On multivalued weakly Picard operators in partial Hausdorff metric spaces

We discuss multivalued weakly Picard operators on partial Hausdorff metric spaces. First, we obtain Kikkawa-Suzuki type fixed point theorems for a new type of generalized contractive conditions. Then, we prove data dependence of a fixed points set theorem. Finally, we present sufficient conditions for well-posedness of a fixed point problem. Our results generalize, complement and extend classical theorems in metric and partial metric spaces.


Introduction and preliminaries
In , von Neumann [] initiated the fixed point theory for multivalued mappings in the study of game theory. Indeed, the fixed point theorems for multivalued mappings are quite useful in control theory and have been frequently used in solving many problems of economics. In , Nadler [] initiated the development of the metric fixed point theory for multivalued mappings. Nadler used the concept of Hausdorff metric to establish the multivalued contraction principle containing the Banach contraction principle as a special case. Also, for the basic problems of fixed point theory for multivalued mappings, we refer to [].
Let (X, d) be a metric space and let CB(X) be the family of all nonempty, closed and bounded subsets of X. For A, B ∈ CB(X), x ∈ X, let Now on, the letters R, R + and N will denote the set of all real numbers, the set of all non-negative real numbers and the set of all positive integers, respectively. Also, CL(X) is the collection of nonempty closed subsets of X.
Definition . Let X be a nonempty set. If T : X → CB(X) is a multivalued operator, then an element x ∈ X is called (i) fixed point of T if x ∈ Tx; (ii) strict fixed point of T if {x} = Tx.
In the sequel, we denote by Fix(T) := {x ∈ X : x ∈ Tx} the set of all fixed points of T and by S Fix(T) := {x ∈ X : {x} = Tx} the set of all strict fixed points of T.
Definition . ([]) Let (X, d) be a metric space and T : X → CL(X) be a multivalued operator. T is called a multivalued weakly Picard operator (briefly MWP operator) if for all x ∈ X and all y ∈ Tx, there exists a sequence {x n } such that: (i) x  = x, x  = y; (ii) x n+ ∈ Tx n for all n ∈ N ∪ {}; (iii) the sequence {x n } is convergent and its limit is a fixed point of T.
A sequence {x n } satisfying (i) and (ii) is also called a sequence of successive approximations (briefly s.s.a.) of T starting from x  .
For interested readers, the theory of MWP operators was presented in [-]. In  Suzuki [] introduced a new type of mappings in order to generalize the wellknown Banach contraction principle. This result has led to some important contributions in metric fixed point theory (see, for instance, [] and the references therein).
As we mentioned above, Nadler proved the following multivalued version of the Banach contraction principle.

Theorem . ([])
Let (X, d) be a complete metric space and T : X → CB(X) be a multivalued mapping satisfying H(Tx, Ty) ≤ kd(x, y) for all x, y ∈ X and k ∈ (, ). Then T has a fixed point.
Assume that there exists r ∈ [, ) such that Then T has a fixed point.
We remark that the right-hand side of the above implication is known as Ćirić type contractive condition, see [, , ]. Also, for our further use, we recall the following refinement of Nadler's theorem, see [].
Theorem . Let η : [, ) → (   , ] be a function defined by η(r) =  +r . Let (X, d) be a complete metric space and T : X → CB(X) be an r-KS multivalued operator, that is, there exists r ∈ [, ) such that for all x, y ∈ X. Then T is an MWP operator.
The other basic notion for the development of our work is the concept of partial metric space, which was introduced by Matthews [] as a part of the study of denotational semantics of dataflow networks. Matthews presented a modified version of the Banach contraction principle, more suitable in this context, see also [, ]. For more reading on interesting approaches to partial metric spaces and related contexts, we refer to [ Consistent with [, -], the following definitions and results will be needed in the sequel.

