Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations
Advances in Difference Equations
Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations
Bashir Ahmad 1
Sotiris K Ntouyas 0
Ioannis K Purnaras 0
0 Department of Mathematics, University of Ioannina , Ioannina, 451 10 , Greece
1 Department of Mathematics, Faculty of Science, King Abdulaziz University , P.O. Box 80203, Jeddah, 21589 , Saudi Arabia
In this paper, we study a nonlinear fractional q-difference equation with nonlocal boundary conditions. The existence of solutions for the problem is shown by applying some well-known tools of fixed-point theory such as Banach's contraction principle, Krasnoselskii's fixed-point theorem, and the Leray-Schauder nonlinear alternative. Some illustrating examples are also discussed. MSC: 34A08; 39A05; 39A12; 39A13
fractional q-difference equations; nonlocal boundary conditions; existence; Leray-Schauder nonlinear alternative; fixed point
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cDqαx(t) = f t, x(t) , ≤ t ≤ , < α ≤ ,
2 Preliminaries on fractional q-calculus
The q analogue of the power (a – b)n is
(a – b)() = ,
(a – b)(n) =
a – bqk , a, b ∈ R, n ∈ N.
The q-gamma function is defined by
( – q)(x–)
q(x) = ( – q)x– , x ∈ R \ {, –, –, . . .}, < q <
and satisfies q(x + ) = [x]q q(x) (see, []).
For < q < , we define the q-derivative of a real valued function f as
Dqf (t) =
Dqf () = tl→im Dqf (t).
The higher order q-derivatives are given by
Dqf (t) = f (t),
Dqnf (t) = DqDqn–f (t), n ∈ N.
Iqf (x) =
f (s) dqs =
x( – q)qnf xqn
provided that the series converges.
f (s) dqs = Iqf (b) – Iqf (a) = ( – q)
qn bf bqn – af aqn ,
DqIqf (x) = f (x),
IqDqf (x) = f (x) – f ().
For more details of the basic material on q-calculus, see the book [].
Definition . ([]) Let α ≥ and f be a function defined on [, ]. The fractional
qintegral of the Riemann-Liouville type is (Iqf )(t) = f (t) and
f (s) dq(s),
Dqαf (t) = D[qα]I[α]–αf (t),
q
cDqαf (t) = Iq[α]–αD[qα]f (t),
Definition . ([]) The fractional q-derivative of the Riemann-Liouville type of order
α ≥ is defined by (Dqf )(t) = f (t) and
Lemma . Let α, β ≥ and let f be a function defined on [, ]. Then the next formulas
hold:
(i) (Iqβ Iqαf )(t) = (Iqα+β f )(t),
(ii) (DqαIqαf )(t) = f (t).
q(k + )
< a < x < b.
⎪⎪⎩ αx() + βDqx() = γx(η),
is given by
x(t) =
g(s) dqs
g(s) dqs
g(s) dqs
g(s) dqs,
g(s) dqs
Proof In view of Lemmas . and ., integrating equation in (.), we have
x(t) =
g(s) dqs + ct + c, t ∈ [, ].
Using the boundary conditions of (.) in (.), we have
g(s) dqs
g(s) dqs,
Solving the above system of equations for c, c, we get
c =
g(s) dqs
g(s) dqs
c =
g(s) dqs
g(s) dqs
g(s) dqs .
Substituting the values of c, c in (.), we obtain (.).
In view of Lemma ., we define an operator F : C([, ], R) → C([, ], R) as
(Fx)(t) =
f s, x(s) dqs
f s, x(s) dqs
f s, x(s) dqs
f s, x(s) dqs
Observe that problem (.)-(.) has a solution if the operator equation Fx = x has a fixed
point, where F is given by (.).
3 Main results
Theorem . Assume that f : [, ] × R → R is continuous and that there exists a
qintegrable function L : [, ] → R such that
Then the boundary value problem (.)-(.) has a unique solution provided
(Fx)(t) ≤
where k is given by (.).
A =
≤ M
L(s) dqs
L(s) dqs
L(s) dqs
L(s) dqs
≤ M
which, in view of (.) and (.), implies that
This shows that FBρ ⊂ Bρ .
Now, for x, y ∈ C, we obtain
(Fx) – (Fy)
f s, x(s) – f s, y(s) dqs
f s, x(s) – f s, y(s) dqs
f s, x(s) – f s, y(s) dqs
f s, x(s) – f s, y(s) dqs
( – qs)(α–)
q(α – ) f s, x(s) – f s, y(s) dqs
L(s) dqs
which, in view of (.), yields
(Fx)(t) – (Fy)(t) ≤ k x – y .
In case L(t) = L (L is a constant), the condition (.) becomes LA < and Theorem .
takes the form of the following result.
Corollary . Assume that f : [, ] × R → R is a continuous function and that
Then the boundary value problem (.)-(.) has a unique solution.
f (t, x) ≤ μ(t)φ |x| , (t, x) ∈ [, ] × R;
(A) there exists a constant r with
r ≥
where μ = supt∈[,T] |μ(t)|.
L(s) dqs
then the boundary value problem (.)-(.) has at least one solution on [, ].
q(α + ) + |γ|δ qη((αα–+)) + |γ|δ qη((αα–+))
+ |α|δ q(α + ) + |β|δ q(α)
≤ r.
(Px)(t) =
f s, x(s) dqs, t ∈ [, ],
f s, x(s) dqs
f s, x(s) dqs
f s, x(s) dqs
( – qs)(α–)
q(α – ) f s, x(s) dqs, t ∈ [, ].
For x, y ∈ Br, we find that
(Px + Qy)(t) ≤
(Px)(t) – (Px)(t)
t (t – qs)(α–) – (t – qs)(α–)
q(α)
f s, x(s) dqs
f s, x(s) dqs –
f s, x(s) dqs
f s, x(s) dqs
t (t – qs)(α–) – (t – qs)(α–)
q(α)
In the special case when φ(u) ≡ , we see that there always exists a positive r so that (.)
holds true, thus we have the following corollary.
∀(t, x) ∈ [, ] × R, and μ ∈ C [, ], R+ .
The next existence result is based on Leray-Schauder nonlinear alternative.
Lemma . (Nonlinear alternative for single valued maps []) Let E be a Banach space,
C a closed, convex subset of E, (...truncated)