Fixed points of (alpha, psi)-contractions in Hausdorff partial metric spaces

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DOI: 10.1002/mma.5251

S P E C I A L I S S U E P A P E R

Fixed points of

(𝛼, 𝜓)-contractions in Hausdorff partial

metric spaces

Muhammad Nazam

1

_{Özlem Acar}

2

1_{Department of Mathematics and}

Statistics, International Islamic University, Islamabad, H-10, Pakistan

2_{Department of Mathematics, Faculty of}

Science, Selcuk University, Selcuklu, Konya, 42003, Turkey

Correspondence

Özlem Acar, Department of Mathematics, Faculty of Science, Selcuk University, Selcuklu, Konya 42003, Turkey. Email: [email protected]

Communicated by: T. Acar

MSC Classification: 47H10; 54H25

The concept of(𝛼, 𝜓)-contractions was introduced by Samet et al. in this paper, we introduce(𝛼, 𝜓)-generalized contractions in a Hausdorff partial metric space. We discuss its significance and obtain some common fixed point theorems for a pair of self-mappings. Some examples are given to support the theory.

K E Y WO R D S

(𝛼, 𝜓)-set valued generalized contraction, complete partial metric space, fixed point

1 I N T RO D U CT I O N

LetΨ represent the class of all functions 𝜓 ∶ [0, ∞) → [0, ∞) satisfying the following conditions: (1) 𝜓 is strictly increasing;

(2) ∑∞_n₌₁𝜓n_{(t) < ∞ for all t > 0, 𝜓}n_{being n}th_{iterate of}_𝜓.

These functions are known in the literature as c-comparison functions. It is easy to prove that if𝜓 is a c-comparison function, then𝜓(t) < t for any t > 0.

Definition 1. (Samet et al1_{) Let}_{𝛼 ∶ X × X → [0, ∞) be a function. The mapping T ∶ X → X is said to be an}

𝛼-admissible if it satisfies the condition:

𝛼(x, 𝑦) ≥ 1implies 𝛼(T(x), T(𝑦)) ≥ 1for all x, 𝑦 ∈ X.

Recently, Samet et al1 _{introduced a meaningful generalization of Banach Contraction Principle using concept of}

𝛼-admissible mappings and control functions.

Definition 2. (Samet et al1_{) Let}(X, d) be a metric space and 𝛼 ∶ X × X → [0, ∞) be a function. The mapping T ∶ X → X is said to be an(𝛼, 𝜓)-contraction mapping if it satisfies the inequality:

𝛼(x, 𝑦)d(T(x), T(𝑦)) ≤ 𝜓(d(x, 𝑦))for all x, 𝑦 ∈ X. (1) Let h ≥ 1 and k ∈ [0, 1), if we define 𝛼(x, y) = h for all x, y ∈ X and 𝜓(t) = kt; t > 0, then inequality 1 reduces to a Banach contraction. Samet1_{presented the following famous theorem.}

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Theorem 1. Let(X, d) be a complete metric space and T ∶ X → X be an (𝛼 − 𝜓)-contractive mapping. If T is an

𝛼-admissible and continuous, then T has a unique fixed point in X.

Asl et al2_{established Theorem 1 for multivalued mappings, and Kumam et al}3_{established the same in a partial metric} space.

In this article, motivated by Samet et al1

and Hassen Aydi et al,4

we prove some common fixed point theorem for a pair of self-mappings satisfying(𝛼, 𝜓)-set valued generalized contraction in a complete partial metric space. We give examples to illustrate main result.

2 P R E L I M I NA R I E S

Throughout this paper, we denote(0, ∞) byR+,[0, ∞) byR+₀,(−∞, +∞) byR, set of natural numbers byNand set of whole numbers byW. The following concepts and results will be required for the proofs of main results.

The notion of a partial metric space was introduced by Steve G. Matthews.5_{The partial metric space is a generalization} of a metric space in which the self-distance is no longer necessarily zero. The notions such as convergence, completeness, and Cauchy sequence in the setting of partial metric spaces can be found in the literature5-10_{and references therein.}

Definition 3. (Matthews5_{) Let M be a nonempty set and if the function p}∶ M×M →R+

0satisfies following properties:

(p1) r1 = r2⇐⇒ p (r1, r1) = p (r1, r2) = p (r2, r2),

(p2) p(r1, r1) ≤ p (r1, r2),

(p3) p(r1, r2) = p (r2, r1),

(p4) p(r1, r3) ≤ p (r1, r2) + p (r2, r3) − p (r2, r2).

for all r1, r2, r3 ∈ M. Then p is called a partial metric on M and the pair (M, p) is known as partial metric space.

Example 1. Consider the function p∶R+₀ ×R+₀ →R+₀ defined by p(r1, r2) = max{r1, r2}. It is easy to check that p

satisfies(p₁) − (p₄), and hence, p is a partial metric onR+₀. However, p does not define a metric onR+₀, since p(a, a) = a for all a∈R+₀.

Example 1 is a classical example of a partial metric. We give, in the following, a new nontrivial example of a partial metric space.

Example 2. Let the set of rational numbers beQ= {r1, r2, · · ·}. We define p ∶R×R→R+by the following:

p(r, s) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 if r= s ∈R−Q; 3 2 if r≠ s ∈R−Q; 1 3 if r= s ∈Q; 1+ 1 m+ 1 n if r= rm, s = rnand m≠ n; 1+ 1 n if{r, s} ∩Q= {rn}and{r, s} −Q≠ 𝜙.

Clearly, p satisfies p₁ − p₃. To prove p₄, let r, s, u ∈R−Qand m ≠ n. Then,

p(r, s) + p(u, u) ≤ p(r, u) + p(s, u); p(rn, s) + p(u, u) = 2 + 1

n ≤ p (rn, u) + p(s, u); p(rn, rn) + p(u, u) = 4

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p(rm, rn) + p(u, u) = 2 + 1 m+ 1 n = p (rm, u) + p (rn, u) ; p(r, s) + p (rk, rk) < 2 < p (r, rk) + p (s, rk) ; p(rn, s) + p (rk, rk) = 4 3 + 1 n < 2 + 1 n + 2 k = p (rn, rk) + p (s, rk) ; p(rn, rn) + p (rk, rk) = 2 3 ≤ p (rn, rk) + p (rn, rk) ; p(rm, rn) + p (rk, rk) = 4 3 + 1 m + 1 n ≤ p (rm, rk) + p (rn, rk) .

