Controllability of nonlinear integral equations of Chandrasekhar type

In this paper, we study the controllability of two problems involving the same Chandrasekhar-type integral equation, but under different kinds of controls. A viability condition is imposed as well. We provide existence results of continuous trajectories coupled to continuous controls. Then, in the non-viable case, we investigate the optimal estimates to be taken in view of the existence of solutions for both problems. The last part of the paper deals with the application of the previous results to the classical Chandrasekhar equation, first showing the existence of a viable continuous solution, then providing also uniqueness and approximability. Two examples of controllability problems governed by this equation are given.


Introduction
In the last decades, a great attention has been given to the quadratic integral equation of Chandrasekhar type where a > 0 and w : [0, T ] → R is a continuous function with w(0) = 0. This equation is a first generalization of the quadratic integral equation introduced by Chandrasekhar in [15], which plays an important role in the radiative transfer theory, in the neutron transport theory, and in the kinetic theory of gases (see, e.g., [8,14,17,19,22,24,26]). Equations of type (1) and some of their generalizations were studied in several papers (see, e.g., [2,3,5,6,[9][10][11]23]). This fact is mainly connected to the applications of the mentioned classes of integral equations to the description of several real-world events which appear in bioengineering, mechanics, physics, mathematical physics, porous media, viscoelasticity, control theory and other important branches of sciences and applied mathematics. Our aim in this paper is to study the controllability of two different generalizations of the equation (1).
In the first case, we work considering the nonlinear integral equation with feedback controls x(t) = a + u(t, x(t)) + g(t, x(t)) ϕ(t) 0 k(t, s)f (s, x(s))ds, t ∈ [0, T ], (2) where a ∈ R, g, f : The presence of the controls naturally leads to the use of multivalued analysis tools. In particular, we provide the existence of an admissible pair trajectory-control by means of a selection theorem for multimaps (see [20,Theorem 1.6.21]), and a version of the Daher fixed point theorem involving a measure of noncompactness on the space C([0, T ]) (see [13,Theorem 3.2]).
The second case is about the integral equation where V is a real multimap defined on [0, T ]. In this other setting, we obtain the existence of an admissible pair by applying the classical Michael selection theorem and contraction fixed point theorem.
The two equations differ from the type of the control. Indeed, in equation (2) a feedback control is considered, while in equation (4) the control function v does not depend on the solution trajectory, but it acts on the integral part.
Notice that in both cases, taking ϕ(t) = t or ϕ(t) = T , the integral term becomes a Volterra or a Fredholm integral, respectively.
The paper is organized as follows. In Sect. 2, we collect the necessary preliminary definitions and results. Then, in Sects. 3 and 4, we state the controllability results for the problems (2)-(3) and (4)- (5), respectively, under a viability condition. The solutions obtained are continuous trajectories coupled to controls, continuous as well. In Sect. 5, we study optimal estimates for the existence of solutions for both nonlinear integral Eqs. (2) and (4) in the nonviable case, and, finally, Sect. 6 is devoted to the application of our results to the classical Chandrasekhar equation (1). At first, we show the existence of a viable continuous solution. Then, under different assumptions on the characteristic function w, we provide existence, uniqueness and approximability again of a viable continuous solution. Finally, we achieve optimal estimates on w, allowing us to improve known results in [6] and [9], and extend those in [5], as explained in Remark 6.1. The paper is completed by two examples of controllability problems governed by the classical Chandrasekhar integral equation.

Notations and preliminary results
Let X, Y be topological spaces. We will use the following notations: If X is also a vector space, then we put P kc (X) = {Ω ∈ P(X) : Ω compact and convex}.
It is well known that a multimap F : A multimap is lower semicontinuous (l.s.c.) if it is lower semicontinuous at every x 0 ∈ X. For more details, we refer, e.g., to [21].
Let (X, d) be a metric space. If A ⊂ X and x ∈ X, the standard dis- Then, given X, Y two metric spaces, we recall that a multimap F : Let (X, · ) be a Banach space. We recall (see, e.g., [13]) that a function β : P b (X) → R + 0 is said to be a measure of noncompactness (MNC, for short) if β(co(Ω)) = β(Ω) , for every Ω ∈ P b (X), where co(Ω) is the closed convex hull of Ω.
In the sequel, we will consider a measure of noncompactness β verifying the following properties: (β 1 ) regularity: β(Ω) = 0 if and only if Ω is compact; (β 2 ) monotonicity: As examples of measures of noncompactness which satisfy all the previous properties, we recall the Hausdorff and the Kuratowski measures of noncompactness.
To state the fixed point theorem we will use in the following, we also recall (see, e.g., [13]) that a function F : D ⊂ X → P(X) is countably condensing if: In our approach, one of the main tools is the next fixed point theorem, which is a version of the Daher fixed point theorem in [16], holding by virtue of Theorem 3.2 in [13]. From now on we will work in the Banach space X = C([0, T ]) of all the real continuous functions defined on [0, T ], endowed with the norm x = max t∈[0,T ] |x(t)|. In X, we will consider the MNC ω 0 defined as follows (cf. [4]).
Let D be a bounded subset of C([0, T ]). For x ∈ D and ε > 0, let ω(x, ε) be the modulus of continuity of the function x on the interval [0, T ], i.e., Put the MNC ω 0 is defined by It is well known that ω 0 is equivalent to the Hausdorff MNC of the space C([0, T ]) (see, e.g., [1]).

