Generalized Hukuhara-Clarke Derivative of Interval-valued Functions and its Properties

Abstract

This paper is devoted to the study of gH-Clarke derivative for interval-valued functions. To find properties of the gH-Clarke derivative, the concepts of limit superior, limit inferior, and sublinear interval-valued functions are studied in the sequel. It is proved that the upper gH-Clarke derivative of a gH-Lipschitz continuous interval-valued function (IVF) always exists. For a convex and gH-Lipschitz IVF, the upper gH-Clarke derivative is found to be identical with the gH-directional derivative. It is observed that the upper gH-Clarke derivative is a sublinear IVF. Several numerical examples are provided to support the entire study.

Appendices

Appendix A Proof of Lemma 1

Proof

Let $\mathbf{Z} =[\underline{z},~\overline{z}]$ and $a, b \in {\mathbb {R}}$.

• (i) If $\mathbf{Z} \succeq \mathbf{0}$, then

  $$\begin{aligned} \underline{z}&\ge 0~\text {and}~\overline{z}\ge 0\\&\quad \implies |a |\underline{z}+|b |\underline{z} \ge |a+b |\underline{z}~\text {and}~|a |\overline{z}+|b |\overline{z} \ge |a+b |\overline{z}\\&\quad \implies |a+b |\odot \mathbf{Z} \preceq |a |\odot \mathbf{Z} \oplus |b |\odot \mathbf{Z} . \end{aligned}$$

• (ii) If $\mathbf{Z} \preceq \mathbf{0}$, then

  $$\begin{aligned}&\underline{z}\le 0~\text {and}~\overline{z}\le 0\\&\quad \implies |a |\underline{z}+|b |\underline{z} \le |a+b |\underline{z}~\text {and}~|a |\overline{z}+|b |\overline{z} \le |a+b |\overline{z}\\&\quad \implies |a+b |\odot \mathbf{Z} \succeq |a |\odot \mathbf{Z} \oplus |b |\odot \mathbf{Z} . \end{aligned}$$

• (iii) If $\mathbf{Z} \nprec \mathbf{0}$, then

  $$\begin{aligned} \overline{z}&\ge 0 \implies |a |\overline{z}+|b |\overline{z} \ge |a+b |\overline{z}\\&\quad \implies |a+b |\odot \mathbf{Z} \nsucc |a |\odot \mathbf{Z} \oplus |b |\odot \mathbf{Z} . \end{aligned}$$

$\square$

Appendix B Proof of Lemma 2

Proof

Let $\mathbf{A} = [\underline{a}, \overline{a}], ~\mathbf{B} = [\underline{b}, \overline{b}],~\mathbf{C} = [\underline{c}, \overline{c}]$ and $\mathbf{D} = [\underline{d}, \overline{d}]$.

• (i) We have the following four possible cases.

  • Case 1. Let $\overline{a}-\overline{c}\ge \underline{a}-\underline{c}$ and $\overline{c}-\overline{b}\ge \underline{c}-\underline{b}$. Then, $\overline{a}-\overline{b}\ge \underline{a}-\underline{b}$ and

    $$\begin{aligned} (\mathbf{A} \ominus _{gH}{} \mathbf{C} )\oplus (\mathbf{C} \ominus _{gH}{} \mathbf{B} )= & {} [\underline{a}-\underline{c}, \overline{a}-\overline{c}]\oplus [\underline{c}-\underline{b},\overline{c}-\overline{b}]\\= & {} [\underline{a}-\underline{b},\overline{a}-\overline{b}]=\mathbf{A} \ominus _{gH}{} \mathbf{B} . \end{aligned}$$

  • Case 2. Let $\overline{a}-\overline{c}\le \underline{a}-\underline{c}$ and $\overline{c}-\overline{b}\le \underline{c}-\underline{b}$. Therefore, $\overline{a}-\overline{b}\le \underline{a}-\underline{b}$ and

    $$\begin{aligned}&(\mathbf{A} \ominus _{gH}{} \mathbf{C} )\oplus (\mathbf{C} \ominus _{gH}{} \mathbf{B} )\\&\quad = [\overline{a}-\overline{c}, \underline{a}-\underline{c}]\oplus [\overline{c}-\overline{b},\underline{c}-\underline{b}]\\&\quad =[\overline{a}-\overline{b},\underline{a}-\underline{b}]=\mathbf{A} \ominus _{gH}{} \mathbf{B} . \end{aligned}$$

