We introduce the notion of modified F-contractive mappings in the setting of complete metric-like spaces and we investigate the existence and uniqueness of fixed point of such mappings. The presented results unify, extend, and improve several results in the related literature.

1. Introduction and Preliminaries

Throughout the paper, N and N0 denote the set of positive integers and the set of nonnegative integers, respectively. Similarly, R, R+, and R0+ represent the set of real, positive real, and nonnegative real numbers, respectively. In what follows, we recall the notion of partial metric which is an interesting generalization of the notion of metric.

Definition 1 (see [<xref ref-type="bibr" rid="B13">1</xref>]).

Let X be a nonempty set. A mapping p:X×X→R0+ is said to be a partial metric on X if for all x,y,z∈X the following conditions are satisfied:

x=y if and only if p(x,x)=p(x,y)=p(y,y);

p(x,x)≤p(x,y);

p(x,y)=p(y,x);

p(x,z)≤p(x,y)+p(y,z)-p(y,y).

In this case, the pair (X,d) is called a partial metric space (PMS).

Notice that the function dp:X×X→R0+ defined by
(1)dp(x,y)=2p(x,y)-p(x,x)-p(y,y)
satisfies the conditions of a metric on X. Each partial metric p on X generates a T0 topology τp on X, whose base is a family of open p-balls Bp(x,ε):x∈X,ε>0 where Bp(x,ε)=y∈X:p(x,y)≤p(x,x)+ε for all x∈X and ε>0. Consequently, several topological concepts can be easily defined as follows.

A sequence {xn} in the PMS (X,p) converges to the limit x if p(x,x)=limn→∞p(x,xn) and is said to be a Cauchy sequence if limn,m→∞p(xn,xm) exists and is finite. A PMS (X,p) is called complete if every Cauchy sequence {xn} in X converges with respect to τp, to a point x∈X such that p(x,x)=limn,m→∞p(xn,xm). For more details, see, for example, [1–12] and the related references therein.

Let (X,p) be a complete PMS. Then consider the follwing.

If p(x,y)=0, then x=y.

If x≠y, then p(x,y)>0.

A sequence {xn} is a Cauchy sequence in the PMS (X,p) if and only if it is a Cauchy sequence in the metric space (X,dp).

A PMS (X,p) is complete if and only if the metric space (X,dp) is complete. Moreover
(2)limn→∞dpx,xn=0⟺p(x,x)=limn→∞p(x,xn)=limn,m→∞pxn,xm.

Assume xn→z as n→∞ in a PMS (X,p) such that p(z,z)=0. Then limn→∞p(xn,y)=p(z,y) for every y∈X.

Now, we state the definition of metric-like (dislocated) function that was first introduced by Hitzler [13] and reintroduced later by Amini-Harandi [14].

Definition 3 (see [<xref ref-type="bibr" rid="B3">13</xref>]).

Let X be a nonempty set. A mapping σ:X×X→R0+ is said to be metric-like (dislocated) on X, if for all x,y,z∈X the following conditions are satisfied:

if σ(x,y)=0 then x=y;

σ(x,y)=σ(y,x);

σ(x,y)≤σ(x,z)+σ(z,y).

The pair (X,σ) is called dislocated (metric-like) space.Remark 4 (see [<xref ref-type="bibr" rid="B10">14</xref>]).

Every partial metric space is a metric-like space.

A sequence {xn}n=1∞, in a metric-like space (X,σ),

converges to x∈X if limn→∞σ(xn,x)=σ(x,x),

is called Cauchy in (X,σ), if limn,m→∞σ(xn,xm) exists and is finite.

A metric-like space(X,σ) is said to be complete if and only if every Cauchy sequence {xn}n=1∞ in X converges to x∈X so that
(3)limn,m→∞σ(xn,xm)=limn→∞σ(xn,x)=σ(x,x).

We recall next some basic definitions and crucial results on the topic. In this paper, we follow the notations of Amini-Harandi [14].

Definition 5 (see [<xref ref-type="bibr" rid="B10">14</xref>]).

Let (X,σ) be a metric-like space and U a subset of X. One says that U is σ-open subset of X, if for all x∈X there exists r>0 such that Bσ(x,r)⊆U. Also, V⊆X is called a σ-closed subset of X if (X∖V) is σ-open subset of X.

