Fixed point iteration processes for asymptotic pointwise nonexpansive mapping in modular function spaces

Let be a uniformly convex modular function space with a strong Opial property. Let be an asymptotic pointwise nonexpansive mapping, where C is a ρ-a.e. compact convex subset of . In this paper, we prove that the generalized Mann and Ishikawa processes converge almost everywhere to a fixed point of T. In addition, we prove that if C is compact in the strong sense, then both processes converge strongly to a fixed point.

MSC: 47H09, 47H10.

In 2008, Kirk and Xu [21] studied the existence of fixed points of asymptotic pointwise nonexpansive mappings , i.e.,

where , for all . Their main result (Theorem 3.5) states that every asymptotic pointwise nonexpansive self-mapping of a nonempty, closed, bounded and convex subset C of a uniformly convex Banach space X has a fixed point. As pointed out by Kirk and Xu, asymptotic pointwise mappings seem to be a natural generalization of nonexpansive mappings. The conditions on can be for instance expressed in terms of the derivatives of iterations of T for differentiable T. In 2009 these results were generalized by Hussain and Khamsi to metric spaces, [9].

In 2011, Khamsi and Kozlowski [18] extended their result proving the existence of fixed points of asymptotic pointwise ρ-nonexpansive mappings acting in modular function spaces. The proof of this important theorem is of the existential nature and does not describe any algorithm for constructing a fixed point of an asymptotic pointwise ρ-nonexpansive mapping. This paper aims at filling this gap.

Let us recall that modular function spaces are natural generalization of both function and sequence variants of many important, from applications perspective, spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces and many others, see the book by Kozlowski [24] for an extensive list of examples and special cases. There exists an extensive literature on the topic of the fixed point theory in modular function spaces, see, e.g., [3-5,8,13,14,17-20,24] and the papers referenced there.

It is well known that the fixed point construction iteration processes for generalized nonexpansive mappings have been successfully used to develop efficient and powerful numerical methods for solving various nonlinear equations and variational problems, often of great importance for applications in various areas of pure and applied science. There exists an extensive literature on the subject of iterative fixed point construction processes for asymptotically nonexpansive mappings in Hilbert, Banach and metric spaces, see, e.g., [1,2,6,7,9,12,16,30-36,38-42] and the works referred there. Kozlowski proved convergence to fixed point of some iterative algorithms of asymptotic pointwise nonexpansive mappings in Banach spaces [25] and the existence of common fixed points of semigroups of pointwise Lipschitzian mappings in Banach spaces [26]. Recently, weak and strong convergence of such processes to common fixed points of semigroups of mappings in Banach spaces has been demonstrated by Kozlowski and Sims [28].

We would like to emphasize that all convergence theorems proved in this paper define constructive algorithms that can be actually implemented. When dealing with specific applications of these theorems, one should take into consideration how additional properties of the mappings, sets and modulars involved can influence the actual implementation of the algorithms defined in this paper.

The paper is organized as follows:

(a) Section 2 provides necessary preliminary material on modular function spaces.

(b) Section 3 introduces the asymptotic pointwise nonexpansive mappings and related notions.

(c) Section 4 deals with the Demiclosedness Principle which provides a critical stepping stone for proving almost everywhere convergence theorems.

(d) Section 5 utilizes the Demiclosedness Principle to prove the almost everywhere convergence theorem for generalized Mann process.

(e) Section 6 establishes the almost everywhere convergence theorem for generalized Ishikawa process.

(f) Section 7 provides the strong convergence theorem for both generalized Mann and Ishikawa processes for the case of a strongly compact set C.

Let Ω be a nonempty set and Σ be a nontrivial σ-algebra of subsets of Ω. Let be a δ-ring of subsets of Ω such that for any and . Let us assume that there exists an increasing sequence of sets such that . By ℰ we denote the linear space of all simple functions with supports from . By we will denote the space of all extended measurable functions, i.e., all functions such that there exists a sequence , and for all . By we denote the characteristic function of the set A.

Definition 2.1 Let be a nontrivial, convex and even function. We say that ρ is a regular convex function pseudomodular if:

(i) ;

(ii) ρ is monotone, i.e., for all implies , where ;

(iii) ρ is orthogonally subadditive, i.e., for any such that , ;

(iv) ρ has the Fatou property, i.e., for all implies , where ;

(v) ρ is order continuous in ℰ, i.e., and implies .

