On Attractivity and Positivity of Solutions for Functional Integral Equations of Fractional Order

f 3 (t, s, x(γ 1 (s)), x(γ 2 (s)))ds: sufficient conditions for the existence, global attractivity, and ultimate positivity of solutions of the equations are derived. The main tools include the techniques of measures of noncompactness and a recent measure theoretic fixed point theorem of Dhage. Our investigations are placed in the Banach space of continuous and bounded real-valued functions defined on unbounded intervals. Moreover, two examples are given to illustrate our results.


Introduction
Nonlinear functional integral equations with bounded intervals have been studied extensively in the literature as regards various qualitative properties.This includes existence, uniqueness, stability, boundedness, monotonicity and extremality of solutions.But the study of nonlinear functional integral equations with unbounded intervals is relatively rare and exploited for the characteristics of attractivity and asymptotic attractivity of solutions.There are two methods for dealing with these characteristics of solutions, namely, classical fixed point theorems involving the hypotheses from analysis and topology and the fixed point theorems involving the use of measure of noncompactness.Each one of these methods has some advantages and disadvantages over the others [1][2][3][4][5][6][7][8][9][10][11].
The aim of this paper is to study the existence, global attractivity, and positivity of solutions for a functional integral equation of fractional order.The mentioned equation has rather general form and contains, as particular cases, a lot of fractional functional equations and nonlinear fractional integral equations of Volterra type.The main technique used in our considerations is the measures of noncompactness and a fixed point theorem of Dhage [3].Our investigations will be situated in the Banach space of real functions which are defined, continuous, and bounded on the right-hand real half axis R + .
The measures of noncompactness used in this paper allow us not only to obtain the existence of solutions of the mentioned fractional functional integral equations but also to characterize those solutions in terms of global attractivity and positivity on unbounded intervals.This assertion means that all possible solutions of the equations in question are globally uniformly attractive and positive in the sense which will be defined further on.
It is worthwhile mentioning that the novelty of our approach consists mainly in the possibility of obtaining the global attractivity, asymptotic attractivity, and positivity of solutions for the considered fractional functional  (, )} (1) satisfies all the conditions of a metric on P() and is called a Hausdorff-Pompeiu metric on , where (, ) = inf{‖ − ‖ :  ∈ }.It is known that the hyperspace (P cl (),   ) is a complete metric space.
The auxiliary way of defining the measures of noncompactness has been adopted in several papers in the literature; see Akhmerov et al. [39], Appell [40], Banaś and Goebel [41], in the works Väth [42] and the references therein.In this paper, we adopt the following axiomatic definition of the measure of noncompactness in a Banach space given by Dhage [3].The other useful forms appear in the work of Banaś and Goebel [41] and the references therein.
Before giving definition of measure of noncompactness, we need the following definitions.
Now we are equipped with the necessary details to define the measures of noncompactness for a bounded subset of the Banach space .
Definition 3 (see [43]).A function  : P  () → R + is called a measure of noncompactness if it satisfies 1  0 ̸ =  −1 (0) ⊂ P rcp (), 2  () = (), where  is the closure of , 3  () = (Conv()), where Conv() is the convex hull of  and 4   is nondecreasing, and 5  if {  } is a decreasing sequence of sets in P bd () such that lim  → ∞ (  ) = 0, then the limiting set  ∞ = lim  → ∞   = ∩ ∞ =0   is nonempty.The family ker  described in 1  is said to be the kernel of  and ker  = { ∈ P bd () |  () = 0} ⊂ P rcp () . ( A measure  is called complete or full if the kernel ker  of  consists of all possible relatively compact subsets of .Next, a measure  is called sublinear if it satisfies There do exist the sublinear measures of noncompactness on Banach spaces .Indeed, the Kuratowskii and Hausdorff measures of noncompactness are sublinear in .A good collection of different types of measures of noncompactness appears in Appell [40].
