Existence of the solution of nonlinear fractional differential equations via new best proximity point results

Abstract

In this paper, we obtain some best proximity point results by introducing the concepts of proximal p -contractions of the first type and proximal p -contractions of the second type on partial metric spaces. Thus, some famous results in the literature such as the main result of Altun et al. (Acta Math Hung 162:393–402, 2020) and Basha (J Approx Theory 163(11):1772–1781, 2011) have been extended. Also, we provide some examples where our results are applicable and the results in Haghi et al. (Topol Appl 160:450–454, 2013) are not. Hence, our results are a real generalization of some results in metric spaces and partial metric spaces. Finally, we obtain sufficient conditions for the existence of the solution of nonlinear fractional differential equations via our results.

Introduction

In 1922, Banach [ 7 ] proved a fundamental theorem, known as the Banach contraction principle, which is considered the beginning of the fixed point theory on metric spaces. Because of its applicability, many authors have studied to generalize this principle [ 15 , 21 ]. One of the interesting and nice generalizations has been obtained by Popescu [ 19 ]. He presented a contractive condition, named p -contraction, which expands Banach contraction as follows:

Theorem 1

Let ϕ:(Ξ,θ)→(Ξ,θ) be a p -contraction mapping, that is, there exists q in [0, 1) satisfying

θ(ϕϖ,ϕς)≤q[θ(ϖ,ς)+θ(ϖ,ϕϖ)-θ(ς,ϕς)],
for all ϖ,ς∈Ξ where (Ξ,θ) is a complete metric space. Hence, there is a point ϖ in Ξ satisfying ϖ=ϕϖ.

Recently, many authors extended the fixed point theory in different way by considering nonself mappings. Let Λ,Π be nonempty subsets of a metric space (Ξ,θ) and ϕ:Λ→Π be a mapping. If Λ∩Π is empty, then there is no a point ϖ in Λ such that ϕϖ=ϖ . Therefore, it is reasonable to investigate whether there is a point ϖ∗∈Λ satisfying θ(ϖ∗,ϕϖ∗) is minimum. In this sense, Basha and Veeramani [ 10 ] gave the notion of best proximity point. We say that the point ϖ∗∈Λ is a best proximity point of ϕ if it satisfies θ(ϖ∗,ϕϖ∗)=θ(Λ,Π) . Best proximity point theorems are logical expansions of fixed point results because every best proximity point turns into a fixed point when Λ=Π=Ξ . Thus, several authors have researched this subject [ 4 , 5 , 9 , 20 , 23 , 24 – 25 ].

We now review several properties linked to the best proximity point theory. We regard the following sets:

Λ0=ϖ∈Λ:θ(ϖ,ς)=θ(Λ,Π)for someς∈Π,
and
Π0=ς∈Π:θ(ϖ,ς)=θ(Λ,Π)for someϖ∈Λ,
where θ(Λ,Π)=infθ(ϖ,ς):ϖ∈Λandς∈Π .

We give the definitions of proximal contractions of the first and second kind [ 8 ].

Definition 1

Let (Ξ,θ) be a metric space. A mapping ϕ:Λ→Π is called a proximal contraction of first kind if there exists q in [0, 1) such that, for all ξ1,ξ2,ϖ1,ϖ2∈Λ ,

θ(ξ1,ϕϖ1)=θ(Λ,Π)θ(ξ2,ϕϖ2)=θ(Λ,Π)⇒θ(ξ1,ξ2)≤qθ(ϖ1,ϖ2).

Definition 2

Let (Ξ,θ) be a metric space. A mapping ϕ:Λ→Π is called a proximal contraction of second kind if there exists q in [0, 1) such that, for all ξ1,ξ2,ϖ1,ϖ2∈Λ ,

θ(ξ1,ϕϖ1)=θ(Λ,Π)θ(ξ2,ϕϖ2)=θ(Λ,Π)⇒θ(ϕξ1,ϕξ2)≤qθ(ϕϖ1,ϕϖ2).

If every sequence {ϖr} in Λ satisfying θ(ς,ϖr)→θ(ς,Λ) for some ς∈Π has a subsequence {ϖrk} such that ϖrk→ϖ for some ϖ∈Λ , then Λ is said to be approximately compact with respect to Π .

For proximal contractions of the first and second kind, the following theorems are the primary results.

Theorem 2

([ 8 ]) Let (Ξ,θ) be a complete metric space, ∅≠Λ,Π⊆ Ξ with Λ0≠∅ where Π is approximatively compact with respect to Λ and Λ is closed. Suppose that ϕ:Λ→Π is a continuous proximal contraction of the first kind and ϕ(Λ0)⊆Π0 . Then, there exists a unique element ϖ in Λ such that θ(ϖ,ϕϖ)=θ(Λ,Π) .