Similarly, as in the case of measure spaces, we say that a set is ρ-null if for every . We say that a property holds ρ-almost everywhere if the exceptional set is ρ-null. As usual, we identify any pair of measurable sets whose symmetric difference is ρ-null as well as any pair of measurable functions differing only on a ρ-null set. With this in mind we define

where each is actually an equivalence class of functions equal ρ-a.e. rather than an individual function. Where no confusion exists we will write ℳ instead of .

Definition 2.2 Let ρ be a regular function pseudomodular.

(1) We say that ρ is a regular convex function semimodular if for every implies ρ-a.e.;

(2) We say that ρ is a regular convex function modular if implies ρ-a.e.;

The class of all nonzero regular convex function modulars defined on Ω will be denoted by ℜ.

Let us denote for , . It is easy to prove that is a function pseudomodular in the sense of Def.2.1.1 in [24] (more precisely, it is a function pseudomodular with the Fatou property). Therefore, we can use all results of the standard theory of modular function spaces as per the framework defined by Kozlowski in [22-24].

Remark 2.1 We limit ourselves to convex function modulars in this paper. However, omitting convexity in Definition 2.1 or replacing it by s-convexity would lead to the definition of nonconvex or s-convex regular function pseudomodulars, semimodulars and modulars as in [24].

Definition 2.3[22-24]

Let ρ be a convex function modular.

(a) A modular function space is the vector space , or briefly , defined by

(b) The following formula defines a norm in (frequently called Luxemurg norm):

In the following theorem, we recall some of the properties of modular spaces that will be used later on in this paper.

Theorem 2.1[22-24]

Let.

(1) , is complete and the normis monotone w.r.t. the natural order in ℳ.

(2) if and only iffor every.

(3) Iffor anthen there exists a subsequenceofsuch thatρ-a.e.

(4) Ifconverges uniformly tofon a setthenfor every.

(5) Letρ-a.e. There exists a nondecreasing sequence of setssuch thatandconverges uniformly tofon every (Egoroff theorem).

(6) wheneverρ-a.e. (Note: this property is equivalent to the Fatou property.)

(7) Definingandwe have:

(a) ,

(b) has the Lebesgue property, i.e., for, and.

(c) is the closure of ℰ (in the sense of).

The following definition plays an important role in the theory of modular function spaces.

Definition 2.4 Let . We say that ρ has the -property if

whenever and .

Theorem 2.2Let. The following conditions are equivalent:

(a) ρhas,

(b) is a linear subspace of,

(c) ,

(d) if, then,

(e) iffor an, then, i.e., the modular convergence is equivalent to the norm convergence.

We will also use another type of convergence which is situated between norm and modular convergence. It is defined, among other important terms, in the following definition.

Definition 2.5 Let .

(a) We say that is ρ-convergent to f and write if and only if .

(b) A sequence where is called ρ-Cauchy if as .

(c) A set is called ρ-closed if for any sequence of , the convergence implies that f belongs to B.

(d) A set is called ρ-bounded if .

(e) A set is called strongly ρ-bounded if there exists such that .

(f) A set is called ρ-compact if for any in C there exists a subsequence and an such that .

(g) A set is called ρ-a.e. closed if for any in C which ρ-a.e. converges to some f, then we must have .

(h) A set is called ρ-a.e. compact if for any in C, there exists a subsequence which ρ-a.e. converges to some .

(i) Let and . The ρ-distance between f and C is defined as

Let us note that ρ-convergence does not necessarily imply ρ-Cauchy condition. Also, does not imply in general , . Using Theorem 2.1, it is not difficult to prove the following:

Proposition 2.1Let.

(i) isρ-complete,

(ii) ρ-ballsareρ-closed andρ-a.e. closed.

Let us compare different types of compactness introduced in Definition 2.5.

Proposition 2.2Let. The following relationships hold for sets:

(i) IfCisρ-compact, thenCisρ-a.e. compact.

(ii) IfCis-compact, thenCisρ-compact.

(iii) Ifρsatisfies, then-compactness andρ-compactness are equivalent in.

Proof

(i) follows from Theorem 2.1 part (3).

(ii) follows from Theorem 2.1 part (2).

(iii) follows from (2.2) and from Theorem 2.2 part (e).

 □

Let us recall the modular definitions of asymptotic pointwise nonexpansive mappings and associated notions, [18].

