Anichini, Giuseppe

Conti, Giuseppe

English

Florence Research

SOME PROPERTIES OF THE SOLUTION SETFOR INTEGRAL DIFFERENTIAL EQUATIONS1 Introduction and NotationsIn this paper we are concerned with the solution sets for Volterra integralequation and integrodierential equations like:x(t) = h(t) +R t0k(t; s)g(s; x(s)dsx(0) = x0;(1)or x(t) = f(t; x(t);R t0k(t; s)g(s; x(s))ds)x(0) = x0;(2)where h : I = [0; T )  ! IRn; k : I  IRn  ! IRn are continuous functions,x0 is a given vector of IRn, I a (possible unbounded) interval of IR.In the following B(x0; r) will denote an r  ball (in the metric space(X; d)) i.e. the set fx 2 X : d(x; x0) < rg where x0 is any point in X; B(0; r)wil denote the closed ball centered in x0 = 0:Let now consider the (Hilbert) space L2(I; IRn) normed, as usually, byjjxjj2 = (RIx2(t)dt)12 and its (ane) subspace E = fx 2 L2(I; IRn) : x(0) =x0g: Let X be some some Banach space; f V  X is some subset then (V )will denote its (topological) closure and V c will denote the complement of V .Finally B(X ) will denote the set of all nonempty and bounded subsets of X.Denition 1 : Let X be a Banach space and A  a subset. A measure : Bd(X)  ! IR+ dened by (V ) = inff > 0 : V 2 B(X ) admits a nitecover by sets of diameter  g where diameter of V is the supfjjx  yjj : x 2V; y 2 V g; is called the (Kuratowski) measure of noncompactness.A measure like  has interesting properties, some of which are listed inthe sequel:1a) (V ) = 0if and only if V is compact;b) (V ) = (V ); (conv(V )) = (V ); (conv(V ) = convex hull ofV );c) ((V1) + (1  )V2)  (V1) + (1  )(V2);  2 [0; 1];d) ifV1  V2 then (V1)  (V2);e) iffVng is a nested sequence of closed sets of Bd(X)and if limn!+1(Vn) = 0 then \1n=1 Vn 6= ;:The analogous measure of noncompactness for an operator is dened by(F (V )) = inffk > 0 : (F (V ))  k(V )g for all bounded subsets V  X:When X is a complete metric space and f : X  ! X is a continuousmapping f is called anmu set contraction if there exists k 2 [0; 1) such that, for all bounded non-compact subsets V of X, the following relation holds: (f(V ))  k(V ) ([?], pag 160).A continuous operator F : X  ! X such that (F (V )) < (V ), for anybounded V  X; is called condensing or densifying.(The concept of measure of noncompactness is considerably dealed within the references [?], [?] or [?].)Let S and S1 be topological spaces and let f : S  ! S1. Then f issaid to be proper if, whenever K1 is a compact subset of S1; f 1(K1) is acompact set in S. It is also known ( [?], pag 160) that if X is a Banach spaceand f : X  ! X is a continuous k set contraction, then I   f is a propermapping.The following result, due to R.K. Juberg ([?]), will be useful in the proofof our main result:Proposition 1 : Let (a; b) be any real (possible unbounded) interval and letLp(a; c); 1  p  +1 be the Lebesgue's space of (the power p) summable2functions over (a; c) for every c 2 (a; b). For u 2 Lp(c; b); v 2 Lq(a; c);where 1p+ 1q= 1; we set = lim!0supf[Z a+xju(y)jpdy] 1p [Z xajv(y)jqdy] 1q ; a < x  a+ g++ lim!0supf[Z bxju(y)jpdy] 1p [Z xb jv(y)jqdy] 1q ; b    x < bg:LetD be the linear operator dened by: D(f(y))(x) =R x0u(x)v(y)f(y)dy;in the sequel wh shall assume that D is a bounded operator in the spaceLp(0; T ): We want to recall that the operator D is bounded (in the Lp(a; b)space) if and only if the function (x) = [R bxju(y)jpdy] 1p [R xajv(y)jqdy] 1q is bounded on (a; b): This operatoris not necessarily a compact operator; as matter of fact it is well known (see[?], for istance), that D is a compact operator if the functions u() e v()belongs to L2(a; b):Furthermore the measure