Some Endpoint Results for β-Generalized Weak Contractive Multifunctions

Abstract

We introduce β-generalized weak contractive multifunctions and give some results about endpoints of the multifunctions. Also, we give some results about role of a point in the existence of endpoints.

1. Introduction

Let (𝑋,𝑑) be a metric space, 𝐶𝐵(𝑋) the collection of all nonempty bounded and closed subsets of 𝑋, and 𝐻 the Hausdorff metric with respect to 𝑑; that is, 𝐻(𝐴,𝐵)=max{sup_𝑥∈𝐴𝑑(𝑥,𝐵),sup_𝑦∈𝐵𝑑(𝑦,𝐴)} for all 𝐴,𝐵∈ 𝐶𝐵(𝑋), where 𝑑(𝑥,𝐵)=inf_𝑦∈𝐵𝑑(𝑥,𝑦). Let 𝑇∶𝑋→2^𝑋 be a multifunction. An element 𝑥∈𝑋 is said to be a fixed point of 𝑇 whenever 𝑥∈𝑇𝑥. Also, an element 𝑥∈𝑋 is said to be an endpoint of 𝑇 whenever 𝑇𝑥={𝑥} [1]. We say that 𝑇 has the approximate endpoint property whenever inf_𝑥∈𝑋sup_{𝑦∈𝑇𝑥}𝑑(𝑥,𝑦)=0 [1]. Let 𝑓∶𝑋→𝑋 be a mapping. We say that 𝑓 has the approximate endpoint property whenever inf_𝑥∈𝑋𝑑(𝑥,𝑓𝑥)=0 [1]. Also, the function 𝑔∶ℝ→ℝ is called upper semicontinuous whenever limsup_𝑛→∞𝑔(𝜆_𝑛)≤𝑔(𝜆) for all sequences {𝜆_𝑛}_𝑛≥1 with 𝜆_𝑛→𝜆 [2]. In 2010, Amini-Harandi defined the concept of approximate endpoint property for multifunctions and proved the following result (see [1]).

Theorem 1. Let 𝜓∶[0,∞)→[0,∞) be an upper semicontinuous function such that 𝜓(𝑡)<𝑡 and liminf_𝑡→∞(𝑡−𝜓(𝑡))>0 for all 𝑡>0, (𝑋,𝑑) a complete metric space, and 𝑇∶𝑋→𝐶𝐵(𝑋) a multifunction satisfing 𝐻(𝑇𝑥,𝑇𝑦)≤𝜓(𝑑(𝑥,𝑦)) for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique endpoint if and only if 𝑇 has the approximate endpoint property.

Then Moradi and Khojasteh introduced the concept of generalized weak contractive multifunctions and improved Theorem 1 by providing the following result [3].

Theorem 2. Let 𝜓∶[0,∞)→[0,∞) be an upper semicontinuous function such that 𝜓(𝑡)<𝑡 and liminf_𝑡→∞(𝑡−𝜓(𝑡))>0 for all 𝑡>0, (𝑋,𝑑) a complete metric space, and 𝑇∶𝑋→𝐶𝐵(𝑋) a generalized weak contractive multifunction; that is, 𝑇 satisfies 𝐻(𝑇𝑥,𝑇𝑦)≤𝜓(𝑁(𝑥,𝑦)) for all 𝑥,𝑦∈𝑋, where 𝑁(𝑥,𝑦)=max{𝑑(𝑥,𝑦),𝑑(𝑥,𝑇𝑥),𝑑(𝑦,𝑇𝑦),(𝑑(𝑥,𝑇𝑦)+𝑑(𝑦,𝑇𝑥))/2}. Then 𝑇 has a unique endpoint if and only if 𝑇 has the approximate endpoint property.

In this paper, we introduce 𝛽-generalized weak contractive multifunctions, and by adding some conditions to assumptions of the results, we give some results about endpoints of 𝛽-generalized weak contractive multifunctions. In 2012, the technique of 𝛼-𝜓-contractive mappings was introduced by Samet et al. [4]. Later, some authors used it for some subjects in fixed point theory (see for example [5–8]) or generalized it by using the method of 𝛽-𝜓-contractive multifunctions (see e.g., [9–12]).

Let (𝑋,𝑑) be a metric space and 𝛽∶2^𝑋×2^𝑋→[0,∞) a mapping. A multifunction 𝑇∶𝑋→2^𝑋 is called 𝛽-generalized weak contraction whenever there exists a nondecreasing, upper, semicontinuous function 𝜓∶[0,+∞)→[0,+∞) such that 𝜓(𝑡)<𝑡 for all 𝑡>0 and 𝛽𝑇𝑥,𝑇𝑦𝐻𝑇𝑥,𝑇𝑦≤𝜓𝑁𝑥,𝑦(1) for all 𝑥,𝑦∈𝑋. We say that 𝑇 is 𝛽-admissible whenever 𝛽(𝐴,𝐵)≥1 implies that 𝛽(𝑇𝑥,𝑇𝑦)≥1 for all 𝑥∈𝐴, and 𝑦∈𝐵, where 𝐴 and 𝐵 are subsets of 𝑋. We say that 𝑇 has the property (𝑅) whenever for each convergent sequence {𝑥_𝑛} in 𝑋 with 𝑥_𝑛→𝑥 and 𝛽(𝑇𝑥_𝑛−1,𝑇𝑥_𝑛)≥1 for all 𝑛≥1, we have 𝛽(𝑇𝑥_𝑛,𝑇𝑥)≥1. One can find idea of the property (𝑅) for mappings in [13]. We say that 𝑇 has the property (𝐾) whenever for each sequence {𝑥_𝑛} in 𝑋 with 𝛽(𝑇𝑥_𝑛−1,𝑇𝑥_𝑛)≥1 for all 𝑛≥1, there exists a natural number 𝑘 such that 𝛽(𝑇𝑥_𝑚,𝑇𝑥_𝑛)≥1 for all 𝑚>𝑛≥𝑘. Finally, we say that 𝑇 has the property (𝐻) whenever for each 𝜀>0, there exists 𝑧∈𝑋 such that sup_{𝑎∈𝑇𝑧}𝑑(𝑧,𝑎)<𝜀 implies that for every 𝑥∈𝑋 there exists 𝑦∈𝑇𝑥 such that 𝐻(𝑇𝑥,𝑇𝑦)=sup_{𝑏∈𝑇𝑦}𝑑(𝑦,𝑏). A multifunction 𝑇∶𝑋→2^𝑋 is called lower semicontinuous at 𝑥₀∈𝑋 whenever for each sequence {𝑥_𝑛} in 𝑋 with 𝑥_𝑛→𝑥₀ and every 𝑦∈𝑇𝑥₀, there exists a sequence {𝑦_𝑛} in 𝑋 with 𝑦_𝑛∈𝑇𝑥_𝑛 for all 𝑛≥1 such that 𝑦_𝑛→𝑦 [14].