Note that p is not a metric onR.

Matthews5_{proved that every partial metric p on M induces a metric p}s_{∶ M × M →}_R+

0 defined by the following:

ps(r1, r2) = 2p (r1, r2) − p (r1, r1) − p (r2, r2) ; (2)

for all r1, r2 ∈ M.

Notice that a metric on a set M is a partial metric p such that p(r, r) = 0 for all r ∈ M and p(r1, r2) = 0 implies r1 = r2

( using(p₁) and (p₂)).

Matthews5_{established that each partial metric p on M generates a T}

0topology𝜏(p) on M. The base of topology 𝜏(p) is

the family of open p-balls{Bp(r, 𝜖) ∶ r ∈ M, 𝜖 > 0

}

, where Bp(r, 𝜖) = {r1∈ M ∶ p (r, r1) < p(r, r) + 𝜖} for all r ∈ M and

𝜖 > 0. A sequence {rn}n∈Nin(M, p) converges to a point r ∈ M if and only if p(r, r) = limn→∞p(r, rn).

Definition 4. (Matthews5_{) Let}(M, p) be a partial metric space.

(1) A sequence{rn}n∈Nin(M, p) is called a Cauchy sequence if limn,m→∞p(rn, rm) exists and is finite.

(2) A partial metric space(M, p) is said to be complete if every Cauchy sequence {rn}n∈Nin M converges, with respect to𝜏(p), to a point r ∈ X such that p(r, r) = limn,m→∞p(rn, rm).

The following lemma will be helpful in the sequel.

Lemma 1.

(1) A sequence rnis a Cauchy sequence5in a partial metric space(M, p) if and only if it is a Cauchy sequence in metric

space(M, ps₎

(2) A partial metric space(M, p) is complete if and only if the metric space (M, ps_{) is complete.}

(3) A sequence{rn}n∈Nin M converges to a point r ∈ M, with respect to 𝜏(ps_{) if and only if lim}

n→∞p(r, rn) = p(r, r) =

limn,m→∞p(rn, rm).

(4) Iflimn→∞rn= 𝜐 such that p(𝜐, 𝜐) = 0 then limn→∞p(rn, r) = p(𝜐, r) for every r ∈ M.

Let CBp(X) be the family of all nonempty, closed and bounded subsets of the partial metric space (X, p) induced by the partial metric p. Note that the closeness is taken from(X, 𝜏p) and the boundedness is given as follows: A is a bounded subset

in(X, p) if there exists x0 ∈ X and M ≥ 0 such that for all a ∈ A, we have a ∈ Bp(x0, M), that is, p(x0, a) < p(x0, x0) + M.

Definition 5. (Aydi et al4_{) For A}_{, B ∈ CB}p_{(X), x ∈ X and the mapping 𝛿}

p∶ CBp(X) × CBp(X) →R+define

(1) Dp(x, A) = inf {p(x, a) ∶ a ∈ A};

(2) 𝛿p(A, B) = sup {p(a, B) ∶ a ∈ A};

(3) 𝛿p(B, A) = sup {p(b, A) ∶ b ∈ B}; (4) Hp(A, B) = max { 𝛿p(A, B), 𝛿p(B, A) } .

It is easy to show that Dp(x, A) = 0 implies Dsp(x, A) = 0 where Dsp(x, A) = inf {ps(x, a) ∶ a ∈ A}.

Lemma 2. Let(X, p) be a PMS and A be any nonempty subset4_{of X then a}∈ ̄A if and only if D

p(a, A) = p(a, a).

Proposition 1. Let(X, p)4_{be PMS. For any A}_{, B, C ∈ CB}p_{(X), we have the following:}

(1) 𝛿p(A, A) = sup {p(a, b) ∶ a, b ∈ A};

(2) 𝛿p(A, B) = 𝛿p(B, A);

(3) 𝛿p(A, A) = 0 ⇒ A ⊆ B;

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Proposition 2.

(1) Hp(A, B) = 0 implies A = B;

(2) Hp(A, A) ≤ Hp(A, B);

(3) Hp(A, B) = Hp(B, A);

(4) Hp(A, B) ≤ Hp(A, C) + Hp(C, B) − infc∈Cp(c, c).

The function Hp ∶ CBp(X) × CBp(X) → [0, ∞) satisfying properties4in Proposition 2 is called partial Hausdorff metric. It

is easy to show that any Hausdorff metric is a partial Hausdorff metric. The converse is not true (see Aydi et al4, example 2.6)_.

Lemma 3. Let(X, p) be PMS , A, B ∈ CBp_{(X) and h > 1. For any a ∈ A}4_{there exists b} = b(a) ∈ B such that p(a, b) < hHp(A, B).

Definition 6. Let(X, p) be a partial metric space and let 𝛼 ∶ X × X → R+₀ be a function. The space(X, p) is said to be𝛼 -regular if for any sequence {xn} ⊂ X such that 𝛼(xn, xn+ 1) ≥ 1 for all n ∈ Nand xn → x as n → ∞, we have

𝛼(xn, x) ≥ 1 for all n ∈N.

3 (𝛼, 𝜓)-SET-VALUED CONTRACTIONS

We begin following definition.

Definition 7. Let(X, p) be a partial metric space. Let S, T ∶ X → CBp_{(X) be two set valued mappings and 𝛼 ∶ X × X →}

[0, + ∞) be a function. The pair (S, T) is said to be a triangular 𝛼∗-admissible if the following conditions hold:

(1) (S, T) is 𝛼_∗-admissible; that is,𝛼(x, y) ≥ 1 implies 𝛼_∗(Sx, Ty) ≥ 1 and 𝛼_∗(Tx, Sy) ≥ 1, where

𝛼∗(A, B) = inf {𝛼(x, 𝑦) ∶ x ∈ A, 𝑦 ∈ B} ,

(2) 𝛼(x, u) ≥ 1 and 𝛼(u, y) ≥ 1 imply 𝛼(x, y) ≥ 1.

