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Existence of solutions for implicit fuzzy differential inclusions

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Abstract

A class of implicit fuzzy differential inclusions (IFDIs) are introduced and studied. Some existence theorems under different conditions are proved with the selection theorems for the open situation and the closed situation, respectively. A viable solution for a closed IFDI is proved to exist under the tangential condition. As an application, an implicit fuzzy differential equation, which comes from the drilling dynamics in petroleum engineering, is analyzed numerically. The obtained results can improve and extend some known results for fuzzy differential inclusions (FDIs) and fuzzy differential equations (FDEs), which might be helpful in the analysis of fuzzy dynamic systems.

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Abbreviations

s :

distance from the measured position to the bottom along the string

T :

axial tension of the component

M t :

torque of the string

N :

distributional stress between the string and the hole wall

N b :

binormal component of N

N n :

principal normal component of N

q :

linear weight of the string submerged in the drill fluid

E :

Young’s modulus

I :

string inertia moment

r :

string radius

µ:

friction coefficient between the string and the hole wall

g :

unit vector in the gravitational direction

t :

tangential unit vector along the borehole

n :

unit vector in the normal direction of the borehole

b :

unit vector in the binormal direction of the borehole

k b :

curvature of the borehole

k n :

torsion of the borehole

M b :

flexure moment of the borehole

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Author information

Authors and Affiliations

  1. State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Chengdu, 610059, China

    Chao Min, Zhibin Liu & Liehui Zhang

  2. School of Science, Southwest Petroleum University, Chengdu, 610500, China

    Chao Min & Zhibin Liu

  3. Department of Mathematics, Sichuan University, Chengdu, 610064, China

    Nanjing Huang

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  1. Chao Min
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  2. Nanjing Huang
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Correspondence to Nanjing Huang.

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Project supported by the National Science Fund for Distinguished Young Scholars of China (No. 51125019), the National Natural Science Foundation of China (No. 11171237), and the Scientific Research Fund of Sichuan Provincial Education Department (No. 11ZA024)

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Min, C., Huang, N., Liu, Z. et al. Existence of solutions for implicit fuzzy differential inclusions. Appl. Math. Mech.-Engl. Ed. 36, 401–416 (2015). https://doi.org/10.1007/s10483-015-1914-6

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  • DOI: https://doi.org/10.1007/s10483-015-1914-6

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