A Note on the Solution Set of a Fractional Integro- Differential Inclusion

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1 Progr. Fract. Differ. Appl. 2, No. 1, (216) 13 Progress in Fractional Differentiation and Applications An International Journal A Note on the Solution Set of a Fractional Integro- Differential Inclusion Aurelian Cernea Faculty of Mathematics and Computer Science, University of Bucharest, Academiei 14, 114 Bucharest, Romania Received: 2 Sep. 215, Revised: 7 Oct 215, Accepted: 19 Oct. 215 Published online: 1 Jan. 216 Abstract: We study an initial value problem associated to a integro-differential inclusion of fractional order defined by a multifunction with nonconvex values and we obtain a qualitative property of its solution set. Keywords: Fractional derivative, differential inclusion, retract. 1 Introduction In the last years one may see a strong development of the theory of differential equations and inclusions of fractional order [1, 2, 3]. The main reason is that fractional differential equations are very useful tools in order to model many physical phenomena. For the last achievements on fractional calculus and fractional differential equations we refer the reader to [4].In the fractional calculus there are several fractional derivatives. From them, the fractional derivative introduced by Caputo in [5], allows to use Cauchy conditions which have physical meanings. Recently, several qualitative results for fractional integro-differential equations were obtained in [6, 7, 8, 9]. This paper is concerned with integro-differential inclusions of fractional order D p c x(t) F(t,x(t),V(x)(t)) a.e.([, )), x()=x, x ()=x 1, (1) where p (1,2], Dc p denotes Caputo s fractional derivative, F : [, ) R R P(R) is a multifunction and x,x 1 R. V : C([, ),R) C([, ),R) is a nonlinear Volterra integral operator defined by V(x)(t) = k(t,s,x(s))ds with k(.,.,.) :[, ) R R R a given function. In the present note we prove that the set of selections of the multifunction F that correspond to the solutions of problem (1) is a retract of L 1 loc ([, ),R). Our main hypothesis is that the multifunction is Lipschitz with respect to the second and third variable and the proof uses a well known selection theorem due to Bressan and Colombo [1] which gives continuous selections for multifunctions that are lower semicontinuous and with decomposable values. We point out that in the classical case of differential inclusions several qualitative properties of solutions exists in the literature [11,12,13]. On one hand, our result may be seen as a generalization of of Theorem 3.4 in [14] proved for problems defined on bounded intervals and on the other hand, Theorem 1 below is a generalization to fractional framework of the main theorem in [13]. 2 Preliminaries In what follows I := [,T], T >, L(I) is the σ-algebra of all Lebesgue measurable subsets of I and (X,. ) is a real separable Banach. C(I,X) denotes the space of continuous functions x : I X with the norm x C = sup t I x(t) and L 1 (I,X) denotes the space of integrable functions x : I X with the norm x 1 = T x(t) dt. The distance between a point x X and a subset A X is defined by d(x,a) = inf x a ;a A} and Pompeiu- Hausdorff distance between the closed subsets A,B X is defined by d H (A,B) = maxd (A,B),d (B,A)}, d (A,B) = supd(a,b); a A}. Corresponding author acernea@fmi.unibuc.ro c 216 NSP

