EXISTENCE THEOREM FOR A FRACTIONAL MULTI-POINT BOUNDARY VALUE PROBLEM

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1 Fixed Point Theory, 5(, No., nodeacj/sfptcj.html EXISTENCE THEOREM FOR A FRACTIONAL MULTI-POINT BOUNDARY VALUE PROBLEM FULAI CHEN AND YONG ZHOU Department of Mathematics, Xiangnan University Chenzhou 3, China cflmath@yahoo.com.cn School of Mathematics and Computational Science Xiangtan University, Xiangtan 5, China yzhou@xtu.edu.cn Abstract. This paper investigates the existence of solutions for a fractional multi-point boundary value problem at resonance by applying the coincidence degree theory. An example is given to illustrate our main result. Key Words and Phrases: Fractional differential equations, multi-point boundary value problem, existence, coincidence degree. Mathematics Subject Classification: 6A33, 3B, 7H.. Introduction Consider the boundary value problem (BVP for short of the following fractional differential equation C D x(t f(t, x(t + e(t, t [, ], < <, (. A i x(ξ i, where C D is the Caputo fractional derivative with < <, f : [, ] R n R n and e : [, ] R n are given functions satisfying some assumptions that will be specified later, A i (i,,, m are constant square matrices of order n, ξ < ξ < < ξ m. We say that BVP (. is a problem at resonance, if the linear equation, C D x(t, t [, ], < <, This work was supported by the Natural Science Foundation of China(9773, the National Natural Science Foundation of Hunan Province (3JJ3, and the Construct Program of the Key Discipline in Hunan Province. Corresponding author. 3

2 FULAI CHEN AND YONG ZHOU with the boundary condition m A ix(ξ i has nontrivial solutions. Otherwise, we call them a problem at nonresonance. In the present work, if m A i, then BVP (. is at resonance, since equation C D x(t with boundary condition A i x(ξ i has nontrivial solutions x c, c R n. The theory of fractional differential equations has been extensively studied since the behavior of many physical systems can be properly described by using the fractional order system theory. There has been a great deal of interest in the solutions of fractional differential equations, see the monographs [6, 7,, 3], and the papers [, 3,, 5, 6, 7,,, 3, 5, 8,,, 5, 6, 9, 3] and the references therein. Tools used to analyze the solvability of these fractional differential equations are mainly focused on fixed point theorems and Leray-Schauder theory. Recently, there have been few studies dealing with the existence for solutions of fractional BVP by using the coincidence degree theory [9,,, 8]. Existence results for fractional BVP with derivative order (, 3 are established in [9, ], and existence results for fractional BVP with derivative order (, are obtained in [, 8]. However, there is no work on the existence of solutions for fractional BVP with derivative order (,. It is clear that the corresponding integral equations of fractional BVP is weakly singular if derivative order < <, and regular for. Motivated by [9], in which the existence to first-order multi-point BVP is proven by using the coincidence degree theory, here we investigate the existence of solutions for fractional multi-point BVP with derivative order (,. Different from the Riemann-Liouville fractional derivative used in [9,,, 8], here we consider the Caputo s one. The outline of the remainder of this paper is as follows. In section, we recall some useful preliminaries. In section 3, we give the existence result of BVP (. at resonance (i.e., m A i. In section, an example is given to illustrate our main results.. Preliminaries and Lemmas Let us recall some notations and an abstract existences result. Let Y, Z be real Banach spaces, L : doml Y Z be a Fredholm map of index zero and P : Y Y, Q : Z Z be continuous projectors such that ImP KerL, KerQ ImL and Y KerL KerP, Z ImL ImQ. It follows that L doml KerP : doml KerP ImL is invertible. We denote the inverse by K P. If Ω is an open bounded subset of Y, and doml Ω, the map N : Y Z will be called L compact on Ω if QN(Ω is bounded and K P (I QN : Ω Y is compact. Theorem. [] Let L : doml Y Z be a Fredholm operator of index zero and let N : Y Z be L compact on Ω. Assume that the following conditions are satisfied: (i Lx λnx for every (x, λ [doml \ KerL Ω] (, ;

