,xn+,xn+)
=ψGfxn,fxn+,fxn
≤βψG(xn,xn+,xn+)
ψG(xn,xn+,xn+)
≤ψG(xn,xn+,xn+)
.
Thus, the sequence{ψ(G(xn,xn+,xn+))}is nonincreasing and bounded below by .
Con-sequently,limn→∞ψ(G(xn,xn+,xn+)) =L≥. We will show thatL= . Assume to the
contrary thatL> . Due to (), we have
ψ(G(fxn,fxn+,fxn))
ψ(G(fxn–,fxn,fxn–))
≤βψG(xn,xn+,xn+)
≤
for eachn≥, which yields
lim
n→∞β
ψG(xn,xn+,xn+)
= .
We show next that{fxn}is aG-Cauchy sequence. Suppose that{fxn}is notG-Cauchy. By Lemma ., there existε> and two sequences{mk}and{nk}of positive integers such that the four sequences
Gfxmk,fxnk,f
x mk
, Gfxmk,fxnk+,f
x mk
,
Gfxmk–,fxnk,f
x mk–
, Gfxmk–,fxnk+,f
x mk–
approachεaskgoes to infinity. Settingx=xmkandy=xnk+in () and regarding (), we
get
ψGfxmk,fxnk+,f
x mk
≤βψG(xmk,xnk+,fxmk)
ψG(xmk,xnk+,fxmk)
=βψGfxmk–,fxnk,f
x mk–
ψGfxmk–,fxnk,f
x mk–
≤ψGfxmk–,fxnk,f
x mk–
,
and thus
ψ(G(fxmk,fxnk+,f
x mk))
ψ(G(fxmk–,fxnk,fxmk–))
≤βψGfxmk–,fxnk,f
x mk–
≤.
This inequality implieslimk→∞β(ψ(G(fxmk–,fxnk,f
x
mk–))) = . Sinceβ∈S, we
con-clude that
lim
k→∞ψ
Gfxmk–,fxnk,f
x mk–
= .
By the fact thatψis a continuous function,ψ(ε) = and henceε= , which contradicts the assumptionε> . Therefore,{fxn}is aG-Cauchy sequence in (X,G). Since (X,G) is a completeG-metric space, there existsz∈Xsuch that
lim
n→∞fxn=z=nlim→∞xn+. ()
Next, we show thatzis a fixed point off. Substitutingx=xn+andy=zin (), by the virtue
of (), we get
ψG(xn+,fz,xn+)
=ψGfxn+,fz,fxn+
≤βψG(xn+,z,fxn+)
ψG(xn+,z,fxn+)
≤ψG(xn+,z,fxn+)
=ψG(xn+,z,xn+)
for eachn≥. Passing to the limitn→ ∞in the above inequality and regarding () and the continuity ofψ, we end up with
that is
fz=z. ()
This completes the proof of the theorem.
Next, we discuss the uniqueness conditions for the fixed point of the map in Theo-rem .. A condition for the uniqueness can be stated as follows:
(i) Every pair of elements inXhas a lower bound or an upper bound.
On the other hand, it can be proved that condition (i) is equivalent to condition
(ii) For everyx,y∈X, there existsz∈Xwhich is comparable to bothxandy.
Accordingly, we prove the following uniqueness theorem.
Theorem .(Uniqueness theorem) Let X satisfies condition(ii)and the hypotheses of Theorem.hold.Ifβ∈S is continuous,then the fixed point of f is unique.
Proof Existence of a fixed point is provided by Theorem .. Assume thatyandzare two different fixed points off. From condition (ii), there existsx∈Xwhich is comparable toy andz. The monotonicity off implies thatfn(x) is comparable tofn(y) =yandfn(z) =zfor n≥. Moreover, using the fact thatzis a fixed point offand condition () of Theorem . we get
ψGz,z,fn(x)=ψGfn(z),fn(z),fn(x)
=ψGffn–(z),ffn–(z),ffn–(x), ()
and thus
ψGz,z,fn(x)
≤βψGfn–(z),fn–(z),ffn–(x)ψGfn–(z),fn–(z),ffn–(x)
≤ψGfn–(z),fn–(z),fn–(x)=ψGz,z,fn–(x). ()
Therefore, the sequence{αn}defined byαn=ψ(G(z,z,fn(x))) is nonnegative and nonin-creasing and hence,
lim
n→∞ψ
Gz,z,fn(x)=α≥.
To show that α = , we assume the contrary, that is, α > . Since β is continuous,
limn→∞β(αn) =β(α) =λ≥. Lettingn→ ∞in (), we get
α≤λα≤α,
which results inλ= . Sinceβ∈S, we deduce
α= lim
n→∞αn=nlim→∞ψ
By similar arguments, we obtain
lim
n→∞ψ
Gy,y,fn(x)= .
