We  prove a Filippov type existence theorem for solutions of a higher order differential inclusion in Banach spaces with nonconvex valued right hand side by applying the contraction principle in the space of the derivatives of solutions instead of the space of solutions

Cernea, A.

Electronic Journal of Qualitative Theory of Differential Equations

Aurelian Cernea

Crossref

Electronic journal of qualitative theory of differential equations

On the existence of solutions for a higher order differential inclusion without convexity

Cernea Aurelian

University of Szeged

Electronic Journal of Qualitative Theory of Differential Equations2007, No. 8, 1-8; http://www.math.u-szeged.hu/ejqtde/On the existence of solutions for ahigher order differential inclusionwithout convexityAurelian CerneaFaculty of Mathematics and Informatics,University of Bucharest,Academiei 14, 010014 Bucharest, Romania,acernea@math.math.unibuc.roAbstractWe prove a Filippov type existence theorem for solutions of ahigher order differential inclusion in Banach spaces with nonconvexvalued right hand side by applying the contraction principle in thespace of the derivatives of solutions instead of the space of solutions.Key words and phrases. Differential inclusion, contractive set-valuedmap, fixed point.2000 AMS Subject Classifications. 34A60.1 IntroductionIn this note we study the n-th order differential inclusionx(n) − λx ∈ F (t, x), a.e. (I) (1.1)with boundary conditions of the formx(i)(0) − x(i)(T ) = µi, i = 0, 1, . . . , n− 1, (1.2)where λ ∈ R, E is real separable Banach space, I = [0, T ], F (·, ·) : I ×E →P(E) and µi ∈ E, i = 0, 1, . . . , n−1. When λ 6= 0 the linear part in equationEJQTDE, 2007 No. 8, p. 1(1.1) is invertible and, in this case, the problem (1.1)-(1.2) is well known asa nonresonance problem. Moreover, if µi = 0, i = 0, 1, . . . , n − 1 then theconditions (1.2) are periodic boundary conditions.The present note is motivated by a recent paper of Benchohra, Graef,Henderson and Ntouyas ([1]) in which several existence results concerningnonresonance impulsive higher order differential inclusions are obtained viafixed point techniques. The aim of our paper is to provide a Filippov typeresult concerning the existence of solutions to problem (1.1)-(1.2). Recallthat for a differential inclusion defined by a lipschitzian set-valued map withnonconvex values, Filippov’s theorem consists in proving the existence of asolution starting from a given ”quasi” solution.Our approach is different from the one in [1] and consists in applying thecontraction principle in the space of derivatives of solutions instead of thespace of solutions. In addition, as usual at a Filippov existence type theorem,our result provides an estimate between the starting ”quasi” solution andthe solution of the differential inclusion. The idea of applying the set-valuedcontraction principle due to Covitz and Nadler ([6]) in the space of derivativesof the solutions belongs to Kannai and Tallos ([7]) and it was already usedfor other results concerning differential inclusions ([3,4,5] etc.).The paper is organized as follows: in Section 2 we recall some preliminaryfacts that we need in the sequel and in Section 3 we prove our main result.2 PreliminariesIn this short section we sum up some basic facts that we are going to uselater.Let (X, d) be a metric space and consider a set valued map T on X withnonempty closed values in X. T is said to be a λ-contraction if there exists0 < λ < 1 such that:dH(T (x), T (y)) ≤ λd(x, y) ∀x, y ∈ X,where dH(·, ·) denotes the Pompeiu-Hausdorff distance. Recall that thePompeiu-Hausdorff distance of the closed subsets A,B ⊂ X is defined bydH(A,B) = max{d∗(A,B), d∗(B,A)}, d∗(A,B) = sup{d(a,B); a ∈ A},where d(x,B) = infy∈B d(x, y).EJQTDE, 2007 No. 8, p. 2If X is complete, then every set valued contraction has a fixed point, i.e.a point z ∈ X such that z ∈ T (z) ([6]).We denote by Fix(T ) the set of all fixed points of the set-valued map T .Obviously, Fix(T ) is closed.Proposition 2.1. ([8]) Let X be a complete metric space and supposethat T1, T2 are λ-contractions with closed values in X. ThendH(Fix(T1), F ix(T2)) ≤11 − λsupz∈Xd(T1(z), T2(z)).In what follows E is a real separable Banach space with norm | · |, C(I, E)is the Banach space of all continuous functions from I to E with the norm‖x(·)‖C = supt∈I |x(t)|, ACi(I, E) is the space of i-times differentiable func-tions x : I → E whose i-th derivative xi(·) is absolutely continuous andL1(I, E) is the Banach space of (Bochner) integrable functions u(·) : I → Eendowed with the norm ‖u(·)‖1 =∫ 10 |u(t)|dt.A function x(·) ∈ ACn−1(I, E) is called a solution of problem (1.1)-(1.2)if there exists a function v(·) ∈ L1(I, E) with v(t) ∈ F (t, x(t)), a.e. (I) suchthat x(n)(t) − λx(t) = v(t), a.e. (I) and x(·) satisfies conditions (1.2).In what follows we consider the Green function G(·, ·) : I × I → Eassociated to the periodic boundary problemx(n) − λx = 0, x(i)(0) − x(i)(T ) = 0, i = 0, 1, . . . , n− 1.For the properties of G(·, ·) we refer to [1].The next result is well known (e.g. [1]).Lemma 2.2. ([1]) If v(·) : [0, T ] → E is an integrable function then theproblemx(n)(t) − λx(t) = v(t) a.e. (I)x(i)(0) − x(i)(T ) = µi, i = 0, 1, . . . , n− 1.has a unique solution x(·) ∈ ACn−1(I, E) given byx(t) = Pµ(t) +∫ T0G(t, s)v(s)ds,wherePµ(t) =n−1∑i=0∂i∂tiG(t, 0)µn−1−i. (2.1)EJQTDE, 2007 No. 8, p. 3In the sequel we assume the following conditions on F .Hypothesis 2.3. (i) F (·, ·) : I ×E → P(E) has nonempty closed valuesand for every x ∈ E F (·, x) is measurable.