Degree theory and existence of positive solutions to coupled systems of multi-point boundary value problems
Shah et al. Boundary Value Problems
Degree theory and existence of positive solutions to coupled systems of multi-point boundary value problems
Kamal Shah
Amjad Ali
Rahmat Ali Khan
= φ(t
t ∈ I = [
⎪⎪⎨ D
t ∈ I = [
⎪⎪⎩ y
= γ y
In this article, we investigate existence and uniqueness of positive solutions to coupled systems of multi-point boundary value problems for fractional order differential equations of the form where α, β ∈ (1, 2], D denotes the Caputo fractional derivative, 0 < δ, γ < 1 are parameters such that δηα < 1, γ ξ β < 1, h, g ∈ C(I, R) are boundary functions and φ, ψ : I × R × R → R are continuous. We use the technique of topological degree theory to obtain sufficient conditions for existence and uniqueness of positive solutions to the system. Finally, we provide an example to illustrate our main results.
coupled system; boundary value problems; fractional differential equations; existence and uniqueness of solutions; topological degree theory
1 Introduction
Recently much attention has been paid to investigate sufficient conditions for existence
of positive solutions to boundary value problems for fractional order differential
equations, we refer to [–] and the references therein. This is because of many applications
of fractional differential equations in various field of science and technology as in [–].
Existence of solutions to boundary value problems for coupled systems of fractional order
differential equations has also attracted some attentions, we refer to [, –]. In these
papers, classical fixed point theorems such as Banach contraction principle and Schauder
fixed point theorem are used for existence of solutions. The use of these results require
stronger conditions on the nonlinear functions involved which restricts the applicability
to limited classes of problems and very specialized systems of boundary value problems.
In order to enlarge the class of boundary value problems and to impose less restricted
conditions, one needs to search for other sophisticated tools of functional analysis. One
such tool is topological degree theory. Few results can be found in the literature which
use degree theory arguments for the existence of solutions to boundary value problems
(BVPs) [–]. However, to the best of our knowledge, the existence and uniqueness of
solutions to coupled systems of multi-point boundary value problems for fractional order
differential equations with topological degree theory approach have not been studied
previously. Wang et al. [] studied the existence and uniqueness of solutions via topological
degree method to a class of nonlocal Cauchy problems of the form
Dqu(t) = f (t, u(t)), t ∈ I = [, T ],
u() + g(u) = u,
where Dq is the Caputo fractional derivative of order q ∈ (, ], u ∈ R, and f : I × R → R is
continuous. The result was extended to the case of a boundary value problem by Chen et
al. [] who studied sufficient conditions for existence results for the following two point
boundary value problem:
Dα+φp(Dβ+u(t)) = f (t, u(t), Dβ+u(t)),
β β
D+u() = D+u() = ,
Dα+φp(Dβ+u(t)) = f (t, u(t), Dβ+u(t)),
u() = , Dβ+u() = Dβ+u(),
where Dα+ and Dβ+ are Caputo fractional derivatives, < α, β ≤ , < α + β ≤ . Wang et
al. [] studied the following two point boundary value problem for fractional differential
equations with different boundary conditions:
where Dα+ and Dβ+ are Caputo fractional derivatives, < α, β ≤ , < α + β ≤ .
Motivated by the work cited above, in this paper, we use a coincidence degree theory
approach for condensing maps to obtain sufficient conditions for the existence and
uniqueness of solutions to more general coupled systems of nonlinear multi-point boundary
value problems. The boundary conditions are also nonlinear. The system is of the form
⎧ Dαx(t) = φ(t, x(t), y(t)), t ∈ I = [, ],
⎪⎪⎪⎨ Dβ y(t) = ψ (t, x(t), y(t)), t ∈ I = [, ],
⎪ x() = g(x), x() = δx(η), < η < ,
⎪⎪⎩ y() = h(y), y() = γ y(ξ ), < ξ < ,
()
where α, β ∈ (, ], D is used for standard Caputo fractional derivative and < δ, γ < such
that δηα < , γ ξ β < , h, g ∈ C(I, R) are boundary functions and φ, ψ : I × R × R → R are
continuous.
Definition . The fractional integral of order ρ ∈ R+ of a function u ∈ L([a, b], R) is
defined as
ρ
I+u(t) =
(ρ) a
t
(t – s)ρ–u(s) ds.
Dρ+u(t) =
(m – ρ) a
t
(t – s)m–ρ–u(m)(s) ds,
Definition . The Caputo fractional order derivative of a function u on the interval [a, b]
is defined by
where m = [ρ] + and [ρ] represents the integer part of ρ.
Lemma . The following result holds for fractional differential equations:
Iρ Dρ u(t) = u(t) + d + dt + dt + · · · + dm–tm–
for arbitrary di ∈ R, i = , , , . . . , m – .
The spaces X = C([, ], R), Y = C([, ], R) of all continuous functions from [, ] →
R are Banach spaces under the topological norms x = sup{|x(t)| : t ∈ [, ]} and y =
sup{|y(t)| : t ∈ [, ]}, respectively. The product space X × Y is a Bana (...truncated)