Definition 3.1 Let and let be nonempty and ρ-closed. A mapping is called an asymptotic pointwise mapping if there exists a sequence of mappings such that

(i) If for every and every , then T is called ρ-nonexpansive or shortly nonexpansive.

(ii) If converges pointwise to , then T is called asymptotic pointwise contraction.

(iii) If for any , then T is called asymptotic pointwise nonexpansive.

(iv) If for any with , then T is called strongly asymptotic pointwise contraction.

Denoting , we note that without loss of generality we can assume that T is asymptotically pointwise nonexpansive if

Define . In view of (3.2), we have

The above notation will be consistently used throughout this paper.

By we will denote the class of all asymptotic pointwise nonexpansive mappings .

In this paper, we will impose some restrictions on the behavior of and . This type of assumptions is typical for controlling the convergence of iterative processes for asymptotically nonexpansive mappings, see, e.g., [25].

Definition 3.2 Define as a class of all such that

We recall the following concepts related to the modular uniform convexity introduced in [18]:

Definition 3.3 Let . We define the following uniform convexity type properties of the function modular ρ: Let , , . Define

Let

and if . We will use the following notational convention: .

Definition 3.4 We say that ρ satisfies if for every , , . Note that for every , , for small enough. We say that ρ satisfies if for every , there exists depending only on s and ε such that

We will need the following result whose proof is elementary. Note that for , this result follows directly from Definition 3.4.

Lemma 3.1Letbeand let. Then for every, there existsdepending only onsandεsuch that

The notion of bounded away sequences of real numbers will be used extensively throughout this paper.

Definition 3.5 A sequence is called bounded away from 0 if there exists such that for every . Similarly, is called bounded away from 1 if there exists such that for every .

We will need the following generalization of Lemma 4.1 from [18] and being a modular equivalent of a norm property in uniformly convex Banach spaces, see, e.g., [36].

Lemma 3.2Letbeand letbe bounded away from 0 and 1. If there existssuch that

then

Proof Assume to the contrary that this is not the case and fix an arbitrary . Passing to a subsequence if necessary, we may assume that there exists an such that

while

Since is bounded away from 0 and 1 there exist such that for all natural n. Passing to a subsequence if necessary, we can assume that . For every and , let us define . Observe that the function is a convex function. Hence that the function

is also convex on , and consequently, it is a continuous function on . Noting that

we conclude that is a continuous function of . Hence

By (3.8) and (3.9)

By (3.12) the left-hand side of (3.13) tends to as while the right-hand side tends to in view of (3.7). Hence

By and by Lemma 3.1, there exists satisfying

Combining (3.14) with (3.15) we get

Letting we get a contradiction which completes the proof. □

Let us introduce a notion of a ρ-type, a powerful technical tool which will be used in the proofs of our fixed point results.

Definition 3.6 Let be convex and ρ-bounded. A function is called a ρ-type (or shortly a type) if there exists a sequence of elements of K such that for any there holds

Note that τ is convex provided ρ is convex. A typical method of proof for the fixed point theorems in Banach and metric spaces is to construct a fixed point by finding an element on which a specific type function attains its minimum. To be able to proceed with this method, one has to know that such an element indeed exists. This will be the subject of Lemma 3.3 below. First, let us recall the definition of the Opial property and the strong Opial property in modular function spaces, [15,17].

Definition 3.7 We say that satisfies the ρ-a.e. Opial property if for every which is ρ-a.e. convergent to 0 such that there exists a for which

the following inequality holds for any not equal to 0

Definition 3.8 We say that satisfies the ρ-a.e. strong Opial property if for every which is ρ-a.e. convergent to 0 such that there exists a for which

the following equality holds for any

Remark 3.1 Note that the ρ-a.e. Strong Opial property implies ρ-a.e. Opial property [15].

Remark 3.2 Also, note that, by virtue of Theorem 2.1 in [15], every convex, orthogonally additive function modular ρ has the ρ-a.e. strong Opial property. Let us recall that ρ is called orthogonally additive if whenever . Therefore, all Orlicz and Musielak-Orlicz spaces must have the strong Opial property.

Note that the Opial property in the norm sense does not necessarily hold for several classical Banach function spaces. For instance, the norm Opial property does not hold for spaces for while the modular strong Opial property holds in for all .