2 14 A. Cernea: A note on the solution set of a fractional... P(X) denotes the family of all nonempty subsets of X and with B(X) the family of all Borel subsets of X. For A I with χ A (.) : I,1} we describe the characteristic function of A. Finally, for any A X cl(a) is its the closure. By definition a subset D L 1 (I,X) it is decomposable if for any u,v D and any subset A L(I) one has uχ A +vχ B D, where B=I\A. We use the notation D(I,X) for the family of all decomposable closed subsets of L 1 (I,X). In the next two results (S,d) is a separable metric space. By definition a set-valued map H : S P(X) is said to be lower semicontinuous (l.s.c.) if for any closed subset G X, the subset s S; H(s) G} is closed. The next two Lemmas are proved in [1]. Lemma 2.1. Consider F : I S P(X) a set-valued map with closed values, L(I) B(S)-measurable and F (t,.) is l.s.c. for any t I. Then the set-valued map H : S D(I,X) defined by H(s)= f L 1 (I,X); f(t) F (t,s) a.e.(i)} is l.s.c. with nonempty closed values if and only if there exists a continuous mapping q : S L 1 (I,X) such that d(,f (t,s)) q(s)(t) a.e.(i), s S. Lemma 2.2. Consider F : S D(I,X) be a l.s.c. set-valued map with closed decomposable values and let ψ : S L 1 (I,X), φ : S L 1 (I,R) be continuous mappings such that the set-valued map H : S D(I,X) given by H(s)=cl f(.) F(s); f(t) ψ(s)(t) <φ(s)(t) a.e.(i)} has nonempty values. Then H admits a continuous selection, i.e. there exists h : S L 1 (I,X) continuous with h(s) H(s) s S. Definition 2.3. [1]. a) The fractional integral of order p> of a Lebesgue integrable function f :(, ) R is defined by I p (t s) p 1 f(t)= f(s)ds, Γ(p) provided the right-hand side is pointwise defined on (, ) and Γ(.) is the (Euler s) Gamma function defined by Γ(p) = t p 1 e t dt. b) Caputo s fractional derivative of order p > of a function f :[, ) R is defined by Dc p 1 t f(t)= (t s) p+n 1 f (n) (s)ds, Γ(n p) where n =[p] + 1. It is assumed implicitly that f is n times differentiable whose n-th derivative is absolutely continuous. Definition 2.4. A mapping x C([, ),R) is said to be a solution of problem (1) if there exists h L 1 loc ([, ),R) with h(t) F(t,x(t),V(x)(t)), a.e.[, ) satisfying Dc p x(t)=h(t) a.e.[, ) and x()=x,x ()=x 1. In the next section we are concerned with the following set associated to problem (1). h(t)=x + tx 1 + (t u) p 1 Γ(p) h(u)du, S(x,x 1 )=h L 1 loc ([, ),R); h(t) F(t, h(t),v( h)(t)) a.e.[, )}. (2) 3 The Result Hypothesis 3.1. i) The set-valued map F(.,.) : [, ) R R P(R) is L([, )) B(R R) measurable and has nonempty closed values. ii) For almost all t I, the set-valued map F(t,.,.) is L(t)-Lipschitz in the sense that there exists L(.) L 1 loc ([, ),R +) with d H (F(t,x 1,y 1 ),F(t,x 2,y 2 )) L(t)( x 1 x 2 + y 1 y 2 ) x 1,x 2,y 1,y 2 R. iii) There exists a locally integrable function q(.) L 1 loc ([, ),R) such that d H (},F(t,,V()(t))) q(t) a.e.([, )). c 216 NSP

