INITIAL AND BOUNDARY VALUE PROBLEMS FOR NONCONVEX VALUED MULTIVALUED FUNCTIONAL DIFFERENTIAL EQUATIONS

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1 Applied Mathematics and Stochastic Analysis, 6:2 23, 9-2. Printed in the USA c 23 by North Atlantic Science Publishing Company INITIAL AND BOUNDARY VALUE PROBLEMS FOR NONCONVEX VALUED MULTIVALUED FUNCTIONAL DIFFERENTIAL EQUATIONS M. BENCHOHRA Université de Sidi Bel Abbès, Laboratoire de Mathématiques BP 89, 22 Sidi Bel Abbès, Algérie E mail: benchohra@univ-sba.dz S.K. NTOUYAS University of Ioannina, Department of Mathematics 45 Ioannina, Greece E mail: sntouyas@cc.uoi.gr Received March 22; Revised March 23 In this note we investigate the existence of solutions of initial and boundary value problems defined on a compact interval to some classes of functional differential inclusions. We shall rely on a fixed point theorem for contraction multivalued maps due to Covitz and Nadler and Schafer s theorem combined with lower semicontinuous multivalued operators with decomposable values. Keywords: Functional Differential Inclusions, Selection, Contraction Multivalued Map, Lower Semicontinuous Multivalued Map, Decomposable Valued Map, Existence, Fixed Point. AMS MOS Subject Classifications: 34A6, 34K. Introduction In this paper we prove existence results for the following functional differential inclusions: ρty t F t, y t, a.e. t [,T],. subject to inital or boundary conditions y = φ, y = η.2 y = φ, yt =η.3 respectively, where F : J C[ r, ], IR n P IR n is a multivalued map, φ C[ r, ], IR n, ρ CJ, IR +,η IR n, PIR n is the family of all subsets of IR n. 9

2 92 M. BENCHOHRA and S.K. NTOUYAS. For any continuous function y defined on the interval [ r, T ] and any t [,T], we denote by y t the element of C[ r, ], IR n defined by y t θ =yt + θ, θ [ r, ]. Here y t represents the history of the state from time t r, up to the present time t. Recently, in [], the authors studied first and second order initial value problems for equation., in the case where ρt =, by using a fixed point theorem for contraction multivalued maps due to Covitz and Nadler [5] see also Deimling [6]. Using a fixed point theorem for condensing multivalued maps due to Martelli, the authors have obtained an existence result for the initial value problem.-.2. Here, by using the fixed point theorem for contraction maps and Schaefer s theorem combined with a selestion theorem of Bressan and Colombo for lower semicontinuous multivalued operators with decomposable values, existence results are proposed for problems.-.2 and Preliminaries In this section, we introduce notations, definitions, and preliminary facts from multivalued analysis which are used throughout this note. C[ r, ], IR n is the Banach space of all continuous functions from [ r, ] into IR n with the norm φ = sup{ φθ : r θ }. By C[,T], IR n, we denote the Banach space of all continuous functions from [,T] into IR n with the norm y [,T ] := sup{ yt : t [,T]}. L [,T], IR n denotes the Banach space of measurable functions y :[,T] IR n which are Lebesgue integrable normed by y L = yt dt for all y L [,T], IR n. AC i [,T], IR n is the space of i-times differentiable functions y :[,T] IR n, whose i th derivative, y i, is absolutely continuous. Let A be a subset of [,T] IR n. A is L B measurable if A belongs to the σ-algebra generated by all sets of the form N D where N is Lebesgue measurable in J and D is Borel measurable in IR n. A subset B of L [,T], IR n is decomposable if, for all u, v B and N [,T] measurable, the function uχ N + vχ J N B, where χ denotes for the characteristic function. Let E be a Banach space, X a nonempty closed subset of E and G : X PE a multivalued operator with nonempty closed values. G is lower semi-continuous l.s.c. if the set {x X : Gx C } is open for any open set C in E. G has a fixed point if there is x X such that x Gx. Definition 2.: Let Y be a separable metric space and let N : Y PL [,T], IR n be a multivalued operator. We say N has property BC if N is lower semi-continuous l.s.c.;

