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Discrete Opial type inequalities for interval-valued functions

Dafang Zhao, Xuexiao You, Delfim F. M. Torres

Abstract

We introduce the forward (backward) gH-difference operator of interval sequences, and establish some new discrete Opial type inequalities for interval-valued functions. Further, we obtain generalizations of classical discrete Opial type inequalities. Some examples are presented to illustrate our results.

Discrete Opial type inequalities for interval-valued functions

Abstract

We introduce the forward (backward) gH-difference operator of interval sequences, and establish some new discrete Opial type inequalities for interval-valued functions. Further, we obtain generalizations of classical discrete Opial type inequalities. Some examples are presented to illustrate our results.
Paper Structure (5 sections, 18 theorems, 66 equations)

This paper contains 5 sections, 18 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Opial type inequalities involving the backward/nabla gH-difference operator
  4. Opial type inequalities involving the forward/delta gH-difference operator
  5. Conclusions

Key Result

Theorem 2.1

Let $F\in C^{1}[0,h]$, $F(0)=F(h)=0$ and $F(t)>0$ for $t\in(0,h)$. Then, where $\frac{h}{4}$ is the best possible.

Theorems & Definitions (32)

  • Theorem 2.1: continuous Opial inequality O60
  • Theorem 2.2: discrete Opial inequality AP95
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3: cf. L68
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 22 more