Solution Sets for Young Differential Inclusions

The paper deals with some properties of solutions of differential inclusions driven by set-valued integrals of a Young type. The existence of solutions, boundedness, closedness of the set of solutions and continuous dependence type results are considered. These inclusions contain as a particular case set-valued stochastic inclusions with respect to a fractional Brownian motion (fBm), and therefore, their properties are crucial for investigation the properties of solutions of fBm stochastic differential inclusions.

where B + C := {b + c : b ∈ B, c ∈ C} denotes the Minkowski sum of B and C. Moreover, for B, C, D ∈ Conv(R n ) the equality holds, see e.g., [19] for details. We use the notation Let β ∈ (0, 1]. For every function f : R n ⊃ [a, b] n → R n we define By C β ([a, b] n , R n ) we denote the space of β-Hölder-continuous functions with a finite norm It can be shown that C β ([a, b] n , R n ) is a Banach space. Similarly, for a set-valued function F : [a, b] n → Conv(R n ) let A set-valued function F is said to be β-Hölder if F β < ∞. The space of β-Hölder set-valued functions having compact and convex values will be denoted by C β [a, b] n , Conv(R n ) . We say that F : R n → Conv(R n ) is locally β-Hölder if for every [a, b] n ⊂ R n a set-valued function F |[a,b] n with its domain restricted to a cube [a, b] n belongs to C β ([a, b] n , Conv(R n )). If F : [0, T ] × R n → Conv(R n ) then we say that F(t, x) ∈ C β×γ ([0, T ] × R n , Conv(R n )) if F is β-Hölder in t and γ -Hölder in x. Let A, B ∈ Conv(R n ). The set C ∈ Conv(R n ) is said to be the Hukuhara difference A ÷ B if A = B + C. Definition 1 Consider a set-valued mapping G : R n → Conv(R n ). For k = 1, . . . n, let e k = (e 1 k , . . . e n k ) be the vector such that e j k = 0 for k = j and e k k = 1. We say that G admits a Hukuhara derivative at x 0 ∈ R n , if there exists a set D H k (G)(x 0 ) such that the limits exist with respect to a Hausdorff metric in Conv(R n ) and are equal to the set D H k (G)(x 0 ), k = 1, . . . , n, (see e.g., [6]).
what together with (4) proves the desired inequality.

Remark 1
In the proof of formula (5) one can take any point x 0 ∈ [a, b] n instead of a point a to obtain the inequality For a detailed discussion of the properties and applications of Hukuhara differentiable set-valued functions we refer the reader to [19].

Hölder Set-Valued Functions and Set-Valued Young Integrals
We recall the notion of a Young integral in a single valued case introduced by L.S. Young in [29]. For details see also [10]. Let f : R 1 → R d and w : R 1 → R 1 be given functions. For the partition m : a = t 0 < t 1 < . . . < t m = b of the interval [a, b] we consider the Riemann sum of f with respect to w Then the following result holds (see e.g., [11] and [26]).
holds for every a ≤ s < t ≤ b, where the constant C(α, β) depends only on β and α. In , R 1 ) and α, β ∈ (0, 1] with α + β > 1, one can express the Young integral by fractional derivatives. Namely, let where I (a,b) (·) denotes the characteristic function of the interval (a, b). The right-sided and left-sided fractional derivatives of order 0 < ρ < 1 for the function f |[a,b] are defined by Then we get by [26] for every ρ ∈ (1 − α, β).
Let F : R n → Conv(R d ) be a measurable set-valued function. For 1 ≤ p < ∞, define the set Elements of the set S L p (F) are called integrable selections of F. Since values of F are closed in R d then S L p (F) is a closed subset of L p (see e.g., [14], formula (1.1)). It is nonempty if F is p-integrably bounded i.e., if there exists h ∈ L p such that F(x) ≤ h(x) for a.e. x ∈ R n ( [14], Theorem 3.2).
A set-valued Aumann integral of F over the measurable set A ⊂ R n was defined in [3] by the formula For properties of measurable set-valued functions and their measurable selections see e.g., [2].
Let G : R 1 → Conv(R d ) be a Hukuhara differentiable set-valued function with p-integrably bounded Hukuhara derivative D H(G). Assume that G belongs to C β (R 1 , Conv(R d )) locally. By S β (G) we denote the set of all locally β-Hölder selections of G, i.e.,

