Nontrivial solutions for fractional q-difference boundary value problems

Transcription

1 Electronic Journal of Qualitative Theory of Differential Equations 21, No. 7, 1-1; Nontrivial solutions for fractional q-difference boundary value problems Rui A. C. Ferreira Department of Mathematics Lusophone University of Humanities and Technologies Lisbon, Portugal November 21, 21 Abstract In this paper, we investigate the existence of nontrivial solutions to the nonlinear q-fractional boundary value problem (D α q y)(x) = f(x,y(x)), < x < 1, y() = = y(1), by applying a fixed point theorem in cones. Keywords: Fractional q-difference equations, boundary value problem, nontrivial solution. 21 Mathematics Subject Classification: 39A13, 34B18, 34A8. 1 Introduction The q-difference calculus or quantum calculus is an old subject that was first developed by Jackson [9, 1]. It is rich in history and in applications as the reader can confirm in the paper [6]. ruiacferreira@ulusofona.pt EJQTDE, 21 No. 7, p. 1

2 The origin of the fractional q-difference calculus can be traced back to the works by Al-Salam [3] and Agarwal [1]. More recently, perhaps due to the explosion in research within the fractional calculus setting (see the books [13, 14]), new developments in this theory of fractional q-difference calculus were made, specifically, q-analogues of the integral and differential fractional operators properties such as q-laplace transform, q-taylor s formula [4, 15], just to mention some. To the best of the author knowledge there are no results available in the literature considering the problem of existence of nontrivial solutions for fractional q-difference boundary value problems. As is well-known, the aim of finding nontrivial solutions is of main importance in various fields of science and engineering (see the book [2] and references therein). Therefore, we find it pertinent to investigate on such a demand within this q-fractional setting. This paper is organized as follows: in Section 2 we introduce some notation and provide to the reader the definitions of the q-fractional integral and differential operators together with some basic properties. Moreover, some new general results within this theory are given. In Section 3 we consider a Dirichlet type boundary value problem. Sufficient conditions for the existence of nontrivial solutions are enunciated. 2 Preliminaries on fractional q-calculus Let q (, 1) and define [a] q = 1 qa 1 q, a R. The q-analogue of the power function (a b) n with n N is n 1 (a b) = 1, (a b) n = (a bq k ), n N, a, b R. More generally, if α R, then k= (a b) (α) = a α n= a bq n a bq α+n. Note that, if b = then a (α) = a α. The q-gamma function is defined by Γ q (x) = (1 q)(x 1), x R\{, 1, 2,...}, (1 q) x 1 EJQTDE, 21 No. 7, p. 2

3 and satisfies Γ q (x + 1) = [x] q Γ q (x). The q-derivative of a function f is here defined by (D q f)(x) = and q-derivatives of higher order by f(x) f(qx) (1 q)x, (D qf)() = lim x (D q f)(x), (D q f)(x) = f(x) and (Dn q f)(x) = D q(d n 1 q f)(x), n N. The q-integral of a function f defined in the interval [, b] is given by (I q f)(x) = f(t)d q t = x(1 q) f(xq n )q n, x [, b]. If a [, b] and f is defined in the interval [, b], its integral from a to b is defined by b a f(t)d q t = b n= f(t)d q t a f(t)d q t. Similarly as done for derivatives, it can be defined an operator I n q, namely, (I q f)(x) = f(x) and (In q f)(x) = I q(i n 1 q f)(x), n N. The fundamental theorem of calculus applies to these operators I q and D q, i.e., (D q I q f)(x) = f(x), and if f is continuous at x =, then (I q D q f)(x) = f(x) f(). Basic properties of the two operators can be found in the book [11]. We point out here four formulas that will be used later, namely, the integration by parts formula f(t)(d q g)td q t = [f(t)g(t)] t=x t= (D q f)(t)g(qt)d q t, and ( i D q denotes the derivative with respect to variable i) [a(t s)] (α) = a α (t s) (α), (1) td q (t s) (α) = [α] q (t s) (α 1), (2) sd q (t s) (α) = [α] q (t qs) (α 1). (3) EJQTDE, 21 No. 7, p. 3

4 Remark 2.1. We note that if α > and a b t, then (t a) (α) (t b) (α). To see this, assume that a b t. Then, it is intended to show that t α n= Let n N. We show that t aq n tα t aqα+n n= t bq n t bqα+n. (4) (t aq n )(t bq α+n ) (t bq n )(t aq α+n ). (5) Indeed, expanding both sides of the inequality (5) we obtain t 2 tbq α+n taq n + aq n bq α+n t 2 taq α+n tbq n + bq n aq α+n q n (aq α + b) q n (bq α + a) b a q α (b a) 1 q α. Since inequality (5) implies inequality (4) we are done with the proof. The following definition was considered first in [1] Definition 2.2. Let α and f be a function defined on [, 1]. The fractional q-integral of the Riemann Liouville type is (Iqf)(x) = f(x) and (I α q f)(x) = 1 (x qt) (α 1) f(t)d q t, α >, x [, 1]. The fractional q-derivative of order α is defined by (Dq f)(x) = f(x) and (Dq α f)(x) = (Dq m Iq m α f)(x) for α >, where m is the smallest integer greater or equal than α. Let us now list some properties that are already known in the literature. Its proof can be found in [1, 15]. Lemma 2.3. Let α, β and f be a function defined on [, 1]. Then, the next formulas hold: 1. (I β q Iα q f)(x) = (Iα+β q f)(x), 2. (D α q I α q f)(x) = f(x). The next result is important in the sequel. Since we didn t find it in the literature we provide a proof here. EJQTDE, 21 No. 7, p. 4

