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The computation of correlation functions in models of statistical mechanics is the key to comparing theoretical results with actual measurements. In the context of integrable lattice models there exists a rich literature on correlators in vertex models and related quantum-spin chains. Less is known for integrable face models and their related anyon chains. Therefore, we define generalised transfer matrices allowing for a solution of the `inverse problem', i.e.\ we express local operators by means of objects from the Yang-Baxter algebra. This motivates the study of reduced density matrices which contain the information of all correlation functions. Instead of directly calculating them, we show that they fulfil a set of functional equations. We use these equations to study density matrices in (R)SOS models and their related anyon chains. In particular, we find integral representations for the two and three-point functions of the $r=4$ RSOS model and calculate those quantities for the $r=5$ model in the thermodynamic limit. In addition we observe a factorisation of the three-point functions into two-point functions and propose an efficient algorithm to factorise reduced density matrices for generic models. In the last section we study density matrices for the $SO(5)_2$ face models. We examine the structure of the two-site reduced density matrices and simplify them for certain topological sectors. Since there exist different inequivalent sets of Boltzmann weights, the latter is done for each choice leading to sets of discrete functional equations.

In this article, a novel method to compute all discrete polyharmonic functions in the quarter plane for models with small steps, zero drift and a finite group is proposed. A similar method is then introduced for continuous polyharmonic functions, and convergence between the discrete and continuous cases is shown.

In this paper we will get a class of functional equations involving a countable set of terms, summed by the well known Bochner–Fejér summation procedure, which are closely associated with the set of almost periodic functions. We will show that the zeros of a prefixed almost periodic function determine analytic solutions of such a functional equation associated with it, and we will obtain other solutions which are analytic or meromorphic on a certain domain.

In this paper we will get a class of functional equations involving a countable set of terms, summed by the well known Bochner–Fejér summation procedure, which are closely associated with the set of almost periodic functions. We will show that the zeros of a prefixed almost periodic function determine analytic solutions of such a functional equation associated with it, and we will obtain other solutions which are analytic or meromorphic on a certain domain.

The aim of this study is to determine through numerical simulations the extent and robustness of the three-dimensional torsional dynamics of the thermal convection in rotating spherical fluids at very low Prandtl numbers. It is known that the kinetic energy of the periodic axisymmetric flows propagates latitudinally on the surface of the sphere. Here, it is shown that when the axisymmetry is broken at a secondary Hopf bifurcation, the flow starts to drift in the azimuthal direction giving rise to a quasiperiodic motion that propagates the energy in latitude and longitude. The double direction of propagation gives rise to a meandering path of the kinetic energy, which is still concentrated on the surface, but highly localized. Several new stable states of convection with different symmetries have been identified in a large range of Rayleigh numbers, all of them retaining the torsional motion of the basic velocity field. Particular attention is paid to their dependence on the Rayleigh number and on the values of the frequencies, of the mean zonal flow, and of the kinetic energy of the fluid., Postprint (author's final draft)

In the present paper we prove that the complex functional equation F(z)+F(2z)+⋯+F(nz)=0, n≥2, z∈C∖(−∞,0], is stable in the generalized Hyers–Ulam sense.

This work is related to inequalities in the approximation theory. Mainly, we study numerical solutions of delay Volterra integral equations by using a collocation method based on sigmoidal function approximation. Error estimation and convergence analysis are provided. At the end of the paper we display numerical simulations verifying our results.

This work is related to inequalities in the approximation theory. Mainly, we study numerical solutions of delay Volterra integral equations by using a collocation method based on sigmoidal function approximation. Error estimation and convergence analysis are provided. At the end of the paper we display numerical simulations verifying our results.

In the present paper we prove that the complex functional equation F(z)+F(2z)+⋯+F(nz)=0, n≥2, z∈C∖(−∞,0], is stable in the generalized Hyers–Ulam sense.

We investigate polyharmonic functions associated to Brownian motions and random walks in cones. These are functions which cancel some power of the usual Laplacian in the continuous setting and of the discrete Laplacian in the discrete setting. We show that polyharmonic functions naturally appear while considering asymptotic expansions of the heat kernel in the Brownian case and in lattice walk enumeration problems. We provide a method to construct general polyharmonic functions through Laplace transforms and generating functions in the continuous and discrete cases, respectively. This is done by using a functional equation approach.

