A critical point approach to multiplicity results for a fractional boundary value problem
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A critical point approach to multiplicity results for a fractional boundary value problem Sougata Dhar1 · Lingju Kong2 Received: 2 August 2019 / Revised: 30 November 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this article, we establish criteria for the existence of at least three solutions of a fractional differential equation with Dirichlet boundary condition using variational methods and critical point theory. We discuss some consequences of our main results. Our work complements existing results in the literature. Keywords Multiple solutions · Fractional boundary value problem · Variational methods · Critical point theorem Mathematics Subject Classification 34B15 · 34A08
1 Introduction In this paper, we investigate the existence of at least three solutions of the fractional boundary value problem (BVP) ⎧ 1 −β ⎨ d 1 −β D (u (t)) + D (u (t)) + λ∇ F(t, u(t)) = 0, t ∈ [0, T ], 0 t dt 2 t 2 T ⎩ u(0) = u(T ) = 0, (1.1)
Communicated by Rosihan M. Ali.
B
Lingju Kong [email protected] Sougata Dhar [email protected]
1
Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA
2
Department of Mathematics, The University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA
123
S. Dhar et al. −β
−β
where T > 0, λ > 0 is a parameter, 0 ≤ β < 1, 0 Dt and t DT are the left and right Riemann–Liouville fractional integrals of order β, respectively, N ≥ 1 is an integer, and F : [0, T ] × R N → R satisfies the following assumption: (A) F(t, x) is measurable in t for each x ∈ R N and continuously differentiable in x for a.e. t ∈ [0, T ], F(t, 0) ≡ 0 on [0, T ], ∇ F(t, x) = (∂ F/∂ x1 , . . . , ∂ F/∂ x N ) is the gradient of F at x, and there exist a ∈ C(R+ , R+ ) and b ∈ L 1 ([0, T ], R+ ) such that |F(t, x)| ≤ a(|x|)b(t) and |∇ F(t, x)| ≤ a(|x|)b(t) for all x ∈ R N and a.e. t ∈ [0, T ]. An absolutely continuous function u : [0, T ] → R N is called a solution of BVP (1.1) if it satisfies the equation in (1.1) a.e. in [0, T ] and the boundary conditions (BCs) in (1.1). The study of BVP (1.1) is motivated by the paper [8] by Ervin and Roop where they studied the fractional advection dispersion equation −β −β − D k ( p 0 Dt + q t DT )Du + b(t)Du + c(t)u = f ,
(1.2)
where D represents a single spatial derivative, 0 ≤ p, q ≤ 1 satisfy p + q = 1, k > 0 is a constant, and b, c, f are functions satisfying some suitable conditions. The relationship of (1.1) and (1.2) was noted and discussed in [15, Remark 1.1]. Moreover, many existence results have been obtained in the literature for the special case of BVP (1.1) with β = 0. In recent years, the theory of fractional calculus has become widely popular among researchers due to its various applications in numerous fields of science and engineering. The monographs [14,16] provide a detailed description and background reading of the origin and development of the theory and applications of fractional calculus. Parallel to the case of ordinary differential equations, the existence and m
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