3 Functional Differential Inclusions 93 2 N has nonempty closed and decomposable values. Let F :[,T] C[ r, ], IR n PIR n be a multivalued map with nonempty compact values. Assign to F the multivalued operator by letting F : C[ r, T ], IR n PL [,T], IR n Fy ={w L [,T], IR n :wt F t, y t for a.e. t [,T]}. The operator F is called the Niemytzki operator associated with F. We say F is of lower semi-continuous type l.s.c. type if its associated Niemytzki operator F is lower semi-continuous and has nonempty closed and decomposable values. Next we state a selection theorem due to Bressan and Colombo. Lemma 2.: [3] Let Y be separable metric space and let N : Y PL [,T], IR n be a multivalued operator which has property BC. Then N has a continuous selection, i.e. there exists a continuous function single-valued g : Y L [,T], IR n such that gy F y for every y Y. Let X, d be a metric space. We use the notations: P X ={Y PX :Y }, P cl X ={Y P X :Y closed}, P b X = {Y P X :Y bounded}. Consider H d : P X P X IR + { }, given by { } H d A, B = max sup a A da, B, sup da, b b B where da, b = inf da, b, da, B = inf da, b. a A b B Then P b,cl X,H d is a metric space and P cl X,H d is a generalized metric space. Definition 2.2: A multivalued operator N : X P cl X is called a γ-lipschitz if and only if there exists γ> such that H d Nx,Ny γdx, y, for each x, y X, b contraction if and only if it is γ-lipschitz with γ<. For more details on multivalued maps and the proof of known results cited in this section we refer to the books of Deimling [6], Górniewicz [8], Hu and Papageorgiou [9] and Tolstonogov []. Our considerations are based on the following fixed point theorem for contraction multivalued operators given by Covitz and Nadler in 97 [5] see also Deimling, [6] Theorem.. Lemma 2.2: Let X, d be a complete metric space. If N : X P cl X is a contraction, then F ixn. 3 Initial Value Problems Now, we are able to state and prove our main theorems. In this section we shall give two results for the IVP..2. Before stating and proving these results, we give the definition of a solution of the IVP..2.,

4 94 M. BENCHOHRA and S.K. NTOUYAS. Definition 3.: A function y :[ r, T ] E is called solution for the IVP.-.2 if y, ρ y C[ r, T ], IR n AC [,T], IR n and satisfies the differential inclusion. a.e. on [,T] and the conditions.2. Theorem 3.: Assume that: H F :[,T] C[ r, ], IR n P cl IR n has the property that F,u:[,T] P cl IR n is measurable for each u C[ r, ], IR n ; H2 H d F t, u,ft, u lt u u, for each t [,T] and u, u C[ r, ], IR n, where l L [,T], IR, and d,ft, lt for a.e. t [,T]. Then the IVP.-.2 has at least one solution on [ r, T ] provided Tρ lsds <, where ρ = min{ρt :t [,T]}. Proof: Transform the problem into a fixed point problem. Consider the multivalued operator N : C[ r, T ], IR n PC[ r, T ], IR n defined by: Ny = where h C[ r, T ], IR n :ht = S F y = Remark 3.: φt, if t [ r, ] φ + ρη + s ds gτdτds, if t [,T],g S F y { } g L [,T], IR n :gt F t, y t for a.e. t [,T]. i It is clear that the fixed points of N are solutions to.-.2. ii for each y C[ r, T ], IR n the set S F y is nonempty since by H F has a measurable selection see [4], Theorem III.6. We shall show that N satisfies the assumptions of Lemma 2.2. The proof will be given in two steps. Step : Ny P cl C r, T ], IR n for each y C[ r, T ], IR n. Indeed, let y n n Ny such that y n ỹ in C[ r, T ], IR n. Then ỹ C[ r, T ], IR n and y n t φ + ρη ds t + s F τ,y τ dτds, t [,T]. Using the fact that F has closed values and from the second part of H2, we can easily show that F s, y s ds is closed for each t [,T]. Then So ỹ Ny. y n t φ + ρη ds t + s F τ,y τ dτds, t [,T].

5 Functional Differential Inclusions 95 Step 2: H d Ny,Ny 2 γ y y 2 [ r,t ] for each y,y 2 C[ r, T ], IR n where γ<. Let y,y 2 C[ r, T ], IR n and h Ny. Then there exists g t F t, y t such that h t =φ + ρη From H2 it follows that Hence there is w F t, y 2t such that ds t + s g τdτds, H d F t, y t,ft, y 2t lt y t y 2t. t [,T]. g t w lt y t y 2t, t [,T]. Consider U :[,T] PIR n, given by Ut ={w IR n : g t w lt y t y 2t }. Since the multivalued operator V t =Ut F t, y 2t is measurable see Proposition III.4 in [4], there exists g 2 t a measurable selection for V. So, g 2 t F t, y 2t and g t g 2 t lt y t y 2t, for each t [,T]. Let us define for each t [,T] Then we have h 2 t =φ + ρη ds t + h t h 2 t T g s g 2 s ds ρ s g 2 τdτds. T ls y s y 2s ds ρ = T ls sup y s θ y 2s θ ds ρ r θ = T ls sup y s + θ y 2 s + θ ds ρ r θ = T ls sup y z y 2 z ds ρ s r z s T ls sup y z y 2 z ds ρ r z T T T lsds y y 2 [ r,t ]. ρ Then h h 2 [ r,t ] T T lsds y y 2 [ r,t ]. ρ