Controllability of a first type nonlinear integral equation
In this Section, we will study the integral equation subject to feedback controls and to the viability condition where g, f : We recall that a problem is said to be controllable if there exists an admissible pair {x, u}, i.e., a continuous function x : [0, T ] → R and a measurable function u : [0, T ] × R → R satisfying (10)- (11).
We will suppose the following hypotheses hold on the control multimap U : (U1) U (t, x) is closed and convex, for every (t, where U (t, x) := sup{|z| : z ∈ U (t, x)}; (U3) U is H-continuous and there exists α To achieve the existence of solutions to the integral inclusion (10), we will use the next result on the existence of a selector for a multimap having the properties of our multimap U ; this result is a particular case of [20, Theorem 1.6.21].
We can now state our controllability result to problem (10)- (12).

be satisfied, and assume that
where K and G are defined in (13). Then there exists an admissible pair {x, u} to problem (10)- (11), with x satisfying the viability condition (12).
Proof. First of all note that U takes values in P kc (R) (see (U1), (U2)); further, U is H-continuous by (U3). Let us consider the multimap U 0 : It is easy to see that U 0 satisfies the hypotheses of the Michael selection theorem. Thus, there exists a continuous function y * : [0, T ] → R such that where L u = 4α < 1. Clearly, by (U2) it is also true that Let us consider now the solution operator Φ : Since ϕ in continuous, using the absolute continuity of the integral and the Lebesgue convergence theorem, we obtain that the right term in the above inequality tends to 0 as n → +∞, that is Φ(x(t n )) → Φ(x(t 0 )) as n → +∞, and Φ(x) is continuous in t 0 . Now, we show that Φ mapsB(0, R) in itself. Indeed, for any t ∈ [0, T ] and x ∈B(0, R) using (17), (g2), (k1), and (f2) we get From (14) we, therefore, obtain Since u, g are uniformly continuous on [0, T ] × [−R, R], we can say that (u(·, x n (·))) n∈N , (g(·, x n (·))) n∈N uniformly converge in [0, T ] to u(·,x(·)) and g(·,x(·)), respectively.
Let D = {x n } n and fix n ∈ N. Taking into account the uniform continuity of the maps u, g, and k, and the Lebesgue integrability of h in (f2), we have that for every ε > 0 there exists δ(ε) ≤ ε such that for every max Now, taking without loss of generality t 1 < t 2 , we obtain the following inequality (recall the monotonicity of ϕ) For each term of the above equation, we have the next estimates: by (16), (7), and (18) similarly, by (g), (7), and (19) and, by (21) too, (26) by (f2) and (20) Hence, by (22) Therefore (recall (8)) Hence, passing to the limit as ε → 0 and using (14), we obtain being L u = 4α < 1. By the arbitrariness of D we get that property (II) of the countably condensivity is proved. Then, we can apply Theorem 2.1 and claim that there exists x ∈B(0, R) such that Φ(x) = x, i.e., x satisfies equation (10) and the viability condition (12). Further, by (15), the pair {x, u} is admissible for problem (10)-(11), with u even continuous.
Remark 3.1. From the proof of Theorem 3.1 it can be observed that condition (14) can be slightly improved as follows: .
then (14) can be replaced by

