On Some Fixed Point Results for 𝐻+-Type Contraction Mappings in Metric Spaces

Research Article On Some Fixed Point Results for 𝐻+-Type Contraction Mappings in Metric Spaces Hemant Kumar Pathak1and Rosana Rodríguez-López2 1School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur, Chhattisgarh 492010, India 2Departamento de An´ alisis Matem´ atico, Facultad de Matem´ aticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain Correspondence should be addressed to Rosana Rodr´ ıguez-L´ opez; [email protected] Received 8 April 2015; Accepted 10 June 2015 Academic Editor: Guang Zhang Copyright © 2015 H. K. Pathak and R. Rodr´ ıguez-L´ opez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We prove some fixed point theorems for 𝐻+-type multivalued contractive mappings in the setting of Banach spaces and metric spaces. The results provided allow recovering different well-known results. 1. Introduction The interest in the study of fixed point theory in the frame of multivalued mappings by using the Hausdorff metric has its origin in the contributions of Markin [1]andNadler[2], which have lead to interesting achievements on this topic with reference to both theoretical results and applications (see the monograph by Singh et al. on fixed point theory [3]andthe references therein). Starting from the work of Nadler [2], many authors have contributed to the huge development of fixed point theory for set-valued contractions. We mention some interesting contributions, such as those made by Reich [4,5], Lami Dozo [6], Singh [7], Lim [8], Kaneko [9], Mizoguchi and Takahashi [10], Dhompongsa et al. [11], Feng and Liu [12], Klim and Wardowski [13], Suzuki [14], and Pathak and Shahzad [15,16] (seealsothemonographsbyGoebelandKirk[17], Petrusel [18], and so forth). We also refer to some recent works on this topic, for instance, those by Hasanzade Asl et al. [19], Samet et al. [20], and Kumam and Sintunavarat [21]. In this paper, we provide an extension of some fixed point results to the context of multivalued 𝐻+-type 𝜑-contraction mappings, by establishing a common frame which allows obtaining as corollaries some well-known fixed point results. 2. Preliminaries For (𝑋,‖⋅‖)anormedlinearspaceand𝐾anonemptysubset of 𝑋, we denote by 2𝐾,CB(𝐾),andK(𝐾)the collection of all nonempty subsets of 𝐾, the collection of all nonempty closed and bounded subsets of 𝐾, and the collection of all compact subsets of 𝐾,respectively. For 𝐴,𝐵∈CB(𝑋), we define the following mappings: 𝐻(𝐴,𝐵)=max {𝜌(𝐴,𝐵),𝜌(𝐵,𝐴)}, 𝐻+(𝐴,𝐵)=1 2{𝜌(𝐴,𝐵)+𝜌(𝐵,𝐴)}, (1) where 𝜌(𝐴,𝐵)=sup𝑥∈𝐴dist(𝑥,𝐵)and dist(𝑥,𝐵)=inf𝑦∈𝐵‖𝑥− 𝑦‖. It is well known that the mapping 𝐻is a metric on CB(𝑋) called the Hausdorff metric induced by the norm in 𝑋. From Proposition 2.1 [16], we also know that 𝐻+is ametriconCB(𝑋).Moreover,𝐻and 𝐻+are equivalent metrics [22], since (1/2)𝐻(𝐴,𝐵) ≤ 𝐻+(𝐴,𝐵) ≤ 𝐻(𝐴,𝐵). By the results in the monograph by Kuratowski [22], we deduce that (CB(𝑋),𝐻+)is complete provided that (𝑋,𝑑) is complete (𝑑denotes the metric induced by the norm ‖⋅‖) and K(𝑋)isaclosedsubspaceof(CB(𝑋),𝐻+)(see Theorem 2.6 [16]). The mapping 𝐻+:CB(𝑋)×CB(𝑋) → Ris Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2015, Article ID 914158, 12 pages http://dx.doi.org/10.1155/2015/914158 2 Discrete Dynamics in Nature and Society also continuous and satisfies the following properties (see Proposition 2.2 [16]): (i) 𝐻+(𝜆𝐴,𝜆𝐵)=|𝜆|𝐻+(𝐴,𝐵),forany𝜆∈Cand 𝐴,𝐵∈ CB(𝑋); (ii) 𝐻+(𝐴+𝑎,𝐵+𝑎)=𝐻+(𝐴,𝐵),forany𝐴,𝐵∈CB(𝑋) and 𝑎∈𝑋. Although we have introduced the definitions of 𝐻and 𝐻+ for normed linear spaces, we can also do it for metric spaces (𝑋,𝑑)just by using the metric 𝑑instead of the norm; this will be enough for our purpose. The notions of multivalued contraction and 𝐻+- contraction mapping are essential to this work and we include it here for completeness. Definition 1. One says that a set-valued mapping 𝑇:𝑋→ CB(𝑋)is a multivalued 𝑘-contraction mapping if there exists afixedrealnumber𝑘,0<𝑘<1, such that 𝐻(𝑇𝑥,𝑇𝑦)≤𝑘󵄩 󵄩 󵄩 󵄩𝑥−𝑦󵄩 󵄩 󵄩 󵄩,∀𝑥,𝑦∈𝑋. (2) Definition 2 (see [7]). Let (𝑋,𝑑)be a