1. Introduction

The study of the linear integral equation in Banach algebra was initiated by Dhage via fixed point theorems ^{[14, 16, 17]}. Recently, many authors are interested in the study of non linear integral equations and integral inclusions, in Banach algebras, by using the argument of measure of noncompactness ^{[1, 2, 5, 6, 7, 13, 15, 18, 19]}. Given a Banach algebra X and a closed bounded interval J = [0, a] in IR, we consider the following functional integral inclusion

(1)

where and The functional integral inclusion (1) can be transformed to the following fixed point problem

where A, B and C are multivalued operators acting on the Banach algebra C(J, X). Note that a particular case of (1) has been studied by B.C Dhage ^[11] where X is the Banach algebra IR.

In order to study problem (1), we look at the nonlinear equation

where S is a non empty closed convex subset (not necessarily bounded) of a Banach algebra X and A, B, C are upper semi-continuous or closed multivalued mappings acting on X.

In ^[11], by using a propriety of the Hausdorff measure of noncompatness, Dhage has examined the nonlinear equation where A and B are multivalued operators acting on a Banach algebra. In ^[8], authors introduced a new class of multivalued mappings of the form where A and C are multivalued operators acting a on Banach algebra. Using this concept, they presented some fixed point theorems for the operator AB + C in weak topology setting.

The present paper is organized as follows. In section 2, we give some useful preliminaries and definitions which will be needed in the sequel. In section 3, by using the propriety (m) of a measure of non compactness on X, we present some hybrid fixed point theorems for the operator AB + C. For this, we use the propriety of D-set lipschitzian with respect to for the multivalued operators A, B and C. On another direction, in section 4, we use the class of multivalued operator to prove some fixed point theorems for the sum AB +C. This is based on the propriety of -condensing of the operato for any measure of non compactness on X. Our results generalize, prove and extend well known results, for single valued mappings ^{[13, 14, 15, 16, 19]} and for multivalued mappings ^[11].

Section 5 is devoted to establishing an existence theorem for the nonlinear integral inclusions (1), where X is a separable Banach algebra, as a consequence of our Theorem 3.2.

2. Preliminaries

Throughout this paper, Let X be a Banach space endowed with the norm For convenience let

We denote by B(x, r) the closed ball centered at x with radius r. For a subset S of X, we write , convS, and to denote the closure, the convex hull and the closed convex hull of the subset S, respectively. We write to denote the convergence with respect to the norm of X.

Let X and Y be two Banach spaces. A correspondence is called a multivalued operator or a multivalued mapping on X into Y. For any subset A of X, define

The range of T is the image T(X) of X. We can identify T with its graph GrT the subset of X × Y given by

A point is called a fixed point of T if

Definition 2.1 Let be a multivalued mapping.

- We say that T is upper semi-continuous if is an open set in X, for all open subset U of Y.

- We say that T is closed if its graph is closed in In other words, if be a sequence in GrphT such that then we have that

- We say that un element x in X is a fixed point of T if

- We say that T is compact if its range T(X) is relatively compact in Y.

- We say that T is completely continuous if it is upper semi-continuous and, for all bounded subset of X, is relatively compact.

In the following proposition we recall some essential properties of upper semi-continuous and closed multivalued mappings in Banach spaces.

Proposition 2.1 ^[26]. Let X and Y be Banach spaces, and let be a multivalued mapping.

1) If T is upper semi-continuous and compact-valued, then for each compact subset K of X, T(K) is compact.

2) If T is upper semi-continuous and compact-valued, then T is closed.

3) If Y is compact, then T is closed if and only if it is upper semi-continuous and closed-valued.

4) If T is compact-valued, then T is upper semi-continuous if and only if for all sequence in GrphT such that the sequence has a limit point in T(x).

We recall the KaKutani-Fan fixed point principle for upper semi-continuous multivalued mappings.

Theorem 2.1 ^[20]. Let K be a compact convex subset of a Banach space E and let be an upper semi-continuous multivalued mapping. Then T has a fixed point.

We cite the following generalizations of the KaKutani-Fan fixed point principle.

Theorem 2.2 (Bohmenblust-Karlin, see ^[21])) Let S be a closed, bounded and convex subset of a Banach space E and let be an upper semi-continuous multivalued mapping with compact range. Then T has a fixed point.

Theorem 2.3 (O'Regan ^[24]). Let S be a closed, bounded and convex subset of a Banach space E and let be closed compact multivalued mapping. Then T has a fixed point.