3 FRACTIONAL MULTI-POINT BOUNDARY VALUE PROBLEM 5 (ii Nx ImL for every x LerL Ω; (iii deg(qn KerL, Ω KerL,, where Q : Z Z is a projection as above with ImL KerQ. Then the equation Lx Nx has at least one solution in doml Ω. We also recall the following known definitions with respect to the fractional integral and derivatives. For more details see [, 3]. Definition. The fractional integral of order γ with the lower limit zero for a function f is defined as I γ f(t Γ(γ t f(s ds, t >, γ >, (t s γ provided that the right side is point-wise defined on [,, where Γ( is the gamma function. Definition. Riemann-Liouville derivative of order γ with the lower limit zero for a function f : [, R can be written as d n D γ f(t Γ(n γ dt n t f(s ds, t >, n < γ < n. (t s γ+ n Definition.3 [6] Caputo s derivative of order γ for a function f : [, R can be written as n C D γ f(t D γ f (k ( f(t Γ(k γ + tk γ, t >, n < γ < n. k Remark. ( If f(t C n ([,, R, then C D γ f(t Γ(n γ t f (n (s (t s γ+ n ds In γ f (n (t, t >, n < γ < n. ( The Caputo derivative of a constant is equal to zero, i.e., if x(t c, then C D c. However, D c ct Γ(. Lemma. [7] Let n < γ < n, then the differential equation C D γ x(t has solutions x(t c + c t + c t + + c n t n, c i R, i,,, n. Lemma. [7] Let n < γ < n, then I γ C D γ x(t x(t + c + c t + c t + + c n t n for some c i R, i,,, n. The following basic inequalities will be used. Lemma.3 [8] Let a, a,, a n, n N, then n ( n r ( n a r i a i n r, r, and n r ( n a r i ( n r a i a r i n a r i, r.

4 6 FULAI CHEN AND YONG ZHOU We denote the n n identity matrix by E, the Banach space of all constant square matrices of order n by M n n with the norm B max i,jn b i,j. For (,, n, define max in i. The L p norm in L p ([, ], R n is defined by x p max in ( x i(t p dt /p for p <. The L norm in C([, ], R n is x max in sup t [,] x i (t. 3. Main Results In this paper, we always assume the following conditions hold. (H ( m A m i and det A iξi. (H f : [, ] R n R n satisfies the following conditions: f(, x is continuous for each fixed x R n, f(t, is Lebesgue measurable for a.e. t [, ], and for each r >, there exists h r L ([, ], R n such that f i (t, x (h r i (t for all x < r, a.e. t [, ], i,,, n. e L ([, ], R n. (H 3 There exists a constant (, such that f L ([, ] R n, R n, e L ([, ], R n. Let Y C([, ], R n, Z L ([, ], R n. Define the linear operator L : doml Y Z with and doml Define N : Y Z by { x C([, ], R n : Then BVP (. can be written as and } A i x(ξ i, C D x L ([, ], R n, Lx C D x, x doml. (3. Nx(t f(t, x(t, t [, ]. Lx Nx. Lemma 3. Let L be defined as (3., then KerL {x doml : x c, c R n }, (3. ImL {y Z : A i (ξ i s y(sds }. (3.3 Proof. By Lemma., C D x(t has solution x(t x( which implies that (3. holds. For y ImL, if the equation C D x(t y(t (3.