Theorem 3

([ 8 ]) Let (Ξ,θ) be a complete metric space and ∅≠Λ,Π⊆ Ξ with Λ0≠∅ where Λ is approximately compact with respect to Π and Λ is closed. Assume that ϕ:Λ→Π is a continuous proximal contraction of the second kind and ϕ(Λ0)⊆Π0 . Then, there exists an element ϖ in Λ such that θ(ϖ,ϕϖ)=θ(Λ,Π) . Moreover, if ϖ∗ is another best proximity point of ϕ , then ϕϖ and ϕϖ∗ are identical.

On the other hand, Matthews [ 18 ] obtained another generalization of the Banach contraction principle by giving a famous notion called a partial metric. Then, numerous fixed point results in partial metric spaces have been obtained in various ways [ 1 , 2 , 6 , 11 , 22 ]. We now remind the concept of the partial metric space and its some properties.

Definition 3

([ 18 ]) Let Ξ≠∅ and κ:Ξ×Ξ→[0,∞) . If κ satisfies the following conditions, for all ϖ,ς,z∈Ξ ,

then κ is called partial metric, and (Ξ,κ) is called partial metric space.

It can be easily seen that a metric space is a partial metric space, but the converse may not happen. In fact, let Ξ=[0,∞) and κ:Ξ×Ξ→[0,∞) be a function defined by κ(ϖ,ς)=max{ϖ,ς} for all ϖ,ς∈Ξ . Although (Ξ,κ) is a partial metric space, it is not a metric space.

Let (Ξ,κ) be a partial metric space. The sets

Bκ(ϖ,ε)=ς∈Ξ:κ(ϖ,ς)<κ(ϖ,ϖ)+ε,
for all ϖ∈Ξ and ε>00$$\end{document}]]> is called open ball.

Let ϖ∈Ξ and {ϖr} be a sequence in Ξ . Hence, {ϖr} converges to ϖ with respect to τκ if and only if

limr→∞κ(ϖr,ϖ)=κ(ϖ,ϖ).
If limr,m→∞κ(ϖr,ϖm) exists and is finite, then the sequence {ϖr} in Ξ is said to be a Cauchy sequence. Further, (Ξ,κ) is said to be complete if every Cauchy sequence {ϖr} in Ξ converges to a point ϖ∈Ξ with respect to τκ such that
κ(ϖ,ϖ)=limm,r→∞κ(ϖr,ϖm).
Let κs:Ξ×Ξ→[0,∞) be a function defined by
κs(ϖ,ς)=2κ(ϖ,ς)-κ(ϖ,ϖ)-κ(ς,ς).
Then, κs is a metric on Ξ.

We now give the following lemma which gives the relationship between κs and κ.

Lemma 1

([ 18 ]) Let (Ξ,κ) be a partial metric space.

The following lemma is important because it gives the characterization of the closed set in partial metric spaces.

Lemma 2

([ 1 ]) Let (Ξ,κ) be a partial metric space and ∅≠Λ⊆Ξ . Define κ(ϖ,Λ)=infκ(ϖ,a):a∈Λ . Then, we have

ϖ∈Λ¯⟺κ(ϖ,Λ)=κ(ϖ,ϖ),
where Λ¯ is closure of Λ with respect to τκ .

Definition 4

([ 18 ]) Let (Ξ,κ) be a partial metric space and ϕ:Ξ→Ξ be a mapping. We say that ϕ is a continuous mapping with respect to τκ if

limr,m→∞κ(ϖr,ϖm)=limr→∞κ(ϖr,ϖ∗)=κ(ϖ∗,ϖ∗),
then
limr,m→∞κ(ϕϖr,ϕϖm)=limr→∞κ(ϕϖr,ϕϖ∗)=κ(ϕϖ∗,ϕϖ∗).

In the present paper, we introduce the new concepts of proximal p -contraction of the first type and proximal p -contraction of the second type on partial metric spaces. Then, we obtain some best proximity point results for such mappings. Hence, we generalize and extend some famous results in the literature such as the main result of [ 3 ] and [ 8 ]. Recently, Haghi et al. [ 13 ] proved that some results on partial metric spaces are equivalent to the results in the context of a metric spaces. But, this case is not valid for our main results. Therefore, we provide some important examples where our results are applicable and their results are not. Hence, our results are a real generalization of some results in metric spaces and partial metric spaces. Finally, with the help of our results, we obtain sufficient conditions for the existence of the solution of nonlinear fractional differential equations.

