In 2019 Gaba, Agyingi and Leko introduced Chatterjea-type contractions on Shukla's partial b-metric spaces and closed their paper with five explicitly posed and still-unanswered questions. This problem reconstructs that question set as a single research task with one common objective: to complete the Chatterjea (and, where stated, Kannan) fixed-point framework on partial b-metric spaces. Concretely: (i) Problem 2.8 (partial ultra-metric metrization), which is void under the source's defective Definition 2.7; (ii) determine for which coefficients 1≤s<21 \le s < \sqrt2 and which contraction constants λ\lambda the Chatterjea fixed point theorem survives, the regime the source leaves open (Problem 3.3); (iii) establish T-stability of Picard iteration for Kannan and Chatterjea contractions (Problem 3.13); (iv) prove or refute the conjectures that joint Chatterjea-Kannan maps have the P property F(T)=F(Tn)F(T)=F(T^n) (Conjectures 3.15-3.16); and (v) carry out the authors' announced theta-phi-Chatterjea program (Section 4).

Background

A partial metric, introduced by Matthews, is a distance p(x,y)≥0p(x,y) \ge 0 on a set in which the self-distance p(x,x)p(x,x) need not vanish - the original motivation is denotational semantics, where p(x,x)p(x,x) measures the information content still missing from a partially specified object xx - subject to: x=yx = y exactly when p(x,x)=p(x,y)=p(y,y)p(x,x) = p(x,y) = p(y,y); p(x,x)≤p(x,y)p(x,x) \le p(x,y); symmetry; and the triangle inequality. Czerwik's b-metric spaces relax the triangle inequality to d(x,y)≤s [d(x,z)+d(z,y)]d(x,y) \le s\,[d(x,z)+d(z,y)] for a coefficient s≥1s \ge 1; combining the two notions, Shukla's partial b-metric spaces (which the source also calls 'partial metric type spaces') replace the partial-metric triangle inequality by p(x,y)+p(z,z)≤s [p(x,z)+p(z,y)]p(x,y) + p(z,z) \le s\,[p(x,z)+p(z,y)] with s≥1s \ge 1 (axioms (pm1)-(pm4) are pinned in the problem statement); the case s=1s = 1 is exactly a partial metric. Convergence is taken in the induced T0T_0-topology: xnx_n p-converges to xx when p(x,xn)→p(x,x)p(x,x_n) \to p(x,x), and a pp-Cauchy sequence is one for which lim⁡n,mp(xn,xm)\lim_{n,m} p(x_n,x_m) exists and is finite; a space is complete when every pp-Cauchy sequence p-converges, and 0-complete when every sequence with p(xn,xm)→0p(x_n,x_m) \to 0 p-converges to a point xx with p(x,x)=0p(x,x) = 0.

On such spaces the source studies Chatterjea contractions, p(Tx,Ty)≤λ [p(x,Ty)+p(y,Tx)]p(Tx,Ty) \le \lambda\,[p(x,Ty)+p(y,Tx)], named after Chatterjea's 1972 theorem that in a complete metric space every such map with λ<1/2\lambda < 1/2 has a unique fixed point; the companion Kannan class d(Tx,Ty)≤λ [d(x,Tx)+d(y,Ty)]d(Tx,Ty) \le \lambda\,[d(x,Tx)+d(y,Ty)] (Kannan 1968) behaves similarly. Both classes are important because their maps need not be continuous. The source's Theorem 3.1 extends Chatterjea's theorem to complete partial b-metric spaces with coefficient s≥2s \ge 2 and constant λ∈[0,1/s2)\lambda \in [0, 1/s^2), and its Theorem 3.4 covers the joint Chatterjea-Kannan inequality (Ch-Ka) - a five-term inequality whose last four terms are rational products normalized by 1+p(x,y)1 + p(x,y), stated in full in item (4) below - on a 0-complete space.