5 FRACTIONAL MULTI-POINT BOUNDARY VALUE PROBLEM 7 has a solution x(t such that m A ix(ξ i, then from (3., we have x(t x( + t Applying condition m A i, we have that is A i x(ξ i x( A i + (t s y(sds. A i (ξ i s y(sds A i (ξ i s y(sds, A i (ξ i s y(sds. (3.5 On the other hand, if (3.5 holds, we can set x(t d + t (t s y(sds, where d R n is arbitrary. Hence, x(t is a solution of (3. and m A ix(ξ i which means that y ImL. Therefore, (3.3 holds. The proof is complete. Lemma 3. L : doml Y Z is a Fredholm operator of index zero. Furthermore, the linear continuous projection operators Q : Z Z and P : Y Y can be defined by Qy ( A i ξi m A i (ξ i s y(sds, for every y Z, P x x(, for every x Y. And the linear operator K P : ImL doml KerP can be written by Also, K p y I y(t t (t s y(sds. K P y ( + ν y, for all y ImL, where ν. Proof. It is easy to know that ImP KerL and P x P x. It follows from x (x P x + P x that Y KerP + KerL. By simple calculation, we have that KerL KerP {}. Thus, Y KerP KerL.

6 8 FULAI CHEN AND YONG ZHOU Since Q y (( Q ( Qy. A i ξi A i ξ i m m A i (ξ i s y(sds A i (ξ i s y(sds For y Z, set y (y Qy+Qy. Then, y Qy KerQ ImL, Qy ImQ. It follows from KerQ ImL and Q y Qy that ImQ ImL {}. Thus, Z ImL ImQ and dim KerL dim R n co dim ImL n. Hence, L is a Fredholm operator of index zero. With definitions of P, K P, it is easy to show that the generalized inverse of L : ImL doml KerP is K P. In fact, for y ImL, one has (LK P y C D I y y, and for x doml KerP which implies that x(, we have (K P Lx(t I C D x(t x(t. This shows that K P (L doml KerP. Let ν, then + ν > since that (,, we have K P y I y t (t s y(sds [ t {( t (t s ds] max in ( + ν t+ν y ( + ν y. } y i (s ds The proof is complete. Lemma 3.3 Assume Ω Y is an open bounded subset and doml Ω, then N is L compact on Ω. Proof. By (H, we have that QN(Ω is bounded. Now we show that K P (I QN : Ω Y is compact. The operator K P (I QN : Y Y is continuous in view of the continuity of f.

7 FRACTIONAL MULTI-POINT BOUNDARY VALUE PROBLEM 9 Since Ω Y is bounded, there exists a positive constant M > such that x M for all x Ω. Set t M max x M (t s [f(s, x(s + e(s]ds ( m m t A i ξi A i (ξ i s [f(s, x(s + e(s]ds, and ( M A i ξi For x Ω, we obtain (K P (I QNx m A i (ξ i s [f(s, x(s + e(s]ds. I {f(t, x(t + e(t Q[(f(t, x + e(t]} t (t s [f(s, x(s + e(s]ds ( m A i ξi A i (ξ i s [f(s, x(s + e(s]ds t (t s [f(s, x(s + e(s]ds ( m m t A i ξi A i (ξ i s [f(s, x(s + e(s]ds M. Thus, K P (I QN(Ω Y is bounded. Let t, t [, ] and t > t. ( + (K P (I QNx(t (K P (I QNx(t I {f(s, x(s + e(s Q[f(s, x(s + e(s]} tt t I {f(s, x(s + e(s Q[f(s, x(s + e(s]} tt t (t s [f(s, x(s + e(s]ds A i ξi t (t s [f(s, x(s + e(s]ds m [ t (t s ds A i (ξ i s [f(s, x(s + e(s]ds t (t s ds] (t s ds

