Existence of solutions for multi-point boundary value problem of fractional q-difference equation

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1 Electronic Journal of Qualitative Theory of Differential Euations 211, No. 92, 1-1; Existence of solutions for multi-point boundary value problem of fractional -difference euation Junchi Ma 1, Jun Yang 1,2 1 College of Science, Yanshan University, Qinhuangdao Hebei 664, China 2 Mathematics Research Center in Hebei Province, Shijiazhuang 5, China Abstract: This paper is mainly concerned with the existence of solutions for a multi-point boundary value problem of nonlinear fractional -difference euations by means of the Banach contraction principle and Krasnoselskii s fixed point theorem. Further, an example is presented to illustrate the main results. Keywords: Fractional -difference euations; Multi-point condition; Fixed point theorem. 2 MR Subject Classification: 34B8, 34B18, 39A13 1 Introduction Fractional differential calculus have recently been addressed by many researchers of various fields of science and engineering such as physics, chemistry, biology, economics, control theory, and biophysics, etc. [1-4]. In particular, the existence of solutions to fractional boundary value problems is under strong research recently, see [5-7] and references therein. The fractional -difference calculus had its origin in the works by Al-Salam [8] and Agarwal [9]. More recently, perhaps due to the explosion in research within the fractional differential calculus setting, new developments in this theory of fractional -difference calculus were made, specifically, -analogues of the integral and differential fractional operators properties such as the -Laplace transform, -Taylor s formula [1,11], just to mention some. The uestion of the existence of solutions for fractional -difference boundary value problems is in its infancy, being few results available in the literature. Ferreira [12] considered the existence of positive solutions to nonlinear -difference boundary value problem: D α u)t) = ft, ut)), < t < 1, 1 < α 2 u) = u1) =. In other paper, Ferreira [13] studied the existence of positive solutions to nonlinear -difference boundary value problem: D α u)t) = ft, ut)), < t < 1, 2 < α 3 u) = D u)) =, D u)1) = β. M.El-Shahed and Farah M.Al-Askar [15] studied the existence of positive solutions to nonlinear -difference euation: C D α u)t) at)fut)) =, t 1, 2 < α 3 u) = D 2 u)) =, γd u1) βd 2 u1) =. This work is supported by the NNSF of China No. 6644), the Special Projects in Mathematics Funded by NSF of Hebei Province No. 7M5), the Scientific and Technological Support Project of Qinhuangdao No. 211A37). Corresponding author. address: majunchi29@163.com EJQTDE, 211 No. 92, p. 1

2 In this paper, we investigate the existence of solutions for nonlinear -difference boundary value problem of the form C D α u)t) = f t, ut), φu)t), ψu)t)), 1 < t < 2, 1 < α 2, u) = u gu), D u1) = u 1 D uξ i ), where < ξ i < 1 i = 1, 2,...,m 2), with ρ = < 1 and C D α represents the standard Caputo fractional -derivative), f : [, 1] X X X X is continuous, for γ, δ : [, 1] [, 1] [, ), φu)t) = γt, s)us)d s, ψu)t) = δt, s)us)d s. Here, X, ) is a Banach space and C = C[, 1], X) denotes the Banach space of all continuous functions from [, 1] X endowed with a topology of uniform convergence with the norm denoted by. 1.1) 2 Preliminaries on fractional -calculus Let, 1) and define [a] = 1 a 1, a R. The -analogue of the power function a b) n with n N is n 1 a b) = 1, a b) n = a b k ), n N, a, b R. More generally, if α R, then a b) α) = a α i= k= a b i a b αi. Note that, if b = then a α) = a α. The -gamma function is defined by Γ x) = 1 )x 1), x R \, 1, 2...}, < < 1, 1 ) x 1 and satisfies Γ x 1) = [x] Γ x). The -derivative of a function fx) is here defined by D f)x) = fx) fx) 1)x x ), D f)) = lim x D f)x), and -derivatives of higher order by fx) if n =, D n f)x) = D D n 1 fx) if n N. The -integral of a function fx) defined in the interval [, b] is given by I f)x) = ft)d t = x1 ) fx n ) n, < 1, x [, b]. n= If a [, b] and f is defined in the interval [, b], its integral from a to s defined by b a ft)d t = b ft)d t a ft)d t. EJQTDE, 211 No. 92, p. 2