(ii) There exists L(·) ∈ L1(I, E) such that for almost all t ∈ I, F (t, ·) isL(t)-Lipschitz in the sense thatdH(F (t, x), F (t, y)) ≤ L(t)|x− y| ∀ x, y ∈ Eand d(0, F (t, 0)) ≤ L(t) a.e. (I).Denote L0 :=∫ T0 L(s)ds and M0 := sup(t,s)∈I×I |G(t, s)|.3 The main resultWe are able now to prove our main result.Theorem 3.1. Assume that Hypothesis 2.3 is satisfied and M0L0 <1. Let y(·) ∈ ACn−1(I, E) be such that there exists q(.) ∈ L1(I, E) withd(y(n)(t) − λy(t), F (t, y(t))) ≤ q(t), a.e. (I). Denote µ̃i = y(i)(0) − y(i)(T ),i = 0, 1, . . . , n− 1.Then for every ε > 0 there exists x(·) a solution of (1.1)-(1.2) satisfyingfor all t ∈ I|x(t) − y(t)| ≤11 −M0L0supt∈I|Pµ(t) − Pµ̃(t)| +M01 −M0L0∫ T0q(t)dt+ ε,where Pµ(t) is defined in (2.1).Proof. For u(·) ∈ L1(I, E) define the following set valued maps:Mu(t) = F(t, Pµ(t) +∫ T0G(t, s)u(s)ds), t ∈ I,T (u) = {φ(.) ∈ L1(I, E); φ(t) ∈Mu(t) a.e. (I)}.It follows from the definition and Lemma 2.2 that x(·) is a solution of(1.1)-(1.2) if and only if x(n)(·) − λx(·) is a fixed point of T (·).We shall prove first that T (u) is nonempty and closed for every u ∈L1(I, E). The fact that the set valued mapMu(·) is measurable is well known.For example the map t → Pµ(t) +∫ T0 G(t, s)u(s)ds can be approximated byEJQTDE, 2007 No. 8, p. 4step functions and we can apply Theorem III. 40 in [2]. Since the values ofF are closed with the measurable selection theorem (Theorem III.6 in [2])we infer that Mu(·) admits a measurable selection φ. One has|φ(t)| ≤ d(0, F (t, 0)) + dH(F (t, 0), F(t, Pµ(t) +∫ T0G(t, s)u(s)ds))≤ L(t)(1 + |Pµ(t)| + supt,s∈I|G(t, s)|∫ T0|u(s)|ds),which shows that φ ∈ L1(I, E) and T (u) is nonempty.On the other hand, the set T (u) is also closed. Indeed, if φn ∈ T (u) and‖φn−φ‖1 → 0 then we can pass to a subsequence φnk such that φnk(t) → φ(t)for a.e. t ∈ I, and we find that φ ∈ T (u).We show next that T (·) is a contraction on L1(I, E).Let u, v ∈ L1(I, E) be given, φ ∈ T (u) and let δ > 0. Consider thefollowing set-valued map:H(t) = Mv(t) ∩{x ∈ E; |φ(t) − x| ≤ L(t)∣∣∣∣∣∫ T0G(t, s)(u(s) − v(s))ds∣∣∣∣∣+ δ}.From Proposition III.4 in [2], H(·) is measurable and from Hypothesis 2.3ii)H(·) has nonempty closed values. Therefore, there exists ψ(·) a measurableselection of H(·). It follows that ψ ∈ T (v) and according with the definitionof the norm we have‖φ− ψ‖1 =∫ T0|φ(t) − ψ(t)|dt≤∫ T0L(t)(∫ T0|G(t, s)||u(s)− v(s)|ds)dt+∫ T0δdt=∫ T0(∫ T0L(t)|G(t, s)|dt)|u(s) − v(s)|ds+ Tδ≤ M0L0‖u− v‖1 + Tδ.Since δ > 0 was chosen arbitrary, we deduce thatd(φ, T (v)) ≤M0L0‖u− v‖1.Replacing u by v we obtaindH(T (u), T (v)) ≤M0L0‖u− v‖1,EJQTDE, 2007 No. 8, p. 5thus T (·) is a contraction on L1(I, E).We consider next the following set-valued mapsF1(t, x) = F (t, x) + q(t)B, (t, x) ∈ I ×E,M1u(t) = F1(t, Pµ̃(t) +∫ T0G(t, s)u(s)ds), t ∈ I, u(·) ∈ L1(I, E),T1(u) = {ψ(·) ∈ L1(I, E); ψ(t) ∈M1u(t) a.e. (I)},where B denotes the closed unit ball in E. Obviously, F1(·, ·) satisfies Hy-pothesis 2.3.Repeating the previous step of the proof we obtain that T1 is also aM0L0-contraction on L1(I, E) with closed nonempty values.We prove next the following estimatedH(T (u), T1(u)) ≤ supt∈I|Pµ(t) − Pµ̃(t)|L0 +∫ T0q(t)dt. (3.1)Let φ ∈ T (u), δ > 0 and defineH1(t) = M1u(t) ∩{z ∈ E; |φ(t) − z| ≤ L(t)|Pµ(t) − Pµ̃(t)| + q(t) +δT}.With the same arguments used for the set valued map H(·), we deducethat H1(·) is measurable with nonempty closed values. Hence let ψ(·) be ameasurable selection of H1(·). It follows that ψ ∈ T1(u) and one has‖φ− ψ‖1 =∫ T0|φ(t) − ψ(t)|dt≤∫ T0[L(t)|Pµ(t) − Pµ̃(t)| + q(t) +δT]dt≤∫ T0L(t)|Pµ(t) − Pµ̃(t)|dt+∫ T0q(t) + δ≤ L0 supt∈I|Pµ(t) − Pµ̃(t)| +∫ T0q(t)dt+ δ.Since δ is arbitrary, as above we obtain (3.1).We apply Proposition 2.1 and we infer thatdH(Fix(T ), F ix(T1)) ≤L01 −M0L0supt∈I|Pµ(t) − Pµ̃(t)| +11 −M0L0∫ T0q(t)dt.EJQTDE, 2007 No. 8, p. 6Since v(·) = y(n)(·) − λy(·) ∈ Fix(T1) it follows that there exists u(·) ∈Fix(T ) such that for any ε > 0‖v − u‖1 ≤L01 −M0L0supt∈I|Pµ(t) − Pµ̃(t)| +11 −M0L0∫ T0q(t)dt+εM0.We define x(t) = Pµ(t) +∫ T0 G(t, s)u(s)ds, t ∈ I and we have|x(t) − y(t)| ≤ |Pµ(t) − Pµ̃(t)| +∫ T0|G(t, s)||u(s)− v(s)|ds≤ supt∈I|Pµ(t) − Pµ̃(t)| + supt∈I|Pµ(t) − Pµ̃(t)|M0L01 −M0L0+11 −M0L0M0∫ T0q(t)dt+ ε≤11 −M0L0supt∈I|Pµ(t) − Pµ̃(t)| +M01 −M0L0∫ T0q(t)dt+ ε,which completes the proof.References[1] M. Benchohra, J. R. Graef, J. Henderson and S. K. Ntouyas, Nonres-onance impulsive higher order functional nonconvex-valued differentialinclusions, Electron. J. Qual. Theory Differ. Equ. (2002), No 13, 13pp(electronic).[2] C. Castaing and M. Valadier, Convex Analysis and Measurable Multi-functions, LNM 580, Springer, Berlin, 1977.[3] A. Cernea, A Filippov type existence theorem for an infinite operationaldifferential inclusion, Stud. Cerc. Mat. 50 (1998), 15-22.[4] A. Cernea, An existence theorem for some nonconvex hyperbolic differ-ential inclusions, Mathematica (Cluj) 45(68) (2003), 121-126.[5] A. Cernea, Existence for nonconvex integral inclusions via fixed points,Arch. Math. (Brno) 39 (2003), 293-298.[6] H. Covitz and S. B. Nadler jr., Multivalued contraction mapping in gen-eralized metric spaces, Israel J. Math. 8 (1970), 5-11.EJQTDE, 2007 No. 8, p. 7[7] Z. Kannai and P. Tallos, Stability of solution sets of differential inclusions,Acta Sci. Math. (Szeged) 61(1995), 197-207.[8] T.C. Lim, On fixed point stability for set valued contractive mappingswith applications to generalized differential equations, J. Math. Anal.Appl. 110 (1985), 436-441.(Received December 15, 2006)EJQTDE, 2007 No. 8, p. 8

