This problem asks for conditions under which a self-mapping TT of a complete U\mathfrak U-metric space XX, together with an auxiliary function φ:X→[0,∞)\varphi:X\to[0,\infty), has a circle or disc all of whose points are simultaneously fixed points of TT and zeros of φ\varphi (a φ\varphi-fixed circle or φ\varphi-fixed disc), and under which such a set is unique. The question was posed by Özgür and Taş (2021) on ordinary metric spaces and was carried to the unifying U\mathfrak U-metric framework (a distance on nn-tuples covering metric, SS-metric, AA-metric, extended bb-metric and parametric metric spaces) by Roy, Dey and Saha (2023), who solved it only for FF-type φx0\varphi_{x_0}-contractions on complete symmetric U\mathfrak U-metric spaces under limit/boundedness hypotheses on the coefficient function α\alpha. It remains open for non-symmetric U\mathfrak U-metrics, unbounded α\alpha, contraction classes beyond FF-type φx0\varphi_{x_0}-contractions, for uniqueness criteria, and for the original metric case itself.

Background

Let (X,d)(X,d) be a metric space and T:X→XT:X\to X a self-mapping with fixed point set Fix(T)={x∈X:Tx=x}Fix(T)=\{x\in X:Tx=x\}. Motivated by partial metric theory, Jleli, Samet and Vetro (2014) introduced the notion of a φ\varphi-fixed point: given a function φ:X→[0,∞)\varphi:X\to[0,\infty), a point xx is a φ\varphi-fixed point of TT if Tx=xTx=x and φ(x)=0\varphi(x)=0, i.e. a fixed point that also lies in the zero set Zφ={x∈X:φ(x)=0}Z_\varphi=\{x\in X:\varphi(x)=0\}. In parallel, the fixed-circle problem, initiated by Özgür and Taş (2019), studies when Fix(T)Fix(T) contains a whole geometric object such as the circle Cx0,r={x:d(x,x0)=r}C_{x_0,r}=\{x:d(x,x_0)=r\} or the disc Dx0,r={x:d(x,x0)≤r}D_{x_0,r}=\{x:d(x,x_0)\le r\}, which is interesting exactly in the non-unique fixed point regime, where Banach-type theorems give no information about the shape of Fix(T)Fix(T). Combining the two viewpoints, Özgür and Taş (2021) defined a φ\varphi-fixed circle (resp. φ\varphi-fixed disc) of TT as a circle Cx0,rC_{x_0,r} (resp. disc Dx0,rD_{x_0,r}) with Cx0,r⊆Fix(T)∩ZφC_{x_0,r}\subseteq Fix(T)\cap Z_\varphi (resp. Dx0,r⊆Fix(T)∩ZφD_{x_0,r}\subseteq Fix(T)\cap Z_\varphi), and posed an open problem: for various classes of self-mappings, determine conditions for the existence and uniqueness of such φ\varphi-fixed circles and discs, suggesting the use of auxiliary comparison numbers such as M(x,y)=max⁡{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty)+d(y,Tx)1+d(x,Tx)+d(y,Ty)}M(x,y)=\max\{d(x,y),d(x,Tx),d(y,Ty),\frac{d(x,Ty)+d(y,Tx)}{1+d(x,Tx)+d(y,Ty)}\}.

Roy, Saha and Dey (2021) introduced the U\mathfrak U-metric space (Fraktur U, also written U-metric space) of nn-tuples: a structure U:Xn×(0,∞)→[0,∞)\mathfrak U:X^n\times(0,\infty)\to[0,\infty) with U(x1,…,xn;θ)=0\mathfrak U(x_1,\dots,x_n;\theta)=0 if and only if x1=⋯=xnx_1=\cdots=x_n, satisfying

U(x1,…,xn;θ)≤α(x1,…,xn;θ)∑i=1nU(xi,…,xi,a;θ)\mathfrak U(x_1,\dots,x_n;\theta)\le\alpha(x_1,\dots,x_n;\theta)\sum_{i=1}^{n}\mathfrak U(x_i,\dots,x_i,a;\theta)

for all x1,…,xn,a∈Xx_1,\dots,x_n,a\in X and θ>0\theta>0, where α:Xn×(0,∞)→[1,∞)\alpha:X^n\times(0,\infty)\to[1,\infty) is a coefficient function. For small nn and constant or specialized α\alpha this specializes to ordinary metrics, SS-metrics (a triple-based distance S(x,y,z)S(x,y,z) that vanishes exactly when the three points coincide), AA-metrics (another distance notion attached to ordered triples), extended bb-metrics (triangle inequality up to a variable factor) and parametric metric spaces, so U\mathfrak U-metrics are a common umbrella for the generalized distances used in fixed point theory. The space is called symmetric when the value of U\mathfrak U is invariant under permutation of its tuple entries (ordinary metrics are automatically symmetric).