Lemma 6 (see [<xref ref-type="bibr" rid="B12">15</xref>]).

Let (X,σ) be a metric-like space. Then,

if σ(x,y)=0 then σ(x,x)=σ(y,y)=0;

if {xn} is a sequence such that limn→∞σ(xn,xn+1)=0, then one has
(4)limn→∞σ(xn,xn)=limn→∞σ(xn+1,xn+1)=0;

if x≠y then σ(x,y)>0;

σ(x,x)≤(2/n)∑i=1i=nσ(x,xi) holds for all xi,x∈X where 1≤i≤n;

if {xn} is a sequence in a σ-closed subset V of X with xn→x as n→∞, then x∈V;

if {xn} is a sequence in X such that xn→x as n→∞ and σ(x,x)=0, then limn→∞σ(xn,y)=σ(x,y) for all y∈X.

Definition 7.

Let (X,σ) and (Y,ρ) be metric-like spaces and {xn}n=1∞ a sequence in X such that xn→x. A mapping f:X→Y is said to be continuous at a point x∈X if f(xn)→f(x).

In this paper, we modify the notion of F-contraction that was introduced by Wardowski [16] and investigate the existence of a fixed point of such modified F-contractive mapping in the context of complete metric-like spaces. We consider also an example to illustrate the main result.

2. Main Result

In this section we present our main theorems. We start with the following definition.

Definition 8.

Let (X,σ) be a metric-like space. A self-mapping T:X→X is said to be modified F-contraction of type I if there exists τ>0 such that
(5)12σx,Tx<σ(x,y)⟹τ+FσTx,Ty≤αFσx,y+βFσx,Tx+γF(σ(y,Ty)),
for all x,y∈X with x≠y where γ∈[0,1) and α,β∈[0,1] are real numbers such that α+β+γ=1 and F:R+→R is a mapping satisfying the following conditions:

F is strictly increasing; that is, for all α,β∈R+ such that α<β, F(α)<F(β),

for any sequence {αn}n=1∞ of positive real numbers limn→∞αn=0 if and only if limn→∞F(αn)=-∞.

Theorem 9.

Let (X,σ) be a complete metric-like space and T a modified F-contraction of type I. Then, T has a fixed point v∈X; that is, Tv=v.

Proof.

For an arbitrary x∈X, we construct a sequence {xn} in the following way:
(6)x=x0,xn+1=Txn,∀n∈N0.
If there exists n0∈N such that σ(xn0,xn0+1)=0, then v=xn0 is the desired fixed point of T which completes the proof. Consequently, we suppose that 0<σ(xn,xn+1) for every n∈N0. Thus, we have
(7)12σ(xn,Txn)=12σ(xn,xn+1)<σ(xn,xn+1),∀n∈N.
By the hypothesis of the theorem, we have
(8)τ+FσTxn,Txn+1≤αFσxn,xn+1+βFσxn,Txn+γFσxn+1,Txn+1,
and hence
(9)τ+1-γFσxn+1,xn+2≤α+βFσxn,xn+1.
Since α+β+γ=1, we get
(10)Fσxn+1,xn+2≤Fσxn,xn+1-τα+β<F(σ(xn,xn+1)).
So from (F1), we conclude that
(11)σ(xn+1,xn+2)<σ(xn,xn+1),∀n∈N.
Therefore, {σ(xn,xn+1)}n=1∞ is a decreasing sequence of real numbers which is bounded below. This implies that {σ(xn,xn+1)}n=1∞ converges and
(12)limn→∞σ(xn,xn+1)=β=inf{σ(xn,xn+1):n∈N}.
We will show that β=0. Suppose, on the contrary, that β>0. For every ε>0 there exists m∈N, such that
(13)σ(xm,Txm)<β+ε.
Hence from (F1), we get
(14)Fσxm,Txm<Fβ+ε.
On the other hand from (7), we have
(15)12σ(xm,Txm)<σ(xm,Txm).
Due to assumption of the theorem, we obtain
(16)τ+FσTxm,T2xm≤αFσxm,Txm+βFσxm,Txm+γFσTxm,T2xm,
which is equivalent to
(17)τ+1-γFσTxm,T2xm<α+βFσxm,Txm.
Consequently, we derive that
(18)FσTxm,T2xm<Fσxm,Txm-τα+β,
since α+β+γ=1. On account of (7), we have (1/2)σ(Txm,T2xm)<σ(Txm,T2xm); thus by assumption of the theorem, we have
(19)τ+FσT2xm,T3xm≤αFσTxm,T2xm+βFσTxm,T2xm+γF(σ(T2xm,T3xm)),
which yields
(20)τ+1-γFσT2xm,T3xm≤α+βFσTxm,T2xm.
Owing to the fact that α+β+γ=1, we obtain that
(21)FσT2xm,T3xm≤FσTxm,T2xm-τα+β.
Now by using (14) and continuing in the same way as in the derivation of (18) and (21), we deduce
(22)FσTnxm,Tn+1xm≤FσTnxm,Tn-1xm-τα+β≤FσTn-1xm,Tn-2xm-2τα+β⋮≤FσTxm,xm-nτα+β<F(β+ε)-nτα+β.
This implies that limn→∞F(σ(Tnxm,Tn+1xm))=-∞. Regarding (F2), we have limn→∞σ(Tnxm,Tn+1xm)=0, and thus, there exists N1∈N such that σ(Tnxm,Tn+1xm)<β, ∀n≥N1. Therefore, from (6), we get
(23)σ(xm+n,Txm+n)<β,∀n≥N1.

