Abstract
We investigate the existence of directional derivatives for strongly coneparaconvex mappings. Our result is an extension of the classical Valadier result on the existence of the directional derivative for cone convex mappings with values in weakly sequentially complete Banach spaces.
Introduction
The concepts of approximate convexity for extended realvalued functions include among others, γparaconvexity [3, 4, 7, 8], γsemiconcavity [1], αparaconvexity, strong αparaconvexity [9], semiconcavity [1], and approximate convexity [6]. Relations between these concepts were investigated by Rolewicz [7,8,9], Daniilidis and Georgiev [2], and Tabor and Tabor [11]. These concepts were used, e.g., in [1], to investigate Hamilton–Jacobi equation. In a series of papers [7,8,9], Rolewicz investigated Gâteaux and Fréchet differentiability of strongly αparaconvex, generalizing in this way the Mazur theorem (1933).
Generalization of the above concepts to vectorvalued mappings with values in a general vector space Y was given by Veselý and Zajicek [13,14,15,16], Valadier [12], and Rolewicz [10]. In the paper [10], Rolewicz defined vectorvalued strongly αk paraconvex mappings and investigated their Gateaux and Fréchet differentiability, where k ∈ K and K is a closed convex cone in a normed vector space Y.
Let \(\alpha :\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}\) be a nondecreasing function satisfying the condition
Let X be a normed space and let k ∈ K. The mapping F : X → Y is strongly αk paraconvex on a convex subset A of X if there exists a constant C > 0 such that for every x_{1},x_{2} ∈ A and every λ ∈ [0,1]
where x≤_{ K }y⇔y − x ∈ K. In the sequel, we use the notation ≤ if the cone K is clear from the context.
The mapping F : X → Y is strongly αK paraconvex on a convex subset A of X if for every k ∈ K there exists a constant C > 0 such that for every x_{1},x_{2} ∈ A and every λ ∈ [0,1]
A strongly α (⋅)K paraconvex mapping F is called strongly coneparaconvex if the cone K and the function α are clear from the context. Since for every λ ∈ [0,1]
condition (1) can be equivalently rewritten as
Strong coneparaconvexity generalizes the cone convexity. The mapping F : X → Y is Kconvex on a convex subset A of X if for every x_{1},x_{2} ∈ A and every λ ∈ [0,1]
In the present paper, we investigate the existence of directional derivatives for strongly coneparaconvex mappings. Our main result (Theorem 2) is a generalization of the theorem of Valadier [12] concerning directional differentiability of cone convex mappings.
Preliminary Facts
Let Y^{∗} be the dual space of Y and K^{∗}⊂ Y^{∗} be the positive dual cone to K,
Clearly, if F is a strongly α (⋅)k paraconvex mapping with constant C > 0, then for every y^{∗}∈ K^{∗}, the function y^{∗}∘ F is a strongly α (⋅)paraconvex function with the constant C ⋅ y^{∗}(k).
In a normed space Y, a cone K is normal (see [12]) if there is a number C > 0 such that
Every normal cone is pointed, i.e., K ∩ (−K) = {0}.
In [13], Veselỳ and Zajiček introduced the concept of d.c. (deltaconvex) mappings acting between Banach spaces X and Y. A mapping F : X → Y is d.c. if there exists a continuous convex function \(g:X\rightarrow \mathbb {R}\) such that for every y^{∗}∈ Y^{∗} the function y^{∗}∘ F + g is a d.c. function, i.e., it is representable as a difference of two convex functions.
According to [15], F is order d.c. if F is representable as a difference of two cone convex mappings on A. Consequently, if the cone K is normal, then F is also weakly order d.c.
Moreover, if the range space Y of an order d.c. mapping F is ordered by a wellbased cone K (and this is true for L_{1}(μ)), it is easy to show (see Proposition 4.1 [15]) that the mapping is then d.c.
In the example below, we show that any strongly ∥⋅∥^{2}k_{0}paraconvex mapping is order d.c.
Example 1
Let X be a Hilbert space. A mapping F : X → Y is strongly ∥⋅∥^{2}k_{0}paraconvex with constant C ≥ 0 on a convex set A if and only if the mapping F + C∥⋅∥^{2}k_{0} is Kconvex on A. Indeed, let x_{1},x_{2} ∈ X. Since
and
we have
The mapping F (⋅) = F (⋅) + C∥⋅∥^{2}k_{0} is clearly order d.c. Furthermore, if K is well based (∃y^{∗}∈ Y^{∗} such that y^{∗}(k) ≥∥k∥ for any k ∈ K), then F is d.c.
