Ekeland’s variational principle is a fundamental theorem in nonconves analysis. Its general statement is as the following:Ekeland’s Variational Principle"’a:. Let V be a complete metric space, and F: F—*-RU{ ...Ekeland’s variational principle is a fundamental theorem in nonconves analysis. Its general statement is as the following:Ekeland’s Variational Principle"’a:. Let V be a complete metric space, and F: F—*-RU{ + °°} a lower semicontinuous function, not identically +00 and bounded from, below. Let s>0 be given, and a point u^V such thatF(u)<infF+e.vThen there exists some point v £ V such that展开更多
In this article, we have two parts. In the first part, we are concerned with the locally Hlder continuity of quasi-minima of the following integral functional ∫Ωf(x, u, Du)dx, (1) where Ω is an open subset of E...In this article, we have two parts. In the first part, we are concerned with the locally Hlder continuity of quasi-minima of the following integral functional ∫Ωf(x, u, Du)dx, (1) where Ω is an open subset of Euclidean N-space (N ≥ 3), u:Ω → R,the Carath′eodory function f satisfies the critical Sobolev exponent growth condition |Du|^p* |u|^p*-a(x) ≤ f(x,u,Du) ≤ L(|Du|^p+|u|^p* + a(x)), (2) where L≥1, 1pN,p^* = Np/N-p , and a(x) is a nonnegative function that lies in a suitable Lp space. In the second part, we study the locally Hlder continuity of ω-minima of (1). Our method is to compare the ω-minima of (1) with the minima of corresponding function determined by its critical Sobolev exponent growth condition. Finally, we obtain the regularity by Ekeland’s variational principal.展开更多
In this paper, by using p-distances on uniform spaces, we establish a general vectorial Ekeland variational principle (in short EVP), where the objective function is defined on a uniform space and taking values in a...In this paper, by using p-distances on uniform spaces, we establish a general vectorial Ekeland variational principle (in short EVP), where the objective function is defined on a uniform space and taking values in a pre-ordered real linear space and the perturbation involves a p-distance and a monotone function of the objective function. Since p-distances are very extensive, such a form of the perturbation in deed contains many different forms of perturbations appeared in the previous versions of EVP. Besides, we only require the objective function has a very weak property, as a substitute for lower semi-continuity, and only require the domain space (which is a uniform space) has a very weak type of completeness, i.e., completeness with respect to a certain p-distance. Such very weak type of completeness even includes local completeness when the uniform space is a locally convex topological vector space. From the general vectorial EVP, we deduce a general vectorial Caristi's fixed point theorem and a general vectorial Takahashi's nonconvex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. We see that the above general vectorial EVP includes many particular versions of EVP, which extend and complement the related known results.展开更多
The main purpose of this paper is to establish the Ekeland’s variational principle andCaristi’s fixed point theorem in probabilistic metric spaces and to give a direct simple proofof the equivalence between these tw...The main purpose of this paper is to establish the Ekeland’s variational principle andCaristi’s fixed point theorem in probabilistic metric spaces and to give a direct simple proofof the equivalence between these two theorems in the probabilistic metric space. The resultspresented in this paper generalize the corresponding results of [9--12].展开更多
We consider optimal birth control for the McKendrick equation of population dyna-mics.It consists of optimizing a system described by a first order partial differential equationwith nonlo-cal bilinear boundary control...We consider optimal birth control for the McKendrick equation of population dyna-mics.It consists of optimizing a system described by a first order partial differential equationwith nonlo-cal bilinear boundary control.Approximate minimum principles are obtained usingEkeland’s vari ational principle.展开更多
文摘Ekeland’s variational principle is a fundamental theorem in nonconves analysis. Its general statement is as the following:Ekeland’s Variational Principle"’a:. Let V be a complete metric space, and F: F—*-RU{ + °°} a lower semicontinuous function, not identically +00 and bounded from, below. Let s>0 be given, and a point u^V such thatF(u)<infF+e.vThen there exists some point v £ V such that
基金Supported by the Program of Fujian Province-HongKong
文摘In this article, we have two parts. In the first part, we are concerned with the locally Hlder continuity of quasi-minima of the following integral functional ∫Ωf(x, u, Du)dx, (1) where Ω is an open subset of Euclidean N-space (N ≥ 3), u:Ω → R,the Carath′eodory function f satisfies the critical Sobolev exponent growth condition |Du|^p* |u|^p*-a(x) ≤ f(x,u,Du) ≤ L(|Du|^p+|u|^p* + a(x)), (2) where L≥1, 1pN,p^* = Np/N-p , and a(x) is a nonnegative function that lies in a suitable Lp space. In the second part, we study the locally Hlder continuity of ω-minima of (1). Our method is to compare the ω-minima of (1) with the minima of corresponding function determined by its critical Sobolev exponent growth condition. Finally, we obtain the regularity by Ekeland’s variational principal.
基金Supported by National Natural Science Foundation of China (Grant No. 10871141)
文摘In this paper, by using p-distances on uniform spaces, we establish a general vectorial Ekeland variational principle (in short EVP), where the objective function is defined on a uniform space and taking values in a pre-ordered real linear space and the perturbation involves a p-distance and a monotone function of the objective function. Since p-distances are very extensive, such a form of the perturbation in deed contains many different forms of perturbations appeared in the previous versions of EVP. Besides, we only require the objective function has a very weak property, as a substitute for lower semi-continuity, and only require the domain space (which is a uniform space) has a very weak type of completeness, i.e., completeness with respect to a certain p-distance. Such very weak type of completeness even includes local completeness when the uniform space is a locally convex topological vector space. From the general vectorial EVP, we deduce a general vectorial Caristi's fixed point theorem and a general vectorial Takahashi's nonconvex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. We see that the above general vectorial EVP includes many particular versions of EVP, which extend and complement the related known results.
基金The project is supported by National Natural Science Foundation of China
文摘The main purpose of this paper is to establish the Ekeland’s variational principle andCaristi’s fixed point theorem in probabilistic metric spaces and to give a direct simple proofof the equivalence between these two theorems in the probabilistic metric space. The resultspresented in this paper generalize the corresponding results of [9--12].
基金This work was supported in part by a grant from the International Development Research Centre Ottawa,Canada
文摘We consider optimal birth control for the McKendrick equation of population dyna-mics.It consists of optimizing a system described by a first order partial differential equationwith nonlo-cal bilinear boundary control.Approximate minimum principles are obtained usingEkeland’s vari ational principle.