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On the Existence of Directional Derivatives for Strongly Cone-Paraconvex Mappings

Abstract

We investigate the existence of directional derivatives for strongly cone-paraconvex 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 real-valued functions include among others, γ-paraconvexity [3, 4, 7, 8], γ-semiconcavity [1], α-paraconvexity, strong α-paraconvexity [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 α-paraconvex, generalizing in this way the Mazur theorem (1933).

Generalization of the above concepts to vector-valued 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 vector-valued strongly α-k paraconvex mappings and investigated their Gateaux and Fréchet differentiability, where kK 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

$$\lim\limits_{t\rightarrow 0^{+}}\frac{\alpha(t)}{t}= 0. $$

Let X be a normed space and let kK. The mapping F : XY is strongly α-k paraconvex on a convex subset A of X if there exists a constant C > 0 such that for every x1,x2A and every λ ∈ [0,1]

$$ F(\lambda x_{1}+(1-\lambda)x_{2})\le_{K} \lambda F(x_{1})+(1-\lambda) F(x_{2})+C\min\{\lambda, 1-\lambda\}\alpha(\|x_{1}-x_{2}\|)k, $$
(1)

where x K yyxK. In the sequel, we use the notation ≤ if the cone K is clear from the context.

The mapping F : XY is strongly α-K paraconvex on a convex subset A of X if for every kK there exists a constant C > 0 such that for every x1,x2A and every λ ∈ [0,1]

$$ F(\lambda x_{1}+(1-\lambda)x_{2})\le_{K} \lambda F(x_{1})+(1-\lambda) F(x_{2})+C\min\{\lambda, 1-\lambda\}\alpha(\|x_{1}-x_{2}\|)k. $$

A strongly α (⋅)-K paraconvex mapping F is called strongly cone-paraconvex if the cone K and the function α are clear from the context. Since for every λ ∈ [0,1]

$$\lambda(1-\lambda)\le\min\{\lambda, 1-\lambda\}\le 2 \lambda(1-\lambda) $$

condition (1) can be equivalently rewritten as

$$ F(\lambda x_{1}+(1-\lambda)x_{2})\le_{K} \lambda F(x_{1})+(1-\lambda) F(x_{2})+ 2C\lambda(1-\lambda)\alpha(\|x_{1}-x_{2}\|)k. $$

Strong cone-paraconvexity generalizes the cone convexity. The mapping F : XY is K-convex on a convex subset A of X if for every x1,x2A and every λ ∈ [0,1]

$$ F(\lambda x_{1}+(1-\lambda)x_{2})\le_{K} \lambda F(x_{1})+(1-\lambda) F(x_{2}). $$

In the present paper, we investigate the existence of directional derivatives for strongly cone-paraconvex 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 KY be the positive dual cone to K,

$$K^{\ast}:=\{y^{\ast}\in Y^{\ast}~ |~ y^{\ast}(y)\ge 0~ \forall~ y\in K\}. $$

Clearly, if F is a strongly α (⋅)-k paraconvex mapping with constant C > 0, then for every yK, the function yF is a strongly α (⋅)-paraconvex function with the constant Cy(k).

In a normed space Y, a cone K is normal (see [12]) if there is a number C > 0 such that

$$0\le_{K} x\le_{K} y \Rightarrow \|x\|\le C \|y\|\quad \text{ for all } x, y \in Y. $$

Every normal cone is pointed, i.e., K ∩ (−K) = {0}.

In [13], Veselỳ and Zajiček introduced the concept of d.c. (delta-convex) mappings acting between Banach spaces X and Y. A mapping F : XY is d.c. if there exists a continuous convex function \(g:X\rightarrow \mathbb {R}\) such that for every yY the function yF + 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 well-based cone K (and this is true for L1(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-k0-paraconvex mapping is order d.c.

Example 1

Let X be a Hilbert space. A mapping F : XY is strongly ∥⋅∥2-k0-paraconvex with constant C ≥ 0 on a convex set A if and only if the mapping F + C∥⋅∥2k0 is K-convex on A. Indeed, let x1,x2X. Since

$$ \lambda \|x_{1}\|^{2} + (1-\lambda) \|x_{2}\|^{2} - \|\lambda x_{1} + (1-\lambda) x_{2})\|^{2}= \lambda(1-\lambda) \|x_{1}-x_{2}\|^{2} $$

and

$$F(\lambda x_{1}+(1-\lambda)x_{2})\le_{K} \lambda F(x_{1})+(1-\lambda) F(x_{2})+{C}\lambda(1-\lambda)\|x_{1}-x_{2}\|^{2}k_{0} $$

we have

$$\begin{array}{@{}rcl@{}} &&F(\lambda x_{1}+(1-\lambda)x_{2}) +C\|\lambda x_{1} + (1-\lambda) x_{2})\|^{2}k_{0}\\ &&\qquad \le_{K} \lambda F(x_{1})+(1-\lambda) F(x_{2})+C(\lambda \|x_{1}\|^{2}k_{0} + (1-\lambda) \|x_{2}\|^{2})k_{0}. \end{array} $$

