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

Abstract

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.

1. Introduction

The problems of boundedness and convergence of sequences of iterative schemes are very important in numerical analysis and the numerical implementation of discrete schemes; see [1–4] and references therein. In particular, [1] describes in detail and with rigor the associated problems linked to the theory of fixed points in various types of spaces like metric spaces, complete and compact metric spaces, and Banach spaces, while it also contains, discusses, and compares results of a number of relevant background references on the subject. In other papers, related problems of fixed point theory or stability are focused on approximations including, in some cases, issues from a computational point of view eventually involving modified numerical methods like, for instance, Aitken’s delta-squared methods or Steffensen’s method [4–11]. Also, a counterpart theory has been also formulated in the framework of common fixed points and coincidence points for several mappings and in the framework of multivalued functions. An important background on fixed, best proximity, and proximal points concerned with nonexpansive, contractive, weakly contractive, and strictly contractive mappings has been developed; see, for instance, [1–4, 8–25] and references therein. In particular, a relevant effort has been also focused on the formulations of extensions of the above problems to the study of existence and uniqueness of fixed and best proximity points in cyclic self-mappings as well to proximal contractions [12–14, 17–20, 24, 25] and to the characterization of approximate fixed and coincidence points [21, 22]. Direct applications of fixed point theory to the study of the stability of dynamic systems including the property of ultimate boundedness for the trajectory solutions having mixed nonexpansive and expansive properties through time or being subject to impulsive controls have been given in [21, 24, 25]. This paper is focused on the study of boundedness and convergence of sequences of distances and iterated points and the characterization of fixed points of a class of composite self-maps in metric spaces. Such maps are built with combinations of sets of elementary self-maps which can be expansive or nonexpansive and the last ones can be contractive (including the case of strict contractions). The composite maps are defined by switching rules which select some self-map (the “active” self-map) on a certain interval of definition of the running index of the sequence of iterates being built. The above-mentioned properties concerning the sequences of iterates being generated from given initial points are investigated under particular constraints for the switching rule. Note, on the other hand, that the properties of controllability, observability, and stability of differential or difference equations as well as the various kinds of dynamic systems are of a wide interest in theory and applications including the cases of presence of disturbances and/or unmodeled dynamics [23–45]; see, for instance, related problems associated with continuous-time, discrete-time, digital, and hybrid systems and those involving delayed dynamics [27, 30, 33, 37–39], hybrid [34–36, 41], and switched dynamic systems [31, 32, 38–43] and references therein. The above problems are often studied in an integrated or combined fashion in the sense that the presence of uncertainties of any nature (basically unmeasurable noise or unmodeled dynamics) is incorporated to the description of differential, difference, or hybrid systems with eventual external delays or delayed dynamics. The stability is studied with different tools as Lyapunov theory, matrix inequalities, or fixed point theory. Fractional calculus has also been widely used in the investigation of the solutions of differential, functional-differential, and dynamic systems; see, for instance, [44, 45] and some references therein.

This paper is firstly devoted to giving a framework for the contractive properties of two general iterative schemes which are constructed via combinations of elementary self-maps in appropriate metric or Banach spaces. The sequences of self-mappings of the first 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. Such weights are nonnegative real sequences in general. The single parameterizations of the first iterative scheme include polytopic-type ones, where a set of real scalar sequences define both the sequence of self-mappings of interest and the individual parameterizations as a particular case. The second iterative scheme is a generalization of De Figueiredo scheme [8], where the sequences of self-mappings are integer powers of a scaled version of a primary elementary self-mapping. Such powers are iteration-dependent, while the scaling weights can be iteration-dependent. A second objective is to describe an application of the developed theoretical framework to study the stability properties of (in general) nonlinear switched dynamic systems under appropriate stabilizing switching rules. The obtained formal results can also be useful to investigate the stability of dynamic systems under combinations of single parameterizations.

1.1. Notation

{𝑇_𝑛}⇉𝑇^∗ (i.e., limsup_𝑛→∞{‖𝑇_𝑛𝑥−𝑇^∗𝑥‖∶𝑥∈Dom𝑇_𝑛}=0; ∀𝑥∈Dom𝑇) and {𝑇_𝑛}→𝑇^∗ (i.e., lim_𝑛→∞𝑇_𝑛𝑥=𝑇^∗𝑥; ∀𝑥∈Dom𝑇_𝑛) for 𝑇^∗,𝑇_𝑛∶𝑋→𝑋, ∀𝑛∈𝐙₀₊; denote, respectively, uniform and point-wise convergence in 𝑋 of  𝑇_𝑛∶𝑋→𝑋 to 𝑇^∗∶𝑋→𝑋 provided that all of them have the same domain.

Fix(𝑇) denotes the set of fixed points of 𝑇∶𝑋→𝑋  and  𝑘={0,1,…,𝑘}.

2. Iterative Scheme 1

Consider the following iterative scheme under a sequence of self-mappings 𝑇_𝑛∶𝑋→𝑋, ∀𝑛∈𝐙₀₊, on a vector space 𝑋: 𝑥_𝑛+1=𝑇_𝑛𝑥_𝑛=^𝑘_𝑖=0𝛼_𝑖^(𝑛)𝑇^𝑖𝑥_𝑛,∀𝑛∈𝐙₀₊, (1) for any given 𝑥₀∈𝑋 with 𝑇∶𝑋→𝑋 and 𝑇_𝑛∶𝑋→𝑋, ∀𝑛∈𝐙₀₊, being defined by 𝑇_𝑛𝑥=(∑^𝑘_𝑖=0𝛼_𝑖^(𝑛)𝑇^𝑖)𝑥 for any 𝑥∈𝑋 and the nonnegative real parameterization sequences being subject to ∑^𝑘_𝑖=0𝛼_𝑖^(𝑛)>0, ∀𝑖∈𝑘={0,1,…,𝑘}, ∀𝑛∈𝐙₀₊.

Theorem 1. Consider the iterative scheme (1) on a vector space 𝑋, with 0∈𝑋, under the following assumptions.(1)Either (𝑋,‖‖) is a normed space endowed with a norm ‖‖ or, respectively, (𝑋,𝑑) is a metric space endowed with a homogeneous translation-invariant metric 𝑑∶𝑋×𝑋→𝐑₀₊.(2)∑^𝑘_𝑖=0𝛼_𝑖^(𝑛)>0 and 0≤𝛼_𝑖^(𝑛+1)=(1+𝛼_𝑖^(𝑛))𝛼_𝑖^(𝑛), ∀𝑖∈𝑘={0,1,…,𝑘}, ∀𝑛∈𝐙₀₊, and inf_{𝑛∈𝐙0+}max_{1≤𝑖≤𝑘}𝛼_𝑖^(𝑛)>0, with the nonnegative real sequences {𝛼_𝑖^(𝑛)}, ∀𝑖∈𝑘, being subject to the constraints |𝛼_𝑖^(𝑛)|≤𝛼_𝑛≤𝑚_𝑛(𝑑(𝑥_𝑛,𝑥_𝑛+1)/𝑑(𝑥_𝑛+1,0)) and 𝑚_𝑛=𝑜[𝑑(𝑥_𝑛+1,0)], ∀𝑖∈𝑘, ∀𝑛∈𝐙₀₊, where the relative one-step increment parameterization sequences are 𝛼_𝑖^(𝑛)=(𝛼_𝑖^(𝑛+1)−𝛼_𝑖^(𝑛))/𝛼_𝑖^(𝑛), ∀𝑖∈𝑘, ∀𝑛∈𝐙₀₊.(3)𝑇∶𝑋→𝑋 possesses the (nonnecessarily contractive) condition 𝑑(𝑇𝑥,𝑇𝑦)≤𝐾𝑑(𝑥,𝑦), ∀𝑥,𝑦∈𝑋, for some 𝐾∈𝐑₊.(4)(1+𝑚_𝑛)(∑^𝑘_𝑖=0𝛼_𝑖^(𝑛)𝐾^𝑖)≤𝜌<1, ∀𝑛∈𝐙₀₊.Then, the following properties hold.(i)There exists the limit lim_𝑛→∞𝑑(𝑥_𝑛,𝑥_𝑛+1)=0 for any given initial point 𝑥₀∈𝑋 of the iterative scheme (1) and the sequence {𝑥_𝑛} is bounded.(ii)If, in addition, {𝑇_𝑛}⇉𝑇^∗ for some limit 𝑇^∗∶𝑋→𝑋 and if either (𝑋,‖‖) is a Banach space or (𝑋,𝑑) is complete, then {𝑥_𝑛} is a Cauchy sequence and thus convergent to some 𝑧 in 𝑋 which is the unique fixed point of 𝑇^∗∶𝑋→𝑋 and thus independent of the initial point 𝑥₀∈𝑋 of the iterative scheme (1). All the self-mappings of the sequence {𝑇_𝑛} as well as 𝑇^∗∶𝑋→𝑋 are strict contractions.(iii)If either (𝑋,‖‖) is a Banach space or (𝑋,𝑑) is complete and {𝛼_𝑖^(𝑛)}→𝛼_𝑖 as 𝑛→∞, ∀𝑖∈𝑘, for some self-mapping  𝑇^∗ on 𝑋, then there is a unique 𝑧^∗∈Fix(𝑇^∗) in 𝑋 such that 𝑧^∗=𝑧 for any given initial point 𝑥₀∈𝑋 of the iterative scheme (1). Also, 𝑇^∗∶𝑋→𝑋 is a strict contraction and thus a strict Picard self-mapping with a unique fixed point 𝑧^∗(=𝑧)∈𝑋 such that 𝑇^∗𝑛𝑥₀→𝑧^∗, 𝑇_𝑛𝑥₀→𝑧^∗ as 𝑛→∞ for any given initial point 𝑥₀∈𝑋, where 𝑇_𝑛∶𝐙₀₊×𝑋→𝑋 is the composite mapping 𝑇_𝑛=𝑇_𝑛𝑇_𝑛−1⋯𝑇₀, ∀𝑛∈𝐙₀₊.(iv)The “a priori” and “a posteriori” error estimates and the convergence rate are, respectively, given by the subsequent relations: 𝑑𝑥_𝑛≤𝜌,𝑧^𝑛𝑑𝑥1−𝜌⁰,𝑥₁,𝑑𝑥_𝑛≤𝜌,𝑧𝑑𝑥1−𝜌^𝑛−1,𝑥_𝑛,𝑑𝑥_𝑛,𝑧≤𝜌^𝑛𝑑𝑥₀,𝑧. (2)(3)(4)

