Dhage_Multi_09.dvi


CUBO A Mathematical Journal

Vol.12, No¯ 03, (139–151). October 2010

Some Generalizations of Mulit-Valued Version

of Schauder’s Fixed Point Theorem

with Applications

BAPURAO C. DHAGE

Kasubai, Gurukul Colony, Ahmedpur – 413515,

Distr. Latur, Maharashtra, India

email: bcdhage@yahoo.co.in

ABSTRACT

In this article, a generalization of a Kakutani-Fan fixed point theorem for multi-valued

mappings in Banach spaces is proved under weaker upper semi-continuity condition and

it is further applied to derive a generalized version of Krasnoselskii’s fixed point theorem

and some nonlinear alternatives of Leray-Schauder type for multi-valued closed mappings

in Banach spaces.

RESUMEN

En este artículo probamos una generalización para el teorema del punto fijo de Kakutani-

Fan para aplicaciones multi-valuadas en espacios de Banach, bajo condición de semi-conti-

nuidad superior debil. Este resultado es aplicado para obtener una versión generalizada

del teorema del punto fijo Krasnoselskii y algunas alternativas de tipo Leray-Schauder

para aplicaciones multi-valuadas cerradas en espacios de Banach.

Key words and phrases: Multi-valued mappings, fixed point theorem, nonlinear alternative.

Math. Subj. Class.: 47H10.



140 Bapurao C. Dhage
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1 Introduction

Throughout this paper, unless otherwise mentioned, let E be a Banach space and let P (E)

denote the class of all subsets of E. Denote

P p(E) = {A ⊂ E | A is non-empty and has a property p}.

Here, p may be the property p = closed (in short cl), or p = compact (in short cp), or p =

convex (in short cv), or p = bounded (in short bd) etc. Thus, P bd (E), P cl (E), P cv(E), P c p (E),

P cl,bd (E), P c p,cv (E) denote the classes of all bounded, closed, convex, compact, closed-bounded

and compact-convex subsets of E respectively. Similarly, P cl,cv,bd (E) and P r c p (E) denote re-

spectively the classes of closed, convex and bounded and relatively compact subsets of E.

A correspondence Q : E → P p(E) is called a multi-valued operator or multi-valued map-

ping on E into E. A point u ∈ E is called a fixed point of Q if u ∈ Qu. For the sake of

convenience, we denote Q(A) =
⋃

x∈A T x for all subsets A of E.

Let E1 and E2 be two Banach spaces and let Q : E1 → P p (E2) be a multi-valued operator.

Then for any non-empty subset A of E2, define

Q
+

(A) = {x ∈ E1 | T x ⊂ A},

Q
−

(A) = {x ∈ E1 | T x ∩ A 6= ;},

and

Q
−1

(A) = {x ∈ E1 | ∪xT x = A}.

Definition 1.1. A multi-valued operator Q : E1 → P p(E2) is called upper semi-continuous

(resp. lower semi-continuous and continuous) if Q+(U) (resp. Q−(U) and Q−1(U)) is open set

in E1 for every open subset U of E2.

In what follows,we confine ourselves only to the fixed point theory related to upper semi-

continuous multi-valued mappings in Banach spaces. The first fixed point theorem in this

direction is due to Kakutani-Fan [11] which is as follows.

Theorem 1.1. Let K be a compact subset of a Banach space E and let Q : E → P c p,cv (E) be an

upper semi-continuous multi-valued operator. Then Q has a fixed point.

Note that following are the main three ingredients for the above Theorem 1.1.

(i) The domain space E,



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Multi-Valued Fixed Point Theory 141

(ii) The domain set K , and

(iii) The nature of the multi-valued operator Q.

Theorem 1.1 has been extended in the literature by generalizing or modifying the above three

hypotheses with the same conclusion. In the following discussion, we do not change the hy-

pothesis on the domain space, and thus keep us in the practical applicability of the so obtained

fixed point theorem to other areas of mathematics. However, the generalizations of the above

Theorem 1.1 with change of domain space may be found in the works of Browder-Fan [11] and

Himmelberg [9] etc.