metric space. A multivalued map 𝑇:𝑋→CB(𝑋)is called 𝐻+-contraction if the following conditions hold: (C1) there exists 𝑘in (0,1)such that 𝐻+(𝑇𝑥,𝑇𝑦)≤𝑘𝑑(𝑥,𝑦),for every 𝑥,𝑦∈𝑋, (3) (C2) for every 𝑥in 𝑋,𝑦in 𝑇(𝑥)and 𝜀>0, there exists 𝑧in 𝑇(𝑦)such that 𝑑(𝑦,𝑧)≤𝐻+(𝑇𝑦,𝑇𝑥)+𝜀. (4) Another important concept is the notion of fixed point for a multivalued map. Definition 3. An element 𝑥∈𝑋is said to be a fixed point of a multivalued map 𝑇:𝐾⊆𝑋→2𝑋if 𝑥∈𝑇(𝑥). One denotes by 𝐹(𝑇)the set of fixed points of 𝑇. Concerning the existence of fixed points for multivalued contractions, a highly relevant result was provided by Nadler (see [2]). Theorem 4 (Theorem 5 [2]). Let (𝑋,𝑑)be a complete metric space and 𝑇:𝑋→CB(𝑋) amultivaluedcontraction mapping. Then 𝑇has a fixed point. Pathak and Shahzad [16] have given a generalization of Theorem 4 under a condition weaker than multivalued contractivity by using the metric 𝐻+,asrecalledbelow. Theorem 5 (Theorem 3.2 [16]). Let (𝑋,𝑑)be a complete metric space. Every 𝐻+-type multivalued contraction mapping 𝑇: 𝑋→CB(𝑋)with Lipschitz constant 𝑘<1has a fixed point. In the next section, we obtain some fixed point results for a more general class of multivalued 𝐻+-type contractions. We provide a common structure which allows obtaining as particular cases some well-known fixed point results. 3. Fixed Point Results for Multivalued 𝐻+-Type Weak 𝜑-Contraction Mappings In this section, we present the main results, in which the existence of fixed points is deduced for multivalued 𝐻+-type weak 𝜑-Lipschitz and 𝜑-contraction mappings, as introduced below. Definition 6. Let (𝑋,𝑑)be a metric space. A multivalued map 𝑇:𝑋→CB(𝑋)is called an 𝐻+-type multivalued weak 𝜑- Lipschitz mapping if condition (C2) in Definition 2 holds and there exist 𝑀:(R+)5→R+and 𝜑:R+→R+such that 𝐻+(𝑇𝑥,𝑇𝑦)≤𝜑(M(𝑑(𝑥,𝑦),𝑑(𝑥,𝑇𝑥),𝑑(𝑦,𝑇𝑦), 𝑑(𝑥,𝑇𝑦),𝑑(𝑦,𝑇𝑥))), for every 𝑥,𝑦∈𝑋. (5) If, moreover, there exists 𝑘∈(0,1)such that 𝜑(𝑡) ≤ 𝑘𝑡,for every 𝑡≥0, we say that 𝑇is an 𝐻+-type multivalued weak 𝜑-contraction. Remark 7. In Definition 6,ifwetake 𝜑(𝑡)=𝑘𝑡, with 𝑘∈(0,1), M(𝑟1,𝑟2,𝑟3,𝑟4,𝑟5)=𝑟1,(6) we get the notion of 𝐻+-type multivalued contraction mapping. On the other hand, if we take 𝜑(𝑡)=𝑘𝑡, with 𝑘∈(0,1), M(𝑟1,𝑟2,𝑟3,𝑟4,𝑟5)=max {𝑟1,𝑟2,𝑟3,𝑟4+𝑟5 2}, (7) then we obtain the notion of 𝐻+-type multivalued weak contractive mapping (see [23, Definition 3.3]), since inequality (5) is reduced to 𝐻+(𝑇𝑥,𝑇𝑦)≤𝑘max {𝑑(𝑥,𝑦),𝑑(𝑥,𝑇𝑥),𝑑(𝑦,𝑇𝑦), 𝑑(𝑥,𝑇𝑦)+𝑑(𝑦,𝑇𝑥) 2}, for every 𝑥,𝑦∈𝑋. (8) Finally, if we take 𝜑(𝑡)=𝑘𝑡, with 𝑘∈(0,1), M(𝑟1,𝑟2,𝑟3,𝑟4,𝑟5)=max {𝑟1,𝑟2,𝑟3,𝑟4,𝑟5}, (9) then we obtain the notion of 𝐻+-type multivalued quasicontraction mapping (compare inequality with [23,equation (2.3)]) 𝐻+(𝑇𝑥,𝑇𝑦)≤𝑘max {𝑑(𝑥,𝑦),𝑑(𝑥,𝑇𝑥),𝑑(𝑦,𝑇𝑦), 𝑑(𝑥,𝑇𝑦),𝑑(𝑦,𝑇𝑥)}, for every 𝑥,𝑦∈𝑋. (10) We start this study by providing a fixed point result for 𝐻+-type multivalued weak 𝜑-Lipschitz mappings. This result Discrete Dynamics in Nature and Society 3 (Theorem 8) will find an extension below in Theorem 18 and further in Theorem 21,bothstatedforpartial𝐻+-type multivalued weak 𝜑-Lipschitz mappings. It is possible to state first these results and, then, obtain Theorem 8 as a corollary. However, by considering Theorem 8 first, we can focus our attention on the properties of function M, considering a general expression which immediately connects with some well-known fixed point results. Therefore, we present the main results in several steps: (s.i) First, in Theorem 8, we consider a general expression for function M, while for function 𝜑it is required that 𝜑(𝑡)≤𝑘𝑡,𝑡≥0. (s.ii) Then, in Theorem 18, we extend this result to partial 𝐻+-type multivalued weak 𝜑-Lipschitz mappings. (s.iii) In Theorem 21,thesametypeofvariabilityisallowed for function Mand we explore some other possible general expressions for function 𝜑,alsoforpartial 𝐻+-type multivalued weak 𝜑-Lipschitz mappings. (s.iv) We include some considerations concerning 𝐻+-type weak 𝜑-Lipschitz mappings with respect to a binary relation (including the case of partial orderings). (s.v) WecompletethisstudybyprovingTheorem 32,which