Endpoints of Multivalued Contraction Operators

Abstract

The existence of the endpoints and approximate endpoints are studied in a general setting for the operators satisfying various contractive conditions. Some recent results are also derived as special cases.

1. Introduction

Among several generalizations of celebrated Banach fixed point theorem, one interesting extension is Nadler’s [1] fixed point theorem for multivalued contraction. He exactly proved that a multivalued contraction has a fixed point in a complete metric space. Subsequently, it received great attention in applicable mathematics and was extended and generalized on various settings. Indeed, these extensions and generalizations have been influenced by the applications of the multivalued fixed point theory in mathematical economics, game theory, differential inclusions, interval arithmetic, Hammerstein equations, convex optimization, duality theory in optimization, variational inequalities and control theory, nonlinear evolution equations and nonlinear semigroups, quasivariational inequalities, and elasticity and plasticity theory (see, for instance, [2–8] and several references thereof). The results related to existence of endpoints or strict fixed-points were first given by Rus [9] in 2003. Thereafter, a number of authors established interesting results concerning existence and uniqueness of endpoints for multivalued contractions in different settings; see, for example, [10–15]. The main purpose of this paper is to establish some existence and uniqueness results for endpoints using different multivalued contractions. Our results include some recent results.

2. Preliminaries

Let 𝑇∶𝑋→2^𝑋 be a multivalued mapping. An element 𝑥∈𝑋 is said to be an endpoint of 𝑇 if 𝑇𝑥={𝑥}. We say that a multivalued mapping 𝑇∶𝑋→2^𝑋 has the approximate endpoint property (AEPP) if inf_𝑥∈𝑋sup_{𝑦∈𝑇𝑥}𝑑(𝑥,𝑦)=0 (also see [3, 10]). Throughout the paper, let (𝑋,𝑑) be a 𝑏-metric space, and let 𝑃(𝑋) denote the family of all nonempty subsets of 𝑋 and 𝐶𝑙(𝑋) the family of all nonempty closed subsets of 𝑋. For any 𝐴,𝐵∈𝑃(𝑋), the Hausdorff metric is defined as 𝐻(𝐴,𝐵)=maxsup_𝑎∈𝐴𝑑(𝑎,𝐵),sup_𝑏∈𝐵𝑑(𝑏,𝐴),(1) where 𝑑(𝑎,𝐵)=inf{𝑑(𝑎,𝑏)∶𝑏∈𝐵} is the distance from the point 𝑎 to the set 𝐵.

Let 𝐼∶𝑋→𝑋 be a single-valued mapping and 𝑇∶𝑋→𝐶𝑙(𝑋) a multivalued contraction. We say that the mappings 𝐼 and 𝑇 have an AEPP provided inf_𝑥∈𝑋sup_{𝑦∈𝑇𝑥}𝑑(𝐼𝑥,𝑦)=0.

A point 𝑥∈𝑋 is called an endpoint of 𝐼 and 𝑇 if 𝑇𝑥={𝐼𝑥}. For each 𝜀>0, let 𝐸_𝜀(𝐼,𝑇)={𝑥∈𝑋∶sup_{𝑦∈𝑇𝑥}𝑑(𝐼𝑥,𝑦)≤𝜀} be the set of all approximate endpoints of the mappings 𝐼 and 𝑇.

Example 1. Let 𝑋=(−∞,∞) with Euclidean norm. Assume that 𝐾=[0,𝑐] and 𝑇∶𝐾→𝐶𝑙(𝐾) defined by []𝑐𝑇𝑥=𝑥,𝑐−𝑥,𝑥∈0,2[]𝑐𝑐−𝑥,𝑥,𝑥∈2,𝑐,(2) where 𝑐 is a positive constant. Clearly, Fix(𝑇)=[0,𝑐] and End(𝑇)={𝑐/2}.

Definition 2 (see [16]). Let 𝑋 be a nonempty set and 𝑏≥1 a given real number. A function 𝑑∶𝑋×𝑋→ℜ₊ (set of nonnegative real numbers) is said to be a 𝑏-metric iff for all 𝑥,𝑦,𝑧∈𝑋 the following conditions are satisfied:(i) 𝑑(𝑥,𝑦)=0iff𝑥=𝑦,(ii) 𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥),(iii) 𝑑(𝑥,𝑧)≤𝑏[𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧)].A pair (𝑋,𝑑) is called a 𝑏-metric space.

The class of 𝑏-metric spaces is effectively larger than that of metric spaces, since a 𝑏-metric space is a metric space when 𝑏=1 in the above condition (iii). The following example shows that a 𝑏-metric on 𝑋 need not be a metric on 𝑋 (see also [16, page 264]).

Example 3 (see [17]). Let 𝑋={𝑥₁,𝑥₂,𝑥₃,𝑥₄} and 𝑑(𝑥₁,𝑥₂)=𝑘≥2,𝑑(𝑥₁,𝑥₃)=𝑑(𝑥₁,𝑥₄)
=𝑑(𝑥₂,𝑥₃)=𝑑(𝑥₂,𝑥₄)=𝑑(𝑥₃,𝑥₄)=1, 𝑑(𝑥_𝑖,𝑥_𝑗)=𝑑(𝑥_𝑗,𝑥_𝑖) for all 𝑖,𝑗=1,2,3,4 and 𝑑(𝑥_𝑖,𝑥_𝑖)=0, 𝑖=1,2,3,4. Then, 𝑑𝑥_𝑖,𝑥_𝑗≤𝑘2𝑑𝑥_𝑖,𝑥_𝑛+𝑑𝑥_𝑛,𝑥_𝑗for𝑛,𝑖,𝑗=1,2,3,4,(3) and if 𝑘>2, the ordinary triangle inequality does not hold.

