Our discrete recurrence relation solutions for impact into layered media, generalize our recent analytical results concerning impact into homogeneous, single-layer targets, as well as prior discrete recurrence relation solutions for stress wave propagation in Goupillaud-type layered elastic media.
In addition, Zhu and Yin [19] derived the stochastic boundedness and positive recurrence of the solution.
To reduce the likelihood of recurrence, 70% phenol solution was used as an adjuvant.
However, theaforementioned quasi-metric fixed-point technique was only able to provide the existenceand uniqueness of solution to recurrence equations and not the complexity class.
Consequently, Theorem 1 guarantees the existence of aunique fixed point f T of Φ T and, thus, the existence and uniqueness of the solution tothe recurrence equation (2).
In order to simplify the analysis for the title problem, the fundamental solutions and recurrence formulas are developed.
The terms of the second-order linear recurrence (R 6,-1,1,6 R 6,-1,1,6solutions of the equareon (z^{2}-8y^{2}=1) for some intheer z.
The distributions of temperature, displacements and stresses in the arch were got by substituting the unknown coefficients back to the recurrence formulae and the solutions for every layer.
For the case d n (x e n (x) > 0, sequences of upper and lower bounds can be obtained by iterating the recurrence relation; for minimal solutions of the recurrence these are convergent sequences.
Using the fundamental solutions and recurrence formulas developed in this paper, the mode shape function of vibration of a non-uniform beam with an arbitrary number of cracks and concentrated masses can be easily determined.
Further iterations are possible and this gives a convergent sequence of upper and lower bounds under the conditions of Theorems 5 and 6 and provided that Perron-Kreuser theorem holds (which implies that the recurrence admits a minimal solutions).
