Dynamic Systems and Applications 26 (2017) 411-424 IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH CAUSAL OPERATORS TAHIRA JABEEN, RAVI P. AGARWAL, DONAL O’REGAN, AND VASILE LUPULESCU Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan Department of Mathematics, Texas A&M University-Kingville, Kingsville, TX, USA School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland Constantin Brancusi University, Republicii 1, 210152 Targu-Jiu, Romania ABSTRACT. In this paper we present an existence result for a class of impulsive differential equationswith causaloperatorsandprovethatthe solutionsetis compactin the space ofregulated functions. The results are obtained under conditions with respect to the Hausdorff measure of noncompactness. An application from optimal control is given to illustrate our main result. KeyWordsandPhrases: Impulsivedifferentialequations,Causaloperators,Regulatedfunctions, Measure of noncompactness, Optimal control AMS (MOS) Subject Classification. 34A37; 34K05 1. Introduction and preliminaries The study of functional equations with causal operators has recently been devel- oped and some results on existence, stability and control are found in the monographs [7, 14, 23]. The term causal operators or Volterra abstract operator was introduced by Tonelli [39] (see also Tikhonov [38], [40]). The theory of these operators has the advantage of unifying some classes of differential equations as: ordinary differential equations, integrodifferential equations, differential equations with finite or infinite delay, Volterra integral equations, and neutral functional equations, and so on. Many papers in the literature address various aspects of the theory of causal operators. Control problems involving causal operators were studied in [4, 8, 18, 36]. A new class of abstract integral equations has been introduced in [17]. We note that differential equations with causal operators were studied by several authors, see [1], [3], [9]–[12], [19], [25]–[33] and the references therein. The properties of the solutions of the differential equations with causal operators were studied in [2, 21, 34, 35, 40]. The existence of solutions for impulsive differential equations with causal operators in finite dimensional spaces were studied in [19, 26]. Received May 22, 2017 1056-2176 $15.00 (cid:13)cDynamic Publishers, Inc. 412 T. JABEEN, R. P. AGARWAL, D. O’REGAN, AND V. LUPULESCU Let E be a real separable Banach space endowed with the norm k·k. For x ∈ E and r > 0 let B (x) := {y ∈ E;ky − xk < r} be the open ball centered at x with r radius r, and let B [x] be its closure. The space of all (classes of) strongly measurable r functions u(·) : [0,b] → E such that b 1/p ku(·)k := ku(t)kp < ∞ p (cid:18)Z (cid:19) 0 for1 ≤ p < ∞andku(·)k := esssup ku(t)k < ∞,willbedenotedbyLp([0,b],E). ∞ t∈[0,b] This is a Banach space with respect to the norm ku(·)k . We denote by PC([0,b],E) p the set of all functions u : [0,b] → E such that u is continuous at t 6= t , left continu- k ous at t = t and the right limit u(t+) exists for k = 1,2,...,m. Then PC([0,b],E) k k is a Banach space with respect to the norm ku(·)k = sup ku(t)k. 0≤t≤b The following definition of causal operator was given by Tonelli [39]. An operator Q : PC([0,b],E) → Lp ([0,b],E) is a causal operator or a Volterra loc operator if, for each τ ∈ [0,b) and for all u(·),v(·) ∈ PC([0,b],E) with u(t) = v(t) for every t ∈ [0,τ], we have Qu(t) = Qv(t) for a.e. t ∈ [0,τ]. In this paper we study the following