Observe that the limiting set  ∞ from 5  is a member of the family ker .In fact, since one infers that ( ∞ ) = 0.This yields that  ∞ ∈ ker .This simple observation will be essential in our further investigations.Now we state a key fixed point theorem of Dhage [3] which will be used in the sequel.Before stating this fixed point result, we give a useful definition.Definition 4 (see [43]).A mapping  :  →  is called -set-Lipschitz if there exists a continuous nondecreasing function  : R + → R + such that (()) ≤ (()) for all  ∈ P bd () with () ∈ P bd (), where (0) = 0. Sometimes we call the function  to be a -function of  on .In the special case, when () = ,  > 0,  is called a -set-Lipschitz mapping, and if  < 1, then  is called a set-contraction on .Further, if () <  for  > 0, then  is called a nonlinear -set-contraction on .
Theorem 5 (see, Dhage [43]).Let  be a nonempty, closed, convex, and bounded subset of a Banach space  and let  :  →  be a continuous and nonlinear -set-contraction.Then  has a fixed point.Remark 6. Denote by Fix() the set of all fixed points of the operator  which belong to .It can be shown that the set Fix() existing in Theorem 5 belongs to the family ker .In fact if Fix() ∉ ker , then (Fix()) > 0 and (Fix()) = Fix().Now from nonlinear -set-contraction it follows that ((Fix())) ≤ ((Fix())) which is a contradiction, since () <  for  > 0. Hence, Fix() ∈ ker .
Our further considerations will be placed in the Banach space (R + , R) consisting of all real functions  = () defined, continuous, and bounded on R + .This space is equipped with the standard supremum norm ‖‖ = sup{|()| :  ∈ R + }.
For our purposes we will use the Hausdorff or ball measure of noncompactness in (R + , R).A handy formula for Hausdorff measure of noncompactness useful in application is defined as follows.Fix a nonempty and bounded subset  of the space (R + , R) and a positive number .For  ∈  and  > 0, denote by   (, ) the modulus of continuity of the function  on the closed and bounded interval [0, ] defined by Next, put It is known that   0 is a measure of noncompactness in the Banach space ([0, ], R) of continuous and real-valued functions defined on a closed and bounded interval [0, ] in R which is equivalent to Hausdorff or ball measure  of noncompactness in it.In fact, one has () = (1/2)  0 () for any bounded subset  of ([0, ], R) (see Banaś and Goebel [41] and the references therein).Finally, define  0 () = lim  → ∞   0 ().Now, for a fixed number  ∈ R + , denote Finally, consider the functions 's defined on the family P cl,bd () by the formulas Let  > 0 be fixed.Then for any Define the functions  ad ,  bd ,  cd : P bd () → R + by the formulas for all  ∈ P cl,bd ().In order to introduce further concepts used in this paper, let us assume that  = (R + , R) and let Ω be a subset of .Let  :  →  be an operator and consider the following operator equation in : Below we give different characterizations of the solutions for ( 14) on R + .Definition 8. We say that solutions of ( 14) are locally attractive if there exists a closed ball   ( 0 ) in the space (R + , R) for some  0 ∈ (R + , R) such that for arbitrary solutions  = () and  = () of ( 14) belonging to   ( 0 ) ∩ Ω one has that lim In the case when the limit ( 15) is uniform with respect to the set   ( 0 ) ∩ Ω, that is, when for each  > 0 there exists  > 0 such that for all ,  ∈   ( 0 ) ∩ Ω being solutions of ( 14) and for all  ≥ , we will say that solutions of ( 14) are uniformly locally attractive on R + .
Definition 9.The solution  = () of ( 14) is said to be globally attractive if (15) holds for each solution  = () of ( 14) on Ω.In other words, we may say that solutions of ( 14) are globally attractive, if for arbitrary solutions () and () of ( 14) on Ω, the condition (15) is satisfied.In the case when the condition ( 15) is satisfied uniformly with respect to the set Ω, that is, if for every  > 0 there exists  > 0 such that the inequality ( 16) is satisfied for all ,  ∈ Ω being the solutions of ( 14) and for all  ≥ , we will say that solutions of ( 14) are uniformly globally attractive on R + .