The mathematical relations between the regular Coulomb function Fηℓ(ρ) and the irregular Coulomb functions H±ηℓ(ρ) and Gηℓ(ρ) are obtained in the complex plane of the variables η and ρ for integer or half-integer values of ℓ. These relations, referred to as “connection formulas,” form the basis of the theory of Coulomb wave functions and play an important role in many fields of physics, especially in the quantum theory of charged particle scattering. As a first step, the symmetry properties of the regular function Fηℓ(ρ) are studied, in particular, under the transformation ℓ ↦ −ℓ − 1, by means of the modified Coulomb function Φηℓ(ρ), which is entire in the dimensionless energy η−2 and the angular momentum ℓ. Then, it is shown that, for integer or half-integer ℓ, the irregular functions H±ηℓ(ρ) and Gηℓ(ρ) can be expressed in terms of the derivatives of Φη,ℓ(ρ) and Φη,−ℓ−1(ρ) with respect to ℓ. As a consequence, the connection formulas directly lead to the description of the singular structures of H±ηℓ(ρ)and Gηℓ(ρ) at complex energies in their whole Riemann surface. The analysis of the functions is supplemented by novel graphical representations in the complex plane of η−1., SCOPUS: ar.j, info:eu-repo/semantics/published

In this thesis, we study the local and global zeta integral of a specific algebra associated with a base field k. In the local case, let k be a local field and char(k) = 0, and B = k[x]=(xn) for a fixed integer n > 1. In this case, we show that the zeta integrals on B have meromorphic continuation and they satisfy certain functional equations. In the global case, let k be an algebraic number field and B(A) = A[x]=(xn) where A is the adele ring of k. In this case, we use Poisson summation formula to study the zeta integrals on B(A), and also prove the meromorphic continuation and certain functional equations of zeta integrals. The corresponding case "n = 1" is studied in detail by John Tate in his thesis[1] where he proved the functional equation and meromorphic continuation of the zeta integrals and the zeta functions for an algebraic number field k. Godement and Jacquet[2] generalized the study of zeta integrals on any simple algebras in 1972. The work in this thesis can be regarded as the study of zeta integrals on certain non-simple algebras B and B(A) and this provides a new example where the theory of zeta integrals in Tate's thesis can be generalized to non-simple algebras.

In this thesis, we study the local and global zeta integral of a specific algebra associated with a base field k. In the local case, let k be a local field and char(k) = 0, and B = k[x]=(xn) for a fixed integer n > 1. In this case, we show that the zeta integrals on B have meromorphic continuation and they satisfy certain functional equations. In the global case, let k be an algebraic number field and B(A) = A[x]=(xn) where A is the adele ring of k. In this case, we use Poisson summation formula to study the zeta integrals on B(A), and also prove the meromorphic continuation and certain functional equations of zeta integrals. The corresponding case "n = 1" is studied in detail by John Tate in his thesis[1] where he proved the functional equation and meromorphic continuation of the zeta integrals and the zeta functions for an algebraic number field k. Godement and Jacquet[2] generalized the study of zeta integrals on any simple algebras in 1972. The work in this thesis can be regarded as the study of zeta integrals on certain non-simple algebras B and B(A) and this provides a new example where the theory of zeta integrals in Tate's thesis can be generalized to non-simple algebras.

In this thesis, we study the local and global zeta integral of a specific algebra associated with a base field k. In the local case, let k be a local field and char(k) = 0, and B = k[x]=(xn) for a fixed integer n > 1. In this case, we show that the zeta integrals on B have meromorphic continuation and they satisfy certain functional equations. In the global case, let k be an algebraic number field and B(A) = A[x]=(xn) where A is the adele ring of k. In this case, we use Poisson summation formula to study the zeta integrals on B(A), and also prove the meromorphic continuation and certain functional equations of zeta integrals. The corresponding case "n = 1" is studied in detail by John Tate in his thesis[1] where he proved the functional equation and meromorphic continuation of the zeta integrals and the zeta functions for an algebraic number field k. Godement and Jacquet[2] generalized the study of zeta integrals on any simple algebras in 1972. The work in this thesis can be regarded as the study of zeta integrals on certain non-simple algebras B and B(A) and this provides a new example where the theory of zeta integrals in Tate's thesis can be generalized to non-simple algebras.