3 Progr. Fract. Differ. Appl. 2, No. 1, (216) / 15 iv) k(.,.,.) :[, ) R R R is a function such that x R, (t,s) k(t,s,x) is measurable. v) k(t,s,x) k(t,s,y) L(t) x y a.e.(t,s) [, ) [, ), x,y R. We use next the following notations M(t) := L(t)(1+ L(u)du), t I, I p M := sup I p M(t). t [, ) Let us note that and for any u 1,u 2 L 1 (I,R) h(t)=x + tx 1 + q (h)(t)= h(t) +q(t)+l(t) ( h(t) + (t s) p 1 h(s)ds, u L 1 (I,R), (3) Γ(p) L(s) h(s) ds), t I (4) d(h(t),f(t, h(t),v( h)(t)) q (h)(t) a.e.(i) (5) q (h 1 ) q (h 2 ) 1 (1+ I p M(T) ) h 1 h 2 1 ; therefore, the mapping q : L 1 (I,R) L 1 (I,R) is continuous. Also define S I (x,x 1 )=h L 1 (I,R); h(t) F(t, h(t),v( h)(t)) a.e.(i)}. I k =[,k], k 1, h 1,k = The proof of the next result may be find in [14]. k h(t) dt, h L 1 (I k,r). Lemma 3.2. Suppose that Hypothesis 3.1 is verified and consider φ : L 1 (I,R) L 1 (I,R) a continuous function with φ(h)=h for all h S I (x,x 1 ). If h L 1 (I,R), we put Ψ(h)=h L 1 (I,R); h(t) F(t, φ(h)(t),v( φ(h))(t)) a.e.(i)}, Φ(h)= h} if h TI (x,x 1 ), Ψ(h) otherwise. Then the set-valued map Φ : L 1 (I,R) P(L 1 (I,R)) is l.s.c. with nonempty closed and decomposable values. Theorem 3.3. Assume that Hypothesis 3.1 is satisfied, there exists I p M <1 and x,x 1 R. Then there exists G : L 1 loc ([, ),R) L1 loc ([, ),R) continuous with the properties (i) G(h) S(x,x 1 ), h L 1 loc ([, ),R), (ii) G(h)=h, h S(x,x 1 ). Proof. The idea of the proof consists in the construction, for every k 1, of a sequence of continuous functions g k : L 1 (I k,r) L 1 (I k,r) satisfying the following conditions (I) g k (h)=h, h S Ik (x,x 1 ) (II) g k (h) S Ik (x,x 1 ), h L 1 (I k,r) (III) g k (h)(t)=g k 1 (h Ik 1 )(t), t I k 1 If this construction is realized, we introduce G : L 1 loc ([, ),R) L1 loc ([, ),R) with L 1 loc G(h)(t)=g k (h Ik )(t), k 1. The continuity of g k (.) and (III) allow to deduce that G(.) is continuous. Taking into account (II), for each h ([, ),R), we get G(h) Ik (t)=gk (h )(t) T Ik Ik (x,x 1 ), k 1, which shows that G(h) T (x,x 1 ). Consider ε > and m. We define ε m = m+2 m+1 ε. If h L1 (I 1,R) and m we put q 1 (h)(t)= h(t) +q(t)+l(t)( h(t) + L(s) h(s) ds), t I 1 c 216 NSP

4 16 A. Cernea: A note on the solution set of a fractional... and q 1 m+1 (h)= I p M m 1 ( Γ(p) q1 (h) 1,1+ ε m+1 ). Since the map q 1 (.)=q (.) is continuous, we find that q 1 m : L1 (I 1,R) L 1 (I 1,R) is also continuous. Set g 1 (h)=h. In what follows, we show that for any m 1there exists g1 m : L1 (I 1,R) L 1 (I 1,R) continuous with the properties g 1 m (h)=h, h S I 1 (x,x 1 ), (a 1 ) If h L 1 (I 1,R), we define g 1 m(h)(t) F(t, g1 m 1 (h)(t),v( g 1 m 1 (h))(t)) a.e.(i 1), (b 1 ) g 1 1 (h)(t) g1 (h)(t) q1 (h)(t)+ε a.e.(i 1 ), (c 1 ) g 1 m(h)(t) g 1 m 1(h)(t) M(t)q 1 m 1(h) a.e.(i 1 ), m 2. (d 1 ) Ψ 1 1 (h)= f L 1 (I 1,R); f(t) F(t, h(t),v( h(t))(t)) a.e.(i 1 )}, Φ1 1 h} if (h)= h SI1 (x,x 1 ), Ψ 1 (h) otherwise. 1 We apply Lemma 3.2 (with φ(h)=h) and we deduce that Φ 1 1 : L1 (I 1,R) D(I 1,R) is l. s. c. Using (5) we obtain that the set H 1 1(h)=cl f Φ 1 1(u); f(t) h(t) <q 1 (h)(t)+ε a.e.(i 1 )} is not empty for any h L 1 (I 1,R). We apply the Lemma 2.2 to obtain a selection g 1 1 of H1 1 which is continuous and verifies (a 1 )-(c 1 ). Assume that g 1 i (.), i = 1,...m satisfying (a 1)-(d 1 ) are already constructed. Therefore, from Hypothesis 1 and (b 1 ), (d 1 ) we infer d(g 1 m(h)(t),f(t, g 1 m(h)(t),v( g 1 m(h))(t)) L(t)( g 1 m 1 (h)(t) g 1 m(h)(t) + L(s) g (6) 1 m 1 (h)(s) g 1 m(h)(s) ds) M(t) I p M q 1 m(h)=m(t)(q 1 m+1 (h) s m)<m(t)q 1 m+1 (h), where s m := I p M m (ε m+1 ε m )>. For h L 1 (I 1,R), we put Ψ 1 m+1 (h)= f L1 (I 1,R); f(t) F(t, g 1 m (h)(t),v( g 1 m (h))(t)) a.e.