EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6846 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Generalised f-Projection Operator over Nonconvex Set Ali Al Tane1,∗, Lee See Keong1 1 School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia. Abstract. This work examines the generalized f -projection operators πf S . By proving the local Lipschitz continuity of the generalized projection operator πf S for S nonempty closed sets that are not necessarily convex, and using convex subdifferential ∂conf , the Fréchet subdifferential ∂F f(x), and the Clarke subdifferential, we extend many properties of πS to πf S . 2020 Mathematics Subject Classifications: 49J52, 49J45 Key Words and Phrases: generalized f -projection, Lipschitz continuity 1. Introduction For X uniformly convex and uniformly smooth Banach spaces, Alber presented the generalized projection operators πS . Additionally, he thoroughly examined their charac- teristics and offered a number of applications for the generalized projections, including approximating the solution to variational-inequalities (see [1]). Li explored the second direction in [2], who expanded the definition of πS to where X is a reflexive Banach space. Also, examined some of its properties and applied them to the solution of variational- inequalities. Since Lie and Alber’s research was based on the ideas that using V -functional, and S is a closed, and convex subset of reflexve Banach space. There have been studies to generalize projection operators by either accepting S as not necessarily convex or by extending V -functional to V f -functional. In [3], K Wu and N Huang presented the gener- alized f -projection operator πf S , which is an extension of the generalized projection opera- tor. It was demonstrated that πf S is well-defined for reflexive Banach spaces by providing certain properties, where they used the FanKKM Theorem to look into the existence of solutions to a few variational-inequality issues as an application of their findings. In their study in [4], M. Bounkhel and R. Al-Yusof used the generalized projection op- erator πS to introduce the new generalized proxmal normal cone in reflexive smooth Ba- nach spaces with S not necessarily convex. However, begin with M. Bounkhel (see [5]) ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6846 Email addresses: alialtane@student.usm.my (A. Al Tane), sklee@usm.my (L. S. Keong) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Al Tane, L. S. Keong / Eur. J. Pure Appl. Math, 18 (4) (2025), 6846 2 of 13 he examined the existence of the generalized projection operator πS for S set that are not necessarily convex. In uniformly convex and smooth Banach spaces, the author demon- strates that πS may be empty for a nonconvex closed set. Nevertheless, he demonstrated that the set of points x∗ ∈ X∗ s.t πS(x ∗) 6= ϕ is dense in X∗ for closed nonconvex sets. In [6], the properties of the generalized projection on nonconvex sets in reflexive smooth Banach spaces with the smooth dual norm are further examined. The local Lipschtz continuity of πS for S not necessarily convex was established by M. Bounkhel and M. Bachar. Additionally, they proved that many of the features of πS on open subsets in X∗ are equivalent. Initiated in the nonconvex case by Bounkhel in [5–7], the present work continues the study of the πf S . By proving the local Lipschtz continuity of the generalized projection operator πf S for S nonempty closed sets not necessarily convex, and extending many properties of πS to πf S on open subsets in X∗. 2. Mathematical Preliminaries Let X∗ be a topological dual space of a Banach space X. In X∗ and X, the closed unit balls are B∗ and B, respectively. For definitions and some results concerning uniformly smooth, uniformly convex Banach spaces, and strictly convex spaces, we refer to ([8]),[9]). For J : X−→ −→X∗, called the normalzed duality mapping defined by J(u) = {u∗ ∈ X∗ : 〈u∗, u〉 = ‖u∗‖.‖u‖ = ‖u∗‖2 = ‖u‖2}. It is clear that ‖J(u)‖ is the norm defined on X∗, and ‖u‖ is the norm defined on X. These are a few of the J(x) map’s features. Consider [10] or [11] for more details. (i) if X smooth Banach spaces, then J continuous operator. (ii) J is a single valued mapping, whenever X∗ is strictly convex . (iii) J is a single valued mapping, whenever X is reflexive smooth Banach space. We now review a number of crucial terms and symbols that are necessary for our work. We begin with the widely recognized concepts of the convex subdifferential ∂conf , the Fréchet subdifferential ∂F f(x), and the Clarke subdifferential (see [12]). (i) Let u ∈ X and consider f to be a convex and continuous function defined on X. The convex subdifferential of f at u is given by the set: ∂conf(u) = {u∗ ∈ X∗ | 〈u∗, u− v〉 ≥ f(u)− f(v), ∀v ∈ X}. (ii) We say that u∗ ∈ ∂F f(u) iff, for every ϵ > 0, ∃ δ > 0 s.t 〈u∗, v − u〉 ≤ f(v)− f(u) + ϵ‖u− v‖, ∀v ∈ u+ δB. Moreover, if f is a convex extended real-valued functional that is lower semi-continuous and defined at u ∈ S, then the Fréchet normal cone of a nonempty closed set S is given by A. Al Tane, L. S. Keong / Eur. J. Pure Appl. Math, 18 (4) (2025), 6846 3 of 13 NF (S, u) = {u∗ ∈ X∗ | ∀ϵ > 0, ∃δ > 0 s.t 〈u∗, v − u〉 ≤ ϵ‖u− v‖, ∀v ∈ u+ δB}. (iii) We say that u∗ ∈ ∂Cf(u) iff 〈u∗, u〉 ≤ lim t→0,v→u sup t−1[f(v + tu)− f(v)], u ∈ X. Definition 1. Let X be a Banach space with dual space X∗ and let f : X → R ∪ {∞} is proper function. Then we define V f : X∗ ×X → R ∪ {∞} as V f (u∗, u) = ‖u∗‖2 + ‖u‖2 − 2〈u∗, u〉+ f(u), u∗ ∈ X∗, u ∈ X. It’s straight to demonstrate that f(u) + (‖u∗ − u‖)2 ≤ V f (u∗, u) ≤ f(u) + (‖u∗ + u‖)2. (1) Definition 2. Given a reflexive Banach space X. Let S be a nonempty closed subset of X. We denote by Mf,S(x ∗) := inf v∈S V f (x∗, v), and we define the operator πf S : X∗−→ −→X