The following definitions appear in the work of Dhage [7].
Definition 10.A line () = , where  is a real number, is called an attractor for a solution  ∈ (R + , R) to ( 14) if lim  → ∞ [() − ] = 0.In this case the solution  to ( 14) is also called to be asymptotic to the line () =  and the line is an asymptote for the solution  on R + .Now we introduce the following definitions which are useful in the sequel.
Definition 11.The solutions of ( 14) are said to be globally asymptotically attractive if for any two solutions  = () and  = () of ( 14), the condition ( 15) is satisfied, and there is a line which is a common attractor to them on R + .In the case when condition (15) is satisfied uniformly, that is, if for every  > 0 there exists  > 0 such that the inequality ( 16) is satisfied for all  ≥  and for all ,  being the solutions of ( 14) and having a line as a common attractor, we will say that solutions of ( 14) are uniformly globally asymptotically attractive on R + .Definition 12.A solution  of ( 14) is called locally ultimately positive if there exists a closed ball In the case when the limit ( 17) is uniform with respect to the solution set of the operator equation (14), that is, when for each  > 0 there exists  > 0 such that for all  being solutions of ( 14) and for all  ≥ , we will say that solutions of ( 14) are uniformly locally ultimately positive on R + .
Definition 13.A solution  ∈ (R + , R) of ( 14) is called globally ultimately positive if (17) is satisfied.In the case when the limit ( 17) is uniform with respect to the solution set of the operator equation (14) in (R + , R), that is, when for each  > 0 there exists  > 0 such that ( 18) is satisfied for all  being solutions of ( 14) and for all  ≥ , we will say that solutions of ( 14) are uniformly globally ultimately positive on R + .
Remark 14.Note that the global attractivity and global asymptotic attractivity imply, respectively, the local attractivity and local asymptotic attractivity of the solutions for the operator equation ( 14) on R + .Similarly, global ultimate positivity implies local ultimate positivity of the solutions for the operator equation ( 14) on unbounded intervals.However, the converse of the above two statements may not be true.
A few details of ultimate positivity are given in the work of Dhage [44].
Finally, we introduce the concept of the fraction integral and the Riemann-Liouville fractional derivative.
Definition 15 (see [45,46]).The fractional integral of order  > 0 with the lower limit  0 for a function  is defined as provided the right-hand side is pointwise on [ 0 , ∞), where Γ() is the Gamma function.
Definition 16 (see [45,46]).The Riemann-Liouville derivative of order  > 0 with the lower limit  0 for a function  : [ 0 , ∞) → R can be written as The first and maybe the most important property of the Riemann-Liouville fractional derivative is that for  >  0 and  > 0, one has   (  ()) = (), which means that the Riemann-Liouville fractional differentiation operator is a left inverse to the Riemann-Liouville fractional integration operator of the same order .
In the following section we prove the main results of this paper.

Attractivity and Positivity Results
In this section we will investigate the following functional integral equation of fractional order with deviating arguments: where , and Γ() denotes the Gamma function.Equation ( 21) has rather general form, when Equation ( 21) reduces to the following quadratic Volterra integral equation of fractional order Equation ( 23) has been studied in the work of Banaś and O'Regan [47] for the existence and local attractivity of solutions via classical hybrid fixed point theory, when Equation ( 21) reduces to the following functional integral equation of fractional order considered in Balachandran et al. [48] for the local attractivity of solutions Therefore, ( 21) is more general and contains as particular cases a lot of fractional functional equations and nonlinear fractional integral equations of Volterra type.By a solution of ( 21) we mean a function  ∈ (R + , R) that satisfies (21), where (R + , R) is the space of continuous real-valued functions defined on R + .
Equation ( 21) will be considered under the following assumptions.
