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But the research of periodic solutions on time scales has not got much attention, see [12 16].
At the same time, interest in obtaining the solutions on time scales has been on-going for several years.
The methods usually used to explore the existence of periodic solutions on time scales are many fixed point theory, upper and lower solutions, Masseras theorem, and so on.
In Figures 5 and 6, we plot the solutions on time scales T 1, T 2 and T 5 introduced in Example 4.10. Figure 5 y RL ( t ) for α = 1 3, y − 2 3 = 3 4. Figure 6 y C ( t ) for α = 1 3, y 0 = 3 4. Fractional calculus on time scales is a new topic providing many directions for further research.
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Four-week old seedlings were watered with 250 mM saline solution (one time on five days).
Finally, we present an asymptotic behavior of solutions depending on time.
Open image in new window Fig. 4 Comparison of two methods to evaluate solutions based on time and sample.
RL solutions based on time difference algorithms such as Q-learning [28] or some actor critic approaches [29] can be proven to converge to the optimal policy when only one agent/decision maker is present in the scenario.
But the now available reduced-order numerical methods as stated above were built by means of the POD basis formulated by the classical numerical solutions on all time nodes, before calculating the reduced-order numerical solutions on the same time nodes, which are some vain reduplicated computations.
Shadowing Methods allow us to estimate differences between exact and approximate solutions on infinite time intervals and to understand the influence of error terms.
Under Hypothesis 2.1 the systems (1) and (4) have unique pathwise solutions on all time interval (0leq tleq T).
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