In this paper the concepts of lower mild and upper mild solutions combined with a fixed point theorem for condensing maps and the semigroup theory are used to investigate the existence of mild solutions for first order impulsive semilinear evolution inclusions

Benchohra, M.

Henderson, Johnny

Ntouyas, S.

Electronic Journal of Qualitative Theory of Differential Equations

Johnny Henderson

Mouffak Benchohra

Sotiris Ntouyas

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Electronic journal of qualitative theory of differential equations

Existence results for first order impulsive semilinear evolution inclusions

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EXISTENCE RESULTS FOR FIRST ORDER IMPULSIVESEMILINEAR EVOLUTION INCLUSIONSM. Benchohra1, J. Henderson2 and S.K. Ntouyas31 Department of Mathematics, University of Sidi Bel AbbesBP 89 2000 Sidi Bel Abbes Algeriae-mail: benchohra@yahoo.com2 Department of Mathematics, Auburn UniversityAuburn, Alabama 36849-5310 USAe-mail: hendej2@mail.auburn.edu3 Department of Mathematics, University of Ioannina451 10 Ioannina Greecee-mail: sntouyas@cc.uoi.grAbstractIn this paper the concepts of lower mild and upper mild solutions combinedwith a fixed point theorem for condensing maps and the semigroup theory areused to investigate the existence of mild solutions for first order impulsive semi-linear evolution inclusions.Key words and phrases: Initial value problem, impulsive differential inclusions,convex multivalued map, condensing map, fixed point, truncation map, upper mildand lower mild solutions.AMS (MOS) Subject Classifications: 34A37, 34A60, 34G20, 35R101. INTRODUCTIONThis paper is concerned with the existence of mild solutions for the impulsive semi-linear evolution inclusion of the form:y′ − A(t)y ∈ F (t, y), t ∈ J = [0, b], t 6= tk, k = 1, . . . , m, (1.1)y(t+k ) = Ik(y(t−k )), k = 1, . . . , m, (1.2)y(0) = a, (1.3)where F : J × E −→ 2E is a closed, bounded and convex valued multivalued map,a ∈ E, A(t), t ∈ J a linear closed operator from a dense subspace D(A(t)) of E intoEJQTDE, 2001 No. 1, p. 1E and E a real ordered Banach space with the norm | · |, 0 = t0 < t1 < . . . < tm <tm+1 = b, Ik ∈ C(E, E) (k = 1, 2, . . . , m), y(t−k ) and y(t+k ) represent the left and rightlimits of y(t) at t = tk, respectively.Impulsive differential equations have become more important in recent years insome mathematical models of real world phenomena, specially in the biological ormedical domain (see the monographs of Lakshmikantham et al [11], and Samoilenkoand Perestyuk [17], and the papers of Agur et al [1], Erbe et al [5], Goldbeter et al [6],Kirane and Rogovchenko [10], Liu et al [13] and Liu and Zhang [14]).This paper will be divided into three sections. In Section 2 we will recall brieflysome basic definitions and preliminary facts from multivalued analysis