**Turkish Journal of Analysis and Number Theory**

## Some Fixed Point Theorems for Multivalued Mappings in Banach Algebras and Application to Integral Inclusions

Higher School of Sciences and Technologies of Hammam Sousse, Street Lamin El Abbassi 4011, Hammam Sousse, TunisiaAbstract | |

1. | Introduction |

2. | Preliminaries |

3. | Fixed Point Theory in Banach Algebras |

4. | Another Direction |

5. | Functional Integral Inclusion |

References |

### Abstract

In this paper, we present new multivalued analogues of the krasnoselskii fixed point theorems, for the sum *AB*+*C*, where the operators *A;B *and *C *are *D**-*set Lipcshitzian with respect to a measure of non-compactness which satisfies condition (*m*). Our results generalize, prove and extend well-known results in the literature. An application to solving non linear integral inclusion is given.

**Keywords:** measure of noncompatness, Banach algebras, condensing multimap, integral equations

Received December 20, 2016; Revised February 27, 2017; Accepted March 20, 2017

**Copyright**© 2017 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Mohamed Boumaiza. Some Fixed Point Theorems for Multivalued Mappings in Banach Algebras and Application to Integral Inclusions.
*Turkish Journal of Analysis and Number Theory*. Vol. 5, No. 2, 2017, pp 69-79. http://pubs.sciepub.com/tjant/5/2/5

- Boumaiza, Mohamed. "Some Fixed Point Theorems for Multivalued Mappings in Banach Algebras and Application to Integral Inclusions."
*Turkish Journal of Analysis and Number Theory*5.2 (2017): 69-79.

- Boumaiza, M. (2017). Some Fixed Point Theorems for Multivalued Mappings in Banach Algebras and Application to Integral Inclusions.
*Turkish Journal of Analysis and Number Theory*,*5*(2), 69-79.

- Boumaiza, Mohamed. "Some Fixed Point Theorems for Multivalued Mappings in Banach Algebras and Application to Integral Inclusions."
*Turkish Journal of Analysis and Number Theory*5, no. 2 (2017): 69-79.

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### 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 **fi**xed 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 **fi**xed 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 Hausdor**ff** measure of noncompactness** ** **de**fi**ned 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** ** **satis**fi**es ** 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 **satis**fi**es** **the condition **(**m**) **if for arbitrary bounded sets** ** **of **X**, the following inequality is satis**fi**ed*

**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 satis**fi**es 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 **fi**xed point*.

**Proof.** Let *x*_{0} 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 Hausdor**ff** 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 le**t ** **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 an**d ** a** subadditive MNC on X which satis**fi**es 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 *L*^{1}*-*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.

(*H*1) The mapping *f *is bounded and there exists a bounded function* ** *such that

(*H*_{2}) The multivalued mapping *F *is* **-*Carathéodory with growth functions* ** *for all *r > *0.

(*H*_{3}) There exists* ** *such that, for all bounded

(*H*4) For all* ** *the mapping* ** *is integrably bounded.

(*H*5) There exists* ** *such that

(*H*_{6}) The function* ** *is continuous with bounded* *

**Theorem 5.1** *Assume that the hypotheses (H*_{1}*)**-**(H*_{6}*) 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 (*H*_{4}),* * 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 (*H*_{1}) and (*H*_{2}), 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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