of noncompactness of D, i.e. (D) satises(12)1+1p  (D)  p 1q q 1p; in the special case when p = q = 2; i.e. when the(Lebesgue) space Lp is a Hilbert space L2, we obtain q18 (D)  2:Denition 2 : An R set is the intersection of a decreasing sequnce fAngof compact AR (metric absolute retracts; see [?] or [?], for a reference.)Moreover it is known (see [?] for istance) that an R set is an acyclic set inthe Cech homology.The following result also will be crucially used in teh sequel:Proposition 2 : ([?], pag 159). Let X be a space and let Y; jjjj be a Banachspace and f : X  ! X be a proper mapping. Assume further that for eachn > 0; n > 0 2 IN a proper mapping fn : X  ! X is given and the coupleof conditions is satised: jjfn(x)  f(x)jj < n; 8x 2 X; for any n > 0 and y 2 E such that jjyjj  n; the equation fn(x) = yhas exactly one solution.3Then the set S = f 1(0) is an R set.Remark: a sequence fn is called an n approximation (of the function f).Proposition 3 : ([?], pag ???). Let F; Fn : B(0; r)  ! Y be condensingoperators such that n = supfFn(x)  F (x)jj; x 2 B(0; r)g ! 0; as n! +1; the equation x = Fn(x) + y has at most one solution if jjyjj  n:Then the set of xed points of F is an R set.Main resultWe are ready to establish out (main) existence result for the (initial valueproblems for) integral equations of the type here introduced.First of all let F : B(0; r)! E be dened as follows:F (y) = h(t) +R t0k(t; s)g(s; y(s)dswhere r is a real number (suitably dened below) and put m0 = jjF (0)jj2:Theorem 1 : Let  the number dened in Proposition 1; then we assumethat:1. i) there are functions ; ; : I ! IRn belonging to L2(I) such thatk(t; s) = (s) for every (t; s) 2 I  I; moreover we assume thatjjkjj2 < 2;2. ii) jjg(t; x)jj  12jjxjj+ b(t); for (t; x) 2 I  IRn; b 2 L2(I); b(t)  0;3. iii) there is a ball B(0; r) such that r > 2m02 jjkjj2 :Then the set of solution of the integral problem (??) is an R set.Remark: The rst part of the assumption i) is satised in many cases:for istance when k(t; s) is a Green function; see, for istance, [?] for similarcases.Proof: Clearly the above operator F is a single value mapping and apossible xed point of F is a solution of the integral problem (??).In order to prove the theorem the following steps in the proof have to beestablished:4a ) F has a closed graph;b) F is a condensing mapping;c) The set of xed point of F is R set.Proof of Step a): in fact, let yn ! y0 and put G(y)(t) = g(t; y(t)). Now, fromassumption ii), it follows that the superposition operator G mapping thespace L2 into L2 is condensing (see [?]); thus we have limn jjG(yn) G(y0)jj2 =0: By using the Holder inequality, we get:jjF (yn)  F (y0)jj2 = [RIjF (yn)(s)  F (y0)(s)j2ds] 12 == [RI[R t)(k(t; s)g(s; yn(s))   k(t; s)g(s; y0(s))ds]2dt] 12  jjkjj2jjjjG(yn)  G(y0)jj2and this quantity is gioing to zero whenever n! +1:Proof of Step b): Always working from B(0; r) into E; we have F (y) =(H G)(y)); whereH(y)(t) =R t0(s)(t)y(s)ds+ h(t):Now, by assumptions i) and ii), we have (see [?]) (G(V ))  12(V );for any bounded set V  L2(I  IRn) and also (H) < 2; so (see [?])(F ) = (H G)(y))  (H)(G) < 1:Proof of Step c): Finally we have to prove that the set of xed points ofthe operator F is an R set (in the sequel we assume that (a; b) = (0; T ):)Let us consider the mappings Fn : L2(0; T )! L2(0; T ) dened as:Fn(x)(t) =(h(t) = if 0  t  Tn;h(t) +R t Tn0(s)(s)g(s; y(s))ds = if Tn t  T: (3)The mappings Fn are continuous mappings; by assumption i) and ii) wehave that they are also condensing. The intervals [0; Tn]; [Tn; 2Tn];    [kTn; (k+1)Tn];    [ (n 1)Tn; T ]are now coming in one after the other: each time the mappings Fn are bijec-tive and their inverses F 1n are continuous. Moreover we have jjFn F jj2 ! 