Definition . ([]
) Let X be any nonempty set. A function p : X × X → R + is said to be a partial metric if and only if for all x, y, z ∈ X the following conditions are satisfied: (P) p(x, x) = p(y, y) = p(x, y) if and only if x = y; The pair (X, p) is called a partial metric space. If p(x, y) = , then (P) and (P) imply that x = y, but the converse does not hold in general. A trivial example of a partial metric space is the pair (R + , p), where p : R + × R + → R + is given by p(x, y) = max{x, y}, see also [].
It is easy to show that the function p : X × X → R + given by p ([a, b], [c, d]) = max{b, d} -min{a, c} defines a partial metric on X.
For further examples, we refer to [, , , , ]. Note that each partial metric p on X generates a T  topology τ p on X which has as a base the family of the open balls (p-balls) {B p (x, ) : x ∈ X, > }, where for all x ∈ X and > , A sequence {x n } in a partial metric space (X, p) is called convergent to a point x ∈ X, with respect to τ p , if and only if p(x, x) = lim n→+∞ p(x, x n ), see [] for details. If p is a partial metric on X, then the function Consistent with [], let (X, p) be a partial metric space and let CB p (X) be the family of all nonempty, closed and bounded subsets of the partial metric space (X, p), induced by the partial metric p. Note that the closedness is taken from (X, τ p ) (τ p is the topology induced by p) and the boundedness is given as follows: A is a bounded subset in (X, p) if there exist x  ∈ X and M ≥  such that for all a ∈ A, we have a ∈ B p (x  , M), that is, where H p : CB p (X) × CB p (X) → R + is called the partial Hausdorff metric induced by p. Also, it is easy to show that p(x, A) =  implies that p S (x, A) = , where

Lemma . ([]) Let (X, p) be a partial metric space and A be any nonempty subset of X, then a ∈ A if and only if p(a, A) = p(a, a).
Lemma . Let (X, p) be a partial metric space and A be any nonempty subset of X. If A is closed in (X, p), then A is closed in (X, p S ).
Proof Let {x n } be a sequence converging to some x ∈ X in (X, p S ). Then we have which implies, by definition, that {x n } converges to x ∈ X also in (X, p). Now, since A is closed in (X, p), then x ∈ A and so we deduce that A is closed also in (X, p S ).

Proposition . ([])
Let (X, p) be a partial metric space. For any A, B, C ∈ CB p (X), we have: Proposition . ([]) Let (X, p) be a partial metric space. For any A, B, C ∈ CB p (X), we have: Notice that each Hausdorff metric is a partial Hausdorff metric but the converse is not true, see Example . in [].
, then T has a fixed point.
In view of the above considerations and following the ideas in [], the aim of this paper is to discuss multivalued weakly Picard operators on partial Hausdorff metric spaces, see also [] for other interesting results. First, we obtain Kikkawa-Suzuki type fixed point theorems for a new type of generalized contractive conditions. Then, we prove data dependence of a fixed points set theorem. Finally, we present sufficient conditions for wellposedness of a fixed point problem. The presented results extend and unify some recently obtained comparable results for multivalued mappings (see [] and the references therein).

Fixed point theorems in partial Hausdorff metric spaces
In this section we present several theorems which characterize MWP operators, defined in the previous section, in terms of different contractive conditions. Results of this section are generalizations of Theorem ., Theorem . (and so Nadler's Theorem .), Ćirić's theorem in [] and others.