Lemma 3.3[27]

Let. Assume thathas theρ-a.e. strong Opial property. Letbe a nonempty, stronglyρ-bounded andρ-a.e. compact convex set. Then anyρ-type defined in C attains its minimum inC.

Let us finish this section with the fundamental fixed point existence theorem which will be used in many places in the current paper.

Theorem 3.1[18]

Assumeis. LetCbe aρ-closedρ-bounded convex nonempty subset. Then anyasymptotically pointwise nonexpansive has a fixed point. Moreover, the set of all fixed pointsisρ-closed.

The following modular version of the Demiclosedness Principle will be used in the proof of our convergence Theorem 5.1. Our proof the Demiclosedness Principle uses the parallelogram inequality valid in the modular spaces with the property (see Lemma 4.2 in [18]). We start with a technical result which will be used in the proof of Theorem 4.1.

Lemma 4.1Let. Letbe a convex set, and let. Ifis aρ-approximate fixed point sequence for T, that is, as, then for every fixedthere holds

as.

Proof It follows from 3.5 that there exists a finite constant such that

Using the convexity of ρ and the ρ-nonexpansiveness of T, we get

as . □

Corollary 4.1If, under the hypothesis of Lemma 4.1, ρsatisfies additionally thecondition, thenas.

The version of the Demiclosedness Principle used in this paper (Theorem 4.1) requires the uniform continuity of the function modular ρ in the sense of the following definition (see, e.g., [17]).

Definition 4.1 We say that is uniformly continuous if to every and , there exists such that

provided and .

Let us mention that the uniform continuity holds for a large class of function modulars. For instance, it can be proved that in Orlicz spaces over a finite atomless measure [37] or in sequence Orlicz spaces [11] the uniform continuity of the Orlicz modular is equivalent to the -type condition.

Theorem 4.1Demiclosedness Principle. Let. Assume that:

(1) ρis,

(2) ρhas strong Opial property,

(3) ρhasproperty and is uniformly continuous.

Letbe a nonempty, convex, stronglyρ-bounded andρ-closed, and let. Let, and. Ifρ-a.e. and, then.

Proof Let us recall that by definition of uniform continuity of ρ to every and , there exists such that

provided and . Fix any . Noting that due to the strong ρ-boundedness of C and that by Corollary (4.1), it follows from (4.5) with and that

as . Hence

Define the ρ-type φ by

By (4.7) we get

Hence, for every there holds

Using (4.10) with and by passing with m to infinity, we conclude that

Since ρ satisfies the strong Opial property, it also satisfies the Opial property. Since -a.e., it follows via the Opial property that for any

which implies that

Combining (4.11) with (4.13), we have

that is,

We claim that

Assume to the contrary that (4.16) does not hold, that is,

By , it follows from (4.17) that does not tend to zero. By passing to a subsequence if necessary, we can assume that there exists such that

for , which implies that

for every . Hence,

for every . Applying the modular parallelogram inequality valid in modular function spaces, see Lemma 4.2 in [18],

where , and for , with , , , , we get

Note that by (4.13)

Combining (4.22) with (4.23), we obtain

which implies

Letting and applying (4.15), we get

Using the properties of Ψ, we conclude that tends to zero itself, which contradicts our assumption (4.17). Hence, as . Clearly, then as , that is, while by ρ-continuity of T. By the uniqueness of the ρ-limit, we obtain , that is, . □

The following elementary, easy to prove, lemma will be used in this paper.

Lemma 5.1[2]

Supposeis a bounded sequence of real numbers andis a doubly-index sequence of real numbers which satisfy

for each. Thenconverges to an.

Following Mann [29], let us start with the definition of the generalized Mann iteration process.

Definition 5.1 Let and let be an increasing sequence of natural numbers. Let be bounded away from 0 and 1. The generalized Mann iteration process generated by the mapping T, the sequence , and the sequence denoted by is defined by the following iterative formula:

Definition 5.2 We say that a generalized Mann iteration process is well defined if

Remark 5.1 Observe that by the definition of asymptotic pointwise nonexpansiveness, for every . Hence we can always select a subsequence such that (5.2) holds. In other words, by a suitable choice of , we can always make well defined.

The following result provides an important technique which will be used in this paper.

Lemma 5.2Letbe. Letbe aρ-closed, ρ-bounded and convex set. Letand let. Assume that a sequenceis bounded away from 0 and 1. Letwbe a fixed point ofTandbe a generalized Mann process. Then there existssuch that

Proof Let . Since

it follows that for every ,

Denote for every and . Observe that since , it follows that . By Lemma 5.1, there exists an such that as claimed. □

The next result will be essential for proving the convergence theorems for iterative process.