Definition 4 (see [16]). Let (𝑋,𝑑) be a 𝑏-metric space. Then, a sequence {𝑥_𝑛}_𝑛∈𝑁 in 𝑋 is called(a)convergent if and only if there exists 𝑥∈𝑋 such that 𝑑(𝑥_𝑛,𝑋)→0 as 𝑛→∞. In this case, one writes lim_𝑛→∞𝑥_𝑛=𝑥,(b)Cauchy if and only if 𝑑(𝑥_𝑛,𝑥_𝑚)→0 as 𝑚,𝑛→∞.

Remark 5 (see [16]). In a 𝑏-metric space (𝑋,𝑑), the following assertions hold.(i)A convergent sequence has a unique limit.(ii)Each convergent sequence is Cauchy.(iii)In general, a 𝑏-metric is not continuous.

Definition 6 (see [16]). Let (𝑋,𝑑) be a 𝑏-metric space. If 𝑌 is a nonempty subset of 𝑋, then the closure 𝑌 of 𝑌 is the set of limits of all convergent sequences of points in 𝑌, i.e., 𝑌=𝑥∈𝑋∶thereexistsasequence𝑥_{𝑛𝑛∈𝑁}suchthatlim_𝑛→∞𝑥_𝑛=𝑥.(4)

Definition 7 (see [16]). Let (𝑋,𝑑) be a 𝑏-metric space. Then, a subset 𝑌⊂𝑋 is called(a)closed if and only if for each sequence {𝑥_𝑛}_𝑛∈𝑁 in 𝑌 which converges to an element 𝑥, one has 𝑥∈𝑌,(b)compact if and only if for every sequence of elements of 𝑌, there exists a subsequence that converges to an element of 𝑌,(c)bounded if and only if 𝛿(𝑌)=sup{𝑑(𝑎,𝑏)∶𝑎,𝑏∈𝑌}<∞.

Definition 8 (see [16]). The 𝑏-metric space (𝑋,𝑑) is complete iff every Cauchy sequence in 𝑋 converges.

Definition 9. Let 𝑋 and 𝑌 be two Hausdorff topological spaces and 𝑇∶𝑋→𝑃(𝑌), a multivalued mapping with nonempty values. Then, 𝑇 is said to be(i)upper semicontinuous (u.s.c.) if, for each closed set 𝐵⊂𝑌, 𝑇⁻¹(𝐵)={𝑥∈𝑋∶𝑇(𝑥)∩𝐵≠𝜙} is closed in 𝑋;(ii)lower semicontinuous (l.s.c.) if, for each open set 𝐵⊂𝑌,  𝑇⁻¹(𝐵)={𝑥∈𝑋∶𝑇(𝑥)∩𝐵≠𝜙} is open in 𝑋;(iii)continuous if it is both u.s.c. and l.s.c.;(iv)closed if its graph Gr(𝑇)={(𝑥,𝑦)∈𝑋×𝑌∶𝑦∈𝑇(𝑥)} is closed;(v)compact if closure of 𝑇(𝑋) is a compact subset of 𝑌.

Definition 10 (see [5]). Let 𝐼∶𝑋→𝑋 be a single-valued mapping and 𝑇∶𝑋→𝐶𝑙(𝑋) a multivalued mapping. Then, 𝑇 is called(i)a multivalued 𝐼-contraction if there exists a number 𝛼∈(0,1)∶𝐻𝑇𝑥,𝑇𝑦≤𝛼𝑑𝐼𝑥,𝐼𝑦,∀𝑥,𝑦∈𝑋,(I-mc)(ii)a multivalued 𝐼-Kannan contraction if there exists a number 𝛽∈(0,1/2)∶𝐻𝑑(),()𝑇𝑥,𝑇𝑦≤𝛽𝐼𝑥,𝑇𝑥+𝑑𝐼𝑦,𝑇𝑦∀𝑥,𝑦∈𝑋,I-mkc(iii)a multivalued 𝐼-Chatterjea contraction if there exists a number 𝛾∈(0,1/2)∶𝐻𝑑,()𝑇𝑥,𝑇𝑦≤𝛾𝐼𝑥,𝑇𝑦+𝑑𝐼𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋,I-mcc(iv)a multivalued 𝐼-quasi-contraction if 𝐻𝑑(),𝑑𝑇𝑥,𝑇𝑦≤𝑘⋅max𝐼𝑥,𝐼𝑦,𝑑𝐼𝑥,𝑇𝑥,𝑑𝐼𝑦,𝑇𝑦𝐼𝑥,𝑇𝑦,𝑑𝐼𝑦,𝑇𝑥I-mqc for some 0≤𝑘<1 and all 𝑥,𝑦 in 𝑋,(v)a multivalued 𝐼-weak or almost contraction if there exist 𝛼∈(0,1) and 𝐿≥0∶𝐻𝑇𝑥,𝑇𝑦≤𝛼𝑑𝐼𝑥,𝐼𝑦+𝐿𝑑𝐼𝑦,𝑇𝑥,∀𝑥,𝑦∈𝑋,(I-mac)(vi)a multivalued generalized 𝐼-almost contraction if there exists a function 𝛼∶[0,∞)→[0,1) satisfying limsup_𝑟→𝑡+𝛼(𝑟)<1 for every 𝑡∈[0,∞) such that 𝐻𝑇𝑥,𝑇𝑦≤𝛼𝑑𝐼𝑥,𝐼𝑦𝑑𝐼𝑥,𝐼𝑦+𝐿𝑑𝐼𝑦,𝑇𝑥,∀𝑥,𝑦∈𝑋.I-gmac

