However, we need a sufficient condition for (1.1) having a bounded nonoscillatory solution.
To prove the next theorem, we need a regularity condition for v such that for 1 ≤ p < ∞, we assume that for a measurable set E ⊂ Q and for σ(E) ≤ θσ(Q) with 0 ≤ θ ≤ 1, we get v ( E ) ≤ 1 - ( 1 - θ ) p v ( Q ).
Since the sets are then bounded, Proposition 4 becomes obsolete as we do not need a condition on the dual variables μ for the dual function to be finite.
For that, we need an additional condition.
For the existence of a common fixed point of four self-mappings of a symmetric space, we need an additional condition, so-called weak compatibility.
we need an additional condition (see [18]) that begin{aligned} -inftyle int_{1}^{+infty}frac{s^{2}b s -kappa}{s},mathrm{+infty}frac{s^{2}b s -kappa}{s}bda_{1}< kappa< +infty.
For the uniqueness, we need an additional condition: (U) For all x, y ∈ Fix ( T ), we have α ( x, y ) ≥ 1, where Fix ( T ) denotes the set of fixed points of T. .
In order to ensure the uniqueness of a fixed point of a generalized α-ψ-contractive mapping of integral type II, we need an additional condition (U) defined in the previous section.
Therefore, we need an additional condition to obtain balance between collective production and consumption rates for S23.
For equation (7) to be consistently estimated by OLS, we need a strict exogeneity condition: Eleft[varDelta {varepsilon}_{is}left| ln left({d}_{it}right),{gamma}_t,{mathit{mathsf{g}}}_iright.right]=0kern0.5em forall s,t.