2. Main Results

Now, we are ready to state and prove our main results.

Theorem 3. Let (𝑋,𝑑) be a complete metric space, 𝛽∶2^𝑋×2^𝑋→[0,∞) a mapping, and 𝑇∶𝑋→𝐶𝐵(𝑋) a 𝛽-admissible, 𝛽-generalized weak contractive multifunction which has the properties (𝑅), (𝐾), and (𝐻). Suppose that there exist a subset 𝐴 of 𝑋 and 𝑥₀∈𝐴 such that 𝛽(𝐴,𝑇𝑥₀)≥1. Then 𝑇 has an endpoint if and only if 𝑇 has the approximate endpoint property.

Proof. It is clear that if 𝑇 has an endpoint, then 𝑇 has the approximate endpoint property. Conversely, suppose that 𝑇 has the approximate endpoint property. Choose 𝐴⊂𝑋 and 𝑥₀∈𝐴 such that 𝛽(𝐴,𝑇𝑥₀)≥1. Since 𝑇 has the approximate endpoint property, for each 𝜀>0, there exists 𝑧∈𝑋 such that sup_{𝑎∈𝑇𝑧}𝑑(𝑧,𝑎)<𝜀. Now by using the condition (𝐻), choose 𝑥₁∈𝑇𝑥₀ such that 𝐻(𝑇𝑥₀,𝑇𝑥₁)=sup_{𝑎∈𝑇𝑥1}𝑑(𝑥₁,𝑎). Also, choose 𝑥₂∈𝑇𝑥₁ such that 𝐻(𝑇𝑥₁,𝑇𝑥₂)=sup_{𝑎∈𝑇𝑥2}𝑑(𝑥₂,𝑎), and by continuing this process, we find a sequence {𝑥_𝑛} in 𝑋 such that 𝑥_𝑛∈𝑇𝑥_𝑛−1 and 𝐻𝑇𝑥_𝑛−1,𝑇𝑥_𝑛=sup_{𝑎∈𝑇𝑥𝑛}𝑑𝑥_𝑛,𝑎,(2) for all 𝑛≥1. Since 𝛽(𝐴,𝑇𝑥₀)≥1 and 𝑇 is 𝛽-admissible, 𝛽(𝑇𝑥₀,𝑇𝑥₁)≥1. By using induction, it is easy to see that 𝛽(𝑇𝑥_𝑛−1,𝑇𝑥_𝑛)≥1 for all 𝑛≥1. Thus, we obtain 𝑑𝑥_𝑛,𝑥_𝑛+1≤sup_{𝑎∈𝑇𝑥𝑛}𝑑𝑥_𝑛,𝑎=𝐻𝑇𝑥_𝑛−1,𝑇𝑥_𝑛≤𝛽𝑇𝑥_𝑛−1,𝑇𝑥_𝑛𝐻𝑇𝑥_𝑛−1,𝑇𝑥_𝑛≤𝜓𝑁𝑥_𝑛−1,𝑥_𝑛(3) for all 𝑛≥1. If 𝑁(𝑥_𝑛−1,𝑥_𝑛)=𝑑(𝑥_𝑛−1,𝑥_𝑛), then 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝜓𝑑𝑥_𝑛−1,𝑥_𝑛.(4) If 𝑁(𝑥_𝑛−1,𝑥_𝑛)=𝑑(𝑥_𝑛−1,𝑇𝑥_𝑛−1), then 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝜓𝑑𝑥_𝑛−1,𝑇𝑥_𝑛−1≤𝜓𝑑𝑥_𝑛−1,𝑥_𝑛.(5) If 𝑁(𝑥_𝑛−1,𝑥_𝑛)=𝑑(𝑥_𝑛,𝑇𝑥_𝑛), then 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝜓𝑑𝑥_𝑛,𝑇𝑥_𝑛≤𝜓𝑑𝑥_𝑛,𝑥_𝑛+1,(6) and so 𝑑(𝑥_𝑛,𝑥_𝑛+1)=0. Thus, 𝑑(𝑥_𝑛,𝑥_𝑛+1)≤𝜓(𝑑(𝑥_𝑛−1,𝑥_𝑛)). If 𝑁𝑥_𝑛−1,𝑥_𝑛=𝑑𝑥_𝑛,𝑇𝑥_𝑛−1+𝑑𝑥_𝑛−1,𝑇𝑥_𝑛2=𝑑𝑥_𝑛−1,𝑇𝑥_𝑛2𝑑𝑥_𝑛−1,𝑇𝑥_𝑛2≤𝑑𝑥_𝑛−1,𝑥_𝑛+12≤𝑑𝑥_𝑛−1,𝑥_𝑛+𝑑𝑥_𝑛,𝑥_𝑛+12≤max𝑑𝑥_𝑛−1,𝑥_𝑛,𝑑𝑥_𝑛,𝑥_𝑛+1,(7) then 𝑑(𝑥_𝑛,𝑥_𝑛+1)≤𝜓(𝑑(𝑥_𝑛−1,𝑥_𝑛)) (other case implies that 𝑑(𝑥_𝑛,𝑥_𝑛+1)=0). Thus, 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝜓𝑑𝑥_𝑛−1,𝑥_𝑛(8) for all 𝑛≥1. We claim that 𝜓(0)=0. If 𝜓(0)>0, then 𝜓²(0)≥𝜓(0)>0 because 𝜓 is nondecreasing. On the other hand, since 𝜓(𝑡)<𝑡 for all 𝑡>0, we have 𝜓²(0)<𝜓(0) which is a contradiction. Hence, 𝜓(0)=0. Let 𝑑_𝑛=𝑑(𝑥_𝑛,𝑥_𝑛+1) for all 𝑛. If there exists a natural number 𝑛₀ such that 𝑑_𝑛0=0, then it is