which will beused throughout Section 3. In Section 3 we establish two existence theorems for (1.1)-(1.3). The first result of this section is new even for the multivalued problem withoutimpulses (see [3]); i.e., Ik(y) = y for k = 1, . . . , m. Our approach is based on theconcepts of upper mild and lower mild solutions combined with a fixed point theoremfor condensing maps due to Martelli [15] and the semigroup theory [16].The notions of lower mild and upper mild solutions for differential equations inordered Banach spaces can be found in the book of Heikkila and Lakshmikantham [8].In our results we do not assume any type of monotonicity condition on Ik, k =1, . . . , m which is usually the situation in the literature, see for instance, [5], [10] and[13].2. PRELIMINARIESWe will briefly recall some basic definitions and facts from multivalued analysis thatwe will use in the sequel.Let the Banach space E be partially ordered by a cone P of E, i.e., u ≤ v if andonly if v − u ∈ P. Let K := {u ∈ C(J, E); u(t) ≥ 0 for all t ∈ J}. Then K is a conein the space C(J, E), and C(J, E) is partially ordered by K, that is u ≤ v if and onlyif v − u ∈ K: i.e., u(t) ≤ v(t) for all t ∈ J . The properties of the cone and the partialorder may found in the monograph by Guo and Lakshmikantham [7].B(E) denotes the Banach space of bounded linear operators from E into E.A measurable function y : J −→ E is Bochner integrable if and only if |y| isLebesgue integrable. For properties of the Bochner integral, we refer to Yosida [18].L1(J, E) denotes the Banach space of functions y : J −→ E which are Bochnerintegrable normed by‖y‖L1 =∫ b0|y(t)|dt for all y ∈ L1(J, E).Let (X, | · |) be a Banach space. A multivalued map G : X −→ 2X is convex(closed) valued if G(x) is convex (closed) for all x ∈ X. G is bounded on bounded setsEJQTDE, 2001 No. 1, p. 2if G(B) = ∪x∈BG(x) is bounded in X for any bounded set B of X (i.e. supx∈B{sup{|y| :y ∈ G(x)}} < ∞). G is called upper semicontinuous (u.s.c.) on X if for each x∗ ∈ X theset G(x∗) is a nonempty, closed subset of X, and if for each open set B of X containingG(x∗), there exists an open neighbourhood V of x∗ such that G(V ) ⊆ B. G is said to becompletely continuous if G(B) is relatively compact for every bounded subset B ⊆ X.If the multivalued map G is completely continuous with nonempty compact values, thenG is u.s.c. if and only if G has a closed graph (i.e. xn −→ x∗, yn −→ y∗, yn ∈ G(xn)imply y∗ ∈ G(x∗). G has a fixed point if there is x ∈ X such that x ∈ G(x).In the following BCC(X) denotes the set of all nonempty bounded, closed andconvex subsets of X.A multivalued map G : J −→ BCC(E) is said to be measurable if for each x ∈ Ethe function Y : J −→ IR defined byY (t) = d(x, G(t)) = inf{|x− z| : z ∈ G(t)}is measurable. An upper semi-continuous map G : X −→ 2X is said to be condensing [2]if for any