# Some Fixed Point Theorems for Nonlinear Set-Valued Contractive Mappings

Authors: Zeqing Liu; Zhihua Wu; Shin Min Kang; Sunhong Lee
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Four fixed point theorems for nonlinear set-valued contractive mappings in complete metric spaces are proved. The results presented in this paper are extensions of a few well-known fixed point theorems. Two examples are also provided to illustrate our results.


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## Body

## 1. Introduction and Preliminaries

The existence of fixed points for various set-valued contractive mappings had been researched by many authors under different conditions, see, for example, [1–9] and the references cited therein. In 1969, Nadler [7] proved a well-known fixed point theorem for the set-valued contraction mapping  (1.1) below.Theorem 1.1 (see [7]).
Let(X,d) be a complete metric space and T:X→CB(X) be a set-valued mapping such that
(1.1)H(Tx,Ty)≤rd(x,y),∀x,y∈X,
where r∈(0,1) is a constant. Then T has a fixed point.In 1972, Reich [8] extended Nadler's result and established an interesting fixed point theorem for the set-valued contraction mapping (1.2) below.Theorem 1.2 (see [8]).
Let(X,d) be a complete metric space and T:X→C(X) satisfy that
(1.2)H(Tx,Ty)≤φ(d(x,y))d(x,y),∀x,y∈X,
where
(1.3)φ:(0,+∞)⟶[0,1)withlim⁡⁡sup⁡r→t+φ(r)<1,∀t∈(0,+∞).
Then T has a fixed point.In [8] Reich posed the question whether Theorem 1.2 is also true for the set-valued contractive mapping T:X→CB(X) with (1.2). The affirmative answer under the hypothesis of limsupr→t+φ(r)<1, for  all t∈[0,+∞) was given by Mizoguchi and Takahashi in [6]. They deduced the following fixed point theorem which is a generalization of the Nadler fixed point theorem.Theorem 1.3 (see [6]).
Let(X,d) be a complete metric space and T:X→CB(X) satisfy (1.2), where
(1.4)φ:(0,+∞)⟶[0,1)withlim⁡sup⁡r→t+φ(r)<1,∀t∈[0,+∞).
Then T has a fixed point.Remark 1.4.
It is clear that the mappingsT in Theorems 1.1–1.3 are continuous on X.Remark 1.5.
Each of Theorems1.2 and 1.3 ensures that T has a fixed point a∈Ta⊆X, which together with (1.2) implies that φ(0)=φ(d(a,a)), that is, φ is defined at 0. Thus the domain of φ in each of (1.3) and (1.4) should be [0,+∞) but not (0,+∞).
The aim of this paper is to present four fixed point theorems for some nonlinear set-valued contractive mappings. Our results extend, improve, and unify the corresponding results in [6–8]. Two nontrivial examples are given to show that our results are genuine generalizations or different from these results in [6–8].
Throughout this paper, we assume thatℝ=(-∞,+∞),ℝ+=[0,+∞), ℕ and ℕ0 denote the sets of all positive integers and nonnegative integers, respectively, and
(1.5)Θ={θ:θ:R+⟶R+satisfies(a)–(d)},
where(a)
θis nondecreasing on ℝ+;(b)
θ(t)>0, for  all t∈(0,+∞);(c)
θ is subadditive in (0,+∞), that is,
(1.6)θ(t1+t2)≤θ(t1)+θ(t2),∀t1,t2∈(0,+∞);(d)
θ(ℝ+)=ℝ+. Clearly (a)–(d) imply that(e)
θ is strictly inverse on ℝ+, that is, if there exist t,s∈ℝ+ satisfying θ(t)<θ(s), then t<s.
Let(X,d) be a metric space, CL(X), CB(X), and C(X) denote the families of all nonempty closed, all nonempty bounded closed, and all nonempty compact subsets of X. For x∈X and A,B∈CL(X), put d(x,A)=inf{d(x,y):y∈A} and
(1.7)H(A,B)={max⁡{sup⁡x∈Ad(x,B),sup⁡y∈Bd(y,A)},if  the  maximum  exists+∞,otherwise.
Such a mapping H is called a generalized Hausdorff metric induced byd in CL(X). It is well known that H is a metric on CB(X). Let T:X→CL(X) be a set-valued mapping, x0∈X and f:X→ℝ+ be defined by
(1.8)f(x)=d(x,Tx),∀x∈X.
