Common fixed point theorems of different compatible type mappings using Ciric\u27s contraction type condition

The purpose of this paper is to establish necessary and sufficient conditions for the existence of common fixed points for a compatible pair of selfmaps under Ciric\u27s contraction type condition. These theorems improve and generalize the results of Mukherjee and Verma [11] and Jungck [9] to a pair of selfmaps. Also established the existence of common fixed points for a pair of compatible mappings of type (B), and obtain a result on the existence of common fixed points for a pair of compatible mappings of type (A) as corollary. Greguš fixed point theorem follows as a special case to our results

Common Fixed Point Theorems under Rational Inequality

In this paper we establish common fixed point theorems for two pairs of self maps in a complete metric space by using occasionally weakly biased maps satisfying the property (E.A.) using contraction  condition involving rational expressions.  These results partially generalize  Pachpatte [10], Jeong and Rhoades [5] and Kameswari [9]. Keywords: Weakly compatible, occasionally weakly compatible, property (E.A), coincidence point, point of coincidence, common fixed point. AMS (2010) Mathematics Subject Classifications: 47H10

Geomagnetic storm effects on GPS based navigation

The energetic events on the sun, solar wind and subsequent effects on the Earth's geomagnetic field and upper atmosphere (ionosphere) comprise space weather. Modern navigation systems that use radio-wave signals, reflecting from or propagating through the ionosphere as a means of determining range or distance, are vulnerable to a variety of effects that can degrade the performance of the navigational systems. In particular, the Global Positioning System (GPS) that uses a constellation of earth orbiting satellites are affected due to the space weather phenomena. <br><br> Studies made during two successive geomagnetic storms that occurred during the period from 8 to 12 November 2004, have clearly revealed the adverse affects on the GPS range delay as inferred from the Total Electron Content (TEC) measurements made from a chain of seven dual frequency GPS receivers installed in the Indian sector. Significant increases in TEC at the Equatorial Ionization anomaly crest region are observed, resulting in increased range delay during the periods of the storm activity. Further, the storm time rapid changes occurring in TEC resulted in a number of phase slips in the GPS signal compared to those on quiet days. These phase slips often result in the loss of lock of the GPS receivers, similar to those that occur during strong(>10 dB) L-band scintillation events, adversely affecting the GPS based navigation