Proof. Define the (𝑘+1) error sequences {𝛼_𝑖^(𝑛)} by 𝛼_𝑖^(𝑛)=𝛼_𝑖^(𝑛+1)−𝛼_𝑖^(𝑛), ∀𝑖∈𝑘={0,1,…,𝑘}, ∀𝑛∈𝐙₀₊. If (𝑋,‖‖) is a normed space, then there is always a metric-induced norm 𝑑(𝑥,𝑦)=‖𝑥−𝑦‖, ∀𝑥,𝑦∈𝑋. On the other hand, if (𝑋,𝑑) is a metric space endowed with a homogeneous translation-invariant metric 𝑑∶𝑋×𝑋→𝐑₀₊ then there is a metric-induced norm ‖𝑥‖=𝑑(𝑥,0), ∀𝑥,𝑦∈𝑋. Both spaces (𝑋,‖‖) and (𝑋,𝑑) are formally identical and they can both deal with a metric-induced norm by using the standard metric properties and its homogeneous and translation-invariance properties. Thus, one gets via recursive calculations that𝑑𝑥_𝑛+2,𝑥_𝑛+1=𝑑^𝑘_𝑖=0𝛼_𝑖^(𝑛)𝑇^𝑖𝑥_𝑛+1+𝛼_𝑖^(𝑛)𝑇^𝑖𝑥_𝑛+1,_𝑘𝑖=0𝛼_𝑖^(𝑛)𝑇^𝑖𝑥_𝑛≤𝑑𝛼₀^(𝑛)𝑥_𝑛+1,𝛼₀^(𝑛)𝑥_𝑛+𝑑𝛼₁^(𝑛)𝑇𝑥_𝑛+1,𝛼₁^(𝑛)𝑇𝑥_𝑛+𝑑^𝑘_𝑖=2𝛼_𝑖^(𝑛)𝑇^𝑖𝑥_𝑛+1,^𝑘_𝑖=2𝛼_𝑖^(𝑛)𝑇^𝑖𝑥_𝑛+𝑑𝛼₀^(𝑛)𝑥_𝑛+1,0+𝑑𝛼₁^(𝑛)𝑇𝑥_𝑛+1,0+𝑑^𝑘_𝑖=2𝛼_𝑖^(𝑛)𝑇^𝑖𝑥_𝑛+1,0≤⋯≤^𝑘_𝑖=0𝑑𝛼_𝑖^(𝑛)𝑇^𝑖𝑥_𝑛+1,𝛼_𝑖^(𝑛)𝑇^𝑖𝑥_𝑛+^𝑘_𝑖=0𝑑𝛼_𝑖^(𝑛)𝑇^𝑖𝑥_𝑛+1≤,01+𝑚_𝑛^𝑘_𝑖=0𝛼_𝑖^(𝑛)𝐾^𝑖𝑑𝑥_𝑛,𝑥_𝑛+1;∀𝑛∈𝐙₀₊. (5) Thus, 𝑑𝑥_𝑛+1,𝑥_𝑛≤^𝑛−1_𝑗=01+𝑚_𝑗^𝑘_𝑖=0𝛼_𝑖^(𝑗)𝐾^𝑖𝑑𝑇₀𝑥₀,𝑥₀≤𝜌𝑑𝑥_𝑛,𝑥_𝑛−1≤𝜌^𝑛𝑑𝑥₁,𝑥₀;∀𝑛∈𝐙₊, (6) so that lim_𝑛→∞𝑑(𝑥_𝑛,𝑥_𝑛+1)=0 for any given 𝑥₀∈𝑋. It follows from (6) that, for any given initial 𝑥₀∈𝑋, 𝑑𝑥_𝑛,𝑥₀≤^𝑛−1_𝑗=0𝑑𝑥_𝑗+1,𝑥_𝑗≤1−𝜌^𝑛𝑑𝑇1−𝜌⁰𝑥₀,𝑥₀<+∞,limsup_𝑛→∞𝑑𝑥_𝑛,𝑥₀≤𝑑𝑥₁,𝑥₀^∞_𝑗=0𝜌^𝑗≤𝑑𝑇₀𝑥₀,𝑥₀1−𝜌<+∞, (7) since 𝜌<1 so that {𝑥_𝑛} is bounded for any given 𝑥₀∈𝑋 and {𝑑(𝑥_𝑛+1,𝑥_𝑛)}→0 from (6). All the self-mappings 𝑇_𝑛∶𝑋→𝑋, ∀𝑛∈𝐙₀₊, are strict contractions by construction from assumption 4. On the other hand, note that, since {𝑇_𝑛}⇉𝑇^∗, one gets 𝑑𝑇^∗𝑥,𝑇^∗𝑦=lim_𝑛→∞𝑑𝑇_𝑛𝑥,𝑇_𝑛𝑦=𝑑lim_𝑛→∞𝑇_𝑛𝑥,lim_𝑛→∞𝑇_𝑛𝑥≤𝜌𝑑𝑥,𝑦;∀𝑥,𝑦∈𝑋, (8) so that 𝑇^∗∶𝑋→𝑋 is a strict contraction. Since 𝑇_𝑛∶𝑋→𝑋, ∀𝑛∈𝐙₀₊, are all strict 𝜌-contractions, {𝑇_𝑛}⇉𝑇^∗, Fix(𝑇^∗)={𝑧}, and Fix(𝑇_𝑛)={𝑧_𝑛}, ∀𝑛∈𝐙₀₊, so that 𝑑(𝑇_𝑛𝑥,𝑇^∗𝑥)<𝜀(1−𝜌), ∀𝑥∈𝑋, so that 𝑑𝑧_𝑛,𝑧=𝑑𝑇_𝑛𝑧_𝑛,𝑇^∗𝑧≤𝑑𝑇_𝑛𝑧_𝑛,𝑇^∗𝑧_𝑛+𝑑𝑇^∗𝑧_𝑛,𝑇^∗𝑧<𝜀1−𝜌+𝜌𝑑𝑧_𝑛,𝑧, (9) and then 𝑑(𝑧_𝑛,𝑧)<𝜀, ∀𝑛≥𝑛₀, so that {𝑧_𝑛}→𝑧. Also, {𝑑(𝑇_𝑛𝑥_𝑛,𝑥_𝑛)}→0 from (6) implies {𝑑(𝑇_𝑛𝑥_𝑛,𝑧_𝑛)}→0 and {𝑇_𝑛𝑧_𝑛}→{𝑇_𝑛𝑧} (since {𝑇_𝑛𝑥_𝑛}→𝑧_𝑛 and {𝑧_𝑛}→𝑧 with Fix(𝑇_𝑛)={𝑧_𝑛}, ∀𝑛∈𝐙₀₊), {𝑑(𝑇_𝑛𝑥_𝑛,𝑧)}→0, {𝑇_𝑛𝑥_𝑛}→𝑧 (since {𝑧_𝑛}→𝑧), and {𝑇_𝑛𝑥_𝑛}→{𝑇^∗𝑥_𝑛} (since {𝑇_𝑛}⇉𝑇^∗). Thus, it follows that {𝑇^∗𝑥_𝑛}→𝑇^∗𝑧(=𝑧) which implies that {𝑥_𝑛}→𝑧. Also, {𝑥_𝑛} is a Cauchy sequence convergent to 𝑧∈𝑋 if (𝑋,‖‖) is a Banach space and if (𝑋,𝑑) is a complete metric space, respectively.
On the other hand, 𝑥_𝑛+1=𝑇_𝑛𝑥_𝑛=𝑇_𝑛𝑥₀→𝑧(∈𝑋) as 𝑛→∞, ∀𝑥₀∈𝑋, where 𝑇_𝑛∶𝐙₀₊×𝑋→𝑋 is the composite mapping 𝑇_𝑛=𝑇_𝑛𝑇_𝑛−1⋯𝑇₀, ∀𝑛∈𝐙₀₊. From (6), the self-mappings 𝑇_𝑛∶𝑋→𝑋, ∀𝑛∈𝐙₊, are all strict contractions. Now, we prove that the limit point 𝑧 is independent of the initial condition 𝑥₀ and thus unique. Assume two distinct initial values 𝑥₀,𝑦₀∈𝑋 such that 𝑇_𝑛𝑥₀→𝑧(=𝑧(𝑥₀)), 𝑇_𝑛𝑦₀→𝜔(=𝜔(𝑦₀)) as 𝑛→∞ for some 𝑧, 𝜔(≠𝑧)∈𝑋. Note from (6) that, since 𝜌∈(0,1) is independent of the sequences {𝑇_𝑛𝑥₀} and {𝑇_𝑛𝑦₀}, one gets 𝑑𝑇_𝑛𝑥₀,𝑇_𝑛𝑦₀≤𝜌^𝑛𝑑𝑥₀,𝑦₀;∀𝑛∈𝐙₊;lim_𝑛→∞𝑑𝑇_𝑛𝑥₀,𝑇_𝑛𝑦₀=0. (10) Since 𝜔≠𝑧, one has the following from the triangle inequality: 0<𝑑(𝜔,𝑧)≤𝑑𝑇_𝑛𝑦₀,𝑇_𝑛𝑥₀≤𝑑𝜔,𝑇_𝑛𝑦₀+𝑑𝑇_𝑛𝑥₀,𝑇_𝑛𝑦₀+𝑑𝑇_𝑛𝑥₀,𝑧;∀𝑛∈𝐙₊, (11) and then one gets the contradiction below to the assumption 𝜔≠𝑧: 0<lim_𝑛→∞𝑑𝜔,𝑇_𝑛𝑦₀+𝑑𝑇_𝑛𝑥₀,𝑇_𝑛𝑦₀+𝑑𝑇_𝑛𝑥₀,𝑧=0, (12) so that 𝜔=𝑧 and 𝑇_𝑛𝑥₀→𝑧 as 𝑛→∞ with 𝑧 being independent of the initial point 𝑥₀ of the iterative scheme (1). Hence, properties (i)-(ii) have been proven.
To prove property (iii), note that the assumption of uniform convergence {𝑇_𝑛}⇉𝑇^∗ in 𝑋 is weakened to point-wise convergence {𝑇_𝑛}→𝑇^∗ in 𝑋 since {𝛼_𝑖^(𝑛)}→𝛼_𝑖 and then {𝛼_𝑖^(𝑛)}→0; ∀𝑖∈𝑘 and 𝑇^∗∶𝑋→𝑋 is a 𝜌-contraction from assumption 4. Thus, {𝑑(𝑥_𝑛+1,𝑥_𝑛)}→0 implies {𝑑(𝑇_𝑛𝑥_𝑛,𝑧)}→0 and {𝑥_𝑛}→𝑧 implies {𝑇_𝑛𝑥_𝑛}→𝑇^∗𝑧(=𝑧). Since (𝑋,𝑑) is complete and 𝑇_𝑛∶𝑋→𝑋 is a strict contraction then 𝑇^∗ is also a strict contraction and thus a strict Picard self-mapping on 𝑋 and there is a unique 𝑧^∗∈Fix(𝑇^∗) in 𝑋. Assume that 𝑇_𝑛𝑥₀→𝑧 as 𝑛→∞ for any given 𝑥₀∈𝑋 and 𝑧^∗≠𝑧. Take the sequence {𝑇_𝑛𝑥_𝑛}≡{𝑇_𝑛𝑥₀}. Define 𝛿𝑇_𝑛=𝑇_𝑛−𝑇^∗𝑛+1 by (𝛿𝑇_𝑛)𝑥=(𝑇_𝑛−𝑇^∗𝑛+1)𝑥 for 𝑥∈𝑋. Then, note that 𝑑𝑇_𝑚+𝑛𝑥_𝑛+1,𝑇^{∗𝑛+𝑚+1}𝑧^∗=𝑑𝑇_𝑚𝑥_𝑛+1,𝑇^∗𝑚+1𝑧^∗=𝑑𝑇^∗𝑚+1+𝛿𝑇_𝑚𝑥_𝑛+1,𝑇^∗𝑚+1𝑧^∗≤𝑑𝑇^∗𝑚+1𝑥_𝑛+1,𝑇^∗𝑚+1𝑧^∗+𝛿𝑇_𝑚𝑥_𝑛+1, (13) and since {𝑥_𝑛} is bounded, 𝑇^∗𝑚𝑧^∗=𝑧^∗, 𝛿𝑇_𝑚→0 as 𝑚→∞, 𝑥_𝑚→𝑧, 𝑇^∗𝑚𝑥_𝑛→𝑧^∗, and 𝑇^∗𝑚𝑧→𝑧^∗ as 𝑚→∞ then the following contradiction holds if 𝑧^∗≠𝑧: lim_{𝑛,𝑚→∞}𝑑𝑇^∗𝑚+1𝑥_𝑛+1,𝑇^∗𝑚+1𝑧^∗+lim_{𝑛,𝑚→∞}𝛿𝑇_𝑚𝑥_𝑛+1=0≥lim_{𝑛,𝑚→∞}𝑑𝑇_𝑚+𝑛𝑥_𝑛+1,𝑇^{∗𝑛+𝑚+1}𝑧^∗=𝑑lim_{𝑛,𝑚→∞}𝑇_𝑚+𝑛𝑥_𝑛+1,𝑧^∗=𝑑𝑧,𝑧^∗>0, (14) and then 𝑧^∗=𝑧. As a result, 𝑇^∗∶𝑋→𝑋 is a strict contraction and thus a strict Picard self-mapping with a unique fixed point 𝑧^∗∈𝑋 such that 𝑇^∗𝑛𝑥₀→𝑧^∗ and 𝑇_𝑛𝑥₀→𝑧^∗ as 𝑛→∞ for any given initial point 𝑥₀∈𝑋, where 𝑇_𝑛∶𝐙₀₊×𝑋→𝑋 is the composite mapping 𝑇_𝑛=𝑇_𝑛𝑇_𝑛−1⋯𝑇₀, ∀𝑛∈𝐙₀₊.
Property (iv) is well known for Picard iterations.