Remark 11. A multivalued mapping 𝑇∶𝑋→𝐶𝑙(𝑋) is called a multivalued 𝐼-Zamfirescu (I-mzc) operator if it satisfies at least one of the conditions (i), (ii), and (iii).

We have used following Cantor’s intersection theorem in our results.

Theorem 12. Let 𝑋 be a compact space, and let 𝐶₁⊃𝐶₂⊃𝐶₃⋯ be a nested chain of nonempty closed subsets of 𝑋. Then, ∩𝐶_𝑛≠𝜙.

3. Main Results

Lemma 13. Let (𝑋,𝑑) be a 𝑏-metric space with the 𝑏-metric as a continuous functional. Let 𝐼∶𝑋→𝑋 be a single-valued mapping such that 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦) for all 𝑥,𝑦∈𝑋, where 𝑟>0 is a constant. If 𝑇∶𝑋→𝐶𝑙(𝑋) satisfies (I-mc) with 𝑟𝛼𝑏²<1, then 𝛿𝐸_𝜀()≤()𝐼,𝑇𝑏𝜀1+𝑏𝑟1−𝛼𝑏²,∀𝜀>0.(5)

Proof. For any 𝑥,𝑦∈𝐸_𝜀(𝐼,𝑇), we have 𝑑{},𝐻})𝐼𝑥,𝐼𝑦=𝐻𝐼𝑥𝐼𝑦≤𝑏({𝐼𝑥,𝑇𝑥+𝐻𝑇𝑥,𝐼𝑦≤𝑏𝜀+𝑏²𝐻𝑇𝑥,𝑇𝑦+𝐻𝑇𝑦,𝐼𝑦≤𝑏𝜀+𝑏²𝐻𝑇𝑥,𝑇𝑦+𝑏²𝜀≤𝑏𝜀+𝑏²𝜀+𝛼𝑏²𝑑𝐼𝑥,𝐼𝑦.(6) So, 𝑑≤()𝐼𝑥,𝐼𝑦𝑏𝜀1+𝑏1−𝛼𝑏².(7) Since 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦), we have 𝛿𝐸_𝜀()≤()𝐼,𝑇𝑏𝜀1+𝑏𝑟1−𝛼𝑏²,∀𝜀>0.(8)

Lemma 14. Let (𝑋,𝑑) be a 𝑏-metric space with the 𝑏-metric as a continuous functional. Let 𝐼∶𝑋→𝑋 be a continuous single-valued mapping. If 𝑇∶𝑋→𝐶𝑙(𝑋) is a lower semicontinuous multivalued mapping. Then, for each 𝜀>0,  𝐸_𝜀(𝐼,𝑇) is closed.

Proof. The proof follows from Lemma 16 of Hussain et al. [11].

Theorem 15. Let (𝑋,𝑑) be a complete 𝑏-metric space with the 𝑏-metric as a continuous functional. Let 𝐼∶𝑋→𝑋 be a continuous single-valued mapping such that 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦), where 𝑟>0 is a constant. Let 𝑇∶𝑋→𝐶𝑙(𝑋) be a lower semicontinuous map satisfying (I-mc). Then, 𝐼 and 𝑇 have a unique endpoint if and only if 𝐼 and 𝑇 have the AEPP.

Proof. It is clear that if 𝐼 and 𝑇 have an endpoint, then 𝐼 and 𝑇 have the AEPP.
Then, 𝐶_𝑛=𝑥∈𝑋∶sup_{𝑦∈𝑇𝑥}𝑑≤1𝐼𝑥,𝑦𝑛≠𝜙,∀𝑛∈𝑁.(9) Also, we have for each 𝑛∈𝑁, 𝐶_𝑛⊇𝐶_𝑛+1. By Lemma 14, 𝐶_𝑛 is closed for each 𝑛∈𝑁. Since 𝐼 and 𝑇 satisfy AEPP, then 𝐶_𝑛≠𝜙 for each 𝑛∈𝑁. Now, we show that lim_𝑛→∞𝛿(𝐶_𝑛)=0. To show this, let 𝑥,𝑦∈𝐶_𝑛. Then, from Lemma 13, 𝛿𝐶_𝑛=𝛿𝐸_1/𝑛()≤𝑏()𝐼,𝑇1+𝑏)(1/𝑛𝑟1−𝛼𝑏²(10) and so lim_𝑛→∞𝛿(𝐶_𝑛)=0. It follows from the Cantor intersection theorem that _𝑛∈𝑁𝐶_𝑛=𝑥₀.(11) Thus, 𝑥₀ is the unique endpoint of 𝐼 and 𝑇.

