A $phi$ - contraction Principle in Partial Metric Spaces with Self-distance Terms

We prove a generalized contraction principle with control function in complete partial metric spaces. The contractive type condition used allows the appearance of self distance terms. The obtained result generalizes some previously obtained results such as the very recent " D. Ili'{c}, V. Pavlovi'{c} and V. Rako\u{c}evi'{c}, Some new extensions of Banach’s contraction principle to partial metric spaces, Appl. Math. Lett. 24 (2011), 1326–1330". An example is given to illustrate the generalization and its properness. Our presented example does not verify the contractive type conditions of the main results proved recently by S. Romaguera in " Fixed point theorems for generalized contractions on partial metric spaces, Topology Appl. 159 (2012), 194-199" and by I. Altun, F. Sola and H. Simsek in “Generalized contractions on partial metric spaces, Topology and Its Applications 157 (18) (2010), 2778–2785”. Therefore, our results have an advantage over the previously obtained.

💡 Research Summary

The paper investigates fixed‑point theory in the setting of partial metric spaces, a generalization of ordinary metric spaces where the self‑distance p(x, x) need not be zero. This feature makes partial metrics particularly suitable for modeling computational processes, data‑flow networks, and other contexts where an element may possess an intrinsic “distance” from itself. The authors introduce a new contraction principle, called a φ‑contraction, that explicitly incorporates self‑distance terms through an auxiliary function ψ.

Main definitions. A partial metric p on a set X satisfies: (i) p(x, x) ≤ p(x, y) for all x, y; (ii) symmetry p(x, y)=p(y, x); (iii) a relaxed triangle inequality p(x, z) ≤ p(x, y)+p(y, z)−p(y, y). The space (X, p) is complete if every Cauchy sequence (in the sense of partial metrics) converges with respect to p. The contraction framework uses two functions: a control function φ :