In this dissertation, we consider modified inexact Newton methods applied to second order nonlinear problems. In the implementation of Newton's method applied to problems with a large number of degrees of freedom, it is often necessary to solve the linear Jacobian system iteratively. Although a general theory for the convergence of modified inexact Newton's methods has been developed, its application to nonlinear problems from nonlinear PDE's is far from complete. The case where the nonlinear operator is a zeroth order perturbation of a fixed linear operator was considered in the paper written by Brown et al.. The goal of this dissertation is to show that one can develop modified inexact Newton's methods which converge at a rate independent of the number of unknowns for problems with higher order nonlinearities. To do this, we are required to first, set up the problem on a scale of Hilbert spaces, and second, to devise a special iterative technique which converges in a higher order Sobolev norm, i.e., H1+alpha(omega) \ H1 0(omega) with 0 < alpha < 1/2. We show that the linear system solved in Newton's method can be replaced with one iterative step provided that the initial iterate is close enough. The closeness criteria can be taken independent of the mesh size. In addition, we have the same convergence rates of the method in the norm of H1 0(omega) using the discrete Sobolev inequalities.

We investigate the existences and qualitative properties of periodic solutions of the following two classes of nonlinear differential equations: I) (Special) Relativistic Pendulum Equations (RPEs); II) (2-coupled) Gross-Pitaevskii Equations (GPEs). The pendulum equation describes the motion of a pendulum. According to Special Relativity, which was published by A. Einstein in 1905, causality is more fundamental than constant time-space, thus time will ow slower and space will distort to keep causality if the speed of motion is near the speed of light. In such high speed situations, the pendulum equation needs to be revised due to Special Relativity. The revised equation is called RPE. Our result answers some open questions about the existence of multiple periodic solutions for RPEs. GPEs are sometimes called coupled nonlinear schrodinger equations. the Schrodinger equation is the fundamental equation of Quantum Mechanics which is the \exotic" probabilistic fundamental physics law of the \micro" world { the world of atoms and molecules. A well-known physicist and Nobel laureate, R. Feynman, said \I think I can safely say that nobody understands quantum mechanics." which indicates the physical/ philosophical difficulty of interpretations. It raises paradoxical problems such the well-known Schrodinger's Cat. Setting aside these difficult, if we combine Special Relativity and Quantum Mechanics as a many-body system, then we have Quantum Field Theory (QFT) which is more deterministic, and governs even elementary particle physics. GPEs are also related to QFT. For example, superconductivity and Bose Einstein Condensates (BEC). These phenomena in condensed matter physics can be thought of as the emergence of the mysterious micro world physics at \macro" level. We study these equations from the viewpoint of mathematical interest. It is generally difficult to solve nonlinear differential equations. It is also generally difficult even to prove the existence of solutions. Although we show there exist solutions, we still do not know how to solve the differential equations analytically. Variational Methods (or Calculus of Variations) are useful tools to show there exist solutions of differential equations. The idea is to convert the problem of solving equations into the problem of finding critical points (i.e. minimum/maximum points or saddle points) of a functional, and each critical point can generally correspond to a weak solution. However, it is also generally difficult to find out such critical points because we look for critical points in an infinite-dimensional functions space. Thus many advanced mathematical theories or tools have been developed and used for decades in nonlinear analysis. We use some topological theories. From information of the functional's shape, these theories deduce if there exists a critical point, or how many critical points exist. The key of these theories is to use the symmetry of the equations. We also investigate bifurcation structures for II), i.e. the connection structures between the solutions. By linearizations which look at the equations \locally," we reduce the problem in the infinite dimension to one in a finite dimension. Furthermore, it allows us to apply Morse Theory, which connects between local and global aspects of the functional's information. In several cases, we show that there are infinitely many bifurcation points that give rise to global bifurcation branches.

Using topological degree methods, we give some existence and multiplicity results for nonlinear differential or difference equations. In Chapter 1 some continuation theorems are presented. Chapter 2 deal with nonlinear difference equations. Using Brouwer degree we obtain upper and lower solutions theorems, Ambrosetti and Prodi type results and sharp existence conditions for nonlinearities which are bounded from below or from above. In Chapter 3, using Leray-Schauder degree, we give various existence and multiplicity result for second order differential equations with $phi$-Laplacian. Such equations are in particular motivated by the one-dimensional mean curvature problems and by the acceleration of a relativistic particle of mass one at rest moving on a straight line. In Chapter 4, using Mawhin continuation theorem, sufficient conditions are obtained for the existence of positive periodic solutions for delay Lotka-Volterra systems. In the last chapter of this work we prove some results concerning the multiplicity of solutions for a class of superlinear planar systems. The results of Chapters 2 and 3 are joint work with Prof. Jean Mawhin. / En utilisant le degré topologique, nous obtenons quelques résultats d'existance et de multiplicité pour des équations non-linéaires différentielles ou aux différences. Quelques théorèmes de continuation sont présentés au Chapitre 1. Le Chapitre 2 concerne des équations aux différences non-linéaires. En utilisant le degré de Brouwer, nous obtenons des résultats de sur et sous-solutions, des résultats de type Ambrosetti-Prodi ainsi que des conditions optimales d'existence pour des non-linéarités bornées inférieurement ou supérieurement. En utilisant le degré de Leray-Schauder, nous donnons au Chapitre 3 des résultats d'existence et de multiplicité pour des équations différentielles du second ordre avec $phi$-Laplacien. De telles équations sont en particulier motivées par le problème de la courbure en dimension un et par l'accélération d'une particule relativisite de masse un sur une droite. Au Chapitre 4, en utilisant le théorème de continuation de Mawhin, des conditions suffisantes sont obtenues pour l'existence de solutions périodiques positives des systhèmes de Lotka-Volterra avec retard. Dans le dernier chapitre de ce travail, nous prouvons certains résultats concernant la multiplicité des solutions pour une classe de systhèmes superlinéaires planaires. Les résultats des Chapitre 2 et 3 sont faits en collaboration avec monsieur le Professeur Jean Mawhin.