It is easy to show that every𝛼_∗-admissible mapping is also an𝛼-admissible mapping but converse is not true (see Minak et al11, example 15_{). Let}_{(M, p) be a partial metric space. Define}

 (r1, r2) = max ⎧ ⎪ ⎨ ⎪ ⎩ p(r1, r2), Dp(r1,T(r1))Dp(r2,T(r2)) 1+p(r1,r2) , Dp(r1,T(r1))Dp(r2,T(r2)) 1+𝛿p(T(r1),T(r2)) ⎫ ⎪ ⎬ ⎪ ⎭ ; (r1, r2) = max ⎧ ⎪ ⎨ ⎪ ⎩ p(r1, r2), Dp(r1,S(r1))Dp(r2,T(r2)) 1+p(r1,r2) , Dp(r1,S(r1))Dp(r2,T(r2)) 1+𝛿p(S(r1),T(r2)) ⎫ ⎪ ⎬ ⎪ ⎭ .

Definition 8. Let(M, p) be a partial metric space and 𝛼 ∶ M × M → [0, ∞) be a function. The mapping T ∶ M →

CBp(M) is called (𝛼, 𝜓)-set valued contraction, if there exists 𝜓 ∈ Ψ such that

𝛼(r1, r2)Hp(T(r1), T(r2)) ≤ 𝜓(p(r1, r2)), (3)

for all r1, r2 ∈ M with 𝛼(r1, r2) ≥ 1.

The following example shows the significance of(𝛼, 𝜓) set-valued-contraction in a partial metric space over (𝛼, 𝜓) set-valued-contraction in a metric space.

Example 3. Let M = [0, 1] and define a partial metric by p(r1, r2) = max {r1, r2} for all r1, r2 ∈ M . The metric

ps_{induced by partial metric p is given by p}s_(r

1, r2) = |r1− r2| for all r1, r2 ∈ M. Define the mappings 𝜓(t) = ₅t,

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T(r) = ⎧ ⎪ ⎨ ⎪ ⎩ { r 5 } ifr∈ [0, 1); { 0,1 7 } ifr= 1 𝛼(r1, r2) = { 1 if r1, r2∈ M; 0 otherwise. Then T is not a(𝛼, 𝜓)-set valued-contraction in a metric space (M, ps_{). Indeed, for r}

1 = 1 and r2= 5₆ Hps ( T(1), T (₅ 6 )) = Hps ({ 0,1 7 } ,{1 6 }) = ps(₁_,5 6 ) = 1 6.

Thus, we deduce that𝛼(r1, r2)Hps(T(r₁), T(r₂))̇𝜓(ps(r₁, r₂)). On the other hand, for r₁ = 1 and r₂= 5

6

Hp(T(r1), T(r2)) ≤ p(r1, r2) implies1

6 ≤ 1, which is true. Similarly, for r1, r2 ∈ [0, 1) with r1 ≤ r2or r2 ≤ r1, we have

Hp(T(r1), T(r2)) ≤

p(r1, r2)

5 . Thus,𝛼(r1, r2)Hp(T(r1), T(r2)) ≤ 𝜓(p(r1, r2)), for all r1, r2 ∈ M.

Since, a set valued Banach contraction is an(𝛼, 𝜓)- set valued contraction. In the following, we show that the Kannan and Chatterjea set valued contractions are(𝛼, 𝜓)-set valued contractions too.

Proposition 3. Let(M, p) be a partial metric space and the mapping T ∶ M → CBp_{(M) be a Kannan-type set valued}

mapping, ie, there exists k∈ [0,1

2) such that

Hp(T(r1), T(r2)) ≤ k[Dp(r1, T(r1)) + Dp(r2, T(r2))], for allr1, r2∈ M.

Then T is an(𝛼, 𝜓)-set valued contraction. Proof. Since, Hp(T(r1), T(r2)) ≤ k[Dp(r1, T(r1)) + Dp(r2, T(r2))] ≤ k[𝛿p({r1}, T(r1)) + 𝛿p({r2}, S(r2))] ≤ k ⎡ ⎢ ⎢ ⎢ ⎣ 𝛿p({r1}, {r2}) + 𝛿p({r2}, T(r1)) − inf r∈{r2} p(r, r)+ 𝛿p({r2}, T(r1)) + 𝛿p(T(r1), T(r2)) − inf r∈T(r1) p(r, r) 𝛿p({r2}, T(r1)) + 𝛿p(T(r1), T(r2)) − inf r∈T(r1) p(r, r) ⎤ ⎥ ⎥ ⎥ ⎦ ≤ k ⎡ ⎢ ⎢ ⎢ ⎣ 𝛿p({r1}, {r2}) + 𝛿p({r2}, T(r1)) − inf r∈{r2} p(r, r)+ 𝛿p({r2}, T(r1)) + Hp(T(r1), T(r2)) − inf r∈T(r1) p(r, r) 𝛿p({r2}, T(r1)) + Hp(T(r1), T(r2)) − inf r∈T(r1) p(r, r) ⎤ ⎥ ⎥ ⎥ ⎦ (1 − k)Hp(T(r1), T(r2)) ≤ kp(r1, r2) + 2k𝛿p({r2}, T(r1)) Hp(T(r1), T(r2)) ≤ k 1− kp(r1, r2) + 2k 1− k𝛿p({r2}, T(r1)). Thus, we obtain the following inequality

( 1− 𝜇 𝛿p({r2}, T(r1)) Hp(T(r1), T(r2)) ) Hp(T(r1), T(r2)) ≤ λp(r1, r2), where𝜇 = 2k 1− k, λ = k 1− k. (4) Define the functions𝛼 ∶ M × M → [0, ∞) and 𝜓 ∶ [0, ∞) → [0, ∞) by

𝛼(r1, r2) = { 1− 𝜇 𝛿p({r2},T(r1)) Hp(T(r1),T(r2)) ifr1, r2∈ M; 0 otherwise, 𝜓(t) = 𝜆t.

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The inequality 4 yields

𝛼(r1, r2)Hp(T(r1), T(r2)) ≤ 𝜓(p(r1, r2)).

Hence, T is an(𝛼, 𝜓) set-valued contraction.

Proposition 4. Let(M, p) be a partial metric space and the mapping T ∶ M → CBp_{(M) be a Chatterjea type set valued}

mapping i.e. there exists k ∈ [0,1

2) such that

Hp(T(r1), T(r2)) ≤ k[Dp({r1}, T(r2)) + Dp({r2}, T(r1))], for allr1, r2∈ M.

Then T is an(𝛼, 𝜓)-set valued contraction.

Proof. The arguments follow the same lines as in proof of Proposition 3 and so, we omit details.