Employing the rectangle inequality (G), we have
G(z,z,y)≤Gz,z,fn(x)+Gfn(x),fn(x),y.
Since the inequalityG(x,x,y)≤G(x,y,y) holds for anyx,y∈Xin both symmetric and asymmetricG-metric spaces, we have
G(z,z,y)≤Gz,z,fn(x)+ Gfn(x),y,y.
From the fact thatψis nondecreasing and sub-additive, we conclude
ψG(z,z,y)≤ψGz,z,fn(x)+ ψGfn(x),y,y.
Letting n → ∞ in the above inequality we obtain ψ(G(z,z,y)) = , which implies G(z,z,y) = and hence,z=y. This completes the proof.
If in Theorem ., we takeψ as the identity function onX we deduce the following particular result.
Corollary . Let(X,)be a partially ordered set(X,G)be a G-complete metric space. Let f :X→X be a nondecreasing map.Suppose that there existsβ∈S such that
Gfx,fy,fx≤βG(x,y,fx)G(x,y,fx)
holds for all x,y∈X with xy.Assume that either f is continuous or that X satisfies the following condition:if an increasing sequence xnin X converges to x,then xnx for each n≥.If in addition,there exists x∈X such that xfxthen f has a fixed point.
Remark . In a recent paper by Karapınar and Samet [] it has been proven that ifd is a metric onXandψ∈, then the functiondψ =ψ(d(x,y)) is also a metric onX. In
a similar way, it can be shown that the functionGψ =ψ(G(x,y,z)), whereψ∈ is also
aG-metric. Employing this definition, the contractive condition in Theorem . can be simplified considerably. More precisely, it becomes
Gψ
fx,fy,fx≤βGψ(x,y,fx)
Gψ(x,y,fx) ()
for allx,y∈Xwithxy.
As a common application of fixed point theorems one can give integral type contractive conditions. In many articles authors apply their results to maps which are defined by inte-grals [, ]. In what follows, we apply our results to maps defined by Lebesgue inteinte-grals. LetYbe a set of functionsχ:R+→R+satisfying the following conditions:
() χis summable on each compact of subset ofR+; () χis sub-additive;
() χ(t)dt> for each> .
A sub-additive integrable function is defined as follows:
Definition . The functionχ:R+→R+is called a sub-additive integrable function if,
for anya,b∈R+, we have
a+b
χ(t)dt≤
a
χ(t)dt+
b
χ(t)dt.
For the class of functions inY, we state the following fixed point theorem.
Theorem . Let(X,)be a partially ordered set and let(X,G)be a complete G-metric space.Let f :X→X be a nondecreasing function.Suppose that there exist functionsβ∈S andψ∈such that forχ∈Y
ψ(G(fx,fy,fx))
χ(t)dt≤β
ψ(G(x,y,fx))
χ(t)dt
ψ(G(x,y,fx))
χ(t)dt ()
holds for all x,y∈X with xy.Assume that either f is continuous or X satisfies the condi-tion:if an increasing sequence{xn}converges to x,then xnx for each n≥.If there exists x∈X such that xfxthen f has a fixed point.
Proof Forχ∈Y, define the function:R+→R+ by (x) =x
χ(t)dt. Observe that ∈. The inequality () can be written as
ψGfx,fy,fx≤βψG(x,y,fx)ψG(x,y,fx). () Let◦ψ=ψ, where clearlyψ∈. Then we have
ψ
Gfx,fy,fx≤βψ
G(x,y,fx)ψ
G(x,y,fx).
Then the conditions of Theorem . are satisfied and thusf has a fixed point, which
com-pletes the proof.
The particular case in which the functionψ is the identity function onXcan be stated as a corollary.
Corollary . Let(X,)be a partially ordered set and, (X,G)be a complete G-metric space.Let f :X→X be a nondecreasing map.Suppose that there existsβ∈S such that for
χ∈Y the inequality
G(fx,fy,fx)
χ(t)dt≤β
G(x,y,fx)
χ(t)dt
G(x,y,fx)
χ(t)dt,
holds for all x,y∈X with xy.Assume that if an increasing sequence{xn}in X converges to x then xnx for each n≥.If there exists x∈X such that xfxthen f has a fixed
3 Application
As an application of the existence and uniqueness Theorems . and ., we consider the problem of existence and uniqueness of an initial value problem defined by a nonlinear heat equation in one dimension. Such an initial value problem is defined as follows:
⎧ ⎨ ⎩
ut(x,t) =uxx(x,t) +F(x,t,u,ux), –∞<x<∞, <t<T,
u(x, ) =ϕ(x), –∞<x<∞, ()
whereϕis assumed to be continuously differentiable,ϕandϕbounded, andF(x,t,u,ux) a continuous function.
