On the Newton-Kantorovich method inK-normed spaces

Nondifferentiable operator equations in spaces equipped with aK-norm — map taking values in the cone of an ordered Banach space — are studied. The majorant method is used to derive the convergence of a modified sequence of Newton-Kantorovich approximations.     

Power Spectrum in Krein Space Quantization

The power spectrum of scalar field and space-time metric perturbations produced in the process of inflation of universe, have been presented in this paper by an alternative approach to field quantization namely, Krein space quantization (Gazeau et al. in Class. Quantum Gravity 17:1415, 2000; Takook in Int. J. Mod. Phys. E 11:509, 2002; Rouhani and Takook in ). Auxiliary negative norm states, the modes of which do not interact with the physical world, have been utilized in this method. Presence of negative norm states play the role of an automatic renormalization device for the theory.     

On Picard iterations for strongly accretive and strongly pseudo-contractive Lipschitz mappings

In this note, we speed up the convergence of the Picard sequence of iterations for strongly accretive and strongly pseudo-contractive mappings. Our results improve the results of Chidume [C.E. Chidume, Picard iteration for strongly accretive and strongly pseudo-contractive Lipschitz maps, ICTP Preprint no. IC2000098; C.E. Chidume, Iterative Algorithms for Non-expansive Mappings and Some of Their Generalizations, in: Nonlinear Analysis and Applications: To V. Lakshmikantham on his 80th Birthday, vol. 1, 2, Kluwer Acad. Publ, Dordrecht, 2003, pp. 383–429], and Liu [L. Liu, Approximation of fixed points of a strictly pseudo-contractive mapping, Proc. Amer. Math. Soc. 125 (2) (1997) 1363–1366], and some other known results. The technique of the proof, presented in this paper, is different from the technique used by Chidume.     

On small homotopies of loops

Two natural questions are answered in the negative:

• “If a space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic?”

• “Can adding arcs to a space cause an essential curve to become nulhomotopic?”

The answer to the first question clarifies the relationship between the notions of a space being homotopically Hausdorff and π₁-shape injective.

Numerical algorithm for parabolic problems with non-classical conditions

The parabolic problems with non-classical conditions are discussed in a reproducing kernel space in this paper. A reproducing kernel space is constructed, in which the non-classical conditions of the parabolic problems are satisfied. Based on the reproducing kernel space, a new technique for solving the non-classical parabolic problems is presented. Some examples are displayed to demonstrate the validity and applicability of the proposed method.     

On positive invertibility of operators and their decompositions

In Banach spaces ordered by a normal cone that contains interior points the positive invertibility of operators is studied. If there exists a uniformly positive functional then any positively invertible operator A possesses a B -decomposition, i.e., a positive decomposition A = U – V with the properties: U^–1 exists, VU^–1 ≥ 0, Ax ≥ 0, U x ≥ 0 imply x ≥ 0 and r (VU^–1) < 1. Earlier it was shown that the existence of a B -decomposition with r (VU^–1) < 1 is sufficient for the positive invertibility of the operator A. Peris' result is obtained as a special case of the main theorem. The decomposition is demonstrated for a positively invertible operator in a Banach space ordered by an ice cream cone (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Torsion-adding and asymptotic winding number for periodic window sequences

In parameter space of nonlinear dynamical systems, windows of periodic states are aligned following the routes of period-adding configuring periodic window sequences. In state space of driven nonlinear oscillators, we determine the torsion associated with the periodic states and identify regions of uniform torsion in the window sequences. Moreover, we find that the measured torsion differs by a constant between successive windows in periodic window sequences. Finally, combining the torsion-adding phenomenon, reported in this work, and the known period-adding rule, we deduce a general rule to obtain the asymptotic winding number in the accumulation limit of such periodic window sequences.     

Size of support of initial heat distribution in the 1D heat equation

The size of support of initial heat distribution is determined by the heat flow observed either at any fixed time or at any fixed position     

Fixed and periodic point theory in certain spaces

This paper proves the existence of periodic and fixed points for contractive conditions in metric (symmetric) space.