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Research Article Fixed Point Results for Dualistic Contractions with an Application Muhammad Nazam , 1 Aiman Mukheimer , 2 Hassen Aydi , 3,4 Muhammad Arshad , 5 and Raheel Riaz 5 1 Department of Mathematics, Allama Iqbal Open University, H-8, Islamabad 44000, Pakistan 2 Université de Sousse, Institut Supérieur d’Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia 3 China Medical University Hospital, China Medical University, Taichung 40402, Taiwan 4 Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia 5 Department of Mathematics and Statistics, International Islamic University, H-10, Islamabad 44000, Pakistan Correspondence should be addressed to Hassen Aydi; [email protected] Received 10 June 2019; Revised 8 August 2019; Accepted 24 August 2019; Published 22 January 2020 Academic Editor: Yong Zhou Copyright © 2020 Muhammad Nazam et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, by introducing a convergence comparison property of a self-mapping, we establish some new fixed point theorems for Bianchini type, Reich type, and Dass-Gupta type dualistic contractions defined on a dualistic partial metric space. Our work generalizes and extends some well known fixed point results in the literature. We also provide examples which show the usefulness of these dualistic contractions. As an application of our findings, we demonstrate the existence of the solution of an elliptic boundary value problem. 1. Introduction e recent study in the metric fixed point theory is due to the Banach Contraction Principle, which has been modified, improved, and generalized in many ways in metric spaces (see for example, [1–17]). is Contraction principle has also been studied in partial metric spaces (PMS) introduced by Matthews [18]. e PMS generalizes the metric space where the self-distance may be not equal to zero. e topological concepts, like convergence, Cauchy sequence, continuity, and completeness in this class can be found in [18–21] and references there in. O’Neill [22] initiated the notion of a dualistic partial met- ric space. is class generalizes the notion of partial metric spaces. In [22], a relationship between a dualistic partial metric and a quasi metric has been obtained. O’Neill [22] also studied various topological properties of a dualistic partial metric space, while fixed point theory on dualistic partial metric spaces was presented by Oltra and Valero [23], who proved a Banach fixed point theorem and gave convergence properties of sequences on complete dualistic partial metric spaces. Later, Nazam et al. [24] ensured fixed point results for rational type contraction mappings in this setting (see also the related paper [25]). In this paper, motivated by [26, 27, 12], we establish some new fixed point theorems in dualistic partial metric spaces, generalizing fixed point results of Bianchini [26], Reich [12], and Dass and Gupta [27]. We also provide examples and an application to show significance of the obtained results involv- ing dualistic contractive conditions. 2. Preliminaries We recall some mathematical basics and definitions. Definition 1 ([18]). A partial metric on a non empty set A is a function : A × A →[0, ∞) so that ( 1 ) = ⇐⇒ (, ) = (, ) = (, ); ( 2 ) (, ) (, ); ( 3 ) (, ) = (, ); Hindawi Discrete Dynamics in Nature and Society Volume 2020, Article ID 6428671, 9 pages https://doi.org/10.1155/2020/6428671

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Page 1: Fixed Point Results for Dualistic Contractions with an Applicationdownloads.hindawi.com/journals/ddns/2020/6428671.pdf · 2019. 6. 10. · Research Article Fixed Point Results for

Research ArticleFixed Point Results for Dualistic Contractions with an Application

Muhammad Nazam ,1 Aiman Mukheimer ,2 Hassen Aydi ,3,4 Muhammad Arshad ,5 and Raheel Riaz5

1Department of Mathematics, Allama Iqbal Open University, H-8, Islamabad 44000, Pakistan2Université de Sousse, Institut Supérieur d’Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia3China Medical University Hospital, China Medical University, Taichung 40402, Taiwan4Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia5Department of Mathematics and Statistics, International Islamic University, H-10, Islamabad 44000, Pakistan

Correspondence should be addressed to Hassen Aydi; [email protected]

Received 10 June 2019; Revised 8 August 2019; Accepted 24 August 2019; Published 22 January 2020

Academic Editor: Yong Zhou

Copyright © 2020 Muhammad Nazam et al. �is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, by introducing a convergence comparison property of a self-mapping, we establish some new �xed point theorems for Bianchini type, Reich type, and Dass-Gupta type dualistic contractions de�ned on a dualistic partial metric space. Our work generalizes and extends some well known �xed point results in the literature. We also provide examples which show the usefulness of these dualistic contractions. As an application of our �ndings, we demonstrate the existence of the solution of an elliptic boundary value problem.

1. Introduction

�e recent study in the metric �xed point theory is due to the Banach Contraction Principle, which has been modi�ed, improved, and generalized in many ways in metric spaces (see for example, [1–17]).

�is Contraction principle has also been studied in partial metric spaces (PMS) introduced by Matthews [18]. �e PMS generalizes the metric space where the self-distance may be not equal to zero. �e topological concepts, like convergence, Cauchy sequence, continuity, and completeness in this class can be found in [18–21] and references there in.

O’Neill [22] initiated the notion of a dualistic partial met-ric space. �is class generalizes the notion of partial metric spaces. In [22], a relationship between a dualistic partial metric and a quasi metric has been obtained. O’Neill [22] also studied various topological properties of a dualistic partial metric space, while �xed point theory on dualistic partial metric spaces was presented by Oltra and Valero [23], who proved a Banach �xed point theorem and gave convergence properties of sequences on complete dualistic partial metric spaces. Later,

Nazam et al. [24] ensured �xed point results for rational type contraction mappings in this setting (see also the related paper [25]).

