Nabla fractional boundary value problem with a non-local boundary condition

In this work, we deal with the following two-point boundary value problem for a finite fractional nabla difference equation with non-local boundary condition: (−(∇ξρ(e) u(z) = p(z, u(z)), z ∈ Nfe+2, u(e) = g(u), u(f) = 0. Here e, f ∈ R, with f −e ∈ N3, 1 < ξ < 2, p : Nfe+2 ×R → R is a continuous function, the functional g ∈ C[Nfe → R] and ∇ξρ(e) denotes the ξth- order Riemann–Liouville backward (nabla) difference operator. First, we derive the associated Green’s function and some of its properties. Using the Guo–Krasnoselskii fixed point theorem on a suitable cone and under appropriate conditions on the non-linear part of the difference equation, we establish sufficient conditions for the existence of at least one positive solution to the boundary value problem. Next, we discuss the uniqueness of the solution to the considered problem. For this purpose, we use Brouwer and Banach fixed point theorem respectively. Finally, we provide an example to illustrate the applicability of established results.Publisher's Versio

O grao topolóxico de Leray-Schauder e aplicacións ás ecuacións diferenciais

Traballo Fin de Grao en Matemáticas. Curso 2021-2022Na primeira década do século XX comezan os primeiros estudos sobre o grao topolóxico, unha ferramenta de utilidade na topoloxía alxébrica e na análise funcional non linear. No presente traballo, desenvolvemos unha pormenorizada introdución á teoría do grao. Partindo de espazos de dimensión finita, definimos o grao de Brouwer para funcións continuamente diferenciables e, posteriormente, para funcións continuas. Presentamos as propiedades máis importantes do grao, que nos permiten, entre outras aplicacións, garantir a existencia de solución dunha ecuación dada. Partindo do grao, probamos teoremas clásicos como o de punto fixo de Brouwer ou o da bóla peluda. En espazos de dimensión infinita, os resultados relativos ó grao de Brouwer non son certos, en xeral. Por este motivo, cómpre redefinir o grao, coa limitación de facelo para unha clase máis restritiva de funcións, as perturbacións compactas da identidade. Construímos, partindo do grao de Brouwer, o grao de Leray-Schauder. Destacamos algunhas das propiedades máis interesantes e xeneralizamos os teoremas de punto fixo para espazos de dimensión infinita. Existen aplicacións da teoría do grao en moitos eidos das matemáticas. Facendo uso do Teorema de punto fixo de Schauder e das propiedades do grao, probamos a existencia de solución local dun problema de valor inicial e a conexidade do seu espazo de soluciónsIn the first decade of the twentieth century began the first studies on the topological degree, a useful tool in algebraic topology and in nonlinear functional analysis. In the present project, we have developed a detailed introduction to the degree theory. Starting from finite dimensional spaces, we define the Brouwer degree for continuously diferentiable functions and subsequently for continuous functions. We present the most important properties of the degree, which allow us, among other applications, to ensure existence of solution of a given equation. Based on the degree, we prove several classic theorems such us the Brouwer fixed point or the one of the hairy ball. In infinite dimensional spaces, the results relating to the Brouwer degree are not true, in general. Therefore it is necessary to redefine the degree, with the limitation of doing so for a more restrictive class of functions, the compact perturbations of the identity. Using the Brouwer degree, we build the Leray-Schauder degree. We highlight some of the most interesting properties and we generalize the xed point theorems for in nite dimensional spaces. Degree theory can be applied in many mathematical elds. We prove, using the Schauder xed point theorem and the degree properties, the existence of local solutions of an initial value problem and the connection of its solution se

The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions

We show that the widely used homotopy method for solving fixpoint problems, as well as the Harsanyi-Selten equilibrium selection process for games, are PSPACE-complete to implement. Extending our result for the Harsanyi-Selten process, we show that several other homotopy-based algorithms for finding equilibria of games are also PSPACE-complete to implement. A further application of our techniques yields the result that it is PSPACE-complete to compute any of the equilibria that could be found via the classical Lemke-Howson algorithm, a complexity-theoretic strengthening of the result in [Savani and von Stengel]. These results show that our techniques can be widely applied and suggest that the PSPACE-completeness of implementing homotopy methods is a general principle.Comment: 23 pages, 1 figure; to appear in FOCS 2011 conferenc

Constant Rank Bimatrix Games are PPAD-hard

The rank of a bimatrix game (A,B) is defined as rank(A+B). Computing a Nash equilibrium (NE) of a rank-$0$, i.e., zero-sum game is equivalent to linear programming (von Neumann'28, Dantzig'51). In 2005, Kannan and Theobald gave an FPTAS for constant rank games, and asked if there exists a polynomial time algorithm to compute an exact NE. Adsul et al. (2011) answered this question affirmatively for rank-$1$ games, leaving rank-2 and beyond unresolved. In this paper we show that NE computation in games with rank $\ge 3$, is PPAD-hard, settling a decade long open problem. Interestingly, this is the first instance that a problem with an FPTAS turns out to be PPAD-hard. Our reduction bypasses graphical games and game gadgets, and provides a simpler proof of PPAD-hardness for NE computation in bimatrix games. In addition, we get: * An equivalence between 2D-Linear-FIXP and PPAD, improving a result by Etessami and Yannakakis (2007) on equivalence between Linear-FIXP and PPAD. * NE computation in a bimatrix game with convex set of Nash equilibria is as hard as solving a simple stochastic game. * Computing a symmetric NE of a symmetric bimatrix game with rank $\ge 6$ is PPAD-hard. * Computing a (1/poly(n))-approximate fixed-point of a (Linear-FIXP) piecewise-linear function is PPAD-hard. The status of rank-$2$ games remains unresolved