On Some Novel Fixed Point Results for Generalized -Contractions in -Metric-Like Spaces with Application

1  Introduction

Fixed point theory has been studied for a long time. Its application relies on the existence of solutions to mathematical problems that are based on the contraction principle. An interesting generalization of the Banach contraction principle was given by Wardowski [1,2] using a different type of contraction called F-contraction and by Karapınar in [3] defining the type of interpolative contractions. These approaches have been extended by weakening the contractive conditions, removing some of the imposed conditions on the used mappings or relaxing axioms of the defined spaces. Starting from these aspects, many researchers have constructed new fixed point theorems in different types of spaces such as metric, b-metric and other generalized metric spaces, as cited in [3–31]. Nevertheless, in the papers of Younis et al. [4,5] the notion of Kanan mappings in the view of F-contraction in the setting of b-metric-like spaces has been expanded and an example related to electrical engineering has been given. In this paper, we introduce general types of (s,ϕ,F) and (s,q,ϕ,F)-contractions, which are variants of Wardowski contractions in the setting of b-metric-like spaces. Using these classes of contractive mappings, we establish unique fixed point theorems that unify and extend recent results on this topic.

2  Preliminaries

In this section, we list some well-known definitions and lemmas in terms of b-metric-like spaces.

Definition 2.1 [9]. Let V be a nonempty set and s≥1 be a given real number. A mapping b:V×V→[0,+∞) is called a b-metric-like if for all γ,δ,ν∈V these conditions are satisfied:

(i)b(γ,δ)=0impliesγ=δ;

(ii)b(γ,δ)=b(δ,γ);

(iii)b(γ,δ)≤s[b(γ,ν)+b(ν,δ)].

The pair (V,b) is called a b-metric-like space (in the sequel we use b−m.l.s for short).

In a b-metric-like space (V,b), if γ,δ∈V and b(γ,δ)=0, then γ=δ. However, the converse need not be true, and b(γ,γ) may be positive for γ∈V.

Definition 2.2 [10]. Let (V,b) be a b−m.l.s with parameter s≥1, {νn} be any sequence in V and ν∈V. Then, the following applies:

(a)   The sequence {νn} is said to be convergent to ν if limn→+∞⁡b(νn,ν)=b(ν,ν);

(b)   The sequence {νn} is said to be a Cauchy sequence in (V,b) if limn,m→+∞⁡b(νn,νm) exists and is finite;

(c)   The pair (V,b) is called a complete b−m.l.s if for every Cauchy sequence {νn}⊂V, there exists ν∈V such that limn,m→+∞⁡b(νn,νm)=limn→+∞⁡b(νn,ν)=b(ν,ν).

Definition 2.3 [10]. Let (V,b) be a b−m.l.s with parameter s≥1 and f be a self-mapping on V. We say that the function f is continuous if and only if limn→+∞⁡b(fνn,fν)=b(fν,fν), for each sequence {νn}⊂V, which satisfies limn→+∞⁡b(νn,ν)=b(ν,ν).

Note that in a b−m.l.s with parameter s≥1, if limn,m→+∞⁡b(νn,νm)=0 then the limit of the sequence {νn} is unique if it exists.

Lemma 2.1 [11,12]. Let (V,b) be a complete b−m.l.s with parameter s≥1 and {νn} be a sequence such that b(νn,νn+1)≤λb(νn−1,νn), for all n∈N, where λ∈[0,1). Then {vn} is a b-Cauchy sequence such that limn,m→+∞⁡b(νn,νm)=0.

Lemma 2.2. [9]. Let (V,b) be a b−m.l.s with parameter s≥1 and suppose that {νn} converges to a ν and b(ν,ν)=0. Then

s−1b(v,z)≤liminfn→+∞⁡b(νn,z)≤limsupn→+∞⁡b(νn,z)≤sb(ν,z),

for all z∈V.