We initiate the study of spectral zeta functions zeta(X) for finite and infinite graphs X, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function zeta(s) is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of the logarithmic derivative of zeta(s). We relate zeta(Z) to Euler's beta integral and show how to complete it giving the functional equation xi(Z)(1 - s) = xi(Z)(s). This function appears in the theory of Eisenstein series although presumably with this spectral intepretation unrecognized. In higher dimensions d we provide a meromorphic continuation of zeta(Zd) (s) to the whole plane and identify the poles. From our aymptotics several known special values of zeta(s) are derived as well as its non-vanishing on the line Re(s) = 1. We determine the spectral zeta functions of regular trees and show it to be equal to a specialization of Appell's hypergeometric function F-1 via an Euler-type integral formula due to Picard.

Through several equivalence binary relations, in this paper we identify, on the one hand, groups of exponential polynomials with the same set of zeros, and on the other, groups of functional equations of the form a1f(1z) + a2f(2z) + : : : + anf(nz) = 0; z 2 C that lead to equivalent exponential polynomials with the same set of zeros.

For given p ∈ [1,∞] and g ∈ Lᵖ(R), we establish the existence and uniqueness of solutions f ∈ Lᵖ(R), to the equation f (x) − a f (bx) = g(x), where a ∈ R, b ∈ R\ {0}, and |a| ̸= |b|1/p. Solutions include well-known nowhere differentiable functions such as those of Bolzano, Weierstrass, Hardy, and many others. Connections and consequences in the theory of fractal interpolation, approximation theory, and Fourier analysis are established.

Through several equivalence binary relations, in this paper we identify, on the one hand, groups of exponential polynomials with the same set of zeros, and on the other, groups of functional equations of the form a1f(1z) + a2f(2z) + : : : + anf(nz) = 0; z 2 C that lead to equivalent exponential polynomials with the same set of zeros.

In this paper, it is showed that, given an integer number n ≥ 2, each zero of an exponential polynomial of the form w1az1+w2az2+⋯+wnazn, with non-null complex numbers w 1,w 2,…,w n and a 1,a 2,…,a n , produces analytic solutions of the functional equation w 1 f(a 1 z) + w 2 f(a 2 z) + ... + w n f(a n z) = 0 on certain domains of C, which represents an extension of some existing results in the literature on this functional equation for the case of positive coefficients a j and w j.

<> Forschung zu Netzwerken und Kommunikationssystemen ist derzeit von großem Interesse. In der Literatur sind dazu mehrere Zugänge zu finden, nämlich experimentelle, numerische und analytische sowie Simulationen - alle mit ihren Vor- und Nachteilen. Mit dem analytischen Zugang können viele Systeme mathematisch beschrieben werden, wobei in vielen Fällen schwierige Funktionalgleichungen in zwei Variablen zu lösen sind. Meist sind dies Gleichungen, in denen die Unbekannten erzeugende Funktionen von Wahrscheinlichkeitsverteilungen der Systeme sind. Viele Funktionalgleichungen entstehen im Kontext unterschiedlichster Anwendungen, wie Populations- und Verhaltensforschung, Sozialwissenschaften, Astronomie, Netzwerk- und Informationstheorie sowie Wirtschaftswissenschaften. Beschreibungen davon können auf eine Weise formuliert werden, die zu Funktionalgleichungen führt, die präzise quantitative Beziehungen beschreiben. Während der letzten sechs Jahrzehnte wurde eine Klasse von Funktionalgleichungen in zwei Variablen untersucht, die aus Kommunikations- und Netzwerkmodellen hervorgehen. Explizite Lösungen dieser Gleichungen sind für Anwendungen von großem Interesse, da sie Informatikern und Ingenieuren Werkzeuge zur Bestimmung von Maßen für das Verhalten von Systemen liefert. Die Bestimmung expliziter Lösungen ist schwierig und abgesehen von wenigen einfachen Fällen sind sie im Allgemeinen nicht bekannt. Häufig wird die Theorie der Randwertprobleme für Lösungsansätze verwendet. Diese Arbeit untersucht die analytischen Lösungen dieser allgemeinen Klasse von Funktionalgleichungen., Investigations of various networks and systems of communications are very important nowadays. There is more than one approach in the literature to such study; namely, the experimental, numerical, simulation, and the analytical approaches. Each of them has its own advantages and disadvantages. The analytical approach describes such systems mathematically and end up in most cases with challenging two-place functional equations. They are mainly equations in which the unknowns are generating functions of systems distributions. Various functional equations arise abundantly in models of widely diverse fields, such as population ethics, behavioral and social sciences, astronomy, networks, information theory, and economics. Specifically, each of these descriptions can be formulated so as to eventually lead to functional equations that can yield precise quantitative relationships. During the last six decades, a certain class of two-place functional equations arose from various models of networks and communication. The explicit closed-form solutions of those equations are very important because such functions will help computer scientists and engineers to find some performance measures of many engineering systems. This is a challenging task to find them and so far there are no such explicit solutions available except for some simple cases. The theory of boundary value problems has been extensively used in solving such interesting class of equations. This thesis is mainly devoted to the investigations of the analytical solutions of such a general class of equations that have many interesting applications in modern disciplines like wireless networks., El-Sayed El-Hady, Innsbruck, Univ., Diss., 2015

