Adv Syst Sci Appl 2021; 03:40–62 Published online at https://ijassa.ipu.ru. A Controllability Problem for Causal Functional Inclusions with an Infinite Delay and Impulse Conditions Maria Afanasova1, Valeri Obukhovskii1, Garik Petrosyan1,2* 1Faculty of Physics and Mathematics, Voronezh State Pedagogical University, Voronezh, Russia 2Research Center, Voronezh State University of Engineering Technologies, Voronezh, Russia Abstract: In this paper we study the controllability problem in a Banach space for various classes of functional inclusions with causal operators with an infinite delay, and impulse effects. Basing on the topological degree theory for condensing multimaps, we prove a global theorem on the existence of trajectories for systems governed by functional inclusions. As an application, we obtain generalizations of existence theorems for the controllability problem for a semilinear first order functional differential inclusions of this type and a semilinear functional differential inclusions of a fractional order 0 < q < 1. Keywords: causal operator, functional inclusion, controllability problem, functional differ- ential inclusion, fractional derivative, measure of non-compactness, fixed point, topological degree, condensing multioperator 1. INTRODUCTION It is well known that the contemporary approach in the theory of control systems and in mathematical physics leads to models which are convenient to be described by using differential equations and inclusions. Recently, the attention of many researchers (see [1], [2], [3] and references therein) has been attracted to generalizations of differential equations and inclusions, namely to the class of functional equations and inclusions with causal operators. The term of a causal or Volterra operator in the sense of A.N. Tikhonov (see [4]), was used in mathematical physics to solve problems of differential equations, integro-differential equations, functional-differential equations with a finite or infinite delay, integral equations of Volterra type, functional equations of a neutral type, etc. (see, for example, [5]). The papers [6], [7], [8], [9] among others are devoted to the study of equations and inclusions with causal operators of various types, theorems on the existence of solutions, description of qualitative properties of solutions and various applications. At the same time, in recent decades the interest to the theory of fractional-order differential equations has significantly increased, thanks to applications in various branches of applied mathematics, physics, engineering, biology, economics, etc. (see, for example, monographs [10], [11] papers [12], [13], [14], [15], [16], etc.). The boundary value problems of various types for fractional differential equations and inclusions were considered in the works [17], [18], [19], [20], [21], [22], [23]. In this paper we develop and generalize the results of papers [2] and [3], and we study the controllability problem in Banach spaces for various classes of functional inclusions with causal operators with an infinite delay and impulse effects. Basing on the topological degree ∗Corresponding author: garikpetrosyan@yandex.ru A CONTROLLABILITY PROBLEM FOR CAUSAL FUNCTIONAL INCLUSIONS WITH AN INFINITE DELAY41 theory for condensing multimaps we prove a global theorem on the existence of trajectories for systems governed by functional inclusions. As an application, we obtain generalizations of existence theorems for solutions of the controllability problem for a first order semilinear functional differential inclusions and a semilinear functional differential inclusions of a fractional order 0 < q < 1. 