Proposition 5. Let(M, p) be a partial metric space. If the mapping T ∶ M → CBp(M) is a Dass-Gupta type set valued

mapping i.e. there exist 𝜆, 𝜇 ≥ 0 with 𝜆 + 𝜇 < 1 such that Hp(T(r1), T(r2)) ≤ 𝜇

Dp(r2, T(r2))(1 + Dp(r1, T(r1)))

1+ p(r1, r2) + λp(r1, r2),

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for all r1, r2 ∈ M. Then T is an (𝛼, 𝜓)-set valued contraction.

Proof. The inequality 5 can be rearranged as

Hp(T(r1), T(r2)) − 𝜇 Dp(r2, T(r2))(1 + Dp(r1, T(r1))) 1+ p(r1, r2) ≤ λp(r1, r2), ( 1− 𝜇Dp(r2, T(r2))(1 + Dp(r1, T(r1))) (1 + p(r1, r2))Hp(T(r1), T(r2)) ) Hp(T(r1), T(r2)) ≤ λp(r1, r2). (6)

Define the functions𝛼 ∶ M × M → [0, ∞) and 𝜓 ∶ [0, ∞) → [0, ∞) by the following:

𝛼(r1, r2) = { 1− 𝜇Dp(r2,T(r2))(1+Dp(r1,T(r1))) (1+p(r1,r2))Hp(T(r1),T(r2)) if r1, r2∈ M; 0 otherwise, 𝜓(t) = λt. The inequality 6 yields

𝛼(r1, r2)Hp(T(r1), T(r2)) ≤ 𝜓(p(r1, r2)).

Hence, T is an(𝛼, 𝜓) set valued contraction.

Proposition 6. Let(M, p) be a partial metric space. If the mapping T ∶ M → CBp(M) is a Jaggi-type set valued mapping

i.e. there exist 𝜆, 𝜇 ≥ 0 with 𝜆 + 𝜇 < 1 such that Hp(T(r1), T(r2)) ≤ 𝜇

Dp(r2, T(r2))Dp(r1, T(r1))

p(r1, r2) + λp(r1, r2),

for all r1, r2 ∈ M. Then T is an (𝛼, 𝜓)-set valued contraction.

Proof. The arguments follow the same lines as in proof of Proposition 5 and so, we omit details.

CBp_{(M) is called (𝛼, 𝜓)-set valued rational type contraction, if there exists 𝜓 ∈ Ψ such that}

𝛼(r1, r2)Hp(T(r1), T(r2)) ≤ 𝜓( (r1, r2)), (7)

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Obviously, Definition 8 is a particular case of Definition 9.

Definition 10. Let(M, p) be a partial metric space and 𝛼 ∶ M × M → [0, ∞) be a function. The mappings S, T ∶ M →

CBp(M) are called a pair of (𝛼, 𝜓)-set valued contraction, if there exists 𝜓 ∈ Ψ such that

𝛼(r1, r2)Hp(T(r1), S(r2)) ≤ 𝜓(p(r1, r2)), (8)

CBp(M) are called a pair of (𝛼, 𝜓)-set valued rational type contraction, if there exists 𝜓 ∈ Ψ such that

𝛼(r1, r2)Hp(T(r1), S(r2)) ≤ 𝜓((r1, r2)), (9)

Observe that Definition 10 is a particular case of Definition 11.

Remark1. It is easy to establish Propositions 3, 4, 5, and 6 for pair of mappings S, T ∶ M → CBp(M).

4 F I X E D P O I N T T H EO R E M S

The following theorems are our main results. We shall give proof of Theorem 5 only.

Theorem 2. Let(M, p) be a complete partial metric space and T ∶ M → CBp(M) be a set-valued mapping such that

(1) T is an(𝛼, 𝜓)-set-valued contraction; (2) T is a triangular𝛼∗-admissible mapping; (3) there exists r0 ∈ M such that 𝛼∗(r0, T(r0)) ≥ 1;

(4) either the mapping T is continuous or M is𝛼-regular. Then there exists a unique fixed point of T in M.

Theorem 3. Let(M, p) be a complete partial metric space and T ∶ M → CBp_{(M) be an (𝛼, 𝜓)-set valued rational type}

contraction complying with assumptions (2),(3), and (4) in Theorem 2. Then there exists a unique fixed point of T in M.

Theorem 4. Let(M, p) be a complete partial metric space and S, T ∶ M → CBp_{(M) be a pair of mappings such that}

(1) (S, T) is a pair of (𝛼, 𝜓)-set-valued contractions; (2) (S, T) is a triangular 𝛼_∗-admissible pair of mappings; (3) there exists r0 ∈ M such that 𝛼∗(r0, S(r0)) ≥ 1;

(4) either the mappings S and T are continuous or M is𝛼-regular. Then there exists a unique common fixed point of the pair(S, T) in M.

Theorem 5. Let(M, p) be a complete partial metric space and S, T ∶ M → CBp(M) be a pair of (𝛼, 𝜓)-set valued rational

type contractions complying with assumptions (2), (3), and (4) in Theorem 4. Then there exists a unique fixed point of T and S in M.

Proof. By assumption (3), there exits r0 ∈ M such that 𝛼∗(r0, S(r0)) ≥ 1. Since, (S, T) is a triangular 𝛼∗-admissible

pair of mappings,

𝛼(r0, r1) ≥ 𝛼∗(r0, S(r0)) ≥ 1implies𝛼∗(S(r0), T(r1)) ≥ 1.

There exist r1 ∈ S(r0), r2 ∈ T(r1) such that 𝛼(r1, r2) ≥ 𝛼∗(S(r0), T(r1)) ≥ 1 which implies 𝛼∗(T(r1), S(r2)) ≥ 1.

There exist r2 ∈ T(r1) and r3 ∈ S(r2) such that 𝛼(r2, r3) ≥ 𝛼∗(T(r1), S(r2)) ≥ 1 which implies 𝛼∗(S(r2), T(r3)) ≥ 1.

Continuing in this way, we obtain that𝛼(r2i + 1, r2i + 2) ≥ 1 and 𝛼(r2i, r2i + 1) ≥ 1. Hence, 𝛼(rn, rn+ 1) ≥ 1 for all n ∈W.