(i 1)}, Φm+1(h)= 1 h} if h SI1 (x,x 1 ), Ψm+1 1 (h) otherwise. Again, Lemma 3.2 (applied for φ(h) = g 1 m (h)) allows to conclude that Φ1 m+1 (.) is is l.s.c. with nonempty closed decomposable values. At the same time, from (6), if h L 1 (I 1,R), the set H 1 m+1(h)=cl f Φ 1 m+1(h); f(t) g 1 m+1(h)(t) <M(t)q 1 m+1(h) a.e.(i 1 )} is nonempty. As above, via Lemma 2.2, it is obtained a selection g 1 m+1 of H1 m+1 continuous with (a 1)-(d 1 ). We conclude that g 1 m+1 (h) g1 m (h) 1,1 I p M m 1 ( Γ(p) q1 (h) 1,1+ ε) which means that the sequence g 1 m(h)} m N is a Cauchy sequence in the Banach space L 1 (I 1,R). Take g 1 (h) L 1 (I 1,R) its limit. Since the mapping s q 1 (h) 1,1 is continuous, thus it is locally bounded and the Cauchy condition is satisfied byg 1 m(h)} m N locally uniformly with respect to h. Therefore, g 1 (.) : L 1 (I 1,R) L 1 (I 1,R) is continuous. Taking into account (a 1 ) we find that g 1 (h)=h, h S I1 (x,x 1 ) and from the hypotheses that the values of F are closed and (b 1 ) we find that g 1 (h)(t) F(t, g 1 (h)(t),v( g 1 (h))(t)), a.e.(i 1 ) h L 1 (I 1,R). c 216 NSP

5 Progr. Fract. Differ. Appl. 2, No. 1, (216) / 17 At the final step of the induction procedure we assume that g i (.) : L 1 (I i,r) L 1 (I i,r), i=2,...,k 1 are constructed and satisfying (I)-(III) and we construct g k (.) : L 1 (I k,r) L 1 (I k,r) continuous with (I)-(III). We introduce the map g k : L1 (I k,r) L 1 (I k,r) Since g k 1 (.) is continuous and for h,h L 1 (I k,r) we have g k (h)(t)=gk 1 (h Ik 1 )(t)χ Ik 1 + h(t)χ Ik \I k 1 (t) (7) k g k (h) gk (h ) 1,k g k 1 (h Ik 1 ) g k 1 (h Ik 1 ) 1,k 1 + h(t) h (t) dt, k 1 and we deduce that g k (.) is continuous. At the same time, the equality g k 1 (h)=h, h S Ik 1 (x,x 1 ) and (7) allow to obtain For h L 1 (I k,r), we define Ψ k g k (h)=h, h S I k (x,x 1 ). 1(h)=l L 1 (I k,r); l(t)=g k 1 (h Ik 1 )(t)χ Ik 1 (t)+n(t)χ Ik \I k 1 (t), n(t) F(t, g k (h)(t),v( g k (h))(t)) a.e.([k 1,k])} Φ1(h)= k h} if h SIk (x,x 1 ), Ψ1 k (h) otherwise. Once again Lemma 3.2 (applied for φ(h) = g k (h)) implies that Φk 1 (.) : L1 (I k,r) D(I k,r) is l.s.c.. In addition, if h L 1 (I k,r) one may write where d(g k (t),f(t, g k (h)(t),v( g k (h))(t))=d(h(t),f(t, g k (h)(t),v( g k (h)(t))χ I k \I k 1 q k (h)(t) a.e.(i k), (8) q k (h)(t)= h(t) +q(t)+l(t)( g k (h)(t) + L(s) g k (h)(s) ds). Obviously, q k : L1 (I k,r) L 1 (I k,r) is continuous. If m we define q k m+1 (h)= I p M m ( kp 1 Γ(p) qk (h) 1,k+ ε m+1 ). and from the continuity of q k (.) we deduce the continuity of qk m : L 1 (I k,r) L 1 (I k,r). Finally, we provide the existence of g k m : L1 (I k,r) L 1 (I k,r) continuous such that Set g k m (h)(t)=gk 1 (h Ik 1 )(t) t I k 1, (a k ) g k m(h)=h h S Ik (x,x 1 ), (b k ) g k m (h)(t) F(t, gk m 1 (h)(t),v( g k m 1 (h))(t)) a.e.(i k), (c k ) g k 1 (h)(t) gk (h)(t) qk (h)(t)+ε a.e.(i k ), (d k ) g k m(h)(t) g k m 1 (h)(t) M(t)qk m 1 (h) a.e.(i k), m 2. (e k ) H k 1 (h)=cl f Φk 1 (h); f(t) gk (h)(t) <qk (h)(t)+ε a.e.(i k )}. Using (8), H1 k(h) / for any h L1 (I 1,R). Taking into account Lemma 2.2 and the fact that the maps g k,qk are continuous we find a continuous selection g k 1 of Hk 1 with (a k)-(d k ). If g k i (.), i=1,...m with (a k)-(e k )are already constructed, from (e k ) one may write d(g k m(h)(t),f(t, g k m(h)(t),v( g k m(h))(t)) L(t)( g k m 1 (h)(t) g k m(h)(t) + L(s) g (9) k m 1 (h)(s) g k m (h)(s) ds) M(t)(qk m+1 (h) s m)<m(t)q k m+1 (h), c 216 NSP

