This paper proves the existence of solutions for a third order initial value nonconvex differential inclusion. We start with an upper semicontinuous compact valued multifunction $F$ which is contained in a lower semicontinuous convex function $\partial V$ and show that,
$$x^{(3)}(t) \in F(x(t),x'(t),x''(t)),\quad x(0)=x_{0}, \quad x'(0)=y_{0}, \quad x''(0)=z_{0}.$

Hopkins, Britney

Electronic journal of qualitative theory of differential equations

Britney Hopkins

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Existence of solutions for nonconvex third order differential inclusions

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Electronic Journal of Qualitative Theory of Differential Equations2005, No. 22, 1-11; http://www.math.u-szeged.hu/ejqtde/Existence of Solutions for Nonconvex Third OrderDifferential Inclusions.Britney HopkinsDepartment of Mathematics and StatisticsUniversity of Arkansas at Little Rock2801 S. UniversityLittle Rock, AR 72204bjhopkins1@ualr.eduAbstractThis paper proves the existence of solutions for a third order initial value nonconvex dif-ferential inclusion. We start with an upper semicontinuous compact valued multifunction Fwhich is contained in a lower semicontinuous convex function ∂V and show that,x(3)(t) ∈ F (x(t), x′(t), x′′(t)), x(0) = x0, x′(0) = y0, x′′(0) = z0.Keywords: Nonconvex Differential InclusionsAMS Subject Classification: 34G20, 47H201 IntroductionThe origins of boundary and initial value problems for differential inclusions are in the theory ofdifferential equations and serve as models for a variety of applications including control theory.Existence results for the second order differential inclusion,x′′ ∈ F (x, x′), x(0) = x0, x′(0) = y0,have been obtained by many authors (see [4], [5] and the references therein). In [5], Lupulescushowed existence for the problemx′′ ∈ F (x, x′) + f(t, x, x′), x(0) = x0, x′(0) = y0for the case in which F is an upper semicontinuous compact valued multifunction such the F (x, y) ⊂∂V (y) and f is a Carathe´odory function.In this paper, we prove an existence result for the third order differential inclusion,x(3)(t) ∈ F (x(t), x′(t), x′′(t)), x(0) = x0, x′(0) = y0, x′′(0) = z0,where F is an upper semicontinuous compact valued multifunction and F (x, y, z) ⊂ ∂V (z) for someproper lower semicontinuous convex function V. Expounding upon the methods used to establishexistence by Lupulescu in [4] and [5], we define a sequence of approximate solutions on a giveninterval and show that the sequence converges to an actual solution.EJQTDE, 2005 No. 22, p. 12 PreliminariesLet Rm be an m dimensional Euclidean space with an inner product 〈·, ·〉 and norm ‖ · ‖.Let x ∈ Rm and r > 0. The open ball centered at x with radius r is defined byBr(x) = {y ∈ Rm : ‖x− y‖ < r},where Br(x) denotes its closure.For the proper lower semicontinuous convex function V : Rm → R, the multifunction ∂V :Rm → 2Rmdefined by∂V (x) = {γ ∈ Rm : V (y)− V (x) ≥ 〈γ, y − x〉, ∀y ∈ Rm} ,is the subdifferential of V.Let L2[a, b] be a Hilbert space with the inner product defined by〈x, y〉 =∫ bax(t)y(t)dt,where y(t) denotes the complex conjugate of y(t), and the norm is defined