considers different restrictions for the selection of functions Mand 𝜑. Theorem 8. Let (𝑋,𝑑)be a complete metric space. Let 𝑇: 𝑋→CB(𝑋)be an 𝐻+-type multivalued weak 𝜑-Lipschitz mapping such that the functions 𝜑,Msatisfy the following: (i) There exists 𝑘>0such that 𝜑(𝑡)≤𝑘𝑡,forevery𝑡≥0. (ii) For every 𝑖=2,3,4,5,thefunctionM(𝑟1,𝑟2,𝑟3,𝑟4,𝑟5) is monotonically increasing in the variable 𝑟𝑖,provided that the other variables 𝑟𝑗,𝑗 =𝑖,remainfixed. (iii) For each 𝑟>0fixed, M(𝑟1,𝑟2,𝑟,𝑟4,𝑟5)is continuous at (𝑟1,𝑟2,𝑟4,𝑟5)=(0,0,𝑟,0). (iv) There exists a function N:R+×R+→R+such that (iv-a) M(𝑟,𝑟,𝑠,𝑟+𝑠,0)≤N(𝑟,𝑠),forevery𝑟,𝑠∈R+, (iv-b) N(⋅,𝑠)is monotonically increasing for each 𝑠∈ R+fixed, (iv-c) N(𝑟,⋅)is monotonically increasing for each 𝑟∈ R+fixed, (iv-d) there exists 𝜏>0such that N(𝑟,𝑟) ≤ 𝜏𝑟,for every 𝑟>0, (iv-e) there exists R≥0such that N(0,𝑠)≤R𝑠,for every 𝑠>0. (v) The constants 𝑘in (i) and 𝜏,Rin (iv) are such that 𝑘<1/max{𝜏,2R}. Then 𝑇has a fixed point. Proof. We denote by 𝜓:𝑋×𝑋 → (R+)5the mapping defined as 𝜓(𝑥,𝑦):=(𝑑(𝑥,𝑦),𝑑(𝑥,𝑇𝑥),𝑑(𝑦,𝑇𝑦),𝑑(𝑥,𝑇𝑦), 𝑑(𝑦,𝑇𝑥)). (11) By (5) and (i), we get 𝐻+(𝑇𝑥,𝑇𝑦)≤𝑘M(𝜓(𝑥,𝑦)), for every 𝑥,𝑦∈𝑋. (12) We construct a sequence in 𝑋in lines similar to [16,Theorem 3.2] or [23, Theorem 3.4] as follows. Let 𝜀>0begivenand take 𝑥0∈𝑋to be arbitrary. We fix an element 𝑥1in 𝑇𝑥0.Now, fromproperty(C2),itispossibletochoose𝑥2∈𝑇𝑥1such that 𝑑(𝑥1,𝑥2)≤𝐻+(𝑇𝑥0,𝑇𝑥1)+𝜀.Atthisstep,wecouldchoose𝜀 depending on 𝑥0and 𝑥1. In general, for 𝑛∈N,if𝑥𝑛is chosen, then we can select 𝑥𝑛+1∈𝑇𝑥𝑛such that 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝐻+(𝑇𝑥𝑛−1,𝑇𝑥𝑛)+𝜀. (13) At this step, we could choose 𝜀depending on 𝑥𝑛−1and 𝑥𝑛. Note that if 𝐻+(𝑇𝑥𝑛−1,𝑇𝑥𝑛)=0forsome𝑛,then𝑇𝑥𝑛−1=𝑇𝑥𝑛 and the proof is complete. By (v), we can take 𝐴,𝐵>0such that 𝐴−𝐵𝑘>0and𝐿:=𝐴−𝐵𝑘+𝑘<1/max{𝜏,2R}.Infact, if we set on each step 𝜀=(𝐴−𝐵𝑘)M(𝜓(𝑥𝑛−1,𝑥𝑛))>0, then, by (13),weobtain,forevery𝑛∈N,that 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝐻+(𝑇𝑥𝑛−1,𝑇𝑥𝑛) +(𝐴−𝐵𝑘)M(𝜓(𝑥𝑛−1,𝑥𝑛)), (14) which implies, for every 𝑛∈N,that 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝑘M(𝜓(𝑥𝑛−1,𝑥𝑛))+(𝐴−𝐵𝑘) ⋅M(𝜓(𝑥𝑛−1,𝑥𝑛))=(𝐴−𝐵𝑘+𝑘) ⋅M(𝜓(𝑥𝑛−1,𝑥𝑛))=𝐿M(𝑑(𝑥𝑛−1,𝑥𝑛), 𝑑(𝑥𝑛−1,𝑇𝑥𝑛−1),𝑑(𝑥𝑛,𝑇𝑥𝑛),𝑑(𝑥𝑛−1,𝑇𝑥𝑛), 𝑑(𝑥𝑛,𝑇𝑥𝑛−1))=𝐿M(𝑑(𝑥𝑛−1,𝑥𝑛),𝑑(𝑥𝑛−1,𝑇𝑥𝑛−1), 𝑑(𝑥𝑛,𝑇𝑥𝑛),𝑑(𝑥𝑛−1,𝑇𝑥𝑛),0)≤𝐿M(𝑑(𝑥𝑛−1,𝑥𝑛), 𝑑(𝑥𝑛−1,𝑥𝑛),𝑑(𝑥𝑛,𝑥𝑛+1),𝑑(𝑥𝑛−1,𝑥𝑛+1),0) ≤𝐿M(𝑑(𝑥𝑛−1,𝑥𝑛),𝑑(𝑥𝑛−1,𝑥𝑛),𝑑(𝑥𝑛,𝑥𝑛+1), 𝑑(𝑥𝑛−1,𝑥𝑛)+𝑑(𝑥𝑛,𝑥𝑛+1),0)≤𝐿N(𝑑(𝑥𝑛−1,𝑥𝑛), 𝑑(𝑥𝑛,𝑥𝑛+1)), (15) wherewehaveused(ii)and(iv-a).Hence, 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝐿N(𝑑(𝑥𝑛−1,𝑥𝑛),𝑑(𝑥𝑛,𝑥𝑛+1)),∀𝑛∈N.(16) If 𝑥𝑛+1=𝑥 𝑛for some 𝑛∈N,then𝑥𝑛=𝑥 𝑛+1∈𝑇𝑥 𝑛,sothat 𝑥𝑛is a fixed point of 𝑇and the proof is concluded. Suppose, therefore, that 𝑑(𝑥𝑛,𝑥𝑛+1)>0, for every 𝑛∈N. Now, assuming that 𝑑(𝑥𝑛−1,𝑥𝑛)<𝑑(𝑥 𝑛,𝑥𝑛+1),forsome 𝑛∈N,by(16) and (iv-b), we get 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝐿N(𝑑(𝑥𝑛,𝑥𝑛+1),𝑑(𝑥𝑛,𝑥𝑛+1)),∀𝑛∈N (17) and, by (iv-d), 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝐿𝜏𝑑(𝑥𝑛,𝑥𝑛+1), ∀𝑛∈N(18) 4 Discrete Dynamics in Nature and Society which is a contradiction since 𝑑(𝑥𝑛,𝑥𝑛+1)>0and𝐿𝜏∈(0,1) (due to inequality 𝐿<1/max{𝜏,2R}≤1/𝜏). Hence, we have proved that 𝑑(𝑥𝑛−1,𝑥𝑛)≥𝑑(𝑥𝑛,𝑥𝑛+1),forevery𝑛∈N,sothat, by (16) and (iv-c), we have 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝐿N(𝑑(𝑥𝑛−1,𝑥𝑛),𝑑(𝑥𝑛,𝑥𝑛+1)) ≤𝐿N(𝑑(𝑥𝑛−1,𝑥𝑛),𝑑(𝑥𝑛−1,𝑥𝑛)),∀𝑛∈N. (19) By (iv-d), we obtain 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝐿𝜏𝑑(𝑥𝑛−1,𝑥𝑛), ∀𝑛∈N.(20) Repeating the same procedure 𝑛-times, we get 𝑑(𝑥𝑛,𝑥𝑛+1)≤ (𝐿𝜏)𝑛𝑑(𝑥0,𝑥1),forall𝑛∈N,where𝐿𝜏<1byhypothesis(v). Hence, {𝑥𝑛}isaCauchysequencesince,for𝑛,𝑚 ∈ Nwith 𝑛<𝑚, 𝑑(𝑥𝑛,𝑥𝑚)≤𝑚−1 ∑ 𝑖=𝑛 𝑑(𝑥𝑖,𝑥𝑖+1)≤𝑚−1 ∑ 𝑖=𝑛 (𝐿𝜏)𝑖𝑑(𝑥0,𝑥1) =(𝐿𝜏)𝑛−(𝐿𝜏)𝑚 1−𝐿𝜏 𝑑(𝑥0,𝑥1) =(𝐿𝜏)𝑛 1−𝐿𝜏(1−(𝐿𝜏)𝑚−𝑛)𝑑(𝑥0,𝑥1). (21) Therefore, by the complete character of 𝑋, there exists 𝑢∈ 𝑋such that lim𝑛→∞𝑥𝑛=𝑢.Supposethat𝑑(𝑢,𝑇𝑢)>0. Then 1 2{𝜌(𝑇𝑥𝑛,𝑇𝑢)+𝜌(𝑇𝑢,𝑇𝑥𝑛)}=𝐻+(𝑇𝑥𝑛,𝑇𝑢) ≤𝑘M(𝜓(𝑥𝑛,𝑢))=𝑘M(𝑑(𝑥𝑛,𝑢),𝑑(𝑥𝑛,𝑇𝑥𝑛), 𝑑(𝑢,𝑇𝑢),𝑑(𝑥𝑛,𝑇𝑢),𝑑(𝑢,𝑇𝑥𝑛))≤𝑘M(𝑑(𝑥𝑛,𝑢), 𝑑(𝑥𝑛,𝑥𝑛+1),𝑑(𝑢,𝑇𝑢),𝑑(𝑥𝑛,𝑇𝑢),𝑑(𝑢,𝑥𝑛+1)). (22) We note that 𝑑(𝑥𝑛,𝑇𝑢)≤𝑑(𝑥𝑛,𝑢)+𝑑(𝑢,𝑇𝑢)and 𝑑(𝑢,𝑇𝑢)≤ 𝑑(𝑢,𝑥𝑛)+𝑑(𝑥𝑛,𝑇𝑢),sothat𝑑(𝑢,𝑇𝑢)−𝑑(𝑢,𝑥𝑛)≤𝑑(𝑥𝑛,𝑇𝑢)≤ 𝑑(𝑥𝑛,𝑢)+𝑑(𝑢,𝑇𝑢), which implies that 𝑑(𝑥𝑛,𝑇𝑢)→𝑑(𝑢,𝑇𝑢) as 𝑛→∞. Hence, from 𝑑(𝑥𝑛,𝑢)→0, 𝑑(𝑥𝑛,𝑥𝑛+1)→0, and 𝑑(𝑥𝑛, 𝑇𝑢) → 𝑑(𝑢,𝑇𝑢),as𝑛→∞, and using hypothesis (iii), we get from (22) 1 2lim inf 𝑛→∞ {𝜌(𝑇𝑥𝑛,𝑇𝑢)+𝜌(𝑇𝑢,𝑇𝑥𝑛)} ≤𝑘M(0,0,𝑑(𝑢,𝑇𝑢),𝑑(𝑢,𝑇𝑢),0).