Main results

Considering partial metric spaces, we begin this section by giving the definition of approximately compact.

Definition 5

Let (Ξ,κ) be a partial metric space, and ∅≠Λ,Π⊆Ξ . If every sequence {ϖr} in Λ satisfying κ(ς,ϖr)→κ(ς,Λ) for some ς∈Π has a subsequence {ϖrk} such that

limk,l→∞κ(ϖrk,ϖrl)=limk→∞κ(ϖrk,ϖ)=κ(ϖ,ϖ)=0,
for some ϖ∈Λ , then Λ is said to be κ -approximately compact with respect to Π .

After that, we introduce the following definition of proximal p -contraction of the first type.

Definition 6

Let (Ξ,κ) be a partial metric space and ∅≠Λ,Π⊆Ξ. A mapping ϕ:Λ→Π is said to be a proximal p -contraction of the first type if there exists q in [0, 1) such that, for all ξ1,ξ2,ϖ1,ϖ2∈Λ with ξ1≠ξ2 ,

κ(ξ1,ϕϖ1)=κ(Λ,Π)κ(ξ2,ϕϖ2)=κ(Λ,Π)⇒κ(ξ1,ξ2)≤qκ(ϖ1,ϖ2)+κ(ξ1,ϖ1)-κ(ξ2,ϖ2).

Now, we give our first main result.

Theorem 4

Let (Ξ,κ) be a complete partial metric space and ∅≠Λ,Π⊆Ξ where Λ is closed and Π is a κ -approximately compact with respect to Λ . Assume that the followings hold:

Then, there exists ϖ∗ in Λ such that κ(ϖ∗,ϕϖ∗)=κ(Λ,Π) .

Proof

Let ϖ0 be an arbitrary point in Λ0. Because of the fact that ϕϖ0∈ϕ(Λ0)⊆Π0, there exists ϖ1∈Λ0 satisfying