In this paper, motivated by [26, 27, 12], we establish some new �xed point theorems in dualistic partial metric spaces, generalizing �xed point results of Bianchini [26], Reich [12], and Dass and Gupta [27]. We also provide examples and an application to show signi�cance of the obtained results involv-ing dualistic contractive conditions.

2. Preliminaries

We recall some mathematical basics and de�nitions.

De�nition 1 ([18]). A partial metric on a non empty set A is a function � : A ×A → [0,∞) so that

(�1) � = �⇐⇒ �(�, �) = �(�, �) = �(�, �);(�2) �(�, �) ≤ �(�, �);(�3) �(�, �) = �(�, �);

HindawiDiscrete Dynamics in Nature and SocietyVolume 2020, Article ID 6428671, 9 pageshttps://doi.org/10.1155/2020/6428671

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Discrete Dynamics in Nature and Society2

(�4) �(�, �) ≤ �(�, �) + �(�, �) − �(�, �),for all �, �, � ∈ A .

�e pair (A , �) is called a partial metric space. O’Neill [22] did one signi�cant change to the de�nition of the partial met-ric � by extending its range from [0,∞) to (−∞,∞). �e par-tial metric � with extended range is called a dualistic partial metric (DPM), denoted by �∗.De�nition 2 ([22]). Let A be a non empty set. If a function �∗ : A ×A → R is such that

(�∗1 ) � = �⇐⇒ �∗(�, �) = �∗(�, �) = �∗(�, �);(�∗2 ) �∗(�, �) ≤ �∗(�, �);(�∗3 ) �∗(�, �) = �∗(�, �);(�∗4 ) �∗(�, �) + �∗(�, �) ≤ �∗(�, �) + �∗(�, �),

for all �, �, � ∈ A , then �∗ is called a dualistic partial metric and the pair (A , �∗) is known as a dualistic partial metric space.

If (A , �∗) is a dualistic partial metric space, then ��∗ : A ×A→ [0,∞) de�ned by

is called a quasi metric on A such that �(�∗) = �(��∗). Moreover, if ��∗ is a dualistic quasi metric on �, then ���∗(�, �) = max{��∗(�, �), ��∗(�, �)} is a metric on A (an induced metric).

Remark 1. Unlike partial metric case, note that if �∗ is a dual-istic partial metric, then �∗(�, �) = 0 may not imply � = �. �e self-distance �∗(�, �) is a feature utilized to describe the amount of information contained in �. �e restriction of �∗to [0,∞) is a partial metric.

Example 1. We de�ne �∗ : R × R→ R as �∗(�, �) = max{�, �}. It is easy to check that �∗ satis�es (�∗1 ) − (�∗4 ) and hence �∗ is a dualistic partial metric on R. Mention that �∗ is not a partial metric on R because that �∗(−�,−�) = −� for each � > 0.Example 2. Let � be a partial metric de�ned on a non empty set A . �e function �∗ : A ×A → R de�ned by

veri�es (�∗1 ) − (�∗4 ) and so it de�nes a dualistic partial metric on A . Note that �∗(�, �) may have negative values.

In the following, a new example of a dualistic partial metric is stated.

Example 3. Take A = R and � > 0. We de�ne �∗ : A ×A → R

as

(1)��∗(�, �) = �∗(�, �) − �∗(�, �), for all�, � ∈ A

(2)�∗(�, �) = �(�, �) − �(�, �) − �(�, �) for all �, � ∈ A

(3)�∗(�, �) = { |� − �| if � ̸= �,−� if � = �.

�e axioms (�∗1 ), (�∗2 ), and (�∗3 ) can be proved immediately. We just prove axiom (�∗4 ) in details, for all �, �, � ∈ A .

Case 1. If � ̸= � = �, then �(�, �) ≤ �(�, �) + �(�, �) − �(�, �) implies |� − �| = |� − �|.Case 2. If � = � ̸= �, then �(�, �) ≤ �(�, �) + �(�, �) − �(�, �) implies |� − �| = |� − �|.Case 3. If � = � = �, then �(�, �) ≤ �(�, �) + �(�, �) − �(�, �) implies −� = −�.Case 4. If � ̸= � ̸= �, then �(�, �) ≤ �(�, �) + �(�, �) − �(�, �) implies |� − �| ≤ |� − �| + |� − �| + �.

�us, the axiom (�∗4 ) holds in all cases. Hence (A , �∗) is a dualistic partial metric space.

O’Neill [22] established that each dualistic partial metric �∗ on A generates a �0 topology �(�∗) on A having a base, the family of �∗-balls {��∗(�, �) : � ∈ A , � > 0} where

De�nition 3 ([22] ). Let (A , �∗) be a dualistic partial metric space, then

(1) A sequence {��}�∈N in (A , �∗) converges to a point � ∈ A if and only if �∗(�, �) = lim�→∞�∗(�, ��).(2) A sequence {��}�∈N in (A , �∗) is called a Cauchy

sequence if lim�,�→∞�∗(��, ��) exists and is �nite.(3) (A , �∗) is said to be complete if every Cauchy sequence {��}�∈N in A converges, with respect to �(�∗), to a

point � ∈ A such that �∗(�, �) = lim�,�→∞�∗(��, ��).�e following lemma will be helpful in the sequel.

Lemma 1 ([22, 22]). (1) A dualistic partial metric space (A , �∗) is complete i� the metric space (A , ���∗) is complete.

(2) A sequence {��}�∈N in A converges to a point � ∈ A , with respect to �(���∗) i�

3. Main Results

We begin with the following useful property.