as‖x‖ =√∫ ba|x(t)|2dt.Let coF (x, y, z) denote the closed convex hull of F and xn ⇒ x denote that xn convergesuniformly to x.We need the following theorems from Aubin and Cellina [1].Theorem 0.3.4 Consider a sequence of absolutely continuous functions xk(·) from an intervalI to a Banach Space X satisfying(i) for every t ∈ I , xk(t)k is a relatively compact subset of X ;(ii) there exists a positive function c(·) ∈ L2(I) such that, for almost all t ∈ I , ‖x′k(t)‖ ≤ c(t).Then there exists a subsequence, again denoted by xk(·), converging to an absolutely continuousfunction x(·) from I to X in the sense that(i) xk(·) converges uniformly to x(·) over compact subsets of I ;(ii) x′k(·) converges weakly to x′k(·) in L2(I, X).EJQTDE, 2005 No. 22, p. 2Theorem 1.1.4 (the Convergence Theorem) Let F be a proper hemicontinuous map froma Hausdorff locally convex space X to the closed convex subsets of a Banach Space Y. Let I bean interval of R and xk(·) and yk(·) be measurable functions from I to Y respectively satisfyingfor almost all t in I and for every neighborhood ℵ of 0 in X × Y , there exists a k0 = k0(t,ℵ) suchthat for every k0 ≤ k, (xk(t), yk(t)) ∈ graph(F ) + ℵ. If,(i) xk(·) converges almost everywhere to a function x(·) from I to X ;(ii) yk(·) belongs to L2(I, Y ) and converges weakly to y(·) in L2(I, Y ),then, for almost all t ∈ I ,(x(t), y(t)) ∈ graph(F ) i.e.y(t) ∈ F (x(t)).We also need the following lemma from Brezis [2].Lemma 3.3 Let u ∈ D(V ) almost everywhere on [0, T ] and suppose g ∈ L2([0, T ], R) suchthat g(t) ∈ ∂V (u(t)) almost everywhere on [0, T ]. Then, the function t 7−→ V (u(t)) is absolutelycontinuous on [0, T ].Also, let t ∈ [0, T ] such that u(t) ∈ D(V ) and let u and V(u) be differentiable. Then for allt ∈ [0, T ]ddtV (u(t)) =〈h,dudt(t)〉∀h ∈ ∂V (u(t)).3 The Main ResultTHEOREM: If F : Ω → 2Rmand V : Rm → R satisfy the assumptions(A1) Ω ⊂ R2m where Ω is open and F : Ω → 2Rmis a compact valued upper semicontinuousmultifunction;(A2) there exists a lower semicontinuous proper convex function V : Rm → R such thatF (x, y, z) ⊂ ∂V (z) for every (x, y, z) ∈ Ω.Then, for every (x0, y0, z0) ∈ Ω, there exists a T > 0 and a solution x : [0, T ] → Rm ofx(3)(t) ∈ F (x(t), x′(t), x′′(t)), x(0) = x0, x′(0) = y0, x′′(0) = z0. (1)EJQTDE, 2005 No. 22, p. 3By a solution we are referring to an absolutely continuous function x : [0, T ] → Rm withabsolutely continuous first and second derivatives with the initial values x(0) = x0, x′(0) = y0,x′′(0) = z0, and x(3)(t) ∈ F (x(t), x′(t), x′′(t)), a.e. on [0, T ].PROOF: Suppose (x0, y0, z0) ∈ Ω. Then, K = B¯r(x0, y0, z0) ⊂ Ω for some r > 0 since Ω is open.By assumption (A1)F (K) =⋃(x,y,z)∈KF (x, y, z)is compact. Then there exists an M > 0 such thatsup{‖v‖ : v ∈ F (x, y, z), (x, y, z) ∈ K)} ≤ M. (2)SetT < min{rM,( rM) 12,( rM) 13,r2‖z0‖,r2‖y0‖,(2r3‖z0‖) 12}. (3)Let n, j be integers where 1 ≤ j ≤ n. Set tjn =jTn. For t ∈[tj−1n , tjn]define,xn(t) = xjn +(t− tjn)yjn +12(t− tjn)2zjn +16(t− tjn)3vjn, (4)where x0n = x0, y0n = y0, and z0n = z0.For 0 ≤ j ≤ n− 1 and vjn ∈ F(xjn, yjn, zjn), definexj+1n = xjn +(Tn)yjn +12(Tn)2zjn +16(Tn)3vjnyj+1n = yjn +(Tn)zjn +12(Tn)2vjnzj+1n = zjn +(Tn)vjn.