κ(ϖ1,ϕϖ0)=κ(Λ,Π).
Similarly, since ϕϖ1∈ϕ(Λ0)⊆Π0, there exists ϖ2∈Λ0 satisfying
κ(ϖ2,ϕϖ1)=κ(Λ,Π).
Performing this process again, one can create a sequence {ϖr} in Λ satisfying
κ(ϖr+1,ϕϖr)=κ(Λ,Π),
for all r≥1. Assume that ϖr0=ϖr0+1 for some r0≥1 , then, from ( 2 ), ϖr0+1 is a best proximity point of ϕ . We now get ϖr≠ϖr+1 for all r≥1 . Because ϕ is a proximal p -contraction of the first type, we have, for all r≥1,
κ(ϖr,ϖr+1)≤qκ(ϖr-1,ϖr)+κ(ϖr,ϖr+1)-κ(ϖr-1,ϖr).
If there exists r0≥1 such that κ(ϖr0-1,ϖr0)≤κ(ϖr0,ϖr0+1) , then we have
κ(ϖr0,ϖr0+1)≤qκ(ϖr0-1,ϖr0)+κ(ϖr0,ϖr0+1)-κ(ϖr0-1,ϖr0)=qκ(ϖr0-1,ϖr0)+κ(ϖr0,ϖr0+1)-κ(ϖr0-1,ϖr0)=qκ(ϖr0,ϖr0+1)<κ(ϖr0,ϖr0+1),
which is a contradiction. Therefore, we can conclude that κ(ϖr,ϖr+1)<κ(ϖr-1,ϖr) for all r≥1 . Hence, we get
κ(ϖr,ϖr+1)≤qκ(ϖr-1,ϖr)+κ(ϖr-1,ϖr)-κ(ϖr,ϖr+1)=2qκ(ϖr-1,ϖr)-qκ(ϖr,ϖr+1),
and so, for all r≥1 ,
κ(ϖr,ϖr+1)≤2qq+1κ(ϖr-1,ϖr).
Using the last inequality, we have, for all r≥1 ,
κ(ϖr,ϖr+1)≤2qq+1κ(ϖr-1,ϖr)≤2qq+12κ(ϖr-2,ϖr-1)⋮≤2qq+1rκ(ϖ0,ϖ1).
Hence, we can obtain, for all m,r≥1 with m>rr$$\end{document}]]> ,
κ(ϖr,ϖm)≤κ(ϖr,ϖr+1)+κ(ϖr+1,ϖr+2)+...+κ(ϖm-1,ϖm)≤2qq+1rκ(ϖ0,ϖ1)+2qq+1r+1κ(ϖ0,ϖ1)+⋯+2qq+1m-1κ(ϖ0,ϖ1)=1+2qq+1+⋯+2qq+1m-r-12qq+1rκ(ϖ0,ϖ1)≤2qq+1r1-2qq+1κ(ϖ0,ϖ1).
Hence, limr,m→∞κ(ϖr,ϖm)=0 , and so {ϖr} is a Cauchy sequence in Λ . Since Λ is a closed subset of complete partial metric space (Ξ,κ) , there is ϖ∗ in Λ satisfying
limr,m→∞κ(ϖr,ϖm)=limr→∞κ(ϖr,ϖ∗)=κ(ϖ∗,ϖ∗)=0.
Moreover, from ( 2 ) and ( 3 ), it can be seen that
κ(ϖ∗,Π)≤κ(ϖ∗,ϕϖr)≤κ(ϖ∗,ϖr+1)+κ(ϖr+1,ϕϖr)=κ(ϖ∗,ϖr+1)+κ(Λ,Π)≤κ(ϖ∗,ϖr+1)+κ(ϖ∗,Π).
Therefore, κ(ϖ∗,ϕϖr)→κ(ϖ∗,Π) as r→∞ . Since Π is a κ -approximately compact with respect to Λ , there exists a subsequence {ϕϖrk} of {ϕϖr} such that
limk,l→∞κ(ϕϖrk,ϕϖrl)=limk→∞κ(ϕϖrk,ς∗)=κ(ς∗,ς∗)=0,
for some ς∗∈Π . Therefore, from ( 2 ) and Lemma 1 (iii), we have
κ(Λ,Π)≤κ(ϖ∗,ς∗)≤κ(ϖ∗,ϖrk+1)+κ(ϖrk+1,ϕϖrk)+κ(ϕϖrk,ς∗)=κ(ϖ∗,ϖrk+1)+κ(Λ,Π)+κ(ϕϖrk,ς∗),
and so taking the limit as k→∞, we have
κ(ϖ∗,ς∗)=κ(Λ,Π).
Hence, we get ϖ∗∈Λ0 . Also, due to the fact that ϕϖ∗∈ϕ(Λ0)⊆Π0 , there exists z∈Λ0 such that
κ(z,ϕϖ∗)=κ(Λ,Π).
We can consider ϖr+1≠z for all r≥1 . Otherwise, since ϖr≠ϖr+1 for all r≥1, we can find subsequences satisfying ϖrk+1≠z for all k≥1 . Then, from ( 2 ) and ( 4 ), we have
κ(ϖr+1,z)≤qκ(ϖr,ϖ∗)+κ(ϖr,ϖr+1)-κ(ϖ∗,z),
for all r≥1. Taking the limit as r→∞ in the last inequality, from ( 3 ) and ( 5 ), we get
κ(ϖ∗,z)≤qκ(ϖ∗,z),
which implies ϖ∗=z . Hence, from ( 4 ), there exists ϖ∗ in Λ such that κ(ϖ∗,ϕϖ∗)=κ(Λ,Π) . □

The following example is important as it shows that the approach of Haghi et al. [ 13 ] cannot be applied to Theorem 4 .

Example 1

Let Ξ={0}∪[1,∞)×{0}∪[1,∞) and κ:Ξ×Ξ→[0,∞) defined by

κ(ϖ,ς)=ϖ12+ϖ2,ϖ=ςϖ1+ς1+max{ϖ2,ς2},otherwise,
where ϖ=(ϖ1,ϖ2) and ς=(ς1,ς2)∈Ξ . Then, (Ξ,κ) is a complete partial metric space. Also, for all ϖ∈Ξ, we have
Bκϖ,ϖ13={ϖ}for allϖ1>0,0, \end{aligned}$$\end{document}]]>
and
Bκ(0,ϖ2),ε={0}×0∪[1,ϖ2+ε).
Consider the sets
Λ={(3,8),(1,11)},
and
Π={(1,7),(0,10)}.
Then, we have κ(Λ,Π)=12, Λ0=Λ and Π0=Π . Further, it can be seen that Λ is closed. If we define a mapping ϕ:Λ→Π by 
ϕ(ϖ1,ϖ2)=(1,7),ϖ1=3(0,10),ϖ1=1,
then we get ϕ(Λ0)⊆Π0 and ϕ is continuous mapping with respect to τκ. Also, the mapping ϕ satisfies ( 1 ) for q=1617 . Indeed, let ξ,v,ϖ,ς be arbitrary points in Λ with ξ≠v satisfying
κ(ξ,ϕϖ)=κ(Λ,Π)κ(v,ϕς)=κ(Λ,Π).
Then, we get ϖ=ξ≠v=ς∈{(3,8),(1,11)}. In this case, we have
κ(ξ,v)=15≤161715+32+8-12-11=qκ(ϖ,ς)+κ(ξ,ϖ)-κ(v,ς).
Thus, all hypotheses of Theorem 4 are satisfied, and so ϕ has a best proximity point in Λ . Now, consider the standard metric θ defined by
θ(ϖ,ς)=0,ϖ=ςκ(ϖ,ς),ϖ≠ς,
as in Proposition 2.1 of [ 13 ]. However, for ξ=ϖ=(3,8) and ς=v=(1,11)∈Λ , we have
θ(ξ,v)=15>15q=qθ(ϖ,ς)+θ(ξ,ϖ)-θ(v,ς),15q=q\left\{ \theta (\varpi ,\varsigma )+\left| \theta (\xi ,\varpi )-\theta (v,\varsigma )\right| \right\} , \end{aligned}$$\end{document}]]>
for all q∈[0,1) . Therefore, ϕ does not satisfy ( 1 ) with respect to θ .

