The goal of this publication is to provide basic tools of differential topology to study systems of nonlinear equations and to apply them to the analysis of general equilibrium models with complete and incomplete markets. The main content of general equilibrium analysis is to study existence, (local) uniqueness and efficiency of equilibria. To study existence Differential Topology and General Equilibrium with Complete and Incomplete Markets combines two features. First order conditions (of agents’ maximization problems) and market clearing conditions, instead of aggregate excess demand function. Then, the application to that “extended system” of a homotopy argument, which is stated and proved in a relatively elementary manner. Local uniqueness and smooth dependence of the endogenous variables from the exogenous ones are studied using a version of a so-called parametric transversality theorem. In a standard general equilibrium model, all equilibria are efficient, but that is not the case if some imperfection, like incomplete markets, asymmetric information, strategic interaction, is added. Then, for almost all economies, equilibria are inefficient, and an outside institution can Pareto Improve upon the market outcome. Those results are proved showing that a well-chosen system of equations has no solutions. \ud \ud The target audience of Differential Topology and General Equilibrium with Complete and Incomplete Markets consists of researchers interested in Economic Theory. The needed background is multivariate analysis, basic linear algebra and basic general topology

In [1] we introduced a concept of orientation and topological degree for nonlinear Fredholm maps between real Banach manifolds. In this paper we study properties of this notion of orientation and we compare it with related results due to Elworthy-Tromba and Fitzpatrick-Pejsachowicz-Rabier.

We define a notion of degree for a class of perturbations of nonlinear Fredholm maps of index zero between infinite-dimensional real Banach spaces. Our notion extends the degree introduced by Nussbaum for locally α-contractive perturbations of the identity, as well as the recent degree for locally compact perturbations of Fredholm maps of index zero defined by the first and third authors

We prove a global bifurcation result for T -periodic solutions of the T -periodic delay differential equation x (t) = λf (t, x(t), x(t − 1)) depending on a real parameter λ 0. The approach is based on the fixed point index theory for maps on ANRs.

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We prove a global bifurcation result for T -periodic solutions of the delay T -periodic differential equation x (t) = λf (t, x(t), x(t − 1)) on a smooth manifold (λ is a nonnegative parameter). The approach is based on the asymptotic fixed point index theory for C 1 maps due to Eells-Fournier and Nussbaum. As an application, we prove the existence of forced oscillations of motion problems on topologically nontrivial compact constraints. The result is obtained under the assumption that the frictional coefficient is nonzero, and we conjecture that it is still true in the frictionless case.

Let H be a real Hilbert space and denote by S its unit sphere. Consider the nonlinear eigenvalue problem Lx + εN (x) = λx, where ε, λ ∈ R, L : H → H is a bounded self-adjoint (linear) operator with nontrivial kernel and closed image, and N : H → H is a (possibly) nonlinear perturbation term. A unit eigenvectorx ∈ S ∩ Ker L of L (corresponding to the eigenvalue λ = 0) is said to be persistent if it is close to solutions x ∈ S of the above equation for small values of the parameters ε = 0 and λ. We give an affirmative answer to a conjecture formulated by R. Chiappinelli and the last two authors in an article published in 2008. Namely, we prove that if N is Lipschitz continuous and the eigenvalue λ = 0 has odd multiplicity, then the sphere S ∩Ker L contains at least one persistent eigenvector. We provide examples in which our results apply, as well as examples showing that if the dimension of Ker L is even, then the persistence phenomenon may not occur.where ε and λ are real parameters. Assume, for the moment, that the eigenvalue λ 0 is simple. Thus, when ε = 0 and λ = λ 0 , one gets exactly two vectors in H satisfying the above problem; namely, the two unit eigenvectors of L corresponding to the eigenvalue λ 0 . Denote by x 1 and x 2 these vectors. Considering small values of ε, R. Chiappinelli in [11] obtained a sort of persistence result of these eigenvectors as well as of the eigenvalue λ 0 . More

Abstract. We give an existence result for a periodic boundary value problem involving mean curvaturelike operators. Following a recent work of R. Manásevich and J. Mawhin, we use an approach based on the Leray-Schauder degree.