easy to see that 𝑑_𝑛=0 for all 𝑛≥𝑛₀, and so lim_𝑛→∞𝑑_𝑛=0. Now suppose that 𝑑_𝑛≠0 for all 𝑛. In this case, we have 𝑑_𝑛≤𝜓(𝑑_𝑛−1)<𝑑_𝑛−1 for all 𝑛. Hence, {𝑑_𝑛} is a decreasing sequence, and so there exists 𝑑≥0 such that lim_𝑛→∞𝑑_𝑛=𝑑. If 𝑑>0, then 𝑑_𝑛>0 for all 𝑛, and so 𝑑_𝑛≤𝜓(𝑑_𝑛−1)<𝑑_𝑛−1 for all 𝑛. Since 𝜓 is upper and semicontinuous, we obtain 𝑑=lim_𝑛→∞𝑑_𝑛≤lim_𝑛→∞𝜓(𝑑_𝑛−1)≤𝜓(lim_𝑛→∞𝑑_𝑛−1)=𝜓(𝑑)<𝑑 which is a contradiction. Thus, lim_𝑛→∞𝑑_𝑛=0. Now, we prove that {𝑥_𝑛} is a Cauchy sequence. If {𝑥_𝑛} is not a Cauchy sequence, then there exist 𝜀>0 and natural numbers 𝑚_𝑘,𝑛_𝑘 such that 𝑚_𝑘>𝑛_𝑘≥𝑘 and 𝑑(𝑥_𝑚𝑘,𝑥_𝑛𝑘)≥𝜀 for all 𝑘≥1. Also, we choose 𝑚_𝑘 as small as possible such that 𝑑𝑥_𝑚𝑘−1,𝑥_𝑛𝑘<𝜀.(9) Thus, 𝜀≤𝑑(𝑥_𝑚𝑘,𝑥_𝑛𝑘)≤𝑑(𝑥_𝑚𝑘,𝑥_𝑚𝑘−1)+𝑑(𝑥_𝑚𝑘−1,𝑥_𝑛𝑘)≤𝑑_𝑚𝑘−1+𝜀 for all 𝑘. Hence, lim_𝑘→∞𝑑(𝑥_𝑚𝑘,𝑥_𝑛𝑘)=𝜀. Since 𝑇 has the property (𝐾), we obtain 𝑑𝑥_𝑚𝑘,𝑥_𝑛𝑘≤𝑑𝑥_𝑚𝑘,𝑥_𝑚𝑘+1+𝑑𝑥_𝑚𝑘+1,𝑥_𝑛𝑘+1+𝑑𝑥_𝑛𝑘+1,𝑥_𝑛𝑘≤𝑑_𝑚𝑘+𝐻𝑇𝑥_𝑚𝑘,𝑇𝑥_𝑛𝑘+𝑑_𝑛𝑘≤𝑑_𝑚𝑘+𝛽𝑇𝑥_𝑚𝑘,𝑇𝑥_𝑛𝑘×𝐻𝑇𝑥_𝑚𝑘,𝑇𝑥_𝑛𝑘+𝑑_𝑛𝑘≤𝑑_𝑚𝑘+𝜓𝑁𝑥_𝑚𝑘,𝑥_𝑛𝑘+𝑑_𝑛𝑘(∗) for all 𝑘. Since lim_𝑘→∞𝑑(𝑥_𝑚𝑘,𝑥_𝑛𝑘)=𝜀, lim_𝑘→∞𝑁(𝑥_𝑚𝑘,𝑥_𝑛𝑘)=𝜀. In fact, 𝑑𝑥_𝑚𝑘,𝑥_𝑛𝑘≤𝑁𝑥_𝑚𝑘,𝑥_𝑛𝑘=max𝑑𝑥_𝑚𝑘,𝑥_𝑛𝑘,𝑑𝑥_𝑚𝑘,𝑇𝑥_𝑚𝑘,𝑑𝑥_𝑛𝑘,𝑇𝑥_𝑛𝑘,𝑑𝑥_𝑚𝑘,𝑇𝑥_𝑛𝑘+𝑑𝑥_𝑛𝑘,𝑇𝑥_𝑚𝑘2≤max𝑑𝑥_𝑚𝑘,𝑥_𝑛𝑘,𝑑𝑥_𝑚𝑘,𝑥_𝑚𝑘+1,𝑑𝑥_𝑛𝑘,𝑥_𝑛𝑘+1,𝑑𝑥_𝑚𝑘,𝑥_𝑛𝑘+1+𝑑𝑥_𝑛𝑘,𝑥_𝑚𝑘+12≤max𝑑𝑥_𝑚𝑘,𝑥_𝑛𝑘,𝑑𝑥_𝑚𝑘,𝑥_𝑚𝑘+1,𝑑𝑥_𝑛𝑘,𝑥_𝑛𝑘+1,𝑑𝑥_𝑚𝑘,𝑥_𝑛𝑘+𝑑𝑥_𝑛𝑘,𝑥_𝑛𝑘+1+𝑑𝑥_𝑛𝑘,𝑥_𝑚𝑘+𝑑𝑥_𝑚𝑘,𝑥_𝑚k+1×(2)⁻¹,(10) and so 𝜀=lim_𝑘→+∞𝑑(𝑥_𝑚𝑘,𝑥_𝑛𝑘)≤lim_𝑘→∞𝑁(𝑥_𝑚𝑘,𝑥_𝑛𝑘)≤𝜀. Since 𝜓 is upper semicontinuous, by using (∗) we obtain 𝜀=lim_𝑘→∞𝑑𝑥_𝑚𝑘,𝑥_𝑛𝑘≤lim_𝑘→∞𝜓𝑁𝑥_𝑚𝑘,𝑥_𝑛𝑘≤𝜓(𝜀)<𝜀,(11) which is a contradiction, and so {𝑥_𝑛} is a Cauchy sequence. Choose 𝑥^∗∈𝑋 such that 𝑥_𝑛→𝑥^∗. Now, note that 𝐻𝑥_𝑛,𝑇𝑥_𝑛=max𝑑𝑥_𝑛,𝑇𝑥_𝑛,sup_{𝑦∈𝑇𝑥𝑛}𝑑𝑥_𝑛,𝑦=𝐻𝑇𝑥_𝑛−1,𝑇𝑥_𝑛(12) for all 𝑛, and so 𝐻𝑥_𝑛,𝑇𝑥_𝑛=𝐻𝑇𝑥_𝑛−1,𝑇𝑥_𝑛≤𝛽𝑇𝑥_𝑛−1,𝑇𝑥_𝑛𝐻𝑇𝑥_𝑛−1,𝑇𝑥_𝑛≤𝜓𝑁𝑥_𝑛−1,𝑥_𝑛≤𝜓𝑑𝑥_𝑛−1,𝑥_𝑛≤𝑑𝑥_𝑛−1,𝑥_𝑛(13) for all 𝑛, and so lim_𝑛→∞𝐻({𝑥_𝑛},𝑇𝑥_𝑛)=0. Since 𝑇 has the property (𝑅), we obtain 𝐻𝑥^∗,𝑇𝑥^∗≤𝑑𝑥^∗,𝑥_𝑛+𝐻𝑥_𝑛,𝑇𝑥_𝑛+𝐻𝑇𝑥_𝑛,𝑇𝑥^∗≤𝑑𝑥^∗,𝑥_𝑛+𝐻𝑥_𝑛,𝑇𝑥_𝑛+𝛽𝑇𝑥_𝑛,𝑇𝑥^∗𝐻𝑇𝑥_𝑛,𝑇𝑥^∗≤𝑑𝑥^∗,𝑥_𝑛+𝐻𝑥_𝑛,𝑇𝑥_𝑛+𝜓𝑁𝑥_𝑛,𝑥^∗(14) for all 𝑛. If 𝑁(𝑥_𝑛,𝑥^∗)=𝑑(𝑥^∗,𝑇𝑥^∗), then we have 𝐻𝑥^∗,𝑇𝑥^∗≤𝑑𝑥^∗,𝑥_𝑛+𝐻𝑥_𝑛,𝑇𝑥_𝑛+𝜓𝐻𝑥^∗,𝑇𝑥^∗(15) for all 𝑛. This implies that 𝐻({𝑥^∗},𝑇𝑥^∗)≤𝜓(𝐻({𝑥^∗},𝑇𝑥^∗)), and so 𝐻𝑥^∗,𝑇𝑥^∗=0.(16) If 𝑁(𝑥_𝑛,𝑥^∗)=𝑑(𝑥_𝑛,𝑥^∗) or 𝑁(𝑥_𝑛,𝑥^∗)≤𝑑(𝑥_𝑛,𝑥_𝑛+1), then it is easy to see that 𝐻({𝑥^∗},𝑇𝑥^∗)=0. Thus, 𝑥^∗ is an endpoint of 𝑇.