Taking Λ=Π=Ξ in Theorem 4 , we obtain the following fixed point result.

Corollary 1

Let ϕ:(Ξ,κ)→(Ξ,κ) be a mapping where (Ξ,κ) is a complete partial metric space. Then, there is a point ϖ∗ in Ξ satisfying ϖ∗=ϕϖ∗ if there is q in [0, 1) satisfying, for all ϖ≠ς ,

κ(ϕϖ,ϕς)≤qκ(ϖ,ς)+κ(ϖ,ϕϖ)-κ(ς,ϕς).

Now, we introduce the following definition of proximal p -contraction of the second type.

Definition 7

Let (Ξ,κ) be a partial metric space and ∅≠Λ,Π⊆Ξ. A mapping ϕ:Λ→Π is said to be proximal p -contraction of the second type if there exists q in [0, 1) such that, for all ξ1,ξ2,ϖ1,ϖ2∈Λ with ϕξ1≠ϕξ2,

κ(ξ1,ϕϖ1)=κ(Λ,Π)κ(ξ2,ϕϖ2)=κ(Λ,Π)⇒κ(ϕξ1,ϕξ2)≤qκ(ϕϖ1,ϕϖ2)+κ(ϕξ1,ϕϖ1)-κ(ϕξ2,ϕϖ2).

Now, we give our second main result.

Theorem 5

Let (Ξ,κ) be a complete partial metric space and ∅≠Λ,Π⊆Ξ where Π is closed and Λ is a κ -approximately compact with respect to Π. Assume that the followings hold:

Then, there exists ϖ∗ in Λ such that κ(ϖ∗,ϕϖ∗)=κ(Λ,Π) .