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Case1. If𝛼(r2i + 1, r2i + 2) = 1 and 𝛼(r2i, r2i + 1) = 1 for all i ∈W, then due to Lemma 3, there exist𝜂 > 1, r1 ∈ S(r0)

and r2 ∈ T(r1) such that p(r1, r2) < 𝜂Hp(S(r0), T(r1)). From contractive condition 9, we get

p(r1, r2) < 𝜂𝛼(r0, r1)Hp(S(r0), T(r1)) ≤ 𝜂𝜓((r0, r1)), where (r0, r1) = max ⎧ ⎪ ⎨ ⎪ ⎩ p(r0, r1), Dp(r0,S(r0))Dp(r1,T(r1)) 1+p(r0,r1) , Dp(r0,S(r0))Dp(r1,T(r1)) 1+𝛿p(S(r0),T(r1)) ⎫ ⎪ ⎬ ⎪ ⎭ ≤ max { p(r0, r1),p(r_1+p(r0,r1)p(r1,r2) 0,r1) , p(r0,r1)p(r1,r2) 1+p(r1,r2) } ≤ max {p(r0, r1), p(r1, r2)} .

If(r0, r1) ≤ p(r1, r2), then p(r1, r2) < 𝜂𝜓(p(r1, r2)), which is a contradiction to definition of 𝜓. Therefore, p(r1, r2) <

𝜂𝜓(p(r0, r1)) implies 𝜓(p(r1, r2)) < 𝜓(𝜂𝜓(p(r0, r1))). Set 𝜂1 = 𝜓(𝜂𝜓(p(r_𝜓(p(r 0,r1)))

1,r2)) > 1 and by Lemma 3, there exist 𝜂1 > 1,

r2 ∈ T(r1) and r3 ∈ S(r2) such that p(r2, r3) < 𝜂1Hp(T(r1), S(r2)) = Hp(S(r2), T(r1)). By contractive condition 9, we

get p(r3, r2) < 𝜂1𝛼(r2, r1)Hp(S(r2), T(r1)) ≤ 𝜂1𝜓((r2, r1)), where (r2, r1) = max ⎧ ⎪ ⎨ ⎪ ⎩ p(r2, r1), Dp(r2,S(r2))Dp(r1,T(r1)) 1+p(r2,r1) , Dp(r2,S(r2))Dp(r1,T(r1)) 1+𝛿p(S(r2),T(r1)) ⎫ ⎪ ⎬ ⎪ ⎭ ≤ max { p(r2, r1), p(r2,r3)p(r1,r2) 1+p(r2,r1) , p(r2,r3)p(r1,r2) 1+p(r3,r2) } ≤ max {p(r2, r3), p(r1, r2)} .

If(r2, r1) ≤ p(r2, r3), then p(r2, r3) < 𝜂1𝜓(p(r2, r3)), which is a contradiction to definition of 𝜓. Therefore, p(r2, r3) <

𝜂1𝜓(p(r1, r2)) = 𝜓(𝜂𝜓(p(r0, r1))) and definition of 𝜓 implies 𝜓(p(r2, r3)) < 𝜓2(𝜂𝜓(p(r0, r1))). Set 𝜂2= 𝜓

2_{(𝜂𝜓(p(r} 0,r1))) 𝜓(p(r2,r3)) > 1

and arguing like above we have

p(r3, r4) < 𝜂2𝜓(p(r2, r3)) = 𝜓2(𝜂𝜓(p(r0, r1))).

Thus, we are able to construct an iterative sequence{rn} of points in M such that r2n− 1 ∈ S(r2n− 2), r2n ∈ T(r2n− 1)

for all n∈Nwith𝛼(r2n− 2, r2n− 1) = 1 and 𝛼(r2n− 1, r2n) = 1 for all n ∈Nsatisfying

p(r2n−1, r2n) < 𝜓2n−2(𝜂𝜓(p(r0, r1)))andp(r2n, r2n+1) < 𝜓2n−1(𝜂𝜓(p(r0, r1))). (10)

Note that if r2n ∈ S(r2n), r2n + 1 ∈ T(r2n + 1), then r2nis a common fixed point of(S, T). The Equation 10 can be

reduced to the following:

p(rn, rn+1) < 𝜓n−1(𝜂𝜓(p(r0, r1))), for alln ∈N. (11)

Now, we show that{rn} is a Cauchy sequence. For m, n ∈Nwith m > n, consider

p(rn, rm) ≤ p(rn, rn+1) + p(rn+1, rn+2) + p(rn+2, rn+3) + ... + p(rm−1, rm) − m_∑−1 k=n+1 p(rk, rk) ≤ p(rn, rn+1) + p(rn+1, rn+2) + p(rn+2, rn+3) + ... + p(rm−1, rm) < m−1 ∑ 𝑗=n−1 𝜓𝑗_{(𝜂𝜓(p(r} 0, r1))) ≤ ∞ ∑ 𝑗=n−1 𝜓𝑗_{(𝜂𝜓(p(r} 0, r1))) → 0as𝑗 → ∞.

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Case2. If𝛼(r2i + 1, r2i + 2) > 1 and 𝛼(r2i, r2i + 1) > 1 for all i ∈ W, due to Lemma 3, we have p(r2i + 1, r2i + 2) <

𝛼(r2i, r2i + 1)Hp(S(r2i), T(r2i + 1)). By contractive condition 9, we get

p(r2i+1, r2i+2) < 𝛼(r2i, r2i+1)Hp(S(r2i), T(r2i+1)) ≤ 𝜓((r2i, r2i+1)),

for all i∈W, where

(r2i, r2i+1) = max ⎧ ⎪ ⎨ ⎪ ⎩ p(r2i, r2i+1), Dp(r2i,S(r2i))Dp(r2i+1,T(r2i+1)) 1+p(r2i,r2i+1) , Dp(r2i,S(r2i))Dp(r2i+1,T(r2i+1)) 1+𝛿p(S(r2i),T(r2i+1)) ⎫ ⎪ ⎬ ⎪ ⎭ ≤ max ⎧ ⎪ ⎨ ⎪ ⎩

p(r2i, r2i+1),p(r2i,r₁_+p(r2i+1)p(r2i+1,r2i+2)

2i,r2i+1) , p(r2i,r2i+1)p(r2i+1,r2i+2)

1+p(r2i+1,r2i+2) ⎫ ⎪ ⎬ ⎪ ⎭ ≤ max {p(r2i, r2i+1), p(r2i+1, r2i+2)} .

If(r2i, r2i+1) ≤ p(r2i+1, r2i+2), then p(r2i + 1, r2i + 2) < 𝜓(p(r2i + 1, r2i + 2)), which is a contradiction to definition of 𝜓.

