Degree theory and existence of positive solutions to coupled systems of multi-point boundary value problems

Boundary Value Problems, Feb 2016

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 { D α x ( t ) = ϕ ( t , x ( t ) , y ( t ) ) , t ∈ I = [ 0 , 1 ] , D β y ( t ) = ψ ( t , x ( t ) , y ( t ) ) , t ∈ I = [ 0 , 1 ] , x ( 0 ) = g ( x ) , x ( 1 ) = δ x ( η ) , 0 < η < 1 , y ( 0 ) = h ( y ) , y ( 1 ) = γ y ( ξ ) , 0 < ξ < 1 , where α , β ∈ ( 1 , 2 ] , D denotes the Caputo fractional derivative, 0 < δ , γ < 1 are parameters such that δ η α < 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. MSC: 34A08, 35R11.

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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: Dα+φp(Dβ+u(t)) = f (t, u(t), Dβ+u(t)), β β D+u() = D+u() = , Dα+φp(Dβ+u(t)) = f (t, u(t), Dβ+u(t)), u() = , Dβ+u() = Dβ+u(), where Dα+ and Dβ+ 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 Dα+ and Dβ+ 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 + dt + dt + · · · + 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)


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Kamal Shah, Amjad Ali, Rahmat Khan. Degree theory and existence of positive solutions to coupled systems of multi-point boundary value problems, Boundary Value Problems, 2016, pp. 43, 2016, DOI: 10.1186/s13661-016-0553-3