Result -I
To provide the first theorem we introduce the notion of (s, r)-contractive multivalued operator in partial Hausdorff metric spaces as follows.
Definition . Let p : X × X → R + be a partial metric and T : X → CB p (X) be a multivalued mapping. T is called an (s, r)-contractive multivalued operator if there exist r ∈ [, ) and s ≥ r such that Now we state and prove our theorem.
Theorem . Let (X, p) be a complete partial metric space and T : X → CB p (X) be an (s, r)-contractive multivalued operator with s ≥ . Then T is an MWP operator.
Proof Let r  be a real number such that  ≤ r < r  < . Let u  ∈ X. As Tu  is nonempty, we can choose u  ∈ Tu  . Clearly, if u  = u  the proof is finished and so we assume u  = u  . Then we get Now, from () and by using condition () we write Thus, we obtain Continuing this process, we can construct a sequence {u n } in X such that u n+ ∈ Tu n , u n+ = u n and for every n > . This shows that lim n→+∞ p(u n , u n+ ) = .
Let >  and pick N ∈ N large enough so that for n ≥ N we have Then, for every positive integer k > n ≥ N , there is some m ∈ N such that k = n + m, and we have It is immediate to deduce that {u n } is a Cauchy sequence in (X, p S ), but by Lemma . {u n } is Cauchy also in (X, p). Moreover, since (X, p) is complete, again by Lemma . we deduce the completeness of (X, p S ). It follows that there exists z ∈ X such that lim n→+∞ u n = z in (X, p S ). Therefore lim n→+∞ p S (u n , z) =  implies Next, we will show that there exists a subsequence {u n(k) } of {u n } such that for all k ∈ N. Reasoning by contradiction, we assume that there exists a positive integer N such that p(z, Tu n ) > sp(z, u n ) for all n ≥ N . This implies p(z, u n+ ) > sp(z, u n ) for all n ≥ N . By induction, for all n ≥ N and m ≥ , we get that for all n ≥ N and m ≥ , then we get Passing to the limit as m → +∞, we have Then we obtain for all n ≥ N and m ≥ . By () and () we get for all n ≥ N and m ≥ . Next, passing to the limit as m → +∞, we obtain that p(z, u n ) =  for all n ≥ N . This contradicts () and therefore there exists a subsequence {u n(k) } of {u n } such that p(z, Tu n(k) ) ≤ sp(z, u n(k) ) for all k ∈ N.
Therefore, we write On passing to the limit as k → +∞, we get Since r < , it follows that p(z, Tz) = .
Therefore p(z, Tz) =  = p(z, z) and hence by Lemma . we deduce that z ∈ Tz, that is, z is a fixed point of T.
The following example illustrates the use of Theorem ..  for all x, y ∈ X. Then T is an MWP operator.
In the case of single-valued mappings, Theorem . reduces to the following significant corollary.
Corollary . Let (X, p) be a complete partial metric space and T : X → X be a singlevalued mapping. Assume that there exist r ∈ [, ) and s ≥  such that We deduce that p(x, y) = , which further implies that p S (x, y) ≤ p(x, y) =  and hence x = y, a contradiction. This completes the proof.
The following two examples, adapted from [], show the validity of Corollary ..
Example . Let X = [, ] be endowed with the partial metric p(x, y) = max{x, y} for all x, y ∈ X. Then (X, p) is a complete partial metric space. Also define T : X → X by Take arbitrary elements x, y ∈ X with y ≤ x. Then we have for all x, y ∈ X and s ≥ . On the other hand, we get Thus the inequality p(Tx, Ty) ≤ rM p (x, y) holds for all x, y ∈ X with y ≤ x and for any r ∈ [   , ). Note that we obtain the same conclusion if we assume that x ≤ y. Hence, all the conditions of Corollary . are satisfied and  is a fixed point of T.
The following example underlines the crucial role of the right-hand side of () in establishing existence of the fixed point. Then (X, p) is a complete partial metric space. Also define T : X → X by Trivially T has no fixed points, but we try to apply Corollary .. It is easy to check that the inequality p(y, Tx) ≤ sp(y, x) certainly holds for all x, y ∈ X with s ≥ . Now we note that, for x =  and y = , we get whatever r ∈ [, ) is chosen. We conclude that Corollary . is not applicable in this case.

Result -II
Another interesting characterization of MWP operators is provided by the following theorem.
Theorem . Let (X, p) be a complete partial metric space and T : X → CB p (X) be a multivalued operator. Assume that there exist r, s ∈ [, ), with r < s, such that for all x, y ∈ X, where M p (x, y) = max p(x, y), p(x, Tx), p(y, Ty), p(x, Ty) + p(y, Tx)  .

