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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.

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Acknowledgements

We extend sincere thanks to the reviewers and editors for their valuable comments to improve the article. The first author is thankful for a research scholarship awarded by the University Grants Commission, Government of India.

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All authors contributed to the study conception and analysis. Material preparation and analysis were performed by Ram Surat Chauhan, Debdas Ghosh, Jaroslav Ramík, and Amit Kumar Debnath. The first draft of the manuscript was written by Ram Surat Chauhan, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Debdas Ghosh.

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Appendices

Appendix A Proof of Lemma 1

Proof

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

  1. (i)

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

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

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

    $$\begin{aligned}&\underline{z}\le 0~\text {and}~\overline{z}\le 0\\&\quad \implies |a |\underline{z}+|b |\underline{z} \le |a+b |\underline{z}~\text {and}~|a |\overline{z}+|b |\overline{z} \le |a+b |\overline{z}\\&\quad \implies |a+b |\odot \mathbf{Z} \succeq |a |\odot \mathbf{Z} \oplus |b |\odot \mathbf{Z} . \end{aligned}$$
  3. (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 \(\mathbf{D} = [\underline{d}, \overline{d}]\).

  1. (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,

    $$\begin{aligned} (\mathbf{A} \ominus _{gH}{} \mathbf{C} )\oplus (\mathbf{C} \ominus _{gH}{} \mathbf{B} ) \nprec \mathbf{A} \ominus _{gH}{} \mathbf{B} . \end{aligned}$$
  2. (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 \(\mathbf{B} \preceq \mathbf{A} \oplus [L, L], ~\text {where}~ L=\Vert \mathbf{B} \ominus _{gH}{} \mathbf{A} \Vert _{I({\mathbb {R}})}.\)

  3. (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:

    1. (a)

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

    2. (b)

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

    3. (c)

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

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

      1. (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.

      2. (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)
      1. (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).

      2. (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.

\(\square \)

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Chauhan, R.S., Ghosh, D., Ramík, J. et al. Generalized Hukuhara-Clarke Derivative of Interval-valued Functions and its Properties. Soft Comput 25, 14629–14643 (2021). https://doi.org/10.1007/s00500-021-06251-w

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Keywords

  • Interval-valued functions
  • Upper gH-Clarke derivative
  • Sublinear IVF
  • gH-Lipschitz function