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1Introduction

Sobolev spaces [7] are the classes of functions defined a.e. on with its derivatives in distributional sense for orders in . One of the most important mathematical discoveries of the XXth century was the concept of Sobolev spaces. This theory is essential in the study of nonlinear partial differential equations in modern analysis. In the early 1970s, Muckenhoupt [14] introduced the class of weights, which are common in applications. Following that, many papers and books have been discussed intensively in Sobolev spaces with Muckenhoupt’s weights.

Measures of noncompactness introduced by Kuratowski [12] are functions that measure the degree of noncompactness of sets in complete metric spaces. These functions play an outstanding role in fixed point theory. In 1955, Darbo presented a fixed point theorem [8] using this notion. Furthermore, several interesting papers on the solvability of various integral equations in Sobolev spaces without weights using the measures of noncompactness have been shown; see, for example, [3, 5, 11, 13].

The coupled system of integral equations describe a phenomenon in biological science, physics, electrodynamics, electromagnetic, and fluid dynamics. In the last decades, the problem of existence solutions of these equations has been taking great interest [1, 4, 16–18]. In particular, Nasiri et al. [16] gave an existing result of the following category of Volterra integral equations system:

in . Here is real Banach algebra, and the entries on system (1) satisfy certain conditions. The idea used here is to prove that system (1) has a coupled fixed point with the help of the measure of noncompactness defined by

where is a convex function from into satisfying if and only if , and is a usual measure of noncompactness.

Another method to ensure the existence of solutions of a coupled system of integral equations is to work on some suitable generalized Banach space in the sense of Perov. In [10], the authors extended Darbo’s fixed point theorem on generalized Banach spaces by replacing the set contraction factor with a matrix convergent to zero and the usual measure of noncompactness of a set A with a generalized (vector) measure of noncompactness

(see Definition 8 and Theorem 1 below).

More recently, authors in [15] constructed a new measure of noncompactness on weighted Sobolev spaces , where is weight, and presented the effectiveness of this measure by studying the existence of solution of some nonlinear convolutiontype integral equations using Darbo’s fixed point theorem.

The organization for the rest of this manuscript is as follows: Section 2 is devoted to the presentation of definitions and some auxiliary results regarding the main objects of the monograph. In Section 3, we present existence results with the help of the so-called generalized measure of noncompactness for the following system of the integral equation (SIE):

where the functions are given and verify some conditions. The functional setting of this system is the generalized Banach space . Finally, an example is given to show the effectiveness of the obtained result.

2 Preliminaries

We recall some concepts that are necessary for this paper. So, this section deals with notations, definitions, and auxiliary results of weighted Sobolev spaces, generalized Banach spaces, generalized measures of noncompactness, and fixed point theory. Beginning with a review of the definition of weights, in particular, weights, for more details, we refer the reader to the following monographs: [9, 14].

Definition 1. (See [19].) A weight on is a locally integrable function such that for a.e..

Definition 2. (See [19].) A weight is said to be an weight if there exists a positive constant such that for every ball ,

here is the Lebesgue measure of the ball . The infimum over all such constants is called the constant of . We denote by the set of all weights.

Let be a weight, and let be open. We define the weighted Lebesgue space as the set of measurable functions on such that

Definition 3. (See [19].) Suppose that the weight is in . Then we define the weighted Sobolev space as the set of functions with weak derivatives . The weighted Sobolev space is a Banach space with the norm

Now, define on the partial order relation as follows. Let and . Put and . Then

Formula

Let be a bounded set of , the supremum bound (resp. the infimun bound) of is the vector

Definition 4. Let be a vector space on . By a generalized norm on we mean a map

satisfying the following properties:

• For all , then ,

• for all and, and

• for all .

The pair is called a vector (generalized) normed space. Furthermore, is called a generalized Banach space (in short, GBS) if the vector metric space generated by its vector metric is complete.

Proposition 1. (See [10].) In a GBS, in the sense of Perov, the definitions of convergence sequence, continuity, open subsets, and closed subsets are similar to those for usual Banach spaces.

Let be a generalized Banach space. Throughout this paper and for , and , we denote bythe open ball centered at with radius (resp. ) and by the closed ball centered at with radius (resp. ). If , we simply denote . Finally, we respectively denote by and co the closure and the convex hull of an arbitrary subset of .

Definition 5. A matrix is said to be convergent to zero if

Lemma 1. See [20].) Let . The following affirmations are equivalent:

• 

• The matrix is invertible, and .

• The spectral radius is strictly less than 1.

