EXISTENCE OF SOLUTIONS FOR SECOND ORDER SEMILINEAR FUNCTIONAL INTEGRODIFFERENTIAL INCLUSIONS IN BANACH SPACES

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1 ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I.CUZA IAŞI Tomul XLVII, s.i a, Matematică, 21, f.1. EXISTENCE OF SOLUTIONS FOR SECOND ORDER SEMILINEAR FUNCTIONAL INTEGRODIFFERENTIAL INCLUSIONS IN BANACH SPACES BY M. BENCHOHRA 1 and S.K. NTOUYAS 2 Abstract. In this paper we investigate the existence of mild solutions on an unbounded real interval to second order initial value problems for a class of semilinear functional integrodifferential inclusions in Banach spaces. We shall make use of a theorem of Ma, which is an extension to multivalued maps on locally convex topological spaces of Schaefer s theorem. Key words and phrases: Initial value problems, convex valued multivalued map, mild solution, functional integrodifferential inclusion, existence, fixed point, abstract space. AMS (MOS) Subject Classifications: 34A6, 34G2, 34K5, 35R1, 47H2 1. Introduction. Existence of solutions on compact intervals for functional differential equations has received much attention in recent years, we refer for instance to the books of ERBE, QINGAI and ZHANG [5] and HEN- DERSON [9] to the paper of HRISTOVA and BAINOV [1], NIETO, JIANG AND JURANG [16] and NTOUYAS [17]. Properties of semigroups theory and additional results on differential and integral equations in abstract spaces using this theory can be found in the books of GOLDSTEIN [8] and CORDUNEANU [2]. For other results on differential inclusions on compact intervals, see for instance, the papers of PAPAGEORGIOU [18], [19] and BENCHOHRA [1] for semilinear differential inclusions on the semi infinite interval. In the above mentioned papers the main tools used for the existence of solutions are the monotone iterative method combined with upper and lower solutions or the topological transversality theory of Granas. For more details on these theories we refer the interesting reader to the book of LADDE,

2 154 M. BENCHOHRA and S.K. NTOUYAS 2 LAKSHMIKANTHAM and VATSALA [13] and to the monograph of DUGUNDJI and GRANAS [4] respectively. In this paper we shall prove a theorem which assures the existence of mild solutions defined on an unbounded real interval J for the initial value problem (IVP for short) of the second order functional integrodifferential inclusion (1.1) y Ay K(t, s)f (s, y s )ds, t J = [, ), (1.2) y = φ, y () = η where F : J C(J, E) 2 E (here J = [ r, ], < r < )) is a bounded, closed, convex valued multivalued map, K : D IR, φ C(J, E), η E, D = {(t, s) J J : t s}, A, is the infinitesimal generator of a strongly continuous cosine family {C(t) : t IR} in a real Banach space E with the norm.. For any continuous function y defined on the interval J 1 = [ r, ) and any t J, we denote by y t the element of C(J, E) defined by y t (θ) = y(t + θ), θ J. Here y t (.) represents the history of the state from time t r, up to the present time t. The method we are going to use is to reduce the existence of solutions to problem (1.1)-(1.2) to the search for fixed points of a suitable multivalued map on the Fréchet space C(J, E). In order to prove the existence of fixed points, we shall rely on a theorem due to MA [15], which is an extension to multivalued maps between locally convex topological spaces of Schaefer s theorem [2]. This paper is a generalization of the previous papers to second order functional differential inclusions on unbounded real intervals. 2. Preliminaries. In this section, we introduce notations, definitions, and preliminary facts from multivalued analysis which are used throughout this paper. J m is the compact real interval [, m] (m IN). C(J, E) is the linear metric Fréchet space of continuous functions from J into E with the metric (see CORDUNEANU [2])