De�nition 4. Let � be a self-mapping on a dualistic partial metric space (A , �∗). If there is a convergent sequence {��} in A with �� → �, such that

then � is said to have the convergence comparison property [in short, (CCP)].

(4)��∗(�, �) = {� ∈ A : �∗(�, �) < � + �∗(�, �)}.

(5)lim�→∞ �∗(�, ��) = �∗(�, �) = lim�, �→∞ �∗(��, ��).

(6)�∗(�, �) ≤ �∗(�(�), �(�)),

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Example 4. Let A = Q. De�ne �∗∨ : A ×A → R by

Clearly, (A , �∗∨) is a dualistic partial metric space. Consider {�� = 1/� − 1, � ≥ 1}�∈N ⊂ A . We have lim�→∞�∗∨(��, −1) =�∗∨(−1, −1). �at is, �� → � = −1 in (A , �∗∨). De�ne �(�) = ��for all � ∈ A . For such � = −1, observe that �∗∨(�, �) =−1 ≤ �−1 = �∗∨(�(�), �(�)), so � has the convergence compar-ison property.

�e following theorem corresponds to the unique �xed point result of Bianchini type dualistic contraction.

Theorem 1. Let � be a self-mapping on a complete dualistic partial metric space (A , �∗) satisfying (CCP). If there is � ∈ [0, 1) so that

for all �, � ∈ A , then � possesses a unique �xed point.

Proof. We generate a Picard iterative sequence {��}�∈N with an initial point �0 ∈ A such that �� = �(��−1) for all � ∈ N. If there exists �0 ∈ N such that ��0 = ��0+1 = �(��0), then ��0is a �xed point of �, so the proof is completed. From now on, assume that �� ̸= ��+1 for all � ∈ N, then by the contractive condition (8), we have

�us,

If

then (10) leads to a contradiction. As a result, we have

Arguing like above, we have

�us, inequality (12) entails

(7)�∗∨(�, �) = max{�, �} for all �, � ∈ A .

(8)�����∗(�(�), �(�))���� ≤ �max{�����∗(�, �(�))����, �����∗(�, �(�))����},

(9)

�����∗(��, ��+1)���� = �����∗(�(��−1), �(��))����≤ � max{�����∗(��−1, �(��−1))����, �����∗(��, �(��))����}= � max{�����∗(��−1, ��)����, �����∗(��, ��+1)����}.

(10)�����∗(��, ��+1)���� ≤ � max{�����∗(��−1, ��)����, �����∗(��, ��+1)����}.

(11)max{�����∗(��−1, ��)����, �����∗(��, ��+1)����} = �����∗(��, ��+1)����,

(12)�����∗(��, ��+1)���� ≤ ������∗(��−1, ��)����,

(13)

�����∗(��−1, ��)���� = �����∗(�(��−2), �(��−1))����≤ � max{�����∗(��−2, �(��−2))����, �����∗(��−1, �(��−1))����}= � max{�����∗(��−2, ��−1)����, �����∗(��−1, ��)����}.

(14)max{�����∗(��−2, ��−1)����, �����∗(��−1, ��)����} = �����∗(��−2, ��−1)����.

(15)�����∗(��, ��+1)���� ≤ �2�����∗(��−2, ��−1)����.

Proceeding further in a similar way, we get

Now consider

the inequality (16) implies

We deduce from (1) that

and using inequalities (16) and (18), we obtain

Now, for � > �, we have

As �, �→∞, ���∗(��, ��) = max{�∗� (��, ��), �∗� (��, ��)}→ 0, thus, {��} is a Cauchy sequence in (A , ���∗). Since (A , �∗) is a complete dualistic partial metric space, by Lemma 1 (1), (A , ���∗)is a complete metric space. �us, there exists �∗ ∈ (A , ���∗) such that �� → �∗ as �→∞, that is lim�→∞��∗(��, �∗) = 0 and by Lemma 1 (2), we know that

Since, lim�, �→∞ ��∗(��, ��) = 0, by (1) and (18), we have

(16)�����∗(��, ��+1)���� ≤ �������∗(�0, �1)����.

(17)

�����∗(��, ��)���� = �����∗(�(��−1), �(��−1))����≤ � max{�����∗(��−1, �(��−1))����, �����∗(��−1, �(��−1))����}= � max{�����∗(��−1, ��)����, �����∗(��−1, ��)����} = ������∗(��−1, ��)����,

(18)�����∗(��, ��)���� ≤ �������∗(�0, �1)����.

(19)��∗(��, ��+1) + �∗(��, ��) ≤ �����∗(��, ��+1)����,

(20)��∗(��, ��+1) ≤ 2�������∗(�0, �1)����.

(21)

��∗(��, ��) ≤ ��∗(��, ��+1) + ��∗(��+1, ��+2)+ ⋅ ⋅ ⋅ + ��∗(��−1, ��)≤ 2�������∗(�0, �1)���� + 2��+1�����∗(�0, �1)����+ ⋅ ⋅ ⋅ + 2��−1�����∗(�0, �1)����≤ 2(�� + ��+1 + ⋅ ⋅ ⋅ + ��−1)�����∗(�0, �1)����≤ 2��(1 − ��−�)1 − � �����∗(�0, �1)����

(22)

��∗(��, ��) ≤ ��∗(��, ��−1) + ��∗(��−1, ��−2)+ ⋅ ⋅ ⋅ + ��∗(��+1, ��)≤ 2��−1�����∗(�0, �1)���� + 2��−2�����∗(�0, �1)����+ ⋅ ⋅ ⋅ + 2�������∗(�0, �1)����≤ 2(�� + ��+1 + ⋅ ⋅ ⋅ + ��−1)�����∗(�0, �1)����≤ 2��(1 − ��−�)1 − � �����∗(�0, �1)����.

