Corrigendum

Abstract

We give the correct statements of Theorem 4.4 and its consequences in Yuan, “Fixed point theorem and related nonlinear analysis by the best approximation method in p-vector spaces” (Numer. Funct. Anal. Optimiz. 44(4): 221 - 295. DOI: 10.1080/01630563.2023.2167088) [1].

Keywords:

• Best approximation
• fixed point
• Leray-Schauder alternative
• locally p-convex spaces
• nonlinear alternative
• optimization
• p-convex
• p-inward and p-outward set
• Schauder theorem

Mathematics Subject Classification:

• 47H04
• 47H10
• 46A16
• 46A55
• 49J27
• 49J35
• 52A07
• 54C60
• 54H25
• 55M20
• 91B50

By following the notation used as in Yuan [1], Theorem 4.4 in [1] should be replaced by the following one (see also Theorem 4.4 and the comments of Remark 4.3 by Yuan [2]), we are sorry for this inconvenience caused by the mistakes due to the manuscript’s preparation and its reading-in-proof.

Theorem 1.

Let K be a non-empty compact s-convex subset of a Hausdorff locally p-convex space X, where $p,s\in (0,1]$. If $T:K\to 2^K$ is an upper semicontinuous set-valued mapping with non-empty closed p-convex values, then T has a fixed point in K.

As the consequence of 1, the statement of Corollary 4.1 in [1] should be replaced by the following one.

Corollary 1.

If K is a nonempty closed s-convex subset of a Hausdorff locally convex space X, where $s\in (0,1]$, then any compact upper semicontinuous set-valued mapping $T:K\to 2^K$ with non-empty closed convex values, has at least one fixed point.

Theorem 4.5 in [1] should be replaced by the following one.

Theorem 2.

If K is a nonempty compact p-convex subset of a Hausdorff locally p-convex space X, where $p\in (0,1]$, then any upper semicontinuous set-valued mapping $T:K\to 2^K$ with nonempty closed p-convex values, has at least one fixed point.

By following the same argument used by Theorem 4.3 in [1], Theorem 4.6 in [1] should be replaced by the following one.

Theorem 3.

If K is a nonempty closed s-convex subset of a Hausdorff locally p-convex space X, where $s,p\in (0,1]$, then any compact upper semicontinuous set-valued mapping $T:K\to 2^K$ with nonempty closed p-convex values, has at least one fixed point.

Theorem 4.9 in [1] should be replaced by the following one.

Theorem 4.

If K is a nonempty closed p-convex subset of a Hausdorff locally p-convex space X, where $p\in (0,1]$, then any upper semicontinuous condensing set-valued mapping $T:K\to 2^K$ with nonempty closed p-convex values, has at least one fixed point.

Corollary 4.6 in [1] should be replaced by the following one.

Corollary 2.

Let K be a non-empty closed convex subset of a Hausdorff locally convex space X. They any upper semicontinuous condensing set-valued mapping $T:K\to 2^K$ with non-empty closed convex values, has at least one fixed point.

Also, the general statement for the conditions “a (Hausdorff) TVS or locally p-convex space E” and “F is with non-empty convex closed values” from Theorem 5.1 to 5.6, Corollary 5.1 to Corollary 5.2; Theorem 6.1 to Theorem 6.6, Corollary 6.1 to Corollary 6.2; Theorem 7.1 to Theorem 7.6, and Corollary 7.1 in [1] should be replaced by “a (Hausdorff) locally p-convex space E” , and “F is with non-empty p-convex closed values”, respectively, whenever it is applicable for $p\in (0,1]$.

Finally, the statement for the condition “F is with non-empty convex closed values” from Theorem 8.1 to Theorem 8.8, and Corollary 8.1 to Corollary 8.2 in [1] should be replaced by “F is with non-empty p-convex closed values” whenever it is applicable for $p\in (0,1]$.

Acknowledgments

The author expresses his thanks to Professor M.Nashed, the chief editor of Numer. Funct. Anal. Optimiz, and the help provided by the production department of the publisher.

Disclosure statement

The author declares that he has no conflict of interest.