Controllability for a second-type nonlinear integral equation
In this Section, for given a ∈ R, f, g : , we consider the integral equation subject to the controls and to the viability condition (12), with fixed In this setting, we can apply the classical Michael's selection theorem joined to the fixed point contraction principle, obtaining the uniqueness of solutions as particular case when the multimap come down to a single-valued function.
where K ∞ is the ess sup of k (which is finite by (k2)) and G is defined in (13).
Proof. First of all, notice that by (V1) and (V3) we can apply the Michael's selection theorem, so that there exists a continuous function v : Let us define the function p : and consider the integral equation Of course, a solution of (33) will be also a solution of (28).
We can, therefore, apply the Banach-Caccioppoli fixed point theorem, leading to the existence of a solution to (33), which is unique inB(0, R). Hence, the integral inclusion (28) has at least one solution, lying inB(0, R).
Finally, noticing that the continuous function v satisfies (32), which is exactly the control condition (29), we can say that the couple {x, v} is an admissible pair to problem (28)-(29) with x satisfying the viability property (12).
Remark 4.1. Notice that, analogously to what observed at the end of Theorem 3.1, see Remark 3.1, condition (31) can be slightly improved by the following:

Optimal estimates for the existence of solutions in the nonviable case
Here we first provide two results on the existence of viable solutions for the two types of nonlinear integral equations, naturally suggested by the study developed in Sects. 3 and 4. Namely, from the proof of Theorem 3.1 we immediately have the following result.
Similarly, Theorem 4.1 leads us to the next result.
whereV = max t∈[0,T ] |v(t)|, then there exists a unique continuous map x : [0, T ] → R satisfying the integral equation and the viability condition (12). Moreover, for any x 0 ∈B(0, R), the sequence (x n ) n , uniformly converges to the solution.
Proof. The assertion is a direct consequence of the application of the classical Banach-Caccioppoli Theorem to the solution operator Φ : In case the viability condition is not assigned, then we can ask ourselves if we can choose the constant R in Theorem 5.1 and in Theorem 5.2 so as to optimize the conditions (14) and (38) Let P 1 : [M + |a|, +∞) → R + be the function Since L g , G are continuous, positive and nondecreasing, then P 1 is continuous and lim R→∞ P 1 (R) = 0. Then, as P 1 (M + |a|) = 0, the function P 1 has maximum on [M + |a|, +∞). LetR > M + |a| be a point of maximum for P 1 , i.e.,P Clearly, the choice ofR in (14) gives the optimal condition for K h 1 . This result can be stated as follows.
has maximum on [|a|, +∞), and this maximum gives the optimal condition in (38). PutP the following holds. We point out that, finding the optimal constantsP 1 ,P 2 , it is possible, in some cases, to compare the existence results stated in Theorems 5.3 and 5.4. An example is given in the next section.
Remark 5.1. Let us note that all the existence theorems proved in Sects. 3, 4, 5 extend in a broad sense analogous results proved in [5,6,9] (see for instance next Remark 6.1).

Existence results for the viability problem
Consider the following Chandrasekhar viability problem We are in position to state the following result.
Then there exists a continuous solution x : [0, T ] → R for the Chandrasekhar viability problem (45).
Proof. Consider the setting Finally, property (46) assures that (14) holds. Indeed, we have being G = R and K = max t,s∈[0,T ] |k(t, s)|. Then Theorem 5.1 can be applied, and (45) has a continuous solution.
On the other hand, from Theorem 5.2, also the following result holds. Theorem 6.2. Let a, R ∈ R, R > |a|, and let w : [0, T ] → R be a L 1 -function such that Then the Chandrasekhar viability problem (45) has a unique continuous solutionx : [0, T ] → R such that for every x 0 ∈ R, |x 0 | ≤ R, the sequence (x n ) n defined by It is easy to verify that the function P 1 has on [|a|, +∞) a unique point of maximumR, andR = 2|a|, .
Since w 1 ≤ T max t∈[0,T ] w(t), then for equation (1), Theorem 5.4 gives a more general condition for the existence of a continuous solution, i.e., the following holds.  [5,6,9]. Corollary 6.1 extends the results for the Chandrasekar equation in [5] and improves the (non-viable) existence results in [6] and [9]. Actually, in [6] the existence of a solution is achieved in the space L 1 (I), whereas here we prove the existence of a (viable) continuous solution. Moreover, to prove the existence of a continuous solution, we require w to be a L 1 -function, instead of the more restrictive condition w ∈ L ∞ (I) used in [5] and [6], or w ∈ C(I) assumed in [9]. Finally, condition sup ess [0,1] |w(t)| < 1/4 required in [6] (or max [0,1] |w(t)| < 1/4 in [9]) implies (49). As a significant example, let us consider the characteristic function Clearly, w α ∈ C(I) ∩ L ∞ (I) ∩ L 1 (I) and max hence, whatever α > 0, the existence theorems proved in [6] and [9] do not apply to the Chandrasekar equation governed by w α . Analogously for the result in [5], since On the other hand, Thus, for every α > 3, the characteristic function w α satisfies all the hypotheses required in Corollary 6.1.

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