The mapping F (⋅) = F (⋅) + C∥⋅∥2k0 is clearly order d.c. Furthermore, if K is well based (∃yY such that y(k) ≥∥k∥ for any kK), 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 GX be an open convex set and let F : GY be a d.c. mapping. Then, the directional derivative F(x0,h) exists whenever x0G and hX .

Let us observe that if the function α (⋅) is not convex, then we cannot expect a strongly α (⋅)-k0 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 KY be a closed convex pointed cone.

For K-convex mappings, the difference quotient is nondecreasing in the sense that

$$\phi(t_{1})-\phi(t_{2}):=\frac{F(x_{0}+t_{1}h)-F(x_{0})}{t_{1}}-\frac{F(x_{0}+t_{2}h)-F(x_{0})}{t_{2}}\in K \quad \text{for } t_{1} \ge t_{2}. $$

For strongly α (⋅)-K paraconvex and strongly α (⋅)-k0 paraconvex mappings, the difference quotient may not be nondecreasing.

Example 2

Let \(Y=\mathbb {R}\), \(K=\mathbb {R}_{+}\), α (x) = x2 and let F (x) = −x2. The mapping F is strongly α (⋅)-K-paraconvex. 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 = −λ2 + λ − 1 ≤ 0, we have

$$\begin{array}{@{}rcl@{}} &(-\lambda^{2}+\lambda-1)({x_{1}^{2}}+{x_{2}^{2}})-2x_{1}x_{2}(-\lambda^{2}+\lambda-1)\le 0,&\\ &{x_{1}^{2}}(-\lambda^{2}+\lambda-1)+x_{1}x_{2}(-2\lambda(1-\lambda)+ 2)+{x_{2}^{2}}(-(1-\lambda)^{2}+ 1-\lambda-1)\le 0,&\\ &-(\lambda x_{1} +(1-\lambda)x_{2})^{2}\le -\lambda {x_{1}^{2}}-(1-\lambda){x_{2}^{2}}+(x_{1}-x_{2})^{2},&\\ &F(\lambda x_{1} + (1-\lambda)x_{2})\le \lambda F(x_{1}) + (1-\lambda) F(x_{2}) + (x_{1}-x_{2})^{2}.& \end{array} $$

The last inequality and Proposition 2.1 from [5] give us paraconvexity of the mapping F.

Let x0 = 0, h = 1. The difference quotient \(\phi (t)=\frac {F(x_{0}+th)-F(x_{0})}{t}\) is decreasing. Indeed, for t1t2, we have ϕ (t1) = −t1 and ϕ (t2) = −t2.

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 : XY be strongly α (⋅)- k0 paraconvex on a convex set AX with constant C ≥ 0, k0K ∖{0}. For any x0A and any hX,h∥ = 1such that x0 + thA for all t sufficiently small, the α (⋅)-difference quotient mapping \(\phi :\mathbb {R}\rightarrow Y\) defined as

$$ \phi(t):=\frac{F(x_{0}+th)-F(x_{0}+t_{0}h)}{t-t_{0}} + C\frac{\alpha(t-t_{0})}{t-t_{0}}k_{0} \quad \text{ for } t_{0}<t, $$
(2)

where \(t_{0}\in \mathbb {R}\) isα (⋅)-nondecreasingin the sense that

$$ \phi(t)-\phi(t_{1})+C\frac{\alpha(t_{1}-t_{0})}{t_{1}-t_{0}}k_{0}\in K\quad \text{ for }t_{0}<t_{1}<t. $$
(3)

