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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

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This work investigates a possibility degree-based micro immune optimization approach to seek the optimal solution of nonlinear interval number programming with constraints. Such approach is designed under the guideline of the theoretical results acquired in the current work, relying upon interval arithmetic rules, interval order relation and immune theory. It involves in two phases of optimization. The first phase, based on a new possibility degree approach, assumes searching efficient solutions of natural interval extension optimization. This executes five modules -constraint bound handling, population division, dynamic proliferation, mutation and selection, with the help of a varying threshold of interval bound. The second phase collects the optimal solution(s) from these efficient solutions after optimizing the bounds of their objective intervals, in terms of the theoretical results. The numerical experiments illustrated that such approach with high efficiency performs well over one recent nested genetic algorithm and is of potential use for complex interval number programming.

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