(5)We claim that(x1n, y1n, z1n)∈ K. Using (5), we have∥∥z1n − z0∥∥ =∥∥∥∥z0n +(Tn)v0n − z0∥∥∥∥ ≤(Tn)M < r.EJQTDE, 2005 No. 22, p. 4As well as,∥∥y1n − y0∥∥ =∥∥∥∥∥y0n +(Tn)z0n +12(Tn)2v0n − y0∥∥∥∥∥≤(Tn)‖z0‖+12(Tn)2M<12r +12r = r.Also,∥∥x1n − x0∥∥ =∥∥∥∥∥x0n +(Tn)y0n +12(Tn)2z0n +16(Tn)3v0n − x0∥∥∥∥∥≤(Tn)‖y0‖+12(Tn)2‖z0‖+16(Tn)3M< T ‖y0‖+12T 2 ‖z0‖+16T 3M<12r +13r +16r = r.Hence the claim holds. Now suppose j ≥ 1. We make the assumption that,xjn = x0n + j(Tn)y0n +12(jTn)2z0n +16(Tn)3[(3j2 − 3j + 1)v0n +(3j2 − 9j + 7)v1n+(3j2 − 15j + 19)v2n + · · ·+ 7vj−2n + vj−1n ],yjn = y0n + j(Tn)z0n +12(Tn)2 [(2j − 1)v0n + (2j − 3)v1n + · · ·+ 3vj−2n + vj−1n],zjn = z0n +(Tn)[v0n + v1n + · · ·+ vj−1n].(6)To see this, let j = 1. Then,z1n = z0n +(Tn)v0n = z0+1n ,y1n = y0n +Tnz0n +12(Tn)2v0n = y0+1n ,x1n = x0n +Tny0n +12(Tn)2z0n +16(Tn)3v0n = x0+1n .EJQTDE, 2005 No. 22, p. 5Thus, (6) holds when j = 1. Let’s suppose assumption (6) holds for j > 1. Using (5) we see that,zj+1n = zjn +(Tn)vjn= z0n +(Tn)[v0n + v1n + · · ·+ vj−1n]+(Tn)vnj= z0n +(Tn)[v0n + v1n + · · ·+ vjn].Thus the assumption holds for zjn. Using this and (5) we have,yj+1n = yjn +(Tn)zjn +12(Tn)2vjn= y0n + j(Tn)z0n +12(Tn)2 [(2j − 1)v0n + (2j − 3)v1n + · · ·+ 3vj−2n + vj−1n]+(Tn)(z0n +Tn[v0n + v1n + · · ·+ vj−1n])+12(Tn)2vjn= y0n + (j + 1)(Tn)z0n +(Tn)2 [(j −12)v0n +(j −32)v1n + · · ·+12vj−1n + v1n + . . . vj−1n]+12(Tn)2vjn= y0n + (j + 1)(Tn)z0n +(Tn)2 [(j +12)v0n +(j −12)v1n + · · ·+32vj−1n +12vjn]= y0n + (j + 1)(Tn)z0n +12(Tn)2 [(2j + 1)v0n + (2j − 1)v1n + · · ·+ vjn]= y0n + (j + 1)(Tn)z0n +12(Tn)2 [(2(j + 1)− 1)v0n + (2(j − 1)− 3)v1n + · · ·+ vjn].Thus the assumption holds for yjn. Finally, with this and (5) we have,xj+1n = xjn +(Tn)yjn +12(Tn)2zjn +16(Tn)3vjn= x0n + j(Tn)y0n +12(jTn)2z0n +16(Tn)3 [(3j2 − 3j + 1)v0n + (3j2 − 9j + 7)v1n + · · ·+ vj−1n]+(Tn)(y0n + j(Tn)z0n +12(Tn)2 [(2j − 1)v0n + (2j − 3)v1n + · · ·+ 3vj−2n + vj−1n])+12(Tn)2 (z0n +Tn[v0n + · · ·+ vj−1n])+16(Tn)3vjnEJQTDE, 2005 No. 22, p. 6= x0n + (j + 1)(Tn)y0n +12(j + 1)2(Tn)2z0n+16(Tn)3 [(3j2 − 3j + 1)v0n +(3j2 − 9j + 7)v1n + · · ·+ vj−1n]+(Tn)3 [jv0n + (j − 1)v1n + · · ·+ vj−1n]+16(Tn)3vjn= x0n + (j + 1)(Tn)y0n +12(j + 1)2(Tn)2z0n+16(Tn)3 [(3j2 − 3j + 1)v0n +(3j2 − 9j + 7)v1n + · · ·+ vj−1n]+16(Tn)3 [6jv0n + (6j − 6)v1n + · · ·+ 6vj−1n]+16(Tn)3vjn= x0n + (j + 1)(Tn)y0n +12(j + 1)2(Tn)2z0n+16(Tn)3 [(3j2 + 3j + 1)v0n +(3j2 − 3j + 1)v1n + · · ·+ 7vj−1n + vjn].Thus the assumption holds for xjn. Using (2), (3) and the relations in (6), we show that(xjn, yjn, zjn)∈K.∥∥zjn − z0∥∥ =∥∥∥∥z0n +(Tn)[v0n + v1n + · · ·+ vj−1n]− z0∥∥∥∥≤ j(Tn)M≤ TM< r.And,∥∥yjn − y0∥∥ =∥∥∥∥∥y0n + j(Tn)z0n +12(Tn)2 [(2j − 1)v0n + (2j − 3)v1n + · · ·+ 3vj−2n + vj−1n]− y0∥∥∥∥∥≤ j(Tn)‖z0‖+12(jTn)2M≤ T ‖z0‖+12T 2M<12r +12r= r.EJQTDE, 2005 No. 22, p. 7Finally,∥∥xjn − x0∥∥ =∥∥∥∥∥x0n + j(Tn)y0n +12(jTn)2z0n+16(Tn)3 [(3j2 − 3j + 1)v0n +(3j2 − 9j + 7)v1n + · · ·+ vj−1n]− x0∥∥∥∥∥≤jTn‖y0‖+12(jTn)2‖z0‖+16(jTn)3M≤ T‖y0‖+12T 2‖z0‖+16T 3M<12r +12(23)r +16r = r.Thus,(xjn, yjn, zjn)∈ K = Br (x0, y0, z0) for 1 ≤ j ≤ n. Now, from the definition of xn in (4) wehave, x′n(t) = yjn +(t− tjn)zjn +12(t− tjn)2vjn,x′′n(t) = zjn +(t− tjn)vjn,x(3)n (t) = vjn.