Proof

Take any t0 < t1 < t. We have \(0<\lambda :=\frac {t_{1}-t_{0}}{t-t_{0}}<1\) and

$$x_{0}+t_{1}h = \lambda(x_{0}+th)+(1-\lambda)(x_{0}+t_{0}h). $$

Let k0K ∖{0}. Since F is strongly α (⋅)-k0 paraconvex with constant C ≥ 0, we have

$$\begin{array}{@{}rcl@{}} F(x_{0}+t_{1}h) &\le_{K}&\lambda F(x_{0}+th)+(1-\lambda)F(x_{0}+t_{0}h)\\ &&+C\min\{\lambda, 1-\lambda\}\alpha(t-t_{0})k_{0}. \end{array} $$

Hence,

$$\begin{array}{@{}rcl@{}} 0&\le_{K}&\lambda[F(x_{0}+th)-F(x_{0}+t_{0}h)]-[F(x_{0}+t_{1}h) - F(x_{0}+t_{0}h)]\\ &&+C\min\{\lambda, 1-\lambda\}\alpha(t-t_{0})k_{0}, \end{array} $$

i.e.,

$$\begin{array}{@{}rcl@{}} &&\left[\frac{F(x_{0}+th)-F(x_{0}+t_{0}h)}{t-t_{0}}\right] - \left[\frac{F(x_{0}+t_{1}h)-F(x_{0}+t_{0}h)}{t_{1}-t_{0}}\right]\\ &&+ C\min\{\lambda, 1-\lambda\}\frac{\alpha(t-t_{0})}{t_{1}-t_{0}}k_{0}\in K. \end{array} $$

We have

  1. (i)

    If λ ≤ 1 − λ, i.e., t1t0tt0, then

    $$ \min\{\lambda, 1-\lambda\} \frac{\alpha(t-t_{0})}{t_{1}-t_{0}}= \frac{\alpha(t-t_{0})}{t-t_{0}}. $$
  2. (ii)

    If λ > 1 − λ, i.e., \(\frac {t_{1}-t_{0}}{t-t_{0}}> \frac {t-t_{1}}{t-t_{0}}\), then

    $$ \min\{\lambda, 1-\lambda\} \frac{\alpha(t-t_{0})}{t_{1}-t_{0}}=\frac{t-t_{1}}{t-t_{0}}\frac{\alpha(t-t_{0})}{t_{1}-t_{0}}<\frac{\alpha(t-t_{0})}{t-t_{0}}. $$

In both cases,

$$\begin{array}{@{}rcl@{}} &&\left[\frac{F(x_{0}+th)-F(x_{0}+t_{0}h)}{t-t_{0}}\right] - \left[\frac{F(x_{0}+t_{1}h)-F(x_{0}+t_{0}h)}{t_{1}-t_{0}}\right]\\ &&+ C\frac{\alpha(t-t_{0})}{t-t_{0}}k_{0}-C\frac{\alpha(t_{1}-t_{0})}{t_{1}-t_{0}}k_{0}+C\frac{\alpha(t_{1}-t_{0})}{t_{1}-t_{0}}k_{0}\in K. \end{array} $$

If int K, then any strongly α (⋅)-k0 paraconvex mapping F is strongly α (⋅)-K paraconvex and for any kK the α (⋅)-difference quotients satisfy formula (3) with different constants C, and in general, one cannot find a single constant C for all 0 ≠ kK.

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 : XY be strongly α (⋅)-k0 paraconvex on a convex set AX with constant C ≥ 0, k0K ∖{0}.

For any x0A and any hX ,h∥ = 1such that x0 + thA for all t sufficiently small, the α (⋅)-difference quotient mapping ϕ : [0, + ) → Y,

$$ \phi(t):=\frac{F(x_{0}+th)-F(x_{0})}{t} + C\frac{\alpha(t)}{t}k_{0} $$

is bounded from below in the sense that there are an elementaY andδ > 0suchthat

$$ \phi(t)-a\in K\quad \text{ for } 0<t<\delta. $$
(4)

Proof

Let us take t0 = −t, t1 = 0. From inclusion (3), we have

$$ \frac{F(x_{0}+th)-F(x_{0}-th)}{2t} + C\frac{\alpha(2t)}{2t}k_{0}- \frac{F(x_{0})-F(x_{0}-th)}{t} - C\frac{\alpha(t)}{t}k_{0}+C\frac{\alpha(t)}{t}k_{0} \in K. $$

Multiplying both sides by 2t > 0, we get

$$F(x_{0}+th)-F(x_{0}-th)+C{\alpha(2t)}k_{0}-2F(x_{0})+ 2F(x_{0}-th)\in K. $$

By simple calculations, we get

$$\frac{F(x_{0}+th)-F(x_{0})}{t}+\frac{F(x_{0}-th)-F(x_{0})}{t}+ 2C\frac{\alpha(2t)}{2t}k_{0}\in K. $$