A sequence {xn}n∈ℕ0 is said to be anorbit ofT if it satisfies that {xn}n∈ℕ0⊂X and xn∈Txn-1 for each n∈ℕ0. The function f:X→ℝ+ is said to be T-orbitallylowersemicontinuousatz∈X if for each orbit {xn}n∈ℕ0⊂X of T with limn→∞xn=z, we have that f(z)≤liminfn→∞f(xn).


## 2. Main Results

The following lemmas play important roles in this paper.Lemma 2.1.
Let(X,d) be a metric space and B∈CL(X). Then for each x∈X and ε>0 there exists b∈B satisfying d(x,b)≤d(x,B)+ε.Proof.
Suppose that there existx0∈X and ε0>0 such that
(2.1)d(x0,b)>d(x0,B)+ε0,∀b∈B,
which yields that
(2.2)d(x0,B)=inf⁡b∈Bd(x0,b)≥d(x0,B)+ε0>d(x0,B),
which is a contradiction. This completes the proof.Lemma 2.2.
Let(X,d) be a metric space, B∈CL(X) and θ∈Θ. Then for each x∈X and q>1 there exists b∈B such that
(2.3)θ(d(x,b))≤qθ(d(x,B)).Proof.
Letx∈X and q>1. Now we consider two possible cases as follows.
Case  1. Suppose that θ(d(x,B))=0. It follows from (b) and (d) that d(x,B)=0. Since B is a closed subset of X, it follows that x∈B. Put b=x. Clearly (2.3) holds.
Case  2. Suppose that θ(d(x,B))>0. Note that (b) and (d) mean that
(2.4)(q-1)θ(d(x,B))∈R+∖{0}=θ(R+∖{0}).
Choose p∈θ-1((q-1)θ(d(x,B))) and ε=p/2>0. Lemma 2.1 ensures that there exists b∈B satisfying d(x,b)≤d(x,B)+ε, which together with (a) and (c) gives that
(2.5)θ(d(x,b))≤θ(d(x,B)+ε)≤θ(d(x,B))+θ(ε)≤θ(d(x,B))+θ(θ-1((q-1)θ(d(x,B))))=qθ(d(x,B)).
That is, (2.3) holds. This completes the proof.Now we prove four fixed point theorems for the nonlinear set-valued contractive mappings (2.6), (2.25), (2.26), and (2.36) below in complete metric spaces.Theorem 2.3.
Let(X,d) be a complete metric space and T:X→CL(X) satisfy that
(2.6)θ(d(y,Ty))≤φ(d(x,y))θ(d(x,y)),∀(x,y)∈X×Tx,
where θ∈Θ and
(2.7)φ:R+⟶[0,1)withlim⁡⁡sup⁡r→t+φ(r)<1,∀t∈R+.
Then for each x0∈X, there exists an orbit {xn}n∈ℕ0 of T and z∈X such that limn→∞xn=z. Furthermore, z∈X is fixed point of T if and only if the function f defined by (1.8) is T orbitally lower semicontinuous at z.Proof.
Letx0∈X be any initial point and choose x1∈Tx0. It follows from (2.6), (2.7) and Lemma 2.2 that for q1=1/max{φ(d(x0,x1)),1/2}>1 there exists x2∈Tx1 satisfying
(2.8)θ(d(x1,x2))≤θ(d(x1,Tx1))max⁡⁡{φ(d(x0,x1)),1/2}≤φ(d(x0,x1))θ(d(x0,x1))max⁡{φ(d(x0,x1)),1/2}≤φ(d(x0,x1))θ(d(x0,x1)),
and for q2=1/max{φ(d(x1,x2)),1/3}>1 there exists x3∈Tx2 satisfying
(2.9)θ(d(x2,x3))≤θ(d(x2,Tx2))max⁡{φ(d(x1,x2)),1/3}≤φ(d(x1,x2))θ(d(x1,x2))max⁡{φ(d(x1,x2)),1/3}≤φ(d(x1,x2))θ(d(x1,x2)).
