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—Many products & systems age, wear, or degrade over time before they fail or break down. Thus, in many engineering reliability experiments, measures of degradation or wear toward failure can often be observed over a period of time before failure occurs. Because the degradation values provide additional information beyond that provided by the failure(More)
—Based on a generalized cumulative damage approach with a stochastic process describing initial damage for a material specimen, a broad class of statistical models for material strength is developed. Plausible choices of stochastic processes for the initial damage include Brownian motion, geometric Brownian motion, and the gamma process; and additive &(More)
A stochastic integral equation of the Volterra type in the form $$x\left( {t;\omega } \right) = h\left( {t;\omega } \right) + \int\limits_0^t k \left( {t,\tau ;\omega } \right)f\left( {\tau ,x\left( {\tau ;\omega } \right)} \right)d\tau , t \geqslant 0,$$ , is formulated in Hilbert space, whereω ∈ Ω, the supporting set of a complete probability measure(More)
Tsokos [12] showed the existence of a unique random solution of the random Volterra integral equation (*)x(t; ω) = h(t; ω) + ∫ o t k(t, τ; ω)f(τ, x(τ; ω)) dτ, whereω ∈ Ω, the supporting set of a probability measure space (Ω,A, P). It was required thatf must satisfy a Lipschitz condition in a certain subset of a Banach space. By using an extension of(More)
A nonlinear stochastic integral equation of the Hammerstein type in the formx(t; ω) = h(t, x(t; ω)) + ∫ s k(t, s; ω)f(s, x(s; ω); ω)dμ(s) is studied wheret ∈ S, a measure space with certain properties,ω ∈ Ω, the supporting set of a probability measure space (Ω,A, P), and the integral is a Bochner integral. A random solution of the equation is defined to be(More)