as πf S(u ∗) = { u ∈ S : V f (u∗, u) = Mf,S(u ∗) } ∀u∗ ∈ X∗. This operator is called a generalized f -projection. If f(x) = 0 for every x ∈ X, then πf S(x ∗) coincides with the generalized projection πS(x ∗), which was introduced and analyzed in Alber [1] and Li [2] for closed convex sets and by [5, 6] for nonempty closed (and not necessarily convex) sets. We would want to draw attention to the fact that, in some situations, it is possible to find a function f(x) 6= 0 for some x ∈ X s.t πS(x ∗) = πf S(x ∗), as demonstrated in the example that follows. Example 1. Jinlu Li [2] proved in the example (1.2) that for X = l1 , and X∗ = l∞ the πS(0) = co{u1, u2}, where 0 = (0, 0, 0, · · · ), u1 = (1, 1, 0, 0, · · · ), u2 = (1, 0, 1, 0, · · · ), and u3 = (2, 0, 0, 1, 0, · · · ) ∈ L1 with S = co{u1, u2, u3}. For us we define f(u) = ‖u‖2 for all u ∈ L1. Then V f (0, u1) = V f (0, u2) = 8, V f (0, u3) = 18. Now let µ ∈ [0, 1] and v ∈ co{u1, u2}. Then v = µu1 + (1− µ)u2 with V f (0, v) = 2‖(1, µ, 1− µ, 0, · · · )‖2 = 8. Now for any y ∈ S and µj ∈ [0, 1]; j = 1, 2, 3 with 3∑ j=1 µj = 1, then y = 3∑ j=1 µjuj. Hence V f (0, y) = 2‖(1 + µ3, µ1, µ2, µ3, 0, · · · )‖2 = 2(2 + µ3) 2 ≥ 8. We can observe from the aforementioned inequality that V f (0, y) = 8 iff µ3 = 0; this implies y ∈ co{u1, u2}. Therefore we get πf S(0) = {v ∈ S : V f (0, v) = inf u∈S V f (0, u) = 8} = co{u1, u2} = πS(0) A. Al Tane, L. S. Keong / Eur. J. Pure Appl. Math, 18 (4) (2025), 6846 4 of 13 Furthermore, Bounkhel (see [5]) shows that πS(x ∗) = ϕ by using an example consid- ering nonconvex closed sets S in Banach spaces that are uniformly smooth and uniformly convex. Using a similar method and by taking X = lp with (p ≥ 1), 0 = (0, 0, 0, . . .) ∈ lp and let S = {s1, s2, . . . , sm, . . .} ; sm = (0, 0, . . . , 1 + 1 m , . . .) define f as f(x) = ‖x‖ the projection πf S(u ∗) may in fact be empty for nonconvex closed sets S in a uniformly smoth and uniformly convex Banach space. This shows that the generalized f -projection over nonconvex sets may be empty even if the function f is convex continuous and the space X is smooth reflexive. Additionally, it recently demonstrated that, whenever the space X is taken to be a reflexive Banach space with the smooth dual norm, the set of points u∗ in X∗ with generalized f -projection is dense in X∗ (see Theorem 2.1 in [7]). Now, we consider the vector space of all convergent sequences of real numbers denoted by C. That equipped with the norm ‖u‖∞ = sup n |un|, and The dual space C∗ = l1 (Note that l1 is neither reflexive nor strictly convex). If u = (u0, u1, . . .) ∈ ℓ1,, and λ = (λ0, λ1, . . .) ∈ C. Then the duality pairing of C∗, C is given by 〈u, λ〉 = u0 lim n→∞ λn + ∞∑ i=1 uiλi. Note that C0 denotes the closed subspace of C that contains all convergent real sequences with limit zero. (For additional details, see[13]). Using C,C0, ℓ1 one can show that if the Banach space is not reflexive, πf S may be empty for some elements x∗ ∈ X∗ even for f(u) 6= 0.