This contradicts the definition of β given in (12). Then we get β=0 and from (12) we conclude
(24)limn→∞σ(xn,xn+1)=0.
In the next step, we claim that
(25)limn,m→∞σ(xn,xm)=0.
Suppose, on the contrary, that there exist ε>0 and sequences {p(n)}n=1∞ and {q(n)}n=1∞ of natural numbers such that
(26)pn>qn>n,σxpn,xqn≥ε,σxpn-1,xqn<ε,∀n∈N.
By triangular inequality, we have
(27)ε≤σ(xp(n),xq(n))≤σ(xp(n),xp(n)-1)+σ(xp(n)-1,xq(n))≤σ(xp(n),xp(n)-1)+ε=σ(xp(n)-1,Txp(n)-1)+ε,∀n∈N.
It follows from (24), (27), and Squeezing Theorem that
(28)limn→∞σ(xp(n),xq(n))=ε.
From (24), (26), and (28), there exists N2∈N such that
(29)12σxpn,Txpn<12ε<ε≤σxpn,xqn<2ε,∀n>N2.
Hence from (29), (F1), and the hypothesis of the theorem, we have
(30)τ+FσTxpn,Txqn≤αFσxpn,xqn+βFσxpn,Txpn+γFσxqn,Txqn<αF(2sε)+βFσxpn,Txpn+γFσxqn,Txqn,∀n>N2.
From (24) and (F2) it follows that
(31)limn→∞FσTxpn,Txqn=-∞,
and hence we get
(32)limn→∞σ(Txp(n),Txq(n))=0⟺limn→∞σ(xp(n)+1,xq(n)+1)=0.
However, this contradicts the relation (26). Hence limm,n→∞σ(xn,xm)=0. Therefore {xn}n=1∞ is a Cauchy sequence in X. By the completeness of (X,σ) there exists v∈X such that
(33)σ(v,v)=limn→∞σ(xn,v)=limn,m→∞σ(xn,xm)=0.
Next, we will prove that, for every n∈N,
(34)12σxn,Txn<σxn,vor12σTxn,T2xn<σTxn,v,∀n∈N.
Arguing by contradiction, we assume that there exists m∈N such that
(35)12σxm,Txm≥σxm,v,12σ(Txm,T2xm)≥σ(Txm,v).
From (18) and (F1), we have
(36)σ(Txm,T2xm)<σ(xm,Txm).
It follows from (35) and (36) that
(37)xm,Txm≤σ(xm,v)+σv,Txm≤12σ(xm,Txm)+12σ(Txm,T2xm)<12σ(xm,Txm)+12σ(xm,Txm)=σ(xm,Txm).
Obviously, this is a contradiction. Hence, inequality (34) is satisfied. Regarding the assumption of the theorem, (34) implies that either
(38)τ+FσTxn,Tv≤αFσxn,v+βFσxn,Txn+γFσv,Tv,
or
(39)τ+FσT2xn,Tv≤αFσTxn,v+βFσTxn,T2xn+γF(σ(v,Tv)),
for every n∈N. In the first case, because of (F2), the limits in (24) and (33) imply
(40)limn→∞Fσxn,v=-∞,limn→∞Fσxn,Txn=-∞.
Thus, letting n→∞ in (38), we conclude that
(41)limn→∞FσTxn,Tv=-∞.
Again by (F2), we observe that
(42)limn→∞σTxn,Tv=0.
On the other hand, from (6), we have
(43)σv,Tv≤σ(v,Txn)+σ(Txn,Tv)=σ(v,xn+1)+σ(Txn,Tv).
It follows from (33) and (42) that σ(v,Tv)=0; therefore v=Tv.