(23) Moreover, by (iv-a) and (iv-e), we obtain 1 2lim inf 𝑛→∞ {𝜌(𝑇𝑥𝑛,𝑇𝑢)+𝜌(𝑇𝑢,𝑇𝑥𝑛)} ≤𝑘N(0,𝑑(𝑢,𝑇𝑢))≤𝑘R𝑑(𝑢,𝑇𝑢).(24) Using that lim inf 𝑛→∞ {𝜌(𝑇𝑥𝑛,𝑇𝑢)+𝜌(𝑇𝑢,𝑇𝑥𝑛)}≤2𝑘R𝑑(𝑢,𝑇𝑢),(25) we deduce that lim inf 𝑛→∞ 𝜌(𝑇𝑥𝑛,𝑇𝑢) ≤lim inf 𝑛→∞ 𝜌(𝑇𝑥𝑛,𝑇𝑢)+lim inf 𝑛→∞ 𝜌(𝑇𝑢,𝑇𝑥𝑛) ≤lim inf 𝑛→∞ {𝜌(𝑇𝑥𝑛,𝑇𝑢)+𝜌(𝑇𝑢,𝑇𝑥𝑛)} ≤2𝑘R𝑑(𝑢,𝑇𝑢). (26) Since 𝑑(𝑢,𝑇𝑢)≤𝑑(𝑢,𝑥𝑛+1)+𝑑(𝑥𝑛+1,𝑇𝑢) ≤𝜌(𝑇𝑥𝑛,𝑇𝑢)+𝑑(𝑥𝑛+1,𝑢) (27) and lim𝑛→∞𝑑(𝑥𝑛+1,𝑢)=0, it follows that 𝑑(𝑢,𝑇𝑢)≤lim inf 𝑛→∞ [𝜌(𝑇𝑥𝑛,𝑇𝑢)+𝑑(𝑥𝑛+1,𝑢)] =lim inf 𝑛→∞ 𝜌(𝑇𝑥𝑛,𝑇𝑢)+lim 𝑛→∞𝑑(𝑥𝑛+1,𝑢) ≤2𝑘R𝑑(𝑢,𝑇𝑢). (28) Hence, if R=0, then 𝑑(𝑢,𝑇𝑢) = 0. On the other hand, if R>0, the assumption 𝑑(𝑢,𝑇𝑢)>0 leads to a contradiction by virtue of hypothesis (v) (2𝑘R<2R/max{𝜏,2R}≤1), which implies that 𝑑(𝑢,𝑇𝑢)=0. Finally, since 𝑇𝑢is closed, it is proved that 𝑢∈𝑇𝑢;thatis,𝑢is a fixed point of 𝑇. Remark 9. Note that if max{𝜏,2R}>1, condition (v) in Theorem 8 implies that 𝑘<1, so that in this case the 𝐻+- type multivalued weak 𝜑-Lipschitz mapping 𝑇in Theorem 8 is in fact an 𝐻+-type multivalued weak 𝜑-contraction. In this sense, both functions 𝜑and Mmay contribute to the contractivity of the multivalued mapping through condition (v) which establishes a relation among the constants involved. Corollary 10. If one takes 𝜑(𝑡) := 𝑘𝑡,with𝑘∈(0,1),and M(𝑟1,𝑟2,𝑟3,𝑟4,𝑟5):=𝑟 1,thenMis monotonically increasing in all the variables and continuous. Besides, for N(𝑟,𝑠) := 𝑟, one has M(𝑟,𝑟,𝑠,𝑟+𝑠,0)=𝑟=N(𝑟,𝑠),𝑓𝑜𝑟𝑒V𝑒𝑟𝑦𝑟,𝑠∈R+, N(𝑟,𝑠)𝑖𝑠𝑚𝑜𝑛𝑜𝑡𝑜𝑛𝑖𝑐𝑎𝑙𝑙𝑦𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑖𝑛𝑟 𝑓𝑜𝑟 𝑠 ∈ R+𝑓𝑖𝑥𝑒𝑑, 𝑁(𝑟,𝑠)𝑖𝑠𝑚𝑜𝑛𝑜𝑡𝑜𝑛𝑖𝑐𝑎𝑙𝑙𝑦𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑖𝑛𝑠 𝑓𝑜𝑟 𝑟 ∈ 𝑅+𝑓𝑖𝑥𝑒𝑑, 𝑁(𝑟,𝑟)=𝑟, 𝑡ℎ𝑢𝑠(𝑖V−𝑑)ℎ𝑜𝑙𝑑𝑠𝑓𝑜𝑟𝜏=1 𝑁(0,𝑠)=0, 𝑓𝑜𝑟 𝑒V𝑒𝑟𝑦𝑠>0, 𝑠𝑜𝑡ℎ𝑎𝑡 (𝑖V−𝑒)ℎ𝑜𝑙𝑑𝑠𝑓𝑜𝑟𝑅=0, max {𝜏,2R}=1, 𝑠𝑜𝑡ℎ𝑎𝑡 (V)𝑖𝑠𝑓𝑢𝑙𝑓𝑖𝑙𝑙𝑒𝑑𝑓𝑜𝑟0<𝑘<1. (29) In this setting of 𝐻+-type multivalued contraction mappings, we have, as a corollary of Theorem 8,Theorem3.2[16](see Theorem 5). Discrete Dynamics in Nature and Society 5 In the context of 𝐻+-type multivalued weak contractive mappings, we obtain the following corollary which corrects Theorem 3.4 [23][seetheproofofTheorem3.4[23], where it was assumed that 𝑑(𝑢,𝑇𝑢)=(1/2)(𝑑(𝑢,𝑇𝑢)+𝑑(𝑢,𝑇𝑢))≤ (1/2)(𝜌(𝑇𝑥𝑛,𝑇𝑢)+𝜌(𝑇𝑢,𝑇𝑥𝑛))+𝑑(𝑢,𝑥𝑛+1)]. Corollary 11. Let (𝑋,𝑑)be a complete metric space and 𝑇: 𝑋→CB(𝑋)an 𝐻+-typemultivaluedweakcontractivemap- ping with 0<𝑘<1/2.Then𝑇has a fixed point. Proof. If we take 𝜑(𝑡):=𝑘𝑡,with𝑘∈(0,1/2),and M(𝑟1,𝑟2,𝑟3,𝑟4,𝑟5):=max {𝑟1,𝑟2,𝑟3,𝑟4+𝑟5 2}, (30) Mis monotonically increasing in all the variables and continuous. Choosing N(𝑟,𝑠):=max{𝑟,𝑠},wehave M(𝑟,𝑟,𝑠,𝑟+𝑠,0)=max {𝑟,𝑟,𝑠,𝑟+𝑠 2}=max {𝑟,𝑠} =N(𝑟,𝑠),for every 𝑟,𝑠∈R+,(31) N(𝑟,𝑠) is monotonically increasing in each variable, N(𝑟, 𝑟)=max{𝑟,𝑟}=𝑟,andthus(iv-d)holdsfor𝜏=1, N(0,𝑠)= max{0,𝑠}=𝑠,forevery𝑠>0, so that (iv-e) holds for R=1 and max{𝜏,2R}=2, so that (v) is fulfilled for the choice 0<𝑘<1/2. Hence Theorem 8 applies. In the context of 𝐻+-type multivalued quasi-contraction mappings, we obtain the following corollary which coincides with Theorem 3.6 [23]. Corollary 12. Let (𝑋,𝑑)be a complete metric space and 𝑇: 𝑋→CB(𝑋) an 𝐻+-type multivalued quasi-contraction mapping with 0<𝑘<1/2.Then𝑇has a fixed point. Proof. Taking 𝜑(𝑡):=𝑘𝑡,with𝑘∈(0,1/2)and M(𝑟1,𝑟2,𝑟3,𝑟4,𝑟5):=max {𝑟1,𝑟2,𝑟3,𝑟4,𝑟5},(32) we have that Mis monotonically increasing in all the variables and also continuous. Taking N(𝑟,𝑠) := 𝑟+𝑠,we get M(𝑟,𝑟,𝑠,𝑟+𝑠,0)=max {𝑟,𝑟,𝑠,𝑟+𝑠,0}=𝑟+𝑠 =N(𝑟,𝑠),for every 𝑟,𝑠∈R+,(33) N(𝑟,𝑠)is monotonically increasing in each variable for the other fixed, N(𝑟,𝑟) = 𝑟+𝑟 = 2𝑟,andthus(iv-d)holdsfor 𝜏=2, N(0,𝑠)=𝑠,forevery𝑠>0, so that (iv-e) holds for R=1andmax{𝜏,2R}=2, so that (v) is fulfilled for 0 <𝑘< 1/2. On the other hand, if a result similar to Theorem 8 was established for 𝐻-multivalued contractions, we would have the following theorem. Theorem 13. Let (𝑋,𝑑)be a complete metric space. Let 𝑇: 𝑋→CB(𝑋)be an 𝐻-type multivalued