Lemma 16. Let (𝑋,𝑑) be a 𝑏-metric space with the 𝑏-metric as a continuous functional. Let 𝐼∶𝑋→𝑋 be a single-valued mapping such that 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦) for all 𝑥,𝑦∈𝑋, where 𝑟>0 is a constant. If 𝑇∶𝑋→𝐶𝑙(𝑋) satisfies (I-mkc), then 𝛿𝐸_𝜀(𝐼,𝑇)≤𝑏𝜀𝑟1+𝑏+2𝛽𝑏,∀𝜀>0.(12)

Proof. For any 𝑥,𝑦∈𝐸_𝜀(𝐼,𝑇), we have 𝑑{},𝐻})𝐼𝑥,𝐼𝑦=𝐻𝐼𝑥𝐼𝑦≤𝑏({𝐼𝑥,𝑇𝑥+𝐻𝑇𝑥,𝐼𝑦≤𝑏𝜀+𝑏²𝐻𝑇𝑥,𝑇𝑦+𝐻𝑇𝑦,𝐼𝑦≤𝑏𝜀+𝑏²𝐻𝑇𝑥,𝑇𝑦+𝑏²𝜀≤𝑏𝜀+𝑏²𝜀+𝑏²𝛽𝑑(𝐼𝑥,𝑇𝑥)+𝑑𝐼𝑦,𝑇𝑦≤𝑏𝜀+𝑏²𝜀+𝑏².2𝛽𝜀≤𝑏𝜀1+𝑏+2𝛽𝑏(13) So, 𝑑𝐼𝑥,𝐼𝑦≤𝑏𝜀1+𝑏+2𝛽𝑏.(14) Since 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦), we have 𝛿𝐸_𝜀(𝐼,𝑇)≤𝑏𝜀𝑟1+𝑏+2𝛽𝑏,∀𝜀>0.(15)

Theorem 17. Let (𝑋,𝑑) be a complete 𝑏-metric space with the 𝑏-metric as a continuous functional. Let 𝐼∶𝑋→𝑋 be a continuous single-valued mapping such that 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦), where 𝑟>0 is a constant. Let 𝑇∶𝑋→𝐶𝑙(𝑋) be a lower semicontinuous map satisfying (I-mkc). Then, 𝐼 and 𝑇 have a unique endpoint if and only if 𝐼 and 𝑇 have the AEPP.

Proof. It is clear that if 𝐼 and 𝑇 have an endpoint, then 𝐼 and 𝑇 have AEPP. Then, 𝐶_𝑛=𝑥∈𝑋∶sup_{𝑦∈𝑇𝑥}𝑑≤1𝐼𝑥,𝑦𝑛≠𝜙,∀𝑛∈𝑁.(16) Also, we have for each 𝑛∈𝑁, 𝐶_𝑛⊇𝐶_𝑛+1. By Lemma 14, 𝐶_𝑛 is closed for each 𝑛∈𝑁. Since 𝐼 and 𝑇 satisfy AEPP, then 𝐶_𝑛≠𝜙 for each 𝑛∈𝑁. Now, we show that lim_𝑛→∞𝛿(𝐶_𝑛)=0. To show this, let 𝑥,𝑦∈𝐶_𝑛. Then, from Lemma 16, 𝛿𝐶_𝑛=𝛿𝐸_1/𝑛()≤𝑏()𝐼,𝑇1/𝑛𝑟1+𝑏+2𝛽𝑏(17) and so lim_𝑛→∞𝛿(𝐶_𝑛)=0. It follows from the Cantor intersection theorem that _𝑛∈𝑁𝐶_𝑛=𝑥₀.(18) Thus, 𝑥₀ is the unique endpoint of 𝐼 and 𝑇.

Lemma 18. Let (𝑋,𝑑) be a 𝑏-metric space with the 𝑏-metric as a continuous functional. Let 𝐼∶𝑋→𝑋 be a single-valued mapping such that 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦) for all 𝑥,𝑦∈𝑋, where 𝑟>0 is a constant. If 𝑇∶𝑋→𝐶𝑙(𝑋) satisfies (I-mcc) with 2𝑏²𝑟𝛾<1, then 𝛿𝐸_𝜀(𝐼,𝑇)≤𝑏𝜀1+𝑏+2𝛾𝑏𝑟1−2𝑏²𝛾,∀𝜀>0.(19)

