Application of Krasnoselskii Theorem for Multivalued Operators

Dr. Ch. Shashi Kumar¹, Dr. B. Rami Reddy², Dr. N. Phani Kumar³

¹Vignan Institute of Technology & Science, Deshmukhi, India- 508284.

²Hindu College, Guntur, India - 522033.

³Vignan Institute of Technology & Science, Deshmukhi, India-508224

*Corresponding Author E-mail skch17@gmail.com

ABSTRACT:

The purpose of this paper is to present some existence results of Krasnoselskii type in generalized Banach spaces for multivalued operators. We provide also an application to a Fredholm-Volterra type integral inclusions system and gives an example in support of our main result. 47H10, 54H25, 34B15 fixed point, generalized contraction multivalued operator, integral inclusion, Krasnoselskii’s theorem.

KEY WORDS: Krasnoselskii Theorem, Fredholm-Volterra.

1. INTRODUCTION:

Let be a nonempty set. A mapping is called a vector-valued metric on if the following properties are satisfied:

(a) for all ; if , then ;

(b) for all ;

A set endowed with a vector-valued metric is called generalized metric space. The notions of convergent sequence, Cauchy sequence, completeness, open subset and closed subset are similar to those for usual metric spaces.

We denote by the set of all matrices with positive elements and by the identity matrix. If , and , then, by definition:

Notice that we will make an identification between row and column vectors in .

Let be a generalized metric space in Perov’s sense. For with for each , we denote by

the open ball centered in with radius and by

the closed ball centered in with radius .

For the proof of the main results we need the following theorems.

Theorem 1 Let . The following assertions are equivalents:

(i) is convergent towards zero;

(ii) as ;

(iii) The eigenvalues of are in the open unit disc, i.e , for every

with ;

(iv) The matrix is nonsingular and

(1.1)

(v) The matrix is nonsingular and has nonnegative

elements;

(vi) and as , for each .

Definition 1 ([29]) Let be a generalized metric space. A subset of is called compact if every open cover of has a finite subcover. A set of a topological space is said to be relatively compact if its closure is compact.

Definition 2 ([23]) Let be two normed generalized spaces, and an operator. Then is called:

i) compact, if for any bounded subset we have that is relatively compact (or equivalently is compact);

ii) complete continuous, if is continuous and compact;

iii) with relatively compact range, if is continuous and is relatively compact.

For the case of multivalued operators in a generalized metric space we recall some notions.

If is a multivalued operator then

is the fixed point set of the operator .

Let us consider now the following sets of subsets of a metric space :

If is a normed space, then we denote:

Let ( be a metric space and we define now the following generalized functionals:

(1)

where is called the gap functional between and

In particular, (where ) is called the distance from the point to the set

(2)

In particular is the diameter of set

(3)

where is called the excess functional of over .

(4)

where is called the generalized Pompeiu-Hausdorff functional of and

If is a generalized metric space with , defined as

then we denote by the generalized gap functional on

We will denote by the Pompeiu-Hausdorff functional, generated by , for

We define as

The following lemma is well-known, see for example S. B. Nadler [10].

Lemma 1 Let be a metric space, and Then for every , there exists such that

We have the following extension.

Lemma 2 Let be a generalized metric space, and Then for each , there exists such that

Lemma 3 Let We suppose that there exists such that

(i) for each there is such that

(ii) for each there is such that

Then

If is a generalized metric space, with , then for , we will denote by the closure of with respect to .

In this framework, a set is said to be closed if and only if Thus, a set is closed if and only if for any convergent (with respect to ) sequence its limit belongs to .

We will denote by the set of all nonempty and closed (with respect to ) subsets of .

Lemma 4 Let and . Then iff

Definition 3 ([4]) Let be a generalized metric space, and be a multivalued operator. Then, is called a multivalued contraction in the sense of Nadler if and only if is a matrix convergent to zero and

If are two generalized metric spaces, we recall that a multivalued operator is said to be:

lower semicontinuos (briefly l.s.c.) in if and only if for any open such that , there exists a neighborhood for such that for any , we have that

Theorem 2 ([28]) Let be a generalized Banach space, a nonempty closed bounded convex subset of and such that:

(i) with completely continuous and

a generalized contraction, i.e. there exists a matrix

convergent to zero, such that

for all ;

(ii) for all .

Then has at least one fixed point in .

Theorem 3 ([16]) Let be a generalized Banach space and . Assume that the operators , satisfy the properties:

(i) , for each ;

(ii) is a multivalued contraction mapping in Nadler’s sense

(iii) is l.s.c. and is relatively compact.