2. PRELIMINARIES 2.1. Multivalued maps and measures of noncompactness Let X be a metric space and Y a normed space. Introduce the following notation: P (Y ) denotes the collection of all non-empty subsets of Y ; Pb(Y ) denotes the collection of all non-empty and bounded subsets of Y ; C(Y ) denotes the collection of all non-empty and closed subsets of Y ; Cv(Y ) denotes the collection of all non-empty, closed and convex subsets of Y ; K(Y ) denotes the collection of all non-empty and compact subsets of Y ; Kv(Y ) denotes the collection of all non-empty, compact and convex subsets of Y. Let us recall some notations (see, for example, [24], [25]). Definition 2.1: A multivalued map (multimap) F : X → P (Y ) is said to be upper semicontinuous (u.s.c.) at a point x ∈ X, if for every open set V ⊂ Y such that F(x) ⊂ V, there exists a neighborhood U(x) of x such that F(U(x)) ⊂ V. Definition 2.2: A multivalued map (multimap) F : X → P (Y ) is called closed if its graph GF = {(x, y) : x ∈ X, y ∈ F(x)} is a closed subset of X × Y. Definition 2.3: A multivalued map (multimap) F : X → P (Y ) is called quasicompact if its restriction to each compact subset A ⊂ X is compact. Lemma 2.1: ( [24], Theorem 1.1.12). If F : X → K (Y ) a closed quasicompact multimap, Then F is u.s.c. Definition 2.4: For a given p ≥ 1, a multifunction G : [0, T ]→ K(Y ) is called: • Lp–integrable if it admits an Lp–Bochner integrable selection, i.e., there exists a function g ∈ Lp ([0, T ];Y ) such that g(t) ∈ G(t) for a.e. t ∈ [0, T ]; • Lp–integrably bounded if there exists a function ξ ∈ Lp([0, T ]) such that ‖G(t)‖ ≤ ξ(t) for a.e. t ∈ [0, T ]. Let E be a Banach space Lemma 2.2: (see [24], Theorem 4.2.1.) Let a sequence of functions {ξn} ⊂ L1([0, T ];E) be L1–integrably bounded. Suppose that χ({ξn} (t)) ≤ α(t) a.e. t ∈ [0, T ] for all n = 1, 2, ..., where α ∈ L1 +([0, T ]). Then for every δ > 0 there exist a compact set Kδ ⊂ E, a set mδ ⊂ [0, T ] of a Lebesgue measure mδ < δ, and a set of functions Gδ ⊂ L1([0, T ];E) with values in Kδ, such that for every n ≥ 1 there exists a function bn ∈ Gδ for Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 42 M. AFANASOVA, V. OBUKHOVSKII, G. PETROSYAN which ‖ξn(t)− bn(t)‖E ≤ 2α(t) + δ, t ∈ [0, T ] \mδ. Moreover, the sequence {bn}may be chosen so that bn ≡ 0 onmδ and this sequence is weakly compact. Definition 2.5: Let (A,≥) be a partially ordered set. A function β : Pb(E)→ A is called the measure of noncompactness (MNC) in E if for each Ω ∈ Pb(E) we have: β(co Ω) = β(Ω), where co Ω denotes the closure of the convex hull of Ω. A measure of noncompactness β is called: 1) monotone if for each Ω0,Ω1 ∈ Pb(E), from Ω0 ⊆ Ω1 follows β(Ω0) ≤ β(Ω1). 2) nonsingular, if for each a ∈ E and each Ω ∈ Pb(E) we have β({a} ∪ Ω) = β(Ω). If A is a cone in a Banach space, the MNC β is called: 3) regular, if β(Ω) = 0 is equivalent to the relative compactness of Ω ∈ Pb(E); 4) real, if A is the set of all real numbers R with the natural ordering. As the example of a real MNC obeying all above properties, we can consider the Hausdorff MNC χ(Ω): χ(Ω) = inf{ε > 0, for which Ω has a finite ε-net in E }. As other examples, consider the measures of noncompactness defined in the space of continuous functions C([a, b];E) with values in the Banach space E: (1) the modulus of fiber noncompactness: ϕ(Ω) = sup t∈[a,b] χE(Ω(t)), where χE is the Hausdorff MNC in E and Ω(t) = {y(t) : y ∈ Ω}; (2) the fading modulus of fiber noncompactness: γ(Ω) = sup t∈[a,b] e−LtχE(Ω(t)), where L > 0 is a given number; (3) the modulus of equicontinuity: modC (Ω) = lim δ→0 sup y∈Ω max |t1−t2|≤δ ‖y (t1)− y (t2)‖ . These measures of noncompactness satisfy all the above properties, except for the regularity. Definition 2.6: A multimap F : X ⊆ E → K(E) is called condensing with respect to a MNC β (or β– condensing) if for each bounded set Ω ⊆ X which is not relatively compact, we have: β(F (Ω)) 6≥ β(Ω). Let D ⊂ E a non-empty closed convex subset, V a non-empty bounded open subset of D, β a monotone nonsingular MNC in E and F : V → Kv (D) be a u.s.c. β-condensing map Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) A CONTROLLABILITY PROBLEM FOR CAUSAL FUNCTIONAL INCLUSIONS WITH AN INFINITE DELAY43 such that x /∈ F (x) for all x ∈ ∂V , where V and ∂V denote the closure and the boundary of the set V in the relative topology of D. In such a setting, the (relative) topological degree degD ( i−F , V ) of the corresponding vector field i−F , satisfying the standard properties is defined (see, for example, [24], [25]). In particular, the condition degD ( i−F , V ) 6= 0 implies that the fixed points set FixF = {x : x ∈ F(x)} is a nonempty subset of V. Application of topological degree theory leads to the following fixed point principles, which will be used in the what follows. Theorem 2.1: ( [24], Corollary 3.3.1). Let M be a convex closed bounded subset of E and F :M→ Kv(M) a β–condensing multimap, where β is a monotone nonsingular MNC in E . Then the fixed point set FixF is non-empty. Theorem 2.2: ( [24], Theorem 3.3.4). Let V ⊂ D be a bounded open neighborhood of a point a ∈ V and F : V → Kv(D) a u.s.c. β-condensing multimap, where β is a monotone nonsingular MNC in E , satisfying the boundary condition x− a /∈ λ(F(x)− a) for all x ∈ ∂V and 0 < λ ≤ 1. Then FixF 6= ∅ is a non-empty compact set. 2.2. Phase space We will use the axiomatic definition of the phase space B, introduced by J.K. Hale and J. Kato (see. [26], [27]).The space B we will be considered as a linear topological space of functions defined on (−∞, 0] with values in a Banach space E endowed with the seminorm ‖ · ‖B. For all function x : (−∞, T ]→ E, where T > 0, and every t ∈ (−∞, T ], xt is a function from (−∞, 0] to E, defined as xt(θ) = x(t+ θ), θ ∈ (−∞, 0]. We will be assume that B satisfies the following axioms: (B1) if the function x : (−∞;T ]→ E is continuous on [0;T ] and x0 ∈ B, then for each t ∈ [0;T ] : (i) xt ∈ B; (ii) the function t 7→ xt is continuous; (iii) ‖xt‖B ≤ K(t) sup 0≤τ≤t ‖x(τ)‖+H(t)‖x0‖B, where the functions K,H : [0;∞)→ [0;∞) are independent of x, K is strictly positive and continuous and H is locally bounded. (B0) there exists l > 0 such that ‖ψ(0)‖E ≤ l‖ψ‖B, for all ψ ∈ B. Notice that under these conditions the space C00 of all continuous functions from (−∞, 0] to E with compact support into phase space B( [27], Proposition 1.2.1). In addition, we will assume that the following condition is satisfied: (BC1) if a uniformly bounded sequence {ψn}+∞ n=1 ⊂ C00 converges to a function ψ compactly (i.e. uniformly on each compact subset (−∞, 0]), then ψ ∈ B and lim n→+∞ ‖ψn − ψ‖B = 0. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 44 M. AFANASOVA, V. OBUKHOVSKII, G. PETROSYAN The condition (BC1) implies that the Banach space of bounded continuous functions BC = BC((−∞, 0];E) is continuously embedded into B. More precisely, the following assertion is true. Theorem 2.3: ( [27], Proposition 7.1.1). (i) BC ⊂ C00, where C00 denote the closure of C00 in B; (ii) if a uniformly bounded sequence {ψn} in BC converges to a function ψ compactly on (−∞, 0], then ψ ∈ B and lim n→+∞ ‖ψn − ψ‖B = 0; (iii) there exists L > 0 such that ‖ψ‖B ≤ L‖ψ‖BC for all ψ ∈ BC. Finally, we will assume that the following