Proof. For any 𝑥,𝑦∈𝐸_𝜀(𝐼,𝑇), we have 𝑑{},𝐻})𝐼𝑥,𝐼𝑦=𝐻𝐼𝑥𝐼𝑦≤𝑏({𝐼𝑥,𝑇𝑥+𝐻𝑇𝑥,𝐼𝑦≤𝑏𝜀+𝑏²𝐻𝑇𝑥,𝑇𝑦+𝐻𝑇𝑦,𝐼𝑦≤𝑏𝜀+𝑏²𝐻𝑇𝑥,𝑇𝑦+𝑏²𝜀≤𝑏𝜀+𝑏²𝜀+𝑏²𝛾𝑑𝐼𝑥,𝑇𝑦+𝑑𝐼𝑦,𝑇𝑥≤𝑏𝜀+𝑏²𝜀+𝑏²𝛾𝑑𝐼𝑥,𝐼𝑦+𝑑𝐼𝑦,𝑇𝑦+𝑏²𝛾𝑑𝐼𝑦,𝐼𝑥+𝑑(𝐼𝑥,𝑇𝑥)=𝑏𝜀+𝑏²𝜀+𝑏²𝛾𝐻{𝐼𝑥},𝐼𝑦+𝐻𝐼𝑦,𝑇𝑦+𝑏²𝛾𝐻𝐼𝑦,{𝐼𝑥}+𝐻({𝐼𝑥},𝑇𝑥)≤𝑏𝜀+𝑏²𝜀+2𝑏²𝛾𝐻{𝐼𝑥},𝐼𝑦+2𝑏²𝛾𝜀=𝑏𝜀+𝑏²𝜀+2𝑏²𝛾𝑑𝐼𝑥,𝐼𝑦+2𝑏²≤𝛾𝜀𝑏𝜀1+𝑏+2𝛾𝑏1−2𝑏²𝛾.(20) So, 𝑑𝐼𝑥,𝐼𝑦≤𝑏𝜀1+𝑏+2𝛾𝑏1−2𝑏²𝛾.(21) Since 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦), we have 𝛿𝐸_𝜀(𝐼,𝑇)≤𝑏𝜀1+𝑏+2𝛾𝑏𝑟1−2𝑏²𝛾,∀𝜀>0.(22)

Theorem 19. Let (𝑋,𝑑) be a complete 𝑏-metric space with the 𝑏-metric as a continuous functional. Let 𝐼∶𝑋→𝑋 be a continuous single-valued mapping such that 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦), where 𝑟>0 is a constant. Let 𝑇∶𝑋→𝐶𝑙(𝑋) be a lower semicontinuous map satisfying (I-mcc). Then, 𝐼 and 𝑇 have a unique endpoint if and only if 𝐼 and 𝑇 have the AEPP.

Proof. It is clear that if 𝐼 and 𝑇 have an endpoint, then 𝐼 and 𝑇 have the AEPP.
Then, 𝐶_𝑛=𝑥∈𝑋∶sup_{𝑦∈𝑇𝑥}𝑑≤1𝐼𝑥,𝑦𝑛≠𝜙,∀𝑛∈𝑁.(23) Also, we have for each 𝑛∈𝑁, 𝐶_𝑛⊇𝐶_𝑛+1. By Lemma 14, 𝐶_𝑛 is closed for each 𝑛∈𝑁. Since 𝐼 and 𝑇 satisfy AEPP, then 𝐶_𝑛≠𝜙 for each 𝑛∈𝑁. Now, we show that lim_𝑛→∞𝛿(𝐶_𝑛)=0. To show this, let 𝑥,𝑦∈𝐶_𝑛. Then, from Lemma 18, 𝛿𝐶_𝑛=𝛿𝐸_1/𝑛()≤𝑏()𝐼,𝑇1/𝑛1+𝑏+2𝛾𝑏𝑟1−2𝑏²𝛾(24) and so lim_𝑛→∞𝛿(𝐶_𝑛)=0. It follows from the Cantor intersection theorem that _𝑛∈𝑁𝐶_𝑛=𝑥₀.(25) Thus, 𝑥₀ is the unique endpoint of 𝐼 and 𝑇.

Lemma 20. Let (𝑋,𝑑) be a  𝑏-metric space with the 𝑏-metric as a continuous functional. Let 𝐼∶𝑋→𝑋 be a single-valued mapping such that 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦)  for all 𝑥,𝑦∈𝑋, where 𝑟>0 is a constant. If  𝑇∶𝑋→𝐶𝑙(𝑋) satisfies (𝐼-𝑚𝑧𝑐) with 𝛿=max{𝛼,𝛽𝑏/(1−𝛽𝑏²),𝛽(1+𝑏²)/2(1−𝛽𝑏²),𝛾𝑏/(1−𝛾𝑏)}, 𝑟𝛿<1, then 𝛿𝐸_𝜀()≤()𝐼,𝑇𝑏𝜀1+𝑏+2𝛿𝑏𝑟(1−𝛿),∀𝜀>0.(26)