(7)By (2) we have that∥∥∥x(3)n (t)∥∥∥ = ∥∥vjn∥∥ ≤ M. Similarly, (2) and (3) give the following,‖x′′n(t)‖ =∥∥zjn + (t− tjn) vjn∥∥=∥∥∥∥z0n +(Tn)[v0n + v1n + · · ·+ vj−1n]+(t− tjn)vjn∥∥∥∥≤ ‖z0‖+(jTn)M +(Tn)M< ‖z0‖+ 2r.As well as,‖x′n(t)‖ =∥∥∥∥yjn + (t− tjn) zjn + 12 (t− tjn)2 vjn∥∥∥∥≤ ‖y0‖+(jTn)‖z0‖+12(jTn)2M +(Tn)‖z0‖+(jTn)2M +12(Tn)2M≤ ‖y0‖+ T ‖z0‖+ 2T2M + T ‖z0‖< ‖y0‖+ 3r.EJQTDE, 2005 No. 22, p. 8And finally,‖xn(t)‖ =∥∥∥∥xjn + (t− tjn) yjn + 12 (t− tjn)2 zjn + 16 (t− tjn)3 vjn∥∥∥∥≤ ‖x0‖+ T ‖y0‖+12T 2 ‖z0‖+16T 3M +(Tn)(‖y0‖+ T ‖z0‖+12T 2M)+12(Tn)2(‖z0‖+ TM) +16(Tn)3M≤ ‖x0‖+ T ‖y0‖+12T 2 ‖z0‖+16T 3M + T ‖y0‖+ T2 ‖z0‖+12T 3M +12T 2M ‖z0‖+12T 3M +16T 3M= ‖x0‖+ 2T ‖y0‖+ 2T2 ‖z0‖+ T3M +13T 3M< ‖x0‖+ r +43r + r +13r< ‖x0‖+ 4r.Since∥∥∥x(3)n (t)∥∥∥ ≤ M ∀t ∈ [0, T ] the sequence (x(3)n (t)) is bounded in L2 ([0, T ], Rm). Furthermore,suppose ε > 0 and ∀t ∈ [0, T ], and ∀τ ∈ [0, T ], |t− τ | < εM. Then,‖x′′n(t)− x′′n(τ)‖ ≤∣∣∣∣∫ tτ‖x(3)n (s)‖ds∣∣∣∣≤∣∣∣∣∫ tτMds∣∣∣∣= M |t− τ |≤ M( εM)= ε.Thus, (x′′n) is equicontinuous. Similarly (x′n) and (xn) are equicontinuous. Theorem 0.3.4 in [1]gives the following:There exists a subsequence, again denoted (xn)n that converges to an absolutely continuousfunction x : [0, T ] → Rm such that:(i) (xn) ⇒ x on [0, T ],(ii) (x′n) ⇒ x′ on [0, T ],(iii) (x′′n) ⇒ x′′ on [0, T ],(iv) (x(3)n ) converges weakly to x3 in L2 ([0, T ], Rm) .EJQTDE, 2005 No. 22, p. 9By the Convergence Theorem, theorem 1.4.1 in [1], we have thatx(3)(t) ∈ coF (x(t), x′(t), x′′(t)) ⊂ ∂V (x′′(t)) a.e., t ∈ [0, T ].Also, by the above and lemma 3.3 in [2],ddtV (x′′(t)) =〈x(3)(t), x(3)(t)〉=∥∥x(3)(t)∥∥2 .Since,∫ T0ddtV (x′′(t)) dt =∫ T0∥∥x(3)(t)∥∥2 dt we have,V (x′′(T ))− V (x′′(0)) =∫ T0∥∥∥x(3)(t)∥∥∥2 dt. (8)However, by (7) we also have x(3)n (t) = vjn ∈ F(xjn, yjn, zjn)⊂ ∂V(x′′n(tjn)), ∀t ∈[tj−1n , tjn]. Which,from the definition of subdifferential, gives the following,V(x′′n(tjn))− V(x′′n(tj−1n))≥〈x(3)n (t), x′′n(tjn)− x′′n(tj−1n)〉=〈x(3)n (t),∫ tjntj−1nx(3)n (s)ds〉=∫ tjntj−1n〈x(3)n (t), x(3)n (t)〉dt=∫ tjntj−1n∥∥∥x(3)n (t)∥∥∥2 dt.Combining the above inequalities with (8), we get the following inequality,V (x′′n(T ))− V (z0) ≥∫ T0∥∥∥x(3)n (t)∥∥∥2 dt.If we let n approach infinity, we haveV (x′′(T ))− V (z0) ≥ lim supn→∞∫ T0∥∥∥x(3)n (t)∥∥∥2 dt∫ T0∥∥∥x(3)(t)∥∥∥2 dt ≥ lim supn→∞∫ T0∥∥∥x(3)n (t)∥∥∥2 dt∥∥∥x(3)(t)∥∥∥2 ≥ lim supn→∞∥∥∥x(3)n (t)∥∥∥2 .However, the weak lower semicontinuity of the norm gives,∥∥∥x(3)(t)∥∥∥2 ≤ lim infn→∞∥∥∥x(3)n (t)∥∥∥2 .EJQTDE, 2005 No. 22, p. 10Thus we have,∥∥∥x(3)(t)∥∥∥2 = limn→∞∥∥∥x(3)n (t)∥∥∥2 .Hence the sequence(x(3)n)→ x(3) pointwise. By assumption (A1), F is closed, implyingx(3)(t) ∈ F (x(t), x′(t), x′′(t)), a.e. t ∈ [0, T ].References[1] J. P. Aubin and A. Cellina; Differential inclusions, Berlin, Springer-Verlag, 1984.[2] H. Brezis; Operateurs Maximaux Monotones et Semigroups de Contractions Dans Les Espacesde Hilbert, Amsterdam, North-Holland, 1973.[3] K. Deimling; Multivalued Differential Equations, Walter de Gruyter, Berlin, New York, 1992.[4] V. Lupulescu; Existence of Solutions for Nonconvex Functional Differential Inclusions, Elec-tronic Journal of Differential Equations, Vol. 2004(2004), No. 141, pp.1-6.[5] V. Lupulescu; Existence of Soutions for Nonconvex Second Order Differential Inclusions, Ap-plied Mathematics E-Notes, 3(2003), 115-123.(Received September 16, 2005)EJQTDE, 2005 No. 22, p. 11