subset B ⊆ X with α(B) 6= 0, we have α(G(B)) < α(B), where α denotes theKuratowski measure of noncompacteness [2]. We remark that a completely continuousmultivalued map is the easiest example of a condensing map. For more details onmultivalued maps see the books of Deimling [4] and Hu and Papageorgiou [9].Definition 2.1 A multivalued map F : J×E −→ 2E is said to be an L1-Carathe´odoryif(i) t 7−→ F (t, y) is measurable for each y ∈ E;(ii) y 7−→ F (t, y) is upper semicontinuous for almost all t ∈ J ;(iii) For each ρ > 0, there exists hρ ∈ L1(J, IR+) such that ‖F (t, y)‖ = sup{|v| : v ∈F (t, y)} ≤ hρ(t) for all |y| ≤ ρ and for almost all t ∈ J .In order to define the solution of (1.1)-(1.3) we shall consider the following spaceΩ = {y : J = [0, b] −→ E : yk ∈ C(Jk, E), k = 0, . . . , m and there existy(t−k ) and y(t+k ), k = 1, . . . , m with y(t−k ) = y(tk)}}which is a Banach space with the norm‖y‖Ω = max{‖yk‖∞, k = 0, . . . , m},where yk is the restriction of y to Jk = [tk, tk+1], k = 0, . . . , m.So let us start by defining what we mean by a solution of problem (1.1)-(1.3).EJQTDE, 2001 No. 1, p. 3Definition 2.2 A function y ∈ Ω is said to be a mild solution of (1.1)-(1.3) (see [16])if there exists a function v ∈ L1(J, E) such that v(t) ∈ F (t, y(t)) a.e. on Jk, andy(t) =T (t, 0)a +∫ t0T (t, s)v(s)ds, t ∈ J0Ik(y(t−k )) +∫ ttkT (t, s)v(s)ds, t ∈ Jk.Now we introduce the notions of lower mild and upper mild solutions for the problem(1.1)-(1.3), which are the basic tools in the approach that follows.Definition 2.3 A function y ∈ Ω is said to be a lower mild solution of (1.1)-(1.3) ifthere exists a function v1 ∈ L1(J, E) such that v1(t) ∈ F (t, y(t)) a.e. on Jk, andy(t) ≤T (t, 0)a +∫ t0T (t, s)v1(s)ds, t ∈ J0Ik(y(t−k )) +∫ ttkT (t, s)v1(s)ds, t ∈ Jk.Similarly a function y ∈ Ω is said to be an upper mild solution of (1.1)-(1.3) if thereexists a function v2 ∈ L1(J, E) such that v2(t) ∈ F (t, y(t)) a.e. on Jk, andy(t) ≥T (t, 0)a +∫ t0T (t, s)v2(s)ds, t ∈ J0Ik(y(t−k )) +∫ ttkT (t, s)v2(s)ds, t ∈ Jk.For the multivalued map F and for each y ∈ C(Jk, E) we define S1F,y byS1F,y = {v ∈ L1(Jk, E) : v(t) ∈ F (t, y(t)) for a.e. t ∈ Jk}.Our main result is based on the following:Lemma 2.1 [12]. Let I be a compact real interval and X be a Banach space. LetF : I × X −→ BCC(X); (t, y) → F (t, y) measurable with respect to t for any y ∈ Xand u.s.c. with respect to y for almost each t ∈ I and S1F,y 6= ∅ for any y ∈ C(I, X)and let Γ be a linear continuous mapping from L1(I, X) to C(I, X), then the operatorΓ ◦ S1F : C(I, X) −→ BCC(C(I, X)), y 7−→ (Γ ◦ S1F )(y) := Γ(S1F,y)is a closed graph operator in C(I, X)× C(I, X).Lemma 2.2 [15]. Let G : X −→ BCC(X) be a condensing map. If the setM := {y ∈ X : λy ∈ G(y) for some λ > 1}is bounded, then G has a fixed point.EJQTDE, 2001 No. 1, p. 43. MAIN RESULTSWe are now in a position to state and prove