Approximating fixed points of $$\left( \lambda ,\rho \right) $$-firmly nonexpansive mappings in modular function spaces

Arabian Journal of Mathematics, Mar 2018

Safeer Hussain Khan

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2Fs40065-018-0204-x.pdf

Approximating fixed points of $$\left( \lambda ,\rho \right) $$-firmly nonexpansive mappings in modular function spaces

Arabian Journal of Mathematics Approximating fixed points of (λ, ρ )-firmly nonexpansive mappings in modular function spaces Mathematics Subject Classification 0 0 S. H. Khan ( In this paper, we first introduce an iterative process in modular function spaces and then extend the idea of a λ-firmly nonexpansive mapping from Banach spaces to modular function spaces. We call such mappings as (λ, ρ)-firmly nonexpansive mappings. We incorporate the two ideas to approximate fixed points of (λ, ρ) -firmly nonexpansive mappings using the above-mentioned iterative process in modular function spaces. - 1 Introduction Fixed point theory has several applications in different disciplines and, therefore, it has been a flourishing area of research. The metric fixed point theory in the framework of Banach spaces usually involves a close link of geometric and topological conditions. Fixed point theory in modular function spaces and metric fixed point theory are near relatives because the former provides modular equivalents of norm and metric concepts. Modular spaces are extensions of the classical Lebesgue and Orlicz spaces, and in many instances conditions cast in this framework are more natural and more easily verified than their metric analogs. For more discussion, see, for example, Khamsi and Kozlowski [ 3 ]. Nowadays, a vigorous research activity is developed in the area of numerical reckoning fixed points for suitable classes of nonlinear operators, see, for example, [ 9,10 ], and applications to image recovery and variational inequalities, see for example, [ 11–14 ]. Existence of fixed points in modular function spaces has been studied by many researchers, for example, see Khamsi and Kozlowski [ 3 ] and the references therein. Dhompongsa et al. [ 2 ] have proved the existence of fixed point of ρ-contractions under certain conditions. Buthina and Kozlowski [ 1 ], for the first time, proved results on approximating fixed points in modular function spaces through Mann and Ishikawa iterative processes. Some work for multivalued mappings in modular function spaces using Mann iterative process was done by Khan and Abbas [ 5 ]. Khan [ 4 ] introduced an iterative process for approximation of fixed points of certain mappings in Banach spaces. This process is independent of both Mann and Ishikawa iterative processes in the sense that neither reduces to the other under the given conditions. Moreover, it is faster than all of Picard, Mann and Ishikawa iterative processes in case of contractions [ 4 ]. We extend this process to the framework of modular function spaces. On the other hand, firmly nonexpansive mappings play an important role in nonlinear analysis due to their correspondence with maximal