Definition . A solution of the initial value problem () is any functionu=u(x,t) de-fined inR×I, whereI= (,T], satisfying the equation and the condition in () and also the conditions:
(a) u∈C(R×I),
(b) ut,uxanduxx∈C(R×I),
(c) uanduxare bounded inR×I.
Consider the spacedefined as
=v(x,t) :v,vx∈C(R×I) andv<∞
,
where the norm on this space is defined as
v= sup
x∈R,t∈I
v(x,t)+ sup
x∈R,t∈I
vx(x,t). ()
The setendowed with the norm·defined in () is a Banach space. Define aG-metric onas follows:
G(u,v,w) = sup
x∈R,t∈I
u(x,t) –v(x,t)+ sup
x∈R,t∈I
ux(x,t) –vx(x,t)
+ sup
x∈R,t∈I
v(x,t) –w(x,t)+ sup
x∈R,t∈I
vx(x,t) –wx(x,t)
+ sup
x∈R,t∈I
u(x,t) –w(x,t)+ sup
x∈R,t∈I
ux(x,t) –wx(x,t).
Then (,G) is a completeG-metric space. Define also a partial orderonas
u,v∈, uv ⇐⇒ u(x,t)≤v(x,t), ux(x,t)≤vx(x,t)
for anyx∈Randt∈I. It can easily be observed that (,) satisfies condition (i) of unique-ness, that is, every pair of elements inhas a lower bound or an upper bound. Indeed, for anyu,v∈,max{u,v}andmin{u,v}are the lower and upper bounds foruandv, respec-tively. Let{vn} ⊆be a monotone nondecreasing sequence which converges tovin. Then, for anyx∈Randt∈I, we have
and
vx(x,t)≤vx(x,t)≤ · · · ≤vnx(x,t)≤ · · ·.
Moreover, since the sequences{vn(x,t)}and{vnx(x,t)}of real numbers converge tov(x,t) andvx(x,t), respectively, we have for allx∈R,t∈Iandn≥ the inequalitiesvn(x,t)≤ v(x,t) andvnx(x,t)≤vx(x,t) hold. Therefore,vn≤vfor alln≥ and hence, (,) with the G-metric defined above satisfies condition ().
We next define a lower solution for the initial value problem, which is needed in the existence-uniqueness proof.
Definition . A lower solution of the initial value problem () is a functionu∈such that
⎧ ⎨ ⎩
ut(x,t)≤uxx(x,t) +F(x,t,u,ux), –∞<x<∞, <t<T,
u(x, )≤ϕ(x), –∞<x<∞,
where the functionϕis continuously differentiable and bothϕandϕare bounded, the set
is the set defined above andF(x,t,u,ux) is a continuous function.
We state the following existence-uniqueness theorem for the solution of the initial value problem ().
Theorem . Consider the problem()and,assume that F:R×I×R×R→Ris a continuous function.Suppose that the following conditions hold:
() For anyc> ,the functionF(x,t,s,p),where|s|<cand|p|<cis uniformly Hölder continuous inxandt,for each compact subset ofR×I.
() There exists a constantcF≤(T+ π–
T)–such that
≤F(x,t,s,p) –F(x,t,s,p)≤cFln(s–s+p–p+ ),
for all(s,p)and(s,p)inR×Rwiths≤sandp≤p. () Fis bounded for boundedsandp.
Then the existence of a lower solution for the initial value problem()provides the exis-tence of the unique solution of the problem().
Proof Observe that the problem () is equivalent to the integral equation
u(x,t) =
∞
–∞k(x–ξ,t)ϕ(ξ)dξ
+
t
∞
–∞
k(x–ξ,t–τ)Fξ,τ,u(ξ,τ),uξ(ξ,τ)
dξdτ,
for allx∈Rand <t≤T, where the functionk(x,t) is the Green’s function of the problem defined as
k(x,t) = √ πtexp
–x
t
for allx∈Rand <t. The initial value problem () has a unique solution if and only if the above integral equation has unique solutionusuch thatuanduxare continuous and bounded for allx∈Rand <t≤T. Define a mappingf:→by
(fu)(x,t) =
∞
–∞k(x–ξ,t)ϕ(ξ)dξ
+
t
∞
–∞
k(x–ξ,t–τ)Fξ,τ,u(ξ,τ),uξ(ξ,τ)
dξdτ
for allx∈Randt∈I. Clearly, the fixed pointu∈off is a solution of the problem (). We will show that the conditions of Theorems . and . are satisfied. Note that the mappingf is nondecreasing since, by condition () in the statement of the theorem, for uv, that is,u≥vandux≥vx, we have
Fx,t,u(x,t),ux(x,t)
≥Fx,t,v(x,t),vx(x,t)
.
