Prederivatives of gamma paraconvex set-valued maps and Pareto optimality conditions for set optimization problems

Prederivatives play an important role in the research of set optimization problems. First, we establish several existence theorems of prederivatives for γ-paraconvex set-valued mappings in Banach spaces with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma>0$\end{document}γ>0. Then, in terms of prederivatives, we establish both necessary and sufficient conditions for the existence of Pareto minimal solution of set optimization problems.


Introduction
Let X and Y be Banach spaces. We say that G : X ⇒ Y is a set-valued mapping if G(x) is a subset of Y for all x ∈ X. Set-valued problems occur in many situations, such as control problems, feasibility problems, optimality problems, equilibrium problems and variational inequality problems. A powerful tool dealing with set-valued problems is set-valued analysis. We refer the reader to the references [-] for more knowledge about set-valued analysis and its applications.
In a pioneering work [], Ioffe introduced a notion of prederivative which can be viewed as an extension of Clarke generalized gradient. It is well known that the prederivative is an effective tool in dealing with nondifferentiable mapping of nonsmooth analysis. In contrast with the derivative, the prederivative may not be unique. However, in terms of prederivatives, one can establish an inverse function theorem and implicit theorem and solve nondifferential inclusion problems []. In the later publication of Pang [, ], and Gaydu, Geoffroy and Jean-Alexis [], some notions of prederivatives were posed and further studied. In , Gaydu, Geoffroy and Marcelin [] studied the existence of some kinds of prederivatives of convex set-valued mappings and established necessary and sufficient optimality conditions for the weak minimizers and the strong minimizers of set optimization problems. γ -paraconvex set-valued mappings are an extension of convex set-valued mappings, and were studied by some researchers [, ]. Moreover, in set optimization problems, Pareto minimizers are more suitable than weak minimizers and strong minimizers in practice [, ]. Now, two natural questions are posed. Can we establish some existence results of some kinds of prederivatives for γ -paraconvex set-valued mappings? Can we give optimality conditions for the Pareto minimizers of set optimization problems by prederivatives?
In this paper, we firstly establish several existence theorems of prederivatives for γparaconvex set-valued mappings and cone-γ -paraconvex set-valued mappings. Then we establish necessary and sufficient optimality conditions for the Pareto minimizers of set optimization problems in terms of prederivatives.

Preliminaries
Throughout this paper, unless stated otherwise, we always assume that X and Y are real Banach spaces and G : X ⇒ Y is a set-valued mapping. The domain of G is defined by The graph of G is defined by We say that G is a closed set-valued mapping if Gr(G) is a closed subset of X × Y . We say that G has convex values if G(x) is a convex subset of Y for any x ∈ X. Let be a subset of X; we use cl( ) to denote the closure of , int( ) to denote the interior of . We use B X and B Y to denote the closed unit ball of X and Y , respectively. Letx ∈ X. We use N(x) to denote all open neighborhoods ofx. Let C ⊆ Y be a nonempty set. We say that C is a cone, if λc ∈ C for any c ∈ C and λ ≥ . We say that C is pointed if C ∩ (-C) = {}. Define G + C : X ⇒ Y as The following definition is needed in the sequel.
Remark . In the special case of η =  and C = {}, C-γ -paraconvex set-valued mappings reduce to convex set-valued mappings.

Definition . ([])
We say that G is Lipschitz continuous atx if there exist l >  and U ∈ N(x) such that If the above equation holds on U = , then we say that G is Lipschitz continuous on .
(i) is called an outer prederivative of G atx, if for any δ >  there exists U ∈ N(x) such that

Prederivatives of gamma paraconvex set-valued mappings
In this section, we establish the existence results of pseudo strict prederivatives for γparaconvex set-valued mappings and strict prederivatives for C-γ -paraconvex set-valued mappings, respectively.
Then, for any y ∈ȳ This implies that Then, for anyδ > , is a neighborhood ofx, which is very close to the openness property of G - at (x,ȳ). However, the coefficients δ and η are fixed in our assumption.