8 5 FULAI CHEN AND YONG ZHOU t (t s [f(s, x(s + e(s]ds t (t s [f(s, x(s + e(s]ds + ( m A i ξi A i (ξ i s [f(s, x(s + e(s]ds t t (t s ds (t s ds t [(t s (t s ][f(s, x(s + e(s]ds + t (t s [f(s, x(s + e(s]ds t + ( m A i ξi A i (ξ i s [f(s, x(s + e(s]ds t t t [(t s (t s ]f(s, x(sds + t [(t s (t s ]e(sds + t (t s f(s, x(sds + t t (t s e(sds t + ( m A i ξi A i (ξ i s [f(s, x(s + e(s]ds t t { t ] } [(t s (t s ds + { t ] } [(t s (t s ds + [ t {[ t (t s ds] max t in + [ t {[ t (t s ds] max e i (s ds t in t { t {[ t ] } max f i (s, x(s ds in {[ t max in f i (s, x(s ds t ] } ] } e i (s ds ] } + M Γ( + (t t [ ]ds} (t s (t s ( f + e + [ t t ( [ t + (t t + ν (t s ds] ( f + e + M t Γ( + (t t ] ( f + e

9 + ( + ν FRACTIONAL MULTI-POINT BOUNDARY VALUE PROBLEM 5 [ (t t ] ( f + e + M ( + ν Γ( + (t t ( f + e (t t + M Γ( + (t t as t t. Then K P (I QN(Ω is equicontinuous. By the Ascoli-Arzela theorem, we have that K P (I QN : Ω Y is compact, then N is L compact on Ω. The proof is complete. To obtain our main results, we also need the following conditions. (H There exists function a, b, r L ([, ], R, and constant θ [, such that f(t, x a(t x + b(t x θ + r(t, (3.6 for all x R n and t [, ] (H 5 There exists a constant M > such that, for x doml, if there exist some i {,,, n} such that x i (t > M for all t [, ], then A i (ξ i s [f(s, x(s + e(s]ds. (3.7 (H 6 There exists a constant M > such that for any c (c, c,, c n R n, if c > M, then either ( m c A i ξi A i (ξ i s [f(s, c + e(s]ds < (3.8 or ( c A i ξi m A i (ξ i s [f(s, c + e(s]ds >. (3.9 Theorem 3. Assume (H (H 6 hold. Then BVP (. has at least one solution x C([, ], R n provided that Proof. Set ( + ν 6 a >. (3. Ω {x doml \ KerL : Lx Nx for some λ [, ]}. Then, for x Ω, Lx λnx, so λ and Nx ImL KerQ. Hence, A i (ξ i s [f(s, x(s + e(s]ds. By (H 5, there exists t i [, ] such that x i (t i M for all i {,,, n}. Since x i ( x i (t i ti (t i s C D x i (sds,

10 5 FULAI CHEN AND YONG ZHOU which implies that Thus, ti x i ( x i (t i + (t i s C D x i (t ds x i (t i + ( ( + ν t+ν i C D x i (t ds ( ( + ν x i (t i + C D x i (t ds. On the other hand, x( M + From (3. and (3., we obtain that ( + ν C D x. (3. C D x Lx Nx. (3. x( M + ( + ν Nx. (3.3 Also for x Ω, x doml \ KerL, then (I P x doml KerP, LP x. Applying Lemma 3.3, we have (I P x K P L(I P x ( + ν Lx ( + ν L(I P x ( + ν Nx. (3. From condition (H and Lemma.3, for i,,, n we have [ [ f i (t, x(t + e i (t dt] ( f i (t, x(t + e i (t dt [ {[ { { 6 [( a(t x 6 [( ( f i (t, x(t + e i (t dt f i (t, x(t dt] + [ ] ] } e i (t dt [ ] } a(t x + b(t x θ + r(t dt + e + ( b(t x θ ( a(t dt x + ( + + r(t r(t dt ] + e ] ]dt} + e b(t dt x θ