Remark 2. Note that the parameterization sequences 0≤𝛼_𝑖^(𝑛)≤1, ∀𝑖∈𝑘={0,1,…,𝑘}, ∀𝑛∈𝐙₀₊, are not necessarily constant in Theorem 1 and 𝛼₁^(𝑛) can be zero for some 𝑛∈𝐙₀₊ and the positive amount ∑^𝑛_𝑖=0𝛼_𝑖^(𝑛) is not necessarily identically equal to one. Furthermore, the constant 𝐾 can be equal to or greater than unity in assumption 3 of Theorem 1. Thus, the iterative scheme generalizes that proposed and analyzed by Cho et al. [1].

Remark 3. Note also that if {𝑇_𝑛}→𝑇^∗ (or if the stronger condition {𝑇_𝑛}⇉𝑇^∗ holds) then 𝑇^∗𝑛𝑥₀→𝑧^∗(∈Fix(𝑇^∗)={𝑧^∗}) and 𝑇_𝑛𝑥₀→𝑧^∗ as 𝑛→∞ irrespective of the given 𝑥₀∈𝑋. However, if the property (𝑇_𝑛−𝑇^∗)→0 as 𝑛→∞ does not hold then 𝑇^𝑛_𝑚𝑥₀→𝑧_𝑚(∈Fix(𝑇^𝑛_𝑚)={𝑧_𝑚}) as 𝑛→∞, ∀𝑚∈𝐙₀₊, for the given 𝑥₀∈𝑋 since all the self-mappings 𝑇_𝑚 on 𝑋 are strict contractions but 𝑧_𝑚 can be distinct of 𝑧^∗.

The following result relaxes condition (3) of strict contraction mappings in the sequence {𝑇_𝑛} of Theorem 1 to weaker condition in terms of those mappings to be contractive in compact metric spaces.

Theorem 4. Consider the iterative scheme (1) on a compact metric space (𝑋,𝑑) endowed with a homogeneous translation-invariant metric 𝑑:𝑋×𝑋→𝐑₀₊, where 𝑋 is a vector space, with 0∈𝑋, under the following assumptions:(1)∑^𝑘_𝑖=0𝛼_𝑖^(𝑛)>0, 𝛼_𝑖^(𝑛)≥0, ∀𝑖∈𝑘={0,1,…,𝑘}, ∀𝑛∈𝐙₀₊, and inf_{𝑛∈𝐙0+}max_{1≤𝑖≤𝑘}𝛼_𝑖^(𝑛)>0, with the nonnegative real sequences {𝛼_𝑖^(𝑛)}, ∀𝑖∈𝑘, and {𝑚_𝑛} being subject to the constraints 𝛼_𝑖^(𝑛+1)=(1+𝛼_𝑖^(𝑛))𝛼_𝑖^(𝑛), |𝛼_𝑖^(𝑛)|≤𝛼_𝑛≤𝑚_𝑛(𝑑(𝑥_𝑛,𝑥_𝑛+1)/𝑑(𝑥_𝑛+1,0)) and 𝑚_𝑛=𝑜[𝑑(𝑥_𝑛+1,0)], ∀𝑖∈𝑘, ∀𝑛∈𝐙₀₊, where 𝛼_𝑖^(𝑛)=(𝛼_𝑖^(𝑛+1)−𝛼_𝑖^(𝑛))/𝛼_𝑖^(𝑛), ∀𝑖∈𝑘, ∀𝑛∈𝐙₀₊.(2)𝑇∶𝑋→𝑋 possesses the weak contractive condition 𝑑(𝑇𝑥,𝑇𝑦)<𝑑(𝑥,𝑦), ∀𝑥,𝑦(≠𝑥)∈𝑋.(3)(1+𝑚_𝑛)(∑^𝑘_𝑖=0𝛼_𝑖^(𝑛))<1, ∀𝑛∈𝐙₀₊.Then, the following properties hold.(i)There exists lim_𝑛→∞𝑑(𝑥_𝑛,𝑥_𝑛+1)=0 for any given initial point 𝑥₀∈𝑋 of the iterative scheme (1).(ii)If, in addition, {𝑇_𝑛}⇉𝑇^∗ for some limit 𝑇^∗:𝑋→𝑋 and Fix(𝑇_𝑛)={𝑧^∗_𝑛}, ∀𝑛∈𝐙₀₊, then the iterated sequence {𝑥_𝑛} is a Cauchy sequence and thus convergent to some 𝑧 in 𝑋. All the self-mappings of the sequence {𝑇_𝑛} as well as 𝑇^∗∶𝑋→𝑋 are contractive.(iii)If  {𝑇_𝑛}→𝑇^∗ for some point-wise limit self-mapping 𝑇^∗ on 𝑋, then there is a unique 𝑧^∗∈Fix(𝑇^∗) in 𝑋 such that 𝑧^∗=𝑧 to which any sequence {𝑥_𝑛} of the iterative scheme (1) converges for any given initial point 𝑥₀∈𝑋. Also, 𝑇^∗∶𝑋→𝑋 is a contractive and thus a Picard self-mapping with a unique fixed point 𝑧^∗(=𝑧)∈𝑋 such that 𝑇^∗𝑛𝑥₀→𝑧^∗, 𝑇_𝑛𝑥₀→𝑧^∗ as 𝑛→∞ for any given initial point 𝑥₀∈𝑋, where 𝑇_𝑛∶𝐙₀₊×𝑋→𝑋 is the composite mapping 𝑇_𝑛=𝑇_𝑛𝑇_𝑛−1⋯𝑇₀, ∀𝑛∈𝐙₀₊.