3 If c j = b j, for j, 1,...,n}, a = c n = b n, < < 1. The restricted -integral of a function fx) defined by b a ft)d t = b n 1 n 1 ft)d t = 1 )b j fb j ) = 1 ) c j fc j ), < < 1, b >, n Z. b n j= Note that the restricted integral b a ft)d t is just a finite sum, so no uestions about convergence arise. Obviously, if f x) g x) on [a, b], then b a ft)d t b a gt)d t. If < k < n, then b ck b ft)d t = ft)d t ft)d t. a c k a The usual Riemann integral can be considered as a limit of the restricted definite -integral in the following way. Since a = b n, = ) 1 a n b. Fix a and b and let n hence, 1). Then, b a ft)d t b a ft)dt assuming that f t) is Riemann integrable on [a, b]. The above formulas were proved by the author [17]. Similarly as done for derivatives, it can be defined an operator I n, namely, fx) if n =, I n f)x) = I I n 1 fx) if n N. The fundamental theorem of calculus applies to these operators I and D, i.e., D I f)x) = fx), and if f is continuous at x =, then I D f)x) = fx) f). and, more generally D n In f)x) = fx), n N, n 1 I n D n x k f)x) = fx) Γ k 1) Dk f)), n N. k= Basic properties of the two operators can be found in the book [14]. We point out here five formulas that will be used later, namely, the integration by parts formula ft)d g)td t = [ft)gt)] t=x t= D f)gt)td t, and i D denotes the derivative with respect to variable i) [at s)] α) = a α t s) α), td t s) α) = [α] t s) α 1), sd t s) α) = [α], x D fx, t)d t)x) = xd fx, t)d t fx, x). Remark 2.1. We note that if α > and a b t, then t a) α) t b) α) [12]. The following definition was considered first in [9]. j= EJQTDE, 211 No. 92, p. 3

4 Definition 2.2. Let α and f be a function defined on [, 1]. The fractional -integral of the Riemann- Liouville type is I f)x) = fx) and I α f)x) = 1 x t) α 1) ft)d t, α >, x [, 1]. Definition 2.3. The fractional -derivative of the Riemann-Liouville type of order α is defined by D f)x) = fx) and D α f)x) = D m I m α f)x), α >, where m is the smallest integer greater than or eual to α. Definition 2.4. [16] The fractional -derivative of the Caputo type of order α is defined by C D α f)x) = I m α D m f)x), α >, where m is the smallest integer greater than or eual to α. Next, let us list some properties that are already know in the literature. Lemma 2.5. Let α, β and f be a function defined on [, 1]. Then, the next formulas hold: 1.I β I α f)x) = I αβ f)x), 2.D α I α f)x) = fx). The next result is important in the seuel. It was proved in a recent work by the author [12]. Theorem 2.1. [12] Let α > and n N. Then, the following euality holds: n 1 I α Dn f)x) =Dn Iα fx) x α nk Γ α k n 1) Dk f)). k= Theorem 2.2. [16] Let α R \ N. Then, the following euality holds: m 1 I α CD α f)x) = fx) k= x k Γ k 1) Dk f)). where m is the smallest integer greater than or eual to α. Lemma 2.6. For a given σ C[, 1] and 1 < α 2, the uniue solution of is given by C D α u)t) = σ t), u) = u gu), D u1) = u 1 D uξ i ), u t) = 1 σ s)d s 1 1 t σ s)d s t σ s)d s u gu) 1 u 1t. Proof. Let us put m = 2. In view of Lemma 2.5 and Theorem 2.2, we have C D α u)t) = σ t) Iα I2 α ut) = CD 2 u)t) = Iα σ t) σ s)d s c c 1 t, 2.1) 2.2) EJQTDE, 211 No. 92, p. 4