our first existence result for problem(1.1)-(1.3).Theorem 3.1 Let t0 = 0, tm+1 = b and assume that F : J × E −→ BCC(E) is anL1-Carathe´odory multivalued map. In addition suppose:(H1) A(t), t ∈ J is continuous such thatA(t)y = limh→0+T (t + h, t)y − yh, y ∈ D(A(t)),where T (t, s) ∈ B(E) for each (t, s) ∈ γ := {(t, s); 0 ≤ s ≤ t ≤ b}, satisfying(i) T (t, t) = I (I is the identity operator in E),(ii) T (t, s)T (s, r) = T (t, r) for 0 ≤ r ≤ s ≤ t ≤ b,(iii) the mapping (t, s) 7−→ T (t, s)y is strongly continuous in γ for each y ∈ E;Suppose that |T (t, s)| ≤ M for (t, s) ∈ γ;(H2) F : J × E −→ BCC(E) is an L1-Carathe´odory multivalued map and for eachfixed y ∈ C(Jk, E) the setS1F,y ={v ∈ L1(Jk, E) : v(t) ∈ F (t, y(t)) for a.e. t ∈ Jk}is nonempty;(H3) There exist y, y repectively lower mild and upper mild solution for (1.1)-(1.3)such that y ≤ y;(H4) y(t+k ) ≤ min[y(t−k),y(t−k)]Ik(y) ≤ max[y(t−k),y(t−k)]Ik(y) ≤ y(t+k ), k = 1, . . . , m.(H5) T (t, s) is order-preserving for all (t, s) ∈ γ;(H6) For each bounded set B ⊆ C(Jk, E) and for each t ∈ Jk the set{∫ ttkT (t, s)v(s)ds : v ∈ S1F,B}is relatively compact in E, where S1F,B = ∪{S1F,y : y ∈ B} and k = 0, . . . , m.EJQTDE, 2001 No. 1, p. 5Then the problem (1.1)-(1.3) has at least one mild solution y ∈ Ω withy(t) ≤ y(t) ≤ y(t) for all t ∈ J.Remark 3.1 (i) If dimE < ∞, then for each y ∈ C(Jk, E), S1F,y 6= ∅ (see Lasotaand Opial [12]).(ii) If dimE = ∞ and y ∈ C(Jk, E) the set S1F,y is nonempty if and only if the functionY : J −→ IR defined byY (t) := inf{|v| : v ∈ F (t, y)}belongs to L1(J, IR) (see Hu and Papageorgiou [9]).(iii) If T (t, s), (t, s) ∈ γ is completely continuous then (H6) is automatically satisfied.Proof. The proof is given in several steps.Step 1. Consider the problem (1.1)-(1.3) on J0 := [0, t1]y′ − A(t)y ∈ F (t, y(t)), a.e. t ∈ J0, (3.1)y(0) = a. (3.2)We transform this problem into a fixed point problem. Let τ : C(J0, E) −→ C(J0, E)be the truncation operator defined by(τy)(t) =y(t), if y < y(t);y(t), if y(t) ≤ y ≤ y(t);y(t)), if y(t) < y.Consider the modified problemy′ − A(t)y ∈ F (t, (τy)(t)), a.e. t ∈ J0, (3.3)y(0) = a. (3.2)SetC0(J0, E) := {y ∈ C(J0, E) : y(0) = a}.A solution to (3.3)-(3.2) is a fixed point of the operator G : C0(J0, E) −→ 2C0(J0,E)defined byG(y) :={h ∈ C0(J0, E) : h(t) = T (t, 0)a +∫ t0T (t, s)v(s)ds : v ∈ S˜1F,τy}whereS˜1F,τy = {v ∈ S1F,τy : v(t) ≥ v1(t) a.e. on A1 and v(t) ≤ v2(t) a.e. on A2}andS1F,τy = {v ∈ L1(J0, E) : v(t) ∈ F (t, (τy)(t)) for a.e. t ∈ J0},A1 = {t ∈ J : y(t) < y(t) ≤ y(t)}, A2 = {t ∈ J : y(t) ≤ y(t) < y(t)}.EJQTDE, 2001 No. 1, p. 6Remark 3.2 For each y ∈ C(J, E) the set S˜1F,τy is nonempty. Indeed, by (H2) thereexists v ∈ S1F,y. Setw = v1χA1 + v2χA2 + vχA3,whereA3 = {t ∈ J : y(t) ≤ y(t) ≤ y(t)}.Then by decomposability w ∈ S˜1F,τy.We shall show that G satisfies the assumptions of Lemma 2.2.Claim 1: G(y) is convex