Existence and Asymptotic Stability of Solutions of a Functional Integral Equation via a Consequence of Sadovskii ’ s Theorem

1 Department of Mathematics, Rzeszów University of Technology, al. Powstańców Warszawy 8, 35-959 Rzeszów, Poland 2Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, Jeddah 21589, Saudi Arabia 3 Department of Mathematics, Faculty of Science, Damanhour University, Damanhour 22511, Egypt 4Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Campus deTafira Baja, 35017 Las Palmas de Gran Canaria, Spain


Introduction
Measures of noncompactness play very important role in nonlinear analysis.They are often applied to the theories of differential and integral equations as well as to the operator theory and geometry of Banach spaces.The concept of a measure of noncompactness was initiated by Kuratowski [1] and Darbo [2].In [2] Darbo, by using the concept of a measure of noncompactness, proved a fixed point theorem.In [3] Sadovskii improved the results obtained in [2].
The purpose of this paper is to present a theorem on the existence and asymptotic stability of solutions of a functional integral equation.Our study will be placed in the Banach space of real functions which are defined, continuous, and bounded on the real half-axis R + .The functional integral equation studied in the paper contains as particular cases a lot of functional and integral equations appearing in the literature.The main tool used in our investigations is a consequence of Sadovskii's fixed point theorem [3].

Notations, Definitions, and Auxiliary Facts
Let  be a given real Banach space with a norm ‖ ⋅ ‖.For a nonempty subset  of  denote by  the closure of  and by Conv the closed convex hull of .For  and  being subsets of , by  +  and ,  ∈ R, we denote the usual algebraic operations on  and .Further, let M  denote the family of all nonempty and bounded subsets of  and N  its subfamily consisting of all relatively compact subsets.If  is a mapping defined on M  with real values, then by ker  we denote the following family: ker  = { ∈ M  :  () = 0} . ( This family will be called the kernel of the mapping .
Following [4], we consider the following definition of the concept of a measure of noncompactness.
Definition 1.A mapping  : M  → R + = [0, ∞) will be called a measure of noncompactness in  if it satisfies the following conditions.
(1) The family ker  is nonempty and ker  ⊂ N  .
In [2] Darbo proved the following fixed point theorem.
Theorem 2. Let Ω be a nonempty, bounded, closed, and convex subset of  and let H : Ω → Ω be a continuous mapping such that there exists a constant  ∈ [0, 1) satisfying for any nonempty subset  of Ω, where  is a measure of noncompactness.
Then H has a fixed point in Ω.

Theorem 3.
Let Ω be a nonempty, bounded, closed, and convex subset of  and let H : Ω → Ω be a continuous mapping such that for any nonempty and noncompact subset  of Ω, where  is a measure of noncompactness in .Then H has a fixed point in Ω.
Notice that in [3] Theorem 3 is proved for a particular measure of noncompactness in , but the same argument serves for an arbitrary measure of noncompactness in  [5,6].
In our study, we will work in the Banach space BC(R + ) consisting of all real, bounded, and continuous functions on R + .This space is furnished with the norm given by the formula In BC(R + ), we will use the measure of noncompactness which appears in [7,8].In order to present this measure of noncompactness, let us fix a nonempty, bounded subset  of BC(R + ) and a number  > 0. For  ∈  and  > 0, we denote by   (, ) the modulus of continuity of the function  on the interval [0, ]; that is, Now, we consider the quantities Further, for a fixed number  ∈ R + , we denote () = {() :  ∈ }.
Finally, the measure of noncompactness  which will be used in our study is defined as where diam () = sup{|() − ()| : ,  ∈ }.In [4], the authors proved that the function  is a measure of noncompactness in BC(R + ).
In order to introduce the concept of asymptotic stability which will be used later, we assume that Ω is a nonempty subset of BC(R + ) and let H : Ω → BC(R + ) be an operator.Also, consider the equation Definition 4. One will say that solutions of ( 8) are locally attractive if there exists a ball   ( 0 ) in BC(R + ) such that, for arbitrary solutions  = () and  = () of ( 8) belonging to In the case when the limit in ( 9) is uniform with respect to the set   ( 0 ) ∩ Ω, that is, when for each  > 0 there exists  > 0 such that      () −  ()     ≤  (10) for all ,  ∈   ( 0 ) ∩ Ω being solutions of (8) and for any  ≥ , one will say that solutions of (8) are asymptotically stable.
We will finish this section with the following generalization of Banach contraction mapping principle due to Geraghty [9] and where the class B of functions  : [0, ∞) → [0, 1) is used satisfying By B 0 we denote the class of functions  : [0, ∞) → [0, 1).
Then H has a unique fixed point in .

Main Result
We start this section with the following result which is a version of Theorem 5 in the context of measure of noncompactness.

Proposition 6.
Let Ω be a nonempty, bounded, closed, and convex subset of a Banach space  and let H : Ω → Ω be a continuous mapping such that  (H) ≤  ( ()) ⋅  () (13) for any nonempty and noncompact subset  of Ω, where  ∈ B 0 and  is an arbitrary measure of noncompactness in .
Then H has at least one fixed point.
Proof.Let  be a nonempty and noncompact subset of Ω.
Since  is an arbitrary nonempty and noncompact subset of Ω, the contractive condition appearing in Theorem 3 is satisfied.Finally, Theorem 3 says that H has a fixed point in Ω.This completes the proof.Now, we present the following result which belongs to the classical metric fixed point theory.
It is easy to see that  is a measure of noncompactness in  [4].Now, we take a nonempty subset  of Ω with () ̸ = 0. Using (14) and the fact that  is nondecreasing, we have When () = diam  = 0, we infer that  is a singleton; thus  is also a singleton.Consequently, (H) = 0. Therefore, ( 16) is also satisfied when () = 0. Since  ∈ B 0 , Proposition 6 gives us the existence of at least one fixed point in Ω.
In order to prove the uniqueness of the fixed point, we take into account Fix H ⊂ ker , since H(Fix H) = Fix H and, consequently, (Fix H) = 0. Finally, since ker  consists of singletons, Fix H is a singleton and this proves the uniqueness of the fixed point.The proof is complete.(b) There exists a continuous and nondecreasing function  : R + → R + with (0) = 0, satisfying