Proof

Let ϖ0∈ Λ0 be any point. Since ϕϖ0∈ϕ(Λ0)⊆Π0, there exists ϖ1∈Λ0 satisfying

κ(ϖ1,ϕϖ0)=κ(Λ,Π).
Similarly, since ϕϖ1∈ϕ(Λ0)⊆Π0, there exists ϖ2∈Λ0 satisfying
κ(ϖ2,ϕϖ1)=κ(Λ,Π).
Repeating this process, we can construct a sequence {ϖr} in Λ such that, for all r≥1,
κ(ϖr+1,ϕϖr)=κ(Λ,Π).
Assume that ϕϖr0=ϕϖr0+1 for some r0≥1 , then from ( 7 ), ϖr0+1 is a best proximity point of ϕ . We now get ϕϖr≠ϕϖr+1 for all r≥1 . Because ϕ is a proximal p -contraction of the second type, we have, for all r≥1,
κ(ϕϖr,ϕϖr+1)≤qκ(ϕϖr-1,ϕϖr)+κ(ϕϖr,ϕϖr+1)-κ(ϕϖr-1,ϕϖr).
Suppose that the inequality κ(ϕϖr0-1,ϕϖr0)≤κ(ϕϖr0,ϕϖr0+1) holds for some r0≥1 , then we have
κ(ϕϖr0,ϕϖr0+1)≤qκ(ϕϖr0-1,ϕϖr0)+κ(ϕϖr0,ϕϖr0+1)-κ(ϕϖr0-1,ϕϖr0)=qκ(ϕϖr0-1,ϕϖr0)+κ(ϕϖr,ϕϖr+1)-κ(ϕϖr0-1,ϕϖr0)=qκ(ϕϖr0,ϕϖr0+1)<κ(ϕϖr0,ϕϖr0+1).
This is a contradiction. Thus, we can conclude that κ(ϕϖr,ϕϖr+1)<κ(ϕϖr-1,ϕϖr) for all r≥1 . Therefore, we get
κ(ϕϖr,ϕϖr+1)≤qκ(ϕϖr-1,ϕϖr)+κ(ϕϖr,ϕϖr-1)-κ(ϕϖr,ϕϖr+1)=2qκ(ϕϖr-1,ϕϖr)-qκ(ϕϖr,ϕϖr+1),
and so, for all r≥1 ,
κ(ϕϖr,ϕϖr+1)≤2qq+1κ(ϕϖr-1,ϕϖr).
Using the last inequality, we have
κ(ϕϖr,ϕϖr+1)≤2qq+1κ(ϕϖr-1,ϕϖr)≤2qq+12κ(ϕϖr-2,ϕϖr-1)⋮≤2qq+1rκ(ϕϖ0,ϕϖ1),
for all r≥1 . Now, for m,r≥1 with m>rr$$\end{document}]]> , we obtain
κ(ϕϖr,ϕϖm)≤κ(ϕϖr,ϕϖr+1)+κ(ϕϖr+1,ϕϖr+2)+⋯+κ(ϕϖm-1,ϕϖm)≤2qq+1rκ(ϕϖ0,ϕϖ1)+2qq+1r+1κ(ϕϖ0,ϕϖ1)+⋯+2qq+1m-1κ(ϕϖ0,ϕϖ1)=1+2qq+1+⋯+2qq+1m-r-12qq+1rκ(ϕϖ0,ϕϖ1)≤2qq+1r1-2qq+1κ(ϕϖ0,ϕϖ1).
Therefore, we have limr,m→∞κ(ϕϖr,ϕϖm)=0 . Thus, {ϕϖr} is a Cauchy sequence in Π . Because of the fact that Π is a closed subset of complete partial metric space (Ξ,κ) , there is ς∗∈Π satisfying
limr,m→∞κ(ϕϖr,ϕϖm)=limr→∞κ(ϕϖr,ς∗)=κ(ς∗,ς∗)=0.
Moreover, from ( 7 ), it can be seen that
κ(ς∗,Λ)≤κ(ς∗,ϖr+1)≤κ(ς∗,ϕϖr)+κ(ϖr+1,ϕϖr)=κ(ς∗,ϕϖr)+κ(Λ,Π)≤κ(ς∗,ϕϖr)+κ(ς∗,Λ).
Hence, we get κ(ς∗,ϖr+1)→κ(ς∗,Λ) as r→∞ . Since Λ is a κ -approximately compact with respect to Π , there exists a subsequence {ϖrk} of {ϖr} such that
limk,l→∞κ(ϖrk,ϖrl)=limk→∞κ(ϖrk,ϖ∗)=κ(ϖ∗,ϖ∗)=0,
for some ϖ∗∈Λ . Since ϕ is a continuous mapping with respect to κ , we get
limk,l→∞κ(ϕϖrk,ϕϖrl)=limr→∞κ(ϕϖrk,ϕϖ∗)=κ(ϕϖ∗,ϕϖ∗)
Hence, we get
κ(Λ,Π)≤κ(ϖ∗,ϕϖ∗)≤κ(ϖ∗,ϖrk+1)+κ(ϖrk+1,ϕϖrk)+κ(ϕϖrk,ϕϖ∗)-κ(ϖrk+1,ϖrk+1)-κ(ϕϖrk,ϕϖrk)=κ(ϖ∗,ϖrk+1)+κ(Λ,Π)+κ(ϕϖrk,ϕϖ∗)-κ(ϖrk+1,ϖrk+1)-κ(ϕϖrk,ϕϖrk),
and so taking the limit as k→∞, we have κ(ϖ∗,ϕϖ∗)=κ(Λ,Π). □

The following example is important as it shows that the approach of Haghi et al. [ 13 ] cannot be applied to Theorem 5 .

