In this paper we develop a new theory for the existence, localization and multiplicity of positive solutions for a class of non-variational, quasilinear, elliptic systems. In order to do this, we provide a fairly general abstract framework for the existence of fixed points of nonlinear operators acting on cones that satisfy an inequality of Harnack type. Our methodology relies on fixed point index theory. We also provide a non-existence result and an example to illustrate the theory.

The paper is devoted to the existence of positive solutions of nonlinear elliptic equations with p-Laplacian. We provide a general topological degree that detects solutions of the problemwhere A : X ⊃ D(A) → X * is a maximal monotone operator in a Banach space X and F : M → X * is a continuous mapping defined on a closed convex cone M ⊂ X. Next, we apply this general framework to a class of partial differential equations with p-Laplacian under Dirichlet boundary conditions. In the paper we employ general ideas from [5], where a setting suitable for the one dimensional p-Laplacian was introduced.

Abstract. In this paper we study the existence, localization and multiplicity of positive solutions for parabolic systems with nonlocal initial conditions. In order to do this, we extend an abstract theory that was recently developed by the authors jointly with Radu Precup, related to the existence of fixed points of nonlinear operators satisfying some upper and lower bounds. Our main tool is the Granas fixed point index theory. We also provide a non-existence result and some examples to illustrate our theory.

We study the existence of constrained fixed points of contractions in arbitrary complete metric spaces from a global and local point of view. In particular, we provide generalizations of results due to Lim, Downing and Kirk and others. Some aspects of the topological transversality in the spirit of Frigon and Granas of contractions under constraints are also considered.

We deal with one dimensional p-Laplace equation of the formunder Dirichlet boundary condition, where p > 2 and f :