weak 𝜑-Lipschitz mapping, that is, such that there exist M:(R+)5→R+and 𝜑:R+→R+with 𝐻(𝑇𝑥,𝑇𝑦)≤𝜑(M(𝑑(𝑥,𝑦),𝑑(𝑥,𝑇𝑥),𝑑(𝑦,𝑇𝑦), 𝑑(𝑥,𝑇𝑦),𝑑(𝑦,𝑇𝑥))), 𝑓𝑜𝑟𝑒V𝑒𝑟𝑦𝑥,𝑦∈𝑋. (34) Suppose that the functions 𝜑,Msatisfy conditions (i)–(iv) in Theorem 8 and (v∗)the constants 𝑘in (i) and 𝜏,Rin (iv) are such that 𝑘< 1/max{𝜏,R}. Then 𝑇has a fixed point. Proof. Identical to the proof of Theorem 8,exceptthelast part, we have lim inf 𝑛→∞ 𝜌(𝑇𝑥𝑛,𝑇𝑢)≤lim inf 𝑛→∞ 𝐻(𝑇𝑥𝑛,𝑇𝑢) ≤𝑘N(0,𝑑(𝑢,𝑇𝑢)) ≤𝑘R𝑑(𝑢,𝑇𝑢). (35) Since 𝑑(𝑢,𝑇𝑢)≤𝑑(𝑢,𝑥𝑛+1)+𝑑(𝑥𝑛+1,𝑇𝑢) ≤𝜌(𝑇𝑥𝑛,𝑇𝑢)+𝑑(𝑥𝑛+1,𝑢) (36) and lim𝑛→∞𝑑(𝑥𝑛+1,𝑢)=0, then 𝑑(𝑢,𝑇𝑢)≤lim inf 𝑛→∞ [𝜌(𝑇𝑥𝑛,𝑇𝑢)+𝑑(𝑥𝑛+1,𝑢)] =lim inf 𝑛→∞ 𝜌(𝑇𝑥𝑛,𝑇𝑢)≤𝑘R𝑑(𝑢,𝑇𝑢).(37) If R=0, then 𝑑(𝑢,𝑇𝑢)=0. If R>0, 𝑑(𝑢,𝑇𝑢)>0leadstoa contradiction in the previous inequality due to (v∗)(𝑘R<1), which implies that 𝑑(𝑢,𝑇𝑢) = 0 and the proof is complete. Remark 14. As a consequence of Theorem 13,usingthe Hausdorff metric 𝐻, the restriction required on 𝑘is (r.i) for multivalued contractions 0 <𝑘<1, (r.ii) for multivalued weak contractive mappings 0 <𝑘<1, (r.iii) for multivalued quasi-contraction mappings 0 <𝑘< 1/2. Remark 15. In Theorems 8and 13, conditions (iv-b), (iv-c), and (iv-d) can be removed, adding the property that (iv-d∗) there exists 𝜃>0suchthatN(𝑟,𝑠) ≤ 𝜃(𝑟+𝑠),for every 𝑟,𝑠>0, while condition (v) (resp., (v∗)) has to be replaced by the fact that (v∗∗)theconstants𝑘in (i) and R,𝜃in (iv) are such that 𝑘<1/max{2𝜃,2R}(for 𝐻+)or𝑘<1/max{2𝜃,R} (for 𝐻). 6 Discrete Dynamics in Nature and Society This comes from the following ideas: from (v∗∗), following the proof of Theorem 8,wecantake𝐴,𝐵 >0suchthat𝐴− 𝐵𝑘 > 0and𝐿:=𝐴−𝐵𝑘+𝑘<1/max{2𝜃,2R}(for the Hausdorff metric 𝐻,wechoosetheminsuchawaythat𝐿:= 𝐴−𝐵𝑘+𝑘<1/max{2𝜃,R}). Supposing that 𝑑(𝑥𝑛,𝑥𝑛+1)>0, for every 𝑛∈N,ifwedonothavemonotonicityofN,from (16) and (iv-d∗), we get 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝐿𝜃(𝑑(𝑥𝑛−1,𝑥𝑛)+𝑑(𝑥𝑛,𝑥𝑛+1)),∀𝑛∈N (38) so that, using 𝐿𝜃<1, 𝑑(𝑥𝑛,𝑥𝑛+1)≤ 𝐿𝜃 1−𝐿𝜃𝑑(𝑥𝑛−1,𝑥𝑛), ∀𝑛∈N.(39) Repeating the same procedure 𝑛-times, we get 𝑑(𝑥𝑛,𝑥𝑛+1)≤ 𝐾𝑛𝑑(𝑥0,𝑥1),forall𝑛∈N,where𝐾=𝐿𝜃/(1−𝐿𝜃)<1by hypothesis (v∗∗). The rest of the proof is valid. Now, following the lines in [21]for𝑞-set-valued quasicontractions, we give the following definition. Definition 16. Let (𝑋,𝑑)be a metric space and let 𝛼:𝑋× 𝑋→[0,∞)be given. A multivalued map 𝑇:𝑋→CB(𝑋) is called a partial 𝐻+-type multivalued weak 𝜑-Lipschitz mapping if condition (C2) in Definition 2 holds and there exist M:(R+)5→R+and 𝜑:R+→R+such that 𝐻+(𝑇𝑥,𝑇𝑦)≤𝜑(M(𝑑(𝑥,𝑦),𝑑(𝑥,𝑇𝑥),𝑑(𝑦,𝑇𝑦), 𝑑(𝑥,𝑇𝑦),𝑑(𝑦,𝑇𝑥))), for every (𝑥,𝑦)∈𝑋×𝑋with 𝛼(𝑥,𝑦)≥1. (40) If, moreover, there exists 𝑘∈(0,1)such that 𝜑(𝑡) ≤ 𝑘𝑡,for every 𝑡≥0, we say that 𝑇is a partial 𝐻+-type multivalued weak 𝜑-contraction. Remark 17. Taking 𝜑(𝑡)=𝑘𝑡,with𝑘∈(0,1),and M(𝑟1,𝑟2,𝑟3,𝑟4,𝑟5)=max {𝑟1,𝑟2,𝑟3,𝑟4,𝑟5}, (41) then we obtain the notion of partial 𝐻+-type 𝑞-set-valued quasi-contraction (compare with [21, Definition 3.1]). In relation with Definitions 2.21, 3.1 and Theorem 3.2 [21], we can establish the following result. Theorem 18. Let (𝑋,𝑑)be a complete metric space, let 𝛼:𝑋× 𝑋→[0,∞)be given, and let 𝑇:𝑋→CB(𝑋)be a partial 𝐻+-type multivalued weak 𝜑-Lipschitz mapping such that (h1) 𝑇is 𝛼-admissible; that is, for each 𝑥∈𝑋and 𝑦∈𝑇𝑥 with 𝛼(𝑥,𝑦)≥1,onehas𝛼(𝑦,𝑧)≥1,forevery𝑧∈𝑇𝑦, (h2) there exist 𝑥0∈𝑋and 𝑥1∈𝑇𝑥0such that 𝛼(𝑥0,𝑥1)≥ 1, (h3) if {𝑥𝑛}is a sequence in 𝑋such that 𝛼(𝑥𝑛,𝑥𝑛+1)≥1,for all 𝑛∈N,andthereexists𝑥∈𝑋such that 𝑥𝑛→𝑥as 𝑛→∞,then𝛼(𝑥𝑛,𝑥)≥1,forevery𝑛, (h4) the functions 𝜑,Mare such that conditions (i), (ii), (iii), (iv) [(a)–(e)], and (v) in Theorem 8 hold. Then 𝑇has a fixed point in 𝑋. Proof. We proceed similarly to the proof of Theorem 8 and also Theorem 3.2 [21]. By (40) and (i), we get 𝐻+(𝑇𝑥,𝑇𝑦)≤𝑘M(𝜓(𝑥,𝑦)), for every (𝑥,𝑦)∈𝑋×𝑋with 𝛼(𝑥,𝑦)≥1.