Document Type : Research Paper

**Authors**

Shahid Bahonar university of Kerman

**Abstract**

In this paper, we study ε-generalized weak subdifferential for vector valued functions defined on a real ordered topological vector space *X*. We give various characterizations of ε-generalized weak subdifferential for this class of functions. It is well known that if the function *f *: *X *→ R is subdifferentiable at *x*0 ∈ *X*, then *f *has a global minimizer at *x*0 if and only if 0 ∈ ∂ *f*(*x*0). We show that a similar result can be obtained for ε-generalized weak subdifferential. Finally, we investigate some relations between ε-directional derivative and ε-generalized weak subdifferential. In fact, in the classical subdifferential theory, it is well known that if the function *f *: *X *→ R is subdifferentiable at *x*0 ∈ *X *and it has directional derivative at *x*0 in the direction *u *∈ *X*, then the relation *f *′(*x*0, *u*) ≥ ⟨*u*, *x*∗⟩, ∀ *x*∗ ∈ ∂ *f*(*x*0) is satisfied. We prove that a similar result can be obtained for ε- generalized weak subdifferential.

**Keywords**

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[3] J.M. Borwein,

[4] R.N. Gasimov,

[5] Guang-ya Chen, Xuexiang Huang and Xiaogi Yang,

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[6] J. Jahn,

[7] Y. Kuc ¨ uk, L. Ataserer and M. K ¨ uc ¨ uk, ¨

[8] R.T. Rockafellar,

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[10] J. Zowe,

69-83.

September 2015

Pages 65-80