In the second case from (6), we have
(44)FσT2xn,Tv<τ+FσT2xn,Tv≤αFσTxn,v+βFσTxn,T2xn+γFσv,Tv=αFσxn+1,v+βFσxn+1,Txn+1+γF(D(v,Tv)).
Then employing (24), (33), and (F2), we conclude that limn→∞F(σ(T2xn,Tv))=-∞. Equivalently, from (F2) we get
(45)limn→∞σT2xn,Tv=0.
Using (6), we obtain
(46)σv,Tv≤σv,T2xn+σT2xn,Tv=σv,xn+2+σT2xn,Tv.
Finally, from (33) and (45) it follows that σ(v,Tv)=0; therefore v=Tv. Hence, v is a fixed point of T.

Definition 10.

Let (X,σ) be a metric-like space. A mapping T:X→X is said to be a modified F-contraction of type II if there exists τ>0 such that
(47)12σ(x,Tx)<σ(x,y)⟹τ+F(σ(Tx,Ty))≤F(σ(x,y)),
for all x,y∈X with x≠y where F:R+→R is a mapping satisfying the conditions (F1) and (F2) stated in Definition 8.

Theorem 11.

Let (X,σ) be a complete metric-like space and T a modified F-contraction of type II. Then, T has a fixed point v∈X; that is, Tv=v.

Proof.

It is sufficient to take α=1 and β=γ=0 in Theorem 9.

Definition 12.

Let (X,p) be a partial metric space. A self-mapping T:X→X is said to be a modified F-contraction of type I if there exists τ>0 such that
(48)12px,Tx<px,y⟹τ+FpTx,Ty≤αF(p(x,y))+βF(p(x,Tx))+γF(p(y,Ty)),
for all x,y∈X with x≠y where γ∈[0,1) and α,β∈[0,1] are real numbers such that α+β+γ=1 and F:R+→R is a mapping satisfying the following conditions:

F is strictly increasing; that is, for all α,β∈R+ such that α<β, F(α)<F(β),

for any sequence {αn}n=1∞ of positive real numbers limn→∞αn=0 if and only if limn→∞F(αn)=-∞.

Theorem 13.

Let (X,p) be a complete partial metric space and T a modified F-contraction of type I. Then, T has a unique fixed point v∈X; that is, Tv=v.

Proof.

Since every partial metric space is metric-like space (see, e.g., Remark 4), the existence of a fixed point v∈X of the mapping T is guaranteed by Theorem 9. Thus, it is sufficient to show that v is the unique fixed point of T. Suppose, on the contrary, that w∈X is another fixed point of T such that v≠w. Then, we have p(v,w)>0. If p(v,v)=0, we have
(49)0=12p(v,v)=12p(v,Tv)<p(v,w).
If p(v,v)>0, then the inequality
(50)12p(v,Tv)=12p(v,v)<p(v,v)≤p(v,w)
follows from the condition (p2) in Definition 1. In any case, the left-hand side of (48) is fulfilled. Hence, we have
(51)τ+FpTv,Tw≤αFpv,w+βFpv,Tv+γFpw,Tw=αF(p(v,w))+βF(p(v,v))+γFpw,w.
On the other hand, from (p2), we have p(v,v)≤p(v,w) and p(w,w)≤p(v,w). Regarding (F2), we get that
(52)Fpv,v≤Fpv,w,F(p(w,w))≤F(p(v,w)).
Combining (51) and (52), we conclude that
(53)Fpv,w<τ+Fpv,w=τ+FpTv,Tw<αFpv,w+βFpv,w+γFpv,w=α+β+γFpv,w=F(p(v,w)),
since τ>0 and α+β+γ=1. This is a contradiction and hence v=w.

Analogously, we conclude a result similar to Theorem 11 by introducing the next definition.

