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Topological degree theory in fuzzy metric spaces

M.H.M. Rashid (2019)

Archivum Mathematicum

The aim of this paper is to modify the theory to fuzzy metric spaces, a natural extension of probabilistic ones. More precisely, the modification concerns fuzzily normed linear spaces, and, after defining a fuzzy concept of completeness, fuzzy Banach spaces. After discussing some properties of mappings with compact images, we define the (Leray-Schauder) degree by a sort of colimit extension of (already assumed) finite dimensional ones. Then, several properties of thus defined concept are proved....

Truncated spectral regularization for an ill-posed non-linear parabolic problem

Ajoy Jana, M. Thamban Nair (2019)

Czechoslovak Mathematical Journal

It is known that the nonlinear nonhomogeneous backward Cauchy problem u t ( t ) + A u ( t ) = f ( t , u ( t ) ) , 0 ≤ t < τ with u ( τ ) = φ , where A is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on φ and f , that a solution of the above problem satisfies an integral equation involving the spectral representation of A , which is also ill-posed. Spectral truncation is used...

Two generic results in fixed point theory

Simeon Reich, Alexander J. Zaslavski (2007)

Banach Center Publications

We give two examples of the generic approach to fixed point theory. The first example is concerned with the asymptotic behavior of infinite products of nonexpansive mappings in Banach spaces and the second with the existence and stability of fixed points of continuous mappings in finite-dimensional Euclidean spaces.

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