functional impulsive differential equation: u′(t) = (Qu(t)), a.e. t ∈ [0,b]r{t ,t ,...,t }, 1 2 m  (1.1) u(t+) = u(t−)+I (u(t−)), k = 1,2,...,m,  k k k k  u(0) = ξ,     where Q : PC([0,b],E) → Lp([0,b],E), 1 ≤ p ≤ ∞, is a continuous causal operator, ξ ∈ E, m ∈ N, 0 = t < t < t < ··· < t < t = bandI : E → E isacontinuous 0 1 2 m m+1 k operator for each k = 1,2,...,m. Now we provide some examples of impulsive differential equations that can be included in impulsive differential equations with causal operators of the form (1.1). The impulsive differential equation u′(t) = F(t,u(t), a.e. t ∈ [0,b]r{t ,t ,...,t }, 1 2 m  u(t+) = u(t−)+I (u(t−)), k = 1,2,...,m,  k k k k  u(0) = ξ,     can be considered as a causal impulsive differential equations by identifying F(t,u(t)) with (Qu)(t). Another example is the general integro-differential equation u′(t) = F(t,u(t), tK(t,s,u(s))ds), a.e. t ∈ [0,b]r{t ,t ,...,t },  0 1 2 m (1.2) u(t+) = u(t−)+IR(u(t−)), k = 1,2,...,m,  k k k k  u(0) = ξ.     IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS 413 Also, the differential equation with “maxima”: u′(t) = F t,u(t), maxu(s) , a.e. t ∈ [0,b]r{t ,t ,...,t },  (cid:18) 0≤s≤t (cid:19) 1 2 m  u(t+k) = u(t−k)+Ik(u(t−k)), k = 1,2,...,m, u(0) = ξ,     is anotherexample of a causal impulsive differential equation. Finally, we remark that the Fredholm operator, given by a (Qu)(t) = K(t,s,u(s))ds, Z 0 is a causal operator if and only if K(t,s,u) ≡ 0 for t < s < a. We denote by χ(A) the Hausdorff measure of non-compactness of a nonempty bounded set A ⊂ E, and it is defined by ([13], [20]): χ(A) = inf{ε > 0;A admits a finite cover by balls of radius ≤ ε}. This is equivalent to the measure of non-compactness introduced by Kuratowski (see [13], [20]). If dim(A) = sup{kx−yk;x,y ∈ A} is the diameter of the bounded set A, then we have that χ(A) ≤ dim(A) and χ(A) ≤ 2d if sup kxk ≤ d. We recall some x∈A properties of χ (see [13], [20]). If A,B are bounded subsets of E and A denotes the closure of A, then (1) χ(A) = 0 if and only if A is compact; (2) χ(A) = χ(A) = χ(co(A)); (3) χ(λA) = |λ|χ(A) for every λ ∈ R; (4) χ(A) ≤ χ(B) if A ⊂ B; (5) χ(A+B) = χ(A)+χ(B); (6) If T : E → E is a bounded linear operator, then γ(TA) ≤ kTkγ(A). If V ⊂ PC([0,b],E) is equicontinuous, then χ (V) := sup χ(V(t)), PC t∈[0,b] where V(t) := {u(t) : u(·) ∈ V}, is the Hausdorff measure of non-compactness in the space PC([0,b],E) (see [13]). We recall the following lemma due to Kisielewicz [22, Lemma 2.2]. Lemma 1.1. Let {u (·);n ≥ 1} be a subset in L1([0,b],E) for which there exists n m(·) ∈ L1([0,b],R ) such that ku (t)k ≤ m(t) for each n ≥ 1 and for a.e. t ∈ [0,b]. + n Then the function t 7→ χ(t) := χ({u (t);n ≥ 1}) is integrable on [0,b] and, for each n t ∈ [0,b], we have t t χ u (t)dt;n ≥ 1 ≤ χ(t)dt. n (cid:18)(cid:26)Z (cid:27)(cid:19) Z 0 0 414 T. JABEEN, R. P. AGARWAL, D. O’REGAN, AND V. LUPULESCU 2. An existence result Consider the following functional impulsive differential equation: u′(t) = (Qu(t)), a.e. t ∈ [0,b]r{t ,t ,...,t }, 1 2 m  (2.1) u(t+) = u(t−)+I (u(t−)), k = 1,2,...,m,  k k k k  u(0) = ξ,    where Q : PC([0,b],E) → Lp([0,b],E), 1 ≤ p ≤ ∞, is a continuous causal operator, ξ ∈ E, m ∈ N, 0 = t < t < t < ··· < t < t = b, I : E → E is continuous for 0 1 2 m m+1 k each k = 1,2,...,m. We consider the following assumptions: (H1) Q : PC([0,b],E) → Lp([0,b],E), 1 ≤ p ≤ ∞, is a continuous causal operator, and I : E → E is continuous for each k = 1,2,...,m. k (H2) For each r > 0 there exist ψ,η ∈ Lp([0,b],R+) such that, for each u(·) ∈ PC([0,b],E) with sup ku(t)k ≤ r, we have 0≤t≤b k(Qu)(t)k ≤ ψ(t) for a.e. t ∈ [0,b], t I (u(t−)) ≤ η(τ)dτ for s,t ∈ [0,b] with s < t. k k Z s 0 (k = 1,2,...,m) such that A B (2.2) χ((QA)(t)) ≤ γ χ(A(t)), A and χ(I (B)) ≤ δkχ(B), k = 1,2,...,m, k B for all t ∈ [0,b], where (QA)(t) := {(Qu)(t) : u(·) ∈ A}. By solution of (2.1) we mean a function u(·) : [0,b] → E such that u(0) = ξ, u(·) is continuous on (t ,t ) for k = 1,2,...,m, u′(t) = (Qu)(t) for a.e. t ∈ [0,b]r k k+1 {t ,t ,...,t }, and u(t+) = u(t−)+I (u(t−)), k = 1,2,...,m. 1 2 m k k k k It is easy to show that (see [13]) a function u(·) ∈ PC([0,b],E) is a solution for (2.1) on [0,b], if and only if t (2.3) u(t) = u(0)+ (Qu)(s)ds+ I (u(t−)) for t ∈ [0,b]. Z k k 0 0 0 be any number and let r := kξk+δ. We choose T ∈ (0,b] such that T T ψ(s)ds ≤ δ/4 and η(s)ds ≤ δ/4 Z Z 0 0 and we consider the set B defined as follows B = {u ∈ PC([0,T],E); ku(t)−ξk ≤ δ}. Further on, we consider the integral operator Λ : B → PC([0,T],E) given by t (Λu)(t) = ξ + (Cu)(s)ds+ I (u(t−)), for t ∈ [0,T] Z k k 0 0 0. On t t the closed set [0,T], the functions t 7→ ψ(s)ds and t 7→ η(s)ds, are uniformly 0 0 continuous, and so there exist η > 0 suchRthat R t t ψ(τ)dτ ≤ ε/4 and η(τ)dτ ≤ ε/4 (cid:12)Z (cid:12) (cid:12)Z (cid:12) (cid:12) s (cid:12) (cid:12) s (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) for every t,s ∈ [0,T] (cid:12)with |t−s|(cid:12)< η. Let t,s(cid:12)∈ [0,T] are(cid:12) such that |t−s| ≤ η. If we suppose that 0 ≤ s ≤ t ≤ T then, for each u(·) ∈ B, we have k(Λu)(t)−(Λu)(s)k t s = (Cu)(τ)dτ + I (u(t−))− (Cu)(τ)dτ − I (u(t−)) (cid:13)(cid:13)(cid:13)Z0 0 1. It is easy to see that if i ∈ {1,2,...,n − 1} and kui(t)k ≤ r for t ∈ n [0,iT/n], then kui+1(t)k ≤ r for t ∈ [0,iT/n] and, by (H ), k(Cui)(t)k ≤ ψ(t) for a.e. n 2 n t ∈ [0,iT/n] and t−T/n kI (ui−1(t−))k ≤ η(s)ds k n k Z 0 0, we can find n(ε) > 0 such that ψ(s)ds < ε/2 for t−T/n t ∈ [0,T] and n ≥ n(ε). Hence we have that R t t χ (CA)(s)ds = χ (Cu )(s)ds;n ≥ n(ε) m (cid:18)Z (cid:19) (cid:18)(cid:26)Z (cid:27)(cid:19) t−T/n t−T/n t ≤ 2 sup ψ(s)ds < ε. Z n≥n(ε) t−T/n Using the last inequality, we obtain that t χ(A(t)) ≤ χ (CA)(s)ds + χ (I A)(t−) . (cid:18)Z (cid:19) k k 0 0 0 such that 3 A (2.5) χ((CA)(t)) ≤ γ sup χ(A(s)) for every t ∈ [0,b]. A 0≤s≤t 3. An optimal control problem In the following, we shall establish necessary conditions for the existence of an optimal solution for the control problem: u′(t) = (Qu)(t), for a.e. t ∈ [0,T]r{t ,t ,...,t } 1 2 m  (3.1) u(t+) = u(t−)+I (u(t−)), k = 1,2,...,m,  k k k k  u(0) = ξ minimize g(u(T)),     where g(·) : E → R is a given function. For this aim, it will need to establish some preliminary results. For a fixed ξ ∈ E we denote by A (ξ) the attainable set of T Cauchy problem (2.1); that is, A (ξ) = {u(T);u(·) ∈ S (ξ)}. T T Lemma 3.1. Assume that Q : PC([0,b],E) → Lp([0,b],E) is a causal operator such that the condition (H )–(H ) hold. Then the multifunction S : E → PC([0,T],E) is 1 3 T upper semicontinuous. 420 T. JABEEN, R. P. AGARWAL, D. O’REGAN, AND V. LUPULESCU Proof. Let K be a closed set in PC([0,T],E) and G = {ξ ∈ E;S (ξ)∩K =6 ∅}. We T must show that G is closed in E. For this, let {ξ } be a sequence in G such that n n≥1 ξ → ξ. Further on, for each n ≥ 1, let u (·) ∈ S (ξ )∩K. Then n n T n t u (t) = ξ + (Qu )(s)ds+ I (u (t−)) n n Z n k n k 0 0

T. Jabeen, R. P. Agarwal, Donal O'Regan, and V. Lupulescu

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