On the existence of solutions for a higher order differential inclusion

without convexity

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Electronic Journal of Qualitative Theory of Differential Equations2007, No. 8, 1-8; http://www.math.u-szeged.hu/ejqtde/On the existence of solutions for ahigher order differential inclusionwithout convexityAurelian CerneaFaculty of Mathematics and Informatics,University of Bucharest,Academiei 14, 010014 Bucharest, Romania,acernea@math.math.unibuc.roAbstractWe prove a Filippov type existence theorem for solutions of ahigher order differential inclusion in Banach spaces with nonconvexvalued right hand side by applying the contraction principle in thespace of the derivatives of solutions instead of the space of solutions.Key words and phrases. Differential inclusion, contractive set-valuedmap, fixed point.2000 AMS Subject Classifications. 34A60.1 IntroductionIn this note we study the n-th order differential inclusionx(n) − λx ∈ F (t, x), a.e. (I) (1.1)with boundary conditions of the formx(i)(0)− x(i)(T ) = µi, i = 0, 1, . . . , n− 1, (1.2)where λ ∈ R, E is real separable Banach space, I = [0, T ], F (·, ·) : I ×E →P(E) and µi ∈ E, i = 0, 1, . . . , n−1. When λ 6= 0 the linear part in equationEJQTDE, 2007 No. 8, p. 1(1.1) is invertible and, in this case, the problem (1.1)-(1.2) is well known asa nonresonance problem. Moreover, if µi = 0, i = 0, 1, . . . , n − 1 then theconditions (1.2) are periodic boundary conditions.The present note is motivated by a recent paper of Benchohra, Graef,Henderson and Ntouyas ([1]) in which several existence results concerningnonresonance impulsive higher order differential inclusions are obtained viafixed point techniques. The aim of our paper is to provide a Filippov typeresult concerning the existence of solutions to problem (1.1)-(1.2). Recallthat for a differential inclusion defined by a lipschitzian set-valued map withnonconvex values, Filippov’s theorem consists in proving the existence of asolution starting from a given ”quasi” solution.Our approach is different from the one in [1] and consists in applying thecontraction principle in the space of derivatives of solutions instead of thespace of solutions. In addition, as usual at a Filippov existence type theorem,our result provides an estimate between the starting ”quasi” solution andthe solution of the differential inclusion. The idea of applying the set-valuedcontraction principle due to Covitz and Nadler ([6]) in the space of derivativesof the solutions belongs to Kannai and Tallos ([7]) and it was already usedfor other results concerning differential inclusions ([3,4,5] etc.).The paper is organized as follows: in Section 2 we recall some preliminaryfacts that we need in the sequel and in Section 3 we prove our main result.2 PreliminariesIn this short section we sum up some basic facts that we are going to uselater.Let (X, d) be a metric space and consider a set valued map T on X withnonempty closed values in X. T is said to be a λ-contraction if there exists0 < λ < 1 such that:dH(T (x), T (y)) ≤ λd(x, y) ∀x, y ∈ X,where dH(·, ·) denotes the Pompeiu-Hausdorff distance. Recall that thePompeiu-Hausdorff distance of the closed subsets A,B ⊂ X is defined bydH(A,B) = max{d∗(A,B), d∗(B,A)}, d∗(A,B) = sup{d(a,B); a ∈ A},where d(x,B) = infy∈B d(x, y).EJQTDE, 2007 No. 8, p. 2If X is complete, then every set valued contraction has a fixed point, i.e.a point z ∈ X such that z ∈ T (z) ([6]).We denote by Fix(T ) the set of all fixed points of the set-valued map T .Obviously, Fix(T ) is closed.Proposition 2.1. ([8]) Let