For d.c. mappings, we have the following result on the existence of directional derivative.
Theorem 1 (Proposition 3.1 of 13)
Let X be a normed linear space and let Y be a Banach space. Let G ⊂ X be an open convex set and let F : G → Y be a d.c. mapping. Then, the directional derivative F^{′}(x_{0},h) exists whenever x_{0} ∈ G and h ∈ X .
Let us observe that if the function α (⋅) is not convex, then we cannot expect a strongly α (⋅)k_{0} paraconvex mapping F to be d.c.
Monotonicity of Difference Quotients
Let X be a normed space. Let Y be a topological vector space and let K ⊂ Y be a closed convex pointed cone.
For Kconvex mappings, the difference quotient is nondecreasing in the sense that
For strongly α (⋅)K paraconvex and strongly α (⋅)k_{0} paraconvex mappings, the difference quotient may not be nondecreasing.
Example 2
Let \(Y=\mathbb {R}\), \(K=\mathbb {R}_{+}\), α (x) = x^{2} and let F (x) = −x^{2}. The mapping F is strongly α (⋅)Kparaconvex. Observe that for any \(x_{1}, x_{2}\in \mathbb {R}\), we have \(t({x^{2}_{1}}+{x_{2}^{2}})2t(x_{1}x_{2})\le 0\) if and only if t ≤ 0. Hence, for t = −λ^{2} + λ − 1 ≤ 0, we have
The last inequality and Proposition 2.1 from [5] give us paraconvexity of the mapping F.
Let x_{0} = 0, h = 1. The difference quotient \(\phi (t)=\frac {F(x_{0}+th)F(x_{0})}{t}\) is decreasing. Indeed, for t_{1} ≤ t_{2}, we have ϕ (t_{1}) = −t_{1} and ϕ (t_{2}) = −t_{2}.
The following two propositions are basic tools for the proof of the main result in the next section. In the proposition below, we investigate the monotonicity properties of the α (⋅)difference quotients for strongly α (⋅)k paraconvex mappings.
Proposition 1
Let X be a normed space and let Y be a vector space and ordered by a convex pointed cone K. Let F : X → Y be strongly α (⋅) k_{0} paraconvex on a convex set A ⊂ X with constant C ≥ 0, k_{0} ∈ K ∖{0}. For any x_{0} ∈ A and any h ∈ X,∥h∥ = 1such that x_{0} + th ∈ A for all t sufficiently small, the α (⋅)difference quotient mapping \(\phi :\mathbb {R}\rightarrow Y\) defined as
where \(t_{0}\in \mathbb {R}\) isα (⋅)nondecreasingin the sense that
Proof
Take any t_{0} < t_{1} < t. We have \(0<\lambda :=\frac {t_{1}t_{0}}{tt_{0}}<1\) and
Let k_{0} ∈ K ∖{0}. Since F is strongly α (⋅)k_{0} paraconvex with constant C ≥ 0, we have
Hence,
i.e.,
We have

(i)
If λ ≤ 1 − λ, i.e., t_{1} − t_{0} ≤ t − t_{0}, then
$$ \min\{\lambda, 1\lambda\} \frac{\alpha(tt_{0})}{t_{1}t_{0}}= \frac{\alpha(tt_{0})}{tt_{0}}. $$ 
(ii)
If λ > 1 − λ, i.e., \(\frac {t_{1}t_{0}}{tt_{0}}> \frac {tt_{1}}{tt_{0}}\), then
$$ \min\{\lambda, 1\lambda\} \frac{\alpha(tt_{0})}{t_{1}t_{0}}=\frac{tt_{1}}{tt_{0}}\frac{\alpha(tt_{0})}{t_{1}t_{0}}<\frac{\alpha(tt_{0})}{tt_{0}}. $$
In both cases,
□
If int K ≠ ∅, then any strongly α (⋅)k_{0} paraconvex mapping F is strongly α (⋅)K paraconvex and for any k ∈ K the α (⋅)difference quotients satisfy formula (3) with different constants C, and in general, one cannot find a single constant C for all 0 ≠ k ∈ K.
In the proposition below, we investigate the boundedness of α (⋅)difference quotient for strongly α (⋅)k paraconvex mappings.
Proposition 2
Let X be a normed space. Let Y be a topological vector space and let Y be ordered by a closed convex pointed cone K. Let F : X → Y be strongly α (⋅)k_{0} paraconvex on a convex set A ⊂ X with constant C ≥ 0, k_{0} ∈ K ∖{0}.