monotone operators. The class of λ-firmly nonexpansive mappings in Banach spaces has attracted many researchers. For a discussion on such mappings, see, for example Ruiz et al. [ 6 ] and the references cited therein. As far as we know, no work has been done until now on this kind of mappings in modular function spaces. We thus introduce the idea of the so-called (λ, ρ)-firmly nonexpansive mappings, in short (λ, ρ)-FNEM. We approximate the fixed points of such mappings using the above-mentioned iterative process in modular function spaces. This will create new results in modular function spaces. 2 Preliminaries Here is a brief note on modular function spaces to make the discussion self-contained. This has mainly been extracted from Khamsi and Kozlowski [ 3 ]. Let be a nonempty set and a nontrivial σ -algebra of subsets of . Let P be a δ-ring of subsets of , such that E ∩ A ∈ P for any E ∈ P and A ∈ . Let us assume that there exists an increasing sequence of sets Kn ∈ P such that = ∪Kn (for instance, P can be the class of sets of finite measure in a σ -finite measure space). By 1A, we denote the characteristic function of the set A in . By E we denote the linear space of all simple functions with supports from P. By M∞ we will denote the space of all extended measurable functions, i.e., all functions f : → [−∞, ∞] such that there exists a sequence {gn} ⊂ E , |gn| ≤ | f | and gn(ω) → f (ω) for all ω ∈ . Definition 2.1 Let ρ : M∞ → [0, ∞] be a nontrivial, convex and even function. We say that ρ is a regular convex function pseudomodular if (1) ρ(0) = 0; (2) ρ is monotone, i.e., | f (ω)| ≤ |g(ω)| for any ω ∈ implies ρ( f ) ≤ ρ(g), where f, g ∈ M∞; (3) ρ is orthogonally sub-additive, i.e., ρ( f 1A∪B ) ≤ ρ( f 1A) + ρ( f 1B ) for any A, B ∈ A ∩ B = φ, f ∈ M∞; (4) ρ has Fatou property, i.e., | fn(ω)| ↑ | f (ω)| for all ω ∈ implies ρ( fn) ↑ ρ( f ), where f ∈ M∞; (5) ρ is order continuous in E , i.e., gn ∈ E , and |gn(ω)| ↓ 0 implies ρ(gn) ↓ 0. such that A set A ∈ is said to be ρ-null if ρ(g1A) = 0 for every g ∈ E . A property p(ω) is said to hold ρ-almost everywhere (ρ-a.e.) if the set {ω ∈ : p(ω) does not hold} 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 M ( , , P, ρ) = { f ∈ M∞ : | f (ω)| < ∞ ρ -a.e.} , where f ∈ M ( , , P, ρ) 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 M( , , P, ρ). It is easy to see that ρ : M →[0, ∞] possess the following properties: 1. ρ(0) = 0 iff f = 0 ρ-a.e. 2. ρ(α f ) = ρ( f ) for every scalar α with |α| = 1 and f ∈ M. 3. ρ(α f + βg) ≤ ρ( f ) + ρ(g) if α + β = 1, α, β ≥ 0 and f, g ∈ M. ρ is called a convex modular if, in addition, the following property is satisfied: 3 . ρ(α f + βg) ≤ αρ( f ) + βρ(g) if α + β = 1, α, β ≥ 0 and f, g ∈ M. Definition 2.2 Let ρ be a regular function pseudomodular. We say that ρ is a regular convex function modular if ρ( f ) = 0 implies f = 0 ρ-a.e. The class of all nonzero regular convex function modulars defined on The convex function modular ρ defines the modular function space Lρ as is denoted by . Lρ = { f ∈ M : ρ(λ f ) → 0 as λ → 0}. Generally, the modular ρ is not sub-additive