Fixed point theory in Fréchet spaces for Volterra type operators

New fixed point theorems for maps (single and multivalued) between Fréchet spaces are presented. The proof relies on fixed point theory in Banach spaces and viewing a Fréchet space as the projective limit of a sequence of Banach spaces.     

Fixed point theorem for generalized Φ -pseudocontractive mappings

Let E$E$ be a real Banach space, C$C$ be a nonempty closed convex subset of E$E$ and T:C→C$T:C\to C$ be a continuous generalized Φ$\Phi$-pseudocontractive mapping. It is proved that T$T$ has a unique fixed point in C$C$.     

Brill–Noether theory for cyclic covers

We recall that the Brill–Noether Theorem gives necessary and sufficient conditions for the existence of a ${\mathfrak{g}}_d^r$. Here we consider a general n-fold, étale, cyclic cover $p:\stackrel{?}{C}\to C$ of a curve C of genus g and investigate for which numbers $r,d$ a ${\mathfrak{g}}_d^r$ exists on $\stackrel{?}{C}$. For $r=1$ this means computing the gonality of $\stackrel{?}{C}$. Using degeneration to a special singular example (containing a Castelnuovo canonical curve) and the theory of limit linear series for tree-like curves we show that the Plücker formula yields a necessary condition for the existence of a ${\mathfrak{g}}_d^r$ which is only slightly weaker than the sufficient condition given by the results of Laksov and Kleimann [24], for all $n,r,d$.     

Fixed point sets for abstract volterra operators on fréchet spaces

In this present article the topological of the solution ser for abstract Volterra equations is studied both in Banach spaces and in Fréchet spaces. It is shown that the solution set for certain nonlinear abstract Volterra equations in the Fréchet spaces C[0,∞) and L^p _loc[0,∞) (l≤p≤∞) are R_δ sets. Applications of the main results to nonlinear classical integral equations are given     

Continuous Selections and Fixed Points for φ-maps with Their Applications to Section Problems

The concept of finitely continuous topological space is introduced andthe basic properties of the space are given.Several continuous selection theorems and fixed point theorems for φ-maps are establish...     

A critical remark on “Fixed point theorems for occasionally weakly compatible mappings”

In the present paper, we show that under contractive conditions, the existence of a common fixed point and occasional weak compatibility are equivalent conditions. We also show that contractive conditions employed by Jungck and Rhoades [Fixed point theorems for occasionally weakly compatible mappings, Fixed Point Theory 7(2) (2006) 287–296; Fixed Point Theory 9 (2008) 383–384 (erratum)] do not provide a nontrivial setting for the application of occasional weak compatible mappings. Finally, we improve the results of Jungck and Rhoades by employing a proper setting.     

Fixed point theorems for Ćirić-type strong almost contractions on partial metric spaces

In the present paper we prove some fixed point theorems for ?iri?-type strong almost contractions on partial metric spaces. We also give an illustrative example.     

The common fixed point of single-valued generalized φf-weakly contractive mappings

Fixed point and coincidence results are presented for single-valued generalized ${\phi }_f$-weakly contractive mappings on complete metric spaces $(X,d)$, where $\phi :[0,\infty )\to [0,\infty )$ is a lower semicontinuous function with $\phi \left(0\right)=0$ and $\phi \left(t\right)>0$ for all $t>0$ and $f:E\to X$ is a function such that $E?X$ is nonempty and closed. Our results extend previous results given by Rhoades (2001) [1] and by Zhang and Song (2009) [2].     

A Remark on the Fixed point Theorem of the Expansion Mappings

Let X be a Complete metric space with distance function d, anp T a mapping from X into X. T is said to be a first type of the expansion mapping, if it satisfies the following condition     

Blow-up analysis for Boltzmann–Poisson equation in Onsager's theory for point vortices with multi-intensities

In this paper we consider the minimizing sequence for some energy functional of an elliptic equation associated with the mean field limit of the point vortex distribution one-sided Borel probability measure. If such a sequence blows up, we derive some estimate which is related to the behavior of solution near the blow-up point. Moreover, we study the two-intensities case to consider the sufficient condition for this estimate. Our main results are new for the standard mean field equation as well.     

Fixed point results for mappings satisfying (ψ φ)-weakly contractive condition in ordered partial metric spaces

In [18], Matthews introduced a new class of metric spaces, that is, the concept of partial metric spaces, or equivalently, weightable quasi-metrics, are investigated to generalize metric spaces (X, d), to develop and to introduce a new fixed point theory. In partial metric spaces, the self-distance for any point need not be equal to zero. In this paper, we study some results for single map satisfying (ψ,φ)-weakly contractive condition in partial metric spaces endowed with partial order. An example is given to support the useability of our results.     

The product formula for the topological degree of strict γ-contractions

Leray and Schauder [8] have defined a topological degree for mappings in Banach spaces of the form I-f, where f is compact. A product formula for this degree, i.e. a formula for the degree of the composed map (I-f)(I-g), has been given by Nagumo [10]. In recent years, this degree has been extended to mappings I-f, where f is a strict -contraction or -condensing with respect to a rather general measure of noncompactness . It has been shown that all the properties of the Leray-Schauder degree are still valid (see [11] and [12]), while the validity of the product formula remained an open question. Nussbaum [11] has announced this formula for the special case of ₁ -spaces and mappings f that are strict - and ß-contractions, simultaneously. Fenske [5] has shown that this formula holds in every Banach space provided f is a differentiable strict -contraction. In this note we shall establish this formula for an arbitrary Banach space and an arbitrary strict -contraction.