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

$$ \frac{F(x_{0}+th)-F(x_{0})}{t}+k_{0}\ge_{K} -\frac{F(x_{0}-th)-F(x_{0})}{t}. $$
(5)

Now, let us take − 1 < −t < 0. We have

$$x_{0}-th=t\underbrace{(x_{0}-h)}_{x_{1}}+(1-t)\underbrace{x_{0}}_{x_{2}}. $$

From the α (⋅)-k0 paraconvexity (1) for λ := t, we get

$$F(x_{0}-th)\le_{K} tF(x_{0}-h)+(1-t)F(x_{0})+C\min\{t, 1-t\}\alpha(1)k_{0} $$

By simple calculation, we get

$$-\frac{F(x_{0}-th)-F(x_{0})}{t}- F(x_{0})+F(x_{0}-h)+C\frac{\min\{t, 1-t\}}{t}\alpha(1)k_{0}\in K. $$

Since \(\frac {\min \{t, 1-t\}}{t}=\frac {1-|2t-1|}{2t}\) and the fact that \(\frac {1-|2t-1|}{2t}\le 1\) is bounded, we get

$$-\frac{F(x_{0}-th)-F(x_{0})}{t}- F(x_{0})+F(x_{0}-h)+ C\alpha(1)k_{0}\in K. $$

Hence,

$$-\frac{F(x_{0}-th)-F(x_{0})}{t}- F(x_{0})+F(x_{0}-h)+C\alpha(1)k_{0}\in K. $$

From (5), we get

$$\frac{F(x_{0}+th)-F(x_{0})}{t}-b\ge_{K} 0, $$

where b := F (x0) − F (x0h) − (Cα (1) + 1) k0. Finally,

$$\phi(t) - b \ge_{K} 0\quad \text{ for } 0<t<\delta. $$

Main Result

The proof of the main theorem is based on the following lemma.

Lemma 1

Let Y be a Banach space. Let KY be a closed convex normal cone. Let\({\Phi }: \mathbb {R}_{+} \rightarrow Y\) satisfy the following conditions

  1. (i)

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

  2. (ii)

    for 0 < t1 < t we have \({\Phi }(t)-{\Phi }(t_{1}) + \frac {\alpha (t_{1})}{t_{1}}k_{0}\in K\) for some k0K,

  3. (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

$$ \|{\Phi}(t_{n})\|>\varepsilon. $$
(6)

Let \(x\in A:=\text {co}({\Phi }(t_{n}), n\in \mathbb {N})\). There are positive numbers λ1,λ2,…,λ m and t1,t2…,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

$$\begin{array}{@{}rcl@{}} {\Phi}(t_{1})-{\Phi}(t_{n}) + \frac{\alpha(t_{n})}{t_{n}}k_{0}&\in &K, \\ {\Phi}(t_{2})-{\Phi}(t_{n})+ \frac{\alpha(t_{n})}{t_{n}}k_{0}&\in &K,\\ &\vdots& \\ {\Phi}(t_{m})-{\Phi}(t_{n})+ \frac{\alpha(t_{n})}{t_{n}}k_{0}&\in &K. \end{array} $$

We get

$$x-{\Phi}(t_{n})+ \frac{\alpha(t_{n})}{t_{n}} k_{0}\in K\quad \text{ for all } n>N. $$

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 xA and n > N.

We show that

$$\mathbb{B}_{\beta/2} \cap (A+k_{0}[0,s])=\emptyset $$

for s > 0 satisfying \(\frac {\alpha (t_{n})}{t_{n}}\le s\). To see this, take any ∈ (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

$$0\le_{K} x+ \frac{\alpha(t_{n})}{t_{n}}k_{0}\le_{K} x+\ell k_{0}. $$

By (6) and the normality of K,

$$\beta/2< \| x+ \frac{\alpha(t_{n})}{t_{n}}k_{0}\|\le\| x+\ell k_{0}\|. $$

From the Hahn–Banach theorem applied to \(\mathbb {B}_{\beta /2}\) and (A + k0[0,s]), there is a linear functional yY and r > 0 such that

$$y^{\ast}(x+ \ell k_{0}) > r \quad \text{ for all } x+\ell k_{0}\in A+k_{0}[0,s]. $$

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 : XY be stronglyα (⋅)-k0 paraconvex on a convex set AX with constant C ≥ 0, k0K ∖{0}. Then, the directional derivative

$$F^{\prime}(x_{0};h):=\lim\limits_{t\rightarrow 0^{+}}\frac{F(x_{0}+th)-F(x_{0})}{t} $$

of F at x0 exists for any x0A and any direction 0 ≠ hX,∥h∥ = 1such that x0 + thA for all t sufficiently small.