A first generalization of Theorem 1.1 is due to Bohnenblust-Karlin as given in Petruşel

[12].

Theorem 1.2 (Bohnenblust-Karlin). Let X be a closed convex and bounded subset of a Banach

algebra E and let Q : X → P c p,cv (X ) be a upper semi-continuous multi-valued operator with a

relatively compact range. Then Q has a fixed point.

A multi-valued map Q : X → P c p (X ) is called compact if Q(X ) is a compact subset of X .

Q is called totally bounded if for any bounded subset A of X , Q(A) =
⋃

x∈A Q x is a totally

bounded subset of X . It is clear that every compact multi-valued operator is totally bounded,

but the converse may not be true. However, these two notions are equivalent on a bounded

subset of X . Finally, Q is called completely continuous if it is upper semi-continuous and

totally bounded on X .

The upper semi-continuity is further weakened to closed graph operators as follows. If

Q : E1 → E2 is a multi-valued operator, then the graph Gr(Q) of the operator Q is defined by

Gr(Q) = {(x, y) ∈ E1 × E2 | y ∈ T x}.

The graph Gr(Q) of the operator Q is said to be closed if {(xn, yn)} be a sequence in Gr(Q) such

that (xn, yn) → (x, y), then we have that (x, y) ∈ Gr(Q).

Definition 1.2. A multi-valued operator Q : E1 → P cl (E2) is called closed if it has a closed

graph in E1 × E2.

The following result concerning the upper semi-continuity of multi-valued mappings in

Banach spaces is very much useful in the study of multi-valued analysis. The details appears

in Deimling [5].

Lemma 1.1. A multi-valued operator Q : E1 → P cl (E2) is upper semi-continuous if and only

if it is closed and has compact range.



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Theorem 1.3 (O’Regan [13]). Let X be a closed convex and bounded subset of a Banach

algebra E and let Q : X → P c p,cv (X ) be a compact and closed multi-valued operator. Then

Q has a fixed point.

The compactness of Q in Theorem 1.3 is further weakened to condensing operators with

the help of measure of noncompactness in the Banach space E. The Kuratowskii measure α

and the ball or Hausdorff measure β of noncompactness of a bounded set in the Banach space

E are the functions α,β : P bd (E) → R
+ defined by

α(A) = inf
{

r > 0 : A ⊂ ∪
n
i=1

Ai , diam(Si ) ≤ r ∀ i
}

, (1.1)

and

β(A) = inf
{

r > 0 | A ⊂
n
⋃

i=1

Br (xi) for some xi ∈ X
}

(1.2)

for all A ∈ P bd (E), where diam (Ai ) = sup{‖x − x‖ : x, y ∈ Ai } and Br (xi ) are the open balls

centered at xi of radius r.

Definition 1.3. A multi-valued operator Q : E → P cl,bd (E) is called β-condensing if for all

bounded sets A in E, Q(A) is bounded and β(Q(A)) < β(A) for β(A) > 0 .

Theorem 1.4. Let X be a closed convex and bounded subset of a Banach space E and let

Q : X → P cl,cv(X ) be a upper semi-continuous and β-condensing multi-valued operator. Then

Q has a fixed point.

In this article, we generalize Theorem 1.1 by weakening the upper semi-continuity as

well as compactness of the multi-valued operator Q in a Banach space E and discuss some of

its applications.

2 Fixed Point Theory

A function dH : P p (E) × P p(E) → R
+ defined by

dH (A, B) = max
{

sup
a∈A

D(a, B) , sup
b∈B

D(b, A)
}

(2.1)

satisfies all the conditions of a metric on P p (E) and is called a Hausdorff-Pompeiu metric

on E, where D(a, B) = inf {‖a − b‖ : b ∈ B}. It is known that the hyperspace
(

P cl (E), dH
)

is a

complete metric space.