Repeating the above argument we obtain a sequence {xn}n∈ℕ0⊂X such that xk∈Txk-1 for 1≤k≤n and for qn=1/max{φ(d(xn-1,xn)),1/(n+1)}>1, there exists xn+1∈Txn satisfying
(2.10)θ(d(xn,xn+1))≤θ(d(xn,Txn))max⁡{φ(d(xn-1,xn)),1/(n+1)}≤φ(d(xn-1,xn))θ(d(xn-1,xn))max⁡{φ(d(xn-1,xn)),1/(n+1)}≤φ(d(xn-1,xn))θ(d(xn-1,xn)),∀n≥1.
Suppose that there exists somen0∈ℕ0 satisfying xn0=xn0+1∈Txn0. It follows from (a), (b), and (2.10) that xn=xn0 for all n≥n0+1. It is clear the conclusion of Theorem 2.3 holds.
Suppose thatxn+1∈Txn∖{xn} for any n∈ℕ0. It follows that d(xn,xn+1)>0 for each n∈ℕ0. Note that (b), (2.7), and (2.10) give that {θ(d(xn,xn+1))}n∈ℕ0 is a positive and decreasing sequence. It follows from (e) that {d(xn,xn+1)}n∈ℕ0 is decreasing. Therefore, there exist constants p and q satisfying
(2.11)lim⁡n→∞θ(d(xn,xn+1))=p≥0,lim⁡n→∞d(xn,xn+1)=q≥0.
Notice that (2.7) implies that there exists a constant r satisfying
(2.12)limsup⁡n→∞φ(d(xn,xn-1))≤limsup⁡t→q+φ(t)=r∈[0,1).
Taking upper limits in (2.10) and by (2.11) and (2.12) we get that
(2.13)p≤limsup⁡n→∞φ(d(xn-1,xn))limsup⁡n→∞θ(d(xn-1,xn))≤rp,
which implies that p=0.
Next we assert thatq=0. Since {d(xn,xn+1)}n∈ℕ0 is a decreasing sequence, it follows from (a) and (2.11) that
(2.14)0≤θ(q)<θ(d(xn,xn+1))⟶p=0asn⟶∞,
that is, θ(q)=0, which together with (b) and (d) yields that q=0.
Putc=(1+r)/2. It follows from (2.12) that c∈(r,1)⊂[0,1), which gives that c2∈(r,1). Notice that (2.11), (2.12), and q=0 ensure that there exist δ>0 and N∈ℕ satisfying
(2.15)φ(t)<c2,∀t∈(0,δ),d(xn,xn+1)<δ,∀n≥N,
which implies that
(2.16)φ(d(xn,xn+1))<c2,∀n≥N.
Note that (2.10) and (2.16) mean that
(2.17)θ(d(xn,xn+1))≤∏k=Nn-1‍φ(d(xk,xk+1))θ(d(xN,xN+1))≤cn-Nθ(d(xN,xN+1)),∀n≥N.
Given ε>0. Since limn→∞cn-Nθ(d(xN,xN+1))=0, it follows from (b) that there exists N1>N satisfying
(2.18)cn-N1-cθ(d(xN,xN+1))<θ(ε),∀n≥N1,
which together with (2.17), (a), and (c) gives that
(2.19)θ(d(xn,xm))≤θ(∑k=nm-1‍d(xk,xk+1))≤∑k=nm-1‍θ(d(xk,xk+1))≤∑k=nm-1‍ck-Nθ(d(xN,xN+1))≤cn-N1-cθ(d(xN,xN+1))<θ(ε),∀m>n≥N1.
In view of (e) and (2.19), we deduce that d(xn,xm)<ε, for all m>n≥N1, which means that {xn}n∈ℕ0 is a Cauchy sequence. Hence there exists z∈X such that limn→∞xn=z by completeness of X.
Suppose thatf is T orbitally lower semicontinuous at z. Since {xn}n≥0 is an orbit of T with limn→∞xn=z, it follows that
(2.20)f(z)≤liminf⁡n→∞f(xn).
Using (2.6) and (2.7), we infer that
(2.21)θ(d(xn,Txn))≤φ(d(xn-1,xn))θ(d(xn-1,xn))<θ(d(xn-1,xn)),∀n∈N,
which together with (e), (2.11), and q=0 implies that
(2.22)0<d(xn,Txn)<d(xn-1,xn)⟶0asn⟶∞,
that is, limn→∞d(xn,Txn)=0, which together with (2.20) yields that
(2.23)0≤d(z,Tz)=f(z)≤liminf⁡n→∞f(xn)=lim⁡n→∞d(xn,Txn)=0,
which gives that d(z,Tz)=0, that is, z∈Tz.