Therefore,

p(r2i+1, r2i+2) < 𝜓(p(r2i, r2i+1)), for alli ∈W. (12)

Similarly, we have

p(r2i+2, r2i+3) < 𝜓(p(r2i+1, r2i+2)), for alli ∈W. (13) Hence, the inequalities (12) and (13) imply

p(rn, rn+1) < 𝜓(p(rn−1, rn)), for alln ∈N. (14)

By (14), we obtain

p(rn−1, rn) < 𝜓 (p(rn−2, rn−1)) .

Repeating these steps, we get

p(rn, rn+1) < 𝜓n(p(r0, r1)) .

Thus, we are able to construct an iterative sequence{rn} of points in M such that r2n− 1 ∈ S(r2n− 2), r2n ∈ T(r2n− 1)

for all n ∈ N with 𝛼(r2n− 2, r2n− 1) > 1 and 𝛼(r2n− 1, r2n) > 1 for all n ∈ N complying with p(rn, rn+1) <

𝜓n_(p(r

0, r1)) , for alln ∈N. Now, we show that{rn} is a Cauchy sequence. For m, n ∈Nwith m > n, consider

p(rn, rm) ≤ p(rn, rn+1) + p(rn+1, rn+2) + p(rn+2, rn+3) + ... + p(rm−1, rm) − m−1 ∑ k=n+1 p(rk, rk) ≤ p(rn, rn+1) + p(rn+1, rn+2) + p(rn+2, rn+3) + ... + p(rm−1, rm) < m−1 ∑ 𝜉=n 𝜓𝜉_(p(r 0, r1)) ≤ ∞ ∑ 𝜉=n 𝜓𝜉_(p(r 0, r1)) → 0as𝜉 → ∞.

Hence,{rn} is a Cauchy sequence in (M, p) in this case as well.

By Lemma 1(1),{rn} is also a Cauchy sequence in (M, ps). Since (M, p) is a complete partial metric space, so (M, ps)

is a complete metric space and as a result there exists𝜐 ∈ M such that limn→∞ps(rn, 𝜐) = 0. By Lemma 1(2), we have

lim

n→∞p(𝜐, rn) = p(𝜐, 𝜐) = limn,m→∞p(rn, rm). (15)

Since limn,m→∞p(rn, rm) = 0, by (15), we deduce that

p(𝜐, 𝜐) = 0 = lim

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Equation 16 implies that r2n + 1→ 𝜐 and r2n + 2→ 𝜐 as n → ∞ with respect to 𝜏(p). We show that 𝜐 is a common fixed

point of pair(S, T). If the functions S, T are continuous, we have

𝜐 = lim n→∞r2n+1 ∈ limn→∞S(r2n) = S ( lim n→∞r2n ) = S(𝜐).

Similarly,𝜐 ∈ T(𝜐) and hence 𝜐 ∈ T(𝜐) and 𝜐 ∈ S(𝜐) that is 𝜐 is common fixed point of S, T. If M is 𝛼-regular, then there exists a subsequence{rnk} of {rn} such that 𝛼(r2nk, 𝜐) ≥ 1 for all k. By (9), we have

Dp(r2nk+1, T(𝜐)) ≤ 𝛼(r2nk, 𝜐)Hp(S(x2nk), T(𝜐))

≤ 𝜓((r2nk, 𝜐)

)

,

which implies that

Dp(r2nk+1, T(𝜐)) < (r2nk, 𝜐). (17) Note that (r2nk, 𝜐) = max ⎧ ⎪ ⎨ ⎪ ⎩ p(r2nk, 𝜐), Dp(r2nk,S(r2nk))Dp(𝜐,T(𝜐)) 1+p(r2nk,𝜐) , Dp(r_2nk,S(r_2nk))Dp(𝜐,T(𝜐)) 1+𝛿p(S(r_2nk),T(𝜐)) ⎫ ⎪ ⎬ ⎪ ⎭ ≤ max ⎧ ⎪ ⎨ ⎪ ⎩ p(r2nk, 𝜐), p(r_2nk,r_2nk+1)Dp(𝜐,T(𝜐)) 1+p(r_2nk,𝜐) , p(r_2nk,r_2nk+1)Dp(𝜐,T(𝜐)) 1+𝛿p(S(r_2nk),T(𝜐)) ⎫ ⎪ ⎬ ⎪ ⎭ . Thus, lim k→∞(r2nk, 𝜐) = 0.

Letting k→ ∞ in (17) implies Dp(𝜐, T(𝜐)) = 0. Thus, 𝜐 ∈ T(𝜐) = T(𝜐) . Similarly, 𝜐 ∈ S(𝜐). Hence, 𝜐 is a common

fixed point of the mappings S and T.

Remark2. In addition to assumptions (1) to (4), if𝛼(·, ·) ≥ 1 for every pair of common fixed point of S and T, then S and T has a unique common fixed point.

Proof. Assume the contrary, that is, there exists𝜔 ∈ M such that 𝜐 ≠ 𝜔 and 𝜔 ∈ T(𝜔), 𝜔 ∈ S(𝜔). If 𝛼(𝜐, 𝜔) = 1, then from Lemma 3, there exists𝜂 > 1 such that p(𝜐, 𝜔) < 𝜂Hp(S(𝜐), T(𝜔)) and contractive condition (9) implies

p(𝜐, 𝜔) < 𝜂𝛼(𝜐, 𝜔)Hp(S(𝜐), T(𝜔)) ≤ 𝜂𝜓((𝜐, 𝜔)), where (18) (𝜐, 𝜔) = max ⎧ ⎪ ⎨ ⎪ ⎩ p(𝜐, 𝜔),Dp(𝜐,S(𝜐))Dp(𝜔,T(𝜔)) 1+p(𝜐,𝑦) , Dp(𝜐,S(𝜐))Dp(𝜔,T(𝜔)) 1+𝛿p(S(𝜐),T(𝜔)) ⎫ ⎪ ⎬ ⎪ ⎭ = p(𝜐, 𝜔). From (18), we have p(𝜐, 𝜔) < 𝜂𝜓(p(𝜐, 𝜔)), (19) The inequality (19) leads to a contradiction. Hence,𝜐 = 𝜔 and 𝜐 is a unique common fixed point of a pair (S, T). Similar arguments can lead to the same result if𝛼(𝜐, 𝜔) > 1.