6 18 A. Cernea: A note on the solution set of a fractional... where s m := I p M m (ε m+1 ε m )>. For h L 1 (I k,r), we define Ψ k m+1 (h)=l L1 (I k,r); l(t)=g k 1 (h Ik 1 )(t)χ Ik 1 (t)+ n(t)χ Ik \I k 1 (t), n(t) F(t, g k m (h)(t),v( g k m (h))(t)) a.e.([k 1,k])}, Φm+1(h)= k h} if h SIk (x,x 1 ), Ψm+1 k (h) otherwise. Applying Lemma 3.2 it is obtained that Φ k m+1 (.) : L1 (I k,r) P(L 1 (I k,r)) has nonempty closed decomposable values and is l.s.c.. As above, the set H k m+1(h)=cl f Φ k m+1(h); f(t) g k m+1(h)(t) <M(t)q k m+1(h) a.e.(i k )} h L 1 (I k,r) is nonempty. Again, Lemma 2.2 allows to obtain a continuous selection g k m+1 of Hk m+1, verifying (a k)-(e k ). By (e k ) one has g k m+1 (h) gk m (h) 1,k I p M m [ kp 1 Γ(p) qk (h) 1,1+ ε]. Repeating the proof done in the first case we get the convergence of g k m (h)} m N to some g k (h) L 1 (I k,r). Moreover, g k (.) : L 1 (I k,r) L 1 (I k,r) is continuous. By (a k ) we have that g k (h)(t)=g k 1 (h Ik 1 )(t) t I k 1, by (b k ) g k (h)=h h S Ik (x,x 1 ) and, finally, since the values of F are closed, from (c k ) we deduce that and the proof is complete. g k (h)(t) F(t, g k (h)(t),v( g k (h))(t)), a.e.(i k ) h L 1 (I k,r), Remark 3.4. By definition, a subspace X of a Hausdorff topological space Y is said to be a retract of Y if there exists a mapping h : Y X continuous with h(x)=x, x X. So, Theorem 3.3 states that for each x,x 1 R, the set S(x,x 1 ) is a retract of the Banach space L 1 loc ([, ),R). References [1] A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 26. [2] J. Klafter, S.-C. Lim and R. Metzler, Fractional calculus: Recent advances, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 211. [3] K. Miller and B. Ross, An introduction to the fractional calculus and differential equations, John Wiley, New York, [4] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional calculus: models and numerical methods, Series on Complexity, Nonlinearity and Chaos, Vol. 3, World Scientific, Singapore, 212. [5] M. Caputo, Elasticità e Dissipazione, Zanichelli, Bologna, [6] A. Cernea, Continuous selections of solutions sets of fractional Integro-differential inclusions, Acta Math. Scient. 35B, (215). [7] D. N. Chalishajar and K. Karthikeyan, Existence and uniqueness results for boundary value problems of higher order fractional integro-differential equations involving Gronwall s inequality in Banach spaces, Acta Math. Scient. 33B, (213). [8] K. Karthikeyan and J. J. Trujillo, Existence and uniqueness results for fractional integro-differential equations with boundary value conditions, Commun Nonlinear Sci. 17, (212). [9] J. R. Wang, W. Wei and Y.Yang, Fractional nonlocal integro-differential equations of mixed type with time varying generating operators and optimal control, Opuscula Math. 3, (21). [1] A. Bressan and G. Colombo, Extensions and selections of maps with decomposable values, Studia Math. 9, (1988). [11] A. Bressan, A. Cellina and A. Fryszkowski, A class of absolute retracts in space of integrable functions, Proc. Amer. Math. Soc. 112, (1991). [12] F. S. De Blasi and G. Pianigiani, Topological properties of nonconvex differential inclusions, Nonlin. Anal. 2, (1993). [13] V. Staicu, On the solution sets to differential inclusions on unbounded interval, Proc. Edinb. Math. Soc. 43, (2). [14] A. Cernea, On a fractional integro-differential inclusion, Electr. J. Qual. Theor. Differ. Equ. 214 (25), 1 11 (214). c 216 NSP