11 which yields that FRACTIONAL MULTI-POINT BOUNDARY VALUE PROBLEM 53 6 ( a x + b x θ + r + e, ( f + e 6 a x + b x θ + r + e. (3.5 Combining (3.3, (3. and (3.5, we have x P x + (I P x x( + (I P x ( + ν Nx + M ( + ν ( + ν Thus, from (3. and (3.6 we have x f + e + M ( 6 a x + 6 b x θ +6 r + e + M. (3.6 6 b x θ ( + ν 6 a 6 r + e + ( + ν M +. ( + ν 6 a Since [,, from above the inequality, there exists M 3 > such that x M 3, which implies that Ω is bounded. Let Ω {x KerL : Nx ImL}. For x Ω, x KerL {x doml : x c, c R n }, and QNx, we can get A i (ξ i s [f(s, c + e(s]ds, and c M. Otherwise, if c > M, from condition (H 6, we have A i (ξ i s [f(s, c + e(s]ds, which is a contradiction. Thus, Ω is bounded. Next, we define the isomorphism J : ImQ KerL by Jc c, c R n.

12 5 FULAI CHEN AND YONG ZHOU According to condition (H 6, for any c R n, if c > M, then either (3.8 or (3.9 holds. If (3.8 holds, set Ω 3 {x KerL : λjx + ( λqnx, λ [, ]}. For any x c Ω 3, we have ( λc ( λ A i ξi m A i (ξ i s [f(s, c + e(s]ds. If λ, then c. If c > M, from (3.8 we have ( m c A i ξi A i (ξ i s [f(s, c + e(s]ds <. Thus, λc c < which contradicts λc c. Therefore, Ω 3 is bounded. If (3.9 holds, set Ω 3 {x KerL : λjx + ( λqnx, λ [, ]}. Similar to the above argument, we also have that Ω 3 is bounded. Let Ω 3 Ω i {} (or Ω Ω i Ω 3 {} be a bounded open subset of Y. It follows from Lemma 3.3 that N is L compact on Ω. Then by the above argument, we have that conditions (i and (ii of Theorem. are satisfied, and we need only prove condition (iii of Theorem. hold. Take H(x, λ ±λjx + ( λqnx. In view of the argument to the sets Ω 3 and Ω 3, we have that H(x, λ for all Ω KerL. By the homotopy of degree, we get that deg(qn KerL, Ω KerL, deg(h(,, Ω KerL, deg(h(,, Ω KerL, deg(±j, Ω KerL,, which means that condition (iii of Theorem. is satisfied. By Theorem., Lx Nx has at least one solution in doml Ω, then BVP (. has at least one solution in C([, ], R n. This completes of the proof.. An Example As the application of our main result, we consider the following example. Example. Consider BVP { C D x x ( + cos x + 3 sin(x 3 + cos t +, C D x x ( + e sin x + 3 sin(x 3 + sin t +, with the boundary condition { 3 x ( + x ( + x (, 3x ( x ( + x ( + x ( + x ( + x (. (. (.

13 FRACTIONAL MULTI-POINT BOUNDARY VALUE PROBLEM 55 From (., we have that f(t, x (f (t, x, f (t, x, e(t (e (t, e (t, where f (t, x x ( + cos x + 3 sin(x 3, f (t, x x ( + e sin x + 3 sin(x 3, e (t cos t +, e (t sin t +. For any x C([, ], R and t [, ], f, e satisfies condition (H. Taking, then <, f L ([, ] R, R and e L ([, ], R. Thus condition (H 3 is satisfied. Let ξ, ξ, ξ 3, (. can be written where we have A (x (ξ, x (ξ + A (x (ξ, x (ξ + A 3 (x (ξ 3, x (ξ 3, ( 3 A 3 A + A + A 3, (, A (, A 3, and det (A ξ + A ξ + A 3 ξ 3 3, then condition (H holds. Taking a(t, b(t 3, then a which implies that condition (H holds. Moreover, where and f(t, x a(t x + b(t x 3, 3 A i (ξ i s f(s, x(sds ( F (x, x, e, e ( + ( ( s [f (s, x(s + e (s]ds ( F (x, x, e, e F (x, x, e, e ( s [f (s, x(s + e (s]ds ( ( s [f (s, x(s + e (s]ds ( s [f (s, x(s + e (s]ds, ( s [f (s, x(s + e (s]ds + F (x, x, e, e + ( s [f (s, x(s + e (s]ds + ( s [f (s, x(s + e (s]ds + ( s [f (s, x(s + e (s]ds, ( s [f (s, x(s + e (s]ds ( s [f (s, x(s + e (s]ds.

