Boundary value problems for fractional differential equations

Hu et al. Boundary Value Problems 2014, 2014:176 http://www.boundaryvalueproblems.com/content/2014/1/176 RESEARCH Open Access Boundary value problems for fractional differential equations Zhigang Hu* , Wenbin Liu and Jiaying Liu * Correspondence: Department of Mathematics, China University of Mining and Technology, Xuzhou, 221008, P.R. China Abstract In this paper we study the existence of solutions of nonlinear fractional differential equations at resonance. By using the coincidence degree theory, some results on the existence of solutions are obtained. MSC: 34A08; 34B15 Keywords: fractional differential equations; boundary value problems; resonance; coincidence degree theory 1 Introduction In recent years, the fractional differential equations have received more and more attention. The fractional derivative has been occurring in many physical applications such as a non-Markovian diffusion process with memory [], charge transport in amorphous semiconductors [], propagations of mechanical waves in viscoelastic media [], etc. Phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry, and material science are also described by differential equations of fractional order (see [–]). Recently boundary value problems (BVPs for short) for fractional differential equations have been studied in many papers (see [–]). In [], by means of a fixed point theorem on a cone, Agarwal et al. considered two-point boundary value problem at nonresonance given by  Dα+ x(t) + f (t, x(t), Dμ+ x(t)) = , x() = x() = , where  < α < , μ >  are real numbers, α – μ ≥  and Dα+ is the Riemann-Liouville fractional derivative. Zhao et al. [] studied the following two-point BVP of fractional differential equations:  Dα+ x(t) = f (t, x(t)), t ∈ (, ), x() = x () = x () = , where Dα+ denotes the Riemann-Liouville fractional differential operator of order α,  < α ≤ . By using the lower and upper solution method and fixed point theorem, they obtained some new existence results. © 2014 Hu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. Hu et al. Boundary Value Problems 2014, 2014:176 http://www.boundaryvalueproblems.com/content/2014/1/176 Page 2 of 11 Liang and Zhang [] studied the following nonlinear fractional boundary value problem:  Dα+ x(t) = f (t, x(t)), t ∈ (, ), x() = x () = x () = x () = , where  < α ≤  is a real number, Dα+ is the Riemann-Liouville fractional differential operator of order α. By means of fixed point theorems, they obtained results on the existence of positive solutions for BVPs of fractional differential equations. In [], Bai considered the boundary value problem of the fractional order differential equation  Dα+ x(t) + a(t)f (t, x(t), x (t)), t ∈ (, ), x() = x () = x () = x () = , where  < α ≤  is a real number, Dα+ is the Riemann-Liouville fractional differential operator of order α. Motivated by the above works, in this paper, we consider the following BVP of fractional equation at resonance  Dα+ x(t) = f (t, x(t), x (t), x (t), x (t)), t ∈ (, ), x () = x (), x() = x () = x () = , (.) where Dα+ denotes the Caputo fractional differential operator of order α,  < α ≤ . f : [, ] × R → ×R is continuous. The rest of this paper is organized as follows. Section  contains some necessary notations, definitions and lemmas. In Section , we establish a theorem on existence of solutions for BVP (.) under nonlinear growth restriction of f , basing on the coincidence degree theory due to Mawhin (see []). Finally, in Section , an example is given to illustrate the main result. 2 Preliminaries In this section, we introduce notations, definitions and preliminary facts which are used throughout this paper. Let X and Y be real Banach spaces and let L : dom L ⊂ X → Y be a Fredholm operator with index zero, and P : X → X, Q : Y → Y be projectors such that Im P = Ker L, Ker Q = Im L, X = Ker L ⊕ Ker P, Y = Im L ⊕ Im Q. It follows that L|dom L∩Ker P : dom L ∩ Ker P → Im L is invertible. We denote the inverse by KP . If  is an open bounded subset of X, and dom L ∩  = ∅, the map N : X → Y will be called L-compact on  if QN() is bounded and KP (I – Q)N :  → X is compact, where I is identity operator. Hu et al. Boundary Value Problems 2014, 2014:176 http://www.boundaryvalueproblems.com/content/2014/1/176 Page 3 of 11 Lemma . ([]) If  is an open bounded set, let L : dom L ⊂ X → Y be a Fredholm operator of index zero and N : X → Y L-compact on . Assume that the following conditions are satisfied: () Lx = λNx for every (x, λ) ∈ [(dom L \ Ker L)] ∩ ∂ × (, ); () Nx ∈/ Im L for every x ∈ Ker L ∩ ∂; () deg(QN|Ker L , Ker L ∩ , ) = , where Q : Y → Y is a projection such that Im L = Ker Q. Then the equation Lx = Nx has at least one solution in dom L ∩ . Definition . The Riemann-Liouville fractional integral operator of order α >  of a function x is given by Iα+ x(t) =  (α)  t (t – s)α– x(s) ds,  provided that the right side integral is pointwise defined on (, +∞). Definition . The Caputo fractional derivative of order α >  of a function x with x(n–) absolutely continuous on [, ] is given by dn x(t) Dα+ x(t) = In–α = + n dt  (n – α)  t (t – s)n–α– x(n) (s) ds,  where n = –[–α]. Lemma . ([]) Let α >  and n = –[–α]. If x(n–) ∈ AC[, ], then α Dα+ x(t) = x(t) – I+ n– (k)  x () k= k! tk . In this paper, we denote X = C  [, ] with the norm x X = max{ x ∞ , x ∞ , x ∞ , x ∞ } and Y = C[, ] with the norm y Y = y ∞ , where x ∞ = maxt∈[,] |x(t)|. Obviously, both X and Y are Banach spaces. Define the operator L : dom L ⊂ X → Y by Lx = Dα+ x, (.) where   dom L = x ∈ X | Dα+ x(t) ∈ Y , x() = x () = x () = , x () = x () . Let N : X → Y be the operator   Nx(t) = f t, x(t), x (t), x (t), x (t) , ∀t ∈ [, ]. Then BVP (.) is equivalent to the operator equation Lx = Nx, x ∈ dom L. Hu et al. Boundary Value Problems 2014, 2014:176 http://www.boundaryvalueproblems.com/content/2014/1/176 Page 4 of 11 3 Main result In this section, a theorem on existence of solutions for BVP (.) will be given. Theorem . Let f : [, ] × R → R be continuous. Assume that (H ) there exist nonnegative functions a, b, c, d, e ∈ C[, ] with (α –)–(b +c +d +e ) >  such that f (t, u, v, w, x) ≤ a(t) + b(t)|u| + c(t)|v| + d(t)|w| + e(t)|x|, ∀t ∈ [, ], (u, v, w, x) ∈ R , where a = a ∞ , b = b ∞ , c = c ∞ , d = d ∞ , e = e ∞ ; (H ) there exists a constant B >  such that for all x ∈ R with |x| > B either xf (t, u, v, w, x) > , ∀t ∈ [, ], (u, v, w) ∈ R xf (t, u, v, w, x) < , ∀t ∈ [, ], (u, v, w) ∈ R . or Then BVP (.) has at le (...truncated)