Hopkins Britney

University of Szeged

Electronic Journal of Qualitative Theory of Differential Equations2005, No. 22, 1-11; http://www.math.u-szeged.hu/ejqtde/Existence of Solutions for Nonconvex Third OrderDifferential Inclusions.Britney HopkinsDepartment of Mathematics and StatisticsUniversity of Arkansas at Little Rock2801 S. UniversityLittle Rock, AR 72204bjhopkins1@ualr.eduAbstractThis paper proves the existence of solutions for a third order initial value nonconvex dif-ferential inclusion. We start with an upper semicontinuous compact valued multifunction Fwhich is contained in a lower semicontinuous convex function ∂V and show that,x(3)(t) ∈ F (x(t), x′(t), x′′(t)), x(0) = x0, x′(0) = y0, x′′(0) = z0.Keywords: Nonconvex Differential InclusionsAMS Subject Classification: 34G20, 47H201 IntroductionThe origins of boundary and initial value problems for differential inclusions are in the theory ofdifferential equations and serve as models for a variety of applications including control theory.Existence results for the second order differential inclusion,x′′ ∈ F (x, x′), x(0) = x0, x′(0) = y0,have been obtained by many authors (see [4], [5] and the references therein). In [5], Lupulescushowed existence for the problemx′′ ∈ F (x, x′) + f(t, x, x′), x(0) = x0, x′(0) = y0for the case in which F is an upper semicontinuous compact valued multifunction such the F (x, y) ⊂∂V (y) and f is a Carathéodory function.In this paper, we prove an existence result for the third order differential inclusion,x(3)(t) ∈ F (x(t), x′(t), x′′(t)), x(0) = x0, x′(0) = y0, x′′(0) = z0,where F is an upper semicontinuous compact valued multifunction and F (x, y, z) ⊂ ∂V (z) for someproper lower semicontinuous convex function V. Expounding upon the methods used to establishexistence by Lupulescu in [4] and [5], we define a sequence of approximate solutions on a giveninterval and show that the sequence converges to an actual solution.EJQTDE, 2005 No. 22, p. 12 PreliminariesLet Rm be an m dimensional Euclidean space with an inner product 〈·, ·〉 and norm ‖ · ‖.Let x ∈ Rm and r > 0. The open ball centered at x with radius r is defined byBr(x) = {y ∈ Rm : ‖x − y‖ < r},where Br(x) denotes its closure.For the proper lower semicontinuous convex function V : Rm → R, the multifunction ∂V :Rm → 2Rmdefined by∂V (x) = {γ ∈ Rm : V (y) − V (x) ≥ 〈γ, y − x〉, ∀y ∈ Rm} ,is the subdifferential of V.Let L2[a, b] be a Hilbert space with the inner product defined by〈x, y〉 =∫ bax(t)y(t)dt,where y(t) denotes the complex conjugate of y(t), and the norm is defined as‖x‖ =√∫ ba|x(t)|2dt.Let coF (x, y, z) denote the closed convex hull of F and xn ⇒ x denote that xn convergesuniformly to x.We need the following theorems from Aubin and Cellina [1].Theorem 0.3.4 Consider a sequence of absolutely continuous functions xk(·) from an intervalI to a Banach Space X satisfying(i) for every t ∈ I , xk(t)k is a relatively compact subset of X ;(ii) there exists a positive function