For any x_{0} ∈ A and any h ∈ X ,∥h∥ = 1such that x_{0} + th ∈ A for all t sufficiently small, the α (⋅)difference quotient mapping ϕ : [0, + ∞) → Y,
is bounded from below in the sense that there are an elementa ∈ Y andδ > 0suchthat
Proof
Let us take t_{0} = −t, t_{1} = 0. From inclusion (3), we have
Multiplying both sides by 2t > 0, we get
By simple calculations, we get
Since \(\lim\limits_{t \rightarrow 0^{+}}\frac {\alpha (t)}{t}= 0\), there exists δ > 0 such that \(2C\frac {\alpha (2t)}{2t} \le 1\) for t ∈ (0,δ). We have
Now, let us take − 1 < −t < 0. We have
From the α (⋅)k_{0} paraconvexity (1) for λ := t, we get
By simple calculation, we get
Since \(\frac {\min \{t, 1t\}}{t}=\frac {12t1}{2t}\) and the fact that \(\frac {12t1}{2t}\le 1\) is bounded, we get
Hence,
From (5), we get
where b := F (x_{0}) − F (x_{0} − h) − (Cα (1) + 1) k_{0}. Finally,
□
Main Result
The proof of the main theorem is based on the following lemma.
Lemma 1
Let Y be a Banach space. Let K ⊂ Y be a closed convex normal cone. Let\({\Phi }: \mathbb {R}_{+} \rightarrow Y\) satisfy the following conditions

(i)
Φ(t) ∈ K for any \(t \in \mathbb {R}_{+}\),

(ii)
for 0 < t_{1} < t we have \({\Phi }(t){\Phi }(t_{1}) + \frac {\alpha (t_{1})}{t_{1}}k_{0}\in K\) for some k_{0} ∈ K,

(iii)
Φ(t) is weakly convergent to 0 when t → 0^{+}.
Then, ∥Φ(t)∥→ 0when t → 0^{+}.
Proof
By contradiction, suppose that \(\{\Phi }(t)\\nrightarrow 0\) when t → 0^{+} and (i) and (ii) are satisfied. We will obtain a contradiction with (iii). By this, there is ε > 0 such that for all δ > 0 one can find 0 < t < δ with ∥Φ(t)∥ > ε. In particular, for \(\delta _{n}=\frac {1}{n}\), there exist \(t_{n}\in (0,\frac {1}{n})\), \(n\in \mathbb {N}\), such that
Let \(x\in A:=\text {co}({\Phi }(t_{n}), n\in \mathbb {N})\). There are positive numbers λ_{1},λ_{2},…,λ_{ m } and t_{1},t_{2}…,t_{ m } such that \(x={\sum }_{i = 1}^{m} \lambda _{i} {\Phi }(t_{i})\), where \({\sum }_{i = 1}^{m}\lambda _{i}= 1\). There exists \(N\in \mathbb {N}\) such that for all n > N, we have
We get
From the fact that Φ(t_{ n }) ∈ K and K is normal, there is some c > 0 such that \(\{\Phi }(t_{n})\\le c \x+ \frac {\alpha (t_{n})}{t_{n}}k_{0}\\). By (6), we obtain \(\x+ \frac {\alpha (t_{n})}{t_{n}}k_{0}\ > \beta := \frac {\varepsilon }{c}\) for all x ∈ A and n > N.
We show that
for s > 0 satisfying \(\frac {\alpha (t_{n})}{t_{n}}\le s\). To see this, take any ℓ ∈ (0,s], where \(\mathbb {B}_{r}:=\{y\in Y: \y\\le r \}\). Since \(\lim _{n\rightarrow +\infty }\frac {\alpha (t_{n})}{t_{n}}= 0\), there exists \(n\in \mathbb {N}\) such that
By (6) and the normality of K,
From the Hahn–Banach theorem applied to \(\mathbb {B}_{\beta /2}\) and (A + k_{0}[0,s]), there is a linear functional y^{∗}∈ Y^{∗} and r > 0 such that
In particular, \(y^{\ast }({\Phi }(t_{n})+ \frac {\alpha (t_{n})}{t_{n}}k_{0})>r>0\), which contradicts (iii). □
We are in a position to prove our main result.
