B-Implicit Contractive Conditions and Unique（Common） Fixed Points on Complex Valued Metric Spaces

A class B of complex functions is introduced and several existence theorems of unique(common) fixed points for mappings satisfying a B-implicit contraction are presented.Moreover, the existence results of common fixed points for two mappings on a nonempty set with two complex valued metrics are provided. Our outcomes generalize and improve some known results, especially, for instance, Banach contraction principle, Chatterjea-type fixed point theorem and the corresponding fixed point theorems.

Resolution of Grandi’s Paradox as Extended to Complex Valued Functions

Grandi’s paradox, which was posed for a real function of the form 1/(1+ x), has been resolved and extended to complex valued functions. Resolution of this approximately three-hundred-year-old paradox is accomplished by the use of a consistent truncation approach that can be applied to all the series expansions of Grandi-type functions. Furthermore, a new technique for improving the convergence characteristics of power series with alternating signs is introduced. The technique works by successively averaging a series at different orders of truncation. A sound theoretical justification of the successive averaging method is demonstrated by two different series expansions of the function 1/(1+ ex ) . Grandi-type complex valued functions such as 1/(i + x) are expressed as consistently-truncated and convergence-improved forms and Fagnano’s formula is established from the series expansions of these functions. A Grandi-type general complex valued function is introduced and expanded to a consistently truncated and successively averaged series. Finally, an unorthodox application of the successive averaging method to polynomials is presented.

Points of Coincidence and Common Fixed Points for Ⅱ-Expansive Mappings on Complex Valued Metric Spaces

Abstract. We use the two mappings satisfying II-expansive conditions on complex valued metric spaces to construct the convergent sequences and prove that the unique limit of the sequences is the point of coincidence or common fixed point of the two mappings. Also, we discuss the uniqueness of points of coincidence or common fixed points and give the existence theorems of unique fixed points. The obtained results generalize and improve the corresponding conclusions in references.

Estimates on the eigenvalues of complex nonlocal Sturm-Liouville problems

The present paper deals with the eigenvalues of complex nonlocal Sturm-Liouville boundary value problems.The bounds of the real and imaginary parts of eigenvalues for the nonlocal Sturm-Liouville differential equation involving complex nonlocal potential terms associated with nonlocal boundary conditions are obtained in terms of the integrable conditions of coefficients and the real part of the eigenvalues.

STUDY ON THE QUASI-FOUR YEAR OSCILLATION OF AIR-SEA INTERACTION

Complex Singular Value Decomposition(CSVD)analysis technique was applied to study the Quasi Four year Oscillation(QFO)of air sea interaction and its coupled pattern evolution during different phases.Results show that:(1)CSVD method can better reveal phase relation between two physical fields:(2)Not only northerly anomalies from Northern Hemisphere but also southerly anomalies from Southern Hemisphere contribute to EI Nino.They converge in western equatorial Pacific,leading to outburst of strong equatorial westerly anomalies,and result in strong El Nino event onset:(3)An abnormal subtropical anticyclone circulation appears over northwestern Pacific while El Nino developing.It favors transitions from the warm SST(EI Nino)to the cold SST(La Nina),just as the tropical westerly anomalies produced by abnormal cyclone during a decaying La Nina.which encourage the development of El Nino:(4)The westerly anomalies in equatorial Pacific are mainly induced by eastward abnormal subtropical cyclone pairs,which are located in north and south Pacific respectively,and are not the eastward westerly anomalies from equatorial Indian Ocean.