On Annihilated Points and Approximate Fixed Points of General Higher-Order Nonexpansive Mappings

Abstract

In this paper, we extend the results obtained by Ezearn on annihilated points for his higher-order nonexpansive mappings to the context of general higher-order nonexpansive mappings. Precisely in his thesis, Ezearn introduced the concept of annihilated points, which extends the notion of fixed points, and it is only meaningful in the context of higher-order nonexpansive mappings and gave some mild conditions when the annihilated points could exist in strictly convex Banach spaces. In the last direction, we also extend Ezearn’s result on the approximate fixed point sequence for higher-order nonexpansive mappings to general higher-order nonexpansive mappings.

1. Introduction

Given a complete metric space , the most well-studied examples of such mappings are those that can be immediately put in the form

For all where is a fixed real number. Such mappings are referred to as Lipschitz continuous mappings. Lipschitz continuous mappings are generally classified into three categories: is a (i)contraction mapping if (ii)nonexpansive mapping if (iii)expansive mapping if

In [1], the concept of mean nonexpansive mappings was introduced which is often seen as a generalization of nonexpansive mappings. Thus, let be a nonempty subset of a Banach space , and let be a self-mapping on . Then is called a mean nonexpansive (or -nonexpansive) if For all and for some , we have for all , and .

Clearly, it is seen that all nonexpansive mappings are mean nonexpansive mappings, but the reverse is not always true, as demonstrated in ([2], Examples 2.3 and 2.4). A more general class of -nonexpansive maps was further introduced in [1]. That is, a self-map on a subset of a Banach space is called -nonexpansive if

For all and for some , we have for all , and for some . It is obvious that -nonexpansive map for is also -nonexpansive, but the reverse is not always true, as shown in [3].

Now, given a metric space , a more general class of mappings which extend inequality (1) can be put in the following form: where and , for all . Such mappings are called higher-order Lipschitz mappings (or th-order Lipschitz mappings, for short) which was introduced by Ezearn [4] in 2015.

Now, to every higher-order Lipschitz mapping, Ezearn associated a polynomial which is defined as and for th-order nonexpansive mapping, we have .

Ezearn [5] in his thesis introduced the concept of annihilated points of a higher-order nonexpansive mapping as defined below:

Definition 1 (Annihilated point of ). Let be a higher-order nonexpansive mapping on a subset of a Banach space , and let be the associated polynomial of . Then is an annihilated point (respectively, a totally annihilated point) of if annihilates (respectively, the Picard iterates of ) that is, (respectively, for all ).

Ezearn is denoted by (respectively, ) the set of annihilated (respectively, a totally annihilated) points of . Ezearn, in an attempt to prove a fixed point result for higher-order nonexpansive mappings, proved the following theorems on sufficient conditions for an annihilated point when the Banach space is strictly convex: a strictly convex Banach space is a Banach space such that whenever and , then if and only if for some constant .

Theorem 2. Let be a convex subset of a strictly Banach space , and let an rth-order nonexpansive mapping of the form Suppose and . Then, .

Theorem 3. Let be an rth-order nonexpansive mapping on a convex subset of a strictly convex Banach space . Suppose and . Then .

With a mild condition on the set of totally annihilated points, , Ezearn proved the following fixed point result in a general Banach space.

Theorem 4. Let be an affine higher-order nonexpansive mapping on a convex subset of a Banach space . Then, only if . In particular, the identity holds, and if , then

Finally, Ezearn proved the following approximate fixed point sequence result for his higher-order nonexpansive mapping in a general Banach space.

Theorem 5. Let be a closed bounded star-convex subset of a Banach space, and let be an affine -order nonexpansive self-mapping on . Then, has an approximate fixed point sequence in . That is, there exists such that .

In 2021, the author [6] introduced the following mappings which generalize both inequality (3) and (4).

Definition 6 (General higher-order Lipschitz mappings). Given a metric space , a self-map on is called a -general rth-order Lipschitz mapping if where , , and for all , , and .