Sincek(x,t) > for all (x,t)∈R×I,
(fu)(x,t) =
∞
–∞k(x–
ξ,t)ϕ(ξ)dξ
+
t
∞
–∞
k(x–ξ,t–τ)Fξ,τ,u(ξ,τ),uξ(ξ,τ)
dξdτ
≥ ∞
–∞k(x–
ξ,t)ϕ(ξ)dξ
+
t
∞
–∞
k(x–ξ,t–τ)Fξ,τ,v(ξ,τ),vξ(ξ,τ)
dξdτ
= (fv)(x,t)
for allx∈Randt∈I. In addition, we have
(fu)(x,t) – (fv)(x,t)
≤
t
∞
–∞
k(x–ξ,t–τ)Fξ,τ,u(ξ,τ),uξ(ξ,τ)
–Fξ,τ,v(ξ,τ),vξ(ξ,τ)dξdτ
≤
t
∞
–∞k(x–ξ,t–τ)cF
lnu(ξ,τ) –v(ξ,τ) +uξ(ξ,τ) –vξ(ξ,τ) +
dξdτ
≤cFln
G(u,v,fu) + t
∞
–∞
k(x–ξ,t–τ)dξdτ
≤cFln
G(u,v,fu) + T, ()
where we have used the facts that
lnu(ξ,τ) –v(ξ,τ) +uξ(ξ,τ) –vξ(ξ,τ) +
≤ln
sup ξ∈R,τ∈I
u(ξ,τ) –v(ξ,τ)+ sup ξ∈R,τ∈I
uξ(ξ,τ) –vξ(ξ,τ)+
and
t
∞
–∞
k(x–ξ,t–τ)dξdτ=T. ()
On the other hand, since (,) satisfies condition (i), it also satisfies condition (ii), as they are equivalent. Therefore, eithervandfuare comparable or there existsw∈which is comparable to bothvandfu. In either case, by means of similar calculations, it can be shown that
(fv)(x,t) –fu(x,t)≤cFln
G(u,v,fu) + T ()
and, similarly,
(fu)(x,t) –fu(x,t)≤cFln
G(u,v,fu) + T, ()
for alluv. Moreover, by using differentiation under integral sign and employing again condition () of the theorem, we compute
∂∂fux(x,t) – ∂fv
∂x(x,t)
≤cFln
G(u,v,fu) + t
∞
–∞
∂∂kx(x–ξ,t–τ)dξdτ ≤cFln
G(u,v,fu) + π–T, ()
∂∂fvx(x,t) –∂f
u
∂x (x,t)
≤cFln
G(u,v,fu) + t
∞
–∞
∂∂kx(x–ξ,t–τ)dξdτ ≤cFln
G(u,v,fu) + π–T, ()
and
∂∂fux(x,t) – ∂f
u
∂x (x,t)
≤cFln
G(u,v,fu) + t
∞
–∞
∂∂kx(x–ξ,t–τ)dξdτ ≤cFln
G(u,v,fu) + π–T ()
are satisfied. Combining (), (), and () with (), (), and (), we deduce
Gfu,fv,fu≤cF
T+ π–TlnG(u,v,fu) + ≤lnG(u,v,fu) + ,
which implies
lnGfu,fv,fu+ ≤lnlnG(u,v,fu) + =ln(ln(G(u,v,fu) + ))
ln(G(u,v,fu) + ) ln
G(u,v,fu) + .
xand hence,β∈S. Finally, letα(x,t) be a lower solution for (). We show thatα≤fα. Upon integrating
α(ξ,τ)k(x–ξ,t–τ)τ–αξ(ξ,τ)k(x–ξ,t–τ)
ξ+
α(ξ,τ)kξ(x–ξ,t–τ)
ξ
≤Fξ,τ,α(ξ,τ),αξ(ξ,τ)
k(x–ξ,t–τ)
over –∞<ξ<∞and <τ<t, we obtain
α(x,t)≤
∞
–∞
k(x–ξ,t)ϕ(ξ)dξ
+
t
∞
–∞k(x–
ξ,t–τ)Fξ,τ,α(ξ,τ),αξ(ξ,τ)
dξdτ
= (fα)(x,t)
for allx∈Randt∈(,T]. Therefore, by Theorems . and .,f has a unique fixed point.
This completes the proof.
Competing interests
The authors declare that they have no competing interests. Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript. Author details
1Department of Mathematics, Cumhuriyet University, Sivas, Turkey.2Department of Mathematics, Atilim University, ˙Incek, Ankara 06836, Turkey.
Acknowledgements
The authors are thankful to the referees for careful reading of the manuscript and the valuable comments and suggestions for the improvement of the paper.
Received: 28 November 2013 Accepted: 14 March 2014 Published:03 Apr 2014
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