1288 Words6 Pages

Abstract. The notion of dualistic Geraghty Contraction is introduced.
A new fixed point theorem is proved in the settings of complete dualistic partial metric spaces. The counterpart theorem provided in partial metric spaces is retrieved as a particular case of our new results.
We give example to prove that the contractive conditions in the statement of our new fixed point theorem can not be replaced by those contractive conditions in the statement of the partial metric counterpart fixed point theorem. Moreover, we give an application of our fixed point theorem to show the existence of solution of integral equations.
AMS Subject Classification: 47H09; 47H10; 54H25
Keywords and Phrases: Fixed point, dualistic contraction, dualistic
partial*…show more content…*

Remark 1.3. It is obvious that every partial metric is dualistic partial metric but converse is not true. To support this comment, define D∨ : R × R → R by D∨(j, k) = j ∨ k = sup{j, k} ∀ j, k ∈ R. It is clear that D∨ is a dualistic partial metric. Note that D∨ is not a partial metric, because D∨(−1, −2) = −1 / ∈ R+. However, the restriction of D∨ to R+, D∨|R+ , is a partial metric. Example 1.4. If (M, d) is a metric space and c ∈ R is an arbitrary constant, then D(j, k) = d(j, k) + c. defines a dualistic partial metric on M. Following [8], each dualistic partial metric D on M generates a T0 topology τ (D) on M which has, as a base, the family of D-open balls {BD(j, ) : j ∈ M, > 0} and BD(j, ) = {k ∈ M : D(j, k) < + D(j, j)}. 74 M. NAZAM, M. ARSHAD AND CH. PARK If (M, D) is a dualistic partial metric space, then dD : M × M → R+0 defined by dD(j, k) = D(j, k) − D(j, j). is called quasi metric on M such that τ (D) = τ (dD) for all j, k ∈ M. Moreover, if dD is a quasi metric on M, then ds D(j, k) = max{dD(j, k), dD(k, j)} defines a metric on M. A sequence {jn}n∈N in (M, D) converges to a point j ∈ M if and*…show more content…*

Definition 1.5. [8] Let (M, D) be a dualistic partial metric space, then (1) A sequence {jn}n∈N in (M, D) is called a Cauchy sequence if limn,m→∞ D(jn, jm) exists and is finite. (2) A dualistic partial metric space (M, D) is said to be complete if every Cauchy sequence {jn}n∈N in M converges, with respect to τ (D), to a point j ∈ M such that D(j, j) = limn,m→∞ D(jn, jm). Following lemma will be helpful in the sequel. Lemma 1.6. [8, 9] (1) A dualistic partial metric (M, D) is complete if and only if the metric space (M, ds D) is

Remark 1.3. It is obvious that every partial metric is dualistic partial metric but converse is not true. To support this comment, define D∨ : R × R → R by D∨(j, k) = j ∨ k = sup{j, k} ∀ j, k ∈ R. It is clear that D∨ is a dualistic partial metric. Note that D∨ is not a partial metric, because D∨(−1, −2) = −1 / ∈ R+. However, the restriction of D∨ to R+, D∨|R+ , is a partial metric. Example 1.4. If (M, d) is a metric space and c ∈ R is an arbitrary constant, then D(j, k) = d(j, k) + c. defines a dualistic partial metric on M. Following [8], each dualistic partial metric D on M generates a T0 topology τ (D) on M which has, as a base, the family of D-open balls {BD(j, ) : j ∈ M, > 0} and BD(j, ) = {k ∈ M : D(j, k) < + D(j, j)}. 74 M. NAZAM, M. ARSHAD AND CH. PARK If (M, D) is a dualistic partial metric space, then dD : M × M → R+0 defined by dD(j, k) = D(j, k) − D(j, j). is called quasi metric on M such that τ (D) = τ (dD) for all j, k ∈ M. Moreover, if dD is a quasi metric on M, then ds D(j, k) = max{dD(j, k), dD(k, j)} defines a metric on M. A sequence {jn}n∈N in (M, D) converges to a point j ∈ M if and

Definition 1.5. [8] Let (M, D) be a dualistic partial metric space, then (1) A sequence {jn}n∈N in (M, D) is called a Cauchy sequence if limn,m→∞ D(jn, jm) exists and is finite. (2) A dualistic partial metric space (M, D) is said to be complete if every Cauchy sequence {jn}n∈N in M converges, with respect to τ (D), to a point j ∈ M such that D(j, j) = limn,m→∞ D(jn, jm). Following lemma will be helpful in the sequel. Lemma 1.6. [8, 9] (1) A dualistic partial metric (M, D) is complete if and only if the metric space (M, ds D) is

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