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In this paper, by using the concept of the α-Garaghty contraction, we introduce the new notion of the α-Θ-Garaghty type contraction and prove some fixed point results for this contraction in partial metric spaces. Also, we give some examples and applications to illustrate the main results.

Throughout this paper, Φ and 2 X denote the real field (or the complex field) and the family of all nonempty subsets of a vector space over Φ, respectively. Let E and F be vector spaces over Φ and • , • : F × E → Φ be a bilinear functional. For each x 0 ∈ E and ε > 0, letWe denote by σ(F, E) the topology on F generated by the family {ω(x, ε) : x ∈ E, ε > 0} as a subbase for the neighbourhood system at 0.It is easy to show that, if F possesses the σ(F, E)-topology, F becomes a locally convex topological vector space. The σ(E, F )-topology on E is defined analogously. A subset X of E is said to be σ(E, F )-compact if X is compact related to the σ(E, F )-topology.Let X be a nonempty subset of E. A set-valued mapping T : X → 2 F is said to be monotone relative to the bilinear functional • , • : F × E → Φ (monotone for short) if, for all x, y ∈ X, u ∈ T (x) and w ∈ T (y), Re u − w, x − y ≥ 0.