Lemma 5.3Letbe. Letbe aρ-closed, ρ-bounded and convex set, and. Assume that a sequenceis bounded away from 0 and 1. Letandbe a generalized Mann iteration process. Then

and

Proof By Theorem 3.1, T has at least one fixed point . In view of Lemma 5.2, there exists such that

Note that

and that

Set , , and note that by (5.7), and by (5.8). Observe also that

Hence, it follows from Lemma 3.2 that

which by the construction of the sequence is equivalent to

as claimed. □

In the next lemma, we prove that under suitable assumption the sequence becomes an approximate fixed point sequence, which will provide an important step in the proof of the generalized Mann iteration process convergence. First, we need to recall the following notions.

Definition 5.3 A strictly increasing sequence is called quasi-periodic if the sequence is bounded, or equivalently, if there exists a number such that any block of p consecutive natural numbers must contain a term of the sequence . The smallest of such numbers p will be called a quasi-period of .

Lemma 5.4Letbesatisfying. Letbe aρ-closed, ρ-bounded and convex set, and. Letbe bounded away from 0 and 1. Letbe such that the generalized Mann processis well defined. If, in addition, the set of indicesis quasi-periodic, thenis an approximate fixed point sequence, i.e.,

Proof Let be a quasi-period of . Observe that it is enough to prove that as through . Indeed, let us fix . From as through it follows that

for sufficiently large k. By the quasi-periodicity of , to every positive integer k, there exists such that . Assume that (the proof for the other case is identical). Since T is ρ-Lipschitzian with the constant , there exist a such that

Note that by (5.6) and by , for k sufficiently large. This implies that

and therefore,

which demonstrates that

as . By again, we get .

To prove that as through , observe that, since for such k, there holds

which tends to zero in view of (5.5), (5.6) and (5.2). □

The next theorem is the main result of this section.

Theorem 5.1Let. Assume that:

(1) ρis,

(2) ρhas Strong Opial Property,

(3) ρhasproperty and is uniformly continuous.

Letbe nonempty, ρ-a.e. compact, convex, stronglyρ-bounded andρ-closed, and let. Assume that a sequenceis bounded away from 0 and 1. Letandbe a well-defined generalized Mann iteration process. Assume, in addition, that the set of indicesis quasi-periodic. Then there existssuch thatρ-a.e.

Proof Observe that by Theorem 4.1 in [18], the set of fixed points is nonempty, convex and ρ-closed. Note also that by Lemma 3.1 in [27], it follows from the strong Opial property of ρ that any ρ-type attains its minimum in C. By Lemma 5.4, the sequence is an approximate fixed point sequence, that is,

as . Consider , two ρ-a.e. cluster points of . There exits then , subsequences of such that ρ-a.e. and ρ-a.e. By Theorem 4.1, and . By Lemma 5.2, there exist such that

We claim that . Assume to the contrary that . Then, by the strong Opial property, we have

The contradiction implies that . Therefore, has at most one ρ-a.e. cluster point. Since, C is ρ-a.e. compact it follows that the sequence has exactly one ρ-a.e. cluster point, which means that ρ-a.e. Using Theorem 4.1 again, we get as claimed. □

Remark 5.2 It is easy to see that we can always construct a sequence with the quasi-periodic properties specified in the assumptions of Theorem 5.1. When constructing concrete implementations of this algorithm, the difficulty will be to ensure that the constructed sequence is not “too sparse” in the sense that the generalized Mann process remains well defined. The similar quasi-periodic type assumptions are common in the asymptotic fixed point theory, see, e.g., [2,25,28].

The two-step Ishikawa iteration process is a generalization of the one-step Mann process. The Ishikawa iteration process, [10], provides more flexibility in defining the algorithm parameters, which is important from the numerical implementation perspective.

Definition 6.1 Let and let be an increasing sequence of natural numbers. Let be bounded away from 0 and 1, and be bounded away from 1. The generalized Ishikawa iteration process generated by the mapping T, the sequences , , and the sequence denoted by is defined by the following iterative formula:

Definition 6.2 We say that a generalized Ishikawa iteration process is well defined if

Remark 6.1 Observe that, by the definition of asymptotic pointwise nonexpansiveness, for every . Hence we can always select a subsequence such that (6.2) holds. In other words, by a suitable choice of , we can always make well defined.