  • Case 3. Let $\overline{a}-\overline{c}<\underline{a}-\underline{c}$ and $\overline{c}-\overline{b}>\underline{c}-\underline{b}$. Therefore,

    $$\begin{aligned}&(\mathbf{A} \ominus _{gH}{} \mathbf{C} )\oplus (\mathbf{C} \ominus _{gH}{} \mathbf{B} )\\&\quad = [\overline{a}-\overline{c}, \underline{a}-\underline{c}]\oplus [\underline{c}-\underline{b},\overline{c}-\overline{b}]\\&\quad =[\overline{a}-\overline{c}+\underline{c}-\underline{b}, \underline{a}-\underline{c}+\overline{c}-\overline{b}]. \end{aligned}$$

    If possible, let

    $$\begin{aligned} (\mathbf{A} \ominus _{gH}{} \mathbf{C} )\oplus (\mathbf{C} \ominus _{gH}{} \mathbf{B} ) \prec \mathbf{A} \ominus _{gH}{} \mathbf{B} .\nonumber \\ \end{aligned}$$

    (Appendix B.1)

    If $\overline{a}-\overline{b}\ge \underline{a}-\underline{b}$, then from (Appendix B.1) we get

    $$\begin{aligned}&[\overline{a}-\overline{c}+\underline{c}-\underline{b}, \underline{a}-\underline{c}+\overline{c}-\overline{b}] \prec [\underline{a}-\underline{b},\overline{a}-\overline{b}]\\&\quad \Longrightarrow \underline{a}-\underline{c}+\overline{c}-\overline{b} \le \overline{a}-\overline{b}\\&\quad \Longrightarrow \underline{a}-\underline{c} \le \overline{a}-\overline{c},~\text { which is an impossibility}. \end{aligned}$$

    Further, if $\overline{a}-\overline{b}\le \underline{a}-\underline{b}$, then from (Appendix B.1), we have

    $$\begin{aligned}&[\overline{a}-\overline{c}+\underline{c}-\underline{b}, \underline{a}-\underline{c}+\overline{c}-\overline{b}] \prec [\overline{a}-\overline{b},\underline{a}-\underline{b}]\\&\quad \Longrightarrow \underline{a}-\underline{c}+\overline{c}-\overline{b} \le \underline{a}-\underline{b}\\&\quad \Longrightarrow \overline{c}-\overline{b} \le \underline{c}-\underline{b},~\text {which is an impossibility}. \end{aligned}$$

    Thus, (Appendix B.1) is not true.

  • Case 4. Let $\overline{a}-\overline{c}>\underline{a}-\underline{c}$ and $\overline{c}-\overline{b}<\underline{c}-\underline{b}$. Proceeding as in Case 3 of (i) we can prove that (Appendix B.1) is not true. Hence,

  $$\begin{aligned} (\mathbf{A} \ominus _{gH}{} \mathbf{C} )\oplus (\mathbf{C} \ominus _{gH}{} \mathbf{B} ) \nprec \mathbf{A} \ominus _{gH}{} \mathbf{B} . \end{aligned}$$

• (ii) As ${\Vert \mathbf{B} \ominus _{gH} \mathbf{A} \Vert }_{I({\mathbb {R}})} = \max \{|\underline{b}-\underline{a}|, |\overline{b}-\overline{a}|\},$ we break the proof in two cases.

  • Case 1. If $(L = )~ {\Vert \mathbf{B} \ominus _{gH} \mathbf{A} \Vert }_{I({\mathbb {R}})} = |\underline{b}-\underline{a}|$, then

    $$\begin{aligned} |\underline{b}-\underline{a}| \ge |\overline{b}-\overline{a}|&\implies |\underline{b}-\underline{a}| \ge \overline{b}-\overline{a} \nonumber \\&\implies \overline{b} \le \overline{a}+L.\nonumber \\ \end{aligned}$$

    (Appendix B.2)

    Since $\underline{b}-\underline{a} \le |\underline{b}-\underline{a}|$, then

    $$\begin{aligned} \underline{b} \le \underline{a}+L. \end{aligned}$$

    (Appendix B.3)

    From (Appendix B.2) and (Appendix B.3), we have $\mathbf{B} \preceq \mathbf{A} \oplus [L, L].$

  • Case 2. If $(L = )~ {\Vert \mathbf{B} \ominus _{gH} \mathbf{A} \Vert }_{I({\mathbb {R}})} = |\overline{b}-\overline{a}|$, then

    $$\begin{aligned}&|\underline{b}-\underline{a}| \le |\overline{b}-\overline{a}|\nonumber \\&\quad \implies \underline{b}-\underline{a} \le |\overline{b}-\overline{a}| \implies \underline{b} \le \underline{a}+L. \nonumber \\ \end{aligned}$$