The axiomatic way of defining the measures of noncompactness has been adopted in

several papers in the literature. See Akhmerov et al. [2], Banas and Goebel [3], and the



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Multi-Valued Fixed Point Theory 143

references given therein. In this paper, we define the measure of noncompactness in a Banach

space on the lines of Dhage [6] which is slightly different manner from that given in the above

monographs.

Definition 2.1. A sequence {An} of non-empty sets in P p(E) is said to converge to a set A,

called the limiting set, if dH (An, A) → 0 as n → ∞.

Definition 2.2. A mapping µ : P p (E) → R
+ is continuous if for any sequence {An } in P p(E),

we have that

dH (An, A) → 0 implies |µ(An) −µ(A)| → 0 as n → ∞.

Definition 2.3. A mapping µ : P p(E) → R
+ is called nondecreasing if A, B ∈ P p(E) are any

two sets with A ⊆ B, then µ(A) ≤ µ(B), where ⊆ is a order relation by inclusion in P p(E).

Now we are equipped with the necessary details to define the measures of noncompact-

ness of a bounded subset of the Banach space E.

Definition 2.4. A function µ : P cl,bd (E) → R
+ is called a measure of noncompactness if it

satisfies

(µ1) ; 6= µ
−1(0) ⊂ P r c p (E),

(µ2) µ(A) = µ(A), where A denotes the closure of A,

(µ3) µ(Conv A) = µ(A), where Conv A denotes the convex hull of A,

(µ4) µ is nondecreasing, and

(µ5) if {An } is a decreasing sequence of sets in P cl,bd (E) satisfying lim
n→∞

µ(An) = 0, then the

limiting set A∞ = lim
n→∞

An is non-empty.

Note that the functions α and β defined by (1.1) and (1.2) satisfy the conditions (µ1)

through (µ5). Hence α and β are the measures of noncompactness on E. Moreover, they are

locally Lipschitz and hence are locally continuous on P cl,bd (E). Some nice properties of α and

β have been discussed in Akhmerov et al. [2] and Banas and Goebel [3].

We remark that if (µ4) holds, then A∞ ∈ P r c p (E). To see this, let limn→∞ µ(An) = 0. As

A∞ ⊆ An for each n = 0, 1, 2, ...; by the monotonicity of µ, we obtain

µ(A∞) ≤ lim
n→∞

An = lim
n→∞

µ(An) = 0.

Hence, by assumption (µ1), we get A∞ is nonempty and A∞ ∈ P r c p (E).

A measure µ is called complete or full if the kernel of µ consists of all possible relatively

compact subsets of E. Next, a measure µ is called sublinear if it satisfies



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(µ6) µ(λA) = |λ|µ(A) for λ ∈ R, and

(µ7) µ(A + B) ≤ µ(A) +µ(B) for A, B ∈ P cl,bd (E).

There do exist the sublinear measures of noncompactness in Banach spaces E. Indeed, the

measures α and β of noncompactness defined by (1.1) and (1.2) are sublinear on E.

Now we prove a fixed point theorem for the mappings in Banach spaces involving the

measures of noncompactness. Before going to the main results, we give a useful definition.

Definition 2.5. A multi-valued mapping Q : E → P cl,bd (E) is called D-set-Lipschitz if there

exists a continuous nondecreasing function ψ : R+ → R+ such that µ(Q(A)) ≤ ψ(µ(A)) for all

A ∈ P cl,bd (E) with Q(A) ∈ P cl,bd (E), where ψ(0) = 0. Sometimes we call the function ψ to be a

D-function of Q on E. In the special case, when ψ(r) = kr, k > 0, Q is called a k-set-Lipschitz

mapping and if k < 1, then Q is called a k-set-contraction on E. Further, if ψ(r) < r for r > 0,

then Q is called a nonlinear D-set-contraction on E.

We need the following lemma in the sequel.