Conversely, suppose thatz∈X is a fixed point of T. Let {yn}n∈ℕ0⊂X be an arbitrarily orbit of T with limn→∞yn=z. It is clear that
(2.24)f(z)=d(z,Tz)=0≤liminf⁡n→∞f(yn),
which implies that f is T orbitally lower semicontinuous at z. This completes the proof.Notice thatd(y,Ty)≤H(Tx,Ty) for each y∈Tx. In light of Theorem 2.3, we haveTheorem 2.4.
Let(X,d) be a complete metric space and T:X→CL(X) satisfy that
(2.25)θ(H(Tx,Ty))≤φ(d(x,y))θ(d(x,y)),∀(x,y)∈X×Tx,
where θ∈Θ and φ satisfies (2.7). Then for each x0∈X, there exists an orbit {xn}n∈ℕ0 ofT and z∈X such that limn→∞xn=z. Furthermore, z∈X is fixed point of T if and only if the function f defined by (1.8) is T orbitally lower semicontinuous at z.Ifφ(d(x,y)) in (2.6) is replaced by φ(d(x,Tx)), one hasTheorem 2.5.
Let(X,d) be a complete metric space and T:X→CL(X) satisfy that
(2.26)θ(d(y,Ty))≤φ(d(x,Tx))θ(d(x,y)),∀(x,y)∈X×Tx,
where θ∈Θ and φ satisfies (2.7). Then for each x0∈X, there exists an orbit {xn}n∈ℕ0 of T and z∈X such that limn→∞xn=z. Furthermore, z∈X is fixed point of T if and only if the function f defined by (1.8) is T orbitally lower semicontinuous at z.Proof.
Letx0∈X be any initial point and choose x1∈Tx0. It follows from (2.7), (2.26), and Lemma 2.2 that for q=1/max{φ(d(x0,Tx0)),φ(d(x1,Tx1)),1/2}>1 there exists x2∈Tx1 such that
(2.27)θ(d(x1,x2))≤θ(d(x1,Tx1))max⁡⁡{φ(d(x0,Tx0)),φ(d(x1,Tx1)),1/2}≤φ(d(x0,Tx0))θ(d(x0,x1))max⁡⁡{φ(d(x0,Tx0)),φ(d(x1,Tx1)),1/2}≤φ(d(x0,Tx0))θ(d(x0,x1)),θ(d(x2,Tx2))≤φ(d(x1,Tx1))θ(d(x1,x2))≤φ(d(x1,Tx1))θ(d(x1,Tx1))max⁡⁡{φ(d(x0,Tx0)),φ(d(x1,Tx1)),1/2}≤φ(d(x1,Tx1))θ(d(x1,Tx1)).
Repeating the above argument we obtain a sequence {xn}n∈ℕ0⊂X satisfying xn+1∈Txn for each n∈ℕ0,
(2.28)θ(d(xn,xn+1))≤θ(d(xn,Txn))max⁡{φ(d(xn-1,Txn-1)),φ(d(xn,Txn)),1/(n+1)}≤φ(d(xn-1,Txn-1))θ(d(xn-1,xn))max⁡{φ(d(xn-1,Txn-1)),φ(d(xn,Txn)),1/(n+1)}≤φ(d(xn-1,Txn-1))θ(d(xn-1,xn)),∀n∈N,(2.29)θ(d(xn+1,Txn+1))≤φ(d(xn,Txn))θ(d(xn,xn+1))≤φ(d(xn,Txn))θ(d(xn,Txn))max⁡{φ(d(xn-1,Txn-1)),φ(d(xn,Txn)),1/(n+1)}≤φ(d(xn,Txn))θ(d(xn,Txn)),∀n∈N.
Suppose thatxn0∈Txn0 for some n0∈ℕ0. It is easy to verify that xn=xn0 for all n≥n0 and the conclusion of Theorem 2.5 holds.
