A New Equilibrium Version of Ekeland’s Variational Principle and Its Applications
Round 1
Reviewer 1 Report
Despite is not quite large, the extension of the Ekeland variational principle prodived by the authors is interesting. The paper can be published. My only remark and suggestion is related to the assertion from the abstract that the paper "establishes a new equilibrium form of Ekeland’s variational principle".
It is stronger that neccesary and I reccommend the formulation used in section 3, " a new equilibrium version of EVP".
Author Response
Accroding to the reviewer's suggestions, we change that the paper "establishes a new equilibrium form of Ekeland’s variational principle" into "establishes a new equilibrium version of Ekeland’s variational principle" in the abstract. In addition, some English expressions have been revised.
Thanks a lot.
Reviewer 2 Report
1-Theorem 1.1. It is not explained if x0 is an arbitrary point in M or the initial condition of a sequence.
2- Theorem 1.1, line 8: and every lamda>0-> and, for every lamda>0
3- Theorem 1.1, line 7, the generic notation for the "infimum", without picking up points x in M, although can be considered correct, does not correspond to the notations for" infimum "then used in other parts of the paper.
4-Line before (14): Conversely->Contrarily
5-Proof of Theorem 4.2: The starting point is unclear. It is not obvious that, according to Corollary 2.2, there exists such a convergent sequence to a limit x/epsilon. Corollary 3.2 does not mention such a convergence of a sequence.
6-Theorem 4.2: After the formula, Where-> where
7-In Theorem 3.1, the set C is closed while in Theorem 4.1, C is compact. Those assumptions on C should be clearly addressed through the respective proofs since it is now not clear where they are used.
Author Response
We are grateful to you for your helpful suggestions.
We make the following corrections.
1.We delete the preliminaries section.
2.We modify the expressions one by one according to your suggestions.
Q1- Theorem 1.1. It is not explained if x0 is an arbitrary point in M or the initial condition of a sequence.
A1-Theorem 1.1 is a result of Professor Zhong(see reference [3]), we cite it here to explain the development and generaliztion of EVP. By the proof of theorem1.1, we see that x_0 is a fixed element in X, and it is also the initial point of the iterative sequence {x_n}.
Q2- Theorem 1.1, line 8: and every lamda>0-> and, for every lamda>0.
A2- Yes, we do.
Q3- Theorem 1.1, line 7, the generic notation for the "infimum", without picking up points x in M, although can be considered correct, does not correspond to the notations for" infimum "then used in other parts of the paper.
A3-Yes, we do.
Q4-Line before (14): Conversely->Contrarily
A4-Yes, we do.
Q5-Proof of Theorem 4.2: The starting point is unclear. It is not obvious that, according to Corollary 2.2, there exists such a convergent sequence to a limit x/epsilon. Corollary 3.2 does not mention such a convergence of a sequence.
A5- We correct it.
Q6-Theorem 4.2: After the formula, Where-> where
A6- Yes, we do.
Q7-In Theorem 3.1, the set C is closed while in Theorem 4.1, C is compact. Those assumptions on C should be clearly addressed through the respective proofs since it is now not clear where they are used.
A7-The compactness of $C$ is used to construct a convergent subsequence {x_{n_k}} of {x_n}.
Thank You
Reviewer 3 Report
In this work, the authors prove a new equilibrium form of Ekeland's variational principle. As the authors state, the paper is a modification of previous results available in the literature. Some application to the existence of equilibrium points for binary functions and the existence of fixed points for nonlinear mappings are provided.
MAJOR COMMENTS
As I pointed out above, the paper is an incremental paper with humble results. A lot of space is devoted to provide background and various elementary and well known concepts. I find no need, for example, to provided the definition of a metric space when it is a standard concept. The authors also 'waste space' by recalling the definition of convergent sequence, Cauchy sequence, diameter of a set in a metric space, etc. The only new result is Theorem 3.1, which is a humble improvement of the variational principle. Here, the authors provide all the details of the proof, making it long while it is really straightforward.
In summary, I recommend this paper for REJECTION in view that it is an incremental work.
Author Response
We are grateful to you for your helpful suggestions.
According to your comments, we make corrections as follows:
1.We deleted the preliminaries, since it contains elementary and well known concepts.
2.We deleted the part of the proof （ Theorem 2.1）that had the same conclusion as the previous literature, and focused on the part of the conclusion that was improved.Thank You
Author Response File: 
Author Response.pdf
Round 2
Reviewer 3 Report
The manuscript has been improved. I recommend it for ACCEPTANCE.