It is obvious that inequality (8) reduces to (3) when . In the same vein, inequality (8) reduces to (4) when and .

Now to every -general higher-order Lipschitz mapping, the author associated the following polynomial:

The author classified -general higher-order Lipschitz mappings as follows: (i) is -general higher-order contraction mapping if (ii) is -general higher-order non-expansive mapping if (iii) is -general higher-order expansive mapping if

In this paper, we generalize Theorem 2 and Theorem 3 to -general higher-order nonexpansive mapping when except that in the second case (Theorem 3), it will not be totally annihilated points but just annihilated points because in Ezearn’s case, all the constants are on the right, and therefore, by induction, he could obtain that result for a totally annihilated point. In the other direction, we generalize Theorem 4 to -general higher-order nonexpansive mappings, but in the context of an affine subset of a given Banach space. In the last direction, we generalize Theorem 5 to -general higher-order nonexpansive mappings. That is, in this paper, we prove the following results:

Theorem 7. Let be a convex subset of a strictly convex Banach space , and define to be a (r,1)-general-higher order nonexpansive mapping of the form Suppose and . Then, .

Theorem 8. Let be an (r,1)-general higher-order nonexpansive mapping on a convex subset of a strictly convex Banach space (, ). Suppose and . Then, .

Theorem 9. Let be an affine general higher-order nonexpansive mapping on an affine subset of a Banach space . The only if . In particular, the identity holds and if , then

Theorem 10. Let be a closed, bounded star-convex subset of a Banach space, and let be an affine -general higher-order nonexpansive self-mapping on . Then has an approximate fixed point sequence in . That is, there exists such that .

From Definition 1, for any (r,p)-general higher-order nonexpansive mapping, the fixed point set is always a subset of the annihilated point set and they coincide when and . To see this, for and , we have the following:

Given that , then we have

In the same vein, since for any (r,p)-general higher-order nonexpansive mapping, we have then, we have the following:

Now, since for , then the above equation reduces to

2. Preliminaries

Proposition 11. Define to be an -general higher-order Lipschitz mapping, and let be the associated polynomial for as stated in Definition 6. (i)If , then we can always find a certain , which is unique and positive if , such that (ii)If , then there exists 1 as the only positive root of (iii)If , then we can find a unique positive such that

Now, let us define to be an -general higher-order Lipschitz mapping on a complete metric space as given in inequality (8) and let be the unique root of the polynomial as guaranteed by Proposition 11. Define the following on the space : where and for all and

Corollary 12. stated in equation (17) is non-negative.

Lemma 13. stated in equation (17) is a metric on the space .

Proposition 14. Define in equation (17). Then the following results hold:

Lemma 15. Given a metric space (not necessarily complete) and define to be an -general higher-order Lipschitz mapping. Then Moreover, a sequence is Cauchy in if and only if the sequence is Cauchy in for all .

Theorem 16. Define the mapping, Then, we have

In particular, if is complete, then has a fixed point in if and only if has a fixed point in .

3. Main Result

We prove the main result of this paper, which is already stated in Theorem 7, Theorem 8, Theorem 9, and Theorem 10. The proofs follow similarly as in Ezearn [5] except for few modifications as necessary.

Proof of Theorem 17. Let for some . Then, given that , then the following identity holds: Hence, we have Similarly, one can also have Hence, we have From equation (26), it follows that when , then and that implies that Note also that when , then and it follows that Combining equation (30) and equation (31), we have that giving or equivalently .
Similarly, from equation (28), when , then and that implies that Note also that when , then and it follows that Combining equation (34) and equation (35), we have that giving or equivalently .
Hence, we assume that . We observe that To see this, we note that if , then we have the following leading to the contradiction that .
Also if , then we have leading to the contradiction that .
Now, given that It follows from the above that and since Then from the strict convexity of , there exists such that the following holds: Set , and equation (43) becomes equivalent to the following: Consequently, we have and Now, since then, we have that . Similarly, since and that gives us and therefore we must have . Hence, we have shown that and so we have or equivalently and that completes the proof.

