Definition 1. Let Γ be a nonvoid set and the mapping $$ are defined, with the following properties:

(A1) $$ for all ℵ, ϖ ∈ Γ and ρ(ℵ, ℵ) = ρ(ϖ, ϖ) = ρ(ℵ, ϖ) if and only if ℵ = ϖ

(A2) ρ(ℵ, ℵ)⪯ρ(ℵ, ϖ)

(A3) ρ(ℵ, ϖ) = ρ(ϖ, ℵ) for all ℵ, ϖ ∈ Γ

(A4) ρ(ℵ, ϖ)⪯ρ(ℵ, γ) + ρ(γ, ϖ) − ρ(γ, γ) for all ℵ, ϖ, γ ∈ Γ

Definition 2. A sequence {ℵ_α} in $$ is called convergent (with respect to $$) to a point ℵ ∈ Γ, if for given ε > 0, ∃$$ such that $$.

Definition 3. A sequence {ℵ_α} in $$ is called Cauchy (with respect to $$), if $$ exists, and it is finite.

Definition 4. The triplet $$ is called complete C^∗ -algebra-valued partial metric space if every Cauchy sequence in Γ is convergent to some point ℵ in Γ such that

Definition 5 (see [18].)Let Γ be a nonvoid set. An element (ℵ, ϖ) ∈ Γ × Γ is said to be

• (1)

  A couple fixed point of the mapping φ : Γ × Γ⟶Γ if φ(ℵ, ϖ) = ℵ and φ(ϖ, ℵ) = ϖ

• (2)

  A coupled coincidence point of the mapping φ : Γ × Γ⟶Γ and g : Γ⟶Γ if φ(ℵ, ϖ) = gℵ and φ(ϖ, ℵ) = gϖ. In this case, (gℵ, gϖ) is said to be coupled point of coincidence

• (3)

  A common coupled fixed point of the mapping φ : Γ × Γ⟶Γ and g : Γ⟶Γ if φ(ℵ, ϖ) = gℵ = ℵ and φ(ϖ, ℵ) = gϖ = ϖ

Definition 6 (see [18].)The mappings φ : Γ × Γ⟶Γ and g : Γ⟶Γ is said to be ω-compatible if g(φ(ℵ, ϖ)) = φ(gℵ, gϖ) whenever gℵ = φ(ℵ, ϖ) and gϖ = φ(ϖ, ℵ).

Theorem 7. Let $$ be a complete C^⋆ -algebra-valued partial metric space. Suppose that the mappings φ : Γ × Γ⟶Γ and g : Γ⟶Γ such that

Proof. Let ℵ₀, ϖ₀ ∈ Γ, then g(ℵ₁) = φ(ℵ₀, ϖ₀), and g(ϖ₁) = φ(ϖ₀, ℵ₀). One can obtain two sequences {ℵ_α} and {ϖ_α} by continuing this process such that g(ℵ_α+1) = φ(ℵ_α, ϖ_α), and g(ϖ_α+1) = φ(ϖ_α, ℵ_α). Then,

Similarly,

Let

Using (4) and (5), we have

Let $$, then $$ implies $$ (Theorem 2.2.5 in [27]). Therefore, for each α ∈ ℕ,

If $$, then φ and g have a coupled coincidence point (ℵ₀, ϖ₀). Now, letting $$, then for each α, ℘∈ℕ,

Consequently,

Since ∥r∥⋖(1/√2), we have

Therefore, {gℵ_α} and {gϖ_α} are Cauchy sequences in g(Γ). Since {gϖ_α} is complete, ∃ℵ, ϖ ∈ Γ such that lim_α⪯∞gℵ_α = gℵ and

lim_α⪯∞gϖ_α = gϖ, and

Now, we show that φ(ℵ, ϖ) = gℵ and φ(ϖ, ℵ) = gϖ. For this,

As α⟶∞, we get $$, and hence, φ(ℵ, ϖ) = gℵ. Similarly, φ(ϖ, ℵ) = gϖ. Therefore, φ and g have a coupled coincidence point (ℵ, ϖ).