Proof. Suppose that 𝑇 satisfies (I-mkc), then we have 𝐻𝑑()𝐻()()()𝑇𝑥,𝑇𝑦≤𝛽𝐼𝑥,𝑇𝑥+𝑑𝐼𝑦,𝑇𝑦≤𝛽𝐼𝑥,𝑇𝑥+𝐻𝐼𝑦,𝑇𝑦≤𝛽𝐻𝐼𝑥,𝑇𝑥+𝛽𝑏𝐻𝐼𝑦,𝐼𝑥+𝛽𝑏𝐻𝐼𝑥,𝑇𝑦≤𝛽𝐻𝐼𝑥,𝑇𝑥+𝛽𝑏𝐻𝐼𝑦,𝐼𝑥+𝛽𝑏²𝐻(𝐼𝑥,𝑇𝑥)+𝛽𝑏²𝐻≤𝑇𝑥,𝑇𝑦𝛽+𝛽𝑏²𝐻(𝐼𝑥,𝑇𝑥)+𝛽𝑏𝑑𝐼𝑦,𝐼𝑥+𝛽𝑏²𝐻𝑇𝑥,𝑇𝑦.(27) So, we have 𝐻𝑇𝑥,𝑇𝑦≤𝛽+𝛽𝑏²1−𝛽𝑏²𝐻(𝐼𝑥,𝑇𝑥)+𝛽𝑏1−𝛽𝑏²𝑑𝐼𝑥,𝐼𝑦.(28) If 𝑇 satisfies (I-mcc), then we have 𝐻𝑑𝐻𝐻()𝐻(()𝐻≤𝑇𝑥,𝑇𝑦≤𝛾𝐼𝑥,𝑇𝑦+𝑑𝐼𝑦,𝑇𝑥≤𝛾𝐼𝑥,𝑇𝑦+𝐻𝐼𝑦,𝑇𝑥≤𝛾𝑏𝐼𝑥,𝑇𝑥+𝐻𝑇𝑥,𝑇𝑦+𝛾𝑏𝐼𝑦,𝐼𝑥+𝐻𝐼𝑥,𝑇𝑥)≤2𝛾𝑏𝐻𝐼𝑥,𝑇𝑥+𝛾𝑏𝑑𝐼𝑥,𝐼𝑦+𝛾𝑏𝐻𝑇𝑥,𝑇𝑦𝑇𝑥,𝑇𝑦2𝛾𝑏𝐻()+1−𝛾𝑏𝐼𝑥,𝑇𝑥𝛾𝑏𝑑.1−𝛾𝑏𝐼𝑥,𝐼𝑦(29) Let 𝛿=max{𝛼,(𝛽𝑏)/(1−𝛽𝑏²),(𝛽(1+𝑏²))/(2(1−𝛽𝑏²)),(𝛾𝑏)/(1−𝛾𝑏)}. Then, 𝐻𝑇𝑥,𝑇𝑦≤2𝛿𝐻(𝐼𝑥,𝑇𝑥)+𝛿𝑑𝐼𝑥,𝐼𝑦.(30) Thus, for any 𝑥,𝑦∈𝐸_𝜀(𝐼,𝑇), we have 𝑑{},𝐻})𝐼𝑥,𝐼𝑦=𝐻𝐼𝑥𝐼𝑦≤𝑏({𝐼𝑥,𝑇𝑥+𝐻𝑇𝑥,𝐼𝑦≤𝑏𝜀+𝑏²𝐻𝑇𝑥,𝑇𝑦+𝐻𝑇𝑦,𝐼𝑦≤𝑏𝜀+𝑏²𝐻𝑇𝑥,𝑇𝑦+𝑏²𝜀≤𝑏𝜀+𝑏²𝜀+𝑏²2𝛿𝑑(𝐼𝑥,𝑇𝑥)+𝛿𝑑𝐼𝑥,𝐼𝑦≤𝑏𝜀+𝑏²𝜀+𝑏²2𝛿𝜀+𝛿𝑑𝐼𝑥,𝐼𝑦.(31) So, 𝑑≤()𝐼𝑥,𝐼𝑦𝑏𝜀1+𝑏+2𝛿𝑏.1−𝛿(32) Since 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦), we have 𝛿𝐸_𝜀()≤()𝐼,𝑇𝑏𝜀1+𝑏+2𝛿𝑏𝑟(1−𝛿),∀𝜀>0.(33)

Theorem 21. Let (𝑋,𝑑) be a complete 𝑏-metric space with the 𝑏-metric as a continuous functional. Let 𝐼∶𝑋→𝑋 be a continuous single-valued mapping such that 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦), where 𝑟>0 is a constant. Let 𝑇∶𝑋→𝐶𝑙(𝑋) be a lower semicontinuous map satisfying (𝐼-𝑚𝑧𝑐). Then, 𝐼 and 𝑇 have a unique endpoint if and only if 𝐼 and 𝑇 have the AEPP.

Proof. It is clear that if 𝐼 and 𝑇 have an endpoint, then 𝐼 and 𝑇 have the AEPP. Then, 𝐶_𝑛=𝑥∈𝑋∶sup_{𝑦∈𝑇𝑥}𝑑≤1𝐼𝑥,𝑦𝑛≠𝜙,∀𝑛∈𝑁.(34) Further, we have for each 𝑛∈𝑁, 𝐶_𝑛⊇𝐶_𝑛+1. By Lemma 14, 𝐶_𝑛 is closed for each 𝑛∈𝑁. Since 𝐼 and 𝑇 satisfy AEPP, then 𝐶_𝑛≠𝜙 for each 𝑛∈𝑁. Now, we show that lim_𝑛→∞𝛿(𝐶_𝑛)=0. To show this, let 𝑥,𝑦∈𝐶_𝑛. Then, from Lemma 20, 𝛿𝐶_𝑛=𝛿𝐸_1/𝑛()≤𝑏()𝐼,𝑇1/𝑛)(1+𝑏+2𝛿𝑏𝑟().1−𝛿(35) and so lim_𝑛→∞𝛿(𝐶_𝑛)=0. It follows from the Cantor intersection theorem that _𝑛∈𝑁𝐶_𝑛=𝑥₀.(36) Thus, 𝑥₀ is the unique endpoint of𝐼 and 𝑇.