( Take note of my example, which uses the same reasoning as example 2.6 [14] for πS ). Example 2. Let f(u) = 4 and u∗ = (0, 1, 1 2 , 1 22 , 1 23 , · · · ) ∈ ℓ1. Then πf C0 (u∗) = ϕ. Proof : We define λn ∈ C0 for every positive integer n such that its first n components are two and all others are 0. Then ‖u∗‖ = (0, 1, 1 2 , 1 22 , 1 23 , · · · ) = ∞∑ i=0 1 2i = 2, ‖λn‖ = ‖(2, 2, . . . , 2, 0, 0, . . . )‖ = 2, and so 〈u, λn〉 = x0 lim n→∞ λn + ∞∑ i=1 uiλi = 0× 2 + n∑ i=1 1 2i−1 = 4(1− 2−n+1). Which implies V f (u∗, λn) = 4− 8(1− 2−n+1) + 8 → 4, as, n → ∞ (2) A. Al Tane, L. S. Keong / Eur. J. Pure Appl. Math, 18 (4) (2025), 6846 5 of 13 Now we will prove V f (u∗, λ) > 4, for all λ ∈ C0 so that the infµ∈C0V f (u∗, µ) by 2 not exists . Suppose that V f (u∗, λ) = 4. Then by (1) (‖u∗‖ − ‖λ‖)2 + f(λ) ≤ V f (u∗, λ) ≤ (‖u∗‖+ ‖λ‖)2 + f(λ) ⇔ (‖u∗‖ − ‖λ‖)2 + 4 ≤ 4 ≤ (‖u∗‖+ ‖λ‖)2 + 4 ⇔ 2 = ‖u∗‖ = ‖λ‖ Hence −2 ≤ λn ≤ 2 for n = 1, 2, . . . . But limn→∞λn = 0, so there are infinitely many n such that λn < 2. That implies V f (u∗, λ) = ‖u∗‖ − 2〈u∗, λ〉+ ‖λ‖2 + f(λ) = 4− 2 ∞∑ i=1 1 2i−1 λi + 8 > 4− 2 ∞∑ i=1 1 2i−1 2 + 8 = 4 which is a contraction to our assumption V f (u∗, λ) = 4. Therefor πf C0 (u∗) = ϕ. These observations force us to assume that the space X complies with this assumption going onward; in other words, we will presume that X is a reflexive Banach space with a smooth dual norm for the remainder of the work. 3. On πf S(x ∗) for S nonconvex set In this section, we generalized results related to generalized projection on closed non- convex sets πS to πf S on closed nonconvex sets with f is a proper, lower semi-continuous function (Unless we specified otherwise,). So we start with the following lemma that is analog with Lemma 2.1 [15] for metric projection, proposition 1.3 in [16] for Hilbert spaces, lemma 2.2 [6] for generalized projection on closed nonconvex sets and Recently this lemma proved for Generalized (f, λ)-projection operator see [7]. Lemma 1. Let X∗ be a smooth dual norm space of a reflexive Banach space X. Then for every β ∈ (0, 1) and for all u ∈ πf S(J(v)), we get πf S ((1− β)J(u) + βJ(v)) = {u}. Proof : For proving u ∈ πf S ((1− β)J(u) + βJ(v)) proceeds like Theorem 3.1 Part 3 in [7] by taking the constant λ defined there, equal half. Unfortunately, we can’t proceed only with λ = 1 2 for the uniqueness part, so we assume that u0 ∈ πf S ((1− β)J(u) + βJ(v)), where u 6= u0. Thus, we have two cases: case1: f(u0)− f(u) ≥ 0, we get V f ((1− β)J(u) + βJ(v), u) = V f ((1− β)J(u) + βJ(v), u0). A. Al Tane, L. S. Keong / Eur. J. Pure Appl. Math, 18 (4) (2025), 6846 6 of 13 So V ((1− β)J(u) + βJ(v), u)−V ((1− β)J(u) + βJ(v), u0) = f(u0)−f(u) ≥ β[f(u0)−f(u)]. On the other hand V ((1− β)J(u) + βJ(v), u)− V ((1− β)J(u) + βJ(v), u0) = β‖u‖2 + (1− β)‖u‖2 − β‖u0‖2 − (1− β)‖u0‖2 − 