(23)�∗(�∗, �∗) = lim�→∞ �∗(��, �∗) = lim�, �→∞ �∗(��, ��).

(24)

�∗(�∗, �∗) = lim�→∞ �∗(��, �∗) = lim�, �→∞ �∗(��, ��) = 0.

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then, (A , �∗) is a complete dualistic partial metric space. De�ne a mapping � : A → A by

Note that the mapping � has the convergence comparison property (CCP). Indeed, consider {�� = −1/� − 2|� = 1, 2 , . . .}. Here {��} is a sequence in A . Clearly, we have that �� → � = −2as �→∞ in (A , �∗). We have �∗(−2, −2) = −2 ≤= 0= �∗(�(−2), �(−2)). Choose � = 3/4. On the other hand, the contractive condition (8) is also satis�ed. For this, let �, � ∈ A . Here, we need the following cases:

Case 1. (a) � ̸= � and �, � ∈ (−4, 0]. We have

(b) � ̸= � and �, � ∈ (−∞,−4]. We have

(c) � ̸= � and � ∈ (−∞,−4], � ∈ (−4, 0] and vice versa. In this case,

Case 2. (a) � = � ∈ (−4, 0]. We have

(b) � = � ∈ (−∞,−4]. We have

Hence, � veri�es all conditions of �eorem 1. Note that � = 0is the unique �xed point of �. Example 6. Let A = (−∞, 0]. De�ne the mapping �∗ : A ×A → R by

then (A , �∗) is a complete dualistic partial metric space. De�ne a mapping � : A → A as in Example 5. It is obvious that � veri�es all conditions of �eorem 1. But, classical Bianchini contraction (contraction without the absolute value function) is not applicable. Indeed, for � = −3 and � = −2, we have

for each � ∈ (0, 1). In what follows, we investigate a unique �xed point of Reich

type dualistic contraction mappings.

(33)�(�) = {−1 if � ∈ (−∞,−4],0 if � ∈ (−4, 0].

(34)�����∗(�(�), �(�))���� = 0.

(35)

�����∗(�(�), �(�))���� = 1 ≤ 3� ≤ � max{�����∗(�, �(�))����, �����∗(�, �(�))����}.

(36)

�����∗(�(�), �(�))���� = 1 ≤ 3� ≤ � max{�����∗(�, �(�))����, �����∗(�, �(�))����}.

(37)�����∗(�(�), �(�))���� = 0.

(38)

�����∗(�(�), �(�))���� = 1 ≤ 3� ≤ � max{�����∗(�, �(�))����, �����∗(�, �(�))����}.

(39)�∗(�, �) = {� + � if � ̸= �,� ∨ � if � = �,

(40)�∗(�(�), �(�)) = �∗(0, 0) = 0 > −2� = �max{�∗(�, �(�)), �∗(�, �(�))}

�is shows that {��} is a Cauchy sequence converging to �∗ ∈ (A , �∗). We shall show that �∗ is a �xed point of �. By (8), we have

as �→∞, �����∗(�∗, �(�∗))���� ≤ ������∗(�∗, �(�∗))���� which implies that �∗(�∗, �(�∗)) = 0. Since � has (CCP), we get

On the other hand, by axiom (�∗4 ), we have �∗(�∗, �∗) ≤�∗(�∗, �(�∗)) + �∗(�(�∗), �∗) − �∗(�(�∗), �(�∗)) which implies that

�e inequalities (26) and (27) imply that �∗(�(�∗),�(�∗)) = 0. �us,

By using axiom (�∗1 ), we have �∗ = �(�∗). �is shows that �∗is a �xed point of �. To prove the uniqueness, suppose that �∗is another �xed point of �, then �(�∗) = �∗ and �∗(�∗, �∗) = 0. By (8), we obtain

�is implies that �∗(�∗, �∗) = �∗(�∗, �∗) = �∗(�∗, �∗) = 0and using axiom (�∗1 ), we have �∗ = �∗, which proves the uniqueness of �∗. ☐

Corollary 1. Let (A , �) be a complete partial metric space and let � : A → A be a mapping satisfying

for all �, � ∈ A and 0 ≤ � < 1. �en, � has a unique �xed point.

Proof. Since the restriction of a dualistic partial metric �∗ to [0,∞), which is �∗����[0,∞) = �, is a partial metric, so arguments follow the same lines as in the proof of �eorem 1. ☐

Corollary 2 ([26]). Let (A , �) be a complete metric space and let � : A → A be a mapping satisfying

for all �, � ∈ A and 0 ≤ � < 1. �en, � has a unique �xed point.

Proof. Set �(�, �) = 0 for all � ∈ A , in Corollary 1. ☐

Example 5. Let A = (−∞, 0]. De�ne the mapping �∗ : A ×A → R by

(25)

�����∗(��, �(�∗))���� = �����∗(�(��−1), �(�∗))����≤ � max{�����∗(��−1, �(��−1))����, �����∗(�∗, �(�∗))����},

(26)0 = �∗(�∗, �∗) ≤ �∗(�(�∗), �(�∗)).

(27)�∗(�(�∗), �(�∗)) ≤ 0.

(28)�∗(�∗, �(�∗)) = �∗(�(�∗), �(�∗)) = �∗(�∗, �∗).

