On a local degree for a class of multi-valued vector fields in infinite dimensional Banach spaces.
In this paper we mainly introduce a min-max procedure to prove the existence of positive solutions for certain semilinear elliptic equations in RN.
Let be a uniformly convex Banach space, , a nonexpansive map, and a closed bounded subset such that . If (1) is weakly inward and is star-shaped or (2) satisfies the Leray-Schauder boundary condition, then has a fixed point in . This is closely related to a problem of Gulevich [Gu]. Some of our main results are generalizations of theorems due to Kirk and Ray [KR] and others.
The paper is about a sub-supersolution method for the prescribed mean curvature problem. We formulate the problem as a variational inequality and propose appropriate concepts of sub- and supersolutions for such inequality. Existence and enclosure results for solutions and extremal solutions between sub- and supersolutions are established.
The aim of this paper is to give the proofs of those results that in [4] were only announced, and, at the same time, to propose some possible developments, indicating some of the most significant open problems.
In this paper, we are interested in the dynamic evolution of an elastic body, acted by resistance forces depending also on the displacements. We put the mechanical problem into an abstract functional framework, involving a second order nonlinear evolution equation with initial conditions. After specifying convenient hypotheses on the data, we prove an existence and uniqueness result. The proof is based on Faedo-Galerkin method.
Let X, Y be two separable F-spaces. Let (Ω,Σ,μ) be a measure space with μ complete, non-atomic and σ-finite. Let be the Nemytskiĭ set-valued operator induced by a sup-measurable set-valued function . It is shown that if maps a modular space into subsets of a modular space , then is automatically modular bounded, i.e. for each set K ⊂ N(L(Ω,Σ,μ;X)) such that we have .