Suppose thatxn∉Txn for each n∈ℕ0. It follows that {d(xn,Txn)}n∈ℕ0 and {d(xn,xn+1)}n∈ℕ0 are positive sequences. Combining (2.7), (2.28), (2.29), (b) and (e), we infer that {θ(d(xn,xn+1))}n∈ℕ0 and {θ(d(xn,Txn))}n∈ℕ0 are both positive and decreasing, so do {d(xn,xn+1)}n∈ℕ0 and {d(xn,Txn)}n∈ℕ0. It follows that there exist constants α,β,s and t satisfying
(2.30)lim⁡n→∞θ(d(xn,xn+1))=α≥0,lim⁡n→∞d(xn,xn+1)=β≥0,lim⁡n→∞θ(d(xn,Txn))=s≥0,lim⁡n→∞d(xn,Txn)=t≥0.
Notice that (2.7) implies that there exists a constant r such that
(2.31)limsup⁡n→∞φ(d(xn,Txn))≤limsup⁡l→t+φ(l)=r∈[0,1).
Taking upper limits in (2.29) and by (2.30) and (2.31) we get that
(2.32)s≤limsup⁡n→∞φ(d(xn,Txn))limsup⁡n→∞θ(d(xn,Txn))≤rs,
which implies that s=0, which together with (2.30) and (a) ensures that
(2.33)0≤θ(t)<θ(d(xn,Txn))⟶0,n⟶∞,
that is, θ(t)=0, which gives that t=0 by (b) and (d). It follows from (2.28), (2.30), and (2.31) that
(2.34)α≤limsup⁡n→∞φ(d(xn,Txn))limsup⁡n→∞θ(d(xn-1,xn))≤rα,
which yields that α=0. Notice that (2.30) and (a) guarantee that
(2.35)0≤θ(β)<θ(d(xn,xn+1))⟶0,n⟶∞,
which together with (b) and (d) yields that β=0. The rest of the proof is similar to that of Theorem 2.3 and is omitted. This completes the proof.The result below follows from Theorem2.5.Theorem 2.6.
Let(X,d) be a complete metric space and T:X→CL(X) satisfy that
(2.36)θ(H(Tx,Ty))≤φ(d(x,Tx))θ(d(x,y)),∀(x,y)∈X×Tx,
where θ∈Θ and φ satisfies (2.7). Then for each x0∈X, there exists an orbit {xn}n∈ℕ0 ofT and z∈X such that limn→∞xn=z. Furthermore, z∈X is fixed point of  T if and only if the function f defined by (1.8) is T orbitally lower semicontinuous at z.


## 3. Comparisons and Examples

Now we construct two examples to compare the results in Section2 with the corresponding results in [6–8].Remark 3.1.
Theorems2.3 and 2.4 extend Theorems 1.1–1.3, and Theorems 2.5 and 2.6 are different from Theorems 1.1–1.3, respectively, in the following ways:(1)
the rangesCL(X) of the nonlinear set-valued contractive mappings T in Theorems 2.3–2.6 are more general than the ranges C(X) and CB(X) of the set-valued contraction mappings T in Theorems 1.1–1.3, respectively;(2)
theT orbit lower semicontinuity at some z∈X of the functions f(x)=d(x,Tx) in Theorems 2.3 and 2.4 is weaker than the continuity of the set-valued contraction mappings T in X in Theorems 1.1–1.3, respectively;(3)
the set-valued contraction mappings (1.1) and (1.2) are special cases of the nonlinear set-valued contractive mapping (2.6) with θ≡1 because
(3.1)d(y,Ty)≤H(Tx,Ty),∀(x,y)∈X×Tx.
Example3.2 below shows that Theorems 2.3 and 2.4 extend substantively Theorems 1.1–1.3, respectively.Example 3.2.
LetX=(-∞,3/10] and d be the standard metric in X. Let θ:ℝ+→ℝ+, φ:ℝ+→[0,1) and T:X→CL(X) be defined by
(3.2)θ(t)=t1/2,φ(t)=265,∀t∈R+,Tx={(-∞,14x],∀x∈(-∞,0),[0,2x2],∀x∈[0,310],
respectively. It is clear that θ∈Θ, φ satisfies (2.7) and
(3.3)f(x)=d(x,Tx)={0,∀x∈(-∞,0)x-2x2,∀x∈[0,310]
is T orbitally lower semicontinuous in X. In order to prove (2.6) holds, we consider two possible cases.
Case  1. Let x∈(-∞,0) and y∈Tx=(-∞,(1/4)x]. It is clear that(3.4)θ(d(y,Ty))≤θ(H(Tx,Ty))=12θ(d(x,y))≤φ(d(x,y))θ(d(x,y)).