(29)

�����∗(�∗, �∗)���� = �����∗(�(�∗), �(�∗))����≤ � max{�����∗(�∗), (�(�∗))����, �����∗(�∗, �(�∗))����} = 0.

(30)�(�(�), �(�)) ≤ � max{�(�, �(�)), �(�, �(�))},

(31)�(�(�), �(�)) ≤ � max{�(�, �(�)), �(�, �(�))},

(32)�∗(�, �) = { |� − �| if � ̸= �,� ∨ � if � = �,

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5Discrete Dynamics in Nature and Society

�e equation (1) implies that

Let �� = �� + ��−1 + ���−2 + ⋅ ⋅ ⋅ + ��−1. �en,

Now, for � > �, we have

We conclude that lim�, �→∞ ���∗(��, ��) = 0. �is shows that {��} is a Cauchy sequence in (A , ���∗). Since (A , �∗) is a com-plete dualistic partial metric space, by Lemma 1 (1), (A , ���∗)is a complete metric space. �us, there exists �∗ ∈ (A , ���∗)such that �� → �∗ as �→∞, that is lim�→∞ ��∗(��, �∗) = 0and by Lemma 1 (2), we know that

Since lim�, �→∞ ��∗(��, ��) = 0, by (1) and (50), we have

�is shows that {��} is a Cauchy sequence converging to �∗ ∈ (A , �∗). We show that �∗ is a �xed point of �. By (41), we have

Applying limit as �→∞ and using equation (55), we have �∗(�∗, �(�∗)) = 0. Since � has (CCP), we get

(50)

�����∗(��, ��)���� ≤ �������∗(�0, �0)���� + (�� − ��� − � )(� + �)�����∗(�0, �1)����.

(51)

��∗(��, ��+1) ≤ �����∗(��, ��+1)���� − �∗(��, ��) ≤ �����∗(��, ��+1)����+ �����∗(��, ��)���� ≤ �������∗(�0, �1)���� + �������∗(�0, �0)����+ �� − ��� − � (� + �)�����∗(�0, �1)����≤ (�� + (� + �)(�� − ��� − � ))�����∗(�0, �1)���� + �������∗(�0, �0)����≤ (�� + �� − ��� − � )�����∗(�0, �1)���� + �������∗(�0, �0)����= (��(� − �) + �� − ��� − � )�����∗(�0, �1)���� + �������∗(�0, �0)����= (�� + ��−1 + ���−2 + ⋅ ⋅ ⋅ + ��−1)�����∗(�0, �1)���� + �������∗(�0, �0)����.

(52)��∗(��, ��+1) ≤ �������∗(�0, �1)���� + �������∗(�0, �0)����.

(53)

��∗(��, ��) ≤ ��∗(��, ��+1) + ��∗(��+1, ��+2) + ⋅ ⋅ ⋅ + ��∗(��−1, ��)≤ �������∗(�0, �1)���� + �������∗(�0, �0)���� + ��+1�����∗(�0, �1)����+ ��+1�����∗(�0, �0)����+ ⋅ ⋅ ⋅ + ��−1�����∗(�0, �1)���� + ��−1�����∗(�0, �0)����= (�� + ��+1 + ⋅ ⋅ ⋅ + ��+1)�����∗(�0, �1)����+ (�� + ��+1 + ⋅ ⋅ ⋅ + ��−1)�����∗(�0, �0)����≤ (�� + ��+1 + ⋅ ⋅ ⋅ + ��+1 + ⋅ ⋅ ⋅)�����∗(�0, �1)����+ (�� + ��+1 + ⋅ ⋅ ⋅ + ��−1 + ⋅ ⋅ ⋅)�����∗(�0, �0)����= ��1 − ������∗(�0, �1)���� +

��1 − � �����∗(�0, �0)����.

(54)�∗(�∗, �∗) = lim�→∞ �∗(��, �∗) = lim�, �→∞ �∗(��, ��).

(55)�∗(�∗, �∗) = lim�→∞ �∗(��, �∗) = lim�, �→∞ �∗(��, ��) = 0.

(56)

�����∗(��, �(�∗))���� = �����∗(�(��−1), �(�∗))���� ≤ ������∗(��−1, �∗)����+ ������∗(��−1, �(��−1))���� + ������∗(�∗, �(�∗))����.

Theorem 2. Let (A , �∗) be a complete dualistic partial metric space and let � : A → A be a mapping satisfying (CCP). If there exist �, �, � ≥ 0 with � + � + � < 1 such that

for all �, � ∈ A , then, � has a �xed point.

Proof. Let {��}�∈N be a sequence with an initial point �0 ∈ �such that �� = �(��−1) for all � ∈ N. If there exists �0 ∈ N such that ��0 = ��0+1 = �(��0), then ��0 is a �xed point of �. On the other hand, if �� ̸= ��+1 for all � ∈ N, then by contractive condition (41), we have

Let � = (� + �)/(1 − �), then 0 ≤ � < 1 and repeating argu-ments given above, we have

Now, consider the self-distance

Due to inequality (44), we have

�e inequality (46) implies that

Continuing further, we get

(41)

�����∗(�(�), �(�))���� ≤ ������∗(�, �)���� + ������∗(�, �(�)) + ������∗(�, �(�))����

(42)

�����∗(��, ��+1)���� = �����∗(�(��−1), �(��))����≤ ������∗(��−1, ��)���� + ������∗(��−1, �(��−1))����+ ������∗(��, �(��))����= ������∗(��−1, ��)���� + ������∗(��−1, ��)���� + ������∗(��, ��+1)����,

(43)�����∗(��, ��+1)���� ≤ � + �1 − � �����∗(��−1, ��)����.