Lemma . ([, Lemma ]) Let A, B and D are subsets of X. If B is a closed convex set, D is a bounded set and A
Then the following conclusions hold: Then Since G is a C-γ -paraconvex set-valued mapping with modulus r, it is easy to verify that G + C is a γ -paraconvex set-valued mapping with modulus r. Taking into account inequality (.), we have Due to the convexity of (G + C)(x) for each x ∈ X, we have Adding θ (G + C)(x  ) on both sides of equation (.), and using (.), we get Clearly, C + θ C = C + C = C = C since C is a convex cone. Therefore, the above equation can be rewritten as Therefore z ∈x + αB X . As x  , z ∈x + αB X , combined with (.), we have By the assumption (.), is a closed convex set and A(z) is a bounded set, it follows from Lemma . and (.) that where the last inequality holds since λ = α  . Therefore, G + C is Lipschitz with modulus η α + r( α  ) γ - onx + α  B X since x  and x  are two arbitrary elements ofx + α  B X .
(ii) Let : X ⇒ Y be defined by Clearly, is a positively homogeneous mapping with bounded closed values. By (.), we get for any δ > . This implies that is a strict prederivative of G + C at each x ∈x + α  B X . (iii) Since C is a cone, it follows from (.) that  ∈ ( + C)(), and for any t >  and x ∈ X, and hence + C is positively homogeneous. Letx ∈x + α  B X . Then there existsr >  such thatx +rB X ⊆x + α  B X . Since  ∈ C, it follows from (.) that for any δ > , Therefore, + C is a strict prederivative of G atx.
Remark . In [, Theorem .], Gaydu, Geoffroy and Marcelin proved the following result. Let Y be a finite dimensional Banach space, C ⊆ Y be a nonempty closed convex cone, G : X ⇒ Y be a C-convex set-valued mapping,x ∈ int(dom(G)). Assume that there exist α >  and η >  such that G(x)+cl(C) is a closed set and G(x) ⊆ ηB Y for all x ∈x+αB X . Then there exists U ∈ N(x) such that (i) G + C is Lipschitz on U; (ii) there exists a positively homogeneous mapping : X ⇒ Y with bounded closed values such that is a strict prederivative of G + C at each x ∈ U; (iii) + C is a strict prederivative of G at each x ∈ U.
In contrast with [, Theorem .], Theorem . has some improvements. Firstly, we extend Y from finite dimensional spaces to general Banach spaces. Secondly, we extend G from C-convex set-valued mappings to C-γ -paraconvex set-valued mappings. Thirdly, we do not need the boundedness of G(x).
In the following, we give an example to illustrate Theorem ..
It follows from [, Example .] that G is a C--paraconvex set-valued mapping with modulus , but not a C-convex set-valued mapping. Takex = , A(x) = ||x| -| and E(x) = R + . Then for all x ∈x + B X . All conditions of Theorem . are justified. By Theorem ., G + C is Lipschitz onx +   B X with modulus , and (·) = | · |B Y satisfies (ii) and (iii) of Theorem . onx +   B X .
Corollary . Let C ⊆ Y be a nonempty closed convex cone, η > , γ ≥ , r > , α > , G : X ⇒ Y be a C-γ -paraconvex set-valued mapping with modulus r andx + αB X ⊆ Dom(G). Assume that Let : X ⇒ Y be defined by Then the following conclusions hold: Then the conclusions follow from Theorem . directly.

Pareto minimizer and prederivative
In this section, we always assume that C is a pointed closed convex cone of Y . Consider the following set optimization problem: (SP) : where is a nonempty closed subset of X with Dom(G) ∩ = ∅. First, we establish a necessary condition for Pareto minimizers of the optimization problem (SP).
Theorem . Letū ∈ int( ), (ū,v) ∈ Gr(G). Suppose that is a pseudo strict prederivative of G at (ū,v) and (ū,v) is a Pareto minimizer of the optimization problem (SP). Then, for any δ >  and u ∈ , Proof Let (ū,v) be a Pareto minimizer of the optimization problem (SP). Suppose that the conclusion is not true. Then there exist δ  >  and u  ∈ such that Since is a pseudo strict prederivative of G at (ū,v), there exist η  >  and η  >  such that Combined (.) with (.), we get As G(ū) ⊆v + C, we get It follows from (.) that Therefore, (ū,v) is a Pareto minimizer of the optimization problem (SP) since u is an arbitrary element of .
Remark . In [], Gaydu, Geoffroy and Marcelin established necessary condition and sufficient conditions for the weak minimizers and the strong minimizers of the optimization problem (SP). It is well known that each strong minimizer is a Pareto minimizer and each Pareto minimizer is a weak minimizer, but the converses are not true.
In the following, we give an example to illustrate Theorem ..

Conclusion
In this paper, we establish two existence theorems of prederivatives for γ -paraconvex setvalued mappings, and give optimality conditions for the Pareto minimizers of set optimization problems. These results improve the corresponding one obtained in []. Moreover, the coefficients in Theorems . and . can be calculated. Theorems . and . give sufficient conditions for the existence of of Theorem ..