Lemma 2.3. [9]. Let (V,b), be a b−m.l.s with parameter s≥1. Then, the following applies:

(a)   If b(γ,δ)=0, then b(γ,γ)=b(δ,δ)=0;

(b)   If {νn} is a sequence such that limn→+∞⁡b(νn,νn+1)=0, then we have

limn→+∞⁡b(νn,νn)=limn→+∞⁡b(νn+1,νn+1)=0;

(c)   If γ≠δ, then b(γ,δ)>0.

Lemma 2.4. [13]. Let (V,b) be a complete b−m.l.s with parameter s≥1. Let {νn}⊂V be a sequence such that limn→+∞⁡b(νn,νn+1)=0. If for sequence {νn} holds limn,m→+∞⁡b(νn,νm)≠0, then there exist ε>0 and sequences {mk}k=1+∞ and {nk}k=1+∞ of natural numbers with nk>mk>k, such that

b(νmk,νnk)≥ε,b(νmk,νnk−1)<ε,

εs2≤limsupk→+∞,b(νmk−1,νnk−1)≤εs,

εs≤limsupk→+∞,b(νnk−1,νmk)≤εs2and

εs≤limsupk→+∞,b(νmk−1,νnk)≤εs2.

3  Results

We begin the main section with a definition that is an expanding outlook of Wardowski type (ϕ,F)-contractions in the frame of a generalized metric space such as b−m.l.s.

Definition 3.1. Let (V,b) be a b−m.l.s with parameter s≥1 and f be a self-mapping on V. We say that f is a (s,q,ϕ,F)-contraction if there exist the functions F:(0,+∞)→R and ϕ:(0,+∞)→(0,+∞) such that

(a)   F is strictly increasing;

(b)   liminfr→t+⁡ϕ(r)>0 for all t>0;

(c)   For all ν,δ∈V with fν≠fδ, and for some q>0

ϕ(b(ν,δ))+F(sqb(fν,fδ))≤F(b(ν,δ)).(1)

Remark 3.1. In the above definitions property F and conditions (1) yield

b(fν,fδ)≤sqb(fν,fδ)<b(ν,δ).

The continuity of the mapping f follows from the inequality b(fν,fδ)<b(ν,δ).

Remark 3.2.

•   Our definition generalizes the previous definitions given in [14–16]. It contains a reduced number of conditions compared with the previous definitions.

•   The definition of (s,q,F)-contraction is an immediate consequence of Definition 2.1, if we take ϕ:(0,+∞)→(0,+∞) to be a constant function.

•   If s=1 we get the definition of Wardowski in [1,2] in the case of metric spaces.

•   For s=1 the definition is valid in the framework of a metric space.

The following is the first fixed point theorem for (s,q,ϕ,F)-contraction type mapping.

Theorem 3.1. Let (V,b) be a complete b−m.l.s with parameter s≥1. If f is a (s,q,ϕ,F)-contraction on V, then the function f has a unique fixed point in V.

Proof. Let be ν0∈V and the Picard iterative sequence {νn} defined by νn+1=f(νn) for n∈{0,1,2,…}. The proof is clear in the case that there exists n0∈N, with νn0+1=νn0. So, we will assume that νn+1≠νn, which implies fνn≠fνn−1 and b(fνn,fνn−1)>0, for all n∈N∪{0}. Using inequality (1), we have

F(sqb(νn,νn+1))≤ϕ(b(νn−1,νn))+F(sqb(νn,νn+1))=ϕ(b(νn−1,νn))+F(sqb(fνn−1,fνn))≤F(b(νn−1,νn)).(2)

Further from inequality (2), we get

sqb(νn,νn+1)<b(νn−1,νn),

which implies

b(νn,νn+1)<1sqb(νn−1,νn).