Four auxiliary notions frame the open items. (i) Picard iteration xn+1=Txnx_{n+1} = Tx_n is T-stable (Qing-Rhoades, adapted to p-convergence in the source's Definition 3.8) if, whenever the iterates p-converge to a fixed point qq, every approximate orbit (yn)(y_n) with p(yn+1,Tyn)→0p(y_{n+1}, Ty_n) \to 0 also p-converges to qq. (ii) A map has the P property (Jeong-Rhoades) if F(T)=F(Tn)F(T) = F(T^n) for every n≥1n \ge 1; the source's Theorem 3.14 shows the one-step estimate p(Tx,T2x)≤λ p(x,Tx)p(Tx, T^2x) \le \lambda\,p(x,Tx), λ∈[0,1)\lambda \in [0,1), already implies it. (iii) A partial ultra-metric (the source's Definition 2.7) is intended to strengthen the triangle axiom in the spirit of ultrametrics, where the maximum replaces the sum; the source's printed axiom is defective (a maximum over a single expression), and the source's Definition 2.1 declares two partial metric types equivalent when each is bounded by constant multiples of the other. Frink's 1937 chain construction is the classical device for building an equivalent metric from a relaxed distance function. (iv) A theta-phi contraction of Zheng-Cai-Wang on a complete metric space uses a control function θ:(0,∞)→(1,∞)\theta : (0,\infty) \to (1,\infty) that is nondecreasing, continuous, and satisfies lim⁡nθ(tn)=1\lim_n \theta(t_n) = 1 if and only if lim⁡ntn=0+\lim_n t_n = 0^+ (class Θ\Theta), together with a nondecreasing continuous ϕ:[1,∞)→[1,∞)\phi : [1,\infty) \to [1,\infty) with lim⁡nϕ n(t)=1\lim_n \phi^{\,n}(t) = 1 for every t>1t > 1 (class Φ\Phi), and requires d(Tx,Ty)≠0⇒θ(d(Tx,Ty))≤ϕ(θ(max⁡{d(x,y),d(x,Tx),d(y,Ty)}))d(Tx,Ty) \ne 0 \Rightarrow \theta(d(Tx,Ty)) \le \phi\big(\theta(\max\{d(x,y), d(x,Tx), d(y,Ty)\})\big). Finally, for ordinary b-metric spaces the Chatterjea constant question is classical: Kir-Kiziltunc fixed β<1/(2K)\beta < 1/(2K) for every coefficient K>1K > 1.

Problem Statement

The problem is to settle the five open questions that Gaba-Agyingi-Leko (arXiv:1902.03108) explicitly pose about Chatterjea-type fixed point theory on partial b-metric spaces. The common objective is to complete that framework, and each item has a definite mathematical answer. Throughout, (X,p)(X,p) is a partial b-metric space with coefficient s≥1s \ge 1 (the source uses 'partial metric type space' as a synonym), i.e. p:X×X→[0,∞)p : X \times X \to [0,\infty) satisfies, for all x,y,z∈Xx, y, z \in X: (pm1) x=yx = y if and only if p(x,x)=p(x,y)=p(y,y)p(x,x) = p(x,y) = p(y,y); (pm2) 0≤p(x,x)≤p(x,y)0 \le p(x,x) \le p(x,y); (pm3) p(x,y)=p(y,x)p(x,y) = p(y,x); (pm4) p(x,y)+p(z,z)≤s [p(x,z)+p(z,y)]p(x,y) + p(z,z) \le s\,[p(x,z) + p(z,y)]. F(T)F(T) denotes the set of fixed points of a self-map TT.

(1) Problem 2.8, partial ultra-metric metrization. The source's Definition 2.7 declares a partial ultra-metric to satisfy (pm1)-(pm3) together with an ultrametric-strengthened (pm4); as printed, that axiom reads p(x,y)+p(z,z)≤max⁡{p(x,z)+p(z,y)}p(x,y) + p(z,z) \le \max\{p(x,z) + p(z,y)\}, a maximum over a single expression in which the quantified s≥1s \ge 1 plays no role, so the printed axiom is the s=1s = 1 form of (pm4). Any reading that replaces a sum under the maximum by a maximum is implied by that s=1s = 1 inequality, since a maximum never exceeds a sum; hence every partial ultra-metric is already a partial metric type with coefficient s=1s = 1, the identity ω′=ω\omega' = \omega is equivalent in the sense of the source's Definition 2.1 with α=β=1\alpha = \beta = 1, and Problem 2.8 is void as a construction question, carrying no non-trivial same-direction content.

(2) Problem 3.3, sharp coefficient regime for Chatterjea's theorem. A Chatterjea contraction is a self-map with p(Tx,Ty)≤λ [p(x,Ty)+p(y,Tx)]p(Tx,Ty) \le \lambda\,[p(x,Ty) + p(y,Tx)] for all x,yx, y. The source's Theorem 3.1 states that on a complete partial b-metric space with coefficient s≥2s \ge 2, every Chatterjea contraction with λ∈[0,1/s2)\lambda \in [0, 1/s^2) has a unique fixed point uu with p(u,u)=0p(u,u) = 0; Problem 3.3 asks whether the theorem can be formulated for 1<s<21 < s < 2 with an appropriate interval for λ\lambda, asserts that the theorem 'remains true for the sharp inequality s≥2s \ge \sqrt2', and closes with 'we still have to figure out what happens for 1≤s<21 \le s < \sqrt2', so the open regime the source leaves is 1≤s<21 \le s < \sqrt2 (with constant 1/s21/s^2, the threshold s≥2s \ge \sqrt2 is exactly the regime 1/s2≤1/21/s^2 \le 1/2). Determine whether, for each ss with 1≤s<21 \le s < \sqrt2, there is a nonempty admissible interval of λ\lambda for which the conclusion holds; identify the sharp interval as a function of ss; or, for some such ss, show that no positive λ\lambda is admissible by constructing, for each positive λ\lambda, a complete partial b-metric space with coefficient ss together with a Chatterjea contraction with that constant λ\lambda which has no fixed point.