for each y ∈ C0(J0, E).This is obvious since S˜1F,τy is convex (because F has convex values).Claim 2: G sends bounded sets into relatively compact sets in C0(J0, E).This is a consequence of the boundedness of T (t, s), (t, s) ∈ γ and the L1-Carathe´dorycharacter of F . As a consequence of Claim 2 together with the Ascoli-Arzela theoremwe can conclude that G : C0(J0, E) −→ 2C0(J0,E) is a compact multivalued map, andtherefore, a condensing map.Claim 3: G has a closed graph.Let yn −→ y∗, hn ∈ G(yn) and hn −→ h∗. We shall prove that h∗ ∈ G(y∗).hn ∈ G(yn) means that there exists vn ∈ S˜F,τyn such thathn(t) = T (t, 0)a +∫ t0T (t, s)vn(s)ds, t ∈ J0.We must prove that there exists v∗ ∈ S˜1F,τy∗such thath∗(t) = T (t, 0)a +∫ t0T (t, s)v∗(s)ds, t ∈ J0.Consider the linear continuous operator Γ : L1(J0, E) −→ C(J0, E) defined by(Γv)(t) =∫ t0T (t, s)v(s)ds.We have‖(hn − T (t, 0)a)− (h∗ − T (t, 0)a)‖∞ −→ 0 as n −→∞.From Lemma 2.1, it follows that Γ ◦ S˜1F is a closed graph operator.Also from the definition of Γ we have thathn(t)− T (t, 0)a ∈ Γ(S˜1F,τyn).EJQTDE, 2001 No. 1, p. 7Since yn −→ y∗, it follows from Lemma 2.1 thath∗(t) = T (t, 0)a +∫ t0T (t, s)v∗(s)ds, t ∈ J0for some v∗ ∈ S˜1F,τy∗.Claim 4: Now, we show that the setM := {y ∈ C0(J0, E) : λy ∈ G(y) for some λ > 1}is bounded.Let y ∈ M, then λy ∈ G(y) for some λ > 1. Thus there exists v ∈ S˜1F,τy such thaty(t) = λ−1T (t, 0)a + λ−1∫ t0T (t, s)v(s)ds, t ∈ J0.Thus|y(t)| ≤ M |a| + M∫ t0|v(s)|ds, t ∈ J0.From the definition of τ there exists φ ∈ L1(J, IR+) such that‖F (t, (τy)(t))‖ = sup{|v| : v ∈ F (t, (τy)(t))} ≤ φ(t) for each y ∈ C(J, E).Thus we obtain‖y‖∞ ≤ M |a| + M‖φ‖L1 .This shows that M is bounded. Hence, Lemma 2.2 applies and G has a fixed pointwhich is a mild solution to problem (3.3)-(3.2).Claim 5: We shall show that the solution y of (3.3)-(3.2) satisfiesy(t) ≤ y(t) ≤ y(t) for all t ∈ J0.Let y be a solution to (3.3)-(3.2). We prove thaty(t) ≤ y(t) for all t ∈ J0.Suppose not. Then there exist e1, e2 ∈ J0, e1 < e2 such that y(e1) = y(e1) andy(t) > y(t) for all t ∈ (e1, e2).In view of the definition of τ one hasy(t) ∈ T (t, e1)y(e1) +∫ te1T (t, s)F (s, y(s))ds a.e. on (e1, e2).EJQTDE, 2001 No. 1, p. 8Thus there exists v(t) ∈ F (t, y(t)) a.e. on (e1, e2) with v(t) ≥ v1(t) a.e. on (e1, e2) suchthaty(t) = T (t, e1)y(e1) +∫ te1T (t, s)v(s)ds t ∈ (e1, e2).Since y is a lower mild solution to (1.1)-(1.2), theny(t)− T (t, e1)y(e1) ≤∫ te1T (t, s)v1(s)ds, t ∈ (e1, e2).Since y(e1) = y(e1) and v(t) ≥ v1(t), it follows thaty(t) ≤ y(t) for all t ∈ (e1, e2)which is a contradiction since y(t) < y(t) for all t ∈ (e1, e2). Consequentlyy(t) ≤ y(t) for all t ∈ J0.Analogously, we can prove thaty(t) ≤ y(t) for all t ∈ J0.This shows that the problem (3.1)-(3.2) has a mild solution in the interval [y, y].Since τ(y) = y for all y ∈ [y, y], then y is a mild solution to (1.1)-(1.3). Denote