An example of the function 𝛼 appearing in
(c) The function (, , ) =  : R 2 + × R → R is continuous and there exist continuous functions , V : (d) There exists a positive solution  0 of the inequality ()+  ≤ , where Then (17) has at least one solution  ∈ (R + ).Moreover, solutions of (17) are asymptotically stable.
Proof.Let us consider the operator H defined on BC(R + ) as follows: For convenience, we divide the proof into several steps. Step This proves that sup Therefore, H maps BC(R + ) into itself.
Step 2 (H maps   0 into itself).It follows from assumption (d) that H maps   0 into itself.
In the case  1 () = 0 we have that  is a relatively compact subset of BC(R + ) and, since H is continuous, H is also relatively compact and thus  1 (H) = 0.This proves that (35) is also satisfied when  1 () = 0. Summarizing, for any nonempty subset  of   0 , we have where  ∈ B 0 (assumption (b 2 )) and  1 is a measure of noncompactness in BC(R + ).
On the other hand, in view of (32), we get and, by using induction, where we have used the nondecreasing character of  and where  () denotes the th iteration.Taking into account ( 2 ), since ()/ ∈ B 0 , we have () <  for  > 0 and as  is continuous, it follows that  () () → 0 for  > 0 [10].Therefore, we deduce that lim Now, taking into account Definition 1, we deduce that the set  = ⋂ ∞ =1    0 is nonempty, bounded, closed, and convex.Moreover, since  1 () ≤  1 (   0 ) for any  ∈ N,  is a member of the kernel ker  1 of the measure of noncompactness  1 .Let us also observe that the operator H transforms the set  into itself.
Next, we will prove that H is continuous on the set .To do this let us fix a number  > 0 and we take a sequence (  ) ⊂  and  ∈  such that   → .We have to prove that H  → H.
In fact, since  ∈ ker  1 , we have where we have used the fact that () <  for  > 0 and the nondecreasing character of .
Finally, taking into account that as  ∈ ker  1 and, consequently,  is relatively compact, H :  →  is a continuous operator, applying the classical Schauder fixed point theorem, we infer that the operator H has at least one fixed point in .
In order to prove that solutions of (17) are asymptotically stable, we notice that any solution () of (17) in   0 is a fixed point of H. Now, taking into account that H transforms   0 into itself, we have Since  1 (H) ≤ ( 1 ()) ⋅  1 () for any nonempty subset  of   0 , we have Further, we distinguish two cases.
Case 2 (( 1 (  0 ∩ Fix H)) > 0).In this case, by (43) and taking into account that the range of the function  is [0, 1), we infer Since  1 () = (1/2)() for any nonempty subset , we deduce that (  0 ∩ Fix H) = 0. Taking into account the definition of the measure of noncompactness  (see Section 2), we have lim sup But this means that for any  > 0 we can find As all solutions of (17) in   0 are in   0 ∩ Fix H, by (46) we have that for  > 0 there exists  > 0 such that where ,  ∈   0 and they are solutions of (17).This means that solutions of (17) are asymptotically stable.The proof is complete.

Example
In order to present an example which illustrates our results, we need to prove some properties about the inverse tangent function.
infer that  is concave and, therefore, for any ,  ∈ R + .
An application of Bolzano's theorem gives that this inequality is satisfied by a number  0 ∈ (0, 1).Therefore, assumption (d) of Theorem 8 is satisfied.Finally, by Theorem 8, we conclude that (57) has at least one solution  in BC(R + ) satisfying ‖‖ ≤  0 .

Final Remarks
In [10] the authors proved the following result.Now, we compare the classes of functions B 0 and A appearing in Proposition 6 and Theorem 18, respectively.To do this, we need the following lemma which appears in [10] under weaker assumptions.For the paper to be selfcontained, we present a proof.Proof.(a)⇒(b) Suppose that the conclusion is not true.This means that we can find  0 > 0 such that ( 0 ) ≥  0 .Since  is nondecreasing, we obtain   ( 0 ) ≥  0 > 0 for any  = 1, 2, . . .and the sequence {  ( 0 )} is nondecreasing.Therefore lim  → ∞   ( 0 ) ≥  0 > 0 and this contradicts (a).
of Ω with () > 0. When () = 0, this means that  is a relatively compact subset of  and, since  is continuous,  is also relatively compact subset of  and, therefore, () = 0. Consequently, condition (70) is satisfied for any nonempty subset  of Ω.This tells us that Theorem 2.2 of[10]can be reformulated in the following way.Theorem 18.Let Ω be a nonempty, bounded, closed, and convex subset of a Banach space  and let  : Ω → Ω be a continuous operator satisfying () ≤  ( ()) ⋅  () , (71)for any nonempty subset  of Ω, where  is an arbitrary measure of noncompactness and  belongs to the class A of functions  : R + → R + with () = ()/, where  : R + → R + is a nondecreasing function such that lim  → ∞   () = 0 for each  ∈ R + .Then  has at least one fixed point in Ω.
for any nonempty subset  of Ω, where  is an arbitrary measure of noncompactness and  : R + → R + is a nondecreasing function such that lim  → ∞   () = 0 for each  ∈ R + , where   denotes the -iteration of .Then  has at least one fixed point in Ω.