( 2 ) the function  1 : R + × R × R → R is continuous and there exists a bounded function ℓ : R + → R with bound  and a positive constant  such that for all  ∈ R + and  1 , for all  ∈ R + and ( 5 ) the function  3 : R + ×R + ×R×R → R is continuous and there exists a function  : R + → R + being continuous on R + and a function  : R + → R + being continuous and nondecreasing on R + with (0) = 0 such that for all  ∈ R + and  1 ,  2 ,  1 ,  2 ∈ R.
( 6 ) the functions , , ,  : R + → R + defined by the formulas are bounded on R + and the functions (), (), (), () vanish at infinity, that is, lim Keeping in mind assumption ( 6 ), define the following finite constants: Now we formulate the last assumption.
for all  ∈ R + and  1 ,  2 ,  1 ,  2 ∈ R, where  ≤ , and the function ℓ is defined as in hypothesis ( 2 ) which further yields the usual Lipschitz condition on the function  1 , for all  ∈ R + and  1 ,  2 ,  1 ,  2 ∈ R provided  < .As mentioned in the work of Dhage [11], our hypothesis ( 2 ) is more general than that existing in the literature.Now, consider the operators , , , and  defined on the space (R + , R): Then one has the following lemma.
Lemma 18.Under the above assumptions the operator  transforms the ball   0 in the space (R + , R) into itself.Moreover, all solutions of (21) belonging to the space (R + , R) are fixed points of the operator .
Proof.Observe that for any function  ∈ (R + , R),  and  are continuous on R + .We show that the same holds also for .Take an arbitrary function Without loss of generality one can assume that  1 <  2 .Then, in view of imposed assumptions, one has where Then, keeping in mind the estimate (37) one obtains From the above inequality one can infer that the function  is continuous on the interval [0, ] for any  > 0. This yields the continuity of  on R + .Finally, combining the continuity of the functions , , and , one deduces that the function  is continuous on on R + .Now, taking an arbitrary function  ∈ (R + , R), then, using our assumptions, for a fixed  ∈ R + , one has Linking this estimate with assumption ( 7 ), one deduces that there exists  0 > 0 such that the operator  transforms the ball   0 into itself.Finally, let us notice that the second assertion of our lemma is obvious in the light of the fact that the operator  transforms the space (R + , R) into itself.The proof is complete.Now, we are prepared to state and prove our main theorem of this section.Theorem 19.Under the above assumptions (H 0 )-(H 7 ), (21) has at least one solution in the space (R + , R).Moreover, these solutions are globally uniformly attractive on R + .
Proof.In what follows we will consider the operator  as a mapping from   0 into itself.Now we show that the operator  is continuous on the ball   0 .To do this, fix arbitrarily  > 0 and take ,  ∈   0 such that ‖ − ‖ < . where Moreover, mention that other notations used in the above estimate were introduced earlier.
From the above estimate one can derive the following inequality: Observe that   (, ) → 0,   1 ( 1 , ) → 0,   1 ( 2 , ) → 0 and   1 ( 3 , ) → 0 as  → 0, which is a simple consequence of the uniform continuity of the functions ,  1 ,  2 , and  3 on the sets respectively.Moreover, it is obvious that the constant Thus, linking the established facts with the estimate (46) one gets Now, taking into account our assumptions, for arbitrarily fixed  ∈ R + as well as for  1 ,  2 ,  1 ,  2 ∈  one can deduce the following estimate (cf. the estimate ( 41)-( 44)): In view of assumptions ( 0 ) and ( 6 ) this yields lim sup Further, using the measure of noncompactness   defined by the formula (9) and keeping in mind the estimates (48) and (50), one obtains Since  ≤  in view of assumption ( 2 ), from the above estimate one infers that   () ≤ (  ()), where () = /( + ) <  for  > 0. Hence, apply Theorem 5 to deduce that the operator  has a fixed point  in the ball   0 .On the other hand, from Remark 6 one concludes that the set Fix() belongs to the family ker   .Now, taking into account the description of sets belonging to ker   (given in Section 2) one deduces that all solutions for (21) are globally uniformly attractive on R + .This completes the proof.in [48].