3 3 SEMILINEAR FUNCTIONAL INTEGRODIFFERENTIAL INCLUSIONS 155 d(y, z) = m= 2 m y z m 1 + y z m for each y, z C(J, E), where y m := sup{ y(t) : t J m }. 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 is Lebesgue integrable. For properties of the Bochner integral we refer to YOSIDA [23]. L 1 (J, E) denotes the Banach space of functions y : J E which are Bochner integrable normed by y L 1 = y(t) dt for all y L 1 (J, E). U p denotes the neighbourhood of in C(J, E) defined by U p := {y C(J, E) : y m p for each m N}. The convergence in C(J, E) is the uniform convergence on compact intervals, i.e. y j y in C(J, E) if and only if for each m IN, y j y m in C(J m, E) as j. M C(J, E) is a bounded set if and only if there exists a positive function ϕ C(J, IR) such that y(t) ϕ(t) for all t J and all y M. The Arzela Ascoli theorem says that a set M C(J, E) is compact if and only if for each m IN, M is a compact set in the Banach space (C(J m, E),. m ). We say that a family {C(t) : t IR} of operators in B(E) is a strongly continuous cosine family if (i) C() = I(I is the identity operator in E), (ii) C(t + s) + C(t s) = 2C(t)C(s) for all s, t R, (iii) the map t C(t)y is strongly continuous for each y E; The strongly continuous sine family {S(t) : t IR}, associated to the given strongly continuous cosine family {C(t) : t IR}, is defined by S(t)y = C(s)yds, y E, t IR.

4 156 M. BENCHOHRA and S.K. NTOUYAS 4 The infinitesimal generator A : E E of a cosine family {C(t) : t IR} is defined by Ay = d2 dt 2 C()y. For more details on strongly continuous cosine and sine families, we refer the reader to the book of GOLDSTEIN [8] and to the papers of FATTORINI 6], [7] and TRAVIS and WEBB [21], [22]. Let (X,. ) be a Banach space. A multivalued map G : X 2 X is convex (closed) valued if G(x) is convex (closed) for all x X. G is bounded on bounded sets if G(B) = x B G(x) is bounded in X for any bounded set B of X (i.e. sup x B {sup{ y : y G(x)}} < ). G is called upper semicontinuous (u.s.c.) on X if for each x X the set G(x ) is a nonempty, closed subset of X, and if for each open set B of X containing G(x ), there exists an open neighbourhood V of x such that G(V ) B. G is said to be completely 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, then G is u.s.c. if and only if G has a closed graph (i.e. x n x, y n y, y n Gx n imply y Gx ). G has a fixed point if there is x X such that x Gx. In the following BCC(X) denotes the set of all nonempty bounded, closed and convex subsets of X. A multivalued map G : J BCC(E) is said to be measurable if for each x E the function Y : J IR defined by Y (t) = d(x, G(t)) = inf{ x z : z G(t)} is measurable. For more details on multivalued maps see the books of DEIMLING [3] and HU and PAPAGEORGIOU [11]. Let us list the following hypotheses: (H1) A is an infinitesimal generator of a given strongly continuous and bounded cosine family {C(t) : t J}; (H2) F : J C(J, E) BCC(E); (t, u) F (t, u) is measurable with respect to t for each u C(J, E), u.s.c. with respect to u for each t J and for each fixed u C(J, E) the set S F,u = { g L 1 (J, E) : g(t) F (t, u) for a.e. } t J is nonempty; (H3) for each t J m (m {1, 2,...}), K(t, s) is measurable on [, t] and K(t) = ess sup{ K(t, s), s t}, is bounded on J m ;

5 5 SEMILINEAR FUNCTIONAL INTEGRODIFFERENTIAL INCLUSIONS 157 (H4) the map t K t is continuous from J m to L (J m, IR); here K t (s) = = K(t, s); (H5) F (t, u) := sup{ v F (t, u)} p(t)ψ( u ) for almost all t J and all u C(J, E), where p L 1 (J, IR + ) and ψ : IR + (, ) is continuous and increasing with Mm sup K(t) m p(s)ds < c dτ ψ(τ) for each m IN; where c = M φ + Mm η and M 1 be such that C(t) M for t J; (H6) for each neighbourhood U p of, u U p and t J the set { C(t)φ + S(t)η + is relatively compact in E. K(s, u)g(σ)dσds : g S F,y } Remark. If dime < and J is a compact real interval, then for each u C(J, E) the set S F,u is nonempty (see LASOTA and OPIAL [14]). Definition. A continuous solution y(t) of the integral inclusion y(t) C(t)φ() + S(t)η + with y = φ is called a mild solution on J 1 of (1.1)-(1.2). K(s, u)f (u, y(u))duds The following lemmas are crucial in the proof of our main theorem. Lemma 2.1. [14] Let I be a compact real interval and X be a Banach space. Let F be a multivalued map satisfying (H2) and let Γ be a linear continuous mapping from L 1 (I, X) to C(I, X), then the operator Γ S F : C(I, X) BCC(C(I, X)), y (Γ S F )(y) := Γ(S F,y ) is a closed graph operator in C(I, X) C(I, X). Lemma 2.2. [15] Let X be a locally convex space and N : X X be a compact convex valued, u.s.c. multivalued map such that there exists a closed neighbourhood U p of for which N(U p ) is a relatively compact set for each p IN. If the set Ω := {y X : λy N(y) for some λ > 1} is bounded, then N has a fixed point.