X be a complete metric space and supposethat T1, T2 are λ-contractions with closed values in X. ThendH(Fix(T1), F ix(T2)) ≤11− λsupz∈Xd(T1(z), T2(z)).In what follows E is a real separable Banach space with norm | · |, C(I, E)is the Banach space of all continuous functions from I to E with the norm‖x(·)‖C = supt∈I |x(t)|, ACi(I, E) is the space of i-times differentiable func-tions x : I → E whose i-th derivative xi(·) is absolutely continuous andL1(I, E) is the Banach space of (Bochner) integrable functions u(·) : I → Eendowed with the norm ‖u(·)‖1 =∫ 10 |u(t)|dt.A function x(·) ∈ ACn−1(I, E) is called a solution of problem (1.1)-(1.2)if there exists a function v(·) ∈ L1(I, E) with v(t) ∈ F (t, x(t)), a.e. (I) suchthat x(n)(t)− λx(t) = v(t), a.e. (I) and x(·) satisfies conditions (1.2).In what follows we consider the Green function G(·, ·) : I × I → Eassociated to the periodic boundary problemx(n) − λx = 0, x(i)(0)− x(i)(T ) = 0, i = 0, 1, . . . , n− 1.For the properties of G(·, ·) we refer to [1].The next result is well known (e.g. [1]).Lemma 2.2. ([1]) If v(·) : [0, T ] → E is an integrable function then theproblemx(n)(t)− λx(t) = v(t) a.e. (I)x(i)(0)− x(i)(T ) = µi, i = 0, 1, . . . , n− 1.has a unique solution x(·) ∈ ACn−1(I, E) given byx(t) = Pµ(t) +∫ T0G(t, s)v(s)ds,wherePµ(t) =n−1∑i=0∂i∂tiG(t, 0)µn−1−i. (2.1)EJQTDE, 2007 No. 8, p. 3In the sequel we assume the following conditions on F .Hypothesis 2.3. (i) F (·, ·) : I ×E → P(E) has nonempty closed valuesand for every x ∈ E F (·, x) is measurable.(ii) There exists L(·) ∈ L1(I, E) such that for almost all t ∈ I, F (t, ·) isL(t)-Lipschitz in the sense thatdH(F (t, x), F (t, y)) ≤ L(t)|x− y| ∀ x, y ∈ Eand d(0, F (t, 0)) ≤ L(t) a.e. (I).Denote L0 :=∫ T0 L(s)ds and M0 := sup(t,s)∈I×I |G(t, s)|.3 The main resultWe are able now to prove our main result.Theorem 3.1. Assume that Hypothesis 2.3 is satisfied and M0L0 <1. Let y(·) ∈ ACn−1(I, E) be such that there exists q(.) ∈ L1(I, E) withd(y(n)(t) − λy(t), F (t, y(t))) ≤ q(t), a.e. (I). Denote µ˜i = y(i)(0) − y(i)(T ),i = 0, 1, . . . , n− 1.Then for every ε > 0 there exists x(·) a solution of (1.1)-(1.2) satisfyingfor all t ∈ I|x(t)− y(t)| ≤11−M0L0supt∈I|Pµ(t)− Pµ˜(t)|+M01−M0L0∫ T0q(t)dt+ ε,where Pµ(t) is defined in (2.1).Proof. For u(·) ∈ L1(I, E) define the following set valued maps:Mu(t) = F(t, Pµ(t) +∫ T0G(t, s)u(s)ds), t ∈ I,T (u) = {φ(.) ∈ L1(I, E); φ(t) ∈Mu(t) a.e. (I)}.It follows from the definition and Lemma 2.2 that x(·) is a solution of(1.1)-(1.2) if and only if x(n)(·)− λx(·) is a fixed point of T (·).We shall prove first that T (u) is nonempty and closed for every u ∈L1(I, E). The fact that the set valued mapMu(·) is measurable is well known.For example the map t → Pµ(t) +∫ T0 G(t, s)u(s)ds can be approximated byEJQTDE, 2007 No. 8, p. 4step functions and we can apply Theorem III. 40 in [2]. Since the values ofF are closed with the measurable selection theorem (Theorem III.6 in [2])we infer that Mu(·) admits a measurable selection φ. One has|φ(t)| ≤ d(0, F (t, 0)) + dH(F (t, 0), F(t, Pµ(t) +∫ T0G(t, s)u(s)ds))≤ L(t)(1 + |Pµ(t)|+ supt,s∈I|G(t, s)|∫ T0|u(s)|ds),which shows that φ ∈ L1(I, E) and T (u) is nonempty.On the other hand, the set T (u) is also closed. Indeed, if φn ∈ T (u) and‖φn−φ‖1 → 0 then we can pass to a subsequence φnk such that φnk(t)→ φ(t)for a.e. t ∈ I, and we find that φ ∈ T (u).We show next that T (·) is a contraction on L1(I, E).Let u, v ∈ L1(I, E) be given, φ ∈ T (u) and let δ > 0. Consider thefollowing set-valued map:H(t) =Mv(t) ∩{x ∈ E; |φ(t)− x| ≤ L(t)∣∣∣∣∣∫ T0G(t, s)(u(s)− v(s))ds∣∣∣∣∣+ δ}.From