6 96 M. BENCHOHRA and S.K. NTOUYAS. By the analogous relation, obtained by interchanging the roles of y and y 2, it follows that T T H d Ny,Ny 2 lsds y y 2 [ r,t ]. ρ So, N is a contraction and thus, by Lemma 2.2, it has a fixed point y, which is solution to.-.2. By the help of the Schaefer s theorem combined with the selection theorem of Bressan and Colombo for lower semicontinuous maps with decomposable values, we shall present an existence result for the problem.-.2. Before this, let us introduce the following hypotheses which are assumed hereafter: H3 F :[,T] C[ r, ], IR n P IR n is a nonempty compact valued multivalued map such that a t, u F t, u isl B measurable; b u F t, u is lower semi-continuous for a.e. t [,T]; H4 For each q>, there exists a function h q L [,T], IR + such that F t, u := sup{ v : v F t, u} h q t for a.e. t [,T] and u C[ r, ], IR n with u q. In the proof of our following theorem, we will need the auxiliary result: Lemma 3.: [7]. Let F :[,T] C[ r, ], IR n PIR n be a multivalued map with nonempty, compact values. Assume H3 and H4 hold. Then F is of l.s.c. type. Theorem 3.2: Suppose, in addition to hypotheses H3, H4, the following also holds: H5 F t, u := sup{ v : v F t, u} ptψ u for almost all t [,T] and all u C[ r, ],E, where p L [,T], IR + and ψ : IR +, is continuous and increasing with T T dτ psds < ψτ, where c = φ + ρ η T ρ. ρ Then the initial value problem..2 has at least one solution. Proof: H3 and H4 imply by Lemma 3. that F is of lower semi-continuous type. Then, from Lemma 2., there exists a continuous function f : C[,T], IR n L [,T], IR n such that fy Fy for all y C[,T], IR n. We consider the problem c ρty t = fyt, a.e. t J =[,T], 3. y = φ, y = η. 3.2 Remark 3.2: If y C[ r, T ], IR n is a solution of the problem , then y is a solution to the problem.-.2.

7 Functional Differential Inclusions 97 Transform problem into a fixed point problem. Consider the multivalued map, N : C[ r, T ], IR n C[ r, T ], IR n defined by: φt, if t [ r, ] Nyt = ds t φ + ρη + s fyududs, if t [,T]. Clearly from H4 and the Arzela-Ascoli theorem, the multivalued operator N is continuous and completely continuous. In order to apply Schaefer s theorem, it remains to show that the set EN :={y C[ r, T ], IR n :λy = Ny, for some λ>} is bounded. Let y EN. Then λy = Ny for some λ>. Thus, for each t [,T] yt =λ φ + λ ρη ds t + λ This implies by H5 that for each t [,T] we have s yt φ + ρ η T + T psψ y s ds. ρ ρ We consider the function µ defined by µt = sup{ ys : r s t}, t T. fyτdτds. Let t [ r, t] be such that µt = yt. Ift [,T], by the previous inequality we have for t [,T] µt φ + ρ η T + T psψµsds. ρ ρ If t [ r, ], then µt = φ and the previous inequality holds. Let us take the right-hand side of the above inequality as vt; then we have c = v = φ + ρ η T ρ, µt vt, t [,T] and v t = T ρ ptψµt, t [,T]. Using the nondecreasing character of ψ, we get v t T ρ ptψvt, t [,T]. This implies for each t [,T] that vt du v ψu T du psds < ρ v ψu.

