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A new approach to (\alpha ,\psi )contractive nonself multivalued mappings
Journal of Inequalities and Applications volume 2014, Article number: 71 (2014)
Abstract
In this paper, we introduce the notions of αadmissible and αψcontractive type condition for nonself multivalued mappings. We establish fixed point theorems using these new notions along with a new condition. Moreover, we have constructed examples to show that our new condition is different from the corresponding existing conditions in the literature.
MSC: 47H10, 54H25.
1 Introduction and preliminaries
In the last decades, metric fixed point theory has been appreciated by a number of authors who have extended the celebrated Banach fixed point theorem for various contractive mapping in the context of different abstract spaces; see, for example, [1–32]. Among them, we mention the interesting fixed point theorems of Samet et al. [20]. In this paper [20], the authors introduced the notions of αψcontractive mappings and investigated the existence and uniqueness of a fixed point for such mappings. Further, they showed that several wellknown fixed point theorems can be derived from the fixed point theorem of αψcontractive mappings. Following this paper, Karapınar and Samet [21] generalized the notion αψcontractive mappings and obtained a fixed point for this generalized version. On the other hand, Asl et al. [22] characterized the notions of αψcontractive mapping and αadmissible mappings with the notions of {\alpha}_{\ast}ψcontractive and {\alpha}_{\ast}admissible mappings to investigate the existence of a fixed point for a multivalued function. Afterward, Ali and Kamran [23] generalized the notion of {\alpha}_{\ast}ψcontractive mappings and obtained further fixed point results for multivalued mappings. Some results in this direction in the context of various abstract spaces were also given by the authors [24–28, 33–36]. The purpose of this paper is to prove fixed point theorems for nonself multivalued (\alpha ,\psi )contractive type mappings using a new condition.
Let Ψ be the family of functions \psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}), known in the literature as BianchiniGrandolfi gauge functions (see, e.g., [30–32]), satisfying the following conditions:
({\psi}_{1}) ψ is nondecreasing;
({\psi}_{2}) {\sum}_{n=1}^{+\mathrm{\infty}}{\psi}^{n}(t)<\mathrm{\infty} for all t>0, where {\psi}^{n} is the n th iterate of ψ.
Notice that such functions are also known as (c)comparison functions in some sources (see, e.g., [29]).
It is easily proved that if \psi \in \mathrm{\Psi}, then \psi (t)<t for any t>0 and \psi (0)=0 for t=0 (see, e.g., [20, 29]). Let (X,d) be a metric space. A mapping G:X\to X is called αψcontractive type if there exist two functions \alpha :X\times X\to [0,\mathrm{\infty}) and \psi \in \mathrm{\Psi} such that
for each x,y\in X. A mapping G:X\to X is called αadmissible [20] if
We denote by N(X) the space of all nonempty subsets of X and by \mathit{CL}(X) the space of all nonempty closed subsets of X. For A\in N(X) and x\in X, d(x,A)=inf\{d(x,a):a\in A\}. For every A,B\in \mathit{CL}(X), let
Such a map H is called a generalized Hausdorff metric induced by d. We use the following lemma in our results.
Lemma 1.1 [23]
Let (X,d) be a metric space and B\in \mathit{CL}(X). Then, for each x\in X with d(x,B)>0 and q>1, there exists an element b\in B such that
Let (X,\u2aaf,d) be an ordered metric space and A,B\subseteq X. We say that A{\prec}_{r}B if for each a\in A and b\in B, we have a\u2aafb.
2 Main results
We begin this section with the following definition which is a modification of the notion of αadmissible.
Definition 2.1 Let (X,d) be a metric space and let D be a nonempty subset of X. A mapping G:D\to \mathit{CL}(X) is called αadmissible if there exists a mapping \alpha :D\times D\to [0,\mathrm{\infty}) such that
for each u\in Gx\cap D and v\in Gy\cap D.
Definition 2.2 Let (X,d) be a metric space and let D be a nonempty subset of X. We say that G:D\to \mathit{CL}(X) is an (\alpha ,\psi )contractive type mapping on D if there exist \alpha :D\times D\to [0,\mathrm{\infty}) and \psi \in \mathrm{\Psi} satisfying the following conditions:

(i)
Gx\cap D\ne \mathrm{\varnothing} for all x\in D,

(ii)
for each x,y\in D, we have
\alpha (x,y)H(Gx\cap D,Gy\cap D)\le \psi (M(x,y)),(2.1)
where M(x,y)=max\{d(x,y),\frac{d(x,Gx)+d(y,Gy)}{2},\frac{d(x,Gy)+d(y,Gx)}{2}\}.
Note that if \psi \in \mathrm{\Psi} in the above definition is a strictly increasing function, then G:D\to \mathit{CL}(X) is said to be a strictly (\alpha ,\psi )contractive type mapping on D.
Theorem 2.3 Let (X,d) be a metric space, let D be a nonempty subset of X which is complete with respect to the metric induced by d, and let G be a strictly (\alpha ,\psi )contractive type mapping on D. Assume that the following conditions hold:

(i)
G is an αadmissible map;

(ii)
there exist {x}_{0}\in D and {x}_{1}\in G{x}_{0}\cap D such that \alpha ({x}_{0},{x}_{1})\ge 1;

(iii)
G is continuous.
Then G has a fixed point.
Proof By hypothesis, there exist {x}_{0}\in D and {x}_{1}\in G{x}_{0}\cap D such that \alpha ({x}_{0},{x}_{1})\ge 1. If {x}_{0}={x}_{1}, then we have nothing to prove. Let {x}_{0}\ne {x}_{1}. If {x}_{1}\in G{x}_{1}\cap D, then {x}_{1} is a fixed point. Let {x}_{1}\notin G{x}_{1}\cap D. From (2.1) we have
since \frac{d({x}_{0},G{x}_{1})}{2}\le max\{d({x}_{0},{x}_{1}),d({x}_{1},G{x}_{1})\} and \frac{d({x}_{0},G{x}_{0})+d({x}_{1},G{x}_{1})}{2}\le max\{d({x}_{0},{x}_{1}),d({x}_{1},G{x}_{1})\}. Assume that max\{d({x}_{0},{x}_{1}),d({x}_{1},G{x}_{1})\}=d({x}_{1},G{x}_{1}). Then from (2.2) we have
a contradiction to our assumption. Thus max\{d({x}_{0},{x}_{1}),d({x}_{1},G{x}_{1})\}=d({x}_{0},{x}_{1}). Then from (2.2) we have
For q>1 by Lemma 1.1, there exists {x}_{2}\in G{x}_{1}\cap D such that
Applying ψ in (2.5), we have
Put {q}_{1}=\frac{\psi (q\psi (d({x}_{0},{x}_{1})))}{\psi (d({x}_{1},{x}_{2}))}. Then {q}_{1}>1. Since G is an αadmissible mapping, \alpha ({x}_{1},{x}_{2})\ge 1. If {x}_{2}\in G{x}_{2}\cap D, then {x}_{2} is a fixed point. Let {x}_{2}\notin G{x}_{2}\cap D. From (2.1) we have
since \frac{d({x}_{1},G{x}_{2})}{2}\le max\{d({x}_{1},{x}_{2}),d({x}_{2},G{x}_{2})\} and \frac{d({x}_{1},G{x}_{1})+d({x}_{2},G{x}_{2})}{2}\le max\{d({x}_{1},{x}_{2}),d({x}_{2},G{x}_{2})\}. Assume that max\{d({x}_{1},{x}_{2}),d({x}_{2},G{x}_{2})\}=d({x}_{2},G{x}_{2}). Then from (2.7) we have
a contradiction to our assumption. Thus max\{d({x}_{1},{x}_{2}),d({x}_{2},G{x}_{2})\}=d({x}_{1},{x}_{2}). Then from (2.7) we have
For {q}_{1}>1 by Lemma 1.1, there exists {x}_{3}\in G{x}_{2}\cap D such that
Applying ψ in (2.10), we have
Put {q}_{2}=\frac{{\psi}^{2}(q\psi (d({x}_{0},{x}_{1})))}{\psi (d({x}_{2},{x}_{3}))}. Then {q}_{2}>1. Since G is an αadmissible mapping, \alpha ({x}_{2},{x}_{3})\ge 1. If {x}_{3}\in G{x}_{3}\cap D, then {x}_{3} is a fixed point. Let {x}_{3}\notin G{x}_{3}\cap D. From (2.1) we have
since \frac{d({x}_{2},G{x}_{3})}{2}\le max\{d({x}_{2},{x}_{3}),d({x}_{3},G{x}_{3})\} and \frac{d({x}_{2},G{x}_{2})+d({x}_{3},G{x}_{3})}{2}\le max\{d({x}_{2},{x}_{3}),d({x}_{3},G{x}_{3})\}. Assume that max\{d({x}_{2},{x}_{3}),d({x}_{3},G{x}_{3})\}=d({x}_{3},G{x}_{3}). Then from (2.12) we have
a contradiction to our assumption. Thus max\{d({x}_{2},{x}_{3}),d({x}_{3},G{x}_{3})\}=d({x}_{2},{x}_{3}). Then from (2.12) we have
For {q}_{2}>1 by Lemma 1.1, there exists {x}_{4}\in G{x}_{3}\cap D such that
Applying ψ in (2.15), we have
Continuing the same process, we get a sequence \{{x}_{n}\} in D such that {x}_{n+1}\in G{x}_{n}\cap D, {x}_{n+1}\ne {x}_{n}, \alpha ({x}_{n},{x}_{n+1})\ge 1, and
For m,n\in \mathbb{N}, we have
Since \psi \in \mathrm{\Psi}, it follows that \{{x}_{n}\} is a Cauchy sequence in D. Since D is complete, there exists {x}^{\ast}\in D such that {x}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}. By the continuity of G, we have
□
Theorem 2.4 Let (X,d) be a metric space, D be a nonempty subset of X which is complete with respect to the metric induced by d, and let G be a strictly (\alpha ,\psi )contractive type mapping on D. Assume that the following conditions hold:

(i)
G is an αadmissible map;

(ii)
there exist {x}_{0}\in D and {x}_{1}\in G{x}_{0}\cap D such that \alpha ({x}_{0},{x}_{1})\ge 1;

(iii)
either

(a)
for any sequence \{{x}_{n}\} in D such that {x}_{n}\to x as n\to \mathrm{\infty} and \alpha ({x}_{n},{x}_{n+1})\ge 1 for each n\in \mathbb{N}\cup \{0\}, {lim}_{n\to \mathrm{\infty}}\alpha ({x}_{n},x)\ge 1,
or

(b)
for any sequence \{{x}_{n}\} in D such that {x}_{n}\to x as n\to \mathrm{\infty} and \alpha ({x}_{n},{x}_{n+1})\ge 1 for each n\in \mathbb{N}\cup \{0\}, \alpha ({x}_{n},x)\ge 1 for each n\in \mathbb{N}\cup \{0\}.

(a)
Then G has a fixed point.
Proof Following the proof of Theorem 2.3, there exists a Cauchy sequence \{{x}_{n}\} in D with {x}_{n}\to {x}^{\ast} as n\to \mathrm{\infty} and \alpha ({x}_{n},{x}_{n+1})\ge 1 for each n\in \mathbb{N}\cup \{0\}. Suppose that d({x}^{\ast},G{x}^{\ast})\ne 0. From (2.1) we have
Letting n\to \mathrm{\infty} in (2.18), we have
Since {lim}_{n\to \mathrm{\infty}}\alpha ({x}_{n},{x}^{\ast})\ge 1, by condition (iii)(a), we have
Further, it is clear that d({x}^{\ast},G{x}^{\ast})\le d({x}^{\ast},G{x}^{\ast}\cap D). Then from (2.20) we have
which is impossible. Thus d({x}^{\ast},G{x}^{\ast})=0. If we use (iii)(b), then from (2.1) we have
Letting n\to \mathrm{\infty} in (2.21), we have
which is impossible. Thus d({x}^{\ast},G{x}^{\ast})=0. □
Example 2.5 Let X=(\mathrm{\infty},8)\cup [0,\mathrm{\infty}) be endowed with the usual metric d, and let D=[0,\mathrm{\infty}). Define G:D\to \mathit{CL}(X) by
and \alpha :D\times D\to [0,\mathrm{\infty}) by
Clearly, Gx\cap D\ne \mathrm{\varnothing} for each x\in D. Let \psi (t)=\frac{t}{2} for each t\ge 0. To see that G is a strictly (\alpha ,\psi )contractive type mapping on D, we consider the following cases.
Case (i) When x,y\in [0,4), we have
Case (ii) When x\in [0,4) and y=4, we have
Case (iii) Otherwise, we have
where M(x,y)=max\{d(x,y),\frac{d(x,Gx)+d(y,Gy)}{2},\frac{d(x,Gy)+d(y,Gx)}{2}\}.
Thus, G is a strictly (\alpha ,\psi )contractive type mapping on D. For \alpha (x,y)\ge 1, we have x,y\in [0,4], then Gx\cap D,Gy\cap D\subseteq [0,1], thus \alpha (u,v)=1 for each u\in Gx\cap D and v\in Gy\cap D. Further, for any sequence \{{x}_{n}\} in D such that {x}_{n}\to x as n\to \mathrm{\infty} and \alpha ({x}_{n},{x}_{n+1})=1 for each n\in \mathbb{N}\cup \{0\}, {lim}_{n\to \mathrm{\infty}}\alpha ({x}_{n},x)=1. Therefore, all the conditions of Theorem 2.4 hold and G has a fixed point.
Corollary 2.6 Let (X,\u2aaf,d) be an ordered metric space, let (D,\u2aaf) be a nonempty subset of X which is complete with respect to the metric induced by d. Let G:D\to \mathit{CL}(X) be a mapping such that Gx\cap D\ne \mathrm{\varnothing} for each x\in D and for each x,y\in D with x\u2aafy, we have
where M(x,y)=max\{d(x,y),\frac{d(x,Gx)+d(y,Gy)}{2},\frac{d(x,Gy)+d(y,Gx)}{2}\} and ψ is an increasing function in Ψ. Also, assume that the following conditions hold:

(i)
there exist {x}_{0}\in D and {x}_{1}\in G{x}_{0}\cap D such that {x}_{0}\u2aaf{x}_{1};

(ii)
if x\u2aafy then Gx\cap D{\prec}_{r}Gy\cap D;

(iii)
either

(a)
G is continuous,
or

(b)
for any sequence \{{x}_{n}\} in D such that {x}_{n}\to x as n\to \mathrm{\infty} and {x}_{n}\u2aaf{x}_{n+1} for each n\in \mathbb{N}\cup \{0\}, {x}_{n}\u2aafx as n\to \mathrm{\infty},
or

(c)
for any sequence \{{x}_{n}\} in D such that {x}_{n}\to x as n\to \mathrm{\infty} and {x}_{n}\u2aaf{x}_{n+1} for each n\in \mathbb{N}\cup \{0\}, {x}_{n}\u2aafx for each n\in \mathbb{N}\cup \{0\}.

(a)
Then G has a fixed point.
Proof Define \alpha :D\times D\to [0,\mathrm{\infty}) by
By using condition (i) and the definition of α, we have \alpha ({x}_{0},{x}_{1})=1. Also, from condition (ii), we have that x\u2aafy implies Gx\cap D{\prec}_{r}Gy\cap D; by using the definitions of α and {\prec}_{r}, we have that \alpha (x,y)=1 implies \alpha (u,v)=1 for each u\in Gx\cap D and v\in Gy\cap D. Moreover, it is easy to check that G is a strictly (\alpha ,\psi )contractive type mapping on D. Therefore, all the conditions of Theorem 2.3 (or Theorem 2.4 for (iii)(b), (iii)(c)) hold, hence G has a fixed point. □
Remark 2.7 Condition (a), in the statement of Theorem 2.4, was introduced by Samet et al. [20]. In Theorem 2.4 we introduce a new condition (b). The following examples show that (a) and (b) are independent conditions.
Example 2.8 Let X=\{\frac{1}{n}:n\in \mathbb{N}\}\cup \{0\}. Consider {x}_{n}=\frac{1}{n+1} for each n\in \mathbb{N}\cup \{0\}, then {x}_{n}\to 0={x}^{\ast} as n\to \mathrm{\infty}. Define \alpha :X\times X\to [0,\mathrm{\infty}) by
Now, we have \alpha ({x}_{n},{x}_{n+1})=\alpha (\frac{1}{n+1},\frac{1}{n+2})=n+2>1 for each n\in \mathbb{N}\cup \{0\} and \alpha ({x}_{n},{x}^{\ast})=\alpha (\frac{1}{n+1},0)=n+1\ge 1 for each n\in \mathbb{N}\cup \{0\}. Thus condition (a) holds but {lim}_{n\to \mathrm{\infty}}\alpha ({x}_{n},{x}^{\ast})={lim}_{n\to \mathrm{\infty}}(n+1)=\mathrm{\infty}. Thus condition (b) does not hold.
Example 2.9 Let X=\{\frac{1}{n}:n\in \mathbb{N}\}\cup \{0\}. Consider {x}_{n}=\frac{1}{n+1} for each n\in \mathbb{N}\cup \{0\}, then {x}_{n}\to 0={x}^{\ast} as n\to \mathrm{\infty}. Define \alpha :X\times X\to [0,\mathrm{\infty}) by
Now, we have \alpha ({x}_{n},{x}_{n+1})=\alpha (\frac{1}{n+1},\frac{1}{n+2})=n+2>1 for each n\in \mathbb{N}\cup \{0\} and \alpha ({x}_{n},{x}^{\ast})=\alpha (\frac{1}{n+1},0)=\frac{2n+2}{2n+3}. Then {lim}_{n\to \mathrm{\infty}}\alpha ({x}_{n},{x}^{\ast})={lim}_{n\to \mathrm{\infty}}\frac{2n+2}{2n+3}=1. Thus condition (b) holds but for n=0, we have \alpha ({x}_{n},{x}^{\ast})=\frac{2}{3}; for n=1, we have \alpha ({x}_{n},{x}^{\ast})=\frac{4}{5}; for n=2, we have \alpha ({x}_{n},{x}^{\ast})=\frac{6}{7}, which implies that \alpha ({x}_{n},x)\ngeqq 1 for each n\in \mathbb{N}\cup \{0\}. Thus condition (a) does not hold.
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Ali, M.U., Kamran, T. & Karapınar, E. A new approach to (\alpha ,\psi )contractive nonself multivalued mappings. J Inequal Appl 2014, 71 (2014). https://doi.org/10.1186/1029242X201471
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DOI: https://doi.org/10.1186/1029242X201471
Keywords
 αadmissible maps
 αψcontractive type condition
 nonself αadmissible maps
 nonself (\alpha ,\psi )contractive type condition