thissolution by y0.Step 2. Consider now the following problem on J1 := [t1, t2]y′ − A(t)y ∈ F (t, y(t)), a.e. t ∈ J1, (3.4)y(t+1 ) = I1(y0(t−1 )), (3.5)and the modified problemy′(t) ∈ F (t, (τy)(t)), a.e. t ∈ J1, (3.6)y(t+1 ) = I1(y0(t−1 )). (3.5)Since y0(t−1 ) ∈ [y(t−1 ), y(t−1 )], then (H4) implies thaty(t+1 ) ≤ I1(y0(t−1 )) ≤ y(t+1 ),that isy(t+1 ) ≤ y(t+1 )) ≤ y(t+1 ).Using the same reasoning as that used for problem (3.1)-(3.2) we can conclude tothe existence of at least one mild solution y to (3.6)-(3.5).EJQTDE, 2001 No. 1, p. 9We now show that this solution satisfiesy(t) ≤ y(t) ≤ y(t) for all t ∈ J1.We first show thaty(t) ≤ y(t) on J1.Assume this is false, then since y(t+1 ) ≥ y(t+1 ), there exist e3, e4 ∈ J1 with e3 < e4such that y(e3) = y(e3) and y(t) < y(t) on (e3, e4).Consequently,y(t)− T (e3, t)y(e3) =∫ te3T (t, s)v(s)ds, t ∈ (e3, e4),where v(t) ∈ F (t, y(t)) a.e. on J1 with v(t) ≥ v1(t) a.e. on (e3, e4).Since y is a lower mild solution to (1.1)-(1.3), theny(t)− T (e3, t)y(e3) ≤∫ te3v1(s)ds, t ∈ (e3, e4).It follows thaty(t) ≤ y(t) on (e3, e4),which is a contradiction. Similarly we can show that y(t) ≤ y(t) on J1. Hence y is asolution of (1.1)-(1.3) on J1. Denote this by y1.Step 3. Continue this process and construct solutions yk ∈ C(Jk, E), k = 2, . . . , mtoy′ − A(t) ∈ F (t, (τy)(t)), a.e. t ∈ Jk, (3.7)y(t+k ) = Ik(y(t−k )), (3.8)with y(t) ≤ yk(t) ≤ y(t), t ∈ Jk := [tk, tk+1]. Theny(t) =y0(t), t ∈ [0, t1]y1(t), t ∈ (t1, t2]...ym−1(t), t ∈ (tm−1, tm]ym(t), t ∈ (tm, b]is a mild solution of (1.1)-(1.3). 2Using the same reasoning as that used in the proof of Theorem 3.1 we can obtainthe following result.EJQTDE, 2001 No. 1, p. 10Theorem 3.2 Let t0 = 0, tm+1 = b and assume that F : J × E −→ BCC(E) is anL1-Carathe´odory multivalued map. In addition to (H1), (H2) and (H6) suppose thatthe following hypotheses hold(H7) there exist functions {rk}k=mk=0 and {sk}k=mk=0 with rk, sk ∈ C(Jk, E),s0(0) ≤ a ≤ r0(0) and sk(t) ≤ rk(t) for t ∈ Jk, k = 0, . . . , m andsk+1(t+k+1) ≤ min[sk(t−k+1),rk(t−k+1)]Ik+1(y)≤ max[sk(t−k+1),rk(t−k+1)]Ik+1(y) ≤ rk+1(t+k+1), k = 0, . . . , m− 1.(H8) there exist v1,k, v2,k ∈ L1(Jk, E), with v1,k(t) ∈ F (t, sk(t)), v2,k(t) ∈ F (t, rk(t)) a.e.on Jk such that for each k = 0, . . . , m∫ tzkT (t, s)v1,k(s)ds ≥ sk(t)− sk(zk),∫ tzkT (t, s)v2,k(s)ds ≥ rk(t)− rk(zk), with t, zk,∈ Jk.Then the problem (1.1)-(1.3) has at least one mild solution.REFERENCES[1] Z. Agur, L. Cojocaru, G. Mazur, R. M. Anderson, and Y.L. Danon, Pulse massmeasles vaccination across age cohorts, Proc. Natl. Acad. Sci. USA 90 (1993), 11698-11702.[2] J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces, MarcelDekker, New York, 1980.[3] M. 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Existence results for first order impulsive semilinear evolution inclusions

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