condition is satisfied: (BC2) if ψ ∈ BC and ‖ψ‖BC 6= 0, then ‖ψ‖B 6= 0. This assumption implies that the space BC, endowed with ‖ · ‖B is a normed space. We will denote it as BC. We may consider the following examples of phase spaces satisfying all the above properties: (1) for γ > 0 let B = Cγ be the space of continuous functions ϕ : (−∞; 0]→ E, having a limit lim θ→−∞ eγθϕ(θ) with ‖ϕ‖B = sup −∞<θ≤0 eγθ‖ϕ(θ)‖. (2) (Spaces of ”fading memory”) Let B = Cρ be the space of functions ϕ : (−∞; 0]→ E such that (a) ϕ is continuous on [−r; 0], r > 0; (b) ϕ is Lebesgue measurable on (−∞; r) and there exists a nonnegative Lebesgue integrable function ρ : (−∞;−r)→ R+ such that ρϕ Lebesgue integrable on (−∞; r); moreover, there exists a locally bounded function P : (−∞; 0]→ R+ such that, for all ξ ≤ 0, ρ(ξ + θ) ≤ P (ξ)ρ(θ) a.e. θ ∈ (−∞;−r). Then, ‖ϕ‖B = sup −r≤θ≤0 ‖ϕ(θ)‖+ −r∫ −∞ ρ(θ)‖ϕ(θ)‖dθ. A simple example of such a space is given by ρ(θ) = eµθ, µ ∈ R. 2.3. Causal multioperators with infinite delay Let E be a separable Banach space. By Lp ([0, T ];E) , 1 ≤ p ≤ ∞, we denote the Banach space of all Bochner summable functions f : [0, T ]→ E with the usual norm. For each subset N ⊂ Lp ([0, T ];E) and τ ∈ (0, T ) we define restriction N on [0, τ ] as N |[0,τ ]= {f |[0,τ ]: f ∈ N}. We split the segment [0, T ] by points 0 < t1 < ... < tm < T,m ≥ 1. For a function c : [0, T ]→ E we will denote c(t+k ) = lim ξ→0+ c(tk + ξ), c(t−k ) = lim ξ→0− c(tk + ξ), Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) A CONTROLLABILITY PROBLEM FOR CAUSAL FUNCTIONAL INCLUSIONS WITH AN INFINITE DELAY45 for 1 ≤ k ≤ m. For a function g : (−∞, T ]→ E, we will assume that its restriction to [0, T ] belongs to the space PC([0, T ];E) of functions z : [0, T ]→ E, continuous on [0, T ] \ {t1, · · · , tm} and such that the left and right limits z(t+k ) and z(t−k ), 1 ≤ k ≤ m, exist and z(t−k ) = z(tk). It is easy to see that the space PC([0, T ];E), endowed with the norm ‖z‖PC = sup t∈[0,T ] ‖g(t)‖E, is a Banach space. The classical space of continuous function C([0, T ], E) is its closed subspace. We denote by C((−∞;T ];E) the normed space of piecewise continuous functions x : (−∞;T ]→ E, endowed with the norm ‖x‖C = ‖x0‖BC + ‖x |[0;T ] ‖PC. Definition 2.7: A multivalued mapQ : C ((−∞, T ];E) ( Lp ([0, T ];E) is said to be a causal multioperator, if for each τ ∈ (0, T ) and for every u(·), v(·) ∈ C ((−∞, T ];E) the condition u |(−∞,τ ]= v |(−∞,τ ] implies that Q(u) |[0,τ ]= Q(v) |[0,τ ] . Let us give examples of causal multioperators. Example 2.1: We assume that the multimap F : [0, T ]× BC × E → Kv (E) satisfies the following conditions: (F1) for each (ψ, φ) ∈ BC × E the multifunction F (·, ψ, φ) : [0, T ]→ Kv (E) admits a measurable selection; (F2) for a.e. t ∈ [0, T ] the multifunction F (t, ·, ·) : BC × E → Kv (E) is u.s.c.; (F3) there exists a function α ∈ Lp+[0, T ], 1 ≤ p ≤ ∞, such that ‖F (t, ψ, φ)‖E := sup{‖z‖E : z ∈ F (t, ψ, φ)} ≤ α(t)(1 + ‖ψ‖BC + ‖φ‖E) for a.e. t ∈ [0, T ] and (ψ, φ) ∈ BC × E. From above conditions (F1)− (F3) and (B1) it follows that the multimap PF : C((−∞;T ];E)→ P (Lp([0, T ];E)), given in the following way PF (x) = {f ∈ Lp([0, T ];E) : f(t) ∈ F (t, xt, x(t)) a.e. t ∈ [0, T ]} is well defined (see, for example, [24], [25]). It is clear that the multioperator PF is causal. Example 2.2: Let F : [0, T ]× BC → Kv(E) be a multimap satisfying conditions (F1)− (F3) from Example 2.1. Suppose that {K(t, s) : 0 ≤ s ≤ t ≤ T} is a continuous (with respect to the corresponding norm) family of bounded linear operators in E and m ∈ L1([0, T ];E) is a given function. Consider the Volterra integral multioperator V : C ((−∞, T ];E) ( L1 ([0, T ];E) defined as V(u)(t) = m(t) + ∫ t 0 K(t, s)F (s, us)ds, i.e., V(u) = {y ∈ L1 ([0, T ];E) : y(t) = m(t) + ∫ t 0 K(t, s)f(s)ds : f ∈ PF (u)}. It is also clear that the multioperator V is causal. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 46 M. AFANASOVA, V. OBUKHOVSKII, G. PETROSYAN 3. THE CONTROLLABILITY PROBLEM FOR FUNCTIONAL INCLUSIONS WITH THE CAUSAL OPERATORS We will assume that the causal operator Q : C ((−∞, T ];E)→ C (Lp ([0, T ];E)) satisfies the following conditions: (Q1) Q is weakly closed in the following sense: conditions {un}∞n=1 ⊂ C ((−∞, T ];E) , {fn}∞n=1 ⊂ Lp ([0, T ];E) , 1 ≤ p ≤ ∞, fn ∈ Q(un), n ≥ 1, un → u0, fn L1 ⇀ f0 implies f0 ∈ Q(u0); (Q2) there exists a function α ∈ L∞+ ([0, T ]) such that ‖Q (u) (t) ‖E ≤ α (t) (1 + ‖u‖C) , for a.e. t ∈ [0, T ], for all u ∈ C((−∞, T ];E); (Q3) there exists a function ω : [0, T ]× R+ → R+ such that (ω1) for all x ∈ R+ : ω(·, x) ∈ Lp+([0, T ]), 1 ≤ p ≤ ∞, ; (ω2) for a.e. t ∈ [0, T ] a function ω(t, ·) : R+ → R+ is continuous, nondecreasing and quasihomogeneous in the sense that ω(t, λx) ≤ λω(t, x) for all x ∈ R+ and λ ≥ 0; (ω3) for each bounded set ∆ ⊂ C ((−∞, T ];E) we have χ (Q (∆) (t)) ≤ ω ( t, sup s∈[0,t] ϕ (∆s) ) for a.e. t ∈ [0, T ], where the set ∆s = {ys : y ∈ ∆} ⊂ BC and ϕ is the modulus of fiber noncompactness in BC. Note that the condition (ω2) means that ω(t, 0) = 0 for a.e.t ∈ [0, T ] and as an example of such a function we can consider ω(t, x) = k(t) · x, where k(·) ∈ Lp+([0, T ]). Consider a linear operator S : Lp([0, T ];E)→ C([0, T ];E), which is causal in the sense that for every τ ∈ (0, T ] and f, g ∈ Lp([0, T ];E) condition f(t) = g(t) for a.e. t ∈ [0, τ ] implies (Sf) (t) = (Sg) (t) for all t ∈ [0, τ ]. Following [24], we impose the next conditions on operator S : (S1) for 1 ≤ p <∞ there exist D ≥ 0 such that ‖Sf(t)− Sg(t)‖pE ≤ D ∫ t 0 ‖f(s)− g(s)‖pEds for all f, g ∈ Lp([0, T ];E) and 0 ≤ t ≤ T ; if p =∞ there exist D1 ≥ 0 such that ‖Sf(t)− Sg(t)‖E ≤ D1 ∫ t 0 ‖f(s)− g(s)‖Eds for all f, g ∈ L∞([0, T ];E) and 0 ≤ t ≤ T. (S2) for an arbitrary compact set K ⊂ E and a sequence {fn}∞n=1 ⊂ Lp ([0, T ];E) , 1 ≤ p ≤ ∞, such that {fn(t)}∞n=1 ⊂ K for all t ∈ [0, T ] the weak convergence fn L1 ⇀ f0 implies Sfn → Sf0 in C([0, T ];E). Also we suppose that S satisfies the relation: (S3) (Sf) (0) = 0 for each function f ∈ Lp([0, T ];E). Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) A CONTROLLABILITY PROBLEM FOR CAUSAL FUNCTIONAL INCLUSIONS WITH AN INFINITE DELAY47 Notice, that condition (S1) implies that the operator S satisfies the Lipschitz condition: (S1′) ‖Sf − Sg‖C ≤ D‖f − g‖L1 . Consider following important examples. (i) Let a closed linear operator A : D (A) ⊂ E → E be the infinitesimal generata of a C0- semigroup {eAt}t≥0. The operator L : L1([0, T ];E)→ C([0, T ];E) defined as Lf(t) = ∫ t 0 eA(t−s)f(s)ds is a special case of the operator S. Note that taking A = 0 we obtain, as a special case, the usual integral operatorLI : L1([0, T ];E)→ C([0, T ];E), LIf(t) = ∫ t 0 f(s)ds; (ii) Let A : D(A)→ E be a closed linear operator E generating a C0- semigroup {U(t)}t≥0 . The operator G : Lp([0, T ];E)→ C([0, T ];E), p > 1/q, defined as Gf(t) = ∫ t 0 (t− s)q−1T (t− s)f(s)ds, 0 < q < 1, where T (t) = q ∫ ∞ 0 θξq(θ)U(tqθ)dθ, ξq(θ) = 1 q θ−1− 1 qΨq(θ −1/q), Ψq(θ) = 1 π ∞∑ n=1 (−1)n−1θ−qn−1 Γ(nq + 1) n! sin(nπq), θ ∈ R+, is a special case of the operator S. Lemma 3.1: ( [24], Lemma 4.2.1, [12], Lemma 3.4). The operators L and G satisfy conditions (S1)− (S3). Consider a control system governed by a functional inclusion with causal operatorsQ and S, of the following