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However, the order of combinations may affect our results (Proposition 2).

Contraction self-mappings can also be 2-cyclic -Kannan self-mappings and vice-versa as addressed in the two following results: Proposition 2.3.

In order to define the family of processes of the equations (2.8), we also need the following results: Proposition 2.8 ([8]).

From Figure 4, it can be observed that BPeMRF and the throughput optimal BP demonstrate very similar behavior with respect to capacity region which is in accordance with the theoretical results (Proposition 1), although BPeMRF runs for a finite number of sweeps (i.e., at most 1,000 sweeps) which implies non-perfect convergence.

Regarding the information diffusion dynamics for class 2, from Fig. 6b it is observed that there is an initial increase which later deflates, due to the decreasing with time user interests, eventually yielding zero number of infected nodes for the simulation results and close to zero number of infected nodes for the numerical results (Proposition 2).

For p ⩾ 1, the same type of results (Propositions 6.5, 6.7) has also been obtained for ℓ p -Tikhonov regularization in [31,50].

Now, we state the main result: Proposition 2.3.

We conclude this section with the proof of the main result Proposition 5.2.

Next, our result, Proposition 2.3, was shown with 3 K 0 replacing K. Therefore, if 3 K 0 < K (see Example 7.3), then our result is tighter.

Based on the feasibility problem (17) we can state the following result: Proposition 1. Assume problem (12) has at least one strictly feasible point.

In fact, regarding alternative strategies, we have the following result: Proposition 1 The big kick strategy always yields at least as many spikes as the reset and kick (and equivalently, the threshold kick) strategy.