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Let X be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space X* . Let G be a bounded open subset of X. Let T:X⊃ D(T)⇒ 2X* be maximal monotone and S: X ⇒ 2X* be bounded pseudomonotone and such that 0 notin cl((T+S)(D(T)∩partG)). Chapter 1 gives general introduction and mathematical prerequisites. In Chapter 2 we develop a homotopy invariance and uniqueness results for the degree theory constructed by Zhang and Chen for multivalued (S+) perturbations of maximal monotone operators. Chapter 3 is devoted to the construction of a new topological degree theory for the sum T+S with the degree mapping d(T+S,G,0) defined by d(T+S,G,0)=limepsilondarr 0+ dS+(T+S+ J,G,0), where dS+ is the degree for bounded (S+)-perturbations of maximal monotone operators. The uniqueness and homotopy invariance result of this degree mapping are also included herein. As applications of the theory, we give associated mapping theorems as well as degree theoretic proofs of known results by Figueiredo, Kenmochi and Le. In chapter 4, we consider T:X D(T)⇒ 2X* to be maximal monotone and S:D(S)=K⇒ 2X* at least pseudomonotone, where K is a nonempty, closed and convex subset of X with 0isinKordm. Let Phi:X⇒ ( infin, infin] be a proper, convex and lower-semicontinuous function. Let f* isin X* be fixed. New results are given concerning the solvability of perturbed variational inequalities for operators of the type T+S associated with the function f. The associated range results for nonlinear operators are also given, as well as extensions and/or improvements of known results by Kenmochi, Le, Browder, Browder and Hess, Figueiredo, Zhou, and others.

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Made available in DSpace on 2014-06-11T19:22:18Z (GMT). No. of bitstreams: 0 Previous issue date: 2007-02-18Bitstream added on 2014-06-13T18:07:52Z : No. of bitstreams: 1 verao_gb_me_sjrp.pdf: 363983 bytes, checksum: c59e477b48d1d71a3199f377018eead3 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo deste trabalho é estudar problemas de valor de contorno do tipo {ÿ + f(t) =0 y(0)=0˙ y(1)= ky(η), (1) onde η ∈ (0, 1), k ∈ R e f ∈C([0, 1],R). Para antingirmos nosso objetivo usamosas funções de Green G(t,s)que nos permitem escrever a solução do problema(1)na seguinte forma: w(t)= ∫ 1 0 G(t,s)f(s)ds. Usando esta solução, investigamos através do ponto fixo de Schauder a solvabilidade do problema não linear { y + f(t,y)=0 y(0)=0˙ y(1)= ky(η). / The main goal of this work is study the following boundary value problems {ÿ + f(t) = 0 =0 y(0)=0˙ y(1)= ky(η), (1), where η ∈ (0, 1), k ∈ R e f ∈C([0, 1],R). To achieve our goal we use the Green's function G(t,s) which allow us to write the solution of the problem (2) in the form: w(t)= ∫ 1 0 G(t,s)f(s)ds. Using this solution and the Schauder point theory, also we study the solvability of a nonlinear problem { y + f(t,y)=0 y(0)=0˙ y(1)= ky(η).

Submitted by Marlene Santos (marlene.bc.ufg@gmail.com) on 2014-09-01T19:21:42Z No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dassael Fabricio dos Reis Santos - Dissertação de Mestrado.pdf: 2389476 bytes, checksum: 8ca3d9cabd2862c5e82bc4db0cec4071 (MD5) / Made available in DSpace on 2014-09-01T19:21:42Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dassael Fabricio dos Reis Santos - Dissertação de Mestrado.pdf: 2389476 bytes, checksum: 8ca3d9cabd2862c5e82bc4db0cec4071 (MD5) Previous issue date: 2014-03-28 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this work we study existence and multiplicity of non-negative solutions of the nonlinear elliptic problem −div(A(x,∇u)) = λf(x,u) in Ω, u = 0 in ∂Ω where Ω⊂IRN is a bounded domain with smooth boundary∂Ω,λ≥ 0 is a parameter, f :Ω×[0,∞)−→ IR and A :Ω×IRN−→ IRN satisfy the Carathéodory conditions, A is monotone and f satisfies a growth condition. To this end we use the method of Sub and Supersolutions, Topological Degree Theory, simmetry arguments and variational methods. / Neste trabalho estudaremos existência e multiplicidade de soluções não-negativas do problema elíptico não-linear −div(A(x,∇u)) = λf(x,u) em Ω, u = 0 em ∂Ω, Onde Ω ⊂ IRN é um domínio limitado com fronteira∂Ω suave,λ≥ 0 é um parâmetro, f :Ω×[0,∞)−→ IR e A :Ω×IRN−→ IRN satisfazem as condições de Carathéodory, A é monotônico e f satisfaz uma condição de crescimento. Para este fim utilizaremos o método de Sub e Super Soluções, Teoria do Grau Topológico, argumentos de simetria e métodos variacionais.