Observe that whenever (n=1) and Φ is the identity function in the statements of Theorem 11 and Corollary 12 we retrieve as a particular case Theorem 3.
To attain this goal, we choose (u_{n}^{pdelta-p+1}) as a test function in (2.2) as in the statement of Lemma 3.2, where (frac{p-1+alpha^{p} leqslantdelta<1); however, (nabla u_{n}^{pdelta-p+1}) will be singular at (u_{n}=0).
With B j being fixed we now build the function v j as required in the statement of the lemma.
It is not hard to see that for the function (Phi_{2}) assertions (1) and (2) hold in the statement of Theorem 7 and, thus, it is a partial metric aggregation function.
In order to deal with differential operators having superlinear growth at infinity, we need to strengthen condition (2.5), taking a Nagumo function with sublinear growth at infinity, as in the statement of the following result.
To conclude the proof, we suppose that, contrary to our claim, there exists a constant 0 < c 2 < L ∗ ⋅ Γ Γ ( − β + m ) such that the inequality [ ∑ n = 1 ∞ n ( p − 1 ) ( 1 − p q α 2 ) ( ∫ 0 ∞ K ( x, n ) f ( x ) d x ) p ] 1 p < c 2 [ ∫ 0 ∞ x p ( m − β ) − 1 ( D ± m f ( x ) ) p d x ] 1 p. holds for all non-negative functions f ∈ Λ ± m, as in the statement of theorem.
Uzer [3] extended the multiplicative calculus to the complex-valued functions, was interested in the statements of some fundamental theorems and concepts of multiplicative complex calculus, and demonstrated some analogies between the multiplicative complex calculus and the classical calculus by theoretical and numerical examples.
Quite recently, Uzer [3] has extended the multiplicative calculus to the complex valued functions and was interested in the statements of some fundamental theorems and concepts of multiplicative complex calculus, and demonstrated some analogies between the multiplicative complex calculus and classical calculus by theoretical and numerical examples.
It functions in the same way as the statement of interest prescribed for an admiralty proceeding by Rule C 6), and is not related to the distinctive meaning of "claim" in admiralty practice.
The statement of a Dirichlet type problem in the class of smooth functions is given and, subject to the type of degeneracy, the exact classical solutions are obtained.