    (Appendix B.4)

    Since $\overline{b}-\overline{a} \le |\overline{b}-\overline{a}|$,

    $$\begin{aligned} \overline{b} \le \overline{a}+L. \end{aligned}$$

    (Appendix B.5)

    From (Appendix B.4) and (Appendix B.5), we obtain $\mathbf{B} \preceq \mathbf{A} \oplus [L, L], ~\text {where}~ L=\Vert \mathbf{B} \ominus _{gH}{} \mathbf{A} \Vert _{I({\mathbb {R}})}.$

• (iii) If possible, let there exist $\mathbf{A} ,~\mathbf{B} ,~\mathbf{C}$ and $\mathbf{D}$ in $I({\mathbb {R}})$ such that

  $$\begin{aligned}&{\Vert (\mathbf{A} \ominus _{gH}{} \mathbf{B} )\ominus _{gH} (\mathbf{C} \ominus _{gH}{} \mathbf{D} ) \Vert }_{I({\mathbb {R}})}\nonumber \\&\quad > \Vert \mathbf{A} \ominus _{gH}{} \mathbf{C} \Vert _{I({\mathbb {R}})} + \Vert \mathbf{B} \ominus _{gH}{} \mathbf{D} \Vert _{I({\mathbb {R}})}.\nonumber \\ \end{aligned}$$

  (Appendix B.6)

  According to the definition of gH-difference of two intervals,

  $$\begin{aligned}&\text {either}~~ \mathbf{A} \ominus _{gH} \mathbf{B} = [\underline{a}-\underline{b}, \overline{a}-\overline{b}]\\&\quad ~~\text {or}~~ \mathbf{A} \ominus _{gH} \mathbf{B} = [\overline{a}-\overline{b}, \underline{a}-\underline{b}], \\&\quad \text {either}~~ \mathbf{C} \ominus _{gH} \mathbf{D} = [\underline{c}-\underline{d}, \overline{c}-\overline{d}]~~\\&\quad \text {or}~~\mathbf{C} \ominus _{gH} \mathbf{D} = [\overline{c}-\overline{d}, \underline{c}-\underline{d}], \\&\quad \text {either}~~ \mathbf{A} \ominus _{gH} \mathbf{C} = [\underline{a}-\underline{c}, \overline{a}-\overline{c}]\\&\quad \text {or}~~ \mathbf{A} \ominus _{gH} \mathbf{C} = [\overline{a}-\overline{c}, \underline{a}-\underline{c}], \end{aligned}$$

  and

  $$\begin{aligned}&\text {either}~~ \mathbf{B} \ominus _{gH} \mathbf{D} = [\underline{b}-\underline{d}, \overline{b}-\overline{d}]~~\text {or}~~\mathbf{B} \ominus _{gH} \mathbf{D} \\&\quad = [\overline{b}-\overline{d}, \underline{b}-\underline{d}]. \end{aligned}$$

  Then, one of the following holds true:

  • (a) $(\mathbf{A} \ominus _{gH} \mathbf{B} )\ominus _{gH}(\mathbf{C} \ominus _{gH} \mathbf{D} )= [\underline{a}-\underline{b}-\underline{c}+\underline{d},~ \overline{a}-\overline{b}-\overline{c}+\overline{d} ]$

  • (b) $(\mathbf{A} \ominus _{gH} \mathbf{B} )\ominus _{gH}(\mathbf{C} \ominus _{gH} \mathbf{D} )= [\underline{a}-\underline{b}-\overline{c}+\overline{d},~ \overline{a}-\overline{b}-\underline{c}+\underline{d} ]$

  • (c) $(\mathbf{A} \ominus _{gH} \mathbf{B} )\ominus _{gH}(\mathbf{C} \ominus _{gH} \mathbf{D} )= [ \overline{a}-\overline{b}-\overline{c}+\overline{d},~\underline{a}-\underline{b}-\underline{c}+\underline{d} ]$

  • (d) $(\mathbf{A} \ominus _{gH} \mathbf{B} )\ominus _{gH}(\mathbf{C} \ominus _{gH} \mathbf{D} )= [\overline{a}-\overline{b}-\underline{c}+\underline{d},~ \underline{a}-\underline{b}-\overline{c}+\overline{d} ]$.

  • Case 1. Let $(\mathbf{A} \ominus _{gH} \mathbf{B} )\ominus _{gH}(\mathbf{C} \ominus _{gH} \mathbf{D} )= [\underline{a}-\underline{b}-\underline{c}+\underline{d},~ \overline{a}-\overline{b}-\overline{c}+\overline{d} ]$.