c(·) ∈ L2(I) such that, for almost all t ∈ I , ‖x′k(t)‖ ≤ c(t).Then there exists a subsequence, again denoted by xk(·), converging to an absolutely continuousfunction x(·) from I to X in the sense that(i) xk(·) converges uniformly to x(·) over compact subsets of I ;(ii) x′k(·) converges weakly to x′k(·) in L2(I, X).EJQTDE, 2005 No. 22, p. 2Theorem 1.1.4 (the Convergence Theorem) Let F be a proper hemicontinuous map froma Hausdorff locally convex space X to the closed convex subsets of a Banach Space Y. Let I bean interval of R and xk(·) and yk(·) be measurable functions from I to Y respectively satisfyingfor almost all t in I and for every neighborhood ℵ of 0 in X × Y , there exists a k0 = k0(t,ℵ) suchthat for every k0 ≤ k, (xk(t), yk(t)) ∈ graph(F ) + ℵ. If,(i) xk(·) converges almost everywhere to a function x(·) from I to X ;(ii) yk(·) belongs to L2(I, Y ) and converges weakly to y(·) in L2(I, Y ),then, for almost all t ∈ I ,(x(t), y(t)) ∈ graph(F ) i.e.y(t) ∈ F (x(t)).We also need the following lemma from Brezis [2].Lemma 3.3 Let u ∈ D(V ) almost everywhere on [0, T ] and suppose g ∈ L2([0, T ], R) suchthat g(t) ∈ ∂V (u(t)) almost everywhere on [0, T ]. Then, the function t 7−→ V (u(t)) is absolutelycontinuous on [0, T ].Also, let t ∈ [0, T ] such that u(t) ∈ D(V ) and let u and V(u) be differentiable. Then for allt ∈ [0, T ]ddtV (u(t)) =〈h,dudt(t)〉∀h ∈ ∂V (u(t)).3 The Main ResultTHEOREM: If F : Ω → 2Rmand V : Rm → R satisfy the assumptions(A1) Ω ⊂ R2m where Ω is open and F : Ω → 2Rmis a compact valued upper semicontinuousmultifunction;(A2) there exists a lower semicontinuous proper convex function V : Rm → R such thatF (x, y, z) ⊂ ∂V (z) for every (x, y, z) ∈ Ω.Then, for every (x0, y0, z0) ∈ Ω, there exists a T > 0 and a solution x : [0, T ] → Rm ofx(3)(t) ∈ F (x(t), x′(t), x′′(t)), x(0) = x0, x′(0) = y0, x′′(0) = z0. (1)EJQTDE, 2005 No. 22, p. 3By a solution we are referring to an absolutely continuous function x : [0, T ] → Rm withabsolutely continuous first and second derivatives with the initial values x(0) = x0, x′(0) = y0,x′′(0) = z0, and x(3)(t) ∈ F (x(t), x′(t), x′′(t)), a.e. on [0, T ].PROOF: Suppose (x0, y0, z0) ∈ Ω. Then, K = B̄r(x0, y0, z0) ⊂ Ω for some r > 0 since Ω is open.By assumption (A1)F (K) =⋃(x,y,z)∈KF (x, y, z)is compact. Then there exists an M > 0 such thatsup{‖v‖ : v ∈ F (x, y, z), (x, y, z) ∈ K)} ≤ M. (2)SetT < min{rM,( rM)12,( rM)13,r2‖z0‖,r2‖y0‖,(2r3‖z0‖)12}. (3)Let n, j be integers where 1 ≤ j ≤ n. Set tjn =jTn. For t ∈[tj−1n , tjn]define,xn(t) = xjn +(t − tjn)yjn +12(t − tjn)2zjn +16(t − tjn)3vjn, (4)where x0n = x0, y0n = y0, and z0n = z0.For 0 ≤ j ≤ n − 1 and vjn ∈ F(xjn, yjn, zjn), definexj+1n = xjn +(Tn)yjn +12(Tn)2zjn +16(Tn)3vjnyj+1n = yjn +(Tn)zjn +12(Tn)2vjnzj+1n = zjn +(Tn)vjn.(5)We claim that(x1n, y1n, z1n)∈ K. Using (5), we have∥∥z1n − z0∥∥ =∥∥∥∥z0n +(Tn)v0n − z0∥∥∥∥≤(Tn)M < r.EJQTDE, 2005 No. 22, p. 4As well as,∥∥y1n − y0∥∥ =∥∥∥∥∥y0n +(Tn)z0n +12(Tn)2v0n − y0∥∥∥∥∥≤(Tn)‖z0‖ +12(Tn)2M<12r +12r = r.Also,∥∥x1n − x0∥∥ =∥∥∥∥∥x0n +(Tn)y0n +12(Tn)2z0n +16(Tn)3v0n − x0∥∥∥∥∥≤(Tn)‖y0‖ +12(Tn)2‖z0‖ +16(Tn)3M< T ‖y0‖ +12T 2 ‖z0‖ +16T 3M<12r +13r +16r = r.Hence the claim holds. Now suppose j ≥ 1. We make the assumption that,xjn = x0n + j(Tn)y0n +12(jTn)2z0n +16(Tn)3[(3j2 − 3j + 1)v0n +(3j2 − 9j + 7)v1n+(3j2 − 15j + 19)v2n + · · · + 7vj−2n + vj−1n ],yjn = y0n + j(Tn)z0n +12(Tn)2[(2j − 1)v0n + (2j − 3)v1n + · · · + 3vj−2n + vj−1n],zjn = z0n +(Tn)[v0n + v1n + · · · + vj−1n].