Definition 14.

Let (X,p) be a partial space. A mapping T:X→X is said to be a modified F-contraction of type II if there exists τ>0 such that
(54)12p(x,Tx)<p(x,y)⟹τ+F(p(Tx,Ty))≤F(p(x,y)),
for all x,y∈X with x≠y where F:R+→R is a mapping satisfying the conditions (F1) and (F2) stated in Definition 12.

Theorem 15.

Let (X,p) be a complete partial space and T a modified F-contraction of type II. Then, T has a unique fixed point v∈X; that is, Tv=v.

Proof.

It is sufficient to take α=1 and β=γ=0 in Theorem 13.

Definition 16.

Let (X,σ) be a metric-like space. A self-mapping T:X→X is said to be modified F-contraction of type III if there exists τ>0 such that
(55)σTx,Ty>0⟹τ+FσTx,Ty<αF(σ(x,y))+βFσx,Tx+γF(σ(y,Ty)),
for all x,y∈X with x≠y where γ∈[0,1) and α,β∈[0,1] are real numbers such that α+β+γ=1 and F:R+→R is a mapping satisfying the conditions (F1) and (F2) introduced in Definition 8.

Theorem 17.

Let (X,σ) be a complete metric-like space and T a continuous modified F-contraction of type III. If σ(Tx,Tx)≤σ(x,x) for all x∈X, then T has a fixed point v∈X; that is, Tv=v.

Proof.

As in the proof of Theorem 9, we construct an iterative sequence {xn} in the following way. Take and arbitrary x∈X and set x=x0 and
(56)xn+1=Txn,∀n∈N0.
Notice that if σ(xn0,xn0+1)=0 for some n0∈N0, the proof is completed. Suppose that
(57)σ(xn-1,xn)>0
for all n∈N. Thus, (55) yields that
(58)τ+Fσxn,Txn+1=τ+FσTxn-1,Txn≤αFσxn-1,xn+βFσxn-1,Txn-1+γF(σ(xn,Txn))=αF(σ(xn-1,xn))+βF(σ(xn-1,xn))+γF(σ(xn,xn+1)),
which can be written as
(59)τ+1-γFσxn,xn+1≤α+βFσxn-1,xn.
Regarding the assumption α+β+γ=1, we get
(60)Fσxn,xn+1≤Fσxn-1,xn-τα+β<F(σ(xn-1,xn)).
From (F1), we conclude that
(61)σ(xn,xn+1)<σ(xn-1,xn),∀n∈N.
Therefore, {σ(xn,xn+1)}n=1∞ is a decreasing sequence of real numbers which is bounded from below. Hence, it converges and
(62)limn→∞σ(xn,xn+1)=β=inf{σ(xn,xn+1):n∈N}.
We will show that β=0 by method of Reductio ad absurdum. Suppose that β>0. Thus, for every ε>0 there exists m∈N, such that σ(xm,xm+1)<β+ε. Because of (F1), we have
(63)F(σ(xm,Txm))<F(β+ε).
On the other hand, it follows from (57) that 0<σ(xm,xm+1)=σ(Txm-1,Txm), which implies
(64)τ+FσTxm-1,Txm≤αFσxm-1,xm+βFσxm-1,Txm-1+γFσxm,Txm=αFσxm-1,xm+βFσxm-1,xm+γF(σ(Txm-1,Txm))
due to (55). Consequently, we have
(65)τ+1-γFσTxm-1,Txm<α+βFσxm-1,xm.
Since α+β+γ=1, we obtain that
(66)F(σ(Txm-1,Txm))≤F(σ(xm-1,xm))-τα+β.
Again from (57), we have 0<σ(Txm,Txm+1); thus, by the hypothesis of the theorem
(67)τ+FσTxm,Txm+1≤αFσxm,xm+1+βFσxm,Txm+γFσxm+1,Txm+1=αFσxm,xm+1+βFσxm,xm+1+γFσTxm,Txm+1,
which results in τ+(1-γ)F(σ(Txm,Txm+1))≤(α+β)F(σ(xm,xm+1)). Taking into account that α+β+γ=1, we derive
(68)FσTxm,Txm+1≤Fσxm,xm+1-τα+β.