<> Forschung zu Netzwerken und Kommunikationssystemen ist derzeit von großem Interesse. In der Literatur sind dazu mehrere Zugänge zu finden, nämlich experimentelle, numerische und analytische sowie Simulationen - alle mit ihren Vor- und Nachteilen. Mit dem analytischen Zugang können viele Systeme mathematisch beschrieben werden, wobei in vielen Fällen schwierige Funktionalgleichungen in zwei Variablen zu lösen sind. Meist sind dies Gleichungen, in denen die Unbekannten erzeugende Funktionen von Wahrscheinlichkeitsverteilungen der Systeme sind. Viele Funktionalgleichungen entstehen im Kontext unterschiedlichster Anwendungen, wie Populations- und Verhaltensforschung, Sozialwissenschaften, Astronomie, Netzwerk- und Informationstheorie sowie Wirtschaftswissenschaften. Beschreibungen davon können auf eine Weise formuliert werden, die zu Funktionalgleichungen führt, die präzise quantitative Beziehungen beschreiben. Während der letzten sechs Jahrzehnte wurde eine Klasse von Funktionalgleichungen in zwei Variablen untersucht, die aus Kommunikations- und Netzwerkmodellen hervorgehen. Explizite Lösungen dieser Gleichungen sind für Anwendungen von großem Interesse, da sie Informatikern und Ingenieuren Werkzeuge zur Bestimmung von Maßen für das Verhalten von Systemen liefert. Die Bestimmung expliziter Lösungen ist schwierig und abgesehen von wenigen einfachen Fällen sind sie im Allgemeinen nicht bekannt. Häufig wird die Theorie der Randwertprobleme für Lösungsansätze verwendet. Diese Arbeit untersucht die analytischen Lösungen dieser allgemeinen Klasse von Funktionalgleichungen., Investigations of various networks and systems of communications are very important nowadays. There is more than one approach in the literature to such study; namely, the experimental, numerical, simulation, and the analytical approaches. Each of them has its own advantages and disadvantages. The analytical approach describes such systems mathematically and end up in most cases with challenging two-place functional equations. They are mainly equations in which the unknowns are generating functions of systems distributions. Various functional equations arise abundantly in models of widely diverse fields, such as population ethics, behavioral and social sciences, astronomy, networks, information theory, and economics. Specifically, each of these descriptions can be formulated so as to eventually lead to functional equations that can yield precise quantitative relationships. During the last six decades, a certain class of two-place functional equations arose from various models of networks and communication. The explicit closed-form solutions of those equations are very important because such functions will help computer scientists and engineers to find some performance measures of many engineering systems. This is a challenging task to find them and so far there are no such explicit solutions available except for some simple cases. The theory of boundary value problems has been extensively used in solving such interesting class of equations. This thesis is mainly devoted to the investigations of the analytical solutions of such a general class of equations that have many interesting applications in modern disciplines like wireless networks., El-Sayed El-Hady, Innsbruck, Univ., Diss., 2015

In this paper, it is showed that, given an integer number n ≥ 2, each zero of an exponential polynomial of the form w1az1+w2az2+⋯+wnazn, with non-null complex numbers w 1,w 2,…,w n and a 1,a 2,…,a n , produces analytic solutions of the functional equation w 1 f(a 1 z) + w 2 f(a 2 z) + ... + w n f(a n z) = 0 on certain domains of C, which represents an extension of some existing results in the literature on this functional equation for the case of positive coefficients a j and w j.