Definition 2
We define the set IS(G) = {g : g ∈ S β (G) and g ∈ S L p (D H(G))} and a set-valued Young integral of a locally β-Hölder and Hukuhara differentiable set-valued function G with respect to a function w ∈ C α (R 1 , R 1 ), β + α > 1, by the formula The above definition of set-valued integral is proper if the set IS(G) is nonempty. Conditions assuring the nonemptiness of this set can be found in [23].
By Proposition 2 we obtain and R d , respectively. It was proved in [23] Theorem 7 that the set The following result is a variant of Lemma 1 from [22]. holds.

Young Differential Inclusion, Existence and Properties of Solutions
be set-valued functions. Assume the following hypotheses on set-valued functions F and G: (H 1) F is a Carathéodory set-valued function, i.e., product measurable in (t, x) and lower semicontinuous in x.
(H 2) There exists a function b ∈ L 1 1−α ([0, T ]) and a constant L > 0 such that for We define the set This integral is nonempty under condition (H 3) above.
. Then, for every δ ∈ (0, 1), G |A(x) is δ-Hölder with a Hölder constant dependent only on (x) and for every 0 ≤ s < t ≤ T . First we will establish the estimation ofH x . For simplicity of notation let us denote the variable t as a "zero" variable i.e., let g(t, Let us note that for every differentiable function f : holds. Let us take an arbitrary g 1 ∈ (IS)(G 1 ) and Since the formula y(u) = y(u 0 ) + u u 0 y (s)ds uniquely determines the function y then inf g 2 ∈IS(G 2 ) is the same as inf g 2 ∈S L p (D H(G 2 )) infȳ 0 ∈G 2 (0,x(0)) , whereȳ 0 means g 2 (0, x(0)). Therefore, because (g 1 ) and (g 1 ) i are arbitrary elements from G 1 and D H i (G 1 ), respectively. Hencē By formula (19) we have . Then by (13) and (11) we obtain desired inequality (10).
Proof Let G 1 := G, G 2 := {0} ⊂ R d×m and 0 ≤ s ≤ t ≤ T . Then by Proposition 4 we get and θ ∈ (0, 1] are the same as in Proposition 4. Since G has compact values in R d×m , then G(0, x 0 ) < ∞. Moreover, since it has also a locally bounded Hukuhara derivative, then sup as n → ∞.
Proof Let us note that for every n ≥ 1 we have
Consider the Young differential inclusion:

Theorem 7 Under conditions (H 1) − (H 3) the Young differential inclusion (YDI) admits solutions.
Proof Since F satisfies (H 1) then there exist selections f : [0, T ] × R d → R d of F which are of a Carathéodory type (i.e., measurable in (t, x) and continuous in x). Moreover, for every (t, x) : f are Carathéodory selections of F} by [12]. For every continuous function x : It means that every f (·, x(·)) is 1/(1 − α)-integrably bounded and therefore, f • x ∈ S L 1 (F • x). From this we deduce that the Aumann integral t s F(u, x(u))du is a nonempty set for every 0 ≤ s < t ≤ T .
Let C ∈ Conv(R d×m ) and let σ C ( p) := σ (C, p) := sup y∈C < p, y >∈ R 1 ∪ {+∞}. The function σ C : R d×m → R 1 ∪ {+∞} is a support function of C. Let denote the unit sphere in R d×m and let V denote a Lebesgue measure of a closed unit ball B(0, 1) in R d×m , i.e., V = π dm/2 / (1 + dm/2) with being the Euler function. Let p V be a normalized Lebesgue measure on B(0, 1), i.e., dp V = dp/V . Let M = μ : μ is a probability measure on B(0, 1) having the C 1 −density dμ/dp V with respect to measure p V .
Let ξ μ := dμ/dp V and let ∇ξ μ denote the gradient of ξ μ . By ω we denote a Lebesgue measure on . The function St μ : Conv(R d×m ) → R d×m called a generalized Steiner center, and given by the formula for every μ ∈ M, has the following properties. For A, B, C ∈ Conv(R d×m ) and a, b ∈ R 1 the following properties hold where L μ = dm max p∈ ξ μ ( p) + max p∈B(0,1) | ξ μ ( p) | (see, [5], [9]). Moreover, it was proved in [9] that every set C ∈ Conv(R d×m ) has a representation (G(t, x)).