6 158 M. BENCHOHRA and S.K. NTOUYAS 6 3. Main result. Now, we are able to state and prove our main theorem. Theorem 3.1. Assume that hypotheses (H1)-(H6) hold. Then the IVP (1.1)-(1.2) has at least one mild solution on J 1. Proof. Let C(J 1, E) be the Fréchet space provided with the seminorms y m := sup{ y(t) : t [ r, m]}, for y C(J 1, E), m IN. Transform the problem into a fixed point problem. Consider the multivalued map, N : C(J 1, E) 2 C(J 1,E) defined by: φ(t), t J C(t)φ() + S(t)η+ Ny = h C(J 1, E) : h(t) = + K(s, u)g(u)duds, t J where g S F,y = { } g L 1 (J, E) : g(t) F (t, y t ) for a.e. t J. Remark 3.1. It is clear that the fixed points of N are mild solutions to (1.1)-(1.2). We shall show that N(U q ) is relatively compact for each neighbourhood U q of C(J, E) with q IN and the multivalued map N has bounded, closed and convex values and it is u.s.c. The proof will be given in several steps. Step 1. Ny is convex for each y C(J, E). Indeed, if h 1, h 2 belong to Ny, then there exist g 1, g 2 S F,y such that for each t J we have and h 1 (t) = C(t)φ() + S(t)η + K(s, u)g 1 (u)duds, h 2 (t) = C(t)φ() + S(t)η + Let α 1. Then for each t J we have (αh 1 + (1 α)h 2 )(t) = C(t)φ() + S(t)η+ + K(s, u)g 2 (u)duds. K(s, u)[αg 1 (u) + (1 α)g 2 (u)]duds. Since S F,y is convex (because F has convex values) then αh 1 +(1 α)h 2 Ny.

7 7 SEMILINEAR FUNCTIONAL INTEGRODIFFERENTIAL INCLUSIONS 159 Step 2: N(U q ) is bounded in C(J, E) for each q IN. Indeed, it is enough to show that for each m IN there exists a positive constant l m such that for each h Ny, y U q one has h l m. If h Ny, then there exists g S F,y such that for each t J we have h(t) = C(t)φ() + S(t)η + K(s, u)g(u)duds. By (H3), (H4) and (H5) we have for each t J m that h(t) C(t)φ() + S(t). η + M φ + Mm η + M K(s, u)g(u) duds K(s, u) p(u)ψ( y(u) )duds. M φ + Mm η + mm sup K(t) p L 1 sup ψ( y(t) ) = l m. Step 3: For each q IN, N(U q ) is equicontinuous for U q C(J, E). Let t 1, t 2 J m, t 1 < t 2 and U q be a neighbourhood of in C(J, E) for q IN. For each y U q and h Ny, there exists g S F,y such that Thus h(t) = C(t)φ() + S(t)η + K(s, u)g(u)duds. h(t 2 ) h(t 1 ) C(t 2 )φ() C(t 1 )φ() + S(t 2 )η S(t 1 )η [S(t 2 s) S(t 1 s)] + S(t 1 s) t 1 K(s, u)g(u)duds K(s, u)g(u)duds C(t 2 )φ() C(t 1 )φ() + S(t 2 )η S(t 1 )η +M(t 2 t 1 ) sup K(t) +Mm sup K(t) 2 t 1 m g(u) du. g(u) du As t 2 t 1 the right-hand side of the above inequality tends to zero.