Coincidence and common fixed point theorems for a class of Suzuki hybrid contractions involving two pairs of single-valued and multivalued maps in a metric space are obtained. In addition, the existence of a common solution for a certain class of functional equations arising in a dynamic programming is also discussed.

Основним об’єктом дослідження статті є структура мно­жини неперервних розв’язків різницево-функціональних рівнянь вигляду \[x\left ( qt \right )=a\left ( t \right )x\left ( t \right )+b\left ( t \right )x\left ( t+1 \right )+f\left ( t \right ),\] де \[a\left ( t \right ),b\left ( t \right ),f\left ( t \right )\] – деякі дійсні функції і q – дійсна стала. З використанням методів теорії диференціальних і різницевих рівнянь досліджено питання існування неперервних роз­в’яз­ків лінійних різницево-функціональних рівнянь зі сталими коефіцієнтами, а також запропоновано метод їх побудови. Зокрема, для однорідних рівнянь побудовано сім’ю неперервних обмежених розв’язків, що залежить від довільної неперервної 1-періодичної функції при \[0< q< 1,a> 1\] і \[0< a< 1,q> 1,\] t – додатне. Крім цього, для \[0< q< 1,a> 1,\] доведено існування неперервних розв’язків для неоднорідних рівнянь при довільному дійсному значенні t та побудовано сім’ю неперервних обмежених розв’язків неоднорідного рівняння при довільному невід’ємному t., The main object of research in this article is a study of the continuous solutions set structure of difference-functional equations of the form \[x\left ( qt \right )=a\left ( t \right )x\left ( t \right )+b\left ( t \right )x\left ( t+1 \right )+f\left ( t \right ),\] where \[a\left ( t \right ),b\left ( t \right ),f\left ( t \right )\]– some real functions and q – real constant. Using methods of the theory of differential and difference equations a question of continuous solutions existence of linear difference-functional equations with constant coefficients was investigated and a building method for them was proposed. In particular for homogeneous equations, a continuous bounded solutions family that depends on an arbitrary continuous 1-periodic function with \[0< q< 1,a> 1\] and \[0< a< 1,q> 1,\] t – positive was built. Furthermore, if \[0< q< 1,a> 1,\] then continuous solutions existence for heterogeneous equations with an arbitrary real value t is proved and continuous bounded solutions family of a heterogeneous equation with an arbitrary nonnegative t are built., Основной объект исследования данной статьи – структура множества непрерывных решений разностно-функцио­наль­ных уравнений вида \[x\left ( qt \right )=a\left ( t \right )x\left ( t \right )+b\left ( t \right )x\left ( t+1 \right )+f\left ( t \right ),\] где \[a\left ( t \right ),b\left ( t \right ),f\left ( t \right )\] – некоторые действительные функции и q – действительная постоянная. С использованием методов теории дифференциальных уравнений исследован вопрос о существовании непрерывных решений линейных разностно-функциональных уравнений с постоянными коэффициентами, а также предложен метод их построения. В частности, для однородных уравнений построено семейство непрерывных ограниченных решений, что зависит от любой непрерывной 1-периодической функции \[0< q< 1,a> 1\] и \[0< a< 1,q> 1,\] t – положительное. Кроме этого, для \[0< q< 1,a> 1,\] доказано существование непрерывных решений для неоднородных уравнений при любом действительном значении t и построено семейство непрерывных ограниченных решений неоднородного уравнения при любом неотрицательном t.

Coincidence and common fixed point theorems for a class of Suzuki hybrid contractions involving two pairs of single-valued and multivalued maps in a metric space are obtained. In addition, the existence of a common solution for a certain class of functional equations arising in a dynamic programming is also discussed.

We investigate the existence of a solution of linear functional equations.

