Properties of interval-valued function space under the gH-difference and their application to semi-linear interval differential equations

Abstract

The conventional subtraction arithmetic on interval numbers makes studies on interval systems difficult because of irreversibility on addition, whereas the gH-difference as a popular concept can ensure interval analysis to be a valuable research branch like real analysis. However, many properties of interval numbers still remain open. This work focuses on developing a complete normed quasi-linear space composed of continuous interval-valued functions, in which some fundamental properties of continuity, differentiability, and integrability are discussed based on the gH-difference, the gH-derivative, and the Hausdorff-Pompeiu metric. Such properties are adopted to investigate semi-linear interval differential equations. While the existence and uniqueness of the (i)-or (ii)-solution are studied, a necessary condition that the (i)-and the (ii)-solutions to be strong solutions is obtained. For such a kind of equation it is demonstrated that there exists at least a strong solution under certain assumptions. MSC: 65G40; 46C99; 34A12

Stability of Set Differential Equations in Fréchet Spaces

“…where X i 0 denotes the projection of X 0 . From condition B2, (10), and the previously stated Property 2.1, we have…”

“…Nowadays, the theory of SDEs has been developed into an independent subject area. There are a few results on the existence, stability, and other properties of solutions for various equations, such as SDEs [2][3][4][5][6][7][8][9][10][11], set functional differential equations [12][13][14][15][16], set integrodifferential equations [17][18][19][20], SDEs on time scales [21,22], SDEs with causal operators [23][24][25][26][27], and others [15,[28][29][30][31], and references are given therein. Systematic development of set differential equations has been provided by Lakshmikantham et al [32] and Martynyuk [33].…”

Solvability Criteria for Uncertain Differential Equations and Their Applicability in an Economic Lot-Size Model with a Type-2 Interval Phenomenon

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On differential equations with interval coefficients

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