Erratum to “Existence of Solutions for Generalized Vector Quasi-Equilibrium Problems by Scalarization Method with Applications”

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We found a mistake in Example  15 in the published paper after its publication. The corrected version is as follows.

Example 15. Let 𝑋=𝑌=𝑍=ℝ, 𝐸=ℝ₊, and 𝐹=[−10,10]⊂𝑍, and let 𝑒(𝑥)=1, for all 𝑥∈𝐸, and 𝐶(𝑥)=ℝ₊, for all 𝑥∈𝐸.
(1) Define 𝐺(𝑢)=[](],{0}[].𝑢,𝑢+1,𝑢∈0,10,𝑢∈−10,0 (17) Evidently, 𝐺 is not usc on 𝐹. After simply calculating, 𝜉_𝐺(𝑥,𝑢)=𝑢+1,𝑥∈𝐸,0<𝑢≤10,0,𝑥∈𝐸,−10≤𝑢≤0. (18)𝜉_𝐺 is not usc on 𝐸×𝐹 due to the fact that {(𝑥,𝑢)∈𝐸×𝐹∶𝜉(𝑥,𝑢)≥1}=ℝ₊×(0,10] is not closed.
(2) Consider the following mapping: 𝐺(𝑢)=[],{0}[).𝑢,𝑢+1,𝑢∈0,10,𝑢∈−10,0 (19) Obviously, 𝐺 is not lsc on 𝐹. Also, 𝜉_𝐺 fails to be lsc on 𝐸×𝐹, where 𝜉_𝐺(𝑥,𝑢)=𝑢+1,𝑥∈𝐸,0≤𝑢≤10,0,𝑥∈𝐸,−10≤𝑢<0. (20)

Furthermore, the following misprints should be noted.

With regard to the conceptions of 𝐷-closeness and 𝐷-bounded, in page 1, the correct text should be: 𝐴 is called 𝐷-closed [11] if 𝐴+𝑐𝑙𝐷 is closed and 𝐷-bounded [11] if for each neighborhood 𝑈 of zero in 𝑌, there exists 𝜆>0 such that 𝐴⊂𝜆𝑈+𝐷.

In page 2, left, line 5, the correct text should be: “Incidentally, every TVS such that any singleton is closed is Hausdorff (see [12]).”

In page 2, right, line −18, the correct text should be: “From now on, unless otherwise specified, let 𝑋, 𝑌 and 𝑍 be Hausdorff real TVSs and …"