Next example shows that a 𝛽-generalized weak contractive multifunction is not necessarily a generalized weak contractive multifunction.

Example 4. Let 𝑋=ℝ. Define 𝑇∶𝑋→𝐶𝐵(𝑋) by 𝑇𝑥=[𝑥,𝑥+2] for all 𝑥∈𝑋. Suppose that 𝜓∶[0,+∞)→[0,+∞) is an arbitrary upper semicontinuous function such that 𝜓(𝑡)<𝑡 for all 𝑡>0. If 𝑥=0 and 𝑦=2, then 𝐻(𝑇𝑥,𝑇𝑦)=𝐻([0,2],[2,4])=2 and 𝑁(𝑥,𝑦)=2. Hence, 𝐻𝑇𝑥,𝑇𝑦=2≩𝜓(2)=𝜓𝑁𝑥,𝑦.(17) Thus, 𝑇 is not a generalized weak contractive multifunction. Now, suppose that 𝜓(𝑡)=𝑡/2 for all 𝑡≥0 and define 𝛽∶2^𝑋×2^𝑋→[0,∞) by 𝛽(𝐴,𝐵)=1/2 for all subsets 𝐴 and 𝐵 of 𝑋. Then, we have 𝛽𝐻=1𝑇𝑥,𝑇𝑦𝑇𝑥,𝑇𝑦2𝑑𝑑𝑁𝑥,𝑦=𝜓𝑥,𝑦=𝜓𝑥,𝑦(18) for all 𝑥,𝑦∈ℝ. Thus, 𝑇 is a 𝛽-generalized weak contractive multifunction.

Next example shows that there are multifunctions which satisfy the conditions of Theorem 3, while they are not generalized weak contractive multifunctions.