Lemma 2.1 (Dhage [8]). If ψ is a D-function with ψ(r) < r for r > 0, then lim
n→∞

ψ
n
(t) = 0 for all

t ∈ [0,∞).

Theorem 2.1. Let X be a non-empty, closed, convex and bounded subset of a Banach space

E and let Q : X → P cl,cv(X ) be a closed and nonlinear D-set-contraction. Then Q has a fixed

point.

Proof. Define a sequence {X n} of sets in P cl,bd (E) by

X0 = X , X n+1 = Conv Q(X n), n = 0, 1, ...

Clearly,

X0 ⊃ X1 ⊃ ··· ⊃ X n ⊃ X n+1 ··· .

and so, {X n} is a decreasing sequence of subsets of E. Since

µ(X n+1) = µ
(

Conv Q(X n)
)

= µ(Q(X n)) ≤ ψ(µ(X n))

for all n = 0, 1, 2, . . ., we have

µ(X n+1) ≤ ψ
n
(µ(X0)).

Therefore

lim sup
n→∞

µ(X n+1) ≤ lim sup
n→∞

ψ
n
(µ(X0)) = 0.



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Multi-Valued Fixed Point Theory 145

From the monotonicity of µ it follows that lim
n→∞

X n = X∞ is a compact subset of E. As X n+1 ⊂

X n and Q : X n → X n for all n = 0, 2, . . ., we have

X∞ = lim
n→∞

X n =
∞
⋂

n=1

X n 6= ;

is a convex subset of E and Q : X∞ → P c p,cv (X∞) which is upper-semi-continuous in view

of Lemma 1.1. Now the desired conclusion follows by an application of Theorem 1.1 to the

operator Q on X∞. This completes the proof.

Remark 2.1. The fixed point set Fix(Q) of the multi-valued operator Q in above Theorem

2.1 is compact. In fact if µ(Fix(Q)) > 0, then from nonlinear D-set-contraction it follows that

µ(Fix(Q)) = µ(Q(Fix(Q))) ≤ ψ(µ(Fix(Q))) which is a contradiction since ψ(r) < r for r > 0.

As a consequence of Theorem 2.1 we obtain a fixed point theorem of Darbo [3] type for

linear set-contractions,

Corollary 2.1. Let X be a closed, convex and bounded subset of a Banach space E and let

Q : X → P cl,cv (X ) be a closed and k-set-contraction. Then Q has a fixed point.

Before stating the generalization of Theorem 2.1of Sadovskii [14] type, we give a useful

definition.

Definition 2.6. A multi-valued mapping Q : E → P (E) is called µ-condensing if for any

bounded subset A of E, Q(A) is bounded and µ(Q(A)) < µ(A) for µ(A) > 0.

Theorem 2.2. Let X be a nonempty, closed, convex and bounded subset of a Banach space E

and let Q : X → P cl,cv (X ) be a closed and µ-condensing mapping. Then Q has a fixed point.

Thus, we have a one way implication that Sadovskii’s type theorem ⇒ Theorem 2.1 ⇒

Darbo’s type theorem. However, it is rather difficult to find the operators satisfying the condi-

tions on Banach spaces given in Sadovskii’s type fixed point theorem.

3 Applications

3.1 Hybrid fixed point theory

First, we derive a Krasnoselskii type fixed point theorem for the sum of two multi-valued

mappings in Banach spaces. Before stating this result, we need the following definition.

Definition 3.1. A multi-valued mapping Q : E → P cl,cv (E) is said to be nonlinear D-contraction

if there is a D-function ψ such that

dH (Q x, Q y) ≤ ψ(d(x, y))



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for all x, y ∈ E, where ψ(r) < r.

Theorem 3.1. Let X be a closed, convex and bounded subset of a Banach space E and let µ

be a sublinear measure of noncompactness in it. Let S, T : X → P cl,cv (E) be two operators such

that

(a) S is closed and nonlinear D-set-contraction,

(b) T is compact and closed, and

(c) Sx + T x ⊂ X for all x ∈ X .