5 Theorem 2.4. Let α > and p be a positive integer. Then, the following equality holds: p 1 (Iq α Dp q f)(x) = (Dp q Iα q f)(x) x α p+k Γ q (α + k p + 1) (Dk qf)(). (6) k= Proof. Let α be any positive number. We will do a proof using induction on p. Suppose that p = 1. Using formula (3) we get: td q [(x t) (α 1) f(t)] = (x qt) (α 1) td q f(t) [α 1] q (x qt) (α 2) f(t). Therefore, (I α q D q f)(x) = 1 = [α 1] q (x qt) (α 1) (D q f)(t)d q t = (D q I α q f)(x) xα 1 f(). Suppose now that (6) holds for p N. Then, (Iq α Dq p+1 f)(x) = (Iq α DqD p q f)(x) = D p q = (D p qi α q D q f)(x) (x qt) (α 2) f(t)d q t + 1 [(x t)(α 1) f(t)] t=x t= p 1 k= [(D q I αq f)(x) xα 1 f() = (D p+1 q I α q f)(x) xα 1 p Γ q (α p) f() The theorem is proved. k=1 x α p+k Γ q (α + k p + 1) (Dk+1 q f)() ] p 1 x α p+k Γ q (α + k p + 1) (Dk+1 q f)() k= p x α (p+1)+k Γ q (α + k (p + 1) + 1) (Dk qf)() p = (Dq p+1 Iq α f)(x) x α (p+1)+k Γ q (α + k (p + 1) + 1) (Dk q f)(). k= EJQTDE, 21 No. 7, p. 5

6 3 Fractional boundary value problem We shall consider now the question of existence of nontrivial solutions to the following problem: subject to the boundary conditions (D α q y)(x) = f(x, y(x)), < x < 1, (7) y() =, y(1) =, (8) where 1 < α 2 and f : [, 1] R R is a nonnegative continuous function (this is the q-analogue of the fractional differential problem considered in [5]). To that end we need the following theorem (see [8, 12]). Theorem 3.1. Let B be a Banach space, and let C B be a cone. Assume Ω 1, Ω 2 are open disks contained in B with Ω 1, Ω 1 Ω 2 and let T : C (Ω 2 \Ω 1 ) C be a completely continuous operator such that Ty y, y C Ω 1 and Ty y, y C Ω 2. Then T has at least one fixed point in C (Ω 2 \Ω 1 ). Let us put p = 2. In view of item 2 of Lemma 2.3 and Theorem 2.4 we see that (Dq α y)(x) = f(x, y(x)) (Iα q D2 q I2 α q y)(x) = Iq α f(x, y(x)) y(x) = c 1 x α 1 + c 2 x α 2 1 (x qt) (α 1) f(t, y(t))d q t, for some constants c 1, c 2 R. Using the boundary conditions given in (8) we take c 1 = 1 1 (1 Γ q(α) qt)(α 1) f(t, y(t))d q t and c 2 = to get y(x) = 1 = 1 [ (1 qt) (α 1) x α 1 f(t, y(t))d q t 1 x (x qt) (α 1) f(t, y(t))d q t ( [x(1 qt)] (α 1) (x qt) (α 1)) f(t, y(t))d q t ] + [x(1 qt)] (α 1) f(t, y(t))d q t. EJQTDE, 21 No. 7, p. 6

7 If we define a function G by G(x, t) = 1 { (x(1 t)) (α 1) (x t) (α 1), t x 1, (x(1 t)) (α 1), x t 1, then, the following result follows. Lemma 3.2. y is a solution of the boundary value problem (7)-(8) if, and only if, y satisfies the integral equation y(x) = G(x, qt)f(t, y(t))d q t. Remark 3.3. If we let α = 2 in the function G, then we get a particular case of the Green function obtained in [16], namely, { t(1 x), t x 1 G(x, t) = x(1 t), x t 1. Some properties of the function G needed in the sequel are now stated and proved. Lemma 3.4. Function G defined above satisfies the following conditions: G(x, qt) and G(x, qt) G(qt, qt) for all x, t 1. (9) Proof. We start by defining two functions g 1 (x, t) = (x(1 t)) (α 1) (x t) (α 1), t x 1 and g 2 (x, t) = (x(1 t)) (α 1), x t 1. It is clear that g 2 (x, qt). Now, in view of Remark 2.1 we get, g 1 (x, qt) = x α 1 (1 qt) (α 1) x α 1 (1 q t x )(α 1) Moreover, for t (, 1] we have that x α 1 (1 qt) (α 1) x α 1 (1 qt) (α 1) =. xd q g 1 (x, t) = x D q [(x(1 t)) (α 1) (x t) (α 1) ] = [α 1] q (1 t) (α 1) x α 2 [α 1] q (x t) (α 2) ( = [α 1] q x [(1 α 2 t) (α 1) 1 t ) ] (α 2) x [ [α 1] q x α 2 (1 t) (α 1) (1 t) (α 2)], EJQTDE, 21 No. 7, p. 7