(44)�����∗(��, ��+1)���� ≤ �������∗(�0, �1)����.

(45)

�����∗(��, ��)���� = �����∗(�(��−1), �(��−1))���� ≤ ������∗(��−1, ��−1)����+ ������∗(��−1, �(��−1))���� + ������∗(��−1, �(��−1))����= ������∗(��−1, ��−1)���� + ������∗(��−1, ��)���� + ������∗(��−1, ��)����= ������∗(��−1, ��−1)���� + (� + �)�����∗(��−1, ��)����.

(46)

�����∗(��, ��)���� ≤ ������∗(��−1, ��−1)���� + (� + �)��−1�����∗(�0, �1)����,

(47)

�����∗(��−1, ��−1)���� = �����∗(�(��−2), �(��−2))���� ≤ ������∗(��−2, ��−2)����+ ������∗(��−2, �(��−2))���� + ������∗(��−2, �(��−2))����≤ ������∗(��−2, ��−2)���� + (� + �)�����∗(��−2, ��−1)����≤ ������∗(��−2, ��−2)���� + (� + �)��−2�����∗(�0, �1)����.

(48)

�����∗(��, ��)���� ≤ �2�����∗(��−2, ��−2)���� + �(� + �)��−2�����∗(�0, �1)����+ (� + �)��−1�����∗(�0, �1)���� ≤ �3�����∗(��−3, ��−3)����+ (�2��−3 + ���−2 + ��−1)(� + �)�����∗(�0, �1)����.

(49)

�����∗(��, ��)���� ≤ �������∗(�0, �0)���� + (��−1 + ��−2� + ��−3�2+ ⋅ ⋅ ⋅ + ���−2 + ��−1)(� + �)�����∗(�0, �1)����,

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Discrete Dynamics in Nature and Society6

� : A → A as in Example 7. It is obvious that (3.36) holds for all �, � ∈ A , while, the classical Reich �xed point theorem is not applicable. Indeed, for � = � = −1, we have

for all �, �, � ∈ (0, 1).Corollary 3. Let (A , �) be a complete partial metric space and let � : A → A be a mapping satisfying

for all �, � ∈ A and �, �, � ≥ 0 such that � + � + � < 1. �en, �has a �xed point.

Proof. Since the restriction of a dualistic partial metric �∗ to [0,∞), �∗����[0,∞) = �, is a partial metric, so arguments follow the same lines as in the proof of �eorem 2. ☐

Corollary 4 ([12]). Let (A , �) be a complete metric space and let � : A → A be a mapping satisfying

for all �, � ∈ A and �, �, � ≥ 0 such that � + � + � < 1. �en, �has a �xed point.

Proof. Set �(�, �) = 0 for all � ∈ A , in Corollary 3. ☐�e following theorem implies the uniqueness of the �xed

point of a new Rational type dualistic contraction.

(61)

�∗(�(�), �(�)) = 0 > −� − � − �= ��∗(�, �) + ��∗(�, �(�)) + ��∗∨(�, �(�))

(62)�(�(�), �(�)) ≤ ��(�, �) + ��(�, �(�)) + ��(�, �(�))

(63)�(�(�), �(�)) ≤ ��(�, �) + ��(�, �(�)) + ��(�, �(�))

By axiom (�∗2 ), we have

�e inequalities (57) and (58) imply that �∗(�(�∗), �(�∗)) = 0. �us,

By using axiom (�∗1 ), we have �∗ = �(�∗). �is shows that �∗is a �xed point of �. ☐

Example 7. Let A = {0,−1,−2,−3,−0.3,−0.1} and �∗ : A ×A → R be as in Example 5. �en (A , �∗) is a complete dualistic partial metric space. De�ne the mapping � : A → A by

�e mapping � satis�es (CCP) for any convergent sequence in A . Indeed, for a convergent sequence {��} in A such that �� → �, � ∈ A due to completeness of (A , �∗). �us for every such � we have �∗(�, �) ≤ �∗(�(�), �(�)). Observe that for each case (as shown in Table 1) there exist some �, �, � ≥ 0with � + � + � < 1 (however, assuming diªerent values in each case) for which the contractive condition (41) is satis�ed.

Example 8. Let A = {0,−1,−2,−3,−0.3,−0.1} and �∗ : A ×A → R be as in Example 6. �en (A , �∗) is a complete dualistic partial metric space. De�ne the mapping

(57)0 = �∗(�∗, �∗) ≤ �∗(�(�∗), �(�∗)).

(58)�∗(�(�∗), �(�∗)) ≤ �∗(�∗, �(�∗)) = 0.

(59)�∗(�∗, �(�∗)) = �∗(�(�∗), �(�∗)) = �∗(�∗, �∗).

(60)�(�) = {{{0 if � ∈ {0,−1,−0.3, −0.1},−0.3 if � = −2,−0.1 if � = −3.

Table 1: Evaluation of contractive condition.