(6)To see this, let j = 1. Then,z1n = z0n +(Tn)v0n = z0+1n ,y1n = y0n +Tnz0n +12(Tn)2v0n = y0+1n ,x1n = x0n +Tny0n +12(Tn)2z0n +16(Tn)3v0n = x0+1n .EJQTDE, 2005 No. 22, p. 5Thus, (6) holds when j = 1. Let’s suppose assumption (6) holds for j > 1. Using (5) we see that,zj+1n = zjn +(Tn)vjn= z0n +(Tn)[v0n + v1n + · · · + vj−1n]+(Tn)vnj= z0n +(Tn)[v0n + v1n + · · · + vjn].Thus the assumption holds for zjn. Using this and (5) we have,yj+1n = yjn +(Tn)zjn +12(Tn)2vjn= y0n + j(Tn)z0n +12(Tn)2[(2j − 1)v0n + (2j − 3)v1n + · · · + 3vj−2n + vj−1n]+(Tn)(z0n +Tn[v0n + v1n + · · · + vj−1n])+12(Tn)2vjn= y0n + (j + 1)(Tn)z0n +(Tn)2 [(j −12)v0n +(j −32)v1n + · · · +12vj−1n + v1n + . . . vj−1n]+12(Tn)2vjn= y0n + (j + 1)(Tn)z0n +(Tn)2 [(j +12)v0n +(j −12)v1n + · · · +32vj−1n +12vjn]= y0n + (j + 1)(Tn)z0n +12(Tn)2[(2j + 1)v0n + (2j − 1)v1n + · · · + vjn]= y0n + (j + 1)(Tn)z0n +12(Tn)2[(2(j + 1) − 1)v0n + (2(j − 1) − 3)v1n + · · · + vjn].Thus the assumption holds for yjn. Finally, with this and (5) we have,xj+1n = xjn +(Tn)yjn +12(Tn)2zjn +16(Tn)3vjn= x0n + j(Tn)y0n +12(jTn)2z0n +16(Tn)3[(3j2 − 3j + 1)v0n + (3j2 − 9j + 7)v1n + · · · + vj−1n]+(Tn)(y0n + j(Tn)z0n +12(Tn)2[(2j − 1)v0n + (2j − 3)v1n + · · · + 3vj−2n + vj−1n])+12(Tn)2 (z0n +Tn[v0n + · · · + vj−1n])+16(Tn)3vjnEJQTDE, 2005 No. 22, p. 6= x0n + (j + 1)(Tn)y0n +12(j + 1)2(Tn)2z0n+16(Tn)3[(3j2 − 3j + 1)v0n +(3j2 − 9j + 7)v1n + · · · + vj−1n]+(Tn)3[jv0n + (j − 1)v1n + · · · + vj−1n]+16(Tn)3vjn= x0n + (j + 1)(Tn)y0n +12(j + 1)2(Tn)2z0n+16(Tn)3[(3j2 − 3j + 1)v0n +(3j2 − 9j + 7)v1n + · · · + vj−1n]+16(Tn)3[6jv0n + (6j − 6)v1n + · · · + 6vj−1n]+16(Tn)3vjn= x0n + (j + 1)(Tn)y0n +12(j + 1)2(Tn)2z0n+16(Tn)3[(3j2 + 3j + 1)v0n +(3j2 − 3j + 1)v1n + · · · + 7vj−1n + vjn].Thus the assumption holds for xjn. Using (2), (3) and the relations in (6), we show that(xjn, yjn, zjn)∈K.∥∥zjn − z0∥∥ =∥∥∥∥z0n +(Tn)[v0n + v1n + · · · + vj−1n]− z0∥∥∥∥≤ j(Tn)M≤ TM< r.And,∥∥yjn − y0∥∥ =∥∥∥∥∥y0n + j(Tn)z0n +12(Tn)2[(2j − 1)v0n + (2j − 3)v1n + · · · + 3vj−2n + vj−1n]− y0∥∥∥∥∥≤ j(Tn)‖z0‖ +12(jTn)2M≤ T ‖z0‖ +12T 2M<12r +12r= r.EJQTDE, 2005 No. 22, p. 7Finally,∥∥xjn − x0∥∥ =∥∥∥∥∥x0n + j(Tn)y0n +12(jTn)2z0n+16(Tn)3[(3j2 − 3j + 1)v0n +(3j2 − 9j + 7)v1n + · · · + vj−1n]− x0∥∥∥∥∥≤jTn‖y0‖ +12(jTn)2‖z0‖ +16(jTn)3M≤ T‖y0‖ +12T 2‖z0‖ +16T 3M<12r +12(23)r +16r = r.Thus,(xjn, yjn, zjn)∈ K = Br (x0, y0, z0) for 1 ≤ j ≤ n. Now, from the definition of xn in (4) wehave,x′n(t) = yjn +(t − tjn)zjn +12(t − tjn)2vjn,x′′n(t) = zjn +(t − tjn)vjn,x(3)n (t) = vjn.