The compactness of the operator T is weakened to condensing operator by using the notion of measure of noncompactness in Banach spaces.

Definition 2.2 A function is said to be a measure of noncompactness on X (MNC, for short), if it satisfies the following properties:

1. for all bounded subsets

2. Monotonicity: For any bounded subsets of X we have

3. Nonsingularity: for all bounded set of

4. if and only if is relatively compact in X.

5. If is a decreasing sequence of sets in satisfying then the limiting set is non empty.

The MNC is said to be positive homogeneous provided

The MNC is said to be subadditive if

The above notion is a generalization of the Hausdorff measure of noncompactness defined on each bounded set of X by

It is well known that is homogenous, subadditive and satisfies the set additivity property. The details of measure of noncompactness appear in Deimling ^[10] and Zeidler ^[25].

Definition 2.3 Let be a MNC on X, we say that the multivalued mapping is -condensing, if for any is bounded and for .

We say that the multivalued mapping is a -set-Lipschitzian (with respect to ) if there exists a continuous nondecreasing function with such that

for all bounded subset with If we say that T is a k-set-Lipschitzian and if k < 1, then T is called a k-set-contraction. If for r > 0, then T is called a nonlinear -set-contraction.

For any let

For and let

The function defined by

is a metric on and is called the Hausdorff metric on X (see ^[26]). It is clear that for any

Proposition 2.2 ^[26] If then is equivalent to the following assertion: and

Definition 2.4 A multivalued mapping is called -Lipschitzian if there exists a continuous nondecreasing function with such that

for all The function is called a -function of T on X. If for r > 0, then T is called a Lipschitzian multivalued mapping. In particular, if K < 1, then T is called a multivalued contraction. If satisfies then T is called a nonlinear contraction multivalued mapping with contraction function

3. Fixed Point Theory in Banach Algebras

Definition 3.1 An algebra X is a vector space endowed with an inner operation noted by (.) which is associative and bilinear.

A normed algebra is an algebra endowed with a norm such that

A complete normed algebra is called a Banach algebra.

Definition 3.2 ^[5]. We state that a measure of noncompactness on a Banach algebra X satisfies the condition (m) if for arbitrary bounded sets of X, the following inequality is satisfied

Lemma 3.1 ^[3]. For any bounded subsets and of X, we have

According to Lemma 3.1, the Hausdorff measure of noncompactness satisfies condition (m).

We note that condition (m) was used for the first time in ^[3] for measures of noncompactness defined on the Banach algebra C[a, b].

Example 3.1 Let X = BC(IR₊) the Banach space of continuous and bounded functions on IR₊ equipped with the standard norm Obviously BC(IR₊) has also the structure of Banach algebra with the standard multiplication of functions. For all L > 0 and we pose

According to ^[4], is a measure of noncompactness on BC(IR₊)which satisfies condition (m) on the family of nonnegative functions in BC(IR₊) (see ^[1]).

We begin this section by proving the following fixed point theorem which extends Theorem 2.2 in ^[12].

Theorem 3.1 Let X be a Banach space, a MNC on X and S a closed convex subset of X. Let be a closed an -condensing mapping such that T(S) is bounded. Then T has a fixed point.

Proof. Let x₀ be an arbitrary element in S. We pose

The set since Let We Show that . Clearly L is a closed convex subset of S and Thus, This implies Hence,

Consequently, Hence, As a result

Suppose that we have

Then and consequently L is compact and convex. By Theorem 2.3, the multivalued mapping has a fixed point.

Theorem 3.2 Let S be a closed convex subset of a Banach algebra X and a subadditive MNC on X satisfying condition (m). Let and be multivalued mappings such that:

1. A, B are upper semi-continuous and C is closed,

2. A, B and C are D-set Lipschitzian (with respect to ) with D-function , and respectively,

3. for all is a convex subset of S,

4. A(S), B(S) and C(S) are bounded.

Then the equation has at least one solution provided

Proof. Let

Since A, B have compact values and C has closed values, assumption 3) guarantees that T is well defined. We show that T has closed graph. Let and

Let such that and Since A and B are upper semicontinous, by Proposition 2.1 we can suppose that It yields that Since C has closed graph, we get then Hence AB + C has closed graph. Let be a bounded subset of S such that It is clear that is bounded and we have

Thus, the multivalued mapping AB + C is -condensing. From Theorem 3.1, AB + C has a fixed point.