We investigate the existence of a solution of linear functional equations.

Coincidence and common fixed point theorems for a class of Suzuki hybrid contractions involving two pairs of single-valued and multivalued maps in a metric space are obtained. In addition, the existence of a common solution for a certain class of functional equations arising in a dynamic programming is also discussed.

This paper proves that every zero of any n th , n ≥ 2, partial sum of the Riemann zeta function provides a vector space of basic solutions of the functional equation f(x)+f(2x)+⋯+f(nx)=0,x∈R . The continuity of the solutions depends on the sign of the real part of each zero.

This paper proves that every zero of any n th , n ≥ 2, partial sum of the Riemann zeta function provides a vector space of basic solutions of the functional equation f(x)+f(2x)+⋯+f(nx)=0,x∈R . The continuity of the solutions depends on the sign of the real part of each zero.

The subject of this work is functional equations with direction towards linear functional equations. The .rst part describes function sets where iterates of the functions converge to a .xed point. In the second part the convergence property is used to provide solutions to linear functional equations by de.ning solutions as in.nite sums. Furthermore, this work contains some transforms to linear form, examples of functions that belong to di¤erent classes and corresponding linear functional equations. We use Mathematica to generate solutions and solve itera- tively equations.

In this thesis a class of nonlinear oscillator equations is studied. Asymptotic approximations of first integrals for nonlinear difference equations are constructed by using the recently developed perturbation method based on invariance vectors. The asymptotic approximations of the solutions of the oscillator equations can be derived from these asymptotic approximations of the first integrals. It is shown that all invariance factors have to satisfy a functional equation. One of the main difficulties in finding first integrals for a system of first order difference equations is solving the aforementioned functional equation. In this thesis, we consider a functional equation which is related to a system of two first order, linear ordinary difference equations. Linear transformations and an adapted version of the method of separation of variables is used to construct the general solution of this functional equation. A perturbation method based on invariance factors and multiple scales is also presented for weakly nonlinear, regularly perturbed systems of ordinary difference equations. Asymptotic approximations of first integrals are constructed on long iteration-scales, that is, on iteration-scales of order ?-1, where ? is a small parameter. To show how this perturbation method works, the method is applied to a Van der Pol equation, and a Rayleigh equation. We also apply an improved version of the multiple scales perturbation method to a general system of weakly nonlinear, regularly perturbed ordinary difference equations including linear, quadratic, and qubic terms. Such systems arise as a result of the discretization of a system of nonlinear differential equations, or as a result in the stability analysis of nonlinear oscillations. As an example, in the study of the forced vibrations of a (damped) linear sdofo with a time-varying mass a system of two nonlinear ordinary difference equations is obtained to describe the stability properties of the oscillator. In such os, Mathematical Physics Department, Electrical Engineering, Mathematics and Computer Science

Es bien conocido que la resolución de ecuaciones no lineales de la forma F(x) = 0, donde F es un operador no lineal definido entre espacios de Banach, es un problema habitual en las ciencias e ingenierías. Habitualmente se buscan aproximaciones numéricas de las raíces de la ecuación anterior, puesto que encontrar raíces exactas suele ser difícil. Para aproximar una raíz de la ecuación anterior se utilizan habitualmente métodos iterativos, de entre los que destaca el método de Newton por su sencillez, fácil aplicación y eficiencia. El primer resultado de convergencia semilocal para el método de Newton en espacios de Banach fue dado por Kantorovich. En esta memoria se analiza la convergencia semilocal del método de Newton en espacios de Banach y cuyo principal objetivo es darle una mayor generalidad al problema de aproximar las raíces de una ecuación no lineal mediante el método de Newton, de manera que se pueda extender la aplicabilidad de este método a situaciones en las que la teoría clásica de Kantorovich no la puede garantizar. Para ello, se utiliza el conocido principio de la mayorante, que se basa en la construcción de sucesiones reales mayorizantes, y que, bajo nuevas condiciones de convergencia semilocal de tipo Kantorovich, permite generalizar las condiciones clásicas de Kantorovich. Aquí juega un papel importante la construcción ad hoc de sucesiones mayorizantes a partir de la resolución de problemas de valor inicial. Se ilustra todo lo anterior con diferentes tipos de ecuaciones no lineales, destacando las ecuaciones integrales de tipo Hammerstein mixto y la ecuación de Bratu, que tienen su origen en diversos problemas del mundo real, tal y como se pone de manifiesto a largo de la Memoria., It is well known that solving nonlinear equations of the form F(x)=0, where F is a nonlinear operator defined between Banach spaces, is a common problem in science and engineering. Usually numerical approximations of the roots of the previous equation are looking for, since finding exact roots is often difficult. To approximate a root of the previous equation is commonly used iterative methods, among which Newton's method is important for its simplicity, easy implementation and efficiency. The first result of semilocal convergence for Newton's method in Banach spaces was given by Kantorovich. In this dissertation it is analyze the semilocal convergence of Newton's method in Banach spaces, whose principal aim is to give greater generality to the problem of approximating the roots of a nonlinear equation by Newton's method, so that it can extend the applicability of this method to situations where the classical theory of Kantorovich cannot. To do this, we use the majorant principle, based on the construction of majorizing sequences, and, under new semilocal convergence conditions of Kantorovich-type, that allows us to generalize the classical conditions of Kantorovich. This plays an important role in the ad hoc construction of majorizing sequences from solving initial value problems. We illustrate the above with different types of nonlinear equations, highlighting the Hammerstein integral equations of mixed type and Bratu's equation, which have their origin in various real-world problems, such as it is evidenced in this dissertation.

