A pre-order principle and set-valued Ekeland variational principle

Qiu, Jinghui

doi:10.1016/j.jmaa.2014.05.027

Cited by 31 publications

(29 citation statements)

References 41 publications

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“…Using Remark 3.3, (A Γ , F ) verifies condition (C'1) if and only if S(u) := {x ∈ X | x u} is -lower closed for every u ∈ X, if and only if (X, d, ) verifies (A'1). This shows that condition (C'1) extends the dynamic closedness of a set-valued mapping as defined in[25] and elsewhere. Also notice that (A Γ , F ) verifies condition (Ca) if and only if (X, d, ) verifies (Aa).…”

mentioning

confidence: 61%

“…(ii) Observe that Theorem 5.4 extends [25,Cor. 3.4] in the case in which Λ is a singleton because condition (C'1) is verified when (X, d) is complete and the sets S(x) are dynamically closed for x ∈ S(x 0 ) (assumed to be nonempty).…”

Section: Ekeland's Variational Principles Of Ha-hamel-löhne's Typementioning

confidence: 95%

“…Proof. The implications in (26), (27) and (28) are almost obvious. The fact that the implication (C'1) ⇒ (C1) in general is not true when (X, d) is not complete is shown in Example 3.6 (a).…”

Section: The Brezis-browder Principlementioning

confidence: 96%

“…(b) Theorems 2.1 in [25], [26] and [27] follow applying Theorem 3.4 for W := S(x 1 ) with As O. Cârjȃ mentions in [2, p. 120], the use of strict monotonicity of the sequence (S(x n )) in the condition corresponding to (Ab) of the usual version of the BB-principle is due to Corneliu Ursescu.…”

Section: The Brezis-browder Principlementioning

confidence: 99%

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Minimal element theorems revisited

Hamel

Zălinescu

2020

Journal of Mathematical Analysis and Applications

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Starting with the Brezis-Browder principle, we give stronger versions of many variational principles and minimal element theorems which appeared in the recent literature. Relationships among the elements of different sets of assumptions are discussed and clarified, i.e., assumptions to the metric structure of the underlying space and boundedness assumptions. New results involving set-valued maps and the increasingly popular set relations are obtained along the way. applications consist in a mere check if the order relation in question satisfies the required assumptions which links it to the metric structure. Therefore, the mentioned results can be proven without involving the Brezis-Browder principle as shown in the alternative proof of Theorem 3.7 below.The second line of research is concerned with order relations on arbitrary sets (without a metric structure, for example). Of course, alternative requirements have to be added. A prominent result of this type is the Brezis-Browder principle [1] (BB-principle) where the existence of a real-valued function is assumed which is bounded from below and increasing wrt the order relation. Further examples for such results can be found in [25,27] and also in [31]. Of course, BB-type theorems can also be used to obtain the results on metric spaces discussed in the previous paragraph with a suitable monotone function; the proof of the BB-principle also involves a countable induction argument (see the first proof of Theorem 3.2 below).A third goal is to lift the results from order relations on a set X to such relations on a product set X × Z. Results in this direction have been obtained first by Göpfert and Tammer (see [5, Section 3.10] and the references 140-144 therein) and are usually called minimal element theorems.In this note, all three of the above lines are followed providing very general results and discussing the relationships between different sets of assumptions in detail. Consequently, most relevant results in the literature are obtained as special cases. In particular, it is shown that variational principles and minimal element theorems involving (very general) set relations can be obtained (see, for example, Corollary 3.14 below). Such results can be found in [8] for complete metric spaces (manuscript version [7] from 2002) and more general ones in [12] (preprint version [11] from 2002), the latter reference already giving a proof based on the BB-principle and nonlinear scalarization.

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mentioning

confidence: 61%

Section: Ekeland's Variational Principles Of Ha-hamel-löhne's Typementioning

confidence: 95%

Section: The Brezis-browder Principlementioning

confidence: 96%

Section: The Brezis-browder Principlementioning

confidence: 99%

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Minimal element theorems revisited

Hamel

Zălinescu

2020

Journal of Mathematical Analysis and Applications

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“…In the last four decades, the famous EVP emerged as one of the most important results of nonlinear analysis and it has significant applications in optimization, optimal control theory, game theory, fixed point theory, nonlinear equations, dynamical systems, etc. Motivated by its wide applications, many authors have been interested in extending EVP to the case with vector-valued maps or set-valued maps, see for example [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and the references therein. Motivated by its wide applications, many authors have been interested in extending EVP to the case with vector-valued maps or set-valued maps, see for example [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and the references therein.…”

Section: Introductionmentioning

confidence: 99%