Example 2

Let (Ξ,κ) be as in Example 1 . Let Λ=(4,9),(2,12) and Π=(2,9),(1,11) . Then, we have κ(Λ,Π)=15 , Λ0=Λ and Π0=Π . Since the single point {(ϖ1,ϖ2)} is closed whenever ϖ1>00$$\end{document}]]> for all ϖ=(ϖ1,ϖ2)∈Π , Π is closed. Also, it can be seen that Λ is κ -approximately compact with respect to Π . Define a mapping ϕ:Λ→Π by

ϕϖ=(2,9),ϖ=(4,9)(1,11),ϖ=(2,12).,
Hence, ϕ is a continuous mapping with respect to τκ . Further, we have ϕ(Λ0)⊆Π0 . We will demonstrate that ϕ is a proximal p -contraction of the second type for q=2931 . Let ξ,v,ϖ,ς be arbitrary points in Λ with ϕξ≠ϕv satisfying
κ(ξ,ϕϖ)=κ(Λ,Π)κ(v,ϕς)=κ(Λ,Π).
Then, we get ϕϖ=ϕξ≠ϕv=ϕς∈{(2,9),(1,11)}. In this case, we have
κ(ϕξ,ϕv)=14≤293114+12+11-10=qκ(ϕϖ,ϕς)+κ(ϕξ,ϕϖ)-κ(ϕv,ϕς).
Thus, all hypotheses of Theorem 5 are satisfied, and so ϕ has a best proximity point in Λ . Now, consider the metric θ defined by
θ(ϖ,ς)=0,ϖ=ςκ(ϖ,ς),ϖ≠ς,
as in Proposition 2.1 of [ 13 ]. However, for ϕϖ=ϕξ=(2,9) and ϕv=ϕς=(1,11) , we have
θ(ϕξ,ϕv)=14>14q=qθ(ϕϖ,ϕς)+θ(ϕξ,ϕϖ)-θ(ϕv,ϕς),14q=q\left\{ \theta (\phi \varpi ,\phi \varsigma )+\left| \theta (\phi \xi ,\phi \varpi )-\theta (\phi v,\phi \varsigma )\right| \right\} , \end{aligned}$$\end{document}]]>
for all q∈[0,1) . Therefore, ϕ does not satisfy ( 1 ) with respect to θ .

If we take Λ=Π=Ξ in Theorem 5 , then we present the following fixed point result.

Corollary 2

Let ϕ:(Ξ,κ)→(Ξ,κ) be a continuous mapping with respect to τκ where (Ξ,κ) is a complete partial metric space. Then, there is a point ϖ∗ in Ξ satisfying ϖ∗=ϕϖ∗ if there exists q in [0, 1) such that for all ϖ,ς∈Ξ with ϕϖ≠ϕς ,

κ(ϕ2ϖ,ϕ2ς)≤qκ(ϕϖ,ϕς)+κ(ϕ2ϖ,ϕϖ)-κ(ϕ2ς,ϕς).

Nonlinear fractional differential equations

The existence and uniqueness of the solution to nonlinear fractional differential equations are examined in this section. First, let us recall some basic definitions of fractional calculus (see [ 12 , 14 , 16 , 17 , 26 , 27 ]). The Caputo derivative of h of order α>00$$\end{document}]]> is defined by

CDα(h(β))=1Γ(r-α)∫0β(β-ȷ)r-α-1h(r)(ȷ)dȷ,
where h:[0,∞)→R is a continuous function and r=α+1 with [α] denoting the integer part of a positive real number α , and Γ is the gamma function

The following nonlinear fractional differential equation of Caputo type

CDα(h(β))=h(β,ϖ(β)),
with integral boundary conditions
ϖ(0)=0andϖ(1)=∫0δϖ(ȷ)dȷ,
where h:[0,1]×R→R is a continuous function, and 1<α≤2 , 0< δ<1 , ϖ∈C[0,1] . Since h is a continuous, it is clear that Eq. ( 10 ) is equal to the integral equation.
ϖ(β)=1Γ(α)∫0β(β-ȷ)α-1h(ȷ,ϖ(ȷ))dȷ-2β(2-δ2)Γ(α)∫01(1-ȷ)α-1h(ȷ,ϖ(ȷ))dȷ+2β(2-δ2)Γ(α)∫0δ∫0ȷ(ȷ-m)α-1h(m,ϖ(m))dmdȷ.

Theorem 6

Assume the following conditions are true:

(i) for all ϖ∈C[0,1] and β∈[0,1], the mapping ϕ:Ξ→Ξ

ϕϖ(β)=1Γ(α)∫0β(β-ȷ)α-1h(ȷ,ϖ(ȷ))dȷ-2β(2-δ2)Γ(α)∫01(1-ȷ)α-1h(ȷ,ϖ(ȷ))dȷ+2β(2-δ2)Γ(α)∫0δ∫0ȷ(ȷ-m)α-1h(m,ϖ(m))dmdȷ,
is a continuous mapping.

(ii) there is q∈[0,1) such that

h(ȷ,ϖ(ȷ))+h(ȷ,ς(ȷ))≤Γ(α+1)5qϖ(ȷ)+ς(ȷ)+supȷ∈0,1ϖ(ȷ)+supȷ∈0,1ϕϖ(ȷ)-supȷ∈0,1ς(ȷ)-supȷ∈0,1ϕς(ȷ).
Then, the problem ( 10 ) has a solution.