Proof. Note that a metric space is compact if and only if it is complete and totally bounded. Note also that (𝑋,‖‖) is a Banach space formally identical to the compact (and then complete) metric space (𝑋,𝑑) when endowed with a homogeneous and translation-invariant metric 𝑑∶𝑋×𝑋→𝐑₀₊ if ‖‖ is the norm-induced metric. Thus, one concludes that 𝑑𝑥_𝑛+1,𝑥_𝑛<𝑑𝑥_𝑛,𝑥_𝑛−1<⋯<𝑑𝑥₁,𝑥₀;∀𝑛∈𝐙₊, (15) which implies that {𝑑(𝑥_𝑛+1,𝑥_𝑛)} is a convergent sequence with lim_𝑛→∞𝑑(𝑥_𝑛,𝑥_𝑛+1)=0 for any given 𝑥₀∈𝑋. Hence, property (i) follows. On the other hand, since the metric space (𝑋,𝑑) is a compact metric space (and thus complete) then the iterated sequence {𝑥_𝑛}, with 𝑥_𝑛+1=𝑇_𝑛𝑥_𝑛 and the point-wise convergence of {𝑇_𝑛} to 𝑇^∗∶𝑋→𝑋, is a Cauchy sequence {𝑇_𝑛}⇉𝑇^∗ and Fix(𝑇_𝑛)={𝑧^∗_𝑛}, ∀𝑛∈𝐙₀₊. Assume that {𝑧^∗_𝑛}→𝑧^∗ is untrue. Then, 𝑑𝑧^∗_𝑛,𝑧=𝑑𝑇_𝑛𝑧^∗_𝑛,𝑇^∗𝑧≤𝑑𝑇_𝑛𝑧^∗_𝑛,𝑇^∗𝑧^∗_𝑛+𝑑𝑇^∗𝑧^∗_𝑛,𝑇^∗𝑧=𝑑𝑇_𝑛𝑧^∗_𝑛−𝑇^∗𝑧^∗_𝑛,0+𝑑𝑇^∗𝑧^∗_𝑛,𝑇^∗𝑧<𝑑𝑇_𝑛𝑧^∗_𝑛−𝑇^∗𝑧^∗_𝑛,0+𝑑𝑧^∗_𝑛,𝑧 (16) so that the contradiction 0=liminf_𝑛→∞𝑑(𝑇_𝑛𝑧^∗_𝑛−𝑇^∗𝑧^∗_𝑛,0)>0 since the metric is homogeneous and translation-invariant, {𝑇_𝑛}⇉𝑇^∗, so that 𝑇_𝑛𝑧^∗_𝑛→𝑇^∗𝑧^∗_𝑛 as 𝑛→∞ since 𝑧^∗_𝑛∈𝑋, and 𝑇^∗∶𝑋→𝑋 is contractive. Hence, {𝑧^∗_𝑛}→𝑧 for some 𝑧 in 𝑋 and any given 𝑥₀∈𝑋, all the self-mappings 𝑇_𝑛∶𝑋→𝑋; ∀𝑛∈𝐙₀₊ in the sequence {𝑇_𝑛} are contractive, and then Picard mappings (since (𝑋,𝑑) is a compact metric space) so that the composite mapping 𝑇_𝑛∶𝑋→𝑋 is also a Picard mapping. As a result, 𝑇_𝑛𝑥₀→𝑧^∗ as 𝑛→∞ for any given initial point 𝑥₀∈𝑋 and 𝑇^𝑚_𝑛𝑥₀→𝑧^∗_𝑛 as 𝑚→∞, ∀𝑛∈𝐙₀₊, ∀𝑥₀∈𝑋 with Fix(𝑇_𝑛)={𝑧_𝑛}, ∀𝑛∈𝐙₀₊, for any 𝑇_𝑛(:𝑋→𝑋)∈{𝑇_𝑛}. If {𝑇_𝑛}→𝑇^∗ then 𝑇^∗𝑛𝑥₀→𝑧^∗, 𝑇_𝑛𝑥₀→𝑧^∗ as 𝑛→∞, and Fix(𝑇^∗)={𝑧^∗}. Hence, properties (ii)-(iii) have been proven.

Remark 5. Note that a metric space is compact if and only if it is complete and totally bounded. Equivalently, a metric space is compact if and only if every family of closed subsets of 𝑋 with the finite intersection property (i.e., the intersection of any finite collection of sets in the family is nonempty) has a nonempty intersection.

An extension of Theorem 1 follows below by admitting the failure of the contractive condition of assumption 4 of Theorem 1 within connected subsets of finite length of 𝐙₀₊ which are adjacent to connected subsets where the contractive condition holds.

Theorem 6. Consider the iterative scheme (1) on a vector space 𝑋 under the assumptions (1)–(3) and (5) of Theorem 1 and, furthermore, ^𝑚𝑗_𝑛=01+𝑚_𝑛^𝑘_𝑖=0𝛼_𝑖^(𝑛)𝐾^𝑖≤𝜌<1;∀𝑝_𝑗∈𝑆⊆𝐙₀₊,∀𝑗∈𝐙₀₊, (17) where 𝑆={𝑝_𝑘∶𝑘∈𝐙₀₊} is a strictly increasing sequence of nonnegative integer numbers subject to 𝑝₀≤𝑝^∗<+∞ and 𝑝_𝑘+1−𝑝_𝑘≤𝑝<+∞, ∀𝑘∈𝐙₀₊.(i)There exists the limit lim_𝑛→∞𝑑(𝑥_𝑛,𝑥_𝑛+1)=0 for any given initial point 𝑥₀∈𝑋 of the iterative scheme (1) and the sequence {𝑥_𝑛} is bounded.(ii)If, in addition, {𝑇_𝑛}⇉𝑇^∗ for some limit self-mapping 𝑇^∗∶𝑋→𝑋 and if either (𝑋,‖‖) is a Banach space or (𝑋,𝑑) is complete, then {𝑥_𝑛} is a Cauchy sequence and thus convergent to some 𝑧 in 𝑋 which is unique and thus independent of the initial point 𝑥₀∈𝑋 of the iterative scheme (1). Also, all the self-mappings in the sequence {𝑇_𝑛} and 𝑇^∗∶𝑋→𝑋 are strict contractions.(iii)If either (𝑋,‖‖) is a Banach space or (𝑋,𝑑) is complete and {𝑇_𝑛}→𝑇^∗ as 𝑛→∞ for some self-mapping 𝑇^∗ on 𝑋, then there is a unique 𝑧^∗∈Fix(𝑇^∗) in 𝑋 such that 𝑧^∗=𝑧 for any given initial point 𝑥₀∈𝑋 of the iterative scheme (1). Also, 𝑇^∗∶𝑋→𝑋 is a strict contraction and thus a strict Picard self-mapping with a unique fixed point 𝑧^∗(=𝑧)∈𝑋 such that 𝑇^∗𝑛𝑥₀→𝑧^∗, 𝑇_𝑛𝑥₀→𝑧^∗ as 𝑛→∞ for any given initial point 𝑥₀∈𝑋, where 𝑇_𝑛∶𝐙₀₊×𝑋→𝑋 is the composite mapping 𝑇_𝑛=𝑇_𝑛𝑇_𝑛−1⋯𝑇₀, ∀𝑛∈𝐙₀₊.(iv)The “a priori” and “a posteriori” error estimates and the convergence rate are, respectively, given by the subsequent relations: 𝑑𝑥_𝑝𝑛≤𝜌,𝑧^𝑛𝑑𝑥1−𝜌^𝑝0,𝑥_𝑝1≤𝑀𝜌^𝑛𝑑𝑥1−𝜌⁰,𝑥₁;𝑑𝑥_{𝑝𝑛+𝑗}≤𝑀,𝑧²𝜌^𝑛𝑑𝑥1−𝜌⁰,𝑥₁,𝑑𝑥_𝑝𝑛≤𝜌,𝑧𝑑𝑥1−𝜌^𝑝𝑛−1,𝑥_𝑝𝑛≤𝑀𝜌𝑑𝑥1−𝜌⁰,𝑥₁;𝑑𝑥_{𝑝𝑛+𝑗}≤𝑀,𝑧²𝜌𝑑𝑥1−𝜌⁰,𝑥₁,𝑑𝑥_𝑝𝑛,𝑧≤𝜌^𝑛𝑑𝑝_𝑛0,𝑧≤𝑀𝜌^𝑛𝑑𝑝_𝑛0;𝑑𝑥,𝑧_{𝑝𝑛+𝑗},𝑧≤𝑀²𝜌^𝑛𝑑𝑥₀,𝑥₁, (18)for some 𝑀∈𝐑₊ and any integer 𝑗∈(1,𝑝_𝑛+1−𝑝_𝑛), ∀𝑛∈𝐙₀₊.