If we suppose that the multivalued mapping B is completely continuous, we obtain the following result.

Theorem 3.3 Let S be a closed convex subset of a Banach algebra X and a subadditive MNC on X satisfying condition (m). Let and be multivalued mappings such that:

1. A is upper semi-continuous and C is closed,

2. A and C are D-set Lipschitzian (with respect to ) with D-functions and , respectively,

3. B is completely continuous,

4. for all is a convex subset of S,

5. A(S), B(S) and C(S) are bounded.

Then the equation has at least one solution provided

Proof. As in the proof of Theorem 3.2, the operator

is well defined and has a closed graph. Let be a bounded subset of S such that Since B is completely continuous and we have

Then T is -condensing and the proof is concluded by Theorem 3.1.

In the following result, we interested in the case where A, B and C are D-Lipschitzian. We need the following lemmas which are essential for the proof.

Lemma 3.2 Let X be a Banach space, S be a non-empty subset of X and be a -Lipschitzian multivalued mapping with -function Then for any bounded subset of S, is bounded.

Proof. Let be a bounded subset of S. Then there is constant r > 0 such that for all Fix Since T is -Lipschitzian, for all we have

Hence is bounded.

Lemma 3.3 Let X be a Banach space and let be a D-Lipschitzian multivalued mapping with D-function . Then T is upper semi-continuous.

Proof. Let be a sequence in X converging to a point and let such that for all integer n. We have

Then Let there exists an integer N such that, for all This implies that

and consequently is relatively compact. Then there exists a subsequence which converges to According to Proposition 2.1, T is upper semi-continuous.

Lemma 3.4 Let X be a Banach space and be a -Lipschitzian multivalued mapping with a -function Then T is -set-Lipschitzian with respect to the Hausdorff measure of noncompactness.

Proof. Let be a bounded subset of X. From Lemma 3.2, is bounded. Let r > 0 such that there exits a finite subset of such that Let there exists such that Since T is D-Lipschitzian, we have

According to Proposition 2.2, Consequently

On the other hand, from Lemma 3.3 T is upper semi-continous and, by Proposition 2.1, is compact. For each there exists in such that

Consequently,

That is for each Letting and by the continuity of , we deduce that Hence T is D-set Lipchitzian with D-function

Now we are ready to prove the following result.

Theorem 3.4 Let S be a closed convex subset of a Banach algebra X. Let be multivalued mappings such that:

1. A, B and C are D-Lipschitzian with D-functions and respectively,

2. for all is a convex subset of S,

3. A(S), B(S) and C(S) are bounded.

Then the equation has at least one solution provided

Proof. From Lemma 3.3, the mappings A, B and C are upper semi-continuous, in particular, by Proposition 2.1, the operator C is closed. Further, by Lemma 3.4, the mappings A, B and C are D-set Lipschitzian with respect to . Since the measure satisfies condition (m), all assumptions of Theorem 3.2 are satisfies and the proof is concluded.

The following result is a direct consequence of Theorem 3.3 and Lemma 3.4.

Theorem 3.5 Let S be a closed convex subset of a Banach algebra X. Let and three multivalued mappings such that:

1. A and C are D-Lipschitzian with D-functions and , respectively,

2. B is completely continuous,

3. for all is a convex subset of S,

4. A(S), B(S) and C(S) are bounded.

Then the equation has at least one solution provided

In the particular case where A and C are Lipschitzian, we obtain the following corollary which extends Theorem 3.5 in ^[11].

Corollary 3.1 Let S be a closed convex subset of a Banach algebra X. Let be multivalued mappings and let such that:

1. A and C are Lipschitzian with Lipschitz constant and , respectively,

2. B is completely continuous,

3. for all is a convex subset of S,

4. A(S), B(S) and C(S) are bounded.

Then the equation has at least one solution provided

In the particular case where A, B and C are single valued mappings, we obtain the following result which generalizes Theorem 1.4 in ^[19].

Corollary 3.2 Let S be a closed convex subset of a Banach algebra X and let such that:

1. A and C are D-Lipschitzian with D-functions and respectively,

2. B is completely continuous,

3. for all is a an element of S,

4. A(S), B(S) and C(S) are bounded.

Then the equation has at least one solution provided

4. Another Direction

In the following, we introduce the operator for multivalued mappings and we well use it to prove existence theorems of the equation .