Es bien conocido que la resolución de ecuaciones no lineales de la forma F(x) = 0, donde F es un operador no lineal definido entre espacios de Banach, es un problema habitual en las ciencias e ingenierías. Habitualmente se buscan aproximaciones numéricas de las raíces de la ecuación anterior, puesto que encontrar raíces exactas suele ser difícil. Para aproximar una raíz de la ecuación anterior se utilizan habitualmente métodos iterativos, de entre los que destaca el método de Newton por su sencillez, fácil aplicación y eficiencia. El primer resultado de convergencia semilocal para el método de Newton en espacios de Banach fue dado por Kantorovich. En esta memoria se analiza la convergencia semilocal del método de Newton en espacios de Banach y cuyo principal objetivo es darle una mayor generalidad al problema de aproximar las raíces de una ecuación no lineal mediante el método de Newton, de manera que se pueda extender la aplicabilidad de este método a situaciones en las que la teoría clásica de Kantorovich no la puede garantizar. Para ello, se utiliza el conocido principio de la mayorante, que se basa en la construcción de sucesiones reales mayorizantes, y que, bajo nuevas condiciones de convergencia semilocal de tipo Kantorovich, permite generalizar las condiciones clásicas de Kantorovich. Aquí juega un papel importante la construcción ad hoc de sucesiones mayorizantes a partir de la resolución de problemas de valor inicial. Se ilustra todo lo anterior con diferentes tipos de ecuaciones no lineales, destacando las ecuaciones integrales de tipo Hammerstein mixto y la ecuación de Bratu, que tienen su origen en diversos problemas del mundo real, tal y como se pone de manifiesto a largo de la Memoria., It is well known that solving nonlinear equations of the form F(x)=0, where F is a nonlinear operator defined between Banach spaces, is a common problem in science and engineering. Usually numerical approximations of the roots of the previous equation are looking for, since finding exact roots is often difficult. To approximate a root of the previous equation is commonly used iterative methods, among which Newton's method is important for its simplicity, easy implementation and efficiency. The first result of semilocal convergence for Newton's method in Banach spaces was given by Kantorovich. In this dissertation it is analyze the semilocal convergence of Newton's method in Banach spaces, whose principal aim is to give greater generality to the problem of approximating the roots of a nonlinear equation by Newton's method, so that it can extend the applicability of this method to situations where the classical theory of Kantorovich cannot. To do this, we use the majorant principle, based on the construction of majorizing sequences, and, under new semilocal convergence conditions of Kantorovich-type, that allows us to generalize the classical conditions of Kantorovich. This plays an important role in the ad hoc construction of majorizing sequences from solving initial value problems. We illustrate the above with different types of nonlinear equations, highlighting the Hammerstein integral equations of mixed type and Bratu's equation, which have their origin in various real-world problems, such as it is evidenced in this dissertation.