Now we employ (63) and applying a procedure similar to that used in derivation of (66) and (68), we obtain
(69)Fσxm+n,xm+n+1=FσTxm+n-1,Txm+n≤Fσxm+n-1,xm+n-τα+β=FσTxm+n-2,Txm+n-1-τα+β≤Fσxm+n-2,xm+n-1-2τα+β=FσTxm+n-3,Txm+n-2-2τα+β≤Fσxm+n-3,xm+n-2-3τα+β⋮≤F(σ(xm+1,xm+2))-(n-1)τα+β=F(σ(Txm,Txm+1))-(n-1)τα+β≤F(β+ε)-nτα+β.

This implies that
(70)limn→∞F(σ(xm+n,xm+n+1))=-∞.
Then, from (F2) we have limn→∞σ(xm+n,xm+n+1)=0, so that there exists N1∈N such that
(71)σ(xm+n,xm+n+1)<β,∀n≥N1.

However, this contradicts the definition of β given in (62). Thus, β=0 and from (62) we conclude
(72)limn→∞σ(xn,xn+1)=0.

In the sequel, we will show that limn,m→∞σ(xn,xm)=0. Assume the contrary; that is, let there exist ε>0 and sequences {p(n)}n=1∞ and {q(n)}n=1∞ of natural numbers such that
(73)pn>qn>n,σxpn,xqn≥ε,σxpn-1,xqn<ε,∀n∈N.
Observe that by the triangle inequality we have
(74)ε≤σxpn,xqn≤σxpn,xpn-1+σxpn-1,xqn≤σxpn,xpn-1+ε=σ(xp(n)-1,Txp(n)-1)+ε,∀n∈N.
It follows from (72), (74), and the Squeeze Theorem that
(75)limn→∞σ(xp(n),xq(n))=ε.
Therefore, there exists N2∈N such that
(76)ε≤σ(xpn,xqn)<2ε∀n>N2.
Hence from (76), (F1), and assumption of theorem, we have
(77)τ+FσTxpn-1,Txqn-1≤αFσxpn-1,xqn-1+βFσxpn-1,Txpn-1+γFσxqn-1,Txqn-1<αF(2sε)+βFσxpn-1,Txpn-1+γFσxqn-1,Txqn-1,∀n>N2.
Now, using (72), (75), and (F2), we obtain
(78)limn→∞FσTxpn-1,Txqn-1=-∞.
However, due to (F2) it follows that
(79)limn→∞σTxpn-1,Txqn-1=0⟺limn→∞σxpn,xqn=0.

This is a contradiction with the relation (73). Hence limm,n→∞σ(xn,xm)=0. By the completeness of (X,σ) there exists v∈X such that
(80)σv,v=limn→∞σxn,v=limn→∞σxn,xm=0.
Since xn→v and T is continuous, we deduce Txn→Tv by Definition 7. Consequently, we have
(81)σ(Tv,Tv)=limn→∞σ(Txn,Tv)=limn→∞σ(xn+1,Tv).
According to the assumption of the theorem σ(Tv,Tv)≤σ(v,v). Then, from (80) and (81) it follows that
(82)limn→∞σ(xn,v)=limn→∞σ(xn,Tv)=0.
Since σ(v,Tv)≤σ(v,xn)+σ(xn,Tv), we obtain σ(v,Tv)=0 and thus the condition (σ1) gives v=Tv, which completes the proof.

In the following result, we proved that Theorem 17 is valid in the context of partial metric space.

Theorem 18.

Let (X,p) be complete partial metric space and let T be a continuous self-mapping on X such that, for all x∈X, p(Tx,Tx)≤p(x,x). Let γ∈[0,1) and α,β∈[0,1] be real numbers such that α+β+γ=1. Assume that there exists τ>0 such that, for all x,y∈X,
(83)pTx,Ty>0⟹τ+F(p(Tx,Ty))≤αF(p(x,y))+βF(p(x,Tx))+γF(p(y,Ty)),
where F:R+→R satisfies the conditions (F1) and (F2). Then, T has a unique fixed point v∈X; that is, Tv=v.

Proof.