Definition 4.1 ^[8] Let X be a Banach algebra and be multivalued mappings. We say that the mapping is well defined on and we write

Theorem 4.1 Let X be a Banach algebra, a MNC on X and a subadditive MNC on X which satisfies condition (m). Let S be a non-empty closed convex subset of X and let be three multivalued mappings satisfying the following properties:

1. A, B are upper semi-continuous and C is closed,

2. A and C are -set-Lipschitzian (with respect to ) with -functions and , respectively,

3. A(X), C(X) and B(S) are bounded,

4. for all y is convex and

5. is -condensing.

Then the equation has at least one solution in S if

Proof. Fix and let Consider

Since A(x) is compact and C(x) is closed, it is clear that is well defined. We claim that has closed graph. Let be a sequence converging to and such that with There exist and such that Since A is upper semi-continuous and has compact values, according to Proposition 2.1, there exists a subsequence, we note also such that Consequently Since C has closed graph, we deduce that which implies that Hence has closed graph.

We show that is -condensing. Fixing a bounded subset of X such that It is clear that is bounded. Since is subadditive and satisfies condition (m), we have

So, is -condensing. Now all assumptions of Theorem 3.1 are satisfied for the operator then there exists such that Thus and, so, Consequently the multivalued mapping is well defined on B(S). Note that, for all is equivalent to By assumption 4), we deduce that

is well defined. We show that T has closed graph. Let be a sequence converging to and such that with We have then where , and Since A and B are upper semi-continuous with compact values, by Proposition 2.1, we can suppose that and Then

and so

According to Theorem 3.1, it suffices to verify that T is bounded. In fact

then assumption 3) guarantees that T(S) is bounded and the proof is concluded.

As consequence of Theorem 4.1, we derive the following corollary.

Corollary 4.1 Let X be a Banach algebra, S be a nonempty closed convex subset of X and a MNC on X. Let and be three multivalued mappings satisfying the following properties:

1. B is upper semi-continuous,

2. A and C are -Lipschitzian with -functions and respectively,

3. A(X), C(X) and B(S) are bounded,

4. for all is convex and ,

5. is -condensing.

Then the equation has at least one solution in S if

Proof. By Lemma 3.3 and Lemma 3.4, the multivalued mappings A and C are upper semi-continuous and D-set Lipschitzian with respect to . Further, by Proposition 2.1, C has closed graph. All assumption of Theorem 4.1 are satisfied and the proof is concluded.

In the case where B is completely continuous, we can omit assumption 5) in Theorem 4.1 and we get the following result.

Theorem 4.2 Let X be a Banach algebra, S be a non-empty closed convex subset of X and a subadditive MNC on X which satisfies condition (m). Let and be three multivalued mappings satisfying the following properties:

1. A is upper semi continuous, B is completely continuous and C have closed graph,

2. A and C are -set-Lipschitzian (with respect to ) with D-functions and , respectively,

3. A(X), C(X) and B(S) are bounded,

4. for all is convex and . Then the equation has at least one solution in S if

Proof. As in the proof of Theorem 4.1, the operator

is well defined and has closed graph. We show that T is -condensing. Let N be a bounded subset of S, it is clear that T(N) is bounded. Further, we have

Then

Hence T is -condensing.

The following result is a direct consequence of Theorem 4.2, Lemma 3.3 and Lemma 3.4.

Corollary 4.2 Let X be a Banach algebra and S be a non-empty closed convex subset of X. Let and be three multivalued mappings satisfying the following properties:

1. B is completely continuous,

2. A and C are -Lipschitzian with -functions and , respectively,

3. A(X), C(X) and B(S) are bounded,

4. for all is convex and . Then the equation has at least one solution in S if

In the particular case where A, B and C are single valued mappings, we obtain the following result which extends Theorem 1.5 in ^[19], Theorem 2.1 in ^[15] and Theorem 2.3 in ^[16].

Theorem 4.3 Let X be a Banach algebra and S be a non-empty closed convex subset of X. Let and be three mappings satisfying the following properties:

1. B is completely continuous,

2. A and C are -Lipschitzian with -functions and , respectively,

3. A(X), C(X) and B(S) are bounded with

4.

Then the equation has at least one solution in S if

Proof. From Theorem 4.2 it suffices to verify that is a single-valued mapping from S into itself. Let be fixed and consider

Let we have

From a fixed point theorem of Boyd and Wong ^[9], there is a unique element such that x =AxBy+Cx which is equivalent to (here is seen as a multivalued mapping). Moreover, bearing in mind 4) we have that there exists a unique such that Hence is well defined as a single-valued mapping.