Example 5. Let 𝑋=[0,9/2] and let 𝑑(𝑥,𝑦)=|𝑥−𝑦|. Define 𝑇∶𝑋→𝐶𝐵(𝑋) by 𝑥𝑇𝑥=230≤𝑥≤14𝑥−231<𝑥≤2{0}329<𝑥≤2.(19) If 𝑥=1 and 𝑦=3/2, then 𝐻1𝑇𝑥,𝑇𝑦=𝐻2,92𝑁,=4>3=𝑁𝑥,𝑦>𝜓𝑥,𝑦(20) where 𝜓∶[0,+∞)→[0,+∞) is an arbitrary upper semicontinuous function such that 𝜓(𝑡)<𝑡 for all 𝑡>0. Thus, 𝑇 is not a generalized weak contractive multifunction. Now, we show that 𝑇 satisfies all conditions of Theorem 3. For this aim, define 𝜓(𝑡)=𝑡/2 and 𝛽(𝐴,𝐵)=1 whenever 𝐴 and 𝐵 are subsets of [0,1] and 𝛽(𝐴,𝐵)=0 otherwise. First suppose that 𝑥∉[0,1] or that 𝑦∉[0,1]. If 𝑥,𝑦∈(3/2,9/2], then 𝑇𝑥,𝑇𝑦⊂[0,1] and 𝛽(𝑇𝑥,𝑇𝑦)=1. But, 𝐻(𝑇𝑥,𝑇𝑦)=0, and so 𝛽(𝑇𝑥,𝑇𝑦)𝐻(𝑇𝑥,𝑇𝑦)≤𝜓(𝑁(𝑥,𝑦)). If 𝑥∈(1,3/2] or 𝑦∈(1,3/2], then 𝑇𝑥⊆[̸0,1], or 𝑇𝑦⊆[̸0,1] and so 𝛽(𝑇𝑥,𝑇𝑦)=0. Hence, 𝛽(𝑇𝑥,𝑇𝑦)𝐻(𝑇𝑥,𝑇𝑦)≤𝜓(𝑁(𝑥,𝑦)). Now, suppose that 𝑥,𝑦∈[0,1]. In this case, we have 𝛽(𝑇𝑥,𝑇𝑦)≥1, 𝐻(𝑇𝑥,𝑇𝑦)=𝐻({𝑥/2},{𝑦/2})=(1/2)𝑑(𝑥,𝑦), and 𝑁(𝑥,𝑦)=max{𝑑(𝑥,𝑦),𝑥/2,𝑦/2,(𝑑(𝑥,𝑦/2)+𝑑(𝑦,𝑥/2))/2}. Thus, 𝑑(𝑥,𝑦)≤𝑁(𝑥,𝑦), and so 𝛽𝐻=1𝑇𝑥,𝑇𝑦𝑇𝑥,𝑇𝑦2𝑑𝑑𝑁.𝑥,𝑦≤𝜓𝑥,𝑦≤𝜓𝑥,𝑦(21) Therefore, 𝑇 is a 𝛽-generalized weak contractive multifunction. Now, we show that 𝑇 is 𝛽-admissible. If 𝛽(𝐴,𝐵)≥1, then 𝐴,𝐵⊂[0,1], and so 𝑇𝑥={𝑥/2}∈[0,1] and 𝑇𝑦={𝑦/2}∈[0,1] for all 𝑥∈𝐴 and 𝑦∈𝐵. Thus, 𝛽(𝑇𝑥,𝑇𝑦)≥1 for all 𝑥∈𝐴 and 𝑦∈𝐵. Now, suppose 𝐴=[0,1/2] and 𝑥₀=1/4. Then, 𝑇𝑥₀={1/8}∈[0,1] and [0,1/2]⊂[0,1]. Hence, 𝛽(𝐴,𝑇𝑥₀)≥1. Now, we show that 𝑇 satisfies the condition (𝐻). First note that, for each 𝜀>0, there exists 𝑧∈𝑋 such that sup_{𝑎∈𝑇𝑧}𝑑(𝑧,𝑎)<𝜀. Now, we show that for each 𝑥∈𝑋 there exists 𝑦∈𝑇𝑥 such that 𝐻(𝑇𝑥,𝑇𝑦)=sup_{𝑏∈𝑇𝑦}𝑑(𝑦,𝑏). If 0≤𝑥≤1, then 𝑇𝑥={𝑥/2}, 𝑇(𝑥/2)={𝑥/4}, and 𝐻𝑥𝑇𝑥,𝑇2𝑥=𝐻2,𝑥4=𝑥4=sup_{()𝑏∈𝑇𝑥/2}𝑑𝑥2,𝑏.(22) Since for 1<𝑥≤3/2 we have 5/2<4𝑥−(3/2)≤9/2, 𝑇(4𝑥−(3/2))={0}. Thus, 𝐻3𝑇𝑥,𝑇4𝑥−23=𝐻4𝑥−2,{03}=4𝑥−2=sup_{()𝑏∈𝑇4𝑥−3/2}𝑑34𝑥−2,𝑏.(23) If 3/2<𝑥≤9/2, then 𝑇𝑥={0} and 𝑇(0)={0}. Hence, 𝐻(𝑇𝑥,𝑇(0))=𝐻({0},{0})=0=sup_{(0)𝑏∈𝑇}𝑑(0,𝑏).(24) It is easy to check that 𝑇 satisfies the conditions (𝑅) and (𝐾). Note that, 0 is the endpoint of 𝑇.

Now, we add an assumption to obtain uniqueness of endpoint. In this way, we introduce a new notion. Let 𝑋 be a set and 𝛽∶2^𝑋×2^𝑋→[0,∞) a map. We say that the set 𝑋 has the property (𝐺_𝛽) whenever 𝛽(𝐴,𝐵)≥1 for all subsets 𝐴 and 𝐵 of 𝑋 with 𝐴⊆𝐵̸ or 𝐵⊆𝐴̸.

Corollary 6. Let (𝑋,𝑑) be a complete metric space, 𝛽∶2^𝑋×2^𝑋→[0,∞) a mapping, and 𝑇∶𝑋→𝐶𝐵(𝑋) a 𝛽-admissible, 𝛽-generalized weak contractive multifunction which has the properties (𝑅), (𝐾), and (𝐻). Suppose that there exist a subset 𝐴 of 𝑋 and 𝑥₀∈𝐴 such that 𝛽(𝐴,𝑇𝑥₀)≥1. If 𝑇 has the approximate endpoint property and 𝑋 has the property (𝐺_𝛽), then 𝑇 has a unique endpoint.

Proof. By using Theorem 3, 𝑇 has a endpoint. If 𝑇 has two distinct endpoints 𝑥^∗ and 𝑦^∗, then 𝛽(𝑇𝑥^∗,𝑇𝑦^∗)=𝛽({𝑥^∗},{𝑦^∗})≥1 because 𝑋 has the property (𝐺_𝛽). Hence, 𝑑𝑥^∗,𝑦^∗≤𝐻𝑇𝑥^∗,𝑇𝑦^∗≤𝛽𝑇𝑥^∗,𝑇𝑦^∗𝐻𝑇𝑥^∗,𝑇𝑦^∗≤𝜓𝑁𝑥^∗,𝑦^∗𝑥<𝑁𝑥∗,𝑦∗=𝑑^∗,𝑦^∗,(25) which is a contradiction. Thus, 𝑇 has a unique endpoint.

In Example 5, 𝑇 has a unique endpoint, while 𝑋 does has not the property (𝐺_𝛽). Also, 𝑇 has the property (𝑅), while 𝑇 is not lower semicontinuous. To see this, consider the sequence {𝑥_𝑛} defined by 𝑥_𝑛=11−𝑛1𝑛=2𝑘1+𝑛𝑛=2𝑘−1(26) for 𝑘≥1 and put 𝑦=1/2 and 𝑥₀=1. Then 𝑥_𝑛→1 and 𝑦∈𝑇𝑥₀={1/2}. Let {𝑦_𝑛} be an arbitrary sequence in 𝑋 such that 𝑦_𝑛∈𝑇𝑥_𝑛 for all 𝑛≥1. Then, 𝑦_2𝑘−1∈𝑇𝑥_2𝑘−1 and 𝑦_2𝑘∈𝑇𝑥_2𝑘 for all 𝑘. But, 𝑦_2𝑘−1=4𝑥_2𝑘−1−(3/2) for sufficiently large 𝑘 and 𝑦_2𝑘=𝑥_2𝑘/2 for all 𝑘 since 𝑦_2𝑘−1→5/2, 𝑦_𝑛↛1/2. This implies that 𝑇 is not lower semicontinuous.

