The aim of the present paper is to introduce the notion of weak reciprocal continuity and obtain fixed point theorems by employing the new notion. The new notion is a proper generalization of reciprocal continuity and is applicable to compatible mappings as well as noncompatible mappings. Our results generalize several fixed point theorems.

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In this paper we introduce a generalization of the concept of compatible mappings, and using that condition, we obtain some new fixed point theorems under both contractive and noncontractive conditions, which may allow the existence of a common fixed point or the existence of multiple fixed or coincidence points. We also manifest that the new concept is a necessary condition for the existence of a common fixed point. MSC: 47H10; 54H25

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Abstract. In this paper, we show that generalized Meir-Keeler type contractive definitions are strong enough to generate a fixed point but do not force the mapping to be continuous at the fixed point. Thus we provide more answers to the open question posed by B.E. Rhoades in the paper Contractive definitions and continuity, Contemporary Mathematics 72(1988), 233-245.

The aim of the present paper is to study an anisotropic spherically symmetric core-envelope model of a super dense star in which core is equipped with linear equation of state, consistent with the quark matter while the envelope is considered to be of quadratic equation of state by adopting the philosophy of Takisa et al. (Pramana J Phys 92:40, 2019). We demonstrate that all the physical parameters are realistic within the core as well as envelope of the stellar object and continuous at the junction. Our model is shown to be physically viable and substantiate with the strange stars SAX J1808.4-3658 and 4U1608-52. Further, We infer that if the mass of the star increases then central density results to higher values and core shrinks, which justifies the dominating effect of gravity for higher mass celestial objects.

We explore a new relativistic anisotropic solution of the Einstein field equations for compact stars based on embedding class one condition. For this purpose, we use the embedding class one methodology by employing the Karmarkar condition. Employing this methodology, we obtain a particular differential equation that connects both the gravitational potentials and . We solve this particular differential equation choosing a simple form of generalized gravitational potential to describe a complete structure of the space-time within the stellar configuration. After determining this space-time geometry for the stellar models, we discuss thermodynamical observables including radial and tangential pressures, matter density, red-shift, velocity of sound, etc., in the stellar models. We also perform a complete graphical analysis, which shows that our models satisfy all the physical and mathematical requirements of ultra-high dense collapsed structures. Further, we discuss the moment of inertia and M-R curve for rotating and non-rotating stars.