form: y(t) ∈ G(t) ( ψ(0) + Σtk 0 such that ‖Ikx‖ ≤ N for all x ∈ E. Lemma 3.2: (see [12]) The operator functions G and T possess the following properties: 1) For each t ∈ [0, T ], G(t) and T (t) are linear bounded operators, more precisely, for each x ∈ E we have ‖G(t)x‖E ≤M ‖x‖E , ‖T (t)x‖E ≤ qM Γ(1 + q) ‖x‖E , where M = supt≥0 ‖U(t)‖ 2) the operator functions G(·) and T (·) are strongly continuous, i.e., functions t ∈ [0, T ]→ G(t)x and t ∈ [0, T ]→ T (t)x are continuous for each x ∈ E. Suppose ψ ∈ BC is a given function. For a function y ∈ PC([0, T ];E) such that y(0) = ψ(0) we define the function y[ψ] ∈ C((−∞, T ];E) as y[ψ](t) = { ψ(t), −∞ ≤ t < 0, y(t), 0 ≤ t ≤ T . We denote by D the closed convex subset of PC([0, T ];E), consisting of all functions y satisfying the condition y(0) = ψ(0). Definition 3.1: A function y ∈ C((−∞, T ];E) is called a mild solution of problem (3.1)-(3.3), if it satisfies conditions: (1) the function y|[0,T ] ∈ D and satisfies inclusion (3.1); (2) y(t) = ψ(t), for t ∈ (−∞, 0]; (3) Iky(tk) = y(t+k )− y(tk), k = 1, ...,m. Now, the controllability problem which we solve in this paper may be formulated in the following way: for a given initial function ψ ∈ BC and a given x1 ∈ E we consider the existence of a mild solution y of problem (3.1)-(3.3) and a control u such that y(t) = ψ(t), t ∈ (−∞, 0], and y(T ) = x1. (3.4) Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) A CONTROLLABILITY PROBLEM FOR CAUSAL FUNCTIONAL INCLUSIONS WITH AN INFINITE DELAY49 The pair (y, u) consisting of an integral solution y of problem (3.1)-(3.3) and a control u ∈ L∞([0, T ];U) will be called the solution of controllability problem (3.1)-(3.4). Consider the multioperator Γ : D( D defined as Γ(y) = {x ∈ D : x(t) = G(t) ( ψ(0) + Σtk 0 we take δ ∈ (0, ε), such that for every m ⊂ [0, T ], with the measure meas(m) < δ, we have: ∫ m |υ(s)|p < ε, for 1 ≤ p <∞, and respectively for p =∞ : ∫ m |υ(s)| < ε. Taking mδ and bn corresponding to {fn} from Lemma 2.2, we, by using property (S1), obtain that the sequence {S(bn)} is relatively compact in C([0, T ];E). Let 1 ≤ p <∞, then the following estimates hold: ‖S(fn)(t)− S(bn)(t)‖pE ≤ D ∫ t 0 ‖fn(s)− bn(s)‖pE ds ≤ D ∫ [0,t]\mδ ‖fn(s)− bn(s)‖pE ds+D ∫ [0,t]∩mδ ‖fn(s)‖pE ds ≤ D ∫ [0,t]\mδ [2υ(s)− δ]p ds+D ∫ mδ |υ(s)|p ds ≤ D ∫ t 0 |2υ(s) + ε|p ds+ εD ≤ D ∫ t 0 2p |2υ(s)|p ds+D ∫ t 0 2pεpds+ εD ≤ 4pD ∫ t 0 υp(s)ds+ 2pεpTD + εD. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) A CONTROLLABILITY PROBLEM FOR CAUSAL FUNCTIONAL INCLUSIONS WITH AN INFINITE DELAY51 Therefore, the relatively compact set SGδ(t) is a ( 4pD ∫ t 0 υp(s)ds+ 2pεpTD + εD )1/p - net for the set {S(fn)(t)} . Since ε > 0 is arbitrary, we obtain the conclusion of the lemma for the case 1 ≤ p <∞. If p =∞, then the following estimates hold: ‖S(fn)(t)− S(bn)(t)‖E ≤ D1 ∫ t 0 ‖fn(s)− bn(s)‖E ds ≤ D1 ∫ [0,t]\mδ ‖fn(s)− bn(s)‖E ds+D1 ∫ [0,t]∩mδ ‖fn(s)‖E ds ≤ D1 ∫ [0,t]\mδ |2υ(s)− δ|ds+D1 ∫ mδ |υ(s)| ds ≤ 2D1 ∫ t 0 υ(s)ds+ εTD1 + εD1. Thus, the relatively compact set SGδ(t) is a 2D1 ∫ t 0 υ(s)ds+ εD1(T + 1) - net for the set {S(fn)(t)} . Since ε > 0 is arbitrary, we obtain the conclusion of the lemma also for the case p =∞. Let M1, M2 be positive constants, such that ‖B‖ ≤M1, ∥∥W−1 ∥∥ ≤M2. (3.5) Consider the measure of noncompactness ν in the space PC([0, T ];E) with values in the cone R2 +. On a bounded subset of Ω ⊂ PC([0, T ];E) we define the values of ν as follows: ν(Ω) = (γ (Ω) ,modC (Ω)) , where modC is the modulus of equicontinuity, γ is the fading modulus of fiber noncompactness γ(Ω) = sup t∈[0,T ] e−Ltχ(Ω(t)). The constant L > 0 is chosen so that max{q1, q2} < 1, where