Fractional q-Difference Equations
As in the classical theory of ordinary fractional differential equations, q-difference equations of fractional order are divided into linear, nonlinear, homogeneous, and inhomogeneous equations with constant and variable coefficients. This chapter is devo
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Fractional q-Difference Equations
Abstract As in the classical theory of ordinary fractional differential equations, q-difference equations of fractional order are divided into linear, nonlinear, homogeneous, and inhomogeneous equations with constant and variable coefficients. This chapter is devoted to certain problems of fractional q-difference equations based on the basic Riemann–Liouville fractional derivative and the basic Caputo fractional derivative. In this chapter, we investigate questions concerning the solvability of these equations in a certain space of functions. A special class of Cauchy type qfractional problems is also developed at the end of this chapter.
8.1 Equations with the Riemann–Liouville Fractional q-Derivatives In this section we shall study the existence and uniqueness of solutions of the qCauchy type problem Dq˛ y.x/ D f .x; y.x// Dq˛k y.0C / D bk ;
.˛ > 0/;
bk 2 R .k D 1; : : : ; d˛e/ :
(8.1) (8.2)
The result of this section is an extension of the results derived by Kilbas et al. in [169, Chap. 3].
8.1.1 Solutions in the Space Lq1 Œ0; a Theorem 8.1. Let ˛ > 0, n D d˛e. Let G be an open set in C and let f W .0; a G ! R be a function such that f .x; y/ 2 Lq;1 Œ0; a for any y 2 G. Assume that
M.H. Annaby and Z.S. Mansour, q-Fractional Calculus and Equations, Lecture Notes in Mathematics 2056, DOI 10.1007/978-3-642-30898-7 8, © Springer-Verlag Berlin Heidelberg 2012
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8 Fractional q-Difference Equations
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y.x/ 2 Lq1 Œ0; a. Then y.x/ satisfies (8.1)–(8.2) for all x 2 .0; a if and only if y.x/ satisfies the q-integral equation y.x/ D
n X kD1
x ˛1 bk x ˛k C q .˛ k C 1/ q .˛/
Z
x
.qt=xI q/˛1 f .t; y.t// dq t; (8.3) 0
for all x 2 .0; a. Proof. First, we prove the necessity condition. Let y.x/ satisfy (8.1)–(8.2), then Dq˛ y.x/ 2 Lq1 Œ0; a. Hence, applying Lemma 4.17 gives Iq˛ Dq˛ y.x/ D y.x/
n X kD1
bk x ˛k q .˛ k C 1/
.0 < x a/:
(8.4)
On the other hand, Iq˛ Dq˛ y.x/ D Iq˛ f .x; y.x// D
x ˛1 q .˛/
Z
x
.qt=xI q/˛1 f .t; y.t// dq t;
(8.5)
0
for all x 2 .0; a. Consequently, combining (8.4) and (8.5) proves the necessity condition. Now we prove the sufficiency. Let y.x/ satisfy (8.3) for all x 2 .0; a, then from Lemma 4.9, y.x/ 2 Lq1 Œ0; a. Moreover, Iqn˛ y.x/ D
n X kD1
bk x n1 x nk C q .n k C 1/ q .n/
Z
x
.qt=xI q/n1 f .t; y.t// dq t; 0 .n/
for all x 2 .0; a. Hence, from Theorem 4.6, Iqn˛ y.x/ 2 A Cq Œ0; a. That is Dq˛ y.x/ exists for all x 2 .0; a. Since Dq˛
x ˛k x ˛k x nk WD Dqn Iqn˛ D Dqn D 0; q .˛ k C 1/ q .˛ k C 1/ q .n k C 1/
for k D 1; 2; : : : ; n, then applying the operator Dq˛ on the two sides of (8.3) gives Dq˛ y.x/ D Dq˛ Iq˛ f .x; y/ D f .x; y.x//; for all x > 0: Now Dq˛k y.0C / D lim Dq˛k y.xq j / D bk C lim Iqk f .xq j ; y.xq j // j !1
.xq j /k1 D bk C lim j !1 q .k/
j !1
Z
(8.6)
xq j
.qt=xI q/k1 f .t; y.t// dq t; 0
8.1 Equations with the Riemann–Liouville Fractional q-Derivatives
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where the limit on the most left hand side of (8.6) vanishes because of f .x; y.x// 2 Lq1
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