Definition 6. Let be a GBS, and let K be a subset of . Then is said to be G-bounded if there is a vector . such that for all , and we write

Definition 7.Let be a GBS. A subset of is called G-compact if every open cover of has a finite subcover. is said relatively G-compact if its closure is G-compact.

We denote by () the family of all relatively G-compact subsets of .

Now, we present a definition of an axiomatic measure of noncompactness for generalized Banach spaces similar to that introduced in 1980 by Banas´ and Goebel [6].

Definition 8. Let be a GBS, and let be the family of G-bounded subsets of . A map

is called a generalized measure of noncompactness (for short G-MNC) defined on if it satisfies the following conditions:

• The family is nonempty, and ker.

• Monotonicity: for all .

• Invariance under closure and convex hull: for all.

• Convexity: for all and .

• Generalized Cantor intersection property: if is a sequence of nonempty, closed subsets of such that is G-bounded and and , then the set is nonempry and is G-compact.

Definition 9. Let be a GBS, and let be a G-MNC. A self-mapping : is said to be a -set contractive with respect to if maps G-bounded sets into G-bounded sets and there exists a matrix such that

for every nonempty G-bounded subset of If the matrix converges to zero, then we say that satisfies the generalized Darbo condition.

Theorem 1. (See[10].) Let be a GBS. Then every nonempty G-bounded, closed, convex subset of has the fixed point property for continuous mappings satisfying the generalized Darbo condition.

Theorem 2. (See [15].) Let be a bounded subset of the space . For and , let us denote

where for , and

The function is a measure of noncompactness on the weighted Sobolev space , and moreover,

Proposition 2. The space define a generalized Banach space equipped with the generalized norm

for each . Furthermore, the function :defined as

is generalized measure of noncompactness on .

Definition 10. (See [2].) A function is said to have the Carathéodory property if

• the function is measurable for any ,

• the function is continuous for almost all .

3 Main results

In this section, we study the existence of solutions for the system of integral equation (SIE) (2). Problem (2) will be discussed under the following assumptions:

(H1)

(H.) The functions satisfy the Carathéodory conditions and have the continuous derivatives of order 1 with respect to the first variable, and

• There exists a matrix such that for each and for , we have

  

• There exists a matrix such that for each and for and

  

• There exists a matrix such that for each and for and , we have

  

• For each the functions .

  (H3) The functions satisfy the Carathéodory conditions and have the continuous derivatives of order 1 with respect to the first variable, and

  • There exists a matrix such that for each and for , we have

    

  • There exists a matrix such that for each and for and, we have

    

  • There exists a matrix such that for each and for and, we have

    

• For each , the functions exist, and

Theorem 3. Suppose that assumptions (H1)-(H3) are satisfied. Then the system of integral equation (SIE) (2) has at least one solution in if there is such that

and the matrix

converges to zero. Here

Proof. We define the operator by

The proof will be broken up into several steps.

Step 1. Our first claim is to show that the operator is well defined. Looking that for each , the function is measurable for any . Also, for any and , we have

Then has a measurable derivative. Now, we shall show that for any . Using our hypotheses, for arbitrarily fixed and , we obtain

hence,

Also,

then

thus

this means that the operator maps into .

Keeping in the mind that the vector fulfills inequality (3), thus for all ,

Due to (4), we derive that is a mapping from into .

Step 2. Our claim here is to proof the continuity of To this end, let be a convergence sequence to some in

Then for each ,

On the other hand, we have for each ,

Step 3. The operator is G-set contractive with respect to . Indeed, let be a nonempty and bounded subset of , and let be such that and , and by applying the same procedure of the previous step we get

then

it follows that

thus

Since for each are compact in and is compact in , we have . Then we obtain

Next, let us fix an arbitrary number . Then, taking into account our hypotheses, for an arbitrary function , we have

But for each when , hence,

then

By the same way we find for ,

hence,

so

Now, by combining (5), (6) we find

Formula

Therefore, by the generalized Darbo fixed point Theorem 1 system (2) has at least one solution in .

Example 1. Consider the following coupled functional integral equation:

Then

and we have

and we simply check that

and

Thus,

with

Furthermore,

then we can verify easily that

and

Hence,

It is easy to see that for each satisfies assumption (H2)(d). Since we have . Then we obtain

By the same way we get

and

Furthermore, condition (H2)(d) can be easily verified. Moreover, . Finally, the matrix

has two eigenvalues: . Therefore, converges to zero. All the conditions in Theorem 3 are satisfied, so system (7) has at least one solution in the space , where .

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