8 16 M. BENCHOHRA and S.K. NTOUYAS 8 The equicontinuity for the cases t 1 < t 2 and t 1 t 2 follows from the uniform continuity of φ on the interval J and from the relation h(t 2 ) h(t 1 ) = h(t 2 ) φ(t 1 ) h(t 2 ) h() + φ() φ(1) respectively. As a consequence of Step 2, Step 3 and (H6) together with the Arzela- Ascoli theorem we can conclude that N(U q ) is relatively compact in C(J, E). Step 4: N has a closed graph. Let y n y, h n Ny n, and h n h. We shall prove that h Ny. h n N(y n ) means that there exists g n S F,yn such that h n (t) = C(t)φ() + S(t)η + We must prove that there exists g S F,y (3.1) h (t) = C(t)φ() + S(t)η + The idea is then to use the facts that such that (i) h n h ; (ii) h n C(t)φ() S(t)η Γ(S F,yn ) where defined by (Γg)(t) := K(s, u)g n (u)duds. Γ : L 1 (J, E) C(J, E) K(s, u)g (u)duds. K(s, u)g(u)duds. If Γ S F is a closed graph operator, we would be done. But we do not know whether Γ S F is a closed graph operator. So, we cut the functions y n, h n C(t)φ() S(t)η, g n and we consider them defined on the interval [k, k+1] for any k IN. Then, using Lemma 2.1, in this case we are able to affirm that (3.1) is true on the compact interval [k, k + 1], i.e. h (t) = C(t)φ() + S(t)η + [k,k+1] K(s, u)g k (u)duds

9 9 SEMILINEAR FUNCTIONAL INTEGRODIFFERENTIAL INCLUSIONS 161 for a suitable L 1 -selection g k of F (t, y t ) on the interval [k, k + 1]. At this point we can paste the functions g k obtaining the selection g defined by g (t) = g k (t) for t [k, k + 1). We obtain then that g is an L 1 -selection and (3.1) will satisfied. We now give the details. Clearly we have that (h n C(t)φ() S(t)η) (h C(t)φ() S(t)η), as n. Now, we consider for all k IN, the mapping S k F : C([k, k + 1], E) L 1 ([k, k + 1], E) u S k F,u := {f L 1 ([k, k+1], E) : f(t) F (t, u(t)) for a.e. t [k, k+1]}. Also, we consider the linear continuous operators Γ k : L 1 ([k, k + 1], E) C([k, k + 1], E) g Γ k (g)(t) = K(s, u)g(u)duds. From Lemma 2.1, it follows that Γ k SF k is a closed graph operator for all k IN. Moreover, we have that ( ) h n (t) C(t)φ() S(t)η Γ k (SF,y [k,k+1] k n ). Since y n y, it follows from Lemma 2.1 that ( ) h (t) C(t)φ() S(t)η = [k,k+1] for some g k S k F,y. So the function g defined on J by g (t) = g k (t) for t [k, k + 1) is in S F,y since g (t) F (t, y t ) for a.e. t J. K(s, u)g k (u)duds

10 162 M. BENCHOHRA and S.K. NTOUYAS 1 Now it remains to show that the set Ω := {y C(J, E) : λy Ny, for some λ > 1} is bounded. Let y Ω. Then λy Ny for some λ > 1. Thus there exists g S F,y such that y(t) = λ 1 C(t)φ + λ 1 S(t)η + λ 1 K(s, u)g(u)duds, t J. This implies by (H3), (H4) and (H5) that for each t J m we have y(t) M φ + Mm η + M M φ + Mm η + M M φ + Mm η + Mm sup K(t) We consider the function µ defined by K(s, u)g(u)duds K(s, u) p(u)ψ( y u )duds p(s)ψ( y s )ds. µ(t) = sup{ y(s) : r s t}, t m. Let t [ r, t] be such that µ(t) = y(t ). If t [, m], by the previous inequality we have for t [, m] µ(t) M φ + Mm η + Mm sup K(t) M φ + Mm η + Mm sup K(t) p(s)ψ( y s )ds p(s)ψ(µ(s))ds. If t J then µ(t) = φ and the previous inequality holds since M 1. Let us take the right-hand side of the above inequality as v(t), then we have v() = M φ + Mm η = c and y(t) v(t). Using the increasing character of ψ we get v (t) Mm sup K(t)p(t)ψ(v(t)).