To prove next result concerning the asymptotic positivity of the attractive solutions, we need the following hypothesis in the sequel.Proof.Consider the closed ball   0 in the Banach space (R + , R), where the real number  0 is given as in the proof of Theorem 19, and define a mapping  : (R + , R) → (R + , R) by (36).Then it is shown as in the proof of Theorem 19 that  defines a continuous mapping from the space (R + , R) into   0 .In particular,  maps   0 into itself.Next we show that  is a nonlinear -set-contraction with respect to the measure  ad of noncompactness in (R + , R).We know that, for any Since  ≤  in view of assumption ( 2 ), from the above estimate one infers that  ad () ≤ ( ad ()), where () = /( + ) <  for  > 0. Hence, applying Theorem 5 to deduce that the operator  has a fixed point  in the ball   0 .
Obviously  is a solution of (21).Now, taking into account the description of sets belonging to ker  ad (given in Section 2) one deduces that all solutions of (21) are uniformly globally attractive and ultimately positive on R + .This completes the proof.
Next we prove the global asymptotic attractivity results for (21).We need the following hypotheses in the sequel.
Proof.Consider the closed ball   0 in the Banach space (R + , R), where the real number  0 is given as in the proof of Theorem 19 and define a mapping  :   0 →   0 by (36).Then  is continuous and maps the space (R + , R) and, in particular,   0 into   0 .We show that  is a nonlinear -set-contraction with respect to the measure   of noncompactness in (R + , R).Let  ∈   0 be arbitrary.V () . (59) Further, using the measure of noncompactness   defined by the formula (11) and keeping in mind the estimates ( 48 Since  ≤  in view of assumption ( 2 ), from the above estimate one infers that   () ≤ (  ()), where () = /( + ) <  for  > 0. Hence, apply Theorem 5 to deduce that the operator  has a fixed point  in the ball   0 .
Obviously  is a solution of the fractional functional integral equation (21).Now, taking into account the description of sets belonging to ker   (given in Section 2) one deduces that all solutions of (21) are uniformly globally asymptotically attractive on R + .This completes the proof.Proof.The proof is similar to Theorem 21 with appropriate modifications.Now the desired conclusion follows an application of the measure of noncompactness  cd in (R + , R).This completes the proof.

Applications
In what follows, we show that the assumptions imposed in  (63) Observe that the above equation is a special case of the fractional functional integral equation (21).Indeed, if we put  = 2/3 and

Theorem 23 .
Under the hypotheses of Theorem 22 and (H 8 ),(21) has at least one solution on R + .Moreover, solutions of (21) are uniformly globally asymptotically attractive and ultimately positive on R + .

𝛼 1 (
) =  1 () =  1 () = ,  2 () =  2 () =  2 () = 2,  () = 1 2  − 2 /2 , [41]rk 7. It can be shown as in Banaś and Goebel[41]that the functions   ,   ,   ,  ad ,  bd , and  cd are measures of noncompactness in the space (R + , R).The kernels ker   , ker   , and ker   of the measures   ,   , and   consist of nonempty and bounded subsets  of (R + , R) such that functions from  are locally equicontinuous on R + and the thickness of the bundle formed by functions from  tends to zero at infinity.Moreover, the functions from ker   come closer along a line () =  and the functions from ker   come closer along the line () = 0 as  increases to ∞ through R + .A similar situation is also true for the kernels ker  ad , ker  bd , and ker  cd of the measures of noncompactness  ad ,  bd , and  cd .Moreover, these measures  ad ,  bd , and  cd characterize the ultimate positivity of the functions belonging to the kernels of ker  ad , ker  bd , and ker  cd .The above expressed property of ker   , ker   , ker   , and ker  ad , ker  bd , ker  cd permits us to characterize solutions of the fractional functional integral equations considered in the sequel.