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For a bounded closed convex subset C of a Banach space X with the origin as an interior point, we study the convexity issue of proximinal sets G in Banach spaces in the sense of generalized best approximation determined by the Minkowski functional generated by C. Some sufficient conditions such as the smoothness and compactly locally uniform convexity of C, or the weakly uniform convexity of are provided for a proximinal set G to be convex. [ABSTRACT FROM AUTHOR]

Abstract: We explore in arbitrary Banach spaces the Fréchet type -subdifferentials and the limiting subdifferentials for the perturbed distance function determined by a closed subset and a lower semicontinuous function defined on . In particular, upper and lower estimates for the Fréchet type -subdifferentials and for the limiting subdifferentials are provided in terms of the corresponding subdifferentials of the sum of the associated functions and . [Copyright &y& Elsevier]

Abstract: Let be a complete positive -finite measure space and let be a Banach space. We are concerned with the proximinality problem for the best simultaneous approximations to two functions in . Let be a sub--algebra of and a nonempty locally weakly compact convex subset of such that and its dual have the Radon–Nikodym property. We prove that is -simultaneous proximinal in (with the additional assumption that be finite for the case when ). Furthermore, for the special case when , we show that the assumption that the dual of has the Radon–Nikodym property can be removed. [Copyright &y& Elsevier]

Abstract: Let be a nonempty closed subset (resp. nonempty bounded closed subset) of a metric space and . The nearest point problem (resp. the farthest point problem) w.r.t. considered here is to find a point such that (resp. ). We study the well posedness of nearest point problems and farthest point problems in geodesic spaces. We show that if is a space of curvature bounded below then the complement of the set of all points for which the nearest point problem (resp. the farthest point problem) w.r.t. is well posed is -porous in . Our results extend and/or improve some recent results about proximinality in geodesic spaces as well as the corresponding ones previously obtained in uniformly convex Banach spaces. In particular, the result regarding the nearest point problem answers affirmatively an open problem proposed by Kaewcharoen and Kirk [A. Kaewcharoen, W.A. Kirk, Proximinality in geodesic spaces, Abstr. Appl. Anal. 2006 (2006) 1–10]. [Copyright &y& Elsevier]

Abstract: We present some sufficient conditions ensuring the upper semicontinuity and the continuity of the Bregman projection operator and the relative projection operator in terms of the -approximate (weak) compactness for a nonempty closed set in a Banach space . We next present certain sufficient conditions as well as equivalent conditions for the convexity of a Chebyshev subset of a Banach space . Our results extend the corresponding results of [H.H. Bauschke, X.F. Wang, J. Ye and X. M. Yuan, Bregman distances and Chebyshev sets, J. Approx. Theory 159 (2009) 3–25] to infinite dimensional spaces. [Copyright &y& Elsevier]

Abstract: The relationships between the generalized directional derivative of the distance function and the existence of nearest points as well as some geometry properties in Banach spaces are studied. It is proved in the present paper that the condition that for each closed subset of and , the Clarke, Michel-Penot, Dini or modified Dini directional derivative of the distance function is 1 or −1 implying the existence of the nearest points to from is equivalent to being compactly locally uniformly convex. Similar results for uniqueness of the nearest point are also established. [Copyright &y& Elsevier]

Abstract: Let X be a Banach space and Z a nonempty subset of X. Let be a lower semicontinuous function bounded from below and . This paper is concerned with the perturbed optimization problem of finding such that , which is denoted by . The notions of the J-strictly convex with respect to Z and of the Kadec with respect to Z are introduced and used in the present paper. It is proved that if X is a Kadec Banach space with respect to Z and Z is a closed relatively boundedly weakly compact subset, then the set of all for which every minimizing sequence of the problem has a converging subsequence is a dense -subset of , where is the set of all points such that z is a solution of the problem . If additionally and X is J-strictly convex with respect to Z, then the set of all for which the problem is well-posed is a dense -subset of . [Copyright &y& Elsevier]

Abstract: The present paper is concerned with the problem of weighted best simultaneous approximations in Banach spaces. The weighted best simultaneous approximations to sequences from S- and BS-suns in the Banach space are characterized in view of the Kolmogorov conditions. Applications are provided for weighted best simultaneous approximations from RS-sets and strict RS-sets. Our results obtained in the present paper extend and improve all earlier known results in this direction. [Copyright &y& Elsevier]

Abstract: The present paper is concerned with the convergence problem of the variants of the Chebyshev–Halley iteration family with parameters for solving nonlinear operator equations in Banach spaces. Under the assumption that the first derivative of the operator satisfies the Hölder condition of order , a convergence criterion of order for the iteration family is established. An application to a nonlinear Hammerstein integral equation of the second kind is provided. [Copyright &y& Elsevier]

Abstract: Let X be a Banach space and Z a nonempty closed subset of X. Let be an upper semicontinuous function bounded from above. This paper is concerned with the perturbed optimization problem , which is denoted by -sup. We shall prove in the present paper that if Z is a closed boundedly relatively weakly compact nonempty subset, then the set of all for which the problem -sup has a solution is a dense -subset of X. In the case when X is uniformly convex and J is bounded, we will show that the set of all points x in X for which there does not exist such that is a σ-porous subset of X and the set of all points such that there exists a maximizing sequence of the problem -sup which has no convergent subsequence is a σ-porous subset of , where denotes the set of all such that z is in the solution set of -sup. [Copyright &y& Elsevier]