5 for some constants c, c 1 R. Using the boundary condition u) = u gu), gives c = u gu). Furthermore, differentiation of 2.2) with respect to t produces D ux) = [α 1] t s) α 2) σ s)d s c 1. Using the boundary conditions D u1) = D uξ i ), we get c 1 = 1 1 σ s)d s 1 σ s)d s u 1. Now, substitution of c and c 1 into 2.1) gives u t) = 1 The proof is complete. σ s)d s 1 1 t σ s)d s t σ s)d s u 1t u gu). 3 Main results Define an operator T : C C by Tu)t) =u gu) u 1t t 1 t fs, us), φu)s), ψu)s))d s fs, us), φu)s), ψu)s))d s fs, us), φu)s), ψu)s))d s, t [, 1]. Clearly, the fixed points of the operator T are solutions of problem 1.1). To establish the main results, we need the following assumptions: H 1 ) There exist positive functions L 1 t), L 2 t), L 3 t) such that ft, ut), φu)t), ψu)t)) ft, vt), φv)t), ψv) t)) L 1 t) u v L 2 t) φu φv L 3 t) ψu ψv, H 2 ) g : C X is continuous and there exists a constant b such that Further, gu) gv) b u v, u, v C γ = sup t [,1] I α L = max γt, s)d s, sup t [,1] δ = sup I α L 1 t), sup t [,1] t [,1] δt, s)d s, I α L 2 t), sup t [,1] I α L 3 t) }, t [, 1], u, v X 3.1) L 1) = max I α 1 L 1 1), I α 1 L 2 1), I α 1 L 3 1) }, L ξ i ) = max I α 1 L 1 ξ i ), L 2 ξ i ), L 3 ξ i ) }, ξ i = 1, 2,..., m 2. EJQTDE, 211 No. 92, p. 5

6 H 3 ) There exists a number κ < 1, where = 1 γ δ ) I α L 1 I α 1 L 1) H 4 ) There exists a number, where = b 1 γ δ ) I α L 1 L 1) L ξ i ) L ξ i ) H 5 ) ft, u t), φu)t), ψu)t)) µ t), t, ut), φu),ψu)) [, 1] X X X, µ L 1 [, 1], R ). Theorem 3.1. Assume that f : [, 1] X X X X is jointly continuous function and satisfies the assumptions H 1 ) H 4 ), then problem 1.1) has a uniue solution provided < 1, where is given in the assumption H 3 ). Proof. Let us set sup t [,1] ft,,, ) = M, and choose [ r 1 u G u 1 1 M 1 Γ α 1) 1 1 ).. ξ α 1) i where λ is such that λ < 1. We show that TB r B r, where B r = x C : u r}. So let u B r and set G = sup u C g u). Then we have Tu)t) u G u fs, us), φu)s), ψu)s)) d s fs, us), φu)s), ψu)s)) d s fs, us), φu)s), ψu)s)) d s } )]} u G u 1 fs, us), φu)s), ψu)s)) f s,,, ) f s,,, ) )d s 1 1 fs, us), φu)s), ψu)s)) f s,,, ) f s,,, ) )d s fs, us), φu)s), ψu)s)) f s,,, ) f s,,, ) )d s u G u 1 L 1 s) u s) L 2 s) φu)s) L 3 s) ψu)s) M)d s 1 1 L 1 s) u s) L 2 s) φu)s) L 3 s) ψu)s) M)d s L 1 s) u s) L 2 s) φu)s) L 3 s) ψu)s) M)d s, } } EJQTDE, 211 No. 92, p. 6

7 u G u L 1 s) u s) γ L 2 s) u s) δ L 3 s) u s) M)d s L 1 s) u s) γ L 2 s) u s) δ L 3 s) u s) M)d s L 1 s) u s) γ L 2 s) u s) δ L 3 s) u s) M)d s u G u 1 I α L 1 t) γ I α L 2 t) δ I α L 3 t) ) r Mtα) Γ α 1) I α 1 I α 1 L 1 1) γ I α 1 L 2 1) δ I α 1 L 3 1) ) r M L 1 ξ i ) γ u G u 1 M Iα L 1 γ δ )r Γ α 1) 1 I α 1 L1)1 γ δ )r M Lξ i )1 γ δ ) r Mξα 1) i L 2 ξ i ) δ I α 1 L 3 ξ i ) ) r Mξα 1) i Γ α) u G u 1 1 γ δ ) I α L 1 1 M Γ α 1) 1 1 ξ α 1) i u G u 1 r M 1 Γ α 1) 1 ) Now, for u, v C and for each t [, 1], we obtain L1) 1 Lξ i ) ξ α 1) i Tu)t) Tv)t) g u) g v) fs, us), φu)s), ψu)s)) fs, vs), φv)s), ψv)s)) d s 1 1 fs, us), φu) s), ψu)s)) fs, vs), φv) s), ψv) s)) d s } ξi fs, us), φu)s), ψu)s)) fs, vs), φv)s), ψv) s)) d s g u) g v) L 1 s) u v L 2 s) φu φv L 3 s) ψu ψv )d s 1 1 L 1 s) u v L 2 s) φu φv L 3 s) ψu ψv )d s } ξi L 1 s) u v L 2 s) φu φv L 3 s) ψu ψv )d s EJQTDE, 211 No. 92, p. 7 r r }