and, therefore, does not behave as a norm or a distance. However, the modular space Lρ can be equipped with an F-norm defined by In case ρ is convex modular, f ρ = inf α > 0 : ρ f ρ = inf α > 0 : ρ f α f α ≤ α . ≤ 1 defines a norm on the modular space Lρ , and is called the Luxemburg norm. Define L0ρ = f ∈ Lρ : ρ ( f, .) is order continuous and the linear space Eρ = L0ρ for every λ > 0 . Definition 2.3 ρ ∈ is said to satisfy the 2-condition, if supn≥1 ρ(2 fn, Dk ) → 0 as k → ∞ whenever {Dk } decreases to φ and supn≥1 ρ( fn, Dk ) → 0 as k → ∞. If ρ is convex and satisfies the 2-condition, then Lρ = Eρ . Moreover, ρ satisfies the 2 -condition if and only if F-norm convergence and modular convergence are equivalent. f ∈ Lρ : λ f ∈ Definition 2.4 Let ρ ∈ . (i) Let r > 0, ε > 0. Define Let D1(r, ) = ( f, g) : f, g ∈ Lρ , ρ( f ) ≤ r, ρ(g) ≤ r, ρ( f − g) ≥ εr . 1 δ1(r, ) = inf 1 − r ρ( f + g 2 ) : ( f, g) ∈ D1(r, ) if D1(r, ) = φ, and δ1(r, ) = 1 if D1(r, ) = φ. We say that ρ satisfies (U C1) if for every r > 0, > 0, δ1(r, ) > 0. Note, that for every r > 0, D1(r, ) = φ, for > 0 small enough. (ii) We say that ρ satisfies (U U C1) if for every s ≥ 0, > 0, there exists η1(s, ) > 0 depending only upon s and such that δ1(r, ) > η1(s, ) > 0 for any r > s. Note that (U C1) implies (U U C1). Definition 2.5 Let ρ ∈ . The sequence { fn} ⊂ Lρ is called: • ρ-convergent to f ∈ Lρ if ρ( fn − f ) → 0 as n → ∞. • ρ-Cauchy, if ρ( fn − fm ) → 0 as n and m → ∞. Note that ρ-convergence does not imply ρ-Cauchy since ρ does not satisfy the triangle inequality. In fact, one can show that this will happen if and only if ρ satisfies the 2-condition. Definition 2.6 Let ρ ∈ . A subset D ⊂ Lρ is called • ρ-closed if the ρ-limit of a ρ-convergent sequence of D always belongs to D. • ρ-a.e. closed if the ρ-a.e. limit of a ρ-a.e. convergent sequence of D always belongs to D. • ρ-compact if every sequence in D has a ρ-convergent subsequence in D. • ρ-a.e. compact if every sequence in D has a ρ-a.e. convergent subsequence in D. • ρ-bounded if di amρ(D) = sup{ρ( f − g) : f, g ∈ D} < ∞. A sequence {tn} ⊂ (0, 1) is called bounded away from 0 if there exists a > 0 such that tn ≥ a for every n ∈ N. Similarly, {tn} ⊂ (0, 1) is called bounded away from 1 if there exists b < 1 such that tn ≤ b for every n ∈ N. The following lemma can be seen as an analog of a famous lemma due to Schu [ 7 ] in Banach spaces. Lemma 2.7 [3, Lemma 4.2] Let ρ ∈ If there exists R > 0 such that satisfy (U U C1) and let {tk } ⊂ (0, 1) be bounded away from 0 and 1. and then lim sup ρ( fn) ≤ R, lim sup ρ(gn) ≤ R, n→∞ n→∞ lim ρ(tn fn + (1 − tn)gn) = R, n→∞ lim ρ( fn − gn) = 0. n→∞ A function f ∈ Lρ is called a fixed point of T : Lρ → Lρ if f = T f. The set of all fixed points of T is denoted by Fρ (T ). The ρ-distance from an f ∈ Lρ to a set D ⊂ Lρ is given as follows: di stρ( f, D) = inf{ρ( f − h) : h ∈ D}. The following definition is a modular space version of the condition (I ) of Senter and Dotson [ 8 ]. Let D ⊂ Lρ . A mapping T : D → D is said to satisfy condition (I ) if there exists a nondecreasing function : [0, ∞) → [0, ∞) with (0) = 0, (r ) > 0 for all r ∈ (0, ∞) such that ρ( f − T f ) ≥ (di stρ ( f, Fρ (T )) for all f ∈ D. Definition 2.8 A mapping T : D → D is called ρ-nonexpansive mapping if ρ(T f − T g) ≤ ρ ( f − g) for all f, g ∈ D. The following general theorem ([3, Theorem 5.7]) confirms the existence of fixed points of ρ-nonexpansive mappings. Theorem 2.9 Assume ρ ∈ satisfies (U U C1). Let D be a ρ-closed, ρ-bounded convex and nonempty subset of Lρ . Then, any T : D → D pointwise asymptotically nonexpansive mapping has a fixed point. Moreover, the set of all fixed points F(T ) is ρ-closed and convex. 3 Fixed point approximation of (λ, ρ)-FNEM We first extend the idea of a λ-firmly nonexpansive mapping from Banach spaces to modular function spaces and call it (λ, ρ)-firmly nonexpansive mapping. We define the idea as follows. Definition 3.1 Let D ⊂ Lρ . We say that a mapping T : D → D is called (λ, ρ)-firmly nonexpansive mapping if for given λ ∈ (0, 1), ρ(T f − T g) ≤ ρ [(1 − λ) ( f − g) + λ(T f − T g)] for all f, g ∈ D . For simplicity, we denote a (λ, ρ)-firmly nonexpansive mapping by (λ, ρ)-FNEM. Remark 3.2 (λ, ρ)-firmly nonexpansive implies ρ-nonexpansiveness. To see this, let T : D → D be a (λ, ρ)firmly nonexpansive mapping. Then ρ(T f − T g) ≤ ρ [(1 − λ) ( f − g) + λ(T f − T g)] ≤ (1 − λ)ρ ( f − g) + λρ(T f − T g) for all f, g ∈ D. This implies that (1−λ)ρ(T f − T g) ≤ (1−λ)ρ ( f − g). Since λ = 1, we get ρ(T f − T g) ≤ ρ ( f − g) as desired. Lemma 3.3 The set of fixed points Fρ (T ) of a (λ, ρ)-firmly nonexpansive mapping is nonempty. Moreover, it is ρ-closed and convex. Proof It follows from Remark 3.2 and Theorem 2.9. and Next we introduce the following iterative process in the setting of modular function spaces. For a mapping T : D → D, we define a sequence { fn} by the following iterative process: f1 ∈ D, fn+1 = T gn, gn = (1 − αn) fn + αn T fn, n ∈ N where {αn} ⊂ (0, 1) is bounded away from both 0 and 1. For details on a similar iterative process but in Banach spaces, see [ 4 ]. In this paper, using the above two ideas together, we prove our main result for approximating fixed points in modular function spaces as follows. Theorem 3.4 Let ρ ∈ satisfy (U U C1) and 2-condition. Let D be a nonempty ρ-closed, ρ-bounded and convex subset of Lρ . Let T : D → D be a (λ, ρ)-FNEM. Let { fn} ⊂ D be defined by the iterative process. Then lim ρ( fn − w) exists for all w ∈ Fρ (T ), n→∞ lim ρ( fn − T fn) = 0. n→∞ Proof Since Fρ (T ) = ∅ by Lemma 3.3, let w ∈ Fρ (T ). To prove that limn→∞ ρ( fn − w) exists for all w ∈ Fρ (T ), consider ρ( fn+1 − w) = ρ (T gn − T w) ≤ ρ [(1 − λ) (gn − w) + λ (T gn − T w)] ≤ (1 − λ)ρ (gn − w) + λρ (T gn − T w) by convexity of ρ. This implies ρ (T gn − T w) ≤ ρ (gn − w) and hence Also, because T is a (λ, ρ)-FNEM, implies ρ (T fn − T w) ≤ ρ ( fn − w) ; therefore, ρ( fn+1 − w) ≤ ρ (gn − w) ρ( fn+1 − w) ≤ ρ (gn − w) . ρ (T fn − T w) ≤ (1 − λ)ρ ( fn − w) + λρ (T fn − T w) Thus, limn→∞ ρ( fn − w) exists for each w ∈ Fρ (T ). Suppose that where m ≥ 0. Note that the above calculations also give the following inequality: = ρ[(1 − αn)ρ ( fn − w) + αnρ (T