8 which implies that g 1 (x, t) is decreasing with respect to x for all t (, 1]. Therefore, g 1 (x, qt) g 1 (qt, qt), < x, t 1. (1) Now note that G(, qt) = G(qt, qt) for all t [, 1]. Therefore, by (1) and the definition of g 2 (it is obviously increasing in x) we conclude that G(x, qt) G(qt, qt) for all x, t 1. This finishes the proof. Let B = C[, 1] be the Banach space endowed with norm u = sup t [,1] u(t). Define the cone C B by C = {u B : u(t) }. Remark 3.5. It follows from the nonnegativeness and continuity of G and f that the operator T : C B defined by (Tu)(x) = G(x, qt)f(t, u(t))d q t, satisfies T(C) C and is completely continuous. For our purposes, let us define two constants ( 1 M = G(qt, qt)d q t), N = ( τ2 τ 1 ) 1 G (qt, qt)d q t, where τ 1 {, q m } and τ 2 = q n with m, n N, m > n. Our existence result is now given. Theorem 3.6. Let f(t, u) be a nonnegative continuous function on [, 1] [, ). If there exists two positive constants r 2 > r 1 > such that f(t, u) Mr 2, for (t, u) [, 1] [, r 2 ], (11) f(t, u) Nr 1, for (t, u) [τ 1, τ 2 ] [, r 1 ], (12) then problem (7)-(8) has a solution y satisfying r 1 y r 2. Proof. Since the operator T : C C is completely continuous we only have to show that the operator equation y = Ty has a solution satisfying r 1 y r 2. EJQTDE, 21 No. 7, p. 8

9 Let Ω 1 = {y C : y < r 1 }. For y C Ω 1, we have y(t) r 1 on [, 1]. Using (9) and (12), and the definitions of τ 1 and τ 2, we obtain (see page 282 in [7]), τ2 Ty = max G (x, qt) f(t, y(t))d q t Nr 1 G (qt, qt)d q t = y. x 1 τ 1 Let Ω 2 = {y C : y < r 2 }. For y C Ω 2, we have y(t) r 2 on [, 1]. Using (9) and (11) we obtain, Ty = max x 1 G (x, qt) f(t, y(t))d q t Mr 2 G (qt, qt) d q t = y. Now an application of Theorem 3.1 concludes the proof. References [1] R. P. Agarwal, Certain fractional q-integrals and q-derivatives, Proc. Cambridge Philos. Soc. 66 (1969), [2] R. P. Agarwal, D. O Regan and P. J. Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Acad. Publ., Dordrecht, [3] W. A. Al-Salam, Some fractional q-integrals and q-derivatives, Proc. Edinburgh Math. Soc. (2) 15 (1966/1967), [4] F. M. Atici and P. W. Eloe, Fractional q-calculus on a time scale, J. Nonlinear Math. Phys. 14 (27), no. 3, [5] Z. Bai and H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (25), no. 2, [6] Ernst T, The History of q-calculus and a New Method, U. U. D. M. Report 2:16, ISSN , Department of Mathematics, Uppsala University, 2. [7] H. Gauchman, Integral inequalities in q-calculus, Comput. Math. Appl. 47 (24), no. 2-3, EJQTDE, 21 No. 7, p. 9

10 [8] D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, MA, [9] Jackson F.H.: On q-functions and a certain difference operator. Trans. Roy Soc. Edin. 46, (198), [1] Jackson F.H.: On q-definite integrals. Quart. J. Pure and Appl. Math. 41 (191), [11] V. Kac and P. Cheung, Quantum Calculus, Springer, New York, 22. [12] M. A. Krasnosel skiĭ, Positive Solutions of Operator Equations, Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron, P. Noordhoff Ltd. Groningen, [13] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, [14] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, [15] P. M. Rajković, S. D. Marinković and M. S. Stanković, Fractional integrals and derivatives in q-calculus, Appl. Anal. Discrete Math. 1 (27), no. 1, [16] M. El-Shahed and H. A. Hassan, Positive solutions of q-difference equation, Proc. Amer. Math. Soc. 138 (21), no. 5, (Received August 4, 21) EJQTDE, 21 No. 7, p. 1