2(1− β)〈J(u), u〉+ 2(1− β)〈J(u), u0〉 − 2β〈J(v), u〉+ 2β〈J(v), u0〉 = β (V (J(v), u)− V (J(v), u0))− (1− β)V (J(u), u0). And so β (V (J(v), u)− V (J(v), u0))− (1− β)V (J(u), u0) ≥ β[f(u0)− f(u)]. Hence, (1− β)V (J(u), u0) ≤ β (V (J(v), u) + f(u)− V (J(v), u0)− f(u0)) = β ( V f (J(v), u)− V f (J(v), u0) ) ≤ 0. Since u ∈ πf S(J(v)), and so V f (J(v), u) ≤ V f (J(v), u0). which implies that V (J(u), u0) ≤ 0 and hence V (J(u), u0) = 0, and so u = u0. case2: f(u)− f(u0) ≥ 0, we get V f ((1− β)J(u) + βJ(v), u) = V f ((1− β)J(u) + βJ(v), u0). So V ((1−β)J(u)+βJ(v), u0)−V ((1−β)J(u)+βJ(v), u) = f(u)− f(u0) ≥ β[f(u)− f(u0)]. proceed like case1, we get the same result, and this ends our prove □ Now, using concepts from [17] and [6]. We provide a necessary and sufficient condition for the existence and uniqueness of the Mf,S(x ∗) in the following lemma, in terms of ∂FMf,S(x ∗). Lemma 2. Let x∗ be an element of smooth dual norm space X∗ of reflexive Banach space X. Then πf S(x ∗) is exists and unique, whenever ∂FMf,S(x ∗) 6= ϕ. Also, we get that ∂FMf,S(x ∗) = {2(J∗x∗ − πf S(x ∗))}. Proof : Let ∂FMf,S(x ∗) 6= ϕ and let x1 ∈ ∂FMf,S(x ∗). Then for every ϵ > 0 and applying the concept of ∂F to obtain, ∃δ > 0 such that given any u∗ ∈ B∗ and any µ ∈ (0, δ), we get: 〈x1;x∗ + µu∗ − x∗〉 = 〈x1;µu∗〉 ≤ Mf,S(x ∗ + µu∗)−Mf,S(x ∗) + ϵµ. A. Al Tane, L. S. Keong / Eur. J. Pure Appl. Math, 18 (4) (2025), 6846 7 of 13 Therefore, we can find a sufficiently large β ∈ N s.t: 〈x1;β−1u∗〉 ≤ Mf,S(x ∗ + β−1u∗)−Mf,S(x ∗) + ϵβ−1 , ∀u∗ ∈ B∗. (3) Using definition of Mf,S , ∃vn ∈ S for all n ≥ 1. Mf,S(x ∗) ≤ V f (x∗, vn) ≤ Mf,S(x ∗) + 1 n2 . (4) Hence, for sufficiently large β ∈ N, we can combine inequalities (3) and (4) for all u∗ ∈ B∗. 〈x1;β−1u∗〉 ≤ V f (x∗ + β−1u∗; vβ)− V f (x∗, vβ) + β−2 + β−1ϵ ≤ ‖x∗ + β−1u∗‖2 − ‖x∗‖2 − 2〈β−1u∗; vβ〉+ β−2 + β−1ϵ. Thus 〈x1 + 2vβ ;u ∗〉 ≤ β‖x∗ + β−1u∗‖2 − β‖x∗‖2 + β−1 + ϵ ∀u∗ ∈ B∗. For β large enough, and ∇F ‖.‖2(x∗) = 2J∗x∗, we can write sup u∗∈B∗ β [ ‖x∗ + β−1u∗‖2 − ‖x∗‖2 − 〈2J∗x∗;β−1u∗〉 ] ≤ ϵ. Therefore, for β large enough ‖x1 − 2(J∗x∗ − vβ)‖| = sup u∗∈B∗ 〈x1 − 2(J∗x∗ − vβ);u ∗〉 ≤ 3ϵ. (5) We conclude that this sequence (vβ)β converges to v := J∗x∗ − 1 2x1 by keeping in mind that the sequence(vβ) can be selected arbitrarily and that for arbitrary positive ϵ, the aforementioned inequality (5) is holds. (when β is sufficiently large). Additionally, since V f (x∗, vn) → Mf,S(x ∗), we conclude that v ∈ πf S(x ∗). Now let x̄ ∈ πf S(x ∗). Then carrying on as previously we get x̄ := J∗x∗ − 1 2x1 = v.