(�, �) �����∗(�(�), �(�))���� ������∗(�, �)���� + ������∗(�, �(�))���� + ������∗(�, �(�))����(0,0) 0 0(0,−1) 0 � + �(0,−2) 0.3 2� + 1.7�(0,−3) 0.1 3� + 2.9�(0,−0.3) 0 0.3(� + �)(0,−0.1) 0 0.1(� + �)(−1,−2) 0.3 � + � + 2�(−1,−3) 0.1 2� + � + 2.9�(−1,−0.1) 0 0.9� + � + 0.1�(−1,−0.3) 0 0.7� + � + 0.3�(−1,−1) 0 � + � + �(−2,−2) 0.3 2� + 1.7(� + �)(−2,−3) 0.2 � + 1.7� + 2.9�(−2,−0.3) 0.3 1.7(� + �) + 0.3�(−2,−0.1) 0.3 1.9� + 1.7� + 0.1�(−3,−3) 0.1 3� + 2.9(� + �)(−3,−0.1) 0.1 2.9(� + �) + 0.1�(−3,−0.3) 0.1 2.7� + 2.9� + 0.3�(−0.3,−0.3) 0 0.3(� + � + �)(−0.1,−0.1) 0 0.1(� + � + �)

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7Discrete Dynamics in Nature and Society

Continuing in the same way, we get

�e equation (1) implies

Set �� + ��−2 + ���−3 + ⋅ ⋅ ⋅ + ��−2 = ��. �en,

Now, for � > �, we have

We deduce that lim�, �→∞ ��∗(��, ��) = 0 and similarly, lim�, �→∞ ��∗(��, ��) = 0 which implies that lim�, �→∞ ���∗(��, ��) = 0. �e remaining part of this proof is similar to the proof of �eorem 2. ☐

Corollary 5. Let (A , �) be a complete partial metric space and let � : A → A be a mapping satisfying

for all �, � ∈ A and �, �, � are non negative numbers such that � + � + � < 1. �en, � has a �xed point.

Proof. Since the restriction of a dualistic partial metric �∗ to [0,∞), �∗����[0,∞) = � is a partial metric, the result is obvious. ☐

Corollary 6. Let (A , �) be a complete metric space and let � : A → A be a mapping satisfying

(70)

�����∗(��, ��)���� ≤ (� + �)(��−1 + ���−2)�����∗(�0, �1)���� + �2�����∗(��−2, ��−2)����.

(71)

�����∗(��, ��)���� ≤(� + �)(��−1 + ���−2 + �2��−3 + ⋅ ⋅ ⋅ + ��−1)⋅ �����∗(�0, �1)���� + �������∗(�0, �0)����,

(72)

�����∗(��, ��)���� ≤ (� + �)(��−1 − ��−1� − � )�����∗(�0, �1)���� + �������∗(�0, �0)����.

(73)

��∗(��, ��+1) ≤ �����∗(��, ��+1)���� + �����∗(��, ��)����≤ �������∗(�0, �1)���� + (� + �)(�

�−1 − ��−1� − � )�����∗(�0, �1)����

+ �������∗(�0, �0)����≤ (�� + ��−2 + ���−3 + ⋅ ⋅ ⋅ + ��−2)�����∗(�0, �1)����+ �������∗(�0, �0)����.

(74)��∗(��, ��+1) ≤ �������∗(�0, �1)���� + �������∗(�0, �0)����.

(75)

��∗(��, ��) ≤ ��∗(��, ��+1) + ��∗(��+1, ��+2) + ⋅ ⋅ ⋅ + ��∗(��−1, ��)≤ �������∗(�0, �1)���� + �������∗(�0, �0)���� + ��+1�����∗(�0, �1)����+ ��+1�����∗(�0, �0)���� + ⋅ ⋅ ⋅ + ��−1�����∗(�0, �1)���� + ��−1�����∗(�0, �0)����≤ (�� + ��+1 + ⋅ ⋅ ⋅ + ��+1)�����∗(�0, �1)����+ (�� + ��+1 + ⋅ ⋅ ⋅ + ��−1)�����∗(�0, �1)����.

(76)

�(�(�), �(�)) ≤ ��(�, �(�)) ⋅ �(�, �(�))�(�, �) + ��(�, �(�)) + ��(�, �),

(77)

�(�(�), �(�)) ≤ ��(�, �(�)) ⋅ �(�, �(�))�(�, �) + ��(�, �(�)) + ��(�, �)

Theorem 3. Let (A , �∗) be a complete dualistic partial metric space and let � : A → A be a mapping satisfying (CCP). If there exist �, �, � ≥ 0 with � + � + � < 1 such that

for all �, � ∈ A . �en, � has a �xed point.

Proof. Let {��}�∈N be a sequence with an initial point �0 ∈ Asuch that �� = �(��−1) for all � ∈ N. If there exists �0 ∈ N such that ��0 = ��0+1 = �(��0), then ��0 is a �xed point of �. On the other hand, if �� ̸= ��+1 for all � ∈ N, then from contractive condition (64), we have

Let � = �/(1 − � − �) < 1, so

Now,

As �∗(��−1, ��)/�∗(��−1, ��−1) ≥ 1, we get

Arguing like above, we have

�e inequality (68) leads to

(64)�����∗(�(�), �(�))���� ≤ �

�����∗(�, �(�)) ⋅ �∗(�, �(�))���������∗(�, �)����+ ������∗(�, �(�))���� + ������∗(�, �)����,

(65)

�����∗(��, ��+1)���� = �����∗(�(��−1), �(��))����≤ �����������∗(��−1, �(��−1)) ⋅ �∗(��, �(��))�∗(��−1, ��)

���������+ ������∗(��, �(��))���� + ������∗(��−1, ��)����= �����������∗(��−1, ��) ⋅ �∗(��, ��+1)�∗(��−1, ��)

���������+ ������∗(��, ��+1)���� + ������∗(��−1, ��)����= (� + �)�����∗(��, ��+1)���� + ������∗(��−1, ��)���������∗(��, ��+1)���� ≤ �1 − � − � �����∗(��−1, ��)����.

