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Decomposable measures and nonlinear equations. (English) Zbl 0934.28015

A closed interval in the extended real line is considered with two binary operations \(\oplus\) and \(\odot\) (pseudo-addition and pseudo-multiplication) having some properties analogous to the properties of the usual addition and multiplication [see E. Pap: “Null-additive set functions” (1995; Zbl 0856.28001)]. A \(\oplus\)-decomposable measure is characterized by the equality \(m(A\cup B)= m(A)+ m(B)\) for disjoint \(A\), \(B\). Then the pseudo-integral is defined and a special case of \(\oplus\) is considered, where \(u\oplus v= g^{-1}(g(u)+ g(v))\). In this case the so-called \(g\)-derivative is introduced and the apparatus is applied for solving ordinary differential equations and nonlinear difference equations. The pseudo-Laplace transform is used for optimization and the Burgers partial differential equation is solved.

MSC:

28E10 Fuzzy measure theory
26E50 Fuzzy real analysis

Citations:

Zbl 0856.28001
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References:

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