Proof

Let Ξ=C[0,1] and κ:Ξ×Ξ→[0,∞) be a function defined by

κ(ξ,v)=supβ∈[0,1}ξ(β),ξ=vsupβ∈[0,1]ξ(β)+supβ∈[0,1]v(β),ξ≠v,
for all ξ,v∈Ξ and β∈[0,1]. Hence, (Ξ,κ) is a complete partial metric space. Now, we will demonstrate that ϕ holds ( 6 ). For all ϖ,ς∈Ξ and β∈[0,1], we have
ϕϖ(β)+ϕς(β)=1Γ(α)∫0β(β-ȷ)α-1h(ȷ,ϖ(ȷ))dȷ-2β(2-δ2)Γ(α)∫01(1-ȷ)α-1h(ȷ,ϖ(ȷ))dȷ+2β(2-δ2)Γ(α)∫0δ∫0ȷ(ȷ-m)α-1h(m,ϖ(m))dmdȷ+1Γ(α)∫0β(β-ȷ)α-1h(ȷ,ς(ȷ))dȷ-2β(2-δ2)Γ(α)∫01(1-ȷ)α-1h(ȷ,ς(ȷ))dȷ+2β(2-δ2)Γ(α)∫0δ∫0ȷ(ȷ-m)α-1h(m,ς(m))dmdȷ≤1Γ(α)∫0ββ-ȷα-1h(ȷ,ϖ(ȷ))+h(ȷ,ς(ȷ))dȷ+2β(2-δ2)Γ(α)∫01(1-ȷ)α-1h(ȷ,ϖ(ȷ))+h(ȷ,ς(ȷ))dȷ+2β(2-δ2)Γ(α)∫0δ∫0ȷ(ȷ-m)α-1h(m,ϖ(m))+h(m,ς(m))dmdȷ≤supβ∈[0,1]∫0ββ-ȷα-1Γ(α)Γ(α+1)5×qϖ(ȷ)+ς(ȷ)+supȷ∈0,1ϖ(ȷ)+supȷ∈0,1ϕϖ(ȷ)-supȷ∈0,1ς(ȷ)-supȷ∈0,1ϕς(ȷ)dȷ+2β(2-δ2)∫01(1-ȷ)α-1Γ(α)Γ(α+1)5×qϖ(ȷ)+ς(ȷ)+supȷ∈0,1ϖ(ȷ)+supȷ∈0,1ϕϖ(ȷ)-supȷ∈0,1ς(ȷ)-supȷ∈0,1ϕς(ȷ)dȷ+2β(2-δ2)∫0δ∫0ȷ(ȷ-m)α-1Γ(α)Γ(α+1)5×qϖ(m)+ς(m)+supm∈0,1ϖ(m)+supm∈0,1ϕϖ(m)-supm∈0,1ς(m)-supm∈0,1ϕς(m)dmdȷ≤Γ(α+1)5qκ(ϖ,ς)+κ(ϖ,ϕϖ)-κ(ς,ϕς)×supβ∈[0,1]1Γ(α+1)+2β(2-δ2)1Γ(α+1)+1Γ(α+1)≤qκ(ϖ,ς)+κ(ϖ,ϕϖ)-κ(ς,ϕς)
which implies that
supβ∈[0,1]ϕϖ(β)+supβ∈[0,1]ϕς(β)≤qκ(ϖ,ς)+κ(ϖ,ϕϖ)-κ(ς,ϕς).
Hence, we get
κ(ϕϖ,ϕς)≤qκ(ϖ,ς)+κ(ϖ,ϕϖ)-κ(ς,ϕς).
As a result, all of Corollary 1 ’s hypotheses are met, and thus ϕ has a fixed point. Consequently, there is a solution to nonlinear fractional differential equation of Caputo type ( 10 ). □

Conclusion

Using best proximity point theory on complete partial metric spaces, we aim to expand several famous conclusions that have already been in the literature. We first introduce some new concepts, named proximal p -contractions of the first type and proximal p -contractions of the second type, and then prove some best proximity point theorems for such mappings on partial metric spaces. We also get some notable examples where our results are valid and the results in [ 13 ] are not. Hence, our results are a real generalization of some of the results that exist in standard metric spaces and partial metric spaces. Finally, with the help of our results, we obtain sufficient conditions for the existence of the solution of nonlinear fractional differential equations.

Acknowledgements

The author is grateful to the referees for their valuable comments, which improved how the paper was presented.

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