Remark3. The arguments for the proof of Theorems 2, 3, and 4 follow the same lines as in proof of Theorem 5. In addition, we have to set S = T in the proof of Theorem 5 for the proofs of Theorems 2 and 3.

The following example illustrates Theorem 5 and hence Theorems 2, 3, and 4.

Example 4. Let M = {0, 1, 4} be endowed with partial metric p ∶ X × X → [0, ∞) defined by the following:

p(r1, r2) = 1

4|r1− r2| + 1

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Note that p(0, 0) = 0, p(1, 1) = 1

2 and p(4, 4) = 2, so, p is not a metric on M. As p s_(r

1, r2) = |r1 − r2|, thus, (M, p)

is complete partial metric space. We observe that{0}, {1}, and {0, 1} are closed sets in partial metric space (M, p). Indeed, if r ∈ M, then r∈ {1} ⇐⇒ p(r, {1}) = p(r, r) ⇐⇒ 1 4|r − 1| + 1 2max{r, 1} = r 2 ⇐⇒ r ∈ {1}.

Hence,{1} is closed in (M, p), similarly {0} is closed in (M, p). Also,

r∈ {0, 1} ⇐⇒ p(r, {0, 1}) = p(r, r)

⇐⇒ min {p(r, 0), p(r, 1)} = p(r, r) ⇐⇒ r ∈ {0, 1}.

Hence,{0, 1} is closed in (M, p). Now define the mappings S, T ∶ M → CBp(M) by the following:

T(0) = T(1) = {0}, T(4) = {0, 1} and S(r) =

{

{0} ifr ∈ {0, 1}; {1} ifr = 4. Define the function𝛼 ∶ M × M → [0, ∞) by the following:

𝛼(r1, r2) =

{

1 if r1, r2∈ M;

0 otherwise.

Note that the pair(S, T) is an 𝛼_∗-admissible pair of mappings with𝛼_∗(1, S(1)) ≥ 1. We shall show that the contractive condition (9) is satisfied for all possible cases. If r1, r2 ∈ {0, 1}, we have 𝛼(r1, r2) = 1 and

Hp(S(r1), T(r2)) = Hp({0}, {0})

= p(0, 0) ≤ 3

4(r1, r2). If r1 ∈ {0, 1} and r2 = 4, we have 𝛼(r1, r2) = 1 and

Hp(S(r1), T(r2)) = Hp({0}, {0, 1}) = max{𝛿p({0}, {0, 1}), 𝛿p({0, 1}, {0})} = max{0,3 4 } = 3 4 < 3 4p(r1, r2) = 3 4(r1, r2). If r1 = 4 and r2 = 4, we have 𝛼(4, 4) = 1 and

Hp(S(4), T(4)) = Hp({1}, {0, 1}) = max{𝛿p({1}, {0, 1}), 𝛿p({0, 1}, {1})} = max {p(1, 1), p(0, 1)} = 3 4 < 3 4p(4, 4) = 3 4(4, 4). We conclude that there exist𝜓 ∈ Ψ defined by 𝜓(t) = 3t

4 such that

𝛼(r1, r2)Hp(T(r1), S(r2)) ≤ 𝜓((r1, r2)).

We observe that r = 0 is a common fixed point of mappings S and T. Also, we note that ps(0, 0) = 0, ps(1, 1) = 0,

ps_{(4, 4) = 0, p}s_{(0, 1) = 1 = p}s_{(1, 0) , p}s_{(4, 0) = 4 = p}s_{(0, 4), and p}s_{(1, 4) = 3 = p}s_{(4, 1). For r} 1 = r2 = 4, we have Hps(S(4), T(4)) = H_ps({0}, {0, 1}) = max{𝛿_ps({0}, {0, 1}), 𝛿_ps({0, 1}, {0})} = max{0, 1} = 1 >3 4p s_{(4, 4) =}3 4 s_{(4, 4).}

Thus, Theorem 5 is not applicable for Hps(Housdorff metric associated to ps), as

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Example 5. Let M = [0, 1] and define p(r1, r2) = max {r1, r2}, then (M, p) is a complete partial metric space.

More-over, metric induced by p is given by ps_(r

1, r2) = |r1− r2|, so (M, ps) is a complete metric space. Define the mappings

S, T ∶ M → CBp_M_{as follows:} T(r) = ⎧ ⎪ ⎨ ⎪ ⎩ { r 5 } ifr∈ [0, 1) ; { 0,1 7 } ifr= 1 and S(r) ={3r 7 } for all r∈ M.

Define the functions𝜓(t) =3t

4 for all t > 0 and 𝛼 ∶ M × M → [0, ∞) by the following:

𝛼(r1, r2) =

{

1 if r1, r2∈ M;

0 otherwise. Let r1, r2 ∈ M such that r1 ≤ r2. Then,

(r1, r2) = max ⎧ ⎪ ⎨ ⎪ ⎩ r2, r1r2 1+ r2, r1r2 1+ max { 3r2 7 , r1 5 } ⎫ ⎪ ⎬ ⎪ ⎭ . Since r1 1+r2 < 1 and r1 1+max{3r2 7, r1 5

} < 1, we have that (r1, r2) = r2. In a similar way, if r1 ≥ r2, we obtain that

(r1, r2) = r1, consequently,(r1, r2) = p(r1, r2). Consider 𝛼(r1, r2)Hp(S(r2), T(r1)) = max { 3r2 7 , r1 5 } = max { 3p(r1, r2) 7 , p(r1, r2) 5 } = 3p(r1, r2) 7 < 𝜓(p(r1, r2)) = 𝜓((r1, r2)).

Thus, the contractive condition (9) is satisfied for all r1, r2 ∈ M. Hence, all the hypotheses of the Theorem 5 are

satisfied, note that(S, T) have a unique common fixed point r = 0.