fn − T w)] ≤ (1 − αn)ρ ( fn − w) + αnρ (T fn − T w) ≤ (1 − αn)ρ ( fn − w) + αnρ ( fn − w) = ρ ( fn − w) . lim ρ( fn − w) = m n→∞ ρ (gn − w) ≤ ρ ( fn − w) . Next, we prove that limn→∞ ρ( fn − T fn) = 0. Now using 3.4, 3.2 and 3.3, we have This gives m = nl→im∞ ρ( fn − w) = nl→im∞ ρ( gn − w) ≤ ρ ( fn − w) = m. lim ρ( gn − w) = m. n→∞ (3.1) (3.2) (3.3) (3.4) Moreover, But then ρ( fn+1 − w) ≤ ρ(gn − w) implies that lim sup ρ( T fn − w) ≤ nl→im∞ ρ( fn − w) = m. n→∞ lim ρ [(1 − αn)( fn − w) + αn(T fn − w)] = nl→im∞ ρ [(1 − αn) fn + αn T fn) − w] n→∞ (3.5) = m. = nl→im∞ ρ( gn − w) Now by (3.3) , (3.5) and Lemma 2.7, we have lim ρ( fn − T fn ) = 0. n→∞ as required. Using the above result, we now prove our convergence result for approximating fixed points of (λ, ρ)-firmly nonexpansive mappings in modular function spaces using our iterative process (3.1) as follows. Theorem 3.5 Let ρ ∈ satisfy (U U C1) and 2-condition. Let D be a nonempty ρ-compact and convex subset of Lρ . Let T : D → D be a (λ, ρ) -FNEM. Let { fn} be as defined in Theorem 3.4. Then { fn} ρ-converges to a fixed point of T . Proof Since D is ρ-compact, there exists a subsequence fnk of { fn} such that limk→∞ some z ∈ D. Since T is a (λ, ρ)-FNEM, using convexity of ρ, we have fnk − z = 0 for ρ z − T z 3 z − fnk fnk − T fnk T fnk − T z = ρ 3 + 3 + 3 1 1 1 ≤ 3 ρ(z − fnk ) + 3 ρ( fnk − T fnk ) + 3 ρ(T fnk − T z) ≤ ρ(z − fnk ) + ρ( fnk − T fnk ) + ρ( fnk − z) ≤ 2ρ z − fnk + ρ( fnk − T fnk ). Applying Theorem 3.4, limn→∞ ρ( fnk − T fnk ) = 0, that is, ρ( z−3T z ) = 0. Hence, z is a fixed point of T , that is, { fn} ρ-converges to a fixed point of T . Theorem 3.6 Let ρ ∈ satisfy (U U C1) and 2-condition. Let D be a nonempty ρ-closed, ρ-bounded and convex subset of Lρ . Let T : D → D be a (λ, ρ)-FNEM satisfying condition (I ). Let { fn} be as defined in Theorem 3.4. Then { fn} ρ-converges to a fixed point of T . Proof By Theorem 3.4, limn→∞ ρ( fn − w) exists for all w ∈ Fρ (T ). Suppose that limn→∞ ρ ( fn − w) = m > 0 because otherwise limn→∞ ρ ( fn − w) = 0 means nothing left to prove. Now by Theorem 3.4, we have ρ ( fn+1 − w) ≤ ρ ( fn − w) so that di stρ ( fn+1, Fρ (T )) ≤ di stρ ( fn, Fρ (T )). This means that limn→∞ di stρ ( fn, Fρ (T )) exists. Applying condition (I ) and Theorem 3.4, we have lim (di stρ ( fn, Fρ (T ))) ≤ nl→im∞ ρ( fn − T fn) = 0. n→∞ Since is a nondecreasing function and (0) = 0, To prove that { fn} is a ρ-Cauchy sequence in D,let ε > 0. By (3.6) , there exists a constant n0 such that for all n ≥ n0, (3.6) lim di stρ ( fn, Fρ (T )) = 0. n→∞ ε di stρ ( fn, Fρ (T )) < . 2 ρ fn0 − y < ε. Hence, there exists a y ∈ Fρ (T ) such that ≤ 2 1 ρ ( fn+m − y) + 2 ρ ( fn − y) ≤ ρ fn0 − y < ε. Fρ (T ) is closed, w ∈ Fρ (T ), that is, { fn } ρ-converges to a fixed point of T . Hence, by 2-condition { fn } is a ρ-Cauchy sequence in a ρ-closed subset D of L ρ , and so it converges in D. Let limn→∞ fn = w. Then di stρ (w, Fρ (T )) = nl→im∞ di stρ ( fn , Fρ (T )) = 0 by (3.6) . Since by Lemma 3.3 4 Concluding remarks We have proved some strong convergence results using (λ, ρ)-firmly nonexpansive mappings on a faster iterative algorithm in modular function spaces. In our opinion, using the above ideas, it would be interesting to consider the following: (1) studying the stability and data dependency problems; (2) finding applications to general variational inequalities or equilibrium problems as well as to split feasibility problems. We may suggest to the reader to combine the ideas studied, for example, in [ 9–14 ]. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 1. Dehaish , B.A.B. ; Kozlowski , W.M.: Fixed point iteration processes for asymptotic pointwise nonexpansive mappings in modular function spaces . Fixed Point Theory Appl . 2012 , 118 ( 2012 ) 2. Dhompongsa , S. ; Benavides, T.D.; Kaewcharoen , A. ; Panyanak , B. : Fixed point theorems for multivalued mappings in modular function spaces . Sci. Math. Jpn . 63 ( 2 ), 161 - 169 ( 2006 ) 3. Khamsi , M.A. ; Kozlowski , W.M. : Fixed Point Theory in Modular Function Spaces . Birkhauser, Basel ( 2015 ) 4. Khan , S.H. : A Picard-Mann hybrid iterative process . Fixed Point Theory Appl . 2013 , 69 ( 2013 ). https://doi.org/10.1186/ 1687 -1812-2013-69 5. Khan , S.H. ; Abbas, M. : Approximating fixed points of multivalued rho-nonexpansive mappings in modular function spaces . Fixed Point Theory Appl . 2014 , 34 ( 2014 ) 6. Ruiz , D.A. ; Acedo , G.L. ; Marquez , V.M.: Firmly nonexpansive mappings . J. Nonlinear Convex Anal . 15 ( 1 ), 61 - 87 ( 2014 ) 7. Schu , J.: Weak and strong convergence to fixed points of asymptotically non expansive mappings . Bull. Aust. Math. Soc . 43 , 153 - 159 ( 1991 ) 8. Senter , H.F. ; Dotson , W.G. : Approximating fixed points of nonexpansive mappings . Proc. Am. Math. Soc . 44 ( 2 ), 375 - 380 ( 1974 ) 9. Thakur , B.S. ; Thakur , D. ; Postolache, M.: A new iterative scheme for numerical reckoning xed points of Suzuki's generalized nonexpansive mappings . Appl. Math. Comput . 275 , 147 - 155 ( 2016 ) 10. Thakur , B.S. ; Thakur , D. ; Postolache, M. : New iteration scheme for numerical reckoning xed points of nonexpansive mappings . J. Inequal. Appl . 2014 , 328 ( 2014 ) 11. Yao , Y. ; Postolache , M. ; Liou , Y.C. ; Yao , Z. : Construction algorithms for a class of monotone variational inequalities . Optim. Lett . 10 ( 7 ), 1519 - 1528 ( 2016 ) 12. Yao , Y. ; Agarwal , R.P. ; Postolache , M. ; Liu, Y.C. : Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem . Fixed Point Theory Appl . 2014 , 183 ( 2014 ) 13. Yao , Y. , Liou , Y.C. , Postolache , M. : Self-adaptive algorithms for the split problem of the demicontractive operators . Optimization . https://doi.org/10.1080/02331934. 2017 .1390747 14. Yao , Y. ; Leng , L. ; Postolache , M. ; Zheng , X. : Mann-type iteration method for solving the split common fixed point problem . J. Nonlinear Convex Anal . 18 ( 5 ), 875 - 882 ( 2017 )


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs40065-018-0204-x.pdf

Safeer Hussain Khan. Approximating fixed points of $$\left( \lambda ,\rho \right) $$-firmly nonexpansive mappings in modular function spaces, Arabian Journal of Mathematics, 2018, 1-7, DOI: 10.1007/s40065-018-0204-x