Proof

Let x0A and let 0 ≠ hX, ∥h∥ = 1 be such that x0 + thA for all t sufficiently small. Let t n 0. For t0 = 0, the α (⋅)-difference quotient by (2) takes the form

$$\phi(t_{n})= \frac{F(x_{0}+t_{n}h)-F(x_{0})}{t_{n}} + C\frac{\alpha(t_{n})}{t_{n}}k_{0}. $$

Let yK. By (4), the sequence a n := y(ϕ (t n )), \(n\in \mathbb {N}\) is bounded from below, i.e.,

$$ a_{n}\ge a:=y^{\ast}(b) \quad \text{for all \textit{n} sufficiently large and } b\in Y. $$

Let us take ε > 0. There is N such that

$$ a_{N}< \underline{a} + \frac{\varepsilon}{2}, $$
(7)

where \(\underline {a}:=\inf \{a_{n}: n\in \mathbb {N}\}\). Since {t n } is decreasing, from (3), we get

$$ a_{N}-a_{n} + C\frac{\alpha(t_{n})}{t_{n}}y^{\ast}(k_{0})\ge 0\quad \text{ for } n> N. $$
(8)

Let \(b_{n}:= C\frac {\alpha (t_{n})}{t_{n}}y^{\ast }(k_{0})\). Since b n → 0 there is N1 such that \(b_{n}\le \frac {\varepsilon }{2}\) for n > N1.

From (7) and (8), we get

$$\underline{a} - \varepsilon < \underline{a} \le a_{n} \le a_{N} +b_{n} \le \underline{a} +\frac{\varepsilon}{2} +b_{n}\le \underline{a}+\varepsilon\quad \text{ for } n>\max\{N,N_{1}\}. $$

Hence, the sequence {a n } is convergent and consequently every sequence {y(ϕ (t n ))} is Cauchy for yK.

Let us take any hY. We show that the sequence {h(ϕ (t n ))} is Cauchy. From the fact that K is normal, we have Y = KK and h = gq with g,qK. Since {g(ϕ (t n ))} and {q(ϕ (t n ))} are Cauchy sequences, there exist N1, N2 such that for \(n,m > \bar {N}:=\max (N_{1}, N_{2})\), we have

$$|g^{\ast}(\phi(t_{n}))- g^{\ast}(\phi(t_{m}))| \le \frac \varepsilon 2 \quad \text{ and }\quad |q^{\ast}(\phi(t_{n}))- q^{\ast}(\phi(t_{m}))| \le \frac \varepsilon 2. $$

For \(n> \bar {N}\), we have

$$\begin{array}{@{}rcl@{}} |h^{\ast}(\phi(t_{n}))-h^{\ast}(\phi(t_{m}))|&=&|g^{\ast}(\phi(t_{n}))-q^{*}(\phi(t_{n}))-g^{\ast}(\phi(t_{m}))+q^{\ast}(\phi(t_{m}))|\\ &\le& \frac \varepsilon 2 + \frac \varepsilon 2=\varepsilon. \end{array} $$

We show that ϕ (t) weakly converges when t → 0+, i.e., there is an y0Y such that for arbitrary t n 0, we have

$$\lim\limits_{n\rightarrow \infty} y^{\ast}(\phi(t_{n})) = y^{\ast}(y_{0}) \quad \text{ for any } y^{\ast} \in Y^{\ast} $$

which is equivalent to

$$ \phi(t)\rightharpoonup y_{0} \quad\text{ when } t\rightarrow 0^{+}. $$
(9)

Since Y is weakly sequentially complete, we need only to show that y0 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,

$$y^{\ast}(\phi(\bar{t}_{n}^{2}))\le y^{\ast}(\phi({t_{n}^{1}}))\le y^{\ast}(\phi(\bar{t}_{n}^{1})) $$

which proves that it must be \({y_{0}^{1}}={y_{0}^{2}}\).