In view of Lemma 2.1, the corresponding Picard sequence {νn} with the initial point ν0 is a Cauchy sequence such that limn,m→+∞⁡b(νn.νn)=0. Since (V,b) is a complete b-metric-like space, we conclude that there exists ν∈V such that

limn→+∞⁡b(νn,ν)=b(ν,ν)=limn,m→+∞⁡b(νn,νm)=0.(3)

According to the (1), it follows:

F(sqb(fν,fνn))≤ϕ(b(ν,νn))+F(sqb(fν,fνn))≤F(b(ν,νn)),

that from property of F we get

sqb(fν,fνn)≤b(ν,νn).(4)

From triangular property and (4), we have

b(fν,fν)≤2sb(fν,fνn)≤2sqb(fν,fνn)≤2b(ν,νn).

Since f is continuous and using (3), (4) we obtain

b(fν,fν)=limn→+∞⁡b(fνn,fν)≤limn→+∞⁡2b(νn,ν)=2b(ν,ν)=0.(5)

Since b(ν,fν)≤s[b(ν,fνn)+b(fν,fνn)], as n→+∞ we obtain that b(ν,fν)=0. Thus fν=ν and so f has a fixed point. Also from (3), we have b(ν,ν)=0. To prove the uniqueness of the fixed point, suppose that u∈V is another different fixed point. From u≠ν follows fu≠fν, then

F(sqb(u,ν))=F(sqb(fu,fν))≤ϕ(b(u,ν))+F(sqb(fu,fν))≤F(b(u,ν)),

which implies

b(u,ν)<1sqb(u,ν).

Previous inequality is a contradiction, so b(u,ν)=0 and the fixed point is unique.

Corollary 3.1. Let (V,b) be a b−m.l.s with parameter s≥1 and f be a self-mapping on V. If there exist an increasing function F:(0,+∞)→R and a positive constant τ such that

τ+F(sqb(fν,fδ))≤F(b(ν,δ))(6)

for all ν,δ∈V with fν≠fδ, and for some q>0, then f has a unique fixed point in V.

Proof. Inequality (1) implies (6) if we set ϕ(r)=τ>0.

Example 3.1. Let V=[0,+∞) and b(x,y)=x2+y2+|x−y|2, for all x,y∈V. It is clear that b is a b-metric-like on V, with parameter s=2 and (V,b) is complete. Also, b is not a metric-like nor b-metric (nor a metric on V). Consider the self-mapping f:V→V by fx=ln⁡(1+x)5. For all x,y∈V and constant q=2, we have

s2b(fx,fy)=4(f2x+f2y+|fx−fx|2)=4((ln⁡(x+1)5)2+(ln⁡(y+1)5)2+|ln⁡(x+1)5−ln⁡(y+1)5|2)≤4[x225+y225+|x5−y5|2]=425[x2+y2+|x−y|2]≤15b(x,y).

Taking the logarithms in the above inequality and fixing τ=ln⁡5 and the function F(t)=ln⁡t then the conditions of Corollary 3.1 are satisfied and clearly x=0 is a unique fixed point of f.

With the aim of expanding the initiated Definition 2.1 and starting a result that includes Theorem 3.1 and its respective corollaries, we will use a class of implicit relations, which makes simultaneously effective enormous literature on this topic.

Let Γ4 be the set of all continuous functions g:[0,+∞)4→[0,+∞) satisfying

(a)   g is non-decreasing with respect to each variable:

(b)   g(t,t,t,t)≤t for t∈[0,+∞).

Definition 3.2. Let (V,b) be a b−m.l.s with parameter s≥1 and f:V→V be a self mapping. We say that f is generalized (s,q,ϕ,F)-g-weak contraction, if there exist functions F:(0,+∞)→R, ϕ:(0,+∞)→(0,+∞) and g∈Γ4 such that

(a) F is strictly increasing;

(b) liminfr→t+⁡ϕ(r)>0 for all t>0;

(c)ϕ(b(x,y))+F(sqb(fx,fy))≤F(g(b(x,y),b(x,fx),b(y,fy),b(x,fy)+b(y,fx)4s))(7)

for all x,y∈V with fx≠fy, and for some q≥1.

Remark 3.3.