□ From Lemma (2), it is clearly that if πf S(u ∗) 6= ϕ, then Mf,S(u ∗) ⊂ {2(J∗u∗ − u)} for all u ∈ πf S(u ∗). In the upcoming lemma, we will show that Mf,S is locally Lipschtz continuous. Lemma 3. Let X∗ be the dual space of the reflexive Banach space X and f is a proper function, and bounded from bellow on X, then u∗ 7→ V f (u∗, u) is locally Lipschtz on X∗. Proof : Fix δ > 0 and let u∗ be an element of X∗. Choose a value ϵ such that 0 < ϵ < δ and fix two elements v∗ and z∗ in u∗+ δB∗. By definition of Mf,S there exists rϵ ∈ S such that Mf,S(v ∗) ≤ V f (v∗, rϵ) < Mf,S(v ∗) + ϵ. A. Al Tane, L. S. Keong / Eur. J. Pure Appl. Math, 18 (4) (2025), 6846 8 of 13 So Mf,S(z ∗)−Mf,S(v ∗) ≤ V f (z∗, rϵ)− V f (v∗, rϵ) + ϵ = V (z∗, rϵ)− V (v∗, rϵ) + ϵ ≤ (‖z∗‖+ ‖v∗‖+ 2‖rϵ‖)‖z∗ − v∗‖+ ϵ. Let r = Mf,S(u ∗). Then ∃uϵ ∈ S s.t. Mf,S(u ∗) ≤ V f (u∗, uϵ) < Mf,S(u ∗) + ϵ. Now we can write ‖rϵ‖ ≤ (V (v∗, rϵ)) 1 2 + ‖v∗‖ = ( V f (v∗, rϵ)− f(rϵ) ) 1 2 + ‖v∗‖ < (Mf,S(v ∗)− f(rϵ) + ϵ) 1 2 + ‖v∗‖ ≤ (Mf,S(v ∗)− f(rϵ) + ϵ) 1 2 + ‖u∗‖+ δ ≤ ( V f (v∗, uϵ)− f(rϵ) + ϵ ) 1 2 + ‖u∗‖+ δ ≤ (V (v∗, uϵ) + f(uϵ)− f(rϵ) + ϵ) 1 2 + ‖u∗‖+ δ ≤ ( (‖v∗‖+ ‖uϵ‖)2 + f(uϵ)− f(rϵ) + ϵ ) 1 2 + ‖u∗‖+ δ ≤ (( ‖u∗‖+ δ + (V (u∗, uϵ)) 1 2 + ‖u∗‖ )2 + f(uϵ)− f(rϵ) + ϵ ) 1 2 + ‖u∗‖+ δ ≤ (( 2‖u∗‖+ δ + (V f (u∗, uϵ)− f(uϵ)) 1 2 )2 + f(uϵ)− f(rϵ) + ϵ ) 1 2 + ‖u∗‖+ δ ≤ (( 2‖u∗‖+ δ + (r + ϵ− f(uϵ)) 1 2 )2 + f(uϵ)− f(rϵ) + ϵ ) 1 2 + ‖u∗‖+ δ ≤ ( (2‖u∗‖+ δ)2 + r + 2ϵ+ 2(2‖u∗‖+ δ)(r + ϵ− f(uϵ)) 1 2 − f(rϵ) ) 1 2 + ‖u∗‖+ δ. Since f(u) ≥ L for some L ∈ R, so we can write ‖rϵ‖ ≤ ( (2‖u∗‖+ δ)2 + r + 2ϵ+ 2(2‖u∗‖+ δ)(r + ϵ− L) 1 2 − L ) 1 2 + ‖u∗‖+ δ := Cδ,ϵ,u∗ . Now letting ϵ → 0 and with Kδ,u∗ =: 2(‖u∗‖+ Cδ,u∗ + δ) we get |Mf,S(z ∗)−Mf,S(v ∗)| ≤ 2(‖u∗‖+ Cδ,u∗ + δ)‖z∗ − v∗‖ ≤ Kδ,u∗‖z∗ − v∗‖. A. Al Tane, L. S. Keong / Eur. J. Pure Appl. Math, 18 (4) (2025), 6846 9 of 13 So the proof is complete □ The Fréchet sub-differentiability of Mf,S is equivalent to its Fréchet differentiability, as demonstrated by the following Lemma. Lemma 4. Given that x∗ an element of smooth dual norm space X∗ of reflexive Banach space X. Then, the two statements that follow are equivalent: (i) ∂FMf,S(x ∗) 6= ϕ. (ii) Mf,S is Fréchet differentiable at x∗. Proof : From (2) → (1) clear, so we merely need to demonstrate the opposite im- plication. Suppose ∂FMf,S(x ∗) 6= ϕ and let z ∈ ∂FMf,S(x ∗). By lemma (2) we get z = 2(J∗x∗ − v) with v ∈ πf S(x ∗). For some ϵ > 0. Using concept of ∂F , ∃β1 > 0, and so µ ∈ (0, β1) and all u∗ ∈ B∗, we get 〈2(J∗x∗ − v);µu∗〉 ≤ Mf,S(x ∗ + µu∗)−Mf,S(x ∗) + ϵµ. Which implies µ−1 [Mf,S(x ∗ + µu∗)−Mf,S(x ∗)]− 〈2(J∗x∗ − v);u∗〉 ≥ −ϵ; ∀µ ∈ (0, β1), ∀u∗ ∈ B∗. Also, by definition of Mf,S we have µ−1 [Mf,S(x ∗ + µu∗)−Mf,S(x ∗)] ≤ µ−1 [ V f (x∗ + µu∗, y)− V f (x∗, y) ] . Since ∇FV f (·; v) = 2(J∗x∗ − v), there exists β2 > 0 such that for any µ ∈ (0, β2) and for all u∗ ∈ B∗ we have.∣∣∣µ−1 [ V f (x∗ + µu∗, y)− V f (x∗, y) ] − 〈2(J∗x∗ − v), u∗〉 ∣∣∣ ≤ ϵ. And so µ−1 [Mf,S(x ∗ + µu∗)−Mf,S(x ∗)]− 〈2(J∗x∗ − v);u∗〉 ≤ ϵ. Hence∣∣µ−1 [Mf,S(x ∗ + µu∗)−Mf,S(x ∗)]− 〈2(J∗x∗ − v);u∗〉 ∣∣ ≤ ϵ, µ ∈ (0, β), ∀u∗ ∈ B∗. With β = min{β1, β2}, since ϵ > 0 is arbitrary, so ∇FMf,S(x ∗) = 2(J∗x∗ − v). Thus, our proof is finished. □ The ‖.‖ − ‖.