Proof. Note from (17) and (6) that since 𝑝_𝑘→∞ as 𝑘→∞, one gets𝑑𝑥_𝑝𝑘+1+1,𝑥_𝑝𝑘+1≤𝜌𝑑𝑥_𝑝𝑘+1,𝑥_𝑝𝑘;∀𝑝_𝑘∈𝑆,𝑑𝑥_𝑝𝑘+2,𝑥_𝑝𝑘+1≤1+𝐿+⋯+𝐿^𝑝−1𝜌𝑑𝑥_𝑝𝑘+1,𝑥_𝑝𝑘≤𝜌₀^𝑘+1𝑑𝑥_𝑝0,𝑥_𝑝1, (19a)(19b)and, provided that 𝜌∈(0,1) is small enough for the given 𝑝∈𝐙₊ so that 𝜌₀=𝜌(1+𝐿+⋯+𝐿^𝑝−1)<1, 𝑑(𝑥_𝑝𝑘+2,𝑥_𝑝𝑘+1)→0 as 𝑘→∞ for any given 𝑥₀∈𝑋. Then, {𝑑(𝑥_𝑝𝑘+1,𝑥_𝑝𝑘)} is a convergent sequence with lim_𝑘→∞𝑑(𝑥_𝑝𝑘+1,𝑥_𝑝𝑘)=lim_{𝑝𝑘→∞}𝑑(𝑥_𝑝𝑘+1,𝑥_𝑝𝑘)=0 for any given 𝑥₀∈𝑋. It follows from (19a), (19b), since 𝑝₁≤𝑝+𝑝<+∞ and 𝑝_𝑘+1−𝑝_𝑘≤𝑝<+∞, that for any given initial 𝑥₀∈𝑋, 𝑑𝑥_𝑝𝑘,𝑥_𝑝0≤1−𝜌^𝑘𝑑𝑥1−𝜌^𝑝1,𝑥_𝑝0<+∞;∀𝑘∈𝐙₊,limsup_𝑘→∞𝑑𝑥_𝑝𝑘,𝑥_𝑝0≤𝑑𝑥_𝑝1,𝑥_𝑝0^∞_𝑗=0𝜌^𝑗≤𝑑𝑥_𝑝1,𝑥_𝑝01−𝜌<+∞,𝑑𝑥_{𝑝𝑘+𝑗},𝑥_{𝑝𝑘+𝑗−1}≤^𝑗−1_𝑛=01+𝑚_{𝑝𝑘+𝑛}^𝑘_𝑖=0𝛼^(𝑝𝑘_𝑖^+𝑛)𝐾^𝑖𝑑𝑥_𝑝𝑘,𝑥_𝑝𝑘−1<+∞,𝑑𝑥_{𝑝𝑘+𝑗},𝑥_𝑝𝑘≤𝜌^𝑘𝑗−1_𝑖=0^𝑖−1_𝑛=01+𝑚_{𝑝𝑘+𝑛}×^𝑘₌₀𝛼^{(𝑝𝑘+𝑛)}𝐾𝑑𝑥_𝑝0,𝑥_𝑝1<+∞, (20)(21)(22)(23) for 𝑗=0,1,…,𝑝_𝑘+1−𝑝_𝑘, ∀𝑘∈𝐙₊, and then, since {𝑝_𝑘} is strictly increasing with 𝑘, 𝜌∈[0,1) and 𝑝₁≤𝑝₀+𝑝≤𝑝+𝑝<+∞, one gets lim_𝑘→∞𝑑(𝑥_{𝑝𝑘+𝑗},𝑥_𝑝𝑘)=0 for 𝑗=0,1,…,𝑝_𝑘+1−𝑝_𝑘, ∀𝑘∈𝐙₊. Then, {𝑑(𝑥_𝑛+1,𝑥_𝑛)}→0 as 𝑛→∞ from (23) and {𝑥_𝑛} is bounded for any initial 𝑥₀∈𝑋. However, {𝑥_𝑛} is not a Cauchy sequence, in general, since the constraint 𝑑(𝑥_𝑛+2,𝑥_𝑛+1)<𝑑(𝑥_𝑛+1,𝑥_𝑛) does not necessarily hold for all 𝑛∈𝐙₀₊.
The variation in the proof development of the concerns derived from the assumption {𝑇_𝑛}⇉𝑇^∗ of Theorem 1 (ii) is addressed as follows. Since {𝑇_𝑛}⇉𝑇^∗ and Fix(𝑇_𝑛)={𝑧^∗_𝑛}, ∀𝑛∈𝐙₀₊, then (17) necessarily leads to 𝑇^∗∶𝑋→𝑋 being a strict contraction, {𝑧^∗_𝑛}→𝑧^∗ with 𝑧^∗∈Fix(𝑇^∗)(={𝑧^∗}), and lim_𝑘→∞(𝑝_𝑘+1−𝑝_𝑘)=1. Therefore, the remaining proofs of properties (i)–(iii) follow in a very close way as their counterparts of Theorem 1. Also, note that 𝑑𝑥_𝑝0,𝑥_𝑝0−1≤^𝑝0−1_𝑖=0𝑑𝑥_𝑖+1,𝑥_𝑖≤^𝑝0−1_𝑖=0^𝑖−1_𝑛=01+𝑚_𝑛^𝑘₌₀𝛼^(𝑛)𝐾𝑑𝑥₀,𝑥₁, (24) and then define 𝑀=sup_{𝑘∈𝐙0+}max_{𝑝𝑘≤𝑛<𝑝𝑘+1}^𝑗−1_𝑖=0^𝑖−1_𝑛=01+𝑚_{𝑝𝑘+𝑛}×^𝑘₌₀𝛼^{(𝑝𝑘+𝑛)}𝐾, (25) so that property (iv) follows from (23) and Theorem 1 (iv).

Remark 7. Note that assumption 4 of Theorem 1 is relaxed to the constraint (17) which holds for a set of connected finite intervals within a strictly increasing sequence of points with the difference between any two consecutive ones being upper-bounded by a prescribed bound.

Remark 8. Note that Theorems 1 (i), 4 (ii), and 6 (iii) hold irrespective of the convergence of the sequence of self-mappings to a limit.

3. Iterative Scheme 2 and Some Generalizations

Now, consider the iterative scheme𝑥_𝑛+1=𝑇^𝑓𝑛_𝑛𝑥_𝑛,𝑇_𝑛𝑥=𝑔_𝑛𝑇𝑥;∀𝑛∈𝐙₀₊, (26) for any given 𝑥₀∈𝑋 which is a further generalization of the De Figueiredo iteration [8]. The following result holds.

Theorem 9. Let the iterative scheme (1) with the nonexpansive self-mapping 𝑇∶𝑋→𝑋 on a vector space 𝑋, with 0∈𝑋, under the following additional assumptions.(1)Either (𝑋,‖‖) is a Banach space endowed with a norm ‖‖ or, respectively, (𝑋,𝑑) is a complete metric space endowed with a homogeneous translation-invariant metric 𝑑∶𝑋×𝑋→𝐑₀₊.(2){𝑔_𝑛}⊂(0,1)∩𝐑₀₊ is a real parameterization sequence with 0<𝑔_𝑛<1, ∀𝑛∈𝐙₀₊, and {𝑓_𝑛}⊆𝐙₀₊ is an integer sequence with 𝑓_𝑛>0; ∀𝑛∈𝐙₊.(3)There exist the following limits: lim_𝑛→∞𝑔_𝑛=1, lim_𝑛→∞(𝑓_𝑛/𝑛)=+∞, and either lim_𝑛→∞(𝑓_𝑛+1log𝑔_𝑛+1/𝑓_𝑛log𝑔_𝑛)<1 or |lim_𝑛→∞𝑛(1−(𝑓_𝑛+1log𝑔_𝑛+1/𝑓_𝑛log𝑔_𝑛))|<+∞.Then, the subsequent properties hold.(i){𝑥_𝑛} converges to a fixed point of  𝑇∶𝑋→𝑋.(ii)If  𝑇∶𝑋→𝑋 is a strict contraction then {𝑥_𝑛} converges to the unique fixed point of  𝑇∶𝑋→𝑋.