•   The above definition reduces to a generalized (s,q,F)-g-weak contraction by setting ϕ:(0,+∞)→(0,+∞) to be a constant function ϕ(r)=τ>0.

•   Fixing the parameter s=1 we get the definition of (ϕ,F)-g-weak contraction in the setting of metric and metric-like spaces.

•   Fixing s=1 and ϕ(r)=τ>0 we get the definition of (F-g)-weak contraction in the setting of metric and metric-like spaces.

Theorem 3.2. Let (V,b) be a b−m.l.s with parameter s≥1 and the self mapping f:V→V be a generalized (s,q,ϕ,F)-g-weak contraction. If f or F is continuous, then f has a unique fixed point in V.

Proof. Let u0∈V be arbitrary and construct the Picard iterative sequence {un} as un+1=f(un) for n∈{0,1,2,…}. The proof is clear in the case that there exists n0∈N, with un0+1=un0. Therefore, we assume that un+1≠un, which means fum≠fun−1 or b(fun,fun−1)>0 for all n∈N∪{0}. Using (7) for x=un, y=un−1 we have

ϕ(b(un,un+1))+F(sqb(un,un+1))=ϕ(b(fun−1,fun))+F(sqb(fun−1,fun))≤F(g(b(un−1,un),b(un−1,fun−1),b(un,fun),b(un−1,fun)+b(un,fun−1)4s))=F(g(b(un−1,un),b(un−1,un),b(un,un+1),b(un−1,un+1)+b(un,un)4s))≤F(g(b(un−1,un),b(un−1,un),b(un,un+1),sb(un−1,un)+sb(un,un+1)+2sb(un−1,un)4s))=F(g(b(un−1,un),b(un−1,un),b(un,un+1),b(un,un+1)+3b(un−1,un)4)).(8)

If we assume that b(un−1,un)≤b(un,un+1), then inequality (8) yields

ϕ(b(un,un+1))+F(sqb(un,un+1))≤F(g(b(un,un+1),b(un,un+1),b(un,un+1),b(un,un+1)))≤F(b(un,un+1)),

for all n∈N. So, we obtain

F(sqb(un,un+1)≤F(b(un,un+1))−ϕ(b(un,un+1))<F(b(un,un+1)),

which is a contradiction. Therefore

b(un,un+1)<b(un−1,un),

for all n∈N. Thus, the sequence {b(un−1,un)} is decreasing and bounded below. Consequently, there exists l≥0 such that b(un−1,un)→l as n→+∞. If l>0, then by taking the limit in (8) we get

ϕ(l)+F(sql)≤F(l),

which is a contradiction. Therefore, we conclude that l=0 and

limn→+∞⁡b(un−1,un)=0.(9)

Next, we show that limn,m→∞⁡b(un,um)=0. Suppose the opposite, limn,m→∞⁡b(un,um)>0. Then by Lemma 2.4, there exist ε>0 and sequences {mk} and {nk} of positive integers, with nk>mk>k, such that

b(umk,unk)≥ε,b(umk,unk−1)<ε,εs2≤limsupk→+∞,b(umk−1,unk−1)≤εs,εs≤limsupk→+∞,b(unk−1,umk)≤εs2andεs≤limsupk→+∞,b(umk−1,unk)≤εs2.

From condition (7), we get

ϕ(b(umk,unk))+F(sqb(umk,unk))=ϕ(b(umk,unk))+F(sqb(fumk−1,funk−1))≤F(g(b(umk−1,unk−1),b(umk−1,fumk−1),b(unk−1,funk−1),b(umk−1,funk−1)+b(unk−1,fumk−1)4s))=F(g(b(umk−1,unk−1),b(umk−1,umk),b(unk−1,unk),b(umk−1,unk)+b(unk−1,umk)4s))(10)

Taking the upper limit in (10) as k→+∞ and using Lemma 2.3, Lemma 2.4 and (9), we get

liminfn→+∞,ϕ(b(umk,unk))+F(sqε)≤liminfn→+∞,ϕ(b(umk,unk))+F(limsupn→+∞⁡sqb(umk,unk))≤F(limsupn→+∞⁡g(b(umk−1,unk−1),b(umk−1,umk),b(unk−1,unk),b(umk−1,unk)+b(unk−1,umk)4s))≤F(g(ε,0,0,ε2s))≤F(εs).