Then has a fixed point in

2 Existence results for a system of operatorial inclusions

In this section we will prove Krasnoselskii’s type fixed point theorems in

generalized Banach spaces for multivalued operators.

Let’s consider the following system of operatorial inclusions:

(2.1)

where , ,

, , , , such

that for each

We define

and

where and .

The system (2.1) is equivalent to

where and .

Theorem 4 Let be a generalized Banach space, and , ,

, , , , satisfy the following conditions:

(i) for each

(ii) , are l.s.c and , for all are relatively compact;

(iii)

where is convergent to zero.

Then there exists at least one solution for the system (2.1).

Proof. We apply the fixed point theorem of Krasnoselskii for multivalued operators to the space , endowed with the vector valued norm defined by

The operator is l.s.c. This follows from the fact that , and are l.s.c. From condition (ii) we obtain that is relatively compact, where .

We have to show that the operator and are - contraction (multivalued - contraction in Nadler’s sense). Using the assumption (iii) we get that

and

or equivalently,

for

Thus is a multivalued - contraction in Nadler’s sense.

The invariance condition is satisfied since

and

which guarantee that

Thus we have that

and

or equivalentely

Thus Theorem 3 applies and guarantees the existence in of at least one fixed point for .

3 Applications:

The aim of this section is to present an application of Krasnoselskii’s type fixed point theorem in generalized Banach spaces for a system of integral inclusions.

Theorem 5 Let (with ) be an interval of the real axis and consider the following system of integral equations:

(3.1)

for , where , for

We assume that:

(i) , , , , and are l.s.c. and integrably bounded.

(ii) , and are measurable and integrably bounded.

(iii) there exists a matrix

for each

(iv)

(v)

with

Then, there exists such that the system (3.1) has at least one solution

Proof. Let us denote , , and we define the multivalued operators given by

where

where The system (3.1) can be rewritten as a fixed point equation of the following form

where

and

where

Obviously is a solution for our integral inclusions system if and only if is a fixed point for the operator

We need to show that the multivalued operators , satisfy the assumptions of Theorem 4. We shall prove that and are multivalued - contractions in Nadler’s sense.

Let’s denote , and

Then and

It follows that there are the operators the integrable selections for , and such that

Since

and

we obtain that there exists

such that

Then the multivalued operators defined by

where and

and

have nonempty values and are measurable.

Let be three integrable selections for , and . Then

and

Let’s define , and

It follows that , , and

denotes the Bielecki-type norm on the generalized Banach space

We obtain that

In a similar way we get that

and

These inequalities can be rewritten in a vectorial form

We get that

where

Taking large enough it follows that the matrix is convergent to zero, and thus and are multivalued contractions in Nadler’s sense. By Covitz- Nadler fixed point theorem we get that is a fixed point for

Let with

The operators and are l.s.c. and , and are relatively compact.

We show that we can choose , such that

We have to prove that

We have to show that for each implies that and for each and implies that and

Let , which is equivalent to the fact that there exist , and such that

We will denote by the supremum norm in , where

It follows that

and

Let and such that

We have to show that

Let which is equivalent to It follows that there is a mapping ( and such that

Let , which is equivalent to the fact that there exist , and such that

It follows that

Let and such that

We have to show that

Let which is equivalent to It follows that there is a mapping ( and such that

and

Let and such that

We have to show that

Let which is equivalent to It follows that there is a mapping ( and such that

Since we know that

we get that there exists

such that

Then the multivalued operators

where and

and

have nonempty values and are measurable.

By Kuratowski and Ryll Nardzewski’selection theorem there exist , and

We make the following estimation

Taking the we get that

In a similar way we get that

and

We show now that

i.e.

which is equivalent to, for each there is an element and

and

which is equivalent to, for each and there is an element and

Let , which is equivalent to the fact that there exist , and such that

It follows that

Let such that We have that

Let , which is equivalent to the fact that there exist , and such that

It follows that

Let such that We have that

Let , which is equivalent to the fact that there exist , and such that

It follows that

Let such that We have that

Thus, all the assumptions of Theorem 4 are satisfied. The conclusion follows by Theorem 4.

Next we give an example in support of our Theorem 4.

Example 1 Let be a generalized Banach space and

then

gives

Also all the conditions of Theorem 4 satisfy and system 2.1 has unique solution

Example 2

Let be a generalized Banach space. Consider the system of integral equation 3.1. We take and

then

gives

Also all the conditions of Theorem 4 satisfied and provided a unique solution of system 3.1 .

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Received on 25.06.2019 Modified on 10.07.2019

Research J. Science and Tech. 2019; 11(3):186-200.

DOI: 10.5958/2349-2988.2019.00029.9