Proof. As in the proof of Theorem 1, the following considerations are applicable for the proof.(1)If (𝑋,‖‖) is a normed space then there is always a metric-induced norm 𝑑(𝑥,𝑦)=‖𝑥−𝑦‖; ∀𝑥,𝑦∈𝑋.(2)If (𝑋,𝑑) is a metric space endowed with a homogeneous translation-invariant metric 𝑑∶𝑋×𝑋→𝐑₀₊ then there is a norm-induced metric ‖𝑥‖=𝑑(𝑥,0); ∀𝑥,𝑦∈𝑋.Both spaces (𝑋,‖‖) and (𝑋,𝑑) are formally identical under assumption 1 and both possess either a metric-induced norm by using the standard metric properties and its homogeneous and translation-invariance properties or a norm-induced metric, respectively. Now, note from (26) that 𝑥_𝑛+𝑘=^𝑘−1_𝑖=0𝑔^{𝑓𝑛+𝑖}_𝑛+𝑖𝑇^{∑𝑘−1𝑖=0𝑓𝑛+𝑖}𝑥_𝑛,𝑥_𝑛+𝑘−𝑥_{𝑛+𝑘−1}=^𝑘−1_𝑖=0𝑔^{𝑓𝑛+𝑖}_𝑛+𝑖𝑇^{∑𝑘−1𝑖=0𝑓𝑛+𝑖}−^𝑘−2_𝑖=0𝑔^{𝑓𝑛+𝑖}_𝑛+𝑖𝑇^{∑𝑘−2𝑖=0𝑓𝑛+𝑖}𝑥_𝑛=^𝑘−1_𝑖=0𝑔^{𝑓𝑛+𝑖}_𝑛+𝑖𝑇^{∑𝑘−1𝑖=0𝑓𝑛+𝑖}−𝑇^{∑𝑘−2𝑖=0𝑓𝑛+𝑖}+11−𝑔^{𝑓𝑛+𝑘−1}_{𝑛+𝑘−1}𝑇^{∑𝑘−2𝑖=0𝑓𝑛+𝑖}𝑥_𝑛;∀𝑛∈𝐙₀₊,∀𝑘∈𝐙₊,𝑥_𝑛+𝑘−𝑥_{𝑛+𝑘−1}≤^𝑘−1_𝑖=0𝑔^{𝑓𝑛+𝑖}_𝑛+𝑖𝐼−𝑇^{𝑓𝑛+𝑘−1}×𝑇^{∑𝑘−2𝑖=0𝑓𝑛+𝑖}𝑥_𝑛+^𝑘−1_𝑖=0𝑔^{𝑓𝑛+𝑖}_𝑛+𝑖11−𝑔^{𝑓𝑛+𝑘−1}_{𝑛+𝑘−1}𝑇^{∑𝑘−2𝑖=0𝑓𝑛+𝑖}𝑥_𝑛;∀𝑛∈𝐙₀₊,∀𝑘∈𝐙₊,limsup_𝑛→∞𝑥_𝑛+𝑘−𝑥_{𝑛+𝑘−1}≤2^𝑘−1_𝑖=0𝑔^{𝑓𝑛+𝑖}_𝑛+𝑖limsup_𝑛→∞𝑇^{∑𝑘−2𝑖=0𝑓𝑛+𝑖}𝑥_𝑛+^𝑘−1_𝑖=0𝑔^{𝑓𝑛+𝑖}_𝑛+𝑖×limsup_𝑛→∞11−𝑔^{𝑓𝑛+𝑘−1}_{𝑛+𝑘−1}𝑇^{∑𝑘−2𝑖=0𝑓𝑛+𝑖}𝑥_𝑛=2limsup_𝑛→∞𝐿_𝑛(𝑘)limsup_𝑛→∞𝑇^{∑𝑘−2𝑖=0𝑓𝑛+𝑖}𝑥_𝑛<+∞;∀𝑘∈𝐙₊, (27) since ‖𝐼−𝑇^{𝑓𝑛+𝑘−1}‖≤2, ∀𝑛∈𝐙₀₊, ∀𝑘∈𝐙₊ because 𝑇∶𝑋→𝑋 is nonexpansive, lim_𝑛→∞𝑔^{𝑓𝑛+𝑘−1}_{𝑛+𝑘−1}=1; ∀𝑘∈𝐙₊ and ∏^𝑘−1_𝑖=0[𝑔^{𝑓𝑛+𝑖}_𝑛+𝑖]=𝐿_𝑛(𝑘)≤𝐿_𝑛(∞)<+∞; ∀𝑘∈𝐙₊, ∀𝑛∈𝐙₀₊ because {𝑔_𝑛}⊂(0,1)∩𝐑₀₊ and either lim_𝑛→∞(𝑓_𝑛+1log𝑔_𝑛+1/𝑓_𝑛log𝑔_𝑛)<1 or |lim_𝑛→∞𝑛(1−(𝑓_𝑛+1log𝑔_𝑛+1/𝑓_𝑛log𝑔_𝑛))|<+∞. Then ∏^∞_𝑖=0[𝑔^{𝑓𝑛+𝑖}_𝑛+𝑖] is convergent since the corresponding logarithmic series of positive numbers converges according to either d’Alembert or Raabe convergence criteria of series of nonnegative real numbers. Then, the sequence {𝑥_𝑛+1−𝑥_𝑛} is bounded. In the same way, we get limsup_𝑘→∞𝑥_𝑘−𝑥_𝑘−1≤2^𝑘−1_𝑖=0𝑔^𝑓𝑖_𝑖limsup_𝑘→∞𝑇^{∑𝑘−2𝑖=0𝑓𝑖}𝑥₀+^𝑘−1_𝑖=0𝑔^𝑓𝑖_𝑖×limsup_𝑘→∞11−𝑔^𝑓𝑘−1_𝑘−1𝑇^{∑𝑘−2𝑖=0𝑓𝑖}𝑥₀=2limsup_𝑘→∞𝐿₀(𝑘)limsup_𝑘→∞𝑇^{∑𝑘−2𝑖=0𝑓𝑖}𝑥₀=2limsup_𝑘→∞𝐿₀(𝑘)𝑥^∗=0, (28) since ∃lim_𝑘→∞𝐿₀(𝑘)=∏^𝑘−1_𝑖=0[𝑔^𝑓𝑖_𝑖]=0 since 𝑔_𝑛<1, ∀𝑛∈𝐙₀₊, and lim_𝑛→∞(𝑓_𝑛/𝑛)=+∞ so that ∃lim_𝑛→∞‖𝑇^{∑𝑘−2𝑖=0𝑓𝑛+𝑖}𝑥_𝑛‖=𝑥^∗(∈Fix(𝑇)). Thus, {𝑑(𝑥_𝑛+1,𝑥_𝑛)} converges to zero for any given 𝑥₀∈𝑋 and 𝑑𝑥_𝑛+1,𝑥^∗≤𝑑𝑥_𝑛+1,𝑥_𝑛+𝑑𝑥_𝑛,𝑥^∗ (29) so that, since 𝑑(𝑥_𝑛+1,𝑥_𝑛)→0 as 𝑛→∞, lim_𝑛→∞(𝑑(𝑥_𝑛+1,𝑥^∗)−𝑑(𝑥_𝑛,𝑥^∗))=0. Then, {𝑑(𝑥_𝑛,𝑥^∗)} converges and {𝑥_𝑛} converges as well to some point of 𝑋 since (𝑋,𝑑) is complete so that 𝑥_𝑛→(𝑥^∗+𝑞) as 𝑛→∞ for some 𝑞∈𝑋 and since 𝑇∶𝑋→𝑋 is nonexpansive so that it is 𝐾-Lipschitz-continuous (i.e., continuous with a Lipschitz constant 𝐾≤1), one gets lim_𝑛→∞𝑑𝑥_𝑛+1,𝑥_𝑛=𝑑lim_𝑛→∞𝑥_𝑛+1,lim_𝑛→∞𝑥_𝑛=𝑑𝑥^∗+𝑞,𝑥^∗+𝑞=0. (30) Since {𝑥_𝑛} converges and (𝑋,𝑑) is a metric space then {𝑥_𝑛} is a Cauchy sequence (and a bounded sequence) and there is 𝑎∈𝐑₀₊ such that 𝑎=lim_𝑛→∞𝑑𝑥_𝑛,𝑥^∗=𝑑𝑥^∗+𝑞,𝑥^∗=lim_𝑛→∞𝑑𝑔_𝑛𝑇^𝑓𝑛𝑥_𝑛,𝑥^∗=𝑑lim_𝑛→∞𝑇^𝑓𝑛𝑥^∗+𝑞,𝑥^∗=𝑑𝑦^∗,𝑥^∗, (31) since the metric is translation-invariant, 𝑔_𝑛→1 as 𝑛→∞, and since 𝑇∶𝑋→𝑋 is nonexpansive, it is 𝐾(≤1)-Lipschitz-continuous and 𝑇^𝑓𝑛(𝑥^∗+𝑞)→𝑦^∗ with 𝑦^∗,𝑥^∗∈Fix(𝑇). If 𝑎=0 then 𝑦^∗=𝑥^∗(∈Fix(𝑇)) and we have proven that {𝑥_𝑛} converges to the fixed point 𝑥^∗ of  𝑇∶𝑋→𝑋. Now, assume that 𝑦^∗(≠𝑥^∗)∈Fix(𝑇). The result is again proven since {𝑥_𝑛} converges to a fixed point of  𝑇∶𝑋→𝑋 which is distinct of 𝑥^∗. Finally, assume that 𝑦^∗∉Fix(𝑇) and proceed by contradiction to prove that this assertion is false. Since 𝑇∶𝑋→𝑋 is nonexpansive, one gets 𝑑(𝑇^𝑛𝑦^∗,𝑇^𝑛𝑥^∗)≤⋯≤𝑑(𝑦^∗,𝑥^∗)=𝑎 so that 𝑑(𝑇^𝑛𝑦^∗,𝑥^∗)→𝑎₁(≤𝑎) as 𝑛→∞; then by everywhere Lipschitz continuity of the nonexpansive self-mapping 𝑇∶𝑋→𝑋, 𝑑lim_𝑛→∞𝑇^𝑛𝑇𝑦^∗,𝑥^∗=𝑑𝑇lim_𝑛→∞𝑇^𝑛𝑦^∗,𝑥^∗=𝑑lim_𝑛→∞𝑇^𝑛𝑦^∗,𝑥^∗=𝑑𝑦^∗₁,𝑥^∗=𝑎₁≤𝑎, (32) and 𝑇^𝑛𝑦^∗→𝑦^∗₁(∈Fix(𝑇)) and 𝑇^𝑓𝑛𝑦^∗→𝑦^∗₁. Since 𝑦^∗ is a limit point of 𝑇^𝑓𝑛𝑥_𝑛,  𝑎₁=𝑎 and then 𝑦^∗(=𝑦^∗₁)∈Fix(𝑇), a contradiction to 𝑦^∗∉Fix(𝑇). Thus, {𝑇^𝑓𝑛𝑥_𝑛} converges to a fixed point of  𝑇∶𝑋→𝑋. Property (i) has been proven. Also, 𝑑𝑇^𝑓𝑛_𝑛𝑥_𝑛,𝑇^𝑓𝑛𝑥_𝑛=𝑑𝑇^𝑓𝑛_𝑛𝑥_𝑛−𝑇^𝑓𝑛𝑥_𝑛≤𝑇,0_𝑓𝑛𝑛−𝑇^𝑓𝑛𝑥_𝑛;∀𝑛∈𝐙₀₊. (33) Since ‖𝑇^𝑓𝑛_𝑛−𝑇^𝑓𝑛‖≤(1−𝑔^𝑓𝑛_𝑛)‖𝑇^𝑓𝑛‖, ∀𝑛∈𝐙₀₊ with (1−𝑔^𝑓𝑛_𝑛)→0 as 𝑛→∞, it follows that ‖𝑇^𝑓𝑛_𝑛−𝑇^𝑓𝑛‖→0 as 𝑛→∞ and since {𝑇^𝑓𝑛𝑥_𝑛} converges to a fixed point of  𝑇∶𝑋→𝑋 then {𝑇^𝑓𝑛_𝑛𝑥_𝑛} also converges to the same fixed point of  𝑇∶𝑋→𝑋. Hence, property (i) follows.
On the other hand, if 𝑇∶𝑋→𝑋 is a strict contraction then Fix(𝑇)={𝑥^∗}, since (𝑋,𝑑) is complete so that 𝑎=0 and 𝑦^∗=𝑥^∗ in (31) and, hence, property (ii) follows as well.