q1 = sup t∈[0,T ] ( 4D1/p ( 1 + 4M1σD 1/pT 1/p ) ∫ t 0 e−Lp(t−s)ωp (s, 1) ds )1/p , q2 = sup t∈[0,T ] ( 2D1 (1 + 2M1σD1T ) ∫ t 0 e−L(t−s)ω (s, 1) ds ) , where the constants D,D1 are from condition (S1), ω is a function from condition (Q3). It is easy to see that the MNC ν is monotone, nonsingular, and algebraically semi-additive. It follows from the Arzela–Ascoli theorem that it is also regular. Theorem 3.2: Let a causal multioperatorQ : C((−∞, T ];E) ( Lp ([0, T ];E) satisfy conditions (Q2) and (Q3) and for a causal operator S : Lp ([0, T ];E)→ C ([0, T ];E) the conditions (S1)–(S3) be satisfied. Then, under conditions (I1), (I2), (W ) the multioperator Γ is ν-condensing. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 52 M. AFANASOVA, V. OBUKHOVSKII, G. PETROSYAN Proof By Lemma 3.2 and conditions (I1), (I2), it is suffices to prove the assertion of the theorem for the multioperator S ◦ Q+ S ◦BW−1 ( x1 − G(T )ψ(0)− ζ ◦ S ◦ Q ) . Let Ω ⊂ D be a bounded set such that ν ( S ◦ Q (Ω[ψ]) + S ◦BW−1 ( x1 − G(T )ψ(0)− ζ ◦ S ◦ Q(Ω[ψ]) )) ≥ ν (Ω) . (3.6) Let us show that the set Ω is relatively compact. Inequality (3.6) means that γ({S ◦ Q (Ω[ψ]) + S ◦BW−1 ( x1 − G(T )ψ(0)− ζ ◦ S ◦ Q(Ω[ψ]) ) }) ≥ γ(Ω). (3.7) Applying the condition (Q3) and by using the properties of the function ω, we obtain for a.e. t ∈ [0, T ] χ ({f(t) : f ∈ Q (Ω[ψ])}) ≤ ω ( t, sup s∈[0,t] ϕ ({y[ψ]s : y ∈ Ω}) ) = ω ( t, ϕ ( {y|[0,t] : y ∈ Ω} )) = ω ( t, eLte−Ltϕ ( {y|[0,t] : y ∈ Ω} )) ≤ ω ( t, eLtγ ( {y|[0,t] : y ∈ Ω} )) ≤ ω ( t, eLtγ (Ω) ) ≤ ω ( t, eLt ) · γ (Ω) . At first, we consider the case 1 ≤ p <∞. By Lemma 3.5 we have for each t ∈ [0, T ] : χ ({Sf(t) : f ∈ Q (Ω[ψ])}) ≤ ( 4pD ∫ t 0 ωp ( s, eLs ) ds · γp (Ω) )1/p ≤ 4D1/p (∫ t 0 epLsωp (s, 1) ds )1/p · γ (Ω) . Further, χ ( {BW−1 ( x1 − G(T )ψ(0)− ζ ◦ Sf(t) : f ∈ Q (Ω[ψ])} ) ≤ M1σχ ({ζ ◦ Sf(t) : f ∈ Q (Ω[ψ])}) = M1σχ ({Sf(T ) : f ∈ Q (Ω[ψ])}) ≤M1σ ( 4pD ∫ T 0 epLsωp (s, 1) ds · γp (Ω) )1/p = 4M1σD 1/p (∫ T 0 epLsωp (s, 1) ds )1/p · γ (Ω) , where σ = supt∈[0,T ] σ(t). Using Lemma 3.5 again, we have for each t ∈ [0, T ] : χ ( S ◦BW−1 ( x1 − G(T )ψ(0)− ζ ◦ Sf(t) : f ∈ Q (Ω[ψ]) ) ≤( 4pD ∫ t 0 4pMp 1σ pD (∫ T 0 epLsωp (s, 1) ds ) dτ · γp (Ω) )1/p ≤ Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) A CONTROLLABILITY PROBLEM FOR CAUSAL FUNCTIONAL INCLUSIONS WITH AN INFINITE DELAY53 16D2/pM1σT 1/p (∫ T 0 epLsωp (s, 1) ds )1/p · γ (Ω) . Inequality (3.7) and the last inequality imply the following γ(Ω) ≤ sup t∈[0,T ] ( 4D1/p ( 1 + 4M1σD 1/pT 1/p ) ∫ t 0 e−Lp(t−s)ωp (s, 1) ds )1/p γ (Ω) = q1 · γ (Ω) , therefore γ (Ω) = 0, thus ϕ (Ω[ψ]t) = 0 for all t ∈ [0, T ]. Let us turn to the case p =∞. By Lemma 3.5 we have for each t ∈ [0, T ] : χ ({Sf(t) : f ∈ Q (Ω[ψ])}) ≤ 2D1 ∫ t 0 ω ( s, eLs ) ds · γ (Ω) ≤ 2D1 ∫ t 0 eLsω (s, 1) ds · γ (Ω) ; χ ( {BW−1 ( x1 − G(T )ψ(0)− ζ ◦ Sf(t) : f ∈ Q (Ω[ψ])} ) ≤ M1σχ ({ζ ◦ Sf(t) : f ∈ Q (Ω[ψ])}) = M1σχ ({Sf(T ) : f ∈ Q (Ω[ψ])}) ≤M1σ2D1 ∫ T 0 eLsω (s, 1) ds · γ (Ω) = 2M1σD1 ∫ T 0 eLsω (s, 1) ds · γ (Ω) , where σ = supt∈[0,T ] σ(t). Using Lemma 3.5, we have for each t ∈ [0, T ] : χ ( S ◦BW−1 ( x1 − G(T )ψ(0)− ζ ◦ Sf(t) : f ∈ Q (Ω[ψ]) ) ≤ 2D1 ∫ t 0 2M1σD1 (∫ T 0 eLsω (s, 1) ds ) dτ · γ (Ω) ≤ 4D1M1σT ∫ T 0 eLsω (s, 1) ds · γ (Ω) . Inequality (3.7) and the last inequality imply the following γ(Ω) ≤ sup t∈[0,T ] ( 2D1 (1 + 2M1σD1T ) ∫ t 0 e−L(t−s)ω (s, 1) ds ) γ (Ω) = q2 · γ (Ω) , therefore γ (Ω) = 0, thus ϕ (Ω[ψ]t) = 0 for each t ∈ [0, T ]. Now we will show that the set Ω is equicontinuous. We take sequences {yn}∞n=1 ⊂ Ω, n ≥ 1 and {fn}∞n=1, fn ∈ Q(yn[ψ]). From conditions (Q2) and (Q3) it follows that the sequence {fn}∞n=1 is Lp-semicompact, and