Case  2. Let x∈[0,3/10] and y∈Tx=[0,2x2]. It follows that
(3.5)θ(d(y,Ty))≤θ(H(Tx,Ty))=2|x+y|1/2θ(d(x,y))≤2(310+950)1/2θ(d(x,y))=φ(d(x,y))θ(d(x,y)),
that is, (2.6) holds. Therefore all assumptions of Theorems 2.3 and 2.4 are satisfied. It follows from each of Theorems 2.3 and 2.4 that T has a fixed point in X. However, we cannot invoke any one of Theorems 1.1–1.3 to show the existence of fixed points for the mapping T in X. Indeed, taking x0=3/10 and y0=1/5, we get that
(3.6)H(Tx0,Ty0)=d(2(310)2,2(15)2)=110≰r10=rd(x0,y0),
for any r∈(0,1) and
(3.7)H(Tx0,Ty0)=d(2(310)2,2(15)2)=110≰110φ(110)=φ(d(x0,y0))d(x0,y0),
for any mapping φ:ℝ+→[0,1) with each of (1.3) and (1.4).
Next we construct an example to explain Theorems2.5 and 2.6.Example 3.3.
LetX=[-3/10,+∞) and d be the standard metric in X. Define θ:ℝ+→ℝ+, φ:ℝ+→[0,1) and T:X→CL(X) by
(3.8)θ(t)=t1/2,∀t∈R+,φ(t)={22t1/2,∀t∈(0,18),265,∀t∈{0}∪[18,+∞),Tx={[x4(1+x),+∞),∀x∈(0,+∞),[-2x2,0],∀x∈[-310,0],
respectively. It is easy to see that (2.7) holds and
(3.9)f(x)=d(x,Tx)={0,∀x∈(0,+∞),-2x2-x,∀x∈[-310,0]
is T orbitally lower semicontinuous in X. In order to check (2.26), we have to consider two cases as follows.
Case  1. Let x∈(0,+∞) and y∈Tx=[x/4(1+x),+∞). It is clear that
(3.10)θ(d(y,Ty))=0≤θ(H(Tx,Ty))=|x4(1+x)-y4(1+y)|1/2=θ(d(x,y))2(1+x)1/2(1+y)1/2≤θ(d(x,y))2(1+x)1/2(1+x/4(1+x))1/2=θ(d(x,y))(5x+4)1/2≤θ(d(x,y))2≤265θ(d(x,y))=φ(0)θ(d(x,y))=φ(d(x,Tx))θ(d(x,y)).
Case  2. Let x∈[-3/10,0] and y∈Tx=[-2x2,0]. It follows that
(3.11)θ(d(y,Ty))≤θ(H(Tx,Ty))=2|x+y|1/2θ(d(x,y))≤2|x-2x2|1/2θ(d(x,y)).
For x=0, we have
(3.12)2|x-2x2|1/2θ(d(x,y))=0≤φ(d(x,Tx))θ(d(x,y)).
For x∈[-3/10,-1/4)∪(-1/4,0), we infer that
(3.13)2|x-2x2|1/2θ(d(x,y))≤22(-2x2-x)1/2θ(d(x,y))=φ(d(x,Tx))θ(d(x,y)).
For x=-1/4, we get that
(3.14)2|x-2x2|1/2θ(d(x,y))=32θ(d(x,y))≤φ(18)θ(d(x,y))=φ(d(x,Tx))θ(d(x,y)).
Hence (2.26) holds. Thus all assumptions of Theorems 2.5 and 2.6 are satisfied. It follows from each of Theorems 2.5 and 2.6 that T has a fixed point in X.
Takingx0=1 and y0=-3/10, we deduce that
(3.15)H(Tx0,Ty0)=H([18,+∞),[-950,0])=+∞≰13r10=rd(x0,y0),
for any r∈(0,1), and
(3.16)H(Tx0,Ty0)=+∞≰265⋅1310=φ(d(x0,y0))d(x0,y0),
for any mapping φ:ℝ+→[0,1) with each of (1.3) and (1.4). That is, Theorems 1.1–1.3 are inapplicable in proving the existence of fixed points for the nonlinear set-valued contractive mapping T.



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*Source: 786061-2012-06-26.xml*