Corollary 7. Let (𝑋,𝑑) be a complete metric space, 𝛽∶2^𝑋×2^𝑋→[0,∞) a mapping, and 𝑇∶𝑋→𝐶𝐵(𝑋) a 𝛽-admissible multifunction which has the properties (𝑅), (𝐾), and (𝐻). Suppose that 𝑋 has the property (𝐺_𝛽), and there exist a subset 𝐴 of X, 𝑥₀∈𝐴 and 𝑘∈[0,1) such that 𝛽(𝐴,𝑇𝑥₀)≥1 and 𝛽(𝑇𝑥,𝑇𝑦)𝐻(𝑇𝑥,𝑇𝑦)≤𝑘𝑁(𝑥,𝑦) for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique endpoint if and only if 𝑇 has the approximate endpoint property.

Proof. It is sufficient that we define 𝜓(𝑡)=𝑘𝑡 for all 𝑡≥0. Then, Theorem 3 and Corollary 6 guarantee the result.

It has been proved that lower semicontinuity of the multifunction 𝑇 and the property (𝑅) are independent conditions [9]. We can replace lower semicontinuity of the multifunction instead of the property (𝑅) to obtain the next result. Its proof is similar to the proof of Theorem 3.

Theorem 8. Let (𝑋,𝑑) be a complete metric space, 𝛽∶2^𝑋×2^𝑋→[0,∞) a mapping, and 𝑇∶𝑋→𝐶𝐵(𝑋) a lower semicontinuous, 𝛽-admissible, 𝛽-generalized weak contractive multifunction which has the properties (𝐾) and (𝐻). Suppose that there exist a subset 𝐴 of 𝑋 and 𝑥₀∈𝐴 such that 𝛽(𝐴,𝑇𝑥₀)≥1. Then 𝑇 has the approximate endpoint property if and only if 𝑇 has an endpoint.

Corollary 9. Let (𝑋,𝑑) be a complete metric space, 𝛽∶2^𝑋×2^𝑋→[0,∞) a mapping, and 𝑇∶𝑋→𝐶𝐵(𝑋) a lower semicontinuous, 𝛽-admissible, 𝛽-generalized weak contractive multifunction which has the properties (𝐾) and (𝐻). Suppose that there exist a subset 𝐴 of 𝑋 and 𝑥₀∈𝐴 such that 𝛽(𝐴,𝑇𝑥₀)≥1. If 𝑇 has the approximate endpoint property and 𝑋 has the property (𝐺_𝛽), then 𝑇 has a unique endpoint.

Corollary 10. Let (𝑋,𝑑) be a complete metric space, 𝛽∶2^𝑋×2^𝑋→[0,∞) a mapping, and 𝑇∶𝑋→𝐶𝐵(𝑋) a 𝛽-admissible multifunction which has the properties (𝑅), (𝐾), and (𝐻). Suppose that 𝑋 has the property (𝐺_𝛽), and there exist a subset 𝐴 of 𝑋, 𝑥₀∈𝐴 and 𝑘∈[0,1) such that 𝛽(𝐴,𝑇𝑥₀)≥1 and 𝛽(𝑇𝑥,𝑇𝑦)𝐻(𝑇𝑥,𝑇𝑦)≤𝑘𝑁(𝑥,𝑦) for all 𝑥,𝑦∈𝑋. If 𝑇 has the approximate endpoint property, then Fix(𝑇)=End(𝑇)={𝑥}.

Proof. If we put 𝜓(𝑡)=𝑘𝑡, then, by using Theorem 2.10 in [9], 𝑇 has a fixed point. Since 𝑇 has the approximate endpoint property, by using Corollary 7, 𝑇 has a unique endpoint such 𝑥. Let 𝑦∈Fix(𝑇). If 𝑇𝑥=𝑇𝑦, then 𝑦=𝑥. If 𝑇𝑥≠𝑇𝑦, then 𝛽(𝑇𝑥,𝑇𝑦)≥1 because 𝑋 has the property (𝐺_𝛽). Also, we have 𝑑{𝑥}𝐻.𝑥,𝑦≤𝐻,𝑇𝑦=𝐻𝑇𝑥,𝑇𝑦≤𝛽𝑇𝑥,𝑇𝑦𝑇𝑥,𝑇𝑦≤𝑘𝑁𝑥,𝑦(27) But, 𝑁(𝑥,𝑦)=max{𝑑(𝑥,𝑦),𝑑(𝑥,𝑇𝑥),𝑑(𝑦,𝑇𝑦),(𝑑(𝑥,𝑇𝑦)+𝑑(𝑦,𝑇𝑥))/2}=𝑑(𝑥,𝑦). Thus, 𝑑(𝑥,𝑦)=0, and so Fix(𝑇)=End(𝑇)={𝑥}.

Next corollary shows us the role of a point in the existence of endpoints.

Corollary 11. Let (𝑋,𝑑) be a complete metric space, 𝑥^∗∈𝑋 a fixed element, and 𝑇∶𝑋→𝐶𝐵(𝑋) a multifunction such that 𝑇 has the property (𝐻) and 𝑥^∗∈𝑇𝑥∩𝑇𝑦 for all subsets 𝐴 and 𝐵 of 𝑋 with 𝑥^∗∈𝐴∩𝐵 and all 𝑥∈𝐴 and 𝑦∈𝐵. Assume that 𝐻(𝑇𝑥,𝑇𝑦)≤𝜓(𝑁(𝑥,𝑦)) for all 𝑥,𝑦∈𝑋 with 𝑥^∗∈𝑇𝑥∩𝑇𝑦, where 𝜓∶[0,+∞)→[0,+∞) is a nondecreasing upper semicontinuous function such that 𝜓(𝑡)<𝑡 for all 𝑡>0. Suppose that there exist a subset 𝐴₀ of 𝑋 and 𝑥₀∈𝐴₀ such that 𝑥^∗∈𝐴₀∩𝑇𝑥₀. Assume that for each convergent sequence {𝑥_𝑛} in 𝑋 with 𝑥_𝑛→𝑥 and 𝑥^∗∈𝑇𝑥_𝑛−1∩𝑇𝑥_𝑛, for all 𝑛≥1, one has 𝑥^∗∈𝑇𝑥_𝑛∩𝑇𝑥. Also, for each sequence {𝑥_𝑛} in 𝑋 with 𝑥^∗∈𝑇𝑥_𝑛−1∩𝑇𝑥_𝑛 for all 𝑛≥1, there exists a natural number 𝑘 such that 𝑥^∗∈𝑇𝑥_𝑚∩𝑇𝑥_𝑛 for all 𝑚>𝑛≥𝑘. Then 𝑇 has an endpoint if and only if 𝑇 has the approximate endpoint property.