Hence, the acquired inequality

liminfn→+∞,ϕ(b(umk,unk))+F(εsq)<F(εs),

is a contradiction since ε>0. So limn,m→∞⁡b(un,um)=0, and the sequence {un} is a Cauchy sequence in the complete b-metric-like space (V,b). Thus, there exists u∈V, such that

limn→+∞⁡b(un,u)=b(u,u)=limn,m→+∞⁡b(un,um)=0.(11)

Let n0∈N such that un+1≠fu for all n≥n0 and u≠fu. Now using condition (7) and property F, we have

ϕ(b(un,u))+F(sqb(un+1,fu))=ϕ(b(un,u))+F(sqb(fun,fu))≤F(g(b(un,u),b(un,fun),b(u,fu),b(un,fu)+b(u,fun)4s))=F(g(b(un,u),b(un,un+1),b(u,fu),b(un,fu)+b(u,un+1)4s)),

which implies

ϕ(b(un,fu))+sqb(un+1,fu)<g(b(un,u),b(un,un+1),b(u,fu),b(u,un+1)2s).(12)

Taking the upper limit in (12), and using Lemma 2.1 and result (9), it follows that

liminfn→+∞⁡ϕ(b(un,fu))+sq−1b(u,fu)=sq⋅1sb(u,fu)<g(0,0,b(u,fu),0)≤b(u,fu).(13)

Since q≥1, the inequality (13) implies b(u,fu)=0 and therefore fu=u. Thus, u is a fixed point and

0=b(u,fu)=b(u,u).(14)

Let u and v be two fixed points of f, where fu=u and fv=v. Since u≠v, it implies fu≠fv. By (7) we have

ϕ(u,v)+F(sqb(u,v))=ϕ(u,v)+F(sqb(fu,fv))≤F(g(b(u,v),b(u,fu),b(v,fv),b(u,fv)+b(v,fu)4s))=F(g(b(u,v),b(u,u),b(v,v),b(u,v)+b(v,u)4s))=F(g(b(u,v),b(u,u),b(v,v),b(u,v)2s))=F(g(b(u,v),0,0,b(u,v)2s))≤F(g(b(u,v),b(u,v),b(u,v),b(u,v)))≤F(b(u,v)).(15)

Since this is a contradiction, it implies b(u,v)=0. Therefore, u=v and the fixed point is unique.

Theorem 3.3. Let (V,b) be b−m.l.s with parameter s≥1 and f:V→V be a continuous self-mapping. Assume that there exist the functions F:(0,+∞)→R, g∈Γ4 and the constant τ>0 such that

(a)   F is strictly increasing;

(b)   τ+F(sqb(fx,fy))≤F(g(b(x,y),b(x,fx),b(y,fy),b(x,fy)+b(y,fx)4s)) for all x,y∈V with fx≠fy, for some q≥1.

Then f has a unique fixed point in V.

Proof. The proof follows from Theorem 3.2. by setting ϕ(r)=τ.

Corollary 3.2. Let (V,b) be a b−m.l.s with parameter s≥1 and f:V→V be a continuous self-mapping. Assume that there exist the functions F:(0,+∞)→R, ϕ:(0,+∞)→(0,+∞) such that:

(a)   F is strictly increasing;

(b)   liminfr→t+⁡ϕ(r)>0 for all t>0;

(c)  ϕ(b(x,y))+F(sqb(fx,fy))≤F(max(b(x,y),b(x,fx),b(y,fy),b(x,fy)+b(y,fx)4s))(16)

for all x,y∈V with fx≠fy, and for some q≥1. Then f has unique fixed point in V.