Then T is an MWP operator.
Proof Let r  be a real number such that  ≤ r < r  < s. Also, let u  ∈ X and u  ∈ Tu  be such that Then we have and so, by using condition (), we obtain Thus implies that p(u  , Tu  ) =  and so we obtain which further implies that p S (u  , Tu  ) = . In view of Lemma ., u  ∈ Tu  , and the proof is finished. On the contrary, if max{p(u  , u  ), p(u  , u  )} = p(u  , u  ), then we have It follows that there exists u  ∈ Tu  such that This implies Now, by using condition (), we get p(u  , Tu  ) ≤ rp(u  , u  ). Continuing this process, we can construct a sequence {u n } in X with the following properties: for every n ∈ N. Next, from p(u n+ , u n+ ) ≤ r  p(u n , u n+ ) we deduce that lim n→+∞ p(u n , u n+ ) = .
Proceeding as in the proof of Theorem ., one can show that the sequence {u n } is a Cauchy sequence in (X, p) converging to some z ∈ X with p(z, z) = . Now, since then we have for all n ≥ . Then we assume that there exists a positive integer N such that p(z, u n ) <   + r p(u n , Tu n ) holds for every n ≥ N . Consequently, we have which is a contradiction. Hence, there exists a subsequence {u n(k) } of {u n } such that p(z, u n(k) ) ≥   + r p(u n(k) , Tu n(k) ) holds for every k ≥ N . Since p(z, u n ) ≤  -s p(u n , Tu n ) for all n ≥ , by condition (), we have H p (Tz, Tu n(k) ) ≤ rM p (z, u n(k) ). This implies ≤ p(u n(k)+ , z) + r max p(u n(k) , z), p(u n(k) , Tu n(k) ), p(z, Tz), p(u n(k) , Tz) + p(z, Tu n(k) )  ≤ r max p(u n(k) , z), p(u n(k) , u n(k)+ ), p(z, Tz), p(u n(k) , Tz) + p(z, u n(k)+ )  .
On passing to the limit as k → +∞, we get p(z, Tz) ≤ r max p(z, Tz), p(z, Tz)  .
Therefore p(z, Tz) =  = p(z, z) and hence by Lemma . we have z ∈ Tz, that is, z is a fixed point of T.
In the case of single-valued mappings, Theorem . reduces to the following corollary.
Corollary . Let (X, p) be a complete partial metric space and T : X → X be a singlevalued mapping. Assume that there exists r ∈ [, ) such that for all x, y ∈ X, where M p (x, y) = max p(x, y), p(x, Tx), p(y, Ty), p(x, Ty) + p(y, Tx)  .

Then T has a fixed point.
Proof It is easy to prove that for every u  ∈ X the sequence {u n } defined by u n+ = Tu n satisfies the relationship p(u n+ , u n+ ) ≤ rp(u n , u n+ ). Consequently, the sequence {u n } is Cauchy, and there is some point z ∈ X such that lim n→+∞ u n = z. Proceeding as in the proof of Theorem ., we can show that p(z, z) = lim n→+∞ p(u n , z) = lim n→+∞ p(u n , u n ) = .
On passing to the limit as k → +∞, we get p(z, Tz) ≤ r max p(z, Tz), p(z, Tz)  .
Since r < , it follows that p(z, Tz) =  and so z = Tz, that is, z is a fixed point of T.
The following example shows that Theorem . is proper extension of the respective result in standard metric spaces.  (x, y) for all x, y ∈ X. Then (X, p) is a complete partial metric space. Also define T : X → CB p (X) by Therefore, we get It follows easily that the inequalities hold for all x, y ∈ X with x = y and for some  > s > r ≥   . Also the above inequalities hold for x = y =  with  > s > r ≥   . On the other hand, the above inequalities are not applicable for x = y ∈ {, }. Clearly we have holds true as for all x, y ∈ X with x = y, as for x = y = . Thus all the conditions of Theorem . are satisfied and  is a fixed point of T.
Next, we consider the metric p S induced by the partial metric p. Indeed, we have p S (x, x) =  for all x ∈ X, p S (, ) =   , p S (, ) =   , p S (, ) =   and p S (x, y) = p S (y, x) for all x, y ∈ X.
We show that Theorem . is not applicable in this case. Indeed, for x =  and y = , the inequalities hold true for all r ∈ [, ) with r < s. Therefore, we need to have Unfortunately, this is not the case because

Data dependence theorem in partial Hausdorff metric spaces
The aim of this section is to discuss data dependence of a fixed points set for MWP operators on partial metric spaces. Also, this section is motivated by Popescu [], see also []. Precisely, we will prove a result for (, r)-contractive multivalued operators in partial Hausdorff metric spaces. First, we need the following auxiliary lemma.
Lemma . Let (X, p) be a partial metric space and T : X → CB p (X) be a (, r)-contractive multivalued operator. If z ∈ Tz, then p(z, z) = . and hence we deduce that p(z, z) = .
We recall the following concept.