The BCQ and the Abadie CQ for infinite systems of convex inequalities in Banach spaces are characterized in terms of the upper semi–continuity of the convex cones generated by the subdifferentials of active convex functions. Some relationships with other constraint qualifications such as the CPLV and the Slate condition are also studied. Applications in best approximation theory are provided. [ABSTRACT FROM AUTHOR]

Abstract: Let X be a Banach space and Z a nonempty closed subset of X. Let be a lower semicontinuous function bounded from below. This paper is concerned with the perturbed optimization problem , denoted by -inf for . In the case when X is compactly fully 2-convex, it is proved in the present paper that the set of all points x in X for which there does not exist such that is a σ-porous set in X. Furthermore, if X is assumed additionally to be compactly locally uniformly convex, we verify that the set of all points such that the problem -inf fails to be approximately compact, is a σ-porous set in , where denotes the set of all such that . Moreover, a counterexample to which some results of Ni [R.X. Ni, Generic solutions for some perturbed optimization problem in nonreflexive Banach space, J. Math. Anal. Appl. 302 (2005) 417–424] fail is provided. [Copyright &y& Elsevier]

Abstract: The convergence problem of the family of the deformed Euler–Halley iterations with parameters for solving nonlinear operator equations in Banach spaces is studied. Under the assumption that the second derivative of the operator satisfies the Hölder condition, a convergence criterion of the order of the iteration family is established. An application to a nonlinear Hammerstein integral equation of the second kind is provided. [Copyright &y& Elsevier]

Let G be a strict RS-set (resp. an RS-set) in X and let F be a bounded (resp. totally bounded) subset of X satisfying , where is the restricted Chebyshev radius of F with respect to G. It is shown that the restricted Chebyshev center of F with respect to G is strongly unique in the case when X is a real Banach space, and that, under some additional convexity assumptions, the restricted Chebyshev center of F with respect to G is strongly unique of order in the case when X is a complex Banach space. [Copyright &y& Elsevier]

Let X be a real separable strictly convex Banach space and G a nonempty closed subset of X. Let K(X) (resp. Kb(X)) denote the family of all nonempty boundedly compact (resp. compact) convex subsets of X endowed with the Hρ-topology (resp. the Hausdorff distance), KG(X) (resp. KGb(X)) the closure of the set {A∈K(X): A∩G=} (resp. {A∈Kb(X): A∩G=}), and V(G) (resp. Vb(G)) the family of A∈KG(X) (resp. A∈KGb(X)) such that the minimization problem min(A,G) fails to be well-posed. It is proved that for most (in the sense of the Baire category) closed subsets (resp. bounded closed subsets) G of X, V(G) (resp. Vb(G)) is everywhere uncountable in KG(X) (resp. KGb(X)). A similar result for the mutually furthest point problem is also given. [Copyright &y& Elsevier]

Let X be a real strictly convex and Kadec Banach space and G a nonempty closed relatively boundedly weakly compact subset of X. Let B(X) (resp. K(X)) be the family of nonempty bounded closed (resp. compact) subsets of X endowed with the Hausdorff distance and let BG(X) denote the closure of the set {A∈B(X) : A∩G=∅} and KG(X)=BG(X)∩K(X). We introduce the admissible family A of B(X) and prove that EAo(G) (resp. EoA(G)), the set of all subsets F∈A⊆BG(X) (resp. F∈A⊆B(X)) such that the minimization problem min(F,G) (resp. the maximization problem max(F,G)) is well-posed, is a dense Gδ-subset of A. Furthermore, when X is uniformly convex, we prove that A&z.drule;EAo(G) and A&z.drule;EoA(G) are σ-porous in A. [Copyright &y& Elsevier]

Some results on convergence of Newton's method in Banach spaces are established under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak L average. [ABSTRACT FROM AUTHOR]

Let G be a nonempty closed (resp. bounded closed) subset in a strongly convex Banach space X. Let denote the space of all nonempty bounded closed subsets of X endowed with the Hausdorff distance and let denote the closure of the set . We prove that Eo(G) (resp. Eo(G)), the set of all (resp. ) such that the minimization (resp. maximization) problem min(A, G) (resp. ) is well-posed, is residual in (resp. ). [ABSTRACT FROM AUTHOR]

The relationship between directional derivatives of generalized distance functions and the existence of generalized nearest points in Banach spaces is investigated. Let G be any nonempty closed subset in a compact locally uniformly convex Banach space. It is proved that if the one-sided directional derivative of the generalized distance function associated to G at x equals to 1 or −1, then the generalized nearest points to x from G exist. We also give a partial answer (Theorem 3.5) to the open problem put forward by S. Fitzpatrick (1989, Bull. Austral. Math. Soc.39, 233–238). [Copyright &y& Elsevier]

The paper is concerned with the convergence problem of Newton type methods for finding zeros of nonlinear operators in Banach spaces. Some families of nonlinear operators are defined by different Lipschitz conditions and an "universal constant" is introduced so that a unified convergence determination of these methods is established for the defined families. [ABSTRACT FROM AUTHOR]