14 56 FULAI CHEN AND YONG ZHOU Take M and assume x (t > M for any t [, ], since x is continuous, then either x (t > M or x (t < M hold for any t [, ]. If x (t > M holds for any t [, ], we have > F (x, x, e, e ( [ ] s x ( + cos x + 3 sin(x 3 + cos t + ds + [ ] ( s x ( + cos x + 3 sin(x 3 + cos t + ds ( [ ] s x (t ds + [ ] ( s x (t ds [ ] M ( [ ] s ds + M ( s ds M >. If x (t < M holds for any t [, ], we have Hence, < F (x, x, e, e ( [ ] s x (t + 5 ds + [ ] M + 5 ( s ds + [ ] ( s x (t + 5 ds [ ] M + 5 ( s ds M + 5 <. 3 A i (ξ i s f(s, x(sds, then condition (H 5 holds. Taking M, for any c R, when c > M, then either c c > M or c c > M. If c c > M, then c c. We have ( c 3 A i ξi A i (ξ i s f(s, cds ( ( ( F (c (c, c, c, e, e 3 F (c, c, e, e ( ( (c F (c, c, c, e, e F (c, c, e, e [ (c 3 c F (c, c, e, e + 3 c F (c, c, e, e [ c ( + cos c + 6c sin(c 3 + c + c ] ( s cos sds

15 + c FRACTIONAL MULTI-POINT BOUNDARY VALUE PROBLEM 57 ( s cos sds + c ( + e sin c + 6c sin(c 3 + c + 3 c ( s sin sds + 3 c c c >. Similarly, if c c > M, then c c. We have ( c 3 A i ξi 3 Thus, condition (H 6 is satisfied. On the other hand, ( + ν 6 a ] ( s sin sds A i (ξ i s f(s, cds c c >. ( >, Γ( 6 then (3. is satisfied. Thus, BVP (. has at least one solution x C([, ], R by using Theorem 3.. References [] R.P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 9(, [] R.P. Agarwal, Y. Zhou, Y. He, Existence of fractional neutral functional differential equations, Comput. Math. Appl., 59(, 95. [3] B. Ahmad, S. Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Anal. Hybrid. Sys., 3(9, [] B. Ahmad, S. Sivasundaram, Existence of solutions for impulsive integral boundary value problems of fractional order, Nonlinear Anal. Hybrid. Sys., (, 3. [5] B. Ahmad, J.J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl., 58(9, [6] B. Ahmad, J.J. Nieto, Existence of solutions for nonlocal boundary value problems of higherorder nonlinear fractional differential equations, Abstract and Applied Analysis, 9, Art. No. 97. [7] B. Ahmad, J.J. Nieto, A. Alsaedi, M. El-Shahed, A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear Anal., RWA, doi:.6/j.nonrwa [8] G.A. Anastassiou, Fractional Differentiation Inequalities, Springer, 9. [9] Z. Bai, Y. Zhang, The existence of solutions for a fractional multi-point boundary value problem, Comput. Math. Appl., 6(, [] Z. Bai, On solutions of some fractional m-point boundary value problems at resonance, Electron. J. Qual. Theory Diff. Eq., 37(, 5. [] M. Benchohra, S. Hamani, S.K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal., 7(9, [] F. Chen, J.J. Nieto, Y. Zhou, Global attractivity for nonlinear fractional differential equations, Nonlinear Anal. Real World Appl., 3(,

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