‖ continuity of πf S are related to the continuou ∇FMf,S by the following lemma. A. Al Tane, L. S. Keong / Eur. J. Pure Appl. Math, 18 (4) (2025), 6846 10 of 13 Lemma 5. Given that U∗ is an open set of smooth dual norm space X∗ of reflexive Banach space X. Then, the two claims that follow are equivalent: (i) The function Mf,S is C1 on U∗. (ii) The operator πf S is single-valued and ‖.‖ − ‖.‖ continuous on U∗. proof : (1) ⇒ (2) Assuming that Mf,S is C1 on U∗, πf S is single-value functional on U∗ and for every u∗ ∈ U∗, πf S(u ∗) = J∗u∗ − 1 2∇ FMf,S(u ∗) by lemma (4). Since J∗ and ∇FMf,S are ‖.‖ − ‖.‖ continuous, so πf S is ‖.‖ − ‖.‖ continuous on U∗. (2) ⇒ (1) First, we observe that Mf,S is locally Lipschtz on U∗ (by lemma 3) hence ac- cording to Mordukhovich-Shao (see [18]), for all u∗ ∈ U∗ we get ∂CMf,S(u ∗) = c̄o{weak− lim v∗→u∗ sup∂FMf,S(v ∗)} = c̄o{weak−limun : un ∈ ∂FMf,S(u ∗ n);u ∗ n → u∗}. Now by lemma (2) for any u∗n → u∗ with un ∈ ∂FMf,S(u ∗ n) we have xn = 2(J∗u∗−πf S(u ∗ n)). Therefore ∂CMf,S(u ∗) = c̄o{weak − limun : un = 2(J∗u∗n − πf S(u ∗ n)), u ∗ n → u∗}. We now obtain ∂CMf,S(u ∗) = {2(J∗u∗ − πf S(u ∗)} using the ‖.‖− ‖.‖ continuity of πf S and J∗ on X∗. As a result, Mf,S will definitely be continuously Gâteaux differentiable on U∗. We may finally detect that it is C1 on U∗ since Mf,S is locally Lipschtz on U∗.□ The following lemma proves that, whenever theπf S is single valued on reflexive Banach spaces with the Kadec condition(i.e. For any sequence (un) ⇀ u weakly in X with ‖un‖ → ‖u‖, then ‖un − u‖ → 0, see [8] ), its norm-to-weak continuity and ‖.‖ − ‖.‖ continuity are equivalent. Its proof is based on a concept from Lemma 5.1 proof in [15] for metric projection, and lemma 2.8 [6] for generalized projection operator. Lemma 6. Given that U∗ is an open set in X∗, where X Banach space is reflexive with the Kadec condition. Suppose that πf S is a single-valued function on U∗, and that f : X → R ∪ {∞} is proper continuous function. Then, πf S is ‖.‖ − ‖.‖ continuous on U∗ iff it is ‖.‖ − to− weak continuous on U∗. Proof : πf S is ‖.‖ − ‖.‖ continuous on U∗ follows naturally from the ‖.‖ − to − weak continuity on U∗. All we have to do is demonstrate the opposite. Let πf S(u ∗ n) → πf S(u ∗) weakly in X, and lim‖u∗n − x∗‖ = 0. Then, by Lipschtz continuity of Mf,S we have V f (u∗n, π f S(u ∗ n)) = Mf,S(u ∗ n) → Mf,S(u ∗) = V f (x∗, πf S(u ∗)). Now notice that: ‖πf S(u ∗ n)‖2 − ‖πf S(u ∗)‖2 = [ V f (u∗n, π f S(u ∗ n))− ‖u∗n‖2 + 2〈u∗n;π f S(u ∗ n)〉 − f(πf S(u ∗ n)) ] − [ V f (u∗, πf S(u ∗))− ‖u∗‖2 + 2〈u∗;πf S(u ∗)〉 − f(πf S(u ∗) ] A. Al Tane, L. S. Keong / Eur. J. Pure Appl. Math, 18 (4) (2025), 6846 11 of 13 = [ V f (u∗n, π f S(u ∗ n))− V f (u∗, πf S(u ∗)) ] + [ ‖u∗‖2 − ‖u∗n‖2 ] + 2 [ 〈u∗n;π f S(u ∗ n)〉 − 〈u∗;πf S(u ∗)〉 ] + [ f(πf S(u ∗)− f(πf S(u ∗ n) ] . By continuity of f , we get that |πf S(u ∗ n)‖ → ‖πf S(u ∗)‖ and hence ‖πf S(u ∗ n) − πf S(u ∗)‖ → 0 is guaranteed since the space X has Kadec property, this concluding the proof □. The ∇FMf,S