    • (a) If ${\Vert (\mathbf{A} \ominus _{gH}{} \mathbf{B} )\ominus _{gH} (\mathbf{C} \ominus _{gH}{} \mathbf{D} ) \Vert }_{I({\mathbb {R}})} = |\underline{a}-\underline{b}-\underline{c}+\underline{d} |$, then from equation (Appendix B.6), we have

      $$\begin{aligned} |\underline{a}-\underline{b}-\underline{c}+\underline{d} |> & {} |\underline{a}-\underline{c}|+ |\underline{b}-\underline{d}|\\> & {} |\underline{a}-\underline{b}-\underline{c}+\underline{d} |, \end{aligned}$$

      which is impossible.

    • (b) If ${\Vert (\mathbf{A} \ominus _{gH}{} \mathbf{B} )\ominus _{gH} (\mathbf{C} \ominus _{gH}{} \mathbf{D} ) \Vert }_{I({\mathbb {R}})} = |\overline{a}-\overline{b}-\overline{c}+\overline{d} |$, then from equation (Appendix B.6), we have

      $$\begin{aligned} |\overline{a}-\overline{b}-\overline{c}+\overline{d} |> & {} |\overline{a}-\overline{c}|+ |\overline{b}-\overline{d}|\\> & {} |\overline{a}-\overline{b}-\overline{c}+\overline{d} |, \end{aligned}$$

      which is again impossible.

  • Case 2. Let $(\mathbf{A} \ominus _{gH} \mathbf{B} )\ominus _{gH}(\mathbf{C} \ominus _{gH} \mathbf{D} )= [\overline{a}-\overline{b}-\overline{c}+\overline{d},~ \underline{a}-\underline{b}-\underline{c}+\underline{d} ]$. For this case, two subcases are similar to the Case 1 of (iii) will lead to impossibilities.

  • Case 3. Let $(\mathbf{A} \ominus _{gH} \mathbf{B} )\ominus _{gH}(\mathbf{C} \ominus _{gH} \mathbf{D} )= [\underline{a}-\underline{b}-\overline{c}+\overline{d},~ \overline{a}-\overline{b}-\underline{c}+\underline{d} ]$. Then,

    $$\begin{aligned} \underline{a}-\underline{b} \le \overline{a}-\overline{b}~\text {and}~\overline{c}-\overline{d} \le \underline{c}-\underline{d}.\nonumber \\ \end{aligned}$$

    (Appendix B.7)

    • (a) If ${\Vert (\mathbf{A} \ominus _{gH}{} \mathbf{B} )\ominus _{gH} (\mathbf{C} \ominus _{gH}{} \mathbf{D} ) \Vert }_{I({\mathbb {R}})} = |\overline{a}-\overline{b}-\underline{c}+\underline{d} |$, then $\overline{a}-\overline{b}-\underline{c}+\underline{d} \ge 0.$ From equation (Appendix B.6), we have

      $$\begin{aligned}&|\overline{a}-\overline{b}-\underline{c}+\underline{d} |> |\overline{a}-\overline{c}|+ |\overline{b}-\overline{d}|\\&\quad \implies \overline{c}-\overline{d} > \underline{c}-\underline{d}, \end{aligned}$$

      which is contradictory to (Appendix B.7).

    • (b) If ${\Vert (\mathbf{A} \ominus _{gH}{} \mathbf{B} )\ominus _{gH} (\mathbf{C} \ominus _{gH}{} \mathbf{D} ) \Vert }_{I({\mathbb {R}})} = |\underline{a}-\underline{b}-\overline{c}+\overline{d} |$, then $\underline{a}-\underline{b}-\overline{c}+\overline{d} < 0.$ From equation (Appendix B.6), we have

      $$\begin{aligned}&-(\underline{a}-\underline{b}-\overline{c}+\overline{d}) = |\underline{a}-\underline{b}-\overline{c}+\overline{d} |> |\underline{a}\\&\quad -\underline{c}|+ |\underline{b}-\underline{d}|\implies \overline{c}-\overline{d} > \underline{c}-\underline{d}, \end{aligned}$$

      which is again contradictory to (Appendix B.7).

  • Case 4. Let $(\mathbf{A} \ominus _{gH} \mathbf{B} )\ominus _{gH}(\mathbf{C} \ominus _{gH} \mathbf{D} )= [\overline{a}-\overline{b}-\underline{c}+\underline{d},~\underline{a}-\underline{b}-\overline{c}+\overline{d} ]$. All the two subcases for this case are similar to Case 3 of (iii).

  Hence, (Appendix B.6) is wrong, and thus the result follows.

$\square$