11 11 SEMILINEAR FUNCTIONAL INTEGRODIFFERENTIAL INCLUSIONS 163 This implies for each t J m that v(t) v() du Mm sup K(t) ψ(u) m p(s)ds > v() du ψ(u). This inequality implies that there exists a constant b such that v(t) b, t J m, and hence µ(t) b, t J m. Since for every t [, m], y t µ(t), we have y := sup{ y(t) : r t m} b, where b depends only m and on the functions p and ψ. This shows that Ω is bounded. Set X := C(J 1, E). As a consequence of Lemma 2.2 we deduce that N has a fixed point which is a mild solution of (1.1)-(1.2) on J 1. REFERENCES 1. BENCHOHRA, M. Existence of mild solutions on infinite intervals to first order initial value problems for a class of differential inclusions in Banach spaces, Discuss. Math. Differential Incl., 19 (1999), CORDUNEANU, C. Integral Equations and Applications, Cambridge Univ. Press, New York, DEIMLING, K. Multivalued Differential Equations, Walter de Gruyter, Berlin-New York, DUGUNDJI, J. and GRANAS, A. Fixed Point Theory, Monografie Mat. PWN, Warsaw, ERBE, L.H., KONG, Q. and ZHANG, B.G. Oscillation Theory for Functional Differential Equations, Pure and Applied Mathematics, FATTORINI, H.O. Ordinary differential equations in linear topological spaces, I, J. Differential Equations 5 (1968), FATTORINI, H.O. Ordinary differential equations in linear topological spaces, II, J. Differential Equations 6 (1969), GOLDSTEIN, J.A. Semigroups of Linear Operators and Applications, Oxford Univ. Press, New York, HENDERSON, J. Boundary Value Problems for Functional Differential Equations, World Scientific, HRISTOVA, S.G. and BAINOV, D.D. Application of the monotone iterative techniques of Lakshmikantham to the solution of the initial value problem for functional differential equations, J. Math. Phys. Sci. 24 (199),

12 164 M. BENCHOHRA and S.K. NTOUYAS HU, SH. and PAPAGEORGIOU, N. Handbook of Multivalued Analysis, Kluwer, Dordrecht, Boston, London, KUSANO, T. and OHARU, S. Semilinear evolution equations with singularities in ordered Banach Spaces, Differential Integral Equations 5 (6) (1992), LADDE, G.S., LAKSHMIKANTHAM, V. and VATSALA, A.S. Monotone Itera tive Techniques for Nonlinear Differential Equations, Pitman, Boston M. A., LASOTA, A. and OPIAL, Z. An application of the Kakutani-Ky-Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), MA, T.W. Topological degrees for set-valued compact vector fields in locally convex spaces, Dissertationes Math. 92 (1972), NIETO, J.J., JIANG, Y. and JURANG, Y. Monotone iterative method for functional differential equations, Nonliear Anal. 32 (6) (1998), NTOUYAS, S.K. Initial and boundary value problems for functional differential equations via the topological method: A survey, Bull. Greek Math. Soc. 4 (1998), PAPAGEORGIOU, N.S. Existence of solutions for second order evolution inclusions, J. Appl. Math. Stochastic Anal. 7 (4) (1994), PAPAGEORGIOU, N.S. Boundary value problems for evolution inclusions, Comment. Math. Univ. Carol. 29 (1988), SCHAEFER, H. Uber die methode der a priori schranken, Math. Ann. 129, (1955), TRAVIS, C.C. and WEBB, G.F. Second order differential equations in Banach spaces, Proc. Int. Symp. on Nonlinear Equations in Abstract Spaces, Academic Press, New York (1978), TRAVIS, C.C. and WEBB, G.F. Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hungarica 32 (1978), YOSIDA, K. Functional Analysis, 6 th edn. Springer-Verlag, Berlin, 198. Received: 3.V.2 1 Département de Mathématiques Université de Sidi Bel Abbès BP 89, 22 Sidi Bel Abbès ALGÉRIE benchohra@yahoo.com 2 Department of Mathematics University of Ioannina Ioannina GREECE sntouyas@cc.uoi.gr