Abstract:
In this paper, a sufficient condition of the strictly B-preinvex function is firstly obtained. Then some properties of the strictly B-preinvex function are shown. Finally, some results for the minimization problem which the objective function is strictly B-preinvex are presented.

Abstract:
In this paper, various characterizations of optimal solution sets of nonsmooth B-preinvex optimization problems with inequality constrains are given. Firstly, making use of Clarke’s subdifferential, we establish the optimality condition for this kind of optimization problem; secondly, we presented a property about the solution set S of constrained B-preinvex optimization proble; finally, five equivalent characterizations of the solution set are obtained, that is,* An example is given to illustrate that five solution sets are equal, i.e. S=｛0｝.(* Indicates a formula, please see the full text)

Abstract:
The paper gives a class fo new generalized convex function-strongly G-preinvex functions, it is a true generalization of strong preinvex function. First, three, examples have been got to show that it's existence, and strongly G-preinvex function is different from G-preinvex function and strictly G-preinvex function. Then, we discuses three properties of strongly G-preinvex function. Finally, we give a sufficient condition about strongly G-preinvex function under the case that G-preinvex function,namely, Let the set * is invex set with * is satisfied with condition * is a G-preinvex function. if * have * . Then, * is a strongly G-preinvex function on K with respect to *.(* Indicates a formula, please see the full text)

Abstract:
This note studies the optimality conditions of vector optimization problems involving generalized convexity in locally convex spaces. Based upon the concept of Dini set-valued directional derivatives, the necessary and sufficient optimality conditions are established for Henig proper and strong minimal solutions respectively in generalized preinvex vector optimization problems.

Abstract:
In this paper, an equivalent condition for a clas of re-preinex function is etablished. A characterization for a twice continuously dierentiable r-preinvex function is obtained by the equivalent condition. Under ome suitable condition, the following result has been proved: Let * be open invex et with respect to * and * satisfy condition * defined on X is twice continuously dierentiable and satifies condition D. Then f is r-preinvex function with respect to * is and only if *. Our results improve and generalize some known result.(* Indicates a formula, please see the full text)

Abstract:
Generalized convexity has been playing an important role in mathematical programming . In this paper, an equivalent condition of twice continuously differentiable preinvex function is established by transforming multivariate real-valued function into univariate real-valued function. Suppose that X be open invex set with respect to η,ηsatisfies condition C , f be twice continuously differentiable and satisfies condition D. Then f is preinvex function with respect to η,ηsatisfies condition C, f be twice continuously differentible and satisfiles condition D. then f is preinvex function with respect to η if and only if νx,y∈X,η(x,y)tV2f(x)η(x,y)≥0. Our results provide new thoughts to verify the preinvexity of function and also generalize some known results.

Backstepping technique
usually adopts back
step design to construct the Lyapunov function gradually, and then to design
the corresponding virtue controller. The backstepping technique based on error
also adopts back
step design process, but the design of virtue controllers depends on the corresponding
errors which are designed to satisfy some expected behaviors. Six different error
equations are deduced by changing the results of the virtue controls
arbitrarily while guaranteeing the system behaviors such
as stability, and an example shows the
effectiveness of these six versions. Simulated results illustrate that thesesix versions of backstepping technique based on error are effective.

Abstract:
In this paper we introduce the notion of an -metric, as a function valued distance mapping, on a set X and we investigate the theory of -metrics paces. We show that every metric space may be viewed as an F-metric space and every -metric space (X,δ) can be regarded as a topological space (X,τδ). In addition, we prove that the category of the so-called extended F-metric spaces properly contains the category of metric spaces. We also introduce the concept of an ` -metric space as a completion of an -metric space and, as an application to topology, we prove that each normal topological space is ` -metrizable.

Abstract:
An equivalent condition of r-preinvexity was given hy condition C which was introduced by S.R. Mohan and S.K. Neogy.In this paper another proof is provided about the equivalent condition by the use of the conclusion which upper semicontinuous function has maximum on compact set. That is, let K he an open invex set with respect to η and η satisfies condition C. Let f be a upper semicontinuous function that satisfies f(y+η(x,y))≤f(x)，Ax,Y∈K. Then f is a r-preinvex function for the same ηif and only if Eα∈(0,1),Ax,y∈K,s.t.f(y+aη(x,y))≤log(αerf(x))+(1-α)erf(y))1/r,r≠0。f(y+aη(x,y))≤af(x)+(1-α)f(y)),r=0 The proof is absence of the assumption which set K is open and A is dense on [0,1].