Theorem 9 has the following derived result.

Corollary 10. Let the iterative scheme (1) be under the nonexpansive self-mapping 𝑇∶𝐶→𝐶, where 𝐶 is a nonempty closed and convex subset of a Hilbert space (𝑋,‖‖), with 0∈𝐶, subject to all the assumptions of Theorem 9. Then, the subsequent properties hold.(i){𝑥_𝑛} converges strongly to a fixed point of  𝑇∶𝐶→𝐶.(ii)If  𝑇∶𝐶→𝐶 is a strict contraction then {𝑥_𝑛} converges to the unique fixed point of  𝑇∶𝐶→𝐶.

Proof. Property (i) follows from Theorem 1 since (𝑋,‖‖) is uniformly convex since it is a Hilbert space; 𝑇∶𝐶→𝐶 is nonexpansive and contains a bounded sequence (since 𝐶 is nonempty, closed, and convex) and then it has at least a fixed point. Property (ii) follows since such a fixed point is unique if 𝑇∶𝐶→𝐶 is a strict contraction.

The iterative scheme (26) is now generalized by using some ideas of Section 2 as follows: 𝑥_𝑛+1=𝑇^𝑓𝑛_𝑛𝑥_𝑛,𝑇_𝑛𝑥=𝑔_𝑛^𝑘_𝑖=0𝛼_𝑖𝑇^𝑖𝑥_𝑛;∀𝑛∈𝐙₀₊, (34) for any given 𝑥₀∈𝑋.

Theorem 11. Let the iterative scheme (34) generated by the self-mapping 𝑇∶𝑋→𝑋 on a vector space 𝑋, with 0∈𝑋, and assumptions 1–3 of Theorem 9 hold as well as the following additional assumptions:(1)∑^𝑘_𝑖=0𝛼_𝑖>0 for nonnegative real scalars 𝛼_𝑖, ∀𝑖∈𝑘, and max_{1≤𝑖≤𝑘}𝛼_𝑖>0;(2)𝑇∶𝑋→𝑋 satisfies the condition 𝑑(𝑇𝑥,𝑇𝑦)≤𝐾𝑑(𝑥,𝑦), ∀𝑥,𝑦∈𝑋, for some 𝐾∈𝐑₊;(3)∑^𝑘_𝑖=0𝛼_𝑖𝐾^𝑖≤1; ∀𝑛∈𝐙₀₊.Then, the subsequent properties hold.(i){𝑥_𝑛} converges to a fixed point of the nonexpansive self-mapping 𝑇∶𝑋→𝑋 defined by 𝑇𝑥=(∑^𝑘_𝑖=0𝛼_𝑖𝑇^𝑖)𝑥, ∀𝑥∈𝑋.(ii)If 𝑇∶𝑋→𝑋 is a strict contraction fulfilling ∑^𝑘_𝑖=0𝛼_𝑖𝐾^𝑖≤𝜌<1 then {𝑥_𝑛} converges to the unique fixed point of  𝑇∶𝑋→𝑋.

Proof. As in Theorem 1 and Theorem 9, both spaces (𝑋,‖‖) and (𝑋,𝑑) are formally identical under assumption 1 of Theorem 9 and both possess either a metric-induced norm by using the standard metric properties and its homogeneous and translation-invariance properties or a norm-induced metric, respectively. Now, define the mapping 𝑇∶𝑋→𝑋 by 𝑇𝑥=(∑^𝑘_𝑖=0𝛼_𝑖𝑇^𝑖)𝑥, ∀𝑥∈𝑋. Thus, (27) in the proof of Theorem 9 still holds with the replacement 𝑇→𝑇. Note that ‖𝐼−𝑇^{𝑓𝑛+𝑘−1}‖≤2, ∀𝑛∈𝐙₀₊,∀𝑘∈𝐙₊, since 𝑇∶𝑋→𝑋 is nonexpansive (even if  𝑇∶𝑋→𝑋 is expansive with 𝐾>1 in the assumption 2), from assumptions 2-3, and 𝐾-Lipschitz-continuous from the assumption 3 with 𝐾=∑^𝑘_𝑖=0𝛼_𝑖𝐾^𝑖≤1. Note also that lim_𝑛→∞𝑔^{𝑓𝑛+𝑘−1}_{𝑛+𝑘−1}=1, ∀𝑘∈𝐙₊, and ∏^𝑘−1_𝑖=0[𝑔^{𝑓𝑛+𝑖}_𝑛+𝑖]≤𝐿_𝑛(𝑘)≤𝐿_𝑛(∞)<+∞, ∀𝑘∈𝐙₊, ∀𝑛∈𝐙₀₊, and then the sequence {𝑥_𝑛+1−𝑥_𝑛} obtained from the iterative scheme (34) is bounded for any 𝑥₀∈𝑋. In the same way, (28) holds from assumptions 2-3 of Theorem 9, since 𝑔_𝑛<1, ∀𝑛∈𝐙₀₊, and lim_𝑛→∞(𝑓_𝑛/𝑛)=+∞ so that ∃lim_𝑛→∞‖𝑇^{∑𝑘−2𝑖=0𝑓𝑛+𝑖}𝑥_𝑛‖=𝑥^∗(∈Fix(𝑇)). Thus, {𝑑(𝑥_𝑛+1,𝑥_𝑛)} converges to zero for any given 𝑥₀∈𝑋 since (𝑋,𝑑) is complete. Then, it follows (as it is deduced from (30) in the proof of Theorem 9) that {𝑥_𝑛} converges so that it is a Cauchy and bounded sequence. Finally, it can be proven in a similar way as in Theorem 9 that {𝑇^𝑓𝑛𝑥_𝑛} converges to some fixed point 𝑥^∗ of the nonexpansive self-mapping 𝑇∶𝑋→𝑋 for each given initial point 𝑥₀∈𝑋 of the iteration (34). Such a fixed point is unique if 𝑇∶𝑋→𝑋 is a strict contraction.

In a similar way as Corollary 10 is got from Theorem 9, one gets the following.

Corollary 12. Consider the iterative scheme (34) under the nonexpansive self-mapping 𝑇∶𝐶→𝐶, where 𝐶 is a nonempty closed and convex subset of a Hilbert space (𝑋,‖‖), with 0∈𝐶, subject to all the assumptions of Theorem 9. Then, the subsequent properties hold.(i){𝑥_𝑛} converges strongly to a fixed point of  𝑇∶𝐶→𝐶.(ii)If 𝑇∶𝐶→𝐶 is a strict contraction fulfilling ∑^𝑘_𝑖=0𝛼_𝑖𝐾^𝑖≤𝜌<1 then {𝑥_𝑛} converges to the unique fixed point of  𝑇∶𝐶→𝐶.

Note that in Theorem 9 (i) and Corollary 10 (i), 𝑇∶𝑋→𝑋 and 𝑇^𝑓𝑛_𝑛∶𝑋→𝑋 for 𝑛∈𝐙₊ are not necessarily Picard mappings since the limiting points can be dependent on the initial condition of the iterative schemes. The same conclusion arises for 𝑇∶𝑋→𝑋 and 𝑇^𝑓𝑛_𝑛∶𝑋→𝑋 for 𝑛∈𝐙₀₊ in Theorem 11 (i) and Corollary 12 (i). However, the above self-mappings are Picard iterations in the corresponding parts (ii) of such results since the relevant mappings are strict contractions.