Proof. It is sufficient we define 𝛽∶2^𝑋×2^𝑋→[0,∞) by 𝛽(𝐴,𝐵)=1 whenever 𝑥^∗∈𝐴∩𝐵 and 𝛽(𝐴,𝐵)=0 otherwise, and then we use Theorem 3.

Corollary 12. Let (𝑋,𝑑) be a complete metric space, x^∗∈𝑋 a fixed element and 𝑇∶𝑋→𝐶𝐵(𝑋) a lower semicontinuous multifunction such that 𝑇 has the property (𝐻) and 𝑥^∗∈𝑇𝑥∩𝑇𝑦 for all subsets 𝐴 and 𝐵 of 𝑋 with 𝑥^∗∈𝐴∩𝐵 and all 𝑥∈𝐴 and 𝑦∈𝐵. Assume that 𝐻𝑇𝑥,𝑇𝑦≤𝜓𝑁𝑥,𝑦(28) for all 𝑥,𝑦∈𝑋 with 𝑥^∗∈𝑇𝑥∩𝑇𝑦, where 𝜓∶[0,+∞)→[0,+∞) is a nondecreasing upper semicontinuous function such that 𝜓(𝑡)<𝑡 for all 𝑡>0. Suppose that there exist a subset 𝐴₀ of 𝑋 and 𝑥₀∈𝐴₀ such that 𝑥^∗∈𝐴₀∩𝑇𝑥₀. Assume that for each convergent sequence {𝑥_𝑛} in 𝑋 with 𝑥_𝑛→𝑥 and 𝑥^∗∈𝑇𝑥_𝑛−1∩𝑇𝑥_𝑛 for all 𝑛≥1, we have 𝑥^∗∈𝑇𝑥_𝑛∩𝑇𝑥. Then 𝑇 has an endpoint if and only if 𝑇 has the approximate endpoint property.

Proof. It is sufficient to define 𝛽∶2^𝑋×2^𝑋→[0,∞) by 𝛽(𝐴,𝐵)=1 whenever 𝑥^∗∈𝐴∩𝐵 and 𝛽(𝐴,𝐵)=0 otherwise, and then we use Theorem 8.

Let (𝑋,𝑑,≤) be an ordered metric space. Define the order ⪯ on arbitrary subsets 𝐴 and 𝐵 of 𝑋 by 𝐴⪯𝐵 if and only if for each 𝑎∈𝐴 there exists 𝑏∈𝐵 such that 𝑎≤𝑏. It is easy to check that (𝐶𝐵(𝑋),⪯) is a partially ordered set.

Theorem 13. Let (𝑋,𝑑,≤) be a complete ordered metric space and 𝑇 a closed and bounded valued multifunction on 𝑋 such that 𝑇 has the property (𝐻) and 𝑇𝑥⪯𝑇𝑦 for all subsets 𝐴 and 𝐵 of 𝑋 with 𝐴⪯𝐵 and all 𝑥∈𝐴 and 𝑦∈𝐵. Assume that 𝐻(𝑇𝑥,𝑇𝑦)≤𝜓(𝑁(𝑥,𝑦)) for all 𝑥,𝑦∈𝑋 with 𝑇𝑥⪯𝑇𝑦, where 𝜓∶[0,+∞)→[0,+∞) is a nondecreasing upper semicontinuous function such that 𝜓(𝑡)<𝑡 for all 𝑡>0. Suppose that there exist a subset 𝐴₀ of 𝑋 and 𝑥₀∈𝐴₀ such that 𝐴₀⪯𝑇𝑥₀. Assume that for each convergent sequence {𝑥_𝑛} in 𝑋 with 𝑥_𝑛→𝑥 and 𝑇𝑥_𝑛−1⪯𝑇𝑥_𝑛, for all 𝑛≥1, one has 𝑇𝑥_𝑛⪯𝑇𝑥. Also, for each sequence {𝑥_𝑛} in 𝑋 with 𝑇𝑥_𝑛−1⪯𝑇𝑥_𝑛 for all 𝑛≥1, there exists a natural number 𝑘 such that 𝑇𝑥_𝑚⪯𝑇𝑥_𝑛 for all 𝑚>𝑛≥𝑘. Then 𝑇 has an endpoint if and only if 𝑇 has the approximate endpoint property.

Proof. Define 𝛽(𝐴,𝐵)=1 whenever 𝐴⪯𝐵 and 𝛽(𝐴,𝐵)=0 otherwise, and then we use Theorem 3.

Corollary 14. Let (𝑋,𝑑,≤) be a complete ordered metric space and 𝑇 a closed and bounded valued multifunction on 𝑋 such that 𝑇 has the property (𝐻) and 𝑇𝑥⪯𝑇𝑦 for all subsets 𝐴 and 𝐵 of 𝑋 with 𝐴⪯𝐵, all 𝑥∈𝐴, and 𝑦∈𝐵. Assume that 𝐻(𝑇𝑥,𝑇𝑦)≤𝜓(𝑁(𝑥,𝑦)) for all 𝑥,𝑦∈𝑋 with 𝑇𝑥⪯𝑇𝑦, where 𝜓∶[0,+∞)→[0,+∞) is a nondecreasing upper semicontinuous function such that 𝜓(𝑡)<𝑡 for all 𝑡>0. Suppose that there exist a subset 𝐴₀ of 𝑋 and 𝑥₀∈𝐴₀ such that 𝐴₀⪯𝑇𝑥₀. Assume that for each convergent sequence {𝑥_𝑛} in 𝑋 with 𝑥_𝑛→𝑥 and 𝑇𝑥_𝑛−1⪯𝑇𝑥_𝑛, for all 𝑛≥1, one has 𝑇𝑥_𝑛⪯𝑇𝑥. Also, for each sequence {𝑥_𝑛} in 𝑋 with 𝑇𝑥_𝑛−1⪯𝑇𝑥_𝑛 for all 𝑛≥1, there exists a natural number 𝑘 such that 𝑇𝑥_𝑚⪯𝑇𝑥_𝑛 for all 𝑚>𝑛≥𝑘. If 𝑇 has the approximate endpoint property and 𝐴⪯𝐵 for all subsets 𝐴 and 𝐵 of 𝑋 with 𝐴⊆𝐵̸ or 𝐵⊆𝐴̸, then 𝑇 has a unique endpoint.