Theorem 6. Let(M, p) be a complete partial metric space and S, T ∶ M → CBp(M) be a pair of mappings such that

𝛼(r1, r2)Hp(T(r1), S(r2)) ≤ 𝜓(i(r1, r2)),

for all r1, r2 ∈ M, i = 1, 2, 3, 4, 5, 6, 𝜓 ∈ Ψ and

1(r1, r2) = max { p(r1, r2), Dp(r1, S(r1)), Dp(r2, T(r2)), Dp(r1, T(r2)) + Dp(r2, S(r1)) 2 } ; 2(r1, r2) = a1p(r1, r2) + a2Dp(r1, S(r1)) + a3Dp(r2, T(r2))) + a4[Dp(r1, T(r2)) + Dp(S(r1), r2)],

whereai≥ 0(i = 1, 2, 3, 4) such that a1+ a2+ a3+ 2a4 < 1;

3(r1, r2) = ap(r1, r2) + bDp(r1, S(r1)) + cDp(r2, T(r2))),

wherea, b, care nonnegative numbers such that a + b + c < 1;

4(r1, r2) = max { p(r1, r2), Dp(r1, T(r1)), Dp(r2, S(r2)) } ; 5(r1, r2) = k 2[Dp(r1, T(r1)) + Dp(r2, S(r2))], 0 ≤ k < 1; 6(r1, r2) = k 2[Dp(r2, T(r1)) + Dp(r1, S(r2))], 0 ≤ k < 1.

If assumptions (2), (3), and (4) assumed in Theorem 5 hold, then there exists a unique common fixed point of the pair

(S, T) in M.

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Remark4. In addition to Theorem 6, it is possible to say that the arguments used in the proof of Theorem 5 can be applied to obtain (common) fixed points of mapping(s) satisfying contractive conditions listed (1) to (25) in a nice paper written by Rhoades.12

4.1 Fixed points of

(𝛼

∗, 𝜓) generalized contractions

CBp(M) is called (𝛼_∗, 𝜓)-set valued contraction, if there exists 𝜓 ∈ Ψ such that

𝛼∗(T(r1), T(r2))Hp(T(r1), T(r2)) ≤ 𝜓(p(r1, r2)), (20)

CBp_{(M) is called (𝛼}

∗, 𝜓)-set valued rational type contraction, if there exists 𝜓 ∈ Ψ such that

𝛼∗(T(r1), T(r2))Hp(T(r1), T(r2)) ≤ 𝜓( (r1, r2)), (21)

Obviously, Definition 12 is a particular case of Definition 13.

CBp_{(M) are called a pair of (𝛼}

∗, 𝜓)-set valued contraction, if there exists 𝜓 ∈ Ψ such that

𝛼∗(T(r1), S(r2))Hp(T(r1), S(r2)) ≤ 𝜓(p(r1, r2)), (22)

CBp_{(M) are called a pair of (𝛼}

∗, 𝜓)-set valued rational type contraction, if there exists 𝜓 ∈ Ψ such that

𝛼∗(T(r1), S(r2))Hp(T(r1), S(r2)) ≤ 𝜓((r1, r2)), (23)

Definition 14 can be seen as a particular case of Definition 15. The following theorems are our main results. We shall give proof of Theorem 10 only.

Theorem 7. Let(M, p) be a complete partial metric space and T ∶ M → CBp_{(M) be a an (𝛼}

∗, 𝜓)-set valued contraction

satisfying assumptions (2), (3), and (4) in Theorem 2. Then there exists a unique fixed point of T in M.

Theorem 8. Let(M, p) be a complete partial metric space and T ∶ M → CBp(M) be an (𝛼∗, 𝜓)-set valued rational type

contraction complying with assumptions (2), (3), and (4) in Theorem 3. Then there exists a unique fixed point of T in M.

Theorem 9. Let(M, p) be a complete partial metric space and S, T ∶ M → CBp(M) be a pair of (𝛼_∗, 𝜓)-set valued

contractions complying with assumptions (2), (3), and (4) in Theorem 4. Then there exists a unique fixed point of T and S in M.

Theorem 10. Let(M, p) be a complete partial metric space and S, T ∶ M → CBp(M) be a pair of (𝛼_∗, 𝜓) -set valued

rational type contractions complying with assumptions (2), (3), and (4) in Theorem 5. Then there exists a unique fixed point of Tand S in M.

Proof. Since every𝛼_∗-admissible pair of mappings is also an𝛼-admissible pair of mappings, therefore, arguments follow from the proof of Theorem 5, to avoid repetition we omit the details.

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4.2 Consequences

Corollary 1. Let(M, p) be a complete partial metric space4_{and T}∶ M → CBp_{(M) be a set valued mapping such that}

Hp(T(r1), T(r2)) ≤ kp(r1, r2), (24)

for all r1, r2 ∈ M with 0 ≤ k < 1. Then there exists a unique fixed point of T in M.

Proof. Let h < 1 and 𝜆 ≥ 1. Define 𝛼(r1, r2) = 𝜆 for all r1, r2 ∈ M and 𝜓(t) = ht for all t > 0. The inequality

(24) reduces to inequality (3) and the arguments, for the proof, follow the same lines as in proof of Theorem 2 for

k= h

λ.

The Corollary given below generalizes the fixed point results proved in Nazam et al10_{and La Rosa and Vetro.}13

Corollary 2. Let(M, p) be a complete partial metric space and S, T ∶ M → CBp_{(M) be set valued mappings such that}

𝛼(r1, r2)Hp(T(r1), S(r2)) ≤ 𝛽((r1, r2))(r1, r2), (25)

for all r1, r2 ∈ M. Then there exists a unique common fixed point of T and S in M.

Proof. Let𝛽 ∶ [0, ∞) → [0, 1) be a function which satisfies the condition: lim

n→∞𝛽(tn) = 1 implies limn→∞tn= 0.

Define𝜓(t) = 𝛽(t)t for all t > 0. The inequality (25) reduces to inequality (9) and the arguments, for the proof, follow the same lines as in proof of Theorem 5.

The following Corollary generalizes a result presented in Zhang and Song.14

Corollary 3. Let(M, p) be a complete partial metric space and S, T ∶ M → CBp(M) be set valued mappings such that

Hp(T(r1), S(r2)) ≤ (r1, r2) − 𝜙((r1, r2)), (26)

for all r1, r2 ∈ M. Then there exists a unique common fixed point of T and S in M.

Proof. Let𝜙 ∶ [0, ∞) → [0, ∞) be a lower semi-continuous and non-decreasing function with 𝜙(t) = 0 if and only if t = 0. Define 𝛼(r1, r2) = 1 for all r1, r2 ∈ M and 𝜓(t) = t − 𝜙(t) for all t > 0. The inequality (26) reduces to

inequality (9) and the arguments, for the proof, follow the same lines as in proof of Theorem 5.

O RC I D

Özlem Acar http://orcid.org/0000-0001-6052-4357

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How to cite this article: Nazam M, Acar Ö. Fixed points of(𝛼, 𝜓)-contractions in Hausdorff partial metric