Proof of Theorem 18.  
Let for all and , then we have , and that also follows that Now, if , then , and this means that and since by definition , then it follows that . Similarly, for , then and also follows that . Hence, we may assume that . First and foremost, we may observe that , and to see this, we note that if , then for all and since by assumption , then we have leading to the contradiction that . In the same vein, if , then for all . Since by assumption, , then we have the following: leading to the contradiction that . Given that is an (r,1)-general higher-order nonexpansive mapping, we have which implies that or equivalently Now, given that and , then for all , we have Since , then whenever , it follows from the strict convexity of that there exists such that Now, set . Then, equation (56) becomes Now, when (respectively ) then we choose (respectively ). Comparing equation (57) to the definition , it follows that for all . Now, we show that We observe that and and hence, we have the following evaluation: Hence, we have since . Similarly, we have the following evaluation: Hence, we have again because .
Now, combining equations (60) and (62) and invoking Proposition 14 gives the following: and that gives us Since , then, we have as claimed. Since is an (r,1)-general higher-order nonexpansive mapping, we have that and finally, we have since .
Finally, recall that and for all , thus we have Hence, we have Observe that Hence, we have By combining equations (69) and (71), we get that or , and that completes the proof.

Proof of Theorem 19.  
Clearly, if , then since by definition we have that . We first show that . Since by definition, , we then show that . Indeed, assume that for some , and then, we have Since , then we have By operating both sides of equation (73) under , we obtain the following: where the last identity follows because is affine and that . Hence, we have that and so by induction if , we have for all . Hence, , and that completes the first part of the proof that . Finally, let ; thus the above theorem states that if , then To see this, observe that (from Proposition 11, noting that here ) Since is affine, then we have and so we have Once , and that completes the proof.

Proof of Theorem 20. For , define by where is arbitrary and that is a null sequence. We show that for all .
We prove that equation (80) is true by induction. Now, for the case where , by the affiness of , we have Now, let us assume that equation (80) is true for . Note that and so by the affiness of , we have the following evaluation: and that completes the proof of equation (80). We have the following evaluation: and as a result, we have the following: Now, by taking norms of equation (85) raised to the power , we have By Definition 6, we have Hence, we have Since is an general higher-order nonexpansive mapping, then by Proposition 11, we have Hence, inequality (88) is an general higher-order contraction mapping, and thus by Theorem 16, has a unique fixed point in , thus . Now, consider Hence, as since is bounded and that completes the proof.

4. Conclusion

As for examples of this map, the immediate examples are algebraic operators (see, for instance, [7, 8]). An algebraic operator is a linear operator satisfying a polynomial identity with scalar coefficients. That is, for any Banach space and a given polynomial , then is a map such that

For instance, given the polynomial , then by the above definition, one obtains the following

By taking norm of the above, we have

Which by the subadditivity of norm, we have which is a higher-order Lipschitz mapping, and hence, a general higher-order Lipschitz mapping.

Algebraic operators are intrinsically interesting and do have good and many applications to other fields in most areas of pure mathematics such as the Connes-Moscovici index theorem for foliated manifolds, algebraic quantum field theory, Novokov conjecture, ordinary and partial differential equations, and Jone’s work connecting Von Neumann algebras and geometric topology, which gave rise to a new knot invariant.

Other generalizations (for instance, operators satisfying a polynomial identity with nonscalar coefficients) and their applications can also be found in [9].

Therefore, general higher-order Lipschitz mappings and our current results indeed have the potential of being applied in some mathematical and nonmathematical fields, just like those results mentioned above.

Data Availability

No data was used for this research.

Conflicts of Interest

The author declares that he has no conflicts of interest.

Acknowledgments

We would also like to acknowledge colleagues for their proof reading and other helpful comments regarding this paper.