Let (ℵ^′, ϖ^′) be another coupled coincidence point of φ and g. Then,

Consequently,

Since $$, then ‖ρ(gℵ, gℵ^′) + ρ(gϖ, gϖ^′)‖ = 0. Hence, we get gℵ = gℵ^′ and gϖ = gϖ^′. Similarly, we can prove that gℵ = gϖ^′ and gϖ = gℵ^′. Then, φ and g have a unique coupled point of coincidence (gℵ, gℵ). Moreover, set v = gℵ, then v = gℵ = φ(ℵ, ℵ). Since φ and g are ω-compatible,

Therefore, φ and g have a coupled point of coincidence (gv, gv). We know gv = gℵ, then v = gv = φ(v, v). Therefore, φ and g have a unique common coupled fixed point (v, v).

Example 1. Let $$ and $$, and the map $$ is defined by

Corollary 8. Let $$ be a complete C^⋆ -algebra-valued partial metric space. Suppose that mapping φ : Γ × Γ⟶Γ such that

Lemma 9. Suppose that $$ is a unital C^⋆ -algebra with a unit $$.

• (1)

  If $$ with ∥r∥<(1/2), then $$ is invertible

• (2)

  If $$ and rs = sr, then $$

• (3)

  If $$ and $$ then r⪯s deduces $$, where $$

Theorem 10. Let $$ is a complete C^⋆ -algebra-valued partial metric space. Suppose that the mappings φ : Γ × Γ⟶Γ and g : Γ⟶Γ such that

Proof. Similar to Theorem 7, construct two sequences {ℵ_α} and {ϖ_α} in Γ such that gℵ_α+1 = φ(ℵ_α, ϖ_α) and gϖ_α+1 = φ(ϖ_α, ℵ_α). Then, by applying (25), we have

Since $$ with ∥r∥+∥s∥<1, we have $$ is invertible and $$ Therefore,

Then,

Since,

Therefore, {gℵ_α} and {gϖ_α} are Cauchy sequences in g(Γ). By the completeness of g(Γ), ∃ℵ, ϖ ∈ Γ such that lim_α°∞gℵ_α = gℵ and

lim_α⟶∞gϖ_α = gϖ, and

Since,

Then, $$ or equivalently φ(ℵ, ϖ) = gℵ. Similarly, one can obtain φ(ϖ, ℵ) = gϖ. Let (ℵ^′, ϖ^′) be another coupled coincidence point of φ and g, then

Theorem 11. Let $$ be a complete C^⋆ -algebra-valued partial metric space. Suppose that mappings φ : Γ × Γ⟶Γ and g : Γ⟶Γ such that

Proof. Following similar process given in Theorem 7, we construct two sequences {ℵ_α} and {ϖ_α} in Γ such that g(ℵ_α+1) = φ(ℵ_α, ϖ_α) and g(ϖ_α+1) = φ(ϖ_α, ℵ_α). From (36), we have

Because of the symmetry in (36),

From (38) and (40), we obtain

Since $$ with ∥r + s∥⪯∥r∥+∥s∥⋖1, then $$ which together with Lemma 9 (3), we obtain

Let $$, then $$. The same argument in Theorem 10 tells that {gℵ_α} is a Cauchy sequence in g(Γ). Similarly, we can derive that {gϖ_α} is also a Cauchy sequence in g(Γ). By the completeness of g(Γ), ∃ℵ, ϖ ∈ Γ such that lim_α⪯∞gℵ_α = gℵ and

lim_α⪯∞gϖ_α = gϖ, and

Now, we show that φ(ℵ, ϖ) = gℵ and φ(ϖ, ℵ) = gϖ. For this,

By the continuity of the metric and the norm, we obtain

Since ∥s∥<1; therefore, ∥ρ(φ(ℵ, ϖ), gℵ)∥ = 0. Thus, φ(ℵ, ϖ) = gℵ. Similarly, φ(ϖ, ℵ) = gϖ. Hence, (ℵ, ϖ) is a coupled coincidence point of φ and g. The same reasoning that Theorem 10 tells us that φ and g have unique common coupled fixed point in Γ.

Theorem 12. Suppose that (for all ℵ, ϖ ∈ Γ)

(S1) There exists a continuous function $$ and θ ∈ (0, 1), such that

(S2) $$.

Proof. Define φ : Γ × Γ⟶Γ by:

Set τ = θI, then $$. For any $$, we have

Hence, all the hypotheses of Corollary 8 are verified, and consequently, the integral Equation (49) has a unique solution.