Note also that Theorem 11 and Corollary 12 still hold by replacing 𝛼_𝑖→𝛼_𝑖^(𝑛) for 𝑖∈𝑘 and the replacement of the constraint max_{1≤𝑖≤𝑘}𝛼_𝑖>0 with inf_{𝑛∈𝐙0+}max_{1≤𝑖≤𝑘}𝛼_𝑖^(𝑛)>0.

4. Simulation Examples towards an Application Perspective on Discrete Nonlinear Dynamic Systems

This section contains two numerical examples. The first one is related to the Iterative Scheme 1 introduced in Section 2 while the second one concerns the Iterative Scheme 2 discussed in Section 3.

4.1. Iterative Scheme 1

Consider the iterative scheme defined by (1) with 𝑇(𝑥)=𝑥/2(1+𝑥) on [0,+∞) and 𝑥_𝑛+1=𝛼₃^(𝑛)𝑇³+𝛼₂^(𝑛)𝑇²+𝛼₁^(𝑛)𝑇¹+𝛼₀^(𝑛)𝐼𝑥_𝑛. (35)𝑇 is a strict contraction satisfying the condition 𝑑(𝑇(𝑥),𝑇(𝑦))≤𝐾𝑑(𝑥,𝑦) with 𝐾=1/2 (for the Euclidean distance) and, hence, it possesses a unique fixed point at 𝑥=0. Note that the above description can also be considered as that of a nonlinear discrete time-varying dynamic system where the state evolves from initial conditions according to the sequence {𝑥_𝑛} with initial condition 𝑥₀ while the output is defined by the real map 𝑥→𝑇𝑥. Note that the fixed point 𝑥=0 is also an equilibrium point of the dynamic system which is suited to be globally asymptotically stable. Consider, firstly, the sequence of constant weights 𝛼^(𝑛)=0.20.30.80.9 for all 𝑛≥0. We are now in conditions of applying Theorem 1 since ∑³_𝑖=0𝛼_𝑖^(𝑛)=2.2>0, 𝛼_𝑖^(𝑛)=𝑚_𝑛=0, and ∑³_𝑖=0𝛼_𝑖^(𝑛)𝐾^𝑖=0.6625<1 for all integers 𝑛≥0. In this case, the system parameterization is close to, but more general than, a polytopic-type time-invariant one but, in particular, the usual constraint ∑³_𝑖=0𝛼_𝑖^(𝑛)=1 is not needed. Accordingly, the sequence of iterates {𝑥_𝑛} is bounded for all 𝑛≥0 and converges to the unique fixed point of 𝑇, 𝑥=0. Moreover, the iterates converge to the unique fixed point regardless of the initial value 𝑥₀. These claims are verified through a numerical simulation in Figure 1.

Figure 1 

Evolution of the iterates for different initial conditions.

Furthermore, Theorem 1  (iv) also provides an upper-bound for the rate of convergence of the sequence of iterates to the fixed point. Therefore, one gets from (4) 𝑑(𝑥_𝑛,0)≤𝜌^𝑛𝑑(𝑥₀,0)=0.6625^𝑛𝑑(𝑥₀,0). Figure 2 displays the evolution of iterates along with the calculated upper-bound for the case 𝑥₀=8.

Figure 2 

Upper-bounding of the convergence rate of iteration.

Consider the time-varying parameterization under the time-varying weights given by 𝛼_𝑖^(𝑛+1)=𝜆_𝑖𝛼_𝑖^(𝑛),𝛼_𝑖^(𝑛+1)≥0.1,0.1,otherwise, (36) for all 𝑛≥0 and 0≤𝑖≤3 with 𝛼⁽⁰⁾=[0.20.30.80.9], 𝜆₀=0.95, 𝜆₁=0.9, 𝜆₂=0.85, and 𝜆₃=0.8. The 0.1 lower bound has been included in (36) so as to satisfy the condition inf_{𝑛∈𝐙0+}max_{1≤𝑖≤𝑘}𝛼_𝑖^(𝑛)>0.

As it can be appreciated in Figure 3, the weights are decreasing with time until they reach the constant lower bound of 0.1 where they stop decreasing and become time-invariant. In fact, Figure 3 shows that this happens for 𝑛≥14. Thus, we are in conditions of applying the results stated in Theorem 6 for the case when the stability condition only holds on a subset of the nonnegative integer numbers. In this way, we have 𝛼_𝑖^(𝑛)=𝑚_𝑛=0 for all 𝑛≥14 and (17) of Theorem 6 is satisfied since ³_𝑖=0𝛼_𝑖^(𝑛)𝐾^𝑖=³_𝑖=00.1𝐾^𝑖=0.1875<1, (37) for all 𝑛≥14. Thus, Theorem 6 guarantees the convergence of iterates to the unique fixed point irrespectively of the initial condition. This fact is shown in Figure 4.

Figure 3 

Evolution of the time-varying weights 𝛼0(𝑛), 𝛼1(𝑛), 𝛼2(𝑛), and 𝛼3(𝑛) through (36).

Figure 4 

Iterates for different initial conditions and weights defined by (36).

One advantage of the results in Theorem 6 for the sake of generality is that an arbitrary variation in the weights is admitted on certain subsets of the natural numbers. Thus, the family of admissible time variations for which the stability of the iteration scheme is guaranteed enlarges with respect to other approaches. Consider a time-varying set of weights defined with 𝑚_𝑛=𝑑(𝑥_𝑛+1,0) in condition (2) of Theorem 1 so that 𝛼_𝑛≤𝑑(𝑥_𝑛,𝑥_𝑛+1)≤𝐾𝑑(𝑥_𝑛−1,𝑥_𝑛)=1/2𝑑(𝑥_𝑛−1,𝑥_𝑛) while condition (4) becomes 1+𝑚_𝑛³_𝑖=0𝛼_𝑖^(𝑛)𝐾^𝑖=1+𝑚_𝑛𝛼₀^(𝑛)+12𝛼₁^(𝑛)+12²𝛼₂^(𝑛)+12³𝛼₃^(𝑛)<1. (38) Such constraints are satisfied, for instance, if we take 𝛼⁽⁰⁾=[0.050.10.150.2] and 𝛼_𝑛=0.05sin(2𝜋0.05𝑛)𝑑𝑥_𝑛−1,𝑥_𝑛. (39) The weights evolution is displayed in Figure 5. Note that the weight variation defined by (39) satisfies the condition 𝛼_𝑛≤𝑑(𝑥_𝑛,𝑥_𝑛+1)≤1/2𝑑(𝑥_𝑛−1,𝑥_𝑛) since 0.05sin(2𝜋0.05𝑛)≤0.05, ∀𝑛∈𝐙₀₊.

Figure 5 

Evolution of the weights under (39).

Figure 6 displays the sample-by-sample stability condition evaluation, in terms of the left-hand side of (38), showing that it remains smaller than unity. Therefore, according to Theorem 1, the iterates converge to zero as Figure 7 depicts.

Figure 6 

Numerical verification of the stability condition (38).

Figure 7 

Convergence of the iterates to zero for time-varying weights given by (39).

Also, 𝑑(𝑥_𝑛−1,𝑥_𝑛)→0 while the weights converge to a real constant according to (39). Thus, the given theoretical results are useful to conclude the convergence of iteration schemes of the form (1).

4.2. Iterative Scheme 2

This second example is concerned with the iterative scheme defined by (26). Note that the first equation can describe, in particular, the state and output of a certain nonlinear discrete dynamic system. Thus, consider the linear discrete-time system given by 𝑇𝑥=𝐴𝑥=0.9550.010.0050.0050.960.005−0.0050.010.965𝑥, (40) with sequences 𝑔_𝑛=1−0.1^𝑛+1 and 𝑓_𝑛=𝑛² for all 𝑛≥0. 𝑇 is a strict contraction with eigenvalues {0.95,0.96,0.97}, ‖𝐴‖_∞=0.98, and ‖𝐴‖₂=0.9717. Thus, it has a unique fixed point at 𝑥=0. These sequences satisfy conditions (2) and (3) stated in Theorem 9 since lim_𝑛→∞𝑔_𝑛=1, 0<𝑔_𝑛<1 for all 𝑛≥0, lim_𝑛→∞(𝑓_𝑛/𝑛)=lim_𝑛→∞𝑛=+∞, and lim_𝑛→∞𝜎_𝑛<+∞ with 𝜎_𝑛=|𝑛(1−(𝑓_𝑛+1log𝑔_𝑛+1/𝑓_𝑛log𝑔_𝑛))| and {𝜎_𝑛}→2 (see Figure 8).

Figure 8 

Evolution of 𝜎𝑛 with time.

From Theorem 9, the sequence of iterates converges to the unique fixed point of 𝑇, 𝑥=0, as it is confirmed in the numerical simulation displayed in Figure 9.

Figure 9 

Evolution of the iterates through time.

5. Conclusion

This paper has investigated the boundedness and convergence properties of two general iterative processes built with sequences of self-mappings in either complete metric or Banach spaces. The self-mappings of the first iterative scheme are built with linear combinations of a set of self-mappings each of them being a weighted version of a self-mapping on the same space. Those of the second scheme are powers of an iteration-dependent scaled version of the primary self-mapping. Some applications are given for global stability of a class of nonlinear polytopic-type parameterizations of dynamic systems.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the Spanish and Basque Governments and to UPV/EHU for Grants DPI2012-30651, IT378-10, SAIOTEK S-PE13UN039, and UFI 2011/07. They thank the reviewers for their comments.