Proposition III.4 in [2], H(·) is measurable and from Hypothesis 2.3ii)H(·) has nonempty closed values. Therefore, there exists ψ(·) a measurableselection of H(·). It follows that ψ ∈ T (v) and according with the definitionof the norm we have‖φ− ψ‖1 =∫ T0|φ(t)− ψ(t)|dt≤∫ T0L(t)(∫ T0|G(t, s)||u(s)− v(s)|ds)dt+∫ T0δdt=∫ T0(∫ T0L(t)|G(t, s)|dt)|u(s)− v(s)|ds+ Tδ≤ M0L0‖u− v‖1 + Tδ.Since δ > 0 was chosen arbitrary, we deduce thatd(φ, T (v)) ≤M0L0‖u− v‖1.Replacing u by v we obtaindH(T (u), T (v)) ≤M0L0‖u− v‖1,EJQTDE, 2007 No. 8, p. 5thus T (·) is a contraction on L1(I, E).We consider next the following set-valued mapsF1(t, x) = F (t, x) + q(t)B, (t, x) ∈ I ×E,M1u(t) = F1(t, Pµ˜(t) +∫ T0G(t, s)u(s)ds), t ∈ I, u(·) ∈ L1(I, E),T1(u) = {ψ(·) ∈ L1(I, E); ψ(t) ∈M1u(t) a.e. (I)},where B denotes the closed unit ball in E. Obviously, F1(·, ·) satisfies Hy-pothesis 2.3.Repeating the previous step of the proof we obtain that T1 is also aM0L0-contraction on L1(I, E) with closed nonempty values.We prove next the following estimatedH(T (u), T1(u)) ≤ supt∈I|Pµ(t)− Pµ˜(t)|L0 +∫ T0q(t)dt. (3.1)Let φ ∈ T (u), δ > 0 and defineH1(t) =M1u(t) ∩{z ∈ E; |φ(t)− z| ≤ L(t)|Pµ(t)− Pµ˜(t)|+ q(t) +δT}.With the same arguments used for the set valued map H(·), we deducethat H1(·) is measurable with nonempty closed values. Hence let ψ(·) be ameasurable selection of H1(·). It follows that ψ ∈ T1(u) and one has‖φ− ψ‖1 =∫ T0|φ(t)− ψ(t)|dt≤∫ T0[L(t)|Pµ(t)− Pµ˜(t)|+ q(t) +δT]dt≤∫ T0L(t)|Pµ(t)− Pµ˜(t)|dt+∫ T0q(t) + δ≤ L0 supt∈I|Pµ(t)− Pµ˜(t)|+∫ T0q(t)dt+ δ.Since δ is arbitrary, as above we obtain (3.1).We apply Proposition 2.1 and we infer thatdH(Fix(T ), F ix(T1)) ≤L01−M0L0supt∈I|Pµ(t)− Pµ˜(t)|+11−M0L0∫ T0q(t)dt.EJQTDE, 2007 No. 8, p. 6Since v(·) = y(n)(·) − λy(·) ∈ Fix(T1) it follows that there exists u(·) ∈Fix(T ) such that for any ε > 0‖v − u‖1 ≤L01−M0L0supt∈I|Pµ(t)− Pµ˜(t)|+11−M0L0∫ T0q(t)dt+εM0.We define x(t) = Pµ(t) +∫ T0 G(t, s)u(s)ds, t ∈ I and we have|x(t)− y(t)| ≤ |Pµ(t)− Pµ˜(t)|+∫ T0|G(t, s)||u(s)− v(s)|ds≤ supt∈I|Pµ(t)− Pµ˜(t)|+ supt∈I|Pµ(t)− Pµ˜(t)|M0L01−M0L0+11−M0L0M0∫ T0q(t)dt+ ε≤11−M0L0supt∈I|Pµ(t)− Pµ˜(t)|+M01−M0L0∫ T0q(t)dt+ ε,which completes the proof.References[1] M. Benchohra, J. R. Graef, J. Henderson and S. K. Ntouyas, Nonres-onance impulsive higher order functional nonconvex-valued differentialinclusions, Electron. J. Qual. Theory Differ. Equ. (2002), No 13, 13pp(electronic).[2] C. Castaing and M. Valadier, Convex Analysis and Measurable Multi-functions, LNM 580, Springer, Berlin, 1977.[3] A. Cernea, A Filippov type existence theorem for an infinite operationaldifferential inclusion, Stud. Cerc. Mat. 50 (1998), 15-22.[4] A. Cernea, An existence theorem for some nonconvex hyperbolic differ-ential inclusions, Mathematica (Cluj) 45(68) (2003), 121-126.[5] A. Cernea, Existence for nonconvex integral inclusions via fixed points,Arch. Math. (Brno) 39 (2003), 293-298.[6] H. Covitz and S. B. Nadler jr., Multivalued contraction mapping in gen-eralized metric spaces, Israel J. Math. 8 (1970), 5-11.EJQTDE, 2007 No. 8, p. 7[7] Z. Kannai and P. Tallos, Stability of solution sets of differential inclusions,Acta Sci. Math. (Szeged) 61(1995), 197-207.[8] T.C. Lim, On fixed point stability for set valued contractive mappingswith applications to generalized differential equations, J. Math. Anal.Appl. 110 (1985), 436-441.(Received December 15, 2006)EJQTDE, 2007 No. 8, p. 8

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On the existence of solutions for a higher order differential inclusion without convexity

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