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This Special Issue aims to be a compilation of new results in the areas of differential and difference Equations, covering boundary value problems, systems of differential and difference equations, as well as analytical and numerical methods. The objective is to provide an overview of techniques used in these different areas and to emphasize their applicability to reallife phenomena, by the inclusion of examples. These examples not only clarify the theoretical results presented, but also provide insight on how to apply, for future works, the techniques used.
Legendre wavelets  collocation method  threestep Taylor method  asymptotic stability  timedependent partial differential equations  noninstantaneous impulses  Caputo fractional derivative  differential equations  state dependent delays  lipschitz stability  limitperiodic solutions  difference equations  exponential dichotomy  strong nonlinearities  effective existence criteria  population dynamics  discrete Lyapunov equation  difference equations  Hilbert space  dichotomy  exponential stability  ?Laplacian operator  mean curvature operator  heteroclinic solutions  problems in the real line  lower and upper solutions  Nagumo condition on the real line  fixed point theory  coupled nonlinear systems  functional boundary conditions  Schauder’s fixed point theory  Arzèla Ascoli theorem  lower and upper solutions  first order periodic systems  SIRS epidemic model  mathematical modelling  Navier–Stokes equations  global solutions  regular solutions  a priori estimates  weak solutions  kinetic energy  dissipation  Bäcklund transformation  Clairin’s method  generalized Liouville equation  Miura transformation  Kortewegde Vries equation  secondorder differential/difference/qdifference equation of hypergeometric type  nonuniform lattices  divideddifference equations  polynomial solution  integrodifferentials  Sumudu decomposition method  dynamical system
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A plethora of problems from diverse disciplines such as Mathematics, Mathematical: Biology, Chemistry, Economics, Physics, Scientific Computing and also Engineering can be formulated as an equation defined in abstract spaces using Mathematical Modelling. The solutions of these equations can be found in closed form only in special case. That is why researchers and practitioners utilize iterative procedures from which a sequence is being generated approximating the solution under some conditions on the initial data. This type of research is considered most interesting and challenging. This is our motivation for the introduction of this special issue on Iterative Procedures.
Banach space  weightedNewton method  local convergence  Fréchetderivative  ball radius of convergence  Nondifferentiable operator  nonlinear equation  divided difference  Lipschitz condition  convergence order  local and semilocal convergence  scalar equations  computational convergence order  Steffensen’s method  basins of attraction  nonlinear equations  multipleroot solvers  Traub–Steffensen method  fast algorithms  Multiple roots  Optimal iterative methods  Scalar equations  Order of convergence  simple roots  Newton’s method  computational convergence order  nonlinear equations  split variational inclusion problem  generalized mixed equilibrium problem  fixed point problem  maximal monotone operator  left Bregman asymptotically nonexpansive mapping  uniformly convex and uniformly smooth Banach space  nonlinear equations  multiple roots  derivativefree method  optimal convergence  multiple roots  optimal iterative methods  scalar equations  order of convergence  Newton–HSS method  systems of nonlinear equations  semilocal convergence  local convergence  convergence order  Banach space  iterative method  nonlinear equations  Chebyshev’s iterative method  fractional derivative  basin of attraction  nonlinear equations  iterative methods  general means  basin of attraction
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Solving nonlinear equations in Banach spaces (real or complex nonlinear equations, nonlinear systems, and nonlinear matrix equations, among others), is a nontrivial task that involves many areas of science and technology. Usually the solution is not directly affordable and require an approach using iterative algorithms. This Special Issue focuses mainly on the design, analysis of convergence, and stability of new schemes for solving nonlinear problems and their application to practical problems. Included papers study the following topics: Methods for finding simple or multiple roots either with or without derivatives, iterative methods for approximating different generalized inverses, real or complex dynamics associated to the rational functions resulting from the application of an iterative method on a polynomial. Additionally, the analysis of the convergence has been carried out by means of different sufficient conditions assuring the local, semilocal, or global convergence. This Special issue has allowed us to present the latest research results in the area of iterative processes for solving nonlinear equations as well as systems and matrix equations. In addition to the theoretical papers, several manuscripts on signal processing, nonlinear integral equations, or partial differential equations, reveal the connection between iterative methods and other branches of science and engineering.
point projection  intersection  parametric curve  ndimensional Euclidean space  Newton’s second order method  fixed point theorem  nonlinear equations  multiple zeros  optimal iterative methods  higher order of convergence  nonlinear operator equation  Fréchet derivative  ?continuity condition  Newtonlike method  Frédholm integral equation  nonlinear equations  Padé approximation  iterative method  order of convergence  numerical experiment  fourth order iterative methods  local convergence  banach space  radius of convergence  nonlinear equation  iterative process  nondifferentiable operator  Lipschitz condition  high order  sixteenth order convergence method  local convergence  dynamics  Banach space  Newton’s method  semilocal convergence  Kantorovich hypothesis  iterative methods  Steffensen’s method  Rorder  with memory  computational efficiency  nonlinear equation  basins of attraction  optimal order  higher order method  computational order of convergence  nonlinear equations  multiple roots  Chebyshev–Halleytype  optimal iterative methods  efficiency index  Banach space  semilocal convergence  ?continuity condition  Jarratt method  error bound  Fredholm integral equation  Newton’s method  global convergence  variational inequality problem  split variational inclusion problem  multivalued quasinonexpasive mappings  Hilbert space  sixteenthorder optimal convergence  multipleroot finder  asymptotic error constant  weight function  purely imaginary extraneous fixed point  attractor basin  drazin inverse  generalized inverse  iterative methods  higher order  efficiency index  integral equation  efficiency index  nonlinear models  iterative methods  higher order  nonlinear equations  optimal iterative methods  multiple roots  efficiency index  iterative methods  nonlinear equations  Newtontype methods  smooth and nonsmooth operators  heston model  Hull–White  option pricing  PDE  finite difference (FD)  iteration scheme  Moore–Penrose  rectangular matrices  rate of convergence  efficiency index  nonlinear equations  conjugate gradient method  projection method  convex constraints  signal and image processing  nonlinear monotone equations  conjugate gradient method  projection method  signal processing  nonlinear systems  multipoint iterative methods  divided difference operator  order of convergence  Newton’s method  computational efficiency index  system of nonlinear equations  Newton method  NewtonHSS method  nonlinear HSSlike method  PicardHSS method  convexity  least square problem  accretive operators  signal processing  point projection  intersection  planar algebraic curve  Newton’s iterative method  the improved curvature circle algorithm  systems of nonlinear equations  King’s family  order of convergence  multipoint iterative methods  nonlinear equations  Potra–Pták method  optimal methods  weight function  basin of attraction  engineering applications  Kung–Traub conjecture  multipoint iterations  nonlinear equation  optimal order  basins of attraction
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