8 98 M. BENCHOHRA and S.K. NTOUYAS. This inequality implies that there exists a constant b = bt,p,ψ such that vt b, t [,T], and hence µt b, t [,T]. Since for every t [,T], y t µt, we have y [r,t ] := sup{ yt : r t T } maxb, φ. This shows that EN is bounded. As a consequence of Schaefer s theorem see [], we deduce that N has a fixed point which is a solution of and hence from Remark 3.2, a solution to the problem Boundary Value Problems In the next theorem we give an existence result for the BVP.,.3. Definition 4.: A function y :[ r, T ] IR n is called solution for the BVP.,.3 if y, ρ y C[ r, T ], IR n AC [,T], IR n and satisfies the differential inclusion. a.e. on [,T] and the boundary condition.3. Theorem 4.: Let F satisfies H and H2. Then the BVP.,.3 has at least one solution on [ r, T ]. Proof: As in Theorem 3. we transform the problem into a fixed point problem. Consider the multivalued operator N : C[ r, T ], IR n PC[ r, T ], IR n defined by: N y := h C[ r, T ], IR n :ht = φt, if t [ r, ] φ + η φ ωt ωt T + Gt, sgsds, if t [,T],g S F y where ωt = ds and G is the Green s function for the corresponding homogenuous problem which is given by the formula ωt ωs ωt, if s t T Gt, s = ωs ωt ωt, if t s T. We shall show that N satisfies the assumptions of Lemma 2.. Using the same reasoning as in Step of Theorem 3., we can show that N y P cl C r, T ], IR n, for each y C[ r, T ], IR n. N is a contraction multivalued map. Indeed, let y,y 2 C[ r, T ], IR n and h N y. Then there exists g t F t, y t such that h t =φ + η φ ωt+ Gt, sg sds, ωt From H2 it follows that t [,T]. H d F t, y t,ft, y 2t lt y t y 2t.

9 Functional Differential Inclusions 99 Hence, there is w F t, y 2t such that g t w lt y t y 2t, t [,T]. Consider U :[,T] PIR n, given by Ut ={w IR n : g t w lt y t y 2t }. Since the multivalued operator V t =Ut F t, y 2t is measurable see Proposition III.4 in [4], there exists g 2 t a measurable selection for V. So, g 2 t F t, y 2t and g t g 2 t lt y t y 2t, for each t [,T]. Let us define for each t [,T] Then we have h 2 t =φ + η φ ωt+ Gt, sg 2 sds. ωt h t h 2 t Gt, s g s g 2 s ds sup Gt, s ls y s y 2s ds [,T ] [,T ] sup Gt, s ltdt y y 2 [ r,t ]. [,T ] [,T ] Then h h 2 [ r,t ] sup Gt, s ltdt y y 2 [ r,t ]. [,T ] [,T ] By the analogous relation, obtained by interchanging the roles of y and y 2, it follows that H d N y,n y 2 sup Gt, s J J ltdt y y 2 [ r,t ]. So, if we choose sup Gt, s ltdt <, N is a contraction, and thus, by [,T ] [,T ] Lemma 2.2, it has a fixed point y, which is solution to.,.3. Using the reasoning used in the proof of Theorem 3.2, we can obtain the following result where the details are left to the reader. Theorem 4.2: Let F satisfy H3 and H6 there exists a function h L [,T], IR + such that F t, u := sup{ v : v F t, u} ht for a.e. t [,T] and u C[ r, ], IR n. Then the BVP.,.3 has at least one solution on [ r, T ].

10 2 M. BENCHOHRA and S.K. NTOUYAS. References [] Benchohra, M. and Ntouyas, S.K., Existence results for functional differential inclusions, Electron. J. Differential Equations 4 2, 8. [2] Benchohra, M. and Ntouyas, S.K., Existence results for second order functional differential and integrodifferential inclusions in Banach spaces, Tamkang J. Math. 32:4 2, [3] Bressan, A. and Colombo, G., Extensions and selections of maps with decomposable values, Studia Math , [4] Castaing, C. and Valadier, M., Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 58, Springer-Verlag, New York 977. [5] Covitz, H. and Nadler, Jr., S.B., Multivalued contraction mappings in generalized metric spaces, Israel J. Math. 8 97, 5. [6] Deimling, K., Multivalued Differential Equations, Walter de Gruyter, New York 992. [7] Frigon, M. and Granas, A., Théorèmes d existence pour des inclusions différentielles sans convexité, C. R. Acad. Sci. Paris, Ser. I 3 99, [8] Gorniewicz, L., Topological Fixed Point Theory of Multivalued Mappings, Mathematics and Its Applications 495, Kluwer Academic Publishers, Dordrecht 999. [9] Hu, S. and Papageorgiou, N., Handbook of Multivalued Analysis, Volume I: Theory, Kluwer Academic Publishers, Dordrecht 997. [] Smart, D.R., Fixed Point Theorems, Cambridge Univ. Press, Cambridge 974. [] Tolstonogov, A.A., Differential Inclusions in a Banach Space, Kluwer Academic Publishers, Dordrecht 2.

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