0as n! +1: The latter fact allows us to say that the mappings I   Fn andI   F are proper maps. Finally we can conclude that the set of xed pointsof F is an R set.5Riferimenti bibliograci[1] G. Anichini - G. Conti Existence of Solutions of a Boundary Value Pro-blem through the solution mapping of a linearized type problem, Ren-diconti del Seminario Mate. Univ. Torino, Fascicolo speciale dedicatoa Mathematical theory of dynamical systems and ordinary dierentialequations, 1990, vol 48 (2), p. 149 { 160,[2] G. Anichini - G. Conti - P. Zecca Using solution sets for solving bounda-ry value problems for ordinary dierential equations, Nonlinear AnalysisTheory Meth.& Appl., 1991, vol 5, p. 465{474,[3] G. Anichini - G. Conti A direct approach to the existence of solutionsof a Boundary Value Problem for a second order dierential system,Dierential Equations and Dynamical Systems, 1995, vol 3 (1), p. 23 {34,[4] G. Anichini - G. Conti About the Existence of Solutions of a BoundaryValue Problem for a Caratheodory Dierential System, Zeitschrift furAnalysis und ihre Anwendungen, 1997, vol 16 (3), p. 621 { 630,[5] G. Anichini - G. Conti Boundary Value Problem for Implicit ODE's ina singular case, Dierential Equations and Dynamical Systems, 1999,vol 7 (4), p. 437 { 459,[6] G. Anichini - G. Conti How to make use of the solutions set to solveBoundary Value Problems, Progress in Nonlinear Dierential Equationsand their Applications, Springer Verlag (Basel), 2000, vol 40,[7] G. Anichini - G. Conti Boundary value problems for perturbed die-rential systems on an unbounded interval, International MathematicalJournal, 2002, vol 2 (3), p. 221 { 234 (?),[8] J. Banas - K. Goebel Measures of noncompactness in Banach spaces,Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, NewYork, 1980,[9] F.E. Browder - C.P. Gupta Topological Degree and Nonlinear Mappingsof Analytic Type in Banach spaces, Journal of Mathematical Analysisand Applications, 1969, vol 26 (4), p. 390 { 402 ?),6[10] G. Conti - J. Pejsachowicz Fixed point theorems for multivalued maps,Annali Matem. Pura Appl., 1980, vol 126 (4), p. 319 { 341[11] G. Darbo Punti uniti in trasformazioni a codominio non compatto,Rend. Sem. Matem. Univ. Padova, 1955, vol 24, p. 84 { 92[12] A. Deimling Nonlinear Functional Analysis, Springer Verlag, Berlin,1984bibitem13 L. Gorniewicz, Topological Approach to dierential inclu-sions, NATO-ASI Series, A.Granas { M. Frigon editors, Kluwer, 1990,vol 472, p. 129 { 190,[13] H. Hochstadt Integral Equations, Pure and Applied Matheamtics, Wiley,New York, 1973,[14] V.I. Istratescu Fixed point theory, D. Reidel Publishing Company,Dordrecht, 1981,[15] R.K. Juberg The measure of noncompactness in Lp for a Class ofIntegral Operators, Indiana Math. Journal, 1973/74, vol 23, p. 925 {936,[16] M.A. Krasnoselkii - P.P. Zabreiko Geometrical methods of nonlinearanalysis, Springer Verlag, Berlin, 1984[17] J. Lasry { R.Robert Analyse nonlineare multivoque, U.E.R. Math de laDecision, 1979, vol 249, Paris Dauphine,[18] W.V. Petryshyn Solvability of various boundary value problems for theequation x00 = f(t; x; x0; x00)  y; Pacic Journal of Math. 1986, vol. 122,p. 169 { 195[19] E.Spanier Algebraic Topology, McGraw Hill, New York, 19667

Some properties of the solution set for integral differential equations

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Some properties of the solution set for integral differential equations

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