(66)

�����∗(��, ��+1)���� ≤ ������∗(��−1, ��)���� ≤ �2�����∗(��−2, ��−1)���� ⋅ ⋅ ⋅ �������∗(�0, �1)����.

(67)

�����∗(��, ��)���� = �����∗(�(��−1), �(��−1))����≤ �����������∗(��−1, �(��−1)) ⋅ �∗(��−1, �(��−1))�∗(��−1, ��−1)

���������+ ������∗(��−1, �(��−1))���� + ������∗(��−1, ��−1)����= �����������∗(��−1, ��) ⋅ �∗(��−1, ��)�∗(��−1, ��−1)

���������+ ������∗(��−1, ��)���� + ������∗(��−1, ��−1)����.

(68)

�����∗(��, ��)���� ≤ ������∗(��−1, ��)���� + ������∗(��−1, ��)���� + ������∗(��−1, ��−1)����= (� + �)�����∗(��−1, ��)���� + ������∗(��−1, ��−1)����= (� + �)��−1�����∗(�0, �1)���� + ������∗(��−1, ��−1)����.

(69)

�����∗(��−1, ��−1)���� ≤ (� + �)��−2�����∗(�0, �1)���� + ������∗(��−2, ��−2)����.

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Discrete Dynamics in Nature and Society8

Let �, � ∈ A and � ∈ [0, 1], by assumption (�), we get

Since ∫10�(�, �)�� = −(�2/2) + (�/2) for all � ∈ [0, 1], we have sup�∈[0,1][∫10�(�, �)��] = 1/8, which implies that

Hence, application of �eorem 1 ensures that � has at least one �xed point �(�) ∈ A , that is, �(�(�)) = �(�) which is a solution of (82). ☐

Data Availability

�e data used to support the �ndings of this study are available from the corresponding author upon request.

Conflicts of Interest

�e authors declare that they have no con³icts of interest.

Authors’ Contributions

All authors contributed equally and signi�cantly in writing this article. All authors read and approved the �nal manuscript.

Acknowledgments

�e third author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

References

[1] H. Aydi, E. Karapinar, and A. F. Rold án López de Hierro, “ω-Interpolative Ciric–Reich–Rus type contractions,” Mathematics, vol. 7, no. 1, p. 57, 2019.

[2] E. Karapinar, R. P. Agarwal, and H. Aydi, “Interpolative Reich-Rus-Ciric type contractions on partial metric spaces,” Mathematics, vol. 6, p. 256, 2018.

(84)

|��(�) − ��(�)| = [��������∫1

0�(�, �)[�(�, �(�)) − �(�, �(�))] ���������]

≤ [∫10�(�, �)�����(�, �(�)) − �(�, �(�))���� �]

≤ [8∫10�(�, �) ln(���� ) �]

≤ [8∫10�(�, �) ln(���� ) �]

= 8 ln(���� )( sup�∈[0,1] [∫1

0�(�, �) �]).

(85)

�����∗(�(�), �(�))���� = sup�∈[0,1]|��(�) − ��(�)| + � ≤ ��, where � = ln( )

= � max{�����∗(�, �(�))����, �����∗(�, �(�))����}.

for all �, � ∈ A and �, �, � are non negative numbers such that � + � + � < 1. �en, � has a �xed point.

Proof. Take in Corollary 5, �(�, �) = 0 for all � ∈ A . ☐

4. Application

In this section, we present an application of �eorem 1 to ensure the existence of the solution of the boundary value problem given by

where � : [0, 1] × R→ R is a continuous mapping. �e Green function associated to the boundary value problem (78) is de�ned by

Let �[0, 1] be the space of all continuous mappings de�ned on [0, 1]. Let A = (�[0, 1],R). De�ne the mapping �∗ : A ×A → R by

It is known that (A , �∗) is a complete dualistic partial metric space. De�ne � : A → A by

for all � ∈ [0, 1]. Note that the problem (78) has a solution iª the operator � has a �xed point.

Theorem 4. Let A = �([0, 1],R). De�ne the mapping � : A → A by

where � : [0, 1] ×A → R is a continuous mapping and �(�) is such that �∗(�(�), �(�)) ≤ �∗(�(�(�))), �(�(�)). Assume that

(i) �e mapping � : [0, 1] ×A → R satis�es

for all � ∈ [0, 1], �, � ∈ A , � = max{�����∗(�, �(�))����, �����∗(�, �(�))����}and � > 0;

�en, boundary value problem (78) has a solution.

Proof. We note that �(�) ∈ (�2[0, 1],R) (say) is a solution of (78) if and only if �(�) ∈ A is a solution of the integral equa-tion (82). �e solution of (82) is given by the �xed point of �, i.e., �(�) = �(�(�)).

(78)−�2���2 = �(�, �(�)), � ∈ [0, 1],�(0) = �(1) = 0,

(79)�(�, �) = { �(1 − �), 0 ≤ � ≤ � ≤ 1,�(1 − �), 0 ≤ � ≤ � ≤ 1.

(80)

�∗(�, �) = ‖(� − �)‖∞ + � = sup�∈[0,1]|�(�) − �(�)| + �, � ∈ R.

(81)��(�) = ∫10�(�, �)�(�, �(�))��

(82)��(�) = ∫10�(�, �)�(�, �(�))��,

(83)�����(�, �) − �(�, �)���� ≤ 8 ln(���

� )

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