therefore by Lemma 3.4 the sequence {Sfn}∞n=1 Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 54 M. AFANASOVA, V. OBUKHOVSKII, G. PETROSYAN is relatively compact. Hence modC({Sfn}∞n=1) = 0. From the conditions that the operatorsB,W−1, ζ are bounded and linear, we can conclude that modC ( S ◦BW−1(x1 − G(T )ψ(0)− ζ{Sfn}∞n=1) ) = 0. Thus ν ( {S ◦ Q (Ω[ψ]) + S ◦BW−1 ( x1 − G(T )ψ(0)− ζ ◦ S ◦ Q(Ω[ψ]) ) } ) = (0, 0), but then it follows from the inequality (3.6) that ν(Ω) = (0, 0), and the last expression yields that the set Ω is relatively compact. To prove the main theorem of the paper, we need the following statements, known as the Gronwall - Bellman Lemma and the generalized Gronwall - Bellman Lemma. Lemma 3.6: Let v(t) and f(t) be nonnegative continuous functions on the segment [a, b], moreover v(t) ≤ c+ ∫ t a f(s)v(s)ds, t ∈ [a, b], where c is a positive constant. Then for each t ∈ [a, b] the inequality v(t) ≤ ce ∫ t a f(s)ds, holds. Lemma 3.7: Let h(t), u(t) and v(t) be nonnegative functions integrable on [a, b] satisfying the inequality: v(t) ≤ u(t) + ∫ t a h(s)v(s)ds, t ∈ [a, b]. Then the following inequality holds: v(t) ≤ u(t) + ∫ t a e ∫ t a h(θ)dθh(s)u(s)ds, t ∈ [a, b]. Theorem 3.3: Let a causal multioperator Q : C((−∞, T ];E)→ Cv(Lp([0, T ];E)), 1 ≤ p ≤ ∞, satisfy conditions (Q1)–(Q3) and a linear causal operator S : Lp([0, T ];E)→ C([0, T ];E) satisfy conditions (S1)–(S3). Then, under conditions (I1), (I2), (W ) the set Σψ of all solutions to problem (3.1)-(3.4) is a non-empty compact set. Proof Let us show that the set of all solutions y ∈ D of a one-parameter inclusion y ∈ λΓ(y), λ ∈ [0, 1], (3.8) is a priori bounded. We divide the proof into three cases: p = 1, 1 < p <∞, p =∞. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) A CONTROLLABILITY PROBLEM FOR CAUSAL FUNCTIONAL INCLUSIONS WITH AN INFINITE DELAY55 Let p = 1, if y ∈ D satisfies condition (3.8), then for each t ∈ [0, T ], using assumptions (B0), (S1), (I1), (I2) and (3.5), we have the following estimates: ‖y(t)‖E ≤ ‖G(t) ( ψ(0) + Σtk 0 such that ‖Ikx‖ ≤ N for all x ∈ E. Notice that when q = 1 : G(t) = eAt, T (t) = eAt, therefore, in accordance with [24], a function y ∈ C((−∞, T ];E), is a mild solution of problem (4.10)-(4.12), if it can be represented in the form: y(t) =  eAt ( ψ(0) + Σtk 1/q satisfies condition (Q1) can be verified as in the paper [28]. Conditions (Q2) and (Q3) for PF follow from (F3) and (F4), respectively. Taking into account Lemma 3.1, we can consider the relation (4.13) as a special case of functional inclusion (3.1) with Q = PF , and S = G is the Cauchy type operator. As a direct consequence of Theorem 3.3, we obtain the following result. Theorem 4.2: Suppose that conditions (A), (F1)–(F4), (I1), (I2), (W ) hold. Then the set of solutions to problem (4.13)-(4.15), (3.4) is a non-empty compact subset of the space C((−∞, T ];E). ACKNOWLEDGEMENTS The work of the first author was supported by the State contract of the Russian Ministry of Education as part of the state task (contract FZGF-2020-0009). 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Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) Introduction Preliminaries Multivalued maps and measures of noncompactness Phase space Causal multioperators with infinite delay The controllability problem for functional inclusions with the causal operators Controllability problems for semilinear differential inclusions with a delay and impulse effects Controllability problems for a first order semilinear functional differential inclusions Controllability problems for fractional semilinear functional differential inclusions