Proof. Define 𝛽(𝐴,𝐵)=1 whenever 𝐴⪯𝐵 and 𝛽(𝐴,𝐵)=0 otherwise, and then we use Corollary 6.

Let (𝑋,𝑑) be a metric space and 𝑇∶𝑋→2^𝑋 a multifunction. We say that 𝑇 is an 𝐻𝑆-multifunction whenever for each 𝑥∈𝑋 there exists 𝑦∈𝑇𝑥 such that 𝐻(𝑇𝑥,𝑇𝑦)=sup_𝑏∈𝑇y𝑑(𝑦,𝑏). It is obvious that each 𝐻𝑆-multifunction is an multifunction which has the property (𝐻). Thus, one can conclude similar results to above ones for 𝐻𝑆-multifunctions. Here, we provide some ones. Although by considering 𝐻𝑆-multifunction we restrict ourselves, we obtain strange results with respect to above ones. One can prove the following by reading exactly the proofs of similar above results.

Theorem 15. Let (𝑋,𝑑) be a complete metric space, 𝛽∶2^𝑋×2^𝑋→[0,∞) a mapping, and 𝑇∶𝑋→𝐶𝐵(𝑋) a 𝛽-admissible, 𝛽-generalized weak contractive 𝐻𝑆-multifunction which has the properties (𝑅) and (𝐾). Suppose that there exist a subset 𝐴 of 𝑋 and 𝑥₀∈𝐴 such that 𝛽(𝐴,𝑇𝑥₀)≥1. Then 𝑇 has an endpoint, and so 𝑇 has the approximate endpoint property.

Theorem 16. Let (𝑋,𝑑) be a complete metric space, 𝛽∶2^𝑋×2^𝑋→[0,∞) a mapping, and 𝑇∶𝑋→𝐶𝐵(𝑋) a lower semicontinuous, 𝛽-admissible, and 𝛽-generalized weak contractive 𝐻𝑆-multifunction which has the property (𝐾). Suppose that there exist a subset 𝐴 of 𝑋 and 𝑥₀∈𝐴 such that 𝛽(𝐴,𝑇𝑥₀)≥1. Then 𝑇 has an endpoint, and so 𝑇 has the approximate endpoint property.

The next result is a consequence of Theorem 15.

Corollary 17. Let (𝑋,𝑑) be a complete metric space, 𝑥^∗∈𝑋 a fixed element, and 𝑇∶𝑋→𝐶𝐵(𝑋) an 𝐻𝑆-multifunction such that 𝑥^∗∈𝑇𝑥∩𝑇𝑦 for all subsets 𝐴 and 𝐵 of 𝑋 with 𝑥^∗∈𝐴∩𝐵, all 𝑥∈𝐴, and 𝑦∈𝐵. Assume that 𝐻(𝑇𝑥,𝑇𝑦)≤𝜓(𝑁(𝑥,𝑦)) for all 𝑥,𝑦∈𝑋 with 𝑥^∗∈𝑇𝑥∩𝑇𝑦, where 𝜓∶[0,+∞)→[0,+∞) is a nondecreasing upper semicontinuous function such that 𝜓(𝑡)<𝑡 for all 𝑡>0. Suppose that there exist a subset 𝐴₀ of 𝑋 and 𝑥₀∈𝐴₀ such that 𝑥^∗∈𝐴₀∩𝑇𝑥₀. Assume that for each convergent sequence {𝑥_𝑛} in 𝑋 with 𝑥_𝑛→𝑥 and 𝑥^∗∈𝑇𝑥_𝑛−1∩𝑇𝑥_𝑛 for all 𝑛≥1 one has 𝑥^∗∈𝑇𝑥_𝑛∩𝑇𝑥. Also, for each sequence {𝑥_𝑛} in 𝑋 with 𝑥^∗∈𝑇𝑥_𝑛−1∩𝑇𝑥_𝑛 for all 𝑛≥1, there exists a natural number 𝑘 such that 𝑥^∗∈𝑇𝑥_𝑚∩𝑇𝑥_𝑛 for all 𝑚>𝑛≥𝑘. Then 𝑇 has an endpoint, and so 𝑇 has the approximate endpoint property.

The next result is a consequence of Theorem 16.

Corollary 18. Let (𝑋,𝑑,≤) be a complete ordered metric space and 𝑇 a closed and bounded valued lower semicontinuous 𝐻𝑆-multifunction on 𝑋 such that 𝑇𝑥⪯𝑇𝑦 for all subsets 𝐴 and 𝐵 of 𝑋 with 𝐴⪯𝐵, all 𝑥∈𝐴, and 𝑦∈𝐵. Assume that 𝐻(𝑇𝑥,𝑇𝑦)≤𝜓(𝑁(𝑥,𝑦)) for all 𝑥,𝑦∈𝑋 with 𝑇𝑥⪯𝑇𝑦, where 𝜓∶[0,+∞)→[0,+∞) is a nondecreasing upper semicontinuous function such that 𝜓(𝑡)<𝑡 for all 𝑡>0. Suppose that there exist a subset 𝐴₀ of 𝑋 and 𝑥₀∈𝐴₀ such that 𝐴₀⪯𝑇𝑥₀. Assume that for each sequence {𝑥_𝑛} in 𝑋 with 𝑇𝑥_𝑛−1⪯𝑇𝑥_𝑛 for all 𝑛≥1, there exists a natural number 𝑘 such that 𝑇𝑥_𝑚⪯𝑇𝑥_𝑛 for all 𝑚>𝑛≥𝑘. Then 𝑇 has an endpoint, and so 𝑇 has the approximate endpoint property.

Acknowledgments

This work was completed while the second author (Dr. Gopal) was visiting the Azarbaijan University of Shahid Madani, Azarshahr, Tabriz, Iran, during the summer of 2012. He thanks Professor Sh. Rezapour and the University for their hospitality and support. The second author gratefully acknowledges the support from the CSIR, govternment of India, Grant no.-25(0215)/13/EMR-II.