(42) We take 𝜀>0given.Westartthesequencewiththeterms 𝑥0and 𝑥1∈𝑇𝑥0and then, using (C2), there exists 𝑥2∈𝑇𝑥1 such that 𝑑(𝑥1,𝑥2)≤𝐻+(𝑇𝑥0,𝑇𝑥1)+𝜀.Note that since 𝑇is 𝛼-admissible, we have 𝛼(𝑥1,𝑥2)≥1. In general, for 𝑛∈N,if 𝑥𝑛∈𝑇𝑥𝑛−1is chosen such that 𝛼(𝑥𝑛−1,𝑥𝑛)≥1, then we can select 𝑥𝑛+1∈𝑇𝑥𝑛such that 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝐻+(𝑇𝑥𝑛−1,𝑇𝑥𝑛)+𝜀 (43) and we also have 𝛼(𝑥𝑛,𝑥𝑛+1)≥1. Again, 𝜀could have been chosen at each step depending on 𝑥𝑛−1and 𝑥𝑛.For this, we remark that if 𝐻+(𝑇𝑥𝑛−1,𝑇𝑥𝑛)=0forsome 𝑛,theproofisconcluded,sothatwecanassumethat M(𝜓(𝑥𝑛−1,𝑥𝑛)) > 0, for every 𝑛andtakeoneachstep𝜀= (𝐴−𝐵𝑘)M(𝜓(𝑥𝑛−1,𝑥𝑛)) > 0, where 𝐴,𝐵 > 0arefixed,by (v), in such a way that 𝐴−𝐵𝑘>0and𝐿:=𝐴−𝐵𝑘+𝑘< 1/max{𝜏,2R}.Then,by(43),weget,forevery𝑛∈N,that 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝐻+(𝑇𝑥𝑛−1,𝑇𝑥𝑛) +(𝐴−𝐵𝑘)M(𝜓(𝑥𝑛−1,𝑥𝑛)) ≤𝐿M(𝜓(𝑥𝑛−1,𝑥𝑛)) (44) which implies, for every 𝑛∈N,that 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝐿M(𝜓(𝑥𝑛−1,𝑥𝑛)) ≤𝐿M(𝑑(𝑥𝑛−1,𝑥𝑛),𝑑(𝑥𝑛−1,𝑥𝑛),𝑑(𝑥𝑛,𝑥𝑛+1), 𝑑(𝑥𝑛−1,𝑥𝑛)+𝑑(𝑥𝑛,𝑥𝑛+1),0)≤𝐿N(𝑑(𝑥𝑛−1,𝑥𝑛), 𝑑(𝑥𝑛,𝑥𝑛+1)), (45) using (ii) and (iv-a). As in the proof of Theorem 8,wecan suppose that 𝑑(𝑥𝑛,𝑥𝑛+1)>0, for every 𝑛∈N.Similarlytothe proof of Theorem 8,wededucethat𝑑(𝑥𝑛−1,𝑥𝑛)≥𝑑(𝑥𝑛,𝑥𝑛+1), for every 𝑛∈Nand 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝐿𝜏𝑑(𝑥𝑛−1,𝑥𝑛), ∀𝑛∈N,(46) so that {𝑥𝑛}is a Cauchy sequence and, by the completeness of 𝑋, there exists 𝑢∈𝑋such that lim𝑛→∞𝑥𝑛=𝑢. Finally, if 𝑑(𝑢,𝑇𝑢) > 0, inequality (22) holds; since 𝛼(𝑥𝑛,𝑥𝑛+1)≥1, for all 𝑛∈N,and𝑥𝑛→𝑢as 𝑛→∞, then, by hypotheses, 𝛼(𝑥𝑛,𝑢)≥1, for every 𝑛,sothat 𝐻+(𝑇𝑥𝑛,𝑇𝑢)≤𝑘M(𝜓(𝑥𝑛,𝑢)),for every 𝑛. (47) Similarly to the proof of Theorem 8,wegetthat 𝑑(𝑢,𝑇𝑢)≤𝑑(𝑢,𝑥𝑛+1)+𝑑(𝑥𝑛+1,𝑇𝑢) ≤𝜌(𝑇𝑥𝑛,𝑇𝑢)+𝑑(𝑥𝑛+1,𝑢); (48) Discrete Dynamics in Nature and Society 7 hence 𝑑(𝑢,𝑇𝑢)≤2𝑘R𝑑(𝑢,𝑇𝑢),(49) so that if R=0, then 𝑑(𝑢,𝑇𝑢) = 0andifR>0, the assumption 𝑑(𝑢,𝑇𝑢) > 0 leads to a contradiction again and the proof is complete. The previous result allows formulating a fixed point result for 𝐻+-type partial quasi-contraction mappings while, in [21], the Pompeiu-Hausdorff metric is used in 𝑏-metric spaces. The procedure in metric spaces could be adapted to 𝑏-metric spaces in the lines of [21]. Remark 19. In Theorem 18,ifweconsiderthePompeiu- Hausdorff metric 𝐻, then condition (v) can be relaxed to hypothesis (v∗) in the statement of Theorem 13. In particular, for partial 𝑞-set-valued quasi-contractions, 𝜑(𝑡):=𝑘𝑡,𝑘∈(0,1/2),andM(𝑟1,𝑟2,𝑟3,𝑟4,𝑟5):=max{𝑟1,𝑟2, 𝑟3,𝑟4,𝑟5}, we obtain, as a corollary, the assertion of Theorem 3.2 [21] for metric spaces. Remark 20. Inequality (40) is trivially valid if one of the following conditions hold (see Corollaries 3.4–3.6 [21]): (r.i) 𝛼(𝑥,𝑦)𝐻+(𝑇𝑥,𝑇𝑦)≤𝜑(M(𝜓(𝑥,𝑦))),forevery𝑥,𝑦∈ 𝑋; (r.ii) (𝐻+(𝑇𝑥,𝑇𝑦)+𝜀)𝛼(𝑥,𝑦) ≤𝜑(M(𝜓(𝑥,𝑦)))+𝜀,forevery 𝑥,𝑦∈𝑋,where𝜀≥1; (r.iii) (𝛼(𝑥,𝑦)−1+𝜀)𝐻+(𝑇𝑥,𝑇𝑦) ≤𝜀𝜑(M(𝜓(𝑥,𝑦))),forevery𝑥,𝑦∈ 𝑋,where𝜀>1. Theorem 21. Let (𝑋,𝑑)be a complete metric space, let 𝛼:𝑋× 𝑋→[0,∞)be given, and let 𝑇:𝑋→CB(𝑋)be a partial 𝐻+-type multivalued weak 𝜑-Lipschitz mapping such that conditions (h1), (h2), and (h3) hold. Suppose also that the functions 𝜑,Msatisfy the following: (A1) 𝜑is monotonically increasing and 𝜑(0)=0. (A2) If 𝑧𝑛→𝑧as 𝑛→∞,thenlim inf𝑛→∞𝜑(𝑧𝑛)≤𝜑(𝑧). (A3) Conditions (ii), (iii), and (iv) [(a)–(e)] in Theorem 8 hold. (A4) For Rgiven in (iv-e), one has that 2𝜑(R𝑧) < 𝑧,for every 𝑧>0. (A5) There exists S>0such that the operator LS:= 𝜑+ S𝐼(where 𝐼is the identity mapping on 𝑋)satisfiesthe following properties: (A5.i) LS(𝜏𝑧)<𝑧,forevery𝑧>0; (A5.ii) LS(𝜏𝑧)≤𝜏LS(𝑧),forevery𝑧>0; (A5.iii) ∑∞ 𝑛=1𝜏𝑛LS𝑛(𝑧) < ∞,forevery𝑧>0,where LS𝑛:=LS (𝑛) ∘⋅⋅⋅∘LS. Then 𝑇has a fixed point in 𝑋. Proof. We proceed similarly to the proof of Theorem 18.We take 𝑥0and 𝑥1∈𝑇𝑥0given by (h2), with 𝛼(𝑥0,𝑥1)≥1; then a sequence is obtained by using (C2) and 𝛼-admissibility. The processissummarizedasfollows:for𝑛∈N,if𝑥𝑛∈𝑇𝑥𝑛−1is chosen such that 𝛼(𝑥𝑛−1,𝑥𝑛)≥1, then, by (C2), we can select 𝑥𝑛+1∈𝑇𝑥𝑛such that 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝐻+(𝑇𝑥𝑛−1,𝑇𝑥𝑛)+𝜀, (50) andwealsohave𝛼(𝑥𝑛,𝑥𝑛+1)≥1. We remark that, due to the monotonically increasing character of 𝜑,themappingLSis always strictly increasing. Note that if 𝐻+(𝑇𝑥𝑛−1,𝑇𝑥𝑛)=0forsome𝑛,theproofisconcluded, so that we can assume that M(𝜓(𝑥𝑛−1,𝑥𝑛)) > 0, for every 𝑛,sothatwecantake,oneachstep,𝜀𝑛=SM(𝜓(𝑥𝑛−1, 𝑥𝑛))>0. Then, for every 𝑛∈N, by (ii) and (iv-a), 