Then the operator inclusion x ∈ Sx + T x has a solution and the set of all solutions is compact

in E.

Proof. Define a mapping Q : X → P cl,cv (X ) by

Q x = Sx + T x. (3.1)

We show that Q satisfies all the conditions of Theorem 2.1. Obviously, by hypothesis (c), Q

defines a mapping Q : X → P cl,cv(X ). Since S and T are closed, the sum Q = S + T is also

closed on X . As hypothesis (a) holds, there is a D-function ψ such that ψ(r) < r for r > 0.

Further, let A be a non-empty subset of X .Then A bounded and

Q(A) ⊆ X and Q(A) ⊆ S(A) + T(A),

and hence Q(A) is bounded. By sublinearity of µ, we obtain

µ(Q(A)) ≤ µ(S(A)) +µ(T(A)) ≤ ψ(µ(A))

where, ψ(r) < r for r > 0. This shows that Q is a nonlinear D-set-contraction on X into itself.

Now an application of Theorem 2.1 yields that Q has a fixed point. Consequently, the operator

equation x ∈ Sx + T x has a solution. This completes the proof.

The following lemma is obvious and the proof may be found in the monographs of Deim-

ling [5] and Hu and Papageorgiou [10].

Lemma 3.1. If Q : E → P c p,cv (E) is nonlinear contraction. Then for any bounded subset A of E

with Q(A) bounded, we have β(Q(A)) ≤ ψ(β(A)), where β is a ball measure of noncompactness

in E defined by (1.2).

Theorem 3.2. Let X be a closed, convex and bounded subset of a Banach space E and let

S, T : X → P c p,cv (E) be two multi-valued operators such that



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Multi-Valued Fixed Point Theory 147

(a) S is a nonlinear D-contraction,

(b) T is compact and closed, and

(c) Sx + T x ⊂ X for all x ∈ X .

Then the operator inclusion x ∈ Sx + T x has a solution and the set of all solutions is compact

in E.

Proof. Since S is nonlinear D-contraction, it is closed on X and there is a D-function ψ of S

on X with the properties that ψ(r) < r for r > 0. Again from Lemma 3.1, it follows that it is a

also nonlinear D-set-contraction with respect to the Hausdorff measure of noncompactness β

and with a D-function ψ on X . Now the desired conclusion follows by a direct application of

Theorem 2.1.

3.2 Nonlinear alternative

The following nonlinear alternative for multi-valued mappings in Banach spaces is well-

known in the literature.

Theorem 3.3 (O’Regan [13]). Let U be a open bounded subset of a Banach space E with 0 ∈ U

and let Q : U → P cl,cv(E) be a compact and closed multi-valued operator. Then either

(i) the operator inclusion x ∈ Q x has a solution in U, or

(ii) there is an element u ∈ ∂U such that λu ∈ Qu for some λ > 1, where ∂U is the boundary

of U in E.

A generalization of above Theorem 3.4 is

Theorem 3.4. Let U be a open bounded subset of a Banach space E with 0 ∈ U and let Q :

U → P cl,cv(E) be a µ-condensing and closed multi-valued operator. Then either

(i) the operator inclusion x ∈ Q x has a solution in U and the set of all solutions is compact

in E, or

(ii) there is an element u ∈ ∂U such that λu ∈ Qu for some λ > 1, where ∂U is the boundary

of U in E.

Proof. The proof is similar to that given for Theorem 3.3 in O’Regan [13]( see also Agarwal et

al. [1]) and now the conclusion follows by an application of Theorem 3.2.



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As a consequence of Theorem 3.4, we obtain

Corollary 3.1. Let Br (0) be a open ball in a Banach space E centered at origin 0 ∈ E of radius

r and let Q : Br (0) → P cl,cv(E) be a µ-condensing and closed multi-valued operator. Then

either

(i) the operator inclusion x ∈ Q x has a solution in Br (0) and the set of all solutions is

compact in E, or

(ii) there is an element u ∈ E such that ‖u‖ = r satisfying λu ∈ Qu for some λ > 1.