(7)By (2) we have that∥∥∥x(3)n (t)∥∥∥=∥∥vjn∥∥ ≤ M. Similarly, (2) and (3) give the following,‖x′′n(t)‖ =∥∥zjn +(t − tjn)vjn∥∥=∥∥∥∥z0n +(Tn)[v0n + v1n + · · · + vj−1n]+(t − tjn)vjn∥∥∥∥≤ ‖z0‖ +(jTn)M +(Tn)M< ‖z0‖ + 2r.As well as,‖x′n(t)‖ =∥∥∥∥yjn +(t − tjn)zjn +12(t − tjn)2vjn∥∥∥∥≤ ‖y0‖ +(jTn)‖z0‖ +12(jTn)2M +(Tn)‖z0‖ +(jTn)2M +12(Tn)2M≤ ‖y0‖ + T ‖z0‖ + 2T2M + T ‖z0‖< ‖y0‖ + 3r.EJQTDE, 2005 No. 22, p. 8And finally,‖xn(t)‖ =∥∥∥∥xjn +(t − tjn)yjn +12(t − tjn)2zjn +16(t − tjn)3vjn∥∥∥∥≤ ‖x0‖ + T ‖y0‖ +12T 2 ‖z0‖ +16T 3M +(Tn)(‖y0‖ + T ‖z0‖ +12T 2M)+12(Tn)2(‖z0‖ + TM) +16(Tn)3M≤ ‖x0‖ + T ‖y0‖ +12T 2 ‖z0‖ +16T 3M + T ‖y0‖ + T2 ‖z0‖+12T 3M +12T 2M ‖z0‖ +12T 3M +16T 3M= ‖x0‖ + 2T ‖y0‖ + 2T2 ‖z0‖ + T3M +13T 3M< ‖x0‖ + r +43r + r +13r< ‖x0‖ + 4r.Since∥∥∥x(3)n (t)∥∥∥≤ M ∀t ∈ [0, T ] the sequence (x(3)n (t)) is bounded in L2 ([0, T ], Rm). Furthermore,suppose ε > 0 and ∀t ∈ [0, T ], and ∀τ ∈ [0, T ], |t − τ | < εM. Then,‖x′′n(t) − x′′n(τ)‖ ≤∣∣∣∣∫ tτ‖x(3)n (s)‖ds∣∣∣∣≤∣∣∣∣∫ tτMds∣∣∣∣= M |t − τ |≤ M( εM)= ε.Thus, (x′′n) is equicontinuous. Similarly (x′n) and (xn) are equicontinuous. Theorem 0.3.4 in [1]gives the following:There exists a subsequence, again denoted (xn)n that converges to an absolutely continuousfunction x : [0, T ] → Rm such that:(i) (xn) ⇒ x on [0, T ],(ii) (x′n) ⇒ x′ on [0, T ],(iii) (x′′n) ⇒ x′′ on [0, T ],(iv) (x(3)n ) converges weakly to x3 in L2 ([0, T ], Rm) .EJQTDE, 2005 No. 22, p. 9By the Convergence Theorem, theorem 1.4.1 in [1], we have thatx(3)(t) ∈ coF (x(t), x′(t), x′′(t)) ⊂ ∂V (x′′(t)) a.e., t ∈ [0, T ].Also, by the above and lemma 3.3 in [2],ddtV (x′′(t)) =〈x(3)(t), x(3)(t)〉=∥∥x(3)(t)∥∥2.Since,∫ T0ddtV (x′′(t)) dt =∫ T0∥∥x(3)(t)∥∥2dt we have,V (x′′(T )) − V (x′′(0)) =∫ T0∥∥∥x(3)(t)∥∥∥2dt. (8)However, by (7) we also have x(3)n (t) = vjn ∈ F(xjn, yjn, zjn)⊂ ∂V(x′′n(tjn)), ∀t ∈[tj−1n , tjn]. Which,from the definition of subdifferential, gives the following,V(x′′n(tjn))− V(x′′n(tj−1n))≥〈x(3)n (t), x′′n(tjn)− x′′n(tj−1n)〉=〈x(3)n (t),∫ tjntj−1nx(3)n (s)ds〉=∫ tjntj−1n〈x(3)n (t), x(3)n (t)〉dt=∫ tjntj−1n∥∥∥x(3)n (t)∥∥∥2dt.Combining the above inequalities with (8), we get the following inequality,V (x′′n(T )) − V (z0) ≥∫ T0∥∥∥x(3)n (t)∥∥∥2dt.If we let n approach infinity, we haveV (x′′(T )) − V (z0) ≥ lim supn→∞∫ T0∥∥∥x(3)n (t)∥∥∥2dt∫ T0∥∥∥x(3)(t)∥∥∥2dt ≥ lim supn→∞∫ T0∥∥∥x(3)n (t)∥∥∥2dt∥∥∥x(3)(t)∥∥∥2≥ lim supn→∞∥∥∥x(3)n (t)∥∥∥2.However, the weak lower semicontinuity of the norm gives,∥∥∥x(3)(t)∥∥∥2≤ lim infn→∞∥∥∥x(3)n (t)∥∥∥2.EJQTDE, 2005 No. 22, p. 10Thus we have,∥∥∥x(3)(t)∥∥∥2= limn→∞∥∥∥x(3)n (t)∥∥∥2.Hence the sequence(x(3)n)→ x(3) pointwise. By assumption (A1), F is closed, implyingx(3)(t) ∈ F (x(t), x′(t), x′′(t)), a.e. t ∈ [0, T ].References[1] J. P. Aubin and A. Cellina; Differential inclusions, Berlin, Springer-Verlag, 1984.[2] H. Brezis; Operateurs Maximaux Monotones et Semigroups de Contractions Dans Les Espacesde Hilbert, Amsterdam, North-Holland, 1973.[3] K. Deimling; Multivalued Differential Equations, Walter de Gruyter, Berlin, New York, 1992.[4] V. Lupulescu; Existence of Solutions for Nonconvex Functional Differential Inclusions, Elec-tronic Journal of Differential Equations, Vol. 2004(2004), No. 141, pp.1-6.[5] V. Lupulescu; Existence of Soutions for Nonconvex Second Order Differential Inclusions, Ap-plied Mathematics E-Notes, 3(2003), 115-123.(Received September 16, 2005)EJQTDE, 2005 No. 22, p. 11

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Existence of solutions for nonconvex third order differential inclusions

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