Within this framework Roy, Dey and Saha (2023) proved: (i) a φ\varphi-fixed point theorem for Banach orbital contractions — mappings controlled along their own orbits by a Wardowski-type inequality. Wardowski's FF-contractions (2012) replace the Banach ratio d(Tx,Ty)≤h d(x,y)d(Tx,Ty)\le h\,d(x,y) by F(d(Tx,Ty))≤hF(d(x,y))F(d(Tx,Ty))\le hF(d(x,y)) for a strictly increasing F:[0,∞)→[0,∞)F:[0,\infty)\to[0,\infty); here the three-variable version F(U(x,…,x,Tx;θ),φ(Tx),φ(x))≤hF(U(x,…,x,x0;θ),φ(x),φ(x0))F(\mathfrak U(x,\dots,x,Tx;\theta),\varphi(Tx),\varphi(x))\le hF(\mathfrak U(x,\dots,x,x_0;\theta),\varphi(x),\varphi(x_0)) with h∈(0,1/s)h\in(0,1/s) and FF from a family with coefficient ss defines what that paper calls an FF-type φx0\varphi_{x_0}-contraction (the precise definition — the admissible family of FF, the coefficient ss, and the quantification of θ\theta and x0x_0 — is given in Roy, Dey and Saha 2023, DOI: 10.1515/ms-2023-0036); (ii) a fixed point theorem via asymptotic regularity — the classical property of Browder and Petryshyn (1966) that distances between consecutive iterates tend to 00; (iii) fixed-circle theorems on U\mathfrak U-metric spaces with a uniqueness condition; and (iv) the first φ\varphi-fixed circle and φ\varphi-fixed disc theorems in the U\mathfrak U-metric framework, of the following shape: if ρ=inf⁡{U(x,…,x,Tx;θ):x≠Tx,θ>0}\rho=\inf\{\mathfrak U(x,\dots,x,Tx;\theta):x\ne Tx,\theta>0\} and φ(x)≤U(x,…,x,Tx;θ)\varphi(x)\le\mathfrak U(x,\dots,x,Tx;\theta) on the relevant circle (resp. disc), then the circle Cx0,ρU={x:U(x,…,x,x0;θ)=ρ for all θ>0}C^{\mathfrak U}_{x_0,\rho}=\{x:\mathfrak U(x,\dots,x,x_0;\theta)=\rho\ \text{for all }\theta>0\} (resp. the disc Dx0,ρUD^{\mathfrak U}_{x_0,\rho}) is a φ\varphi-fixed circle (resp. disc). All of these results are restricted to complete symmetric U\mathfrak U-metric spaces whose coefficient function α\alpha satisfies additional limit/boundedness hypotheses. The paper presents these results as new solutions to the Özgür–Taş open problem on the geometric properties of φ\varphi-fixed points and the existence and uniqueness of φ\varphi-fixed circles and φ\varphi-fixed discs for various classes of self-mappings, rather than a complete resolution: the cases of non-symmetric U\mathfrak U-metrics, unbounded α\alpha, contraction types other than FF-type φx0\varphi_{x_0}-contractions, and uniqueness criteria remain open.

Problem Statement

Let (X,U)(X,\mathfrak U) be a complete U\mathfrak U-metric space, where U:Xn×(0,∞)→[0,∞)\mathfrak U:X^n\times(0,\infty)\to[0,\infty) satisfies U(x1,…,xn;θ)=0\mathfrak U(x_1,\dots,x_n;\theta)=0 if and only if x1=⋯=xnx_1=\cdots=x_n and the inequality

U(x1,…,xn;θ)≤α(x1,…,xn;θ)∑i=1nU(xi,…,xi,a;θ)\mathfrak U(x_1,\dots,x_n;\theta)\le\alpha(x_1,\dots,x_n;\theta)\sum_{i=1}^{n}\mathfrak U(x_i,\dots,x_i,a;\theta)

holds for a coefficient function α:Xn×(0,∞)→[1,∞)\alpha:X^n\times(0,\infty)\to[1,\infty); let φ:X→[0,∞)\varphi:X\to[0,\infty) and let T:X→XT:X\to X be a self-mapping with Fix(T)≠XFix(T)\ne X (equivalently, TT is not the identity mapping). Recall that for r>0r>0 a circle Cx0,rU={x∈X:U(x,…,x,x0;θ)=r for all θ>0}C^{\mathfrak U}_{x_0,r}=\{x\in X:\mathfrak U(x,\dots,x,x_0;\theta)=r\ \text{for all }\theta>0\} (resp. a disc Dx0,rU={x∈X:U(x,…,x,x0;θ)≤r for all θ>0}D^{\mathfrak U}_{x_0,r}=\{x\in X:\mathfrak U(x,\dots,x,x_0;\theta)\le r\ \text{for all }\theta>0\}) is a φ\varphi-fixed circle (resp. φ\varphi-fixed disc) of TT if it is nonempty and contained in Fix(T)∩Zφ={x∈X:Tx=x and φ(x)=0}Fix(T)\cap Z_\varphi=\{x\in X:Tx=x\ \text{and}\ \varphi(x)=0\}, the set of fixed points of TT that are simultaneously zeros of φ\varphi. The problem, posed by Özgür and Taş on metric spaces and left open in this generality by Roy, Dey and Saha, is: for various classes of self-mappings, determine conditions under which (1) TT possesses a φ\varphi-fixed circle or φ\varphi-fixed disc, and (2) such a φ\varphi-fixed circle or φ\varphi-fixed disc is unique as a point set (the disc case, whose defining property is monotone in the radius, being read as uniqueness of the largest such disc). The regimes explicitly left open by the source paper include: (i) U\mathfrak U-metric spaces that are not symmetric; (ii) coefficient functions α\alpha that are unbounded on the relevant tuples; and (iii) contraction classes other than the FF-type φx0\varphi_{x_0}-contractions treated there (as defined in Roy, Dey and Saha 2023), for example rational, Meir–Keeler-type, simulation-function-based or multivalued conditions. The regimes (i)–(iii), and the ordinary metric case, are alternatives rather than parts of a single joint problem, and the φ\varphi-fixed circle question and the φ\varphi-fixed disc question are logically independent. The special case of ordinary metric spaces (existence and uniqueness of φ\varphi-fixed circles and discs for various classes of self-mappings, as originally posed in 2021) is part of the same question and remains open beyond scattered sufficient conditions, uniqueness having been explicitly open since then.

References

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