Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions

Boundary Value Problems, Mar 2016

In this paper, the existence and uniqueness of solutions for an impulsive mixed boundary value problem of nonlinear differential equations of fractional order are obtained. Our results are based on some fixed point theorems. Some examples are also presented to illustrate the main results. MSC: 34B15, 34A08.

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Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions

Bai et al. Boundary Value Problems Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions Zhanbing Bai 0 1 2 3 4 6 Xiaoyu Dong 0 1 2 3 4 6 Chun Yin 0 1 2 4 5 0 Technology , Qianwangang Road 1 University of Science 2 System Science , Shandong 3 College of Mathematics 4 Qingdao , 266590 , P.R. China 5 School of Automation Engineering, University of Electronic Science and Technology of China , Chengdu, 611731 , P.R. China 6 College of Mathematics and System Science, Shandong University of Science and Technology , Qianwangang Road, Qingdao, 266590 , P.R. China In this paper, the existence and uniqueness of solutions for an impulsive mixed boundary value problem of nonlinear differential equations of fractional order are obtained. Our results are based on some fixed point theorems. Some examples are also presented to illustrate the main results. fractional differential equations; impulse; mixed boundary value - problem; fixed point theorem 1 Introduction Recently, boundary value problems of nonlinear fractional differential equations have been addressed by several researchers. Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, control theory, biology, economics, blood flow phenomena, signal and image processing, biophysics, aerodynamics, fitting of experimental data, etc. For details, see [–] and the references therein. Impulsive differential equations, which provide a natural description of observed evolution processes, are regarded as important mathematical tools for a better understanding of several real world problems in the applied sciences. Recently, the boundary value problems of impulsive differential equations of integer order have been studied extensively in the literature (see [, –, –]). In [, ], Wang et al. gave a new concept of some impulsive differential equations with fractional derivative, which is a correction of that of piecewise continuous solutions used in [, , –]. This paper is strongly motivated by the above research papers. We investigate the existence and uniqueness of solutions for a mixed boundary value problem of nonlinear impulsive differential equations of fractional order given by u(tk) = Ik(u(tk)), ⎧ CDq+ u(t) = f (t, u(t)), t ∈ J , ⎨⎪ ⎪⎩ u() + u () = , u() + u () = , u (tk) = Jk(u(tk)), k = , , . . . , p, (.) where CDq+ is the Caputo fractional derivative of order q ∈ (, ), f ∈ C(J × R, R). Ik, Jk ∈ C(R, R), J = [, ], J = J \ {t, t, . . . , tp}, the {tk} satisfy  = t < t < t < · · · < tp < tp+ = , p ∈ N , u(tk) = u(tk+) – u(tk–), u (tk) = u (tk+) – u (tk–), where u(tk+) and u(tk–) represent the right and left limits of u(t) at t = tk . A function u ∈ PC(J, R) is said to be a solution of problem (.) if u(t) = uk(t) for t ∈ (tk, tk+) and uk ∈ C([, tk+], R) satisfies CDq+ u(t) = f (t, u(t)) a.e. on (, tk+) with the restriction that uk(t) on [, tk) is just uk–(t) and the conditions u(tk) = Ik(u(tk)), u (tk) = Jk(u(tk)), k = , , . . . , p with u() + u () = , u() + u () = . The rest of this paper is organized as follows. In Section , we give some notations, recall some concepts and preparation results. In Section , we give the main results, the first result based on Banach contraction principle, the second result based on Krasnoselskii’s fixed point theorem. Two examples are given in Section  to demonstrate the application of our main results. 2 Preliminaries In this section, we introduce preliminary facts which are used throughout this paper. Let J = [, t], J = (t, t], . . . , Jp– = (tp–, tp], Jp = (tp, ]. We have PC(J) = u : [, ] → R | u ∈ C J , and u tk+ , u tk– exist, and u tk– = u(tk),  ≤ k ≤ p . Obviously, PC(J) is a Banach space with the norm u PC = sup u(t) . ≤t≤ Iq+f (t) =  t f (s) (q)  (t – s)–q ds, t > , q > , (.) Definition . The fractional integral of order q of a function f : [, ∞) → R is defined as provided the right side is point-wise defined on (, ∞), where (·) is the gamma function. Definition . The Caputo derivative of fractional order q for a function f : [, ∞) → R is defined as CDq+ f (t) =  dn (n – q) dtn  t f (s) – kn=– skk! f (k)() (t – s)q–n+ where [q] denotes the integer part of the real number q. ds, t > , n = –[–q], (.) Remark . In the case f (t) ∈ Cn[, +∞), there is CDq+ f (t) = In+–qf (n)(t). That is to say that Definition . is just the usual Caputo’s fractional derivative. In this paper, we consider an impulsive problem, so Definition . is appropriate. Lemma . ([]) Let M be a closed, convex, and nonempty subset of a Banach space X, and A, B the operators such that () Ax + By ∈ M whenever x, y ∈ M; () A is compact and continuous; () B is a contraction mapping. Then there exists z ∈ M such that z = Az + Bz. Lemma . ([]) The set F ⊂ PC([, T ], Rn) is relatively compact if and only if: (i) F is bounded, that is, x ≤ C for each x ∈ F and some C > ; (ii) F is quasi-equicontinuous in [, T ]. That is to say that for any >  there exists δ >  such that if x ∈ F ; k ∈ N ; τ, τ ∈ (tk–, tk ], and |τ – τ| < δ, we have |x(τ) – x(τ)| < . Lemma . ([]) For q > , the general solution of the fractional differential equation C Dq+ u(t) =  is given by (.) (.) (.) u(t) = c + ct + ct + · · · + cn–tn–, where ci ∈ R, i = , , , . . . , n – , n = –[–q]. In view of Lemma ., it follows that q C Dq+ u (t) = u(t) + c + ct + ct + · · · + cn–tn–, I+ where ci ∈ R, i = , , , . . . , n – , n = –[–q]. Lemma . Let q ∈ (, ) and h : J → R be continuous. A function u given by  t  (q)  (t – s)q–h(s) ds + –(qt)  ( – s)q–h(s) ds + (q––t) ( – s)q–h(s) ds, t ∈ [, t];  t  (q)  (t – s)q–h(s) ds + –(qt)  ( – s)q–h(s) ds –t  + (q–)  ( – s)q–h(s) ds + ( – t) + ( – t) jp= Ij(u(tj)) – (t – tj) jp=k+ Jj(u(tj)) – t ∈ (tk , tk+], k = , , . . . , p – ;  t  (q)  (t – s)q–h(s) ds + –(qt)  ( – s)q–h(s) ds –t  + (q–)  ( – s)q–h(s) ds + ( – t) is a unique solution of the following impulsive problem: ⎪⎪⎨⎧ C Dq+ u(t) = h(t), u(tk ) = Ik (u(tk )), ⎪⎪⎩ u() + u () = , t ∈ J , u (tk ) = Jk (u(tk )), u() + u () = . k = , , . . . , p, Proof With Lemma ., a general solution u of the equation C Dq+ u(t) = h(t) on each interval (tk , tk+] (k = , , , . . . , p) is given by u(t) =  (q)  t (t – s)q–h(s) ds + ak + bk t, for t ∈ (tk , tk+], where t =  and tp+ = . Then we have u (t) = So applying the boundary conditions (.), we have (t – s)q–h(s) ds + bk, for t ∈ (tk, tk+]. (.) a + b = , Combining (.), (.), (.) with (.) yields  (q – )   Hence for k = , , , . . . , p – , (.) and (.) imply ak + bk t = ( – s)q–h(s) ds + j= p j=k+ Ij u(tj) (.) (.) (.) (.) (.)  – t (q)   For k = p, (.) and (.) imply ak + bk t = ( – s)q–h(s) ds + 3 Main results This section deals with the existence and uniqueness of solutions to problem (.). Theorem . Let f : J × R → R be a continuous function. Suppose there exist positive constants L, L, L, M, M such that (A) |f (t, x) – f (t, y)| ≤ L|x – y|, for all t ∈ J , x, y ∈ R; (A) |Ik (x) – Ik (y)| ≤ L|x – y|, |Jk (x) – Jk (y)| ≤ L|x – y|, |Ik (x)| ≤ M, |Jk (x)| ≤ M, x, y ∈ R, k = , , . . . , p, with ( – s)q– f s, u(s) – f (s, ) ds ( – s)q– f (s, ) ds +  (qM+ ) + L(qr) + M(q) ( – s)q– f s, x(s) – f s, y(s) ds +  + p j= p j=k+ +  + j= p j=k+ p j= p j=k+ Jj x(tj) – Jj y(tj) +  Ij x(tj) – Ij y(tj) Jj x(tj) – Jj y(tj) + Ij x(tj) – Ij y(tj) ≤ (qL+ ) x – y PC + p (qL+ ) x – y PC + p L x – y PC +  L x – y PC L(q) x – y PC j= p k=j+ L x – y PC + L x – y PC ≤ (qL+ ) x – y PC + L(q) x – y PC + pL x – y PC + pL x – y PC + pL x – y PC + pL x – y PC T is a contraction mapping. Thus, the conclusion follows by the contraction mapping principle. Define the operators P and Q on Br as r ≥ μ L σ (J)  (q)( q––σσ )–σ + (Qu)(t) = ( – t) Jk u(tk) ( – tk) + ( – t) Ik u(tk) – (t – tk) Jk u(tk) – Ik u(tk) . p k= p k=j+ p k=j+ p k= For any u, v ∈ Br and t ∈ J , using the condition that |f (t, u)| ≤ μ(t) and the Hölder inequality,  t   ≤ t ≤ t ≤ (t – s)q–f s, u(s) ds Therefore,  t   t t  μ(s) σ ds  μ(s) σ ds  μ(s) σ ds σ σ σ μ ≤ ( q––σσL)σ–(σJ) , μ ≤ ( q––σσL)σ–(σJ) , μ L σ (J) . ≤ ( q––σσ– )–σ Pu + Qv PC  μ L σ (J) ≤ (q)( q––σσ )–σ + μ L σ (J) (q – )( q––σσ– )–σ + pM + pM + pM + pM = μ L σ (J)  (q)( q––σσ )–σ + fmax (q)  ( – s)q–f s, u(s) ds – ( – s)q–f s, u(s) ds – (τ – s)q–f s, u(s) ds which tends to zero as τ → τ. This shows that P is quasi-equicontinuous on the interval (tk, tk+]. It is obvious that P is compact by Lemma ., so P is relatively compact on Br. Thus all the assumptions of Lemma . are satisfied and problem (.) has at least one solution on J . 4 Example Example . Consider the following impulsive fractional boundary value problem: ⎪⎪⎩⎪ u() + u () = , ⎪⎪⎧⎪⎨ CDu(+u)(=t)=|+u(|(ut+()|))| ,si+nuu((tt)) , u (t ∈) [=,|+]u,|(ut()=| )| , u() + u () = . Obviously, L = /, L = /, L = /, M = /, M = /, p = , ((q++q)) = √π , L (q+ ) + (q) L < ((q++q)) , + p(L + L) = √ π +  < . Thus, all the assumptions in Theorem . are satisfied. Hence, the impulsive fractional boundary value problem (.) has a unique solution on [, ]. Example . Consider the following impulsive fractional boundary value problem: Set Obviously, f (t, u) ≤ (t +et) . ⎪⎧⎪⎪⎨ CDu(+u)(=t)=++|(u|tu(+e(t)))|| ,+|u|u(t()t|)| , ut(∈ )[=, +]||,uut((=))|| , ⎪⎪⎪⎩ u() + u () = , u() + u () = . f (t, u) = (t +et)  +|u||u| , (t, u) ∈ [, ] × [, ∞). 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Zhanbing Bai, Xiaoyu Dong, Chun Yin. Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions, Boundary Value Problems, 2016, 63, DOI: 10.1186/s13661-016-0573-z