Lemma 6.1Letbe. Letbe aρ-closed, ρ-bounded and convex set. Letand let. Letbe bounded away from 0 and 1, andbe bounded away from 1. Letandbe a generalized Ishikawa process. There exists then ansuch that.

Proof Define by

It is easy to see that and that . Moreover, a straight calculation shows that each satisfies

where

and

Note that , which follows directly from the fact that and from (6.5). Using (6.5) and the fact that , we have

Fix any . Since , it follows that there exists a such that for , . Therefore, using the same argument as in the proof of Lemma 5.2, we deduce that for and

Arguing like in the proof of Lemma 5.2, we conclude that there exists an such that . □

Lemma 6.2Letbe. Letbe aρ-closed, ρ-bounded and convex set. Letand let. Letbe bounded away from 0 and 1, andbe bounded away from 1. Letbe a generalized Ishikawa process. Define

Then

or equivalently

Proof By Theorem 3.1, . Let us fix . By Lemma 6.1, exists. Let us denote it by r. Since , , and by Lemma 6.1, we have the following:

Note that

Applying Lemma 3.2 with and , we obtain the desired equality , while (6.11) follows from (6.10) via the construction formulas for and . □

Lemma 6.3Letbesatisfying. Letbe aρ-closed, ρ-bounded and convex set. Letand let. Letbe bounded away from 0 and 1, andbe bounded away from 1. Letbe a well-defined generalized Ishikawa process. Then

Proof Let . Hence

Since is bounded away from 1, there exists such that for every . Hence,

The right-hand side of this inequality tends to zero because by Lemma 6.2 and ρ satisfies . □

Lemma 6.4Letbesatisfying. Letbe aρ-closed, ρ-bounded and convex set, and. Letbe bounded away from 0 and 1 andbe bounded away from 1. Letbe such that the generalized Ishikawa processis well defined. If, in addition, the setis quasi-periodic, thenis an approximate fixed point sequence, i.e.,

Proof The proof is analogous to that of Lemma 5.4 with (6.11) used instead of (5.6) and (6.14) replacing (5.5). □

Theorem 6.1Let. Assume that

(1) ρis,

(2) ρhas Strong Opial Property,

(3) ρhasproperty and is uniformly continuous.

Letbe nonempty, ρ-a.e. compact, convex, stronglyρ-bounded andρ-closed, and let. Let. Letbe bounded away from 0 and 1, andbe bounded away from 1. Letbe such that the generalized Ishikawa processis well defined. If, in addition, the setis quasi-periodic, thengenerated byconvergesρ-a.e. to a fixed point.

Proof The proof is analogous to that of Theorem 5.1 with Lemma 5.4 replaced by Lemma 6.4, and Lemma 5.2 replaced by Lemma 6.1. □

It is interesting that, provided C is ρ-compact, both generalized Mann and Ishikawa processes converge strongly to a fixed point of T even without assuming the Opial property.

Theorem 7.1Letsatisfy conditionsand. Letbe aρ-compact, ρ-bounded and convex set, and let. Letbe bounded away from 0 and 1, andbe bounded away from 1. Letbe such that the generalized Mann process (resp. Ishikawa process) is well defined. Then there exists a fixed pointsuch that thengenerated by (resp. ) converges strongly to a fixed point ofT, that is

Proof By the ρ-compactness of C, we can select a subsequence of such that there exists with

Note that

which tends to zero by Lemma 5.3 (resp. Lemma 6.4) and by (7.2). By it follows from (7.3) that

Observe that by the convexity of ρ and by ρ-nonexpansiveness of T, we have

which tends to zero by (7.4) and by Lemma 5.3 (resp. Lemma 6.4). Hence which implies that . Applying Lemma 5.2 (resp. Lemma 6.1), we conclude that exists. By (7.4) this limit must be equal to zero which implies that

 □

Remark 7.1 Observe that in view of the assumption, the ρ-compactness of the set C assumed in Theorem 7.1 is equivalent to the compactness in the sense of the norm defined by ρ.

The authors declare that they have no competing interests.

Both authors equally participated in all stages of preparations of this article. Both authors read and approved the final manuscript.

The authors would like to thank MA Khamsi for his valuable suggestions to improve the presentation of the paper.

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