Definition . ([]
) Let (X, p) be a partial metric space and let φ : X → R + be a function on X. Then the function φ is called p-lower semi-continuous on X whenever Now we state and prove our theorem.
Theorem . Let (X, p) be a partial metric space and T  , T  : X → CB p (X) be two multivalued operators. We suppose that:

and T  are MWP operators and
Proof (a) From Theorem . we have that Fix(T i ) is a nonempty set, i ∈ {, }. Let us prove that the fixed point set of a (, r i )-contractive multivalued operator T i is closed. Let x n ∈ Fix(T i ), with n ≥ , be such that lim n→+∞ x n = z in (X, p). In view of (iii) and Lemma ., we have Also, since x n ∈ T i x n , we have p(z, T i x n ) ≤ p(z, x n ) and then Passing to the limit as n → +∞, we obtain that p(z, T i z) = . Therefore and hence by Lemma . and T i z ∈ CL p (X) we get z ∈ T i z, that is, z ∈ Fix(T i ).
(b) From the proof of Theorem . we immediately get that a (, r i )-contractive multivalued operator is an MWP operator. For the second conclusion, let q be a real number such that q > , and x  ∈ Fix(T  ) be arbitrary. Then, by Lemma ., there exists x  ∈ T  x  such that Iterating this process allows us to construct a sequence of successive approximations for T  starting from x  , satisfying the following assertions: x n+ ∈ T  x n and p(x n , x n+ ) ≤ (qr  ) n p(x  , x  ) for all n ∈ N.
Hence, for all n ≥ N and m ≥ , we write Consequently, we get Now, choosing  < q < min{  r  ,  r  } and passing to the limit as n → +∞, we deduce easily that the sequence {x n } is Cauchy in (X, p S ) and, by Lemma ., {x n } is Cauchy in (X, p). Then there exists u ∈ X such that lim n→+∞ x n = u in X, p S . Therefore, () and lim n→+∞ p S (x n , u) =  imply We will prove that u is a fixed point for T  . To this aim, suppose that there exists a positive integer N such that p(u, T  x n ) > p(u, x n ) for all n ≥ N.
This implies that p(u, x n+ ) > p(u, x n ) for all n ≥ N , which leads to contradiction since x n → u as n → +∞. Hence, there exists a subsequence {x n(k) } such that p(u, T  x n(k) ) ≤ p(u, x n(k) ) for all k ∈ N.
Clearly, (b  ) of Definition . implies (b  ) of Definition .. Moreover, from (a  ) and (a  ), that is, Fix(T) = S Fix(T) = {z}, we deduce that if the fixed point problem is well posed for T with respect to p, then it is well posed for T with respect to H p .
Motivated by the above facts, we will prove the following theorem for (s, r)-contractive multivalued operators, with s > , in partial metric spaces.
Theorem . Let (X, p) be a partial metric space and T : X → CB p (X) be a multivalued operator. We suppose that: (b) Now, let x n ∈ X, with n ∈ N, be such that lim n→+∞ p(x n , Tx n ) = . We have to show that lim n→+∞ p(x n , z) = . Suppose this is not the case; suppose that p(x n , z) does not converge to zero. Consequently, there exist >  and a subsequence {x n(k) } such that p(x n(k) , z) ≥ for all k ∈ N. Now, assume that there exists a subsequence {x n(k(j)) } of {x n(k) } with p(z, Tx n(k(j)) ) ≤ sp(z, x n(k(j)) ).
Passing to the limit as j → +∞, since lim n→+∞ p(x n , Tx n ) = , we get the contradiction = . Consequently, we deduce that there exists k  ∈ N such that p(z, Tx n(k) ) > sp(z, x n(k) ) for all k ≥ k  .
Again, since lim n→+∞ p(x n , Tx n ) = , there exists k  ≥ k  such that p(x n(k) , Tx n(k) ) < (s -) for all k ≥ k  .

Application to integral equations
The literature is rich with papers focusing on the study of integral operators of various types: Fredholm, Urysohn, Volterra and others. It is well known that integral operators provide an important subject of numerous mathematical investigations and are often applicable in many scientific disciplines as physics, biology and economics. The papers we refer to essentially present a fixed point approach based on the Banach contraction principle and its constructive proof. These results give sufficient conditions for establishing the existence (and uniqueness) of solution of certain integral operators, see [-].
Here, following this line of research, we prove an existence theorem for the solution of integral equations by using Corollary .. Precisely, we consider the following integral equation: for all x ∈ C(I) and t ∈ I.
for all u, v ∈ C(I). Thus Corollary . is applicable in this case, and hence the operator T has a unique fixed point x * ∈ C(I). Clearly, x * ∈ C(I) is the unique solution of ().