H(G)(t, x) and taking in mind formula (7) we deduce that every St μ (G) ∈ IS(G).
Letf be an arbitrary Carathéodory selection of F and St μ (G) be any generalized Steiner selection of G. It was proved above that St μ (G) ∈ IS(G).
Let Sol(x 0 , F, G, w) denote the set of all solutions of the inclusion (Y DI ). First we will present conditions imposed on F, G : [0, T ] × R d → Conv(R d )(resp., Conv(R d×m ) assuring the closedness of the set Sol(x 0 , F, G Assume the following condition (H 2 ) F is locally δ-Hölder in x for some δ ∈ (0, 1).

Theorem 8 Under conditions (H 1) − (H 3) and (H 2 ), the set Sol(x 0 , F, G, w) of all solutions of the Young differential inclusion (YDI) is closed in C α ([0, T ], R d ).
Proof Suppose that (x k ) ∞ k=1 ⊂ Sol(x 0 , F, G, w) is a sequence convergent to some limit x in C α ([0, T ], R d ). We have to prove that x ∈ Sol(x 0 , F, G, w). Since x k are solutions of (Y DI ) then x k (0) = x 0 for every k. But x k − x ∞ → 0 and from this we deduce If g ∈ IS(G) then Similarly as in the proof of Proposition 1 we deduce that g |A(x) is δ-Hölder with a Hölder constant M δ (g |A(x) ) ≤ N (x) T 1−δ + d(2 (x)) 1−δ . Moreover, Taking σ = αδ and dividing both sides of the above inequality by | t − s | σ we get Therefore, g • x is σ -Hölder. Similarly, g • x k are σ -Hölder and Let G n , G : [0, T ] × R d → R d×m be such that: (H 1 ) G n and G are γ -Hölder for some γ > (1 − α)/α, α > 1/2 and satisfy sup n M γ (G n ) ≤ K 1 for some K 1 > 0, (H 2 ) G n and G have γ -Hölder and bounded Hukuhara derivatives and sup n M γ (D H(G n )) ≤ K 2 for some K 2 > 0, Theorem 10 Let (H 1 ) and (H 2 ) hold. Moreover, assume Then every sequence of solutions (x n ) of (Y DI n ) satisfying (H 2 ). Then, similarly as in Corollary 6, there exist K 3 > 0 and N 0 > 0 such that

Proof D H(G) is bounded by
for n > N 0 by assumption (iii). Therefore, (G n ) have uniformly bounded Hukuhara derivatives for n > N 0 . Similarly as in formula (18)  It was proved in Proposition 1 of [20] that there exist constants R n depending only on T , α, γ, M α (w n ), M γ (St μ (G n )), a n and such that x n ∞ ≤ R n and M α (x n ) ≤ R n . Since sup n M γ (St μ (G n )) ≤ L μ K 1 and sup n | a n |< ∞ then there exists one constant P 1 = sup n R n < ∞ such that x n ∞ ≤ P 1 and M α (x n ) ≤ P 1 for every n. The same holds for solution x of (Y DE), i.e., there exists a constant P 2 depending only on T , α, γ, M α (w), M γ (St μ (G)), a and such that x ∞ ≤ P 2 and M α (x) ≤ P 2 . Moreover, we have St μ (G n ) ∞ ≤ K 3 and sup n M γ (St μ (G n ) ) ≤ L μ K 2 . Taking R = max{P 1 , P 2 , K 3 , L μ K 2 } all assumptions of Proposition 9 are satisfied and we get