8 g u) g v) I α L 1 t) γ I α L 2 t) δ I α L 3 t) ) u v 1 I α 1 L 1 1) γ I α 1 L 2 1) δ I α 1 L 3 1) I α 1 b u v 1 γ δ ) = u v L 1 ξ i ) γ L 2 ξ i ) δ L 3 ξ i ) u v I α L 1 L 1) L ξ i ) u v where is given in the assumption H 4 ). As < 1, therefore T is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle. Now, we state Krasnoselskii s fixed point theorem which is needed to prove the existence of at least one solution of 1.1). Theorem 3.2. Let K be a closed convex and nonempty subset of a Banach space X. Let T, S be the operators such that i) Tu Sv K whenever u, v K; ii) T is compact and continuous; iii) S is a contraction mapping. Then there exists z K such that z = T z Sz. Theorem 3.3. Suppose that f : [, 1] X X X X is jointly continuous and the assumptions H 1 ) H 2 ) and H 5 ) hold with 1 1 = b 1 γ δ ) I α 1 L 1) I α 1 L ξ i ) < 1. Then there exists at least one solution of the boundary value problem 1.1) on [, 1]. Proof. Choose r u G u 1 µ L 1 1 Γ α 1) 1 ) 1 ξ α 1) i and consider Ω r = u C : u r}. We define the operators T 1 and T 2 on Ω r as T 1 u)t) = u T 2 u)t) =gu) u 1t 1 f s, us), φu)s),ψu)s)) d s, 1 t fs, us), φu)s), ψu)s))d s } t fs, us), φu)s),ψu)s))d s Let s observe that if u, v Ω r then T 1 u T 2 v Ω r. Indeed it is easy to check the ineuality T 1 u T 2 v u G u 1 µ 1 L 1 Γ α 1) 1 1 ξ α 1) i r. ) By H 1 ), it is also that T 2 is a contraction mapping for 1 < 1... EJQTDE, 211 No. 92, p. 8

9 Since f is continuous, then T 1 u)t) is continuous. Let s now note that T 1 is uniformly bounded on Ω r. This follows from the ineuality T 1 u)t) u u 1 µ L 1 Γ α 1). Now,we show that T 1 Ω r ) is euicontinuous. The functions T 1 u, u Ω r are euicontinuous at t =. For t 1, t 2 n : n N }, and t 1 < t 2. Using the fact that f is bounded on the compact set [, 1] Ω r Ω r Ω r, therefore, we define sup t,u,φu,ψu) [,1] Ωr Ω r Ω r f t, u, φu, ψu) = f max <. We have 2 t 2 s) α 1) T 1 u)t 2 ) T 1 u)t 1 ) = f s, us), φu)s),ψu)s))d s 1 t 1 s) α 1) f s, us), φu)s),ψu)s)) d s 1 t 2 s) α 1) t 1 s) α 1) = f s, us), φu)s),ψu)s))d s 2 t 1 f max Γ α 1) f s, us), φu)s),ψu)s)) d s 1 t α) 2. α 1) t 2 s) t α) which is independent of u Ω r when t 1 t 2. Indeed, let u Ω r, we have f max lim t α) t 1 t α) 2. 1 t 2 Γ α 1) Therefore, T 1 Ω r ) is relatively compact on Ω r. Hence, by Arzela-Ascoli s Theorem, T 1 is compact on Ω r. Thus all the assumptions of Theorem 3.2 are satisfied and the conclusion of Theorem 3.2 implies that the boundary value problem 1.1) has at least one solution on [, 1]. 4 Examples Example 4.1. Consider the following boundary value problem CD ut) = t u 8 1 u 1 e s t) u s)ds 1 e s t 2 u s)ds, t [, 1], u) =, D.5 u1) = b 1 D.5 uξ 1 ), 4.1) s t Here, γ t, s) = e s t) 5, δ t, s) = e 2 5, G = 1 1, b 1 = 1 1, ξ = 1 2. With γ = e 1 5, δ = 2 e 1), 5 sup I.5L 1.5 t) , I.5L 1) , I.5L 1/2) , t [,1] we find that = e 1 2 ) [ e 1) ] ) < /1 Thus, by Theorem 3.1, the boundary value problem 4.1) has a uniue solution on [, 1]. EJQTDE, 211 No. 92, p. 9

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