Remark 1 Assumption 4) in Theorem 4.3 is satisfied if we suppose that for all Then, we obtain the following corollary which extends and proves a result due to Dhage cited in ^[13] (Theorem 2.3) and proved in ^[14].

Corollary 4.3 Let S be a non-empty closed convex subset of a Banach algebra X and let and be three mappings satisfying: the following properties:

1. B is completely continuous,

2. A and C are -Lipschitzian with -functions and , respectively,

3. A(X), C(X) and B(S) are bounded,

4. for all .

Then the equation x = A(x)B(x) + C(x) has at least one solution in S if

5. Functional Integral Inclusion

In the following, we suppose that is a separable Banach algebra and By a solution of (1) we mean a function that satisfies

for some satisfying and a.e. for

A multivalued mapping is said to be measurable if for any the function is measurable. Further T is said to be integrably bounded if there exists such that a.e for all

For we pose

It is known that this set is nonempty if and only if (see ^[22]). This is the case if T is integrably bounded.

A multifunction is called Carathéodory if

(i) is measurable for each

(ii) is upper semi-continuous a.e. for

A Carathéodory multivalued mapping is called L¹-Carathéodory if for every real number r > 0 there exists a function such that

for all with Denote

To discuss equation (1), we list the following hypotheses.

(H1) The mapping f is bounded and there exists a bounded function such that

(H₂) The multivalued mapping F is -Carathéodory with growth functions for all r > 0.

(H₃) There exists such that, for all bounded

(H4) For all the mapping is integrably bounded.

(H5) There exists such that

(H₆) The function is continuous with bounded

Theorem 5.1 Assume that the hypotheses (H₁)-(H₆) hold. Suppose that there exists r > 0 such that

and

where Then (1) has a solution in

Proof. For all we pose

We show that operators A, B and C satisfy all assumptions of Theorem 3.2. We pose E = C(J, X) and

Step 1. Let and We have

Since x is continuous, then Ax is also continuous. Hence, the operator is well defined as a single valued mapping. For all we have

Then A is Lipschitzian with constant It is clear that

Step 2. We prove that is well defined and D-Lipschitzian. From assumption (H₄), is non empty and, so, Bx is non empty. Let and for all Since it is clear that

We show that Cx is compact, for all . Let (y_n) be a sequence in Cx such that

Since and is compact, then for all the subset is relatively compact in X. The pointwise topology coincides with the product topology on then

is relatively compact in with respect to the pointwise topology. Hence, there exists a subsequence, for simplicity we note also such that for all For all (h the growth function of G(x)). By the convergent dominate theorem, we get and

We deduce that is relatively compact, for all For all we have

It follows that the family is equicontinuous. By Ascoli theorem's, we deduce that is relatively compact in Then, there exists a subsequence which converge uniformly to On the other hand, for all is closed, so Then, and Hence Cx is compact.

We show that C is D-Lipshitzian. Let and such that

Since

by Proposition 2,2 there exists such that for all and

Further the mapping w is measurable (see ^[18]). We pose we have

It follows that By Proposition 2.2, we deduce that

Hence the multivalued mapping C is Lipschitzian with constant By Proposition 2.1 and Lemma 3.3, we deduce that C has closed graph. On the other hand, for all we have

For all with and we get

Step 3. Since the multivalued map F satisfies (H₁) and (H₂), then the multivalued operator B is upper semiccontinuous with compact values (see ^[23], Theorem 5.1.2 and corollary 5.1.2). We show that the multivalued mapping B is Lipschitzian, with respect to the Hausdorff measure of nocompactness on C(J, X), also noted . Let be a subset of S and where

for some For all we have

Then, the subset is equicontinuous in By the properties of , we have

The multivalued mapping is integrably bounded with growth function Further, we have

Hence, the multivalued operator B is lipschitzian with respect to .

For all we have

Then

Step 4. We show that Ax.Bx + Cx is a convex subset of S, for each . Let

and

where and

Then, for all we have

Since and are convex subsets of X, we get

where and

Let with

Then

Since

By Lemma 3.3 and Lemma 3.4, the multivalued mappings A;B and C satisfy all the conditions of Theorem 3.2 and equation has a solution in C(J, X).

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