Corollary 3.2. Let E be a Banach space and let Q : E → P cl,cv(E) be a µ-condensing and

closed multi-valued operator. Then, either

(i) the operator inclusion x ∈ Q x has a solution and the set of all solutions is compact in E,

or

(ii) the set E = {u ∈ E | λu ∈ Qu} is in unbounded for some λ > 1.

The above Corollary 3.1 includes the following fixed point result due to Martelli [10]

which has been used by several authors in the literature for proving the existence theorems

for differential and integral inclusions.

Corollary 3.3. Let E be a Banach space and let Q : E → P cl,cv (E) be a upper semi-continuous

and α-condensing( or β-condensing) multi-valued operator. Then, either

(i) the operator inclusion x ∈ Q x has a solution in X , or

(ii) the set E = {u ∈ E | λu ∈ Qu} is in unbounded for some λ > 1.

Similarly, we can apply Theorem 3.4 to obtain the following nonlinear alternatives for

sum of the two multi-valued operators in Banach spaces.

Theorem 3.5. Let U be a open bounded subset of a Banach space E with 0 ∈ U and let S, T :

U → P cl,cv(E) be two multi-valued operators such that

(a) S is closed and nonlinear D-set-contraction, and

(b) T is compact and closed.

Then, either

(i) the operator inclusion x ∈ Sx + T x has a solution in U and the set of all solutions is

compact in E, or



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Multi-Valued Fixed Point Theory 149

(ii) there is an element u ∈ ∂U such that λu ∈ Su + T u for some λ > 1, where ∂U is the

boundary of U in E.

Theorem 3.6. Let U be a open bounded subset of a Banach space E with 0 ∈ U and let S, T :

U → P c p,cv (E) be two multi-valued operators such that

(a) S is nonlinear D-contraction, and

(b) T is compact and closed.

Then, either

(i) the operator inclusion x ∈ Sx + T x has a solution in U and the set of all solutions is

compact in E, or

(ii) there is an element u ∈ ∂U such that λu ∈ Su + T u for some λ > 1, where ∂U is the

boundary of U in E.

Corollary 3.4. Let Br (0) be a open ball in a Banach space E centered at origin 0 ∈ E of radius

r and let S, T : B r (0) → P c p,cv (E) be two multi-valued operators such that

(a) S is nonlinear D-contraction, and

(b) T is compact and closed.

Then, either

(i) the operator inclusion x ∈ Sx + T x has a solution in Br (0) and the set of all solutions is

compact in E, or

(ii) there is an element u ∈ E such that ‖u‖ = r satisfying λu ∈ Su + T u for some λ > 1.

Corollary 3.5. Let E be a Banach space E and let S, T : E → P c p,cv (E) be two multi-valued

operators such that

(a) S is nonlinear D-contraction, and

(b) T is compact and closed.

Then, either

(i) the operator inclusion x ∈ Sx + T x has a solution and the set of all solutions is compact

in E, or



150 Bapurao C. Dhage
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12, 3 (2010)

(ii) the set E = {u ∈ E | λu ∈ Su + T u} is in unbounded for some λ > 1.

Remark 3.1. Note that our Theorem 3.6 and Corollary 3.4 improve the hybrid fixed point the-

orems for multi-valued mappings proved in Dhage [6, 7] under weaker upper semi-continuity

conditions.

4 The Conclusion

Finally, while concluding, we remark that the multi-valued fixed point theorems of this paper

have some nice applications to differential and integral inclusions for proving the existence

as well as some characterizations of solutions such as global and local asymptotic attractivity

of solutions on bounded and unbounded intervals of real line. The investigations of these and

other similar problems form the scope for further research work in the theory of differential

and integral inclusions under weaker upper semi-continuity conditions. Some of the results

in this direction will be reported elsewhere.

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