Proof. The proof follows from Theorem 3.2 by taking g∈Γ4 as g(t1,t2,t3,t4)=max{t1,t2,t3,t4}.

Corollary 3.3. Let (V,b) be a b−m.l.s with parameter s≥1 and f:V→V be a continuous self-mapping. Assume that there exist functions F:(0,+∞)→R, ϕ:(0,+∞)→(0,+∞) such that:

(a)   F is strictly increasing;

(b)   liminfr→t+⁡ϕ(r)>0 for all t>0;

(c)  ϕ(b(x,y))+F(sqb(fx,fy))≤F(max(a1b(x,y)+a2b(x,fx)+a3b(y,fy)+a4b(x,fy)+b(y,fx)4s))(17)

for all x,y∈V with fx≠fy, and for some q≥1. Then f has unique fixed point in V.

Proof. The proof follows from Theorem 3.2. by taking g∈Γ4 as g(t1,t2,t3,t4)=a1t1+a2t2+a3t3+a4t4 with 0<a1+a2+a3+a4<1.

Recently, many authors have studied new types of contractions known as interpolative contractions and hybrid contractions. The reader can refer to [3,11,17–21]. The rest of the paper deals with this type of contractions extended in the setting of b-metric-like spaces, which can be obtained from our results as a certain special cases.

Theorem 3.4. Let (V,b) be a b−m.l.s with parameter s≥1 and f:V→V be a continuous self-mapping. Assume that there exist the functions F:(0,+∞)→R, ϕ:(0,+∞)→(0,+∞) such that

(a)   F is strictly increasing;

(b)   liminfr→t+⁡ϕ(r)>0 for all t>0;

(c)  ϕ(b(x,y))+F(sqb(fx,fy))≤F([a1(b(x,y))p+a2(b(x,fx))p+a3(b(y,fy))p+a4(b(x,fy)+b(y,fx)4s)p]1p)(18)

for all x,y∈V with fx≠fy, and for some q≥1. Then f has unique fixed point in V.

Proof. The proof follows from Theorem 3.2 by taking g∈Γ4 as

g(t1,t2,t3,t4)=[a1t1p+a2t2p+a3t3p+a4t4p]1p,p>0,

where 0<a1+a2+a3+a4<1.

Theorem 3.5. Let (V,b) be a b−m.l.s with parameter s≥1 and f:V→V be a continuous self-mapping. Assume that there exist the functions F:(0,+∞)→R, ϕ:(0,+∞)→(0,+∞) such that

(a)   F is strictly increasing;

(b)   liminfr→t+⁡ϕ(r)>0 for all t>0;

(c)  ϕ(b(x,y))+F(sqb(fx,fy))≤F([max{(b(x,y))p,(b(x,fx))p,(b(y,fy))p,(b(x,fy)+b(y,fx)4s)p}]1p)(19)

Then f has unique fixed point in V.

Proof. The proof follows from Theorem 3.2 by taking g∈Γ4 as

g(t1,t2,t3,t4)=[max{t1p,t2p,t3p,t4p}]1p,p>0.

Theorem 3.6. Let (V,b) be a b−m.l.s with parameter s≥1 and f:V→V be a continuous self-mapping. Assume that there exist the functions F:(0,+∞)→R, ϕ:(0,+∞)→(0,+∞) such that:

(a)   F is strictly increasing;

(b)   liminfr→t+⁡ϕ(r)>0 for all t>0;

(c)  (b(x,y))+F(sqb(fx,fy))≤F((b(x,y))a1(b(x,fx))a2(b(y,fy))a3(b(x,fy)+b(y,fx)4s)1−(a1+a2+a3))(20)

Then f has unique fixed point in V.

Proof. The proof follows from Theorem 3.2 by taking g∈Γ4 as

g(t1,t2,t3,t4)=t1a1⋅t2a2⋅t3a3⋅t41−(a1+a2+a3),