(3) Problem 3.13, T-stability for Kannan and Chatterjea contractions. In the sense of the source's Definition 3.8 (an adaptation of Qing-Rhoades to p-convergence): the Picard iteration of TT is T-stable if, the Picard iterates xn+1=Txnx_{n+1} = Tx_n p-converging to some q∈F(T)q \in F(T), every sequence (yn)(y_n) with p(yn+1,Tyn)→0p(y_{n+1}, Ty_n) \to 0 also p-converges to qq. Study T-stability for Kannan contractions p(Tx,Ty)≤λ [p(x,Tx)+p(y,Ty)]p(Tx,Ty) \le \lambda\,[p(x,Tx) + p(y,Ty)] and for Chatterjea contractions on complete (or 0-complete) partial b-metric spaces. The source settles T-stability only for its joint Chatterjea-Kannan class (Theorem 3.10, under the additional constraint 2sλ1+2λ3+(s+s2)(λ4+λ5)<22s\lambda_1 + 2\lambda_3 + (s+s^2)(\lambda_4+\lambda_5) < 2).

(4) Conjectures 3.15 and 3.16, the P property. Under the hypotheses of Theorem 3.4 - a 0-complete partial b-metric space with coefficient s≥1s \ge 1 and the joint Chatterjea-Kannan inequality (Ch-Ka) p(Tx,Ty)≤λ1 p(x,y)+λ2 p(x,Tx) p(y,Ty)1+p(x,y)+λ3 p(x,Ty) p(y,Tx)1+p(x,y)+λ4 p(x,Tx) p(x,Ty)1+p(x,y)+λ5 p(y,Ty) p(y,Tx)1+p(x,y)p(Tx,Ty) \le \lambda_1\,p(x,y) + \lambda_2\,\frac{p(x,Tx)\,p(y,Ty)}{1+p(x,y)} + \lambda_3\,\frac{p(x,Ty)\,p(y,Tx)}{1+p(x,y)} + \lambda_4\,\frac{p(x,Tx)\,p(x,Ty)}{1+p(x,y)} + \lambda_5\,\frac{p(y,Ty)\,p(y,Tx)}{1+p(x,y)} for all x,yx, y, with nonnegative λi\lambda_i satisfying λ1+λ2+2sλ3+sλ4+sλ5<1\lambda_1+\lambda_2+2s\lambda_3+s\lambda_4+s\lambda_5 < 1 - and of Corollary 3.6 (the case s=1s = 1 on a complete partial metric space, with λ1+λ2+λ3+λ4+λ5<1\lambda_1+\lambda_2+\lambda_3+\lambda_4+\lambda_5 < 1), does TT necessarily have the P property, i.e. F(T)=F(Tn)F(T) = F(T^n) for every n≥1n \ge 1? The authors note that verifying p(Tx,T2x)≤λ p(x,Tx)p(Tx, T^2x) \le \lambda\,p(x,Tx) for some λ∈[0,1)\lambda \in [0,1) would suffice (Theorem 3.14); prove the conjectures or exhibit counterexamples.

(5) Section 4 program, theta-phi-Chatterjea contractions. The source announces a definition of theta-phi-Chatterjea type contractions held for its unpublished companion paper (their reference [8]) and asks that the class follow the idea of Chatterjea contractions, generalize them in a way that keeps their properties and their relationship with other contractions, and be checked for existence of fixed points, T-stability and the P property. Since that definition was never published, the class is pinned here as the direct Chatterjea analogue of the theta-phi contractions of Zheng-Cai-Wang (defined in the background: control functions θ∈Θ\theta \in \Theta, ϕ∈Φ\phi \in \Phi): a self-map TT of a partial b-metric space (X,p)(X,p) is a theta-phi-Chatterjea contraction if, for all x,yx, y with p(Tx,Ty)≠0p(Tx,Ty) \ne 0, θ(p(Tx,Ty))≤ϕ(θ(p(x,Ty)+p(y,Tx)))\theta\big(p(Tx,Ty)\big) \le \phi\big(\theta\big(p(x,Ty) + p(y,Tx)\big)\big), i.e. the Zheng-Cai-Wang scheme with the Chatterjea combination p(x,Ty)+p(y,Tx)p(x,Ty) + p(y,Tx) in place of max⁡{d(x,y),d(x,Tx),d(y,Ty)}\max\{d(x,y), d(x,Tx), d(y,Ty)\}. Establish, for this class on complete partial b-metric spaces, that it generalizes ordinary Chatterjea contractions and retains their characteristic properties, and determine whether the class has fixed points (and whether such a point is unique), whether the Picard iteration is T-stable, and whether such maps have the P property.

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