and continuous Fréchet differentiability of Mf,S are equivalent on reflex- ive Banach spaces with the Kadec condition. This is shown by the following lemma, based on a concept from Lemma 4.2 in [15] and Lemma 2.9 in [6]. Lemma 7. Given U∗ as an open set in X∗, with X a reflexive Banach space with a smooth dual norm and the Kadec condition, and f is proper, convex, and Fréchet differentiable on U∗, the two assertions that follow are equivalent: (i) Mf,S is C1 in U∗. (ii) Mf,S is Fréchet differentiable on U∗. Proof : From (1) → (2) clear, We need only to prove (2) → (1). Let u∗n be a sequence that converges to u∗ in X. We want to demonstrate that ∇FMf,S(u ∗ n) → ∇FMf,S(u ∗). Notice that Mf,S(v ∗) = inf s∈S V f (v∗, s) = inf s∈S {‖v∗‖2 − 2〈v∗, s〉+ ‖s‖2 + f(s)} = ‖v∗‖2 − sup s∈S {−‖s‖2 + 2〈v∗, s〉 − f(s)}. The function f is convex, and it is clear that both functions v∗ 7→ VS(v ∗) = sup s∈S {−‖s‖2 + 2〈v∗, s〉 − f(s)}. Mf,S(v ∗) = ‖v∗‖2 + VS(v ∗) are convex. f is Fréchet differentiable. Then, VS is func- tional and is both convex and Fréchet differentiable. So the derivative of VS is norm-to- weak continuous (see[19]), which implies ∇FVS(u ∗ n) converges weakly to ∇FVS(u ∗), which means that ∇FMf,S(u ∗ n) approaching ∇FMf,S(u ∗) weakly. Lemma 4 allows us to write ∇FMf,S(u ∗ n) = 2(J∗u∗n − πf S(u ∗ n)) and ∇FMf,S(u ∗) = 2(J∗u∗ − πf S(u ∗)). Thus, πf S(u ∗ n) converges weakly to πf S(u ∗). By 6 we get the Strong convergence of πf S(u ∗ n) to πf S(u ∗).The continuity of ∇FMf,S has been established, Which brings the proof to an end. □ The theorem below establishes an equivalence between the properties of πf S and Mf,S on an open subset of X∗. Theorem 1. Let X be a reflexive Banach space with smooth dual norm and Kadec property. Let U∗ be a subset of X∗ that is open. Let f is proper, convex, continuous, bounded from bellow, and Fréchet differentiable function. The statements that follow are equivalent. A. Al Tane, L. S. Keong / Eur. J. Pure Appl. Math, 18 (4) (2025), 6846 12 of 13 (i) Mf,S is C1 in U∗. (ii) Mf,S is Fréchet differentiable on U∗. (iii) ∂FMf,S 6= ϕ on U∗. (iv) πf S is single-valued and ‖.‖ − to− weak continuous on U∗. (v) πf S is single-valued and ‖.‖ − ‖.‖ continuous on U∗. Proof : Combining the previously proven lemmas to prove this theorem is enough. 4. Conclusion To investigate the πf S , this study shows the efficiency of the subdifferential ∂conf , the Fréchet subdifferential ∂F f(x), and the Clarke subdifferential to extend many properties of πS to πf S In addition to provide a necessary and sufficient condition for the existence and uniqueness and prove locally Lipschtz continuous of the πf S . In the future, we intend to introduce a new set defined in terms of the generalized f -projection, to extend many well-known results on the usual V -proximal normal cone in the setting of Banach spaces. 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