We investigate the connections between UC and UC* properties for ordered pairs of subsets (A,B) in metric spaces, which are involved in the study of existence and uniqueness of best proximity points. We show that the $UC^{*}$ property is included into the UC property. We introduce some new notions: bounded UC (BUC) property and uniformly convex set about a function. We prove that these new notions are generalizations of the $UC$ property and that both of them are sufficient for to ensure existence and uniqueness of best proximity points. We show that these two new notions are different from a uniform convexity and even from a strict convexity. If we consider the underlying space to be a Banach space we find a sufficient condition which ensures that from the UC property it follows the uniform convexity of the underlying Banach space. We illustrate the new notions with examples. We present an example of a cyclic contraction T in a space, which is not even strictly convex and the ordered pair (A,B) has not the UC property, but has the $BUC$ property and thus there is a unique best proximity point of T in A.
On coupled best proximity points and Ulam-Hyers stability
对于一致凸巴拿赫空间$X$的两个非空闭有界凸子集$A$和$B$，考虑满足$T(A,B) \subset B$和$T(B, A) \subset A$的映射$T:(A \times B) \cup (B \times A) \rightarrow A \cup B$.本文证明了当$T$是p-循环压缩映射和p-循环非扩张映射时，耦合最佳邻近点的存在性.最佳邻近点问题的Ulam-Hyers稳定性也进行了研究。
Best proximity point results for almost contraction and application to nonlinear differential equation
Brinde [Approximating不动点of weak contractions using the Picard itration，Nonlinear Anal. Forum 9（2004），43-53]引入了几乎压缩映射，并证明了几乎压缩映射的Banach压缩原理。引入了集值几乎$Θ$ -压缩映射的概念，并给出了这类映射的最佳邻近点结果。作为应用，得到了弱单值$Θ$ -压缩映射的最佳邻近点和不动点的结果。一个例子来支持本文提出的结果。最后给出了一个非线性微分方程的应用。
Best proximity pair results for relatively nonexpansive mappings in geodesic spaces
给定度量空间中的两个非空子集$A$和$B$，映射$T : A \cup B \rightarrow A \cup B$是相对非扩张的，如果$d(Tx,Ty) \leq d(x,y) \text{for every} x\in A, y\in B.$这样的映射的最佳邻近点是点$x \in A \cup B$，使得$d(x,Tx)=\text{dist}(A,B)$。本文推广了[A.A.西弗吉尼亚州埃尔德雷德Kirk，P. Veeramani，Proximal normal structure and relatively nonexpansive mappings，Studia Math.，171（2005），283-293]对于巴拿赫空间到更一般度量空间的相对非扩张映射。即在Busemann凸自反度量空间中，给出了循环和非循环相对非扩张映象最佳邻近点的存在性结果。在CAT（0）和一致凸测地空间的背景下，证明了一些特殊的结果.最后，我们表明，邻近的正常结构是一个充分的，但不是必要条件存在于$A \times B$的一对最佳邻近点。
Best proximity point results of operator type condensing mappings via measure of noncompactness and application to an infinite system of differential equations in $c_0$ and $\ell_1$
Punjab Engineering College (Deemed to be University)（旁遮普工程学院（视为大学））
Some remarks on "Convergence of Picard's iteration using projection algorithm for noncyclic contractions" [Indag. Math. 30 (2019) 227--239]
本文首先利用投影算子证明了在严格凸巴拿赫空间中，循环相对非扩张映象的最佳邻近点的存在性等价于非循环相对非扩张映象的最佳邻近对的存在性.这样，我们就得到了“Proximal normal structure and relatively nonexpansive mappings”，Studia Math.，(171（2005）283- 293）。然后，我们讨论了最佳邻近对非循环压缩的收敛性，通过应用循环压缩的迭代序列的收敛性，并表明最近发表在Indag的一篇论文的收敛方法。数学，30（1）（2019）227- 239是从皮卡德的迭代序列中精确获得的。