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝜑(M(𝜓(𝑥𝑛−1,𝑥𝑛))) +SM (𝜓(𝑥𝑛−1,𝑥𝑛)) =LS(M(𝜓(𝑥𝑛−1,𝑥𝑛))) ≤LS(N(𝑑(𝑥𝑛−1,𝑥𝑛),𝑑(𝑥𝑛,𝑥𝑛+1))). (51) If 𝑥𝑛=𝑥𝑛+1,forsome𝑛,then𝑥𝑛is a fixed point of 𝑇,sowe assume that 𝑥𝑛=𝑥𝑛+1,forevery𝑛∈N. Assuming that 𝑑(𝑥𝑛−1,𝑥𝑛)<𝑑(𝑥𝑛,𝑥𝑛+1),forsome𝑛∈N, by (iv-b) and (iv-d), we obtain 𝑑(𝑥𝑛,𝑥𝑛+1)≤LS(N(𝑑(𝑥𝑛,𝑥𝑛+1),𝑑(𝑥𝑛,𝑥𝑛+1))) ≤LS(𝜏𝑑(𝑥𝑛,𝑥𝑛+1)), ∀𝑛∈N,(52) which is a contradiction due to (A5) since 𝑑(𝑥𝑛,𝑥𝑛+1)>0. This proves that 𝑑(𝑥𝑛−1,𝑥𝑛)≥𝑑(𝑥 𝑛,𝑥𝑛+1),forevery𝑛∈N. Therefore, by (iv-c), we get 𝑑(𝑥𝑛,𝑥𝑛+1)≤LS(N(𝑑(𝑥𝑛−1,𝑥𝑛),𝑑(𝑥𝑛,𝑥𝑛+1))) ≤LS(N(𝑑(𝑥𝑛−1,𝑥𝑛),𝑑(𝑥𝑛−1,𝑥𝑛))) ≤LS(𝜏𝑑(𝑥𝑛−1,𝑥𝑛)) ≤𝜏LS(𝑑(𝑥𝑛−1,𝑥𝑛)), ∀𝑛∈N. (53) This implies, from (A5) and the property that LS(𝑧)>0for every 𝑧>0, that 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝜏LS(𝑑(𝑥𝑛−1,𝑥𝑛)) ≤𝜏LS(𝜏LS(𝑑(𝑥𝑛−2,𝑥𝑛−1))) ≤𝜏2LS 2(𝑑(𝑥𝑛−2,𝑥𝑛−1)) ≤𝜏𝑛LS𝑛(𝑑(𝑥0,𝑥1)), ∀𝑛∈N. (54) To check that {𝑥𝑛}is a Cauchy sequence, we observe that, for 𝑛,𝑚∈Nwith 𝑛<𝑚,weget 𝑑(𝑥𝑛,𝑥𝑚)≤𝑚−1 ∑ 𝑖=𝑛 𝑑(𝑥𝑖,𝑥𝑖+1)≤𝑚−1 ∑ 𝑖=𝑛 𝜏𝑖LS𝑖(𝑑(𝑥0,𝑥1)) 󳨀→ 0,as 𝑛󳨀→∞. (55) 8 Discrete Dynamics in Nature and Society Since 𝑋is complete, there exists 𝑢∈𝑋such that lim𝑛→∞𝑥𝑛= 𝑢.If 𝑑(𝑢,𝑇𝑢) > 0, using that 𝛼(𝑥𝑛,𝑥𝑛+1)≥1, for all 𝑛∈N, and 𝑥𝑛→𝑢as 𝑛→∞,then,by(h3),wehave𝛼(𝑥𝑛,𝑢)≥1, for every 𝑛;hence 𝐻+(𝑇𝑥𝑛,𝑇𝑢)≤𝜑(M(𝜓(𝑥𝑛,𝑢))), for every 𝑛(56) so that lim inf 𝑛→∞ {𝜌(𝑇𝑥𝑛,𝑇𝑢)+𝜌(𝑇𝑢,𝑇𝑥𝑛)} ≤2lim inf 𝑛→∞ 𝜑(M(𝜓(𝑥𝑛,𝑢))). (57) Therefore, similarly to inequality (22),wehave,bythe monotonicity of 𝜑, 𝜑(M(𝜓(𝑥𝑛,𝑢)))=𝜑(M(𝑑(𝑥𝑛,𝑢),𝑑(𝑥𝑛,𝑇𝑥𝑛), 𝑑(𝑢,𝑇𝑢),𝑑(𝑥𝑛,𝑇𝑢),𝑑(𝑢,𝑇𝑥𝑛))) ≤𝜑(M(𝑑(𝑥𝑛,𝑢),𝑑(𝑥𝑛,𝑥𝑛+1),𝑑(𝑢,𝑇𝑢), 𝑑(𝑥𝑛,𝑇𝑢),𝑑(𝑢,𝑥𝑛+1))). (58) Analogously to the proof of Theorem 8 and using (iii), lim 𝑛→∞M(𝑑(𝑥𝑛,𝑢),𝑑(𝑥𝑛,𝑥𝑛+1),𝑑(𝑢,𝑇𝑢),𝑑(𝑥𝑛,𝑇𝑢), 𝑑(𝑢,𝑥𝑛+1))=M(0,0,𝑑(𝑢,𝑇𝑢),𝑑(𝑢,𝑇𝑢),0).(59) By (A2), we have lim inf 𝑛→∞ 𝜑(M(𝜓(𝑥𝑛,𝑢)))≤lim inf 𝑛→∞ 𝜑(M(𝑑(𝑥𝑛,𝑢), 𝑑(𝑥𝑛,𝑥𝑛+1),𝑑(𝑢,𝑇𝑢),𝑑(𝑥𝑛,𝑇𝑢),𝑑(𝑢,𝑥𝑛+1))) ≤𝜑(M(0,0,𝑑(𝑢,𝑇𝑢),𝑑(𝑢,𝑇𝑢),0)). (60) Hence, from this inequality and (57), by (iv-a) and (iv-e), we get lim inf 𝑛→∞ {𝜌(𝑇𝑥𝑛,𝑇𝑢)+𝜌(𝑇𝑢,𝑇𝑥𝑛)} ≤2lim inf 𝑛→∞ 𝜑(M(𝜓(𝑥𝑛,𝑢))) ≤2𝜑(M(0,0,𝑑(𝑢,𝑇𝑢),𝑑(𝑢,𝑇𝑢),0)) ≤2𝜑(N(0,𝑑(𝑢,𝑇𝑢)))≤2𝜑(R𝑑(𝑢,𝑇𝑢)). (61) Similarly to the proof of Theorem 8,wehave𝑑(𝑢,𝑇𝑢) ≤ 2𝜑(R𝑑(𝑢,𝑇𝑢)). Therefore, if R=0, then 𝑑(𝑢,𝑇𝑢) ≤ 𝜑(0)=0 and the proof is concluded. If R>0, we get to a contradiction due to 𝑑(𝑢,𝑇𝑢) > 0 and (A4). Hence, 𝑑(𝑢,𝑇𝑢) = 0 and the proof is complete, since 𝑇𝑢is closed, so that 𝑢∈𝑇𝑢. Remark 22. Theorem 8 is a particular case of Theorem 21. Indeed, if we take 𝜑(𝑡)=𝑘𝑡,where𝑘>0, and assuming that condition (v) holds, then we have the following: (r.i) 𝜑is monotonically increasing and 𝜑(0)=0; hence (A1) holds. (r.ii) 𝜑is continuous (so that (A2) holds). (r.iii) Condition (A4) is valid since 2𝜑(R𝑧) < 𝑧,forevery 𝑧>0 (this is true since it is equivalent to 𝑘<1/2R). (r.iv) Concerning (A5), since we assume that 𝑘𝜏<1, then we can take S>0suchthat(𝑘+S)𝜏 < 1andthe operator LS:= 𝜑+S𝐼is monotonically increasing and (r.a) LS(𝜏𝑧)=(𝑘+S)𝜏𝑧<𝑧,forevery𝑧>0; (r.b) LS(𝜏𝑧)=(𝑘+S)𝜏𝑧=𝜏(𝑘+S)𝑧=𝜏LS(𝑧),for every 𝑧>0; (r.c) ∑∞ 𝑛=1𝜏𝑛LS𝑛(𝑧) = 𝑧∑∞ 𝑛=1(𝑘𝜏)𝑛<∞,forevery 𝑧>0. It is also obvious that if 𝑇is an 𝐻+-type multivalued weak 𝜑- Lipschitz mapping and there exists 𝑘>0suchthat𝜑(𝑡)≤𝑘𝑡, for 𝑡≥0, then 𝑇is also an 𝐻+-type multivalued weak  𝜑- Lipschitz mapping for  𝜑(𝑡)=𝑘𝑡. Remark 23. In the proof of Theorems 18 and 21,itiseasyto observe that a proper combination of condition (C2) with the 𝛼-admissible character of the mapping, (h1), would allow relaxing slightly the definition of partial 𝐻+-type multivalued weak 𝜑-Lipschitz mappings. Indeed, in these theorems, it is possible to replace these hypotheses ((C2) and 𝛼-admissibil- ity of 𝑇)bythefollowing: (2∗)forevery𝑥∈𝑋,𝑦∈𝑇𝑥with 𝛼(𝑥,𝑦)≥1and𝜀>0, there exists 𝑧∈𝑇𝑦such that 𝑑(𝑦,𝑧)≤𝐻+(𝑇𝑦,𝑇𝑥)+𝜀 and 𝛼(𝑦,𝑧)≥1. If we consider self-mappings 𝑇:𝑋→𝑋,aprocedure similar to Theorem 2.3 [19] gives the following result. Theorem 24. Let (𝑋,𝑑)be a complete metric space, let 𝛼:𝑋× 𝑋→[0,∞)be given, and let 𝑇:𝑋→𝑋be such that (t.i) for each 𝑥∈𝑋with 𝛼(𝑥,𝑇𝑥) ≥ 1,onehas 𝛼(𝑇𝑥,𝑇(𝑇𝑥))≥1, (t.ii) there exists 𝑥0∈𝑋such that 𝛼(𝑥0,𝑇𝑥0)≥1, and there exist M:(R+)5→R+and 𝜑:R+→R+with 𝑑(𝑇𝑥,𝑇𝑦)≤𝜑(𝑀(𝑑(𝑥,𝑦),𝑑(𝑥,𝑇𝑥),𝑑(𝑦,𝑇𝑦), 𝑑(𝑥,𝑇𝑦),𝑑(𝑦,𝑇𝑥))), 𝑓𝑜𝑟 𝑒V𝑒𝑟𝑦 (𝑥,𝑦)∈𝑋×𝑋𝑤𝑖𝑡ℎ𝛼(𝑥,𝑦)≥1. (62) Supposealsothat(h3)holdsandthatthefunctions𝜑,Msatisfy the following conditions: (I) 𝜑is monotonically increasing, Mis increasing in the fourth variable, and conditions (iv) [(a)–(d)] in Theorem 8 hold, where 𝜏≤1. (II) Consider ∑∞ 𝑛=1𝜑𝑛(𝑡)<+∞,forevery𝑡>0. (III) One of the following conditions holds: (III.i) For 𝑟3>0fixed, if 𝑟1,𝑟2,𝑟5are small enough and 𝑟4is close enough to 𝑟3,thenM(𝑟1,𝑟2,𝑟3,𝑟4,𝑟5)≤ 𝑟3. Discrete Dynamics in Nature and Society 9 (III.ii) (A2) is satisfied and conditions (iii) and (iv-e) in Theorem 8 hold, where R≤1. Then 𝑇has a fixed point in 𝑋. Proof. We take 𝑥0∈𝑋with 𝛼(𝑥0,𝑇𝑥0)≥1 and define the sequence {𝑥𝑛}such that 𝑥𝑛+1=𝑇𝑥𝑛,𝑛∈N.If𝑥𝑛=𝑥𝑛+1,for some 𝑛,thenwehaveafixedpointof𝑇;thusweassumethat 𝑥𝑛=𝑥𝑛+1,forevery𝑛(i.e., 𝑑(𝑥𝑛,𝑥𝑛+1)>0, for every 𝑛∈N). Using the hypotheses, it is easy to prove that 𝛼(𝑥𝑛,𝑥𝑛+1)≥ 1, for every 𝑛.Then,foreach𝑛∈N, by the nondecreasing character of 𝜑and M(in the fourth variable) and (iv-a), 𝑑(𝑥𝑛,𝑥𝑛+1)=𝑑(𝑇𝑥𝑛−1,𝑇𝑥𝑛)≤𝜑(M(𝜓(𝑥𝑛−1,𝑥𝑛))) =𝜑(M(𝑑(𝑥𝑛−1,𝑥𝑛),𝑑(𝑥𝑛−1,𝑇𝑥𝑛−1), 𝑑(𝑥𝑛,𝑇𝑥𝑛),𝑑(𝑥𝑛−1,𝑇𝑥𝑛),𝑑(𝑥𝑛,𝑇𝑥𝑛−1))) ≤𝜑(M(𝑑(𝑥𝑛−1,𝑥𝑛),𝑑(𝑥𝑛−1,𝑥𝑛),𝑑(𝑥𝑛,𝑥𝑛+1), 𝑑(𝑥𝑛−1,𝑥𝑛)+𝑑(𝑥𝑛,𝑥𝑛+1),0)) ≤𝜑(N(𝑑(𝑥𝑛−1,𝑥𝑛),𝑑(𝑥𝑛,𝑥𝑛+1))). (63) If 𝑑(𝑥𝑛−1,𝑥𝑛)<𝑑(𝑥 𝑛,𝑥𝑛+1),forsome𝑛∈N,bythe monotonicity of 𝜑, (iv-b), (iv-d), 𝜏≤1, and (II), we get 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝜑(N(𝑑(𝑥𝑛,𝑥𝑛+1),𝑑(𝑥𝑛,𝑥𝑛+1))) ≤𝜑(𝜏𝑑(𝑥𝑛,𝑥𝑛+1))≤𝜑(𝑑(𝑥𝑛,𝑥𝑛+1)) <𝑑(𝑥𝑛,𝑥𝑛+1), (64) which is a contradiction. Then, 𝑑(𝑥𝑛−1,𝑥𝑛)≥𝑑(𝑥𝑛,𝑥𝑛+1),for every 𝑛∈N;hence,bythemonotonicityof𝜑, (iv-c), (iv-d), and 𝜏≤1, 𝑑(𝑥𝑛,𝑥𝑛+1)≤𝜑(N(𝑑(𝑥𝑛−1,𝑥𝑛),𝑑(𝑥𝑛−1,𝑥𝑛))) ≤𝜑(𝜏𝑑(𝑥𝑛−1,𝑥𝑛))≤𝜑(𝑑(𝑥𝑛−1,𝑥𝑛)). (65) Using (II), the sequence {𝑥𝑛}is a Cauchy sequence. Then there exists 𝑢such that 𝑥𝑛→𝑢as 𝑛→∞.By(h3),𝛼(𝑥𝑛,𝑢)≥1, for every 𝑛.Supposethat𝑑(𝑢,𝑇𝑢)>0; then 𝑑(𝑢,𝑇𝑢)≤𝑑(𝑢,𝑇𝑥𝑛)+𝑑(𝑇𝑥𝑛,𝑇𝑢)≤𝑑(𝑢,𝑥𝑛+1) +𝜑(M(𝜓(𝑥𝑛,𝑢)))=𝑑(𝑥𝑛+1,𝑢) +𝜑(M(𝑑(𝑥𝑛,𝑢),𝑑(𝑥𝑛,𝑥𝑛+1),𝑑(𝑢,𝑇𝑢), 𝑑(𝑥𝑛,𝑇𝑢),𝑑(𝑢,𝑥𝑛+1))). (66) In case (III.i), for 𝑛being large enough, we have 𝑑(𝑢,𝑇𝑢)≤𝑑(𝑥𝑛+1,𝑢)+𝜑(𝑑(𝑢,𝑇𝑢)),(67) so that 𝑑(𝑢,𝑇𝑢) ≤ 𝜑(𝑑(𝑢,𝑇𝑢)) < 𝑑(𝑢,𝑇𝑢),whichisa contradiction; hence 𝑇𝑢 = 𝑢and the proof is complete. On the other hand, in case (III.ii), by the monotonicity of 𝜑and (II), we have 𝑑(𝑢,𝑇𝑢)≤lim inf 𝑛→∞ 𝜑(M(𝑑(𝑥𝑛,𝑢),𝑑(𝑥𝑛,𝑥𝑛+1), 𝑑(𝑢,𝑇𝑢),𝑑(𝑥𝑛,𝑇𝑢),𝑑(𝑢,𝑥𝑛+1))) ≤𝜑(lim 𝑛→∞M(𝑑(𝑥𝑛,𝑢),𝑑(𝑥𝑛,𝑥𝑛+1),𝑑(𝑢,𝑇𝑢), 𝑑(𝑥𝑛,𝑇𝑢),𝑑(𝑢,𝑥𝑛+1)))≤𝜑(M(0,0,𝑑(𝑢,𝑇𝑢), 𝑑(𝑢,𝑇𝑢),0))≤𝜑(N(0,𝑑(𝑢,𝑇𝑢)))≤𝜑(R𝑑(𝑢, 𝑇𝑢))≤𝜑(𝑑(𝑢,𝑇𝑢))<𝑑(𝑢,𝑇𝑢), (68) a contradiction again and the proof is finished. Remark 25. If M(𝑟1,𝑟2,𝑟3,𝑟4,𝑟5):=max{𝑟1,𝑟2,𝑟3,(𝑟4+𝑟5)/2} in Theorem 24,thenwehaveTheorem2.3[19]. Indeed, all the conditions are satisfied (see the proof of Corollary 11)and, for 𝑟3>0fixed,if𝑟1,𝑟2,𝑟5are small enough and 𝑟4is close enough to 𝑟3,thenM(𝑟1,𝑟2,𝑟3,𝑟4,𝑟5)=𝑟3; hence (III.i) is also fullfilled. Remark 26. Certain conditions in Theorem 24 also extend some hypotheses in [20]. Asindicatedin[21], a function 𝛼can be defined in connection with a binary relation Rin 𝑋(which could be, e.g.,apartialorderingin𝑋). Thus, Theorem 4.7 [21]canbe extended to 𝐻+-type multivalued weak 𝜑-Lipschitz mappings with respect to R,whicharedefinedasfollows.Here,weonly consider the case of metric spaces, but 𝑏-metric spaces could be considered accordingly. Definition 27. Let (𝑋,𝑑)be a metric space and Rabinary relation on 𝑋.Amultivaluedmap𝑇:𝑋→CB(𝑋)is called an 𝐻+-type multivalued weak 𝜑-Lipschitz mapping with respect to Rif condition (C2) in Definition 2 holds and there exist M:(R+)5→R+and 𝜑:R+→R+such that 𝐻+(𝑇𝑥,𝑇𝑦)≤𝜑(M(𝑑(𝑥,𝑦),𝑑(𝑥,𝑇𝑥),𝑑(𝑦,𝑇𝑦), 𝑑(𝑥,𝑇𝑦),𝑑(𝑦,𝑇𝑥))), for every 𝑥,𝑦∈𝑋with 𝑥R𝑦. (69) If, moreover, there exists 𝑘∈(0,1)such that 𝜑(𝑡) ≤ 𝑘𝑡,for every 𝑡≥0, one says that 𝑇is an 𝐻+-type multivalued weak 𝜑-contraction with respect to R. Theorem 28. Let (𝑋,𝑑) be a complete metric space, Ra binary relation on 𝑋,and𝑇:𝑋→CB(𝑋)an 𝐻+-type multivalued weak 𝜑-Lipschitz mapping with respect to Rsuch that (H1) 𝑇is weakly preserving, that is, for each 𝑥∈𝑋and 𝑦∈ 𝑇𝑥with 𝑥R𝑦,onehas𝑦R𝑧,forevery𝑧∈𝑇𝑦, (H2) there exist 𝑥0∈𝑋and 𝑥1∈𝑇𝑥0such that 𝑥0R𝑥1, (H3) if {𝑥𝑛}is a sequence in 𝑋such that 𝑥𝑛R𝑥𝑛+1,forall 𝑛∈N,andthereexists𝑥∈𝑋such that 𝑥𝑛→𝑥as 𝑛→∞,then𝑥𝑛R𝑥,forevery𝑛,