Utilizing Remark 3.1, we derive the claim.
We shall utilize Remark 2.15 in our examples below.
The following lemma utilizes some remarks made in Sect. 3 comparing JWBG2 and JWBG1 models to transfer stability results for JWBG2 established in Olaizola and Valenciano (2015a) to JWBG1.
Based on these remarks and utilizing the constant (c_K) from (16) we can set begin{aligned} h_j:=frac{1}{4c_Kn}left( frac{a^2j^2}{n^2}right) ^{frac{1}{alpha }-frac{1}{2}}!,quad 1le jle m end{aligned} (19)and choose (N_jle c_1h_j^{1-d}) points ({mathbf{y}_{i,quad K_j, 1le ile N_jle=Y_{n,j}) on (K_j) so that begin{aligned} d(K_j, Y_{n,j})le h_j,quad 1le jle m. end{aligned} (20).
Remark 11 By utilizing the technique of Samet et al. [40], we can obtain corresponding coupled fixed point results from our Theorems 4, 6 and 8.
Remark 4.5 By utilizing the technique of Amini-Harandi [10] and Samet et al. [23], we can obtain corresponding coupled fixed point results from our Theorem 4.1 and Corollaries 4.3 and 4.4 on the basis of the following simple lemma.
This means that (0<(log c_{0}) e^{nbeta_{n}}/n<e^{nbeta_{n}}/ncdot(n alpha_{n}-beta_{n}))to0), an alpha_{n}-beta_{n} (e^{nbeto0{n}}/nto0). Now we utilize Theorem 2.1 (Remand 2.1) to elaborate Example 1.2 again.
Using Theorem 10 and Remark 3 for g ≡ f and utilizing elementary calculations, we obtain the following converse of Jensen's operator inequality.
Using Theorem 6 and Remark 1 for g ≡ f and α = 1 and utilizing elementary calculations, we obtain the following converse of Jensen's inequality.
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