Thus, by assumptions
Remark 2 All the results are easily adapted to deal with the stochastic differential inclusion where B H is a fractional Brownian motion. Last theorem needs additional property that Hurst index of fBm satisfies inequality H > 1/2.
Let us conclude with applications to stochastic inclusions. Consider a 1-dimensional fractional Brownian motion B H = (B H (t)) t∈[0,T ] of Hurst index H ∈ (1/2, 1), which is a centered Gaussian process such that for every s, t ∈ [0, T ]. Let ( , , P) be the associated canonical probability space. By the Garcia-Rodemich-Rumsey lemma (see Nualart [25], Lemma A.3.1), the paths of B H are α-Hölder continuous for any α ∈ (0, H ) and therefore, the integral (IS) t s G(u, x(u))d B H (u) can be treated as a Young integral. Let us consider the following medical model. Let x : [0, T ] → R + be a function for which the value x(t) denotes the number of cancerous cells at time t ∈ [0, T ]. The disease is diagnosed if x(t) ≥ a 0 > 0 where a 0 is a given reference level. In practice we may assume that a 0 is a level at which some therapy starts. One can also assume that there is a maximum level of cancer cells a 1 > a 0 at which no further treatment (dose of drugs) can help. So in the presence of active treatment x(t) ∈ [a 0 , a 1 ]. Let u : [0, T ] → [0, c] be a function for which u(t) represents the dose of the drug at time t and let d : [0, c] → R + be a given continuous function describing destroying rate per tumor cell. In [27] the following controlled cancer growth model under the influence of drugs was considered where λ and μ are some positive parameters. In [18] the function d was taken in particular as d(u) = k 1 u/(k 2 + u) with some positive parameters k 1 , k 2 . Since therapy effects may be different in cases of different patients it seems reasonable to generalize model (21) adding a stochastic perturbation governed by the fractional Brownian motion B H with Hurst parameter H ∈ (0, 1), i.e., to consider the model or in its integral form instead of model (1). The term v(u(t)) can be interpreted as unforeseen (random) reaction of the patient to treatment. It is assumed that v is a continuous function. Then the controlled model (22) can be rewritten in the framework of Young integral inclusion as Let us note that both f + F 1 and G take on nonempty, compact and convex values in R 1 . By standard calculations and Mean Value Theorem one can show that f is Lipschitz continuous with a Lipschitz constant K = λ(| ln(μ) | + max{| ln(a 0 ) |, | ln(a 1 ) |} + 1).
It is also easy to see that F 1 is Lipschitz continuous with a Lipschitz constant d ([0, c]) . Thus F is Lipschitz continuous too. It can be also verified that F(x) ≤ L | x | with L = λ max{| ln(μ/a 0 ) |, | ln(μ/a 1 ) |} + d ([0, c] H 3). Therefore, using Theorems 7 and 8 we deduce that inclusion (23) admits solutions and the set of its solutions is closed in C α ([a 0 , a 1 ], R 1 ). It is known, that classical Steiner point can be interpreted as a center of a mass of a given set. Therefore, Theorem 10 applied to a system of equations x n (t) = a n + St G(u(τ )) · x(τ )d B H τ allows us to deduce that a mean unforced reaction of a patient to treatment v(u(t)) is stable with respect to a convergent sequence of individual mean unforced reactions v n (u(t)), initial reference levels a n and a sequence of different fractional Brownian motions B H ,n τ . Author Contributions MM and JM wrote the main manuscript text. All authors reviewed the manuscript.
Funding Not applicable.

Conflict of interest
The authors declare that they have no conflict of interest.
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