Lemma 22. Let (𝑋,𝑑) be a 𝑏-metric space with the 𝑏-metric as a continuous functional. Let 𝐼∶𝑋→𝑋 be a single-valued mapping such that 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦) for all 𝑥,𝑦∈𝑋, where 𝑟>0 is a constant. If 𝑇∶𝑋→𝐶𝑙(𝑋) satisfies I-mqc with 𝑘<1/(𝑟(𝑏²+𝑏)), then 𝛿𝐸_𝜀()≤()𝐼,𝑇𝑏𝜀1+𝑏+𝑘𝑏𝑟1−𝑘𝑏²,∀𝜀>0.(37)

Proof. For any 𝑥,𝑦∈𝐸_𝜀(𝐼,𝑇), we have 𝑑{},𝐻})𝐼𝑥,𝐼𝑦=𝐻𝐼𝑥𝐼𝑦≤𝑏({𝐼𝑥,𝑇𝑥+𝑑𝑇𝑥,𝐼𝑦≤𝑏𝜀+𝑏²𝐻𝑇𝑥,𝑇𝑦+𝐻𝑇𝑦,𝐼𝑦≤𝑏𝜀+𝑏²𝐻𝑇𝑥,𝑇𝑦+𝑏²𝜀≤𝑏𝜀+𝑏²𝜀+𝑘𝑏²𝑑(),𝑑⋅max𝐼𝑥,𝐼𝑦,𝑑𝐼𝑥,𝑇𝑥,𝑑𝐼𝑦,𝑇𝑦𝐼𝑥,𝑇𝑦,𝑑𝐼𝑦,𝑇𝑥≤𝑏𝜀+𝑏²𝜀+𝑘𝑏²𝑑𝐼𝑥,𝐼𝑦+𝜀.(38) So, 𝑑≤()𝐼𝑥,𝐼𝑦𝑏𝜀1+𝑏+𝑘𝑏1−𝑘𝑏².(39) Since 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦), we have 𝛿𝐸_𝜀()≤()𝐼,𝑇𝑏𝜀1+𝑏+𝑘𝑏𝑟1−𝑘𝑏²,∀𝜀>0.(40)

Theorem 23. Let (𝑋,𝑑) be a complete 𝑏-metric space with the 𝑏-metric as a continuous functional. Let 𝐼∶𝑋→𝑋 be a continuous single-valued mapping such that 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦), where 𝑟>0 is a constant. Let 𝑇∶𝑋→𝐶𝑙(𝑋) be a lower semicontinuous map satisfying I-mqc. Then, 𝐼 and 𝑇 have a unique endpoint if and only if 𝐼 and 𝑇 have the AEPP.

Proof. It is clear that if 𝐼 and 𝑇 have an endpoint, then 𝐼 and 𝑇 have the AEPP.
Then, 𝐶_𝑛=𝑥∈𝑋∶sup_{𝑦∈𝑇𝑥}𝑑≤1𝐼𝑥,𝑦𝑛≠𝜙,∀𝑛∈𝑁.(41) Also, it is clear that, for each 𝑛∈𝑁, 𝐶_𝑛⊇𝐶_𝑛+1. By the above Lemma 14, 𝐶_𝑛 is closed for each 𝑛∈𝑁. Since 𝐼 and 𝑇 satisfy AEPP, then 𝐶_𝑛≠𝜙 for each 𝑛∈𝑁. Now, we show that lim_𝑛→∞𝛿(𝐶_𝑛)=0. To show this, let 𝑥,𝑦∈𝐶_𝑛. Then, from Lemma 22, 𝛿𝐶_𝑛=𝛿𝐸_1/𝑛()≤𝑏()𝐼,𝑇1/𝑛)(1+𝑏+𝑘𝑏𝑟1−𝑘𝑏²(42) and so lim_𝑛→∞𝛿(𝐶_𝑛)=0. It follows from the Cantor intersection theorem that _𝑛∈𝑁𝐶_𝑛=𝑥₀.(43) Thus, 𝑥₀ is the unique endpoint of 𝐼 and 𝑇.

Lemma 24. Let (𝑋,𝑑) be a 𝑏-metric space and 𝐼∶𝑋→𝑋 a single-valued mapping such that 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦) for all 𝑥,𝑦∈𝑋, where 𝑟>0 is a constant. If  𝑇∶𝑋→𝐶𝑙(𝑋) satisfies (I-mac) with 𝑟𝑏²(𝛼+𝑏𝐿)<1, then 𝛿𝐸_𝜀(𝐼,𝑇)≤𝑏𝜀1+𝑏+𝐿𝑏²𝑟1−𝑏²(𝛼+𝑏𝐿),∀𝜀>0.(44)