Now, we show that the mapping Φ(t) := ϕ (t) − y0 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 yK such that

$$ y^{\ast}({\Phi}(\bar{t}))=y^{\ast}(\phi(\bar{t})-y_{0})<0. $$
(10)

From inclusion (3) in Proposition 1, we have

$$\phi(\bar{t})-y_{0}-\phi({t})+y_{0} + C\frac{\alpha(t)}{t}k_{0}\in K \quad \text{ for all } t\in (0,\bar{t}). $$

In particular

$$y^{\ast}(\phi(\bar{t})-y_{0}) \ge y^{\ast}\left( \phi({t})-y_{0}-C\frac{\alpha(t)}{t}k_{0}\right)\quad \text{ for all } t\in (0,\bar{t}). $$

And by (10), we get

$$0>y^{\ast}(\phi(\bar{t})-y_{0})\ge y^{\ast}\left( \phi({t})-y_{0}-C\frac{\alpha(t)}{t}k_{0}\right)\quad \text{ for all } t\in (0,\bar{t}). $$

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

$$\lim\limits_{t\rightarrow 0^{+}} \frac{F(x_{0}+th)-F(x_{0})}{t}=y_{0} $$

which completes the proof. □

Remark 1

For K-convex mappings F, i.e., strongly α (⋅)-K paracanovex mappings with constant C = 0 Theorem 2 can be found in [12].

References

  1. Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton–Jacobi Equation and Optimal Control. Progress in Nonlinear Differential Equations and Their Applications, vol. 58. Birkhäuser, Boston (2004)

    MATH  Google Scholar 

  2. Daniilidis, A., Georgiev, P.: Approximate convexity and submonotonicity. J. Math. Anal. Appl. 291, 292–301 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  3. Huang, H., Li, R.: Global error bounds for γ-paraconvex multifunctions. Set-Valued Var. Anal. 19, 487–504 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  4. Huang, H.: Coderivative conditions for error bounds of γ-paraconvex multifunctions. Set-Valued Var. Anal. 20, 567–579 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  5. Jourani, A.: Subdifferentiability and subdifferential monotonicity of γ- paraconvex functions. Control Cybern. 25, 721–737 (1996)

    MathSciNet  MATH  Google Scholar 

  6. Ngai, H.V., Luc, D.T., Théra, M.: Approximate convex functions. J. Nonlinear Convex Anal. 1, 155–176 (2000)

    MathSciNet  MATH  Google Scholar 

  7. Rolewicz, S.: On paraconvex multifunctions. Oper. Res. Verf. (Methods Oper. Res.) 31, 540–546 (1979)

    MathSciNet  MATH  Google Scholar 

  8. Rolewicz, S.: On γ-paraconvex multifunctions. Math. Japonica 24B, 293–300 (1979)

    MathSciNet  MATH  Google Scholar 

  9. Rolewicz, S.: On α (⋅)-paraconvex and strongly α (⋅)-paraconvex functions. Control Cybern. 29, 367–377 (2000)

    MathSciNet  MATH  Google Scholar 

  10. Rolewicz, S.: Differentiability of strongly paraconvex vector-valued functions. Funct. Approx. Comment. Math. 44, 273–277 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  11. Tabor, J., Tabor, J.: Paraconvex, but not strongly, Takagi functions. Control Cybern. 41, 545–559 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Valadier, M.: Sous-Différentiabilité de fonctions convexes à valeurs dans un espace vectoriel ordonné. Math. Scand. 30, 65–74 (1972)

    MathSciNet  Article  MATH  Google Scholar 

  13. Veselý, L., Zajíček, L.: Delta-Convex Mappings between Banach Spaces. Dissertationnes Mathematicae, vol. 289 (1989)

  14. Veselý, L., Zajíček, L.: On D.C. mappings and differences of convex operators. Acta Univ. Carol. Math. Phys. 42, 89–97 (2001)

    MathSciNet  MATH  Google Scholar 

  15. Veselý, L., Zajíček, L.: On connections between delta-convex mappings and convex operators. Proc. Edinb. Math. Soc. 49, 739–751 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  16. Veselý, L., Zajíček, L.: On composition of d.c. functions and mappings. J. Convex Anal. 16, 423–439 (2009)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Krzysztof Leśniewski.

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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 Cone-Paraconvex Mappings. Vietnam J. Math. 46, 381–389 (2018). https://doi.org/10.1007/s10013-018-0290-2

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  • DOI: https://doi.org/10.1007/s10013-018-0290-2

Keywords

  • Directional derivative
  • Normal cones
  • Strongly paraconvex mappings
  • Cone convex mappings
  • Weakly sequentially complete Banach spaces

Mathematics Subject Classification (2010)

  • Primary 49J50
  • Secondary 52A41