Since every partial metric space is a metric-like space, Theorem 17 provides the existence of a fixed point; that is, T has a fixed point v∈X. Therefore, it is sufficient to show the uniqueness of the fixed point of T. Indeed, if there is another fixed point w∈X of T, such that v≠w, due to (p1), we have p(v,w)>0 and equivalently p(Tv,Tw)>0. By the assumption of theorem, we have
(84)τ+FpTv,Tw<αFpv,w+βFpv,Tv+γFpw,Tw=αF(p(v,w))+βFpv,v+γF(p(w,w)).
On the other hand, (p2) implies
(85)pv,v≤pv,w,p(w,w)≤p(v,w).
Moreover, from (F2), we get
(86)Fpv,v≤Fpv,w,Fpw,w≤Fpv,w.
Due to the fact that τ>0 and α+β+γ=1, from (84) and (86), we conclude
(87)Fpv,w=FpTv,Tw<αFpv,w+βFpv,w+γFpv,w=α+β+γFpv,w=Fpv,w.
However, this is a contradiction, and hence, v=w.

Definition 19.

Let (X,σ) be metric-like spaces. A self-mapping T:X→X is said to be modified F-contraction of type IV if there exists τ>0 such that
(88)σTx,Ty>0⟹τ+FσTx,Ty<Fσx,y,
for all x,y∈X with x≠y where F:R+→R is a mapping satisfying the conditions (F1) and (F2) introduced in Definition 8.

Theorem 20.

Let (X,σ) be a complete metric-like space and T a continuous modified F-contraction of type IV. If σ(Tx,Tx)≤σ(x,x) for all x∈X, then T has a fixed point v∈X; that is, Tv=v.

Proof.

The proof is obvious by taking α=1 and β=γ=0 in Theorem 17.

Theorem 21.

Let (X,p) be complete PMS and let T be a continuous self-mapping on X such that, for all x∈X, p(Tx,Tx)≤p(x,x). Assume that there exists τ>0 such that, for all x,y∈X,
(89)0<pTx,Ty⟹τ+FpTx,Ty≤Fpx,y,
where F:R+→R satisfies conditions (F1) and (F2). Then, T has a unique fixed point v∈X; that is, Tv=v.

Proof.

The proof is trivial by taking α=1 and β=γ=0 in Theorem 18.

Last, we provide an example which illustrates our results.

Example 22.

Let k be a real number such that k>1, N∈N, and
(90)S0=0,S1=1,S2=1+2,⋮SN=1+2+⋯+N.
Let X={S0,S1,S2,…,SN} and p be defined as
(91)p:X×X⟶R0+px,y=maxx,yk+x-y.
Define a mapping T:X→X as follows:
(92)TS0=S0,TSn=Sn-1,∀n∈{1,2,…,N}.
Then T satisfies in the conditions of Theorem 21.

Observe that (X,p) is a complete partial metric space, but not a metric space. Define the function F in Theorem 21 as F(α)=lnα+α. Then we get
(93)τ+FpTx,Ty≤F(p(x,y))⟺p(Tx,Ty)p(x,y)ep(Tx,Ty)-p(x,y)≤e-τ.
Note also that, for all 0≤n<m≤N, we have
(94)p(TSn,TSm)p(Sn,Sm)e[p(TSn,TSm)-p(Sn,Sm)]=p(Sn-1,Sm-1)p(Sn,Sm)e[p(Sn-1,Sm-1)-p(Sn,Sm)]=maxSn-1,Sm-1k+Sn-1-Sm-1maxSn,Smk+Sn-Sm·e[max{Sn-1,Sm-1}k+|Sn-1-Sm-1|-max{Sn,Sm}k-|Sn-Sm|]=Sm-1k+(Sm-1-Sn-1)Smk+(Sm-Sn)·e[Sm-1k+(Sm-1-Sn-1)-Smk-(Sm-Sn)].
On the other hand,
(95)Smk+(Sm-Sn)=1+2+⋯+mk+n+1+n+2+⋯+m-1+m>1+2+⋯+m-1k+m+[(n+1)+(n+2)+⋯+(m-1)+m]>1+2+⋯+m-1k+m+[(n+1)+(n+2)+⋯+(m-1)+n]=Sm-1p+(Sm-1-Sn-1)+m.
Hence, we get
(96)p(TSn,TSm)p(Sn,Sm)ep(TSn,TSm)-p(Sn,Sm)<e-1.
Therefore T is an F-contraction satisfying the conditions of Theorem 21, τ=1, and TS0=S0; that is, S0 is the unique fixed point of T.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgments

This research was supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors thank anonymous referees for their remarkable comments, suggestions, and ideas that helped to improve this paper.

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