Proof. For any 𝑥,𝑦∈𝐸_𝜀(𝐼,𝑇), we have 𝑑{},𝐻})𝐼𝑥,𝐼𝑦=𝐻𝐼𝑥𝐼𝑦≤𝑏({𝐼𝑥,𝑇𝑥+𝑑𝑇𝑥,𝐼𝑦≤𝑏𝜀+𝑏²𝐻𝑇𝑥,𝑇𝑦+𝐻𝑇𝑦,𝐼𝑦≤𝑏𝜀+𝑏²𝐻𝑇𝑥,𝑇𝑦+𝑏²𝜀≤𝑏𝜀+𝑏²𝜀+𝑏²𝛼𝑑𝐼𝑥,𝐼𝑦+𝐿𝑑𝐼𝑦,𝑇𝑥≤𝑏𝜀+𝑏²𝜀+𝑏²𝛼𝑑𝐼𝑥,𝐼𝑦+𝑏³𝐿𝑑𝐼𝑥,𝐼𝑦+𝑑(𝐼𝑥,𝑇𝑥)≤𝑏𝜀+𝑏²𝜀+𝑏²𝛼𝑑𝐼𝑥,𝐼𝑦+𝑏³𝐿𝑑𝐼𝑥,𝐼𝑦+𝑏³𝐿𝜀≤𝑏𝜀1+𝑏+𝐿𝑏²+𝑏²(𝛼+𝑠𝐿)𝑑𝐼𝑥,𝐼𝑦.(45) So, 𝑑𝐼𝑥,𝐼𝑦≤𝑏𝜀1+𝑏+𝐿𝑏²1−𝑏²().𝛼+𝑏𝐿(46) Since 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦), we have 𝛿𝐸_𝜀(𝐼,𝑇)≤𝑏𝜀1+𝑏+𝐿𝑏²𝑟1−𝑏²(𝛼+𝑏𝐿),∀𝜀>0.(47)

Theorem 25. Let (𝑋,𝑑) be a complete 𝑏-metric space with the 𝑏-metric as a continuous functional. Let 𝐼∶𝑋→𝑋 be a continuous single-valued mapping such that 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦), where 𝑟>0 is a constant. Let 𝑇∶𝑋→𝐶𝑙(𝑋) be a lower semicontinuous map satisfying (I-mac). Then, 𝐼 and 𝑇 have a unique endpoint if and only if 𝐼 and 𝑇 have the AEPP.

Proof. It is clear that if 𝐼 and 𝑇 have an endpoint, then 𝐼 and 𝑇 have the AEPP. Then, 𝐶_𝑛=𝑥∈𝑋∶sup_{𝑦∈𝑇𝑥}𝑑≤1𝐼𝑥,𝑦𝑛≠𝜙,∀𝑛∈𝑁.(48) Also, we have for each 𝑛∈𝑁, 𝐶_𝑛⊇𝐶_𝑛+1. By Lemma 14, 𝐶_𝑛 is closed for each 𝑛∈𝑁. Since 𝐼 and 𝑇 satisfy AEPP, then 𝐶_𝑛≠𝜙 for each 𝑛∈𝑁. Now, we show that lim_𝑛→∞𝛿(𝐶_𝑛)=0. To show this, let 𝑥,𝑦∈𝐶_𝑛. Then, from Lemma 24, 𝛿𝐶_𝑛=𝛿𝐸_1/𝑛()≤𝑏(𝐼,𝑇1/𝑛)1+𝑏+𝐿𝑏²𝑟1−𝑏²(𝛼+𝑏𝐿)(49) and so lim_𝑛→∞𝛿(𝐶_𝑛)=0. It follows from the Cantor intersection theorem that _𝑛∈𝑁𝐶_𝑛=𝑥₀.(50) Thus, 𝑥₀ is the unique endpoint of 𝐼 and 𝑇.

On putting 𝑏=1 in the above Theorem 25, we obtain the following result of [11].

Corollary 26 (see [11]). Let (𝑋,𝑑) be a complete metric space. Let 𝐼∶𝑋→𝑋 be a continuous single-valued mapping such that 𝑟𝑑(𝑥,𝑦)≤𝑑(𝐼𝑥,𝐼𝑦), where 𝑟>0 is a constant. Let 𝑇∶𝑋→𝐶𝑙(𝑋) be a lower semicontinuous map satisfying (I-mac). Then, 𝐼 and 𝑇 have a unique endpoint if and only if 𝐼 and 𝑇 have the AEPP.

If 𝐼 is the identity mapping on 𝑋 and 𝑏=1, then the above result reduces to the following results:

Corollary 27 (see [11, Corollary 3.5]). Let (𝑋,𝑑) be a metric space, and let 𝑇∶𝑋→𝐶𝑙(𝑋) satisfy (I-mac) with 𝛼+𝐿<1. Then, for each 𝜀>0, 𝛿𝐸_𝜀(𝑇)≤𝜀(2+𝐿)(),1−𝛼+𝐿(51) where 𝐸_𝜀(𝑇)={𝑥∈𝑋∶sup_{𝑦∈𝑇𝑥}𝑑(𝑥,𝑦)≤𝜀}.

Corollary 28 (see [11, Corollary 3.6]). Let (𝑋,𝑑) be a complete metric space. Let 𝑇∶𝑋→𝐶𝑙(𝑋) be a lower semicontinuous map satisfying (I-mac) with 𝛼+𝐿<1. Then, 𝑇 has a unique endpoint if and only if 𝑇 has the AEPP.

If 𝐿=0, in almost contraction, then we have following result in metric space.

Corollary 29 ([10, Corollary 2.2]). Let (𝑋,𝑑) be a complete metric space. Let 𝑇∶𝑋→𝐶𝑙(𝑋) satisfy (𝑚𝑐). Then, 𝑇 has a unique endpoint if and only if 𝑇 has the AEPP.

Acknowledgments

The authors would like to sincerely thank the referees and the editors for their valuable comments to improve the paper in its present form.