Theorem 2
Let X be a normed space. Let Y be a weakly sequentially complete Banach space ordered by a closed convex normal cone K. Let F : X → Y be stronglyα (⋅)k_{0} paraconvex on a convex set A ⊂ X with constant C ≥ 0, k_{0} ∈ K ∖{0}. Then, the directional derivative
of F at x_{0} exists for any x_{0} ∈ A and any direction 0 ≠ h ∈ X,∥h∥ = 1such that x_{0} + th ∈ A for all t sufficiently small.
Proof
Let x_{0} ∈ A and let 0 ≠ h ∈ X, ∥h∥ = 1 be such that x_{0} + th ∈ A for all t sufficiently small. Let t_{ n }↓0. For t_{0} = 0, the α (⋅)difference quotient by (2) takes the form
Let y^{∗}∈ K^{∗}. By (4), the sequence a_{ n } := y^{∗}(ϕ (t_{ n })), \(n\in \mathbb {N}\) is bounded from below, i.e.,
Let us take ε > 0. There is N such that
where \(\underline {a}:=\inf \{a_{n}: n\in \mathbb {N}\}\). Since {t_{ n }} is decreasing, from (3), we get
Let \(b_{n}:= C\frac {\alpha (t_{n})}{t_{n}}y^{\ast }(k_{0})\). Since b_{ n } → 0 there is N_{1} such that \(b_{n}\le \frac {\varepsilon }{2}\) for n > N_{1}.
Hence, the sequence {a_{ n }} is convergent and consequently every sequence {y^{∗}(ϕ (t_{ n }))} is Cauchy for y^{∗}∈ K^{∗}.
Let us take any h^{∗}∈ Y^{∗}. We show that the sequence {h^{∗}(ϕ (t_{ n }))} is Cauchy. From the fact that K is normal, we have Y^{∗} = K^{∗}− K^{∗} and h^{∗} = g^{∗}− q^{∗} with g^{∗},q^{∗}∈ K^{∗}. Since {g^{∗}(ϕ (t_{ n }))} and {q^{∗}(ϕ (t_{ n }))} are Cauchy sequences, there exist N_{1}, N_{2} such that for \(n,m > \bar {N}:=\max (N_{1}, N_{2})\), we have
For \(n> \bar {N}\), we have
We show that ϕ (t) weakly converges when t → 0^{+}, i.e., there is an y_{0} ∈ Y such that for arbitrary t_{ n }↓0, we have
which is equivalent to
Since Y is weakly sequentially complete, we need only to show that y_{0} is the same for all sequences {t_{ n }}, t_{ n }↓0. On the contrary, suppose that there are two different weak limits \({y_{0}^{1}}\), \({y_{0}^{2}}\) corresponding to sequences \({t_{n}^{1}}\) and \({t_{n}^{2}}\), respectively.
We can subtract subsequences \(\{\bar {t}_{n}^{2}\}\subset \{{t_{n}^{2}}\}\) and \(\{\bar {t}_{n}^{1}\}\subset \{{t_{n}^{1}}\}\) such that \(\bar {t}_{n}^{2} \le {t_{n}^{1}} \le \bar {t}_{n}^{1}\). Correspondingly,
which proves that it must be \({y_{0}^{1}}={y_{0}^{2}}\).
Now, we show that the mapping Φ(t) := ϕ (t) − y_{0} satisfies all the assumptions of Lemma 1. From (3) and (9), it is enough to show that Φ(t) ∈ K for all t ≥ 0.
By contradiction, let us assume that there is some \(\bar {t}>0\) such that \({\Phi }(\bar {t})\notin K\). There exists y^{∗}∈ K^{∗} such that
From inclusion (3) in Proposition 1, we have
In particular
And by (10), we get
Then, by letting t → 0^{+}, we get contradiction with (9). By Lemma 1, Φ(t) tends to 0 when t → 0^{+}. Since \(\lim _{t\rightarrow 0^{+}}\frac {\alpha (t)}{t}= 0\), we get
which completes the proof. □
Remark 1
For Kconvex mappings F, i.e., strongly α (⋅)K paracanovex mappings with constant C = 0 Theorem 2 can be found in [12].
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Dedicated to Michel Théra on the occasion of his 70th birthday.
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Bednarczuk, E., Leśniewski, K. On the Existence of Directional Derivatives for Strongly ConeParaconvex Mappings. Vietnam J. Math. 46, 381–389 (2018). https://doi.org/10.1007/s1001301802902
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DOI: https://doi.org/10.1007/s1001301802902
Keywords
 Directional derivative
 Normal cones
 Strongly paraconvex mappings
 Cone convex mappings
 Weakly sequentially complete Banach spaces
Mathematics Subject Classification (2010)
 Primary 49J50
 Secondary 52A41