Approximate Solutions by Truncated Taylor Series Expansions of Nonlinear Differential Equations and Related Shadowing Property with Applications

This paper investigates the errors of the solutions as well as the shadowing property of a class of nonlinear differential equations which possess unique solutions on a certain interval for any admissible initial condition. The class of differential equations is assumed to be approximated by well-posed truncated Taylor series expansions up to a certain order obtained about certain, in general nonperiodic, sampling points t(i) is an element of [t(0), t(J)] for i = 0, 1, . . . , J of the solution. Two examples are provided.

Convergence Properties and Fixed Points of Two General Iterative Schemes with Composed Maps in Banach Spaces with Applications to Guaranteed Global Stability

This paper investigates the boundedness and convergence properties of two general iterative processes which involve sequences of self-mappings on either complete metric or Banach spaces. The sequences of self-mappings considered in the first iterative scheme are constructed by linear combinations of a set of self-mappings, each of them being a weighted version of a certain primary self-mapping on the same space. The sequences of self-mappings of the second iterative scheme are powers of an iteration-dependent scaled version of the primary self-mapping. Some applications are also given to the important problem of global stability of a class of extended nonlinear polytopic-type parameterizations of certain dynamic systems.

On Nonnegative Solutions of Fractional q-Linear Time-Varying Dynamic Systems with Delayed Dynamics

This paper is devoted to the investigation of nonnegative solutions and the stability and asymptotic properties of the solutions of fractional differential dynamic linear time-varying systems involving delayed dynamics with delays. The dynamic systems are described based on q-calculus and Caputo fractional derivatives on any order.

Effectiveness of a Multi-Component Smoking Cessation Support Programme (McSCSP) for Patients with Severe Mental Disorders: Study Design

Only a few studies have examined the efficacy and safety of smoking cessation programmes in patients with mental disorders. The aim of this paper is to describe in detail the methodology used in the study as well as the Multi-component Smoking Cessation Support Programme in terms of pharmacological treatments and psychological interventions. An open-label 9-month follow-up study was conducted in Spain. A total of 82 clinically stable outpatients with schizophrenia, schizoaffective or bipolar disorder were enrolled. Treatment consisted of a programme specifically developed by the research team for individuals with severe mental disorders. The programme consisted of two phases: (1) weekly individual motivational therapy for 4-12 weeks, and (2) a 12-week active treatment phase. During this phase, at each study visit patients received a one- or two-week supply of medication (transdermal nicotine patches, varenicline or bupropion) with instructions on how to take it, in addition to group psychotherapy for smoking cessation. Evaluations were performed: (1) at the time of enrolment in the study, (2) during the 12-week active treatment phase of the study (weekly for the first 4 weeks and then biweekly), and (3) after the end of this phase (two follow-up assessments at weeks 12 and 24). Evaluations included: (1) smoking history, (2) substance use, (3) psychopathology, (4) adverse events, and (5) laboratory tests. The importance of this study lies in addressing a topical issue often ignored by psychiatrists: the unacceptably high rates of tobacco use in patients with severe mental disorders.

A New Type of Coincidence and Common Fixed Point Theorem with Applications

Coincidence and common fixed point theorems for a class of 'Ciric-Suzuki hybrid contractions involving a multivalued and two single-valued maps in a metric space are obtained. Some applications including the existence of a common solution for certain class of functional equations arising in a dynamic programming are also discussed..