Wang and Li Advances in Difference Equations (2016) 2016:186 DOI 10.1186/s13662-016-0913-4
R E S E A R C H Open Access Existence and attractivity of global solutions for a class of fractional quadratic integral equations in Banach space Huiwen Wang* and Fang Li
*Correspondence: [email protected] Abstract School of Mathematics, Yunnan Normal University, Kunming, In this paper, we present some results concerning the existence and attractivity of 650092, P.R. China global solutions for a class of nonlinear fractional integral equations and fractional differential equations in a Banach space X, respectively. These results are new even in thecaseofX = R. Some examples are given to show the applications of the abstract results. MSC: 34K05; 34A12 Keywords: fractional order; integral equations; fractional differential equations; global solutions; attractivity
1 Introduction The theory of integral equations is frequently applicable in other branches of mathemat- ics and mathematical physics, like as radiative transfer, kinetic theory of gases, neutron transport, etc., and engineering, economics, biology as well as in describing problems con- nected with the real world (see [–] and the references therein). On the other hand, the application of fractional calculus is very broad, including the characterization of mechan- ics and electricity, earthquake analysis, the memory of many kinds of material, electronic circuits, electrolysis chemical, etc. [–]. Recently, some basic theory for initial value problems for fractional differential equa- tions and inclusions was discussed by many researchers (see [–] and the references therein). Moreover, there has been a significant development in solving integral equations involving fractional derivatives in X = R (see [–] and the references therein). How- ever, to the best of our knowledge, there are few works on the attractivity of solutions for fractional integral equations in a Banach space. In this paper, let X be a Banach space, we discuss the existence and attractivity of global solutions for fractional integral equation on X of the form g(t, x(β(t))) η(t) K(t, s)h(s, x(γ (s))) x t f t x α t ds t ∈ ∞ ( )= , ( ) + –q , [, + ), (.) (q) (t – s)
where < q <.f , g, h : R+ × X → X are to be specified later. α, β, γ , η : R+ → R+ are con-
tinuous. {K(t, s):t ≥ s, t, s ∈ R+} is a set of bounded linear operators on X.
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In previous work [–], there are different techniques dealing with integral equations. For example, the authors apply the measure of noncompactness and fixed point theorem of Darbo type to deal with quadratic Erdélyi-Kober type integral equations of fractional order with three parameters []; the authors deal with integral equations in an order in- terval of a cone []; in [], the technique used is the measure of noncompactness on space BC(R+) associated with the Schauder fixed point theorem, etc. In [–]andthe references therein, the continuity (even Lipschitz continuity) of f (or g, h)oftwovariables is always required. In this paper, we study a more complicated model than previous ones [–], apply different techniques and obtain results under weaker assumptions. We de- fine a Banach space Cδ (X) and investigate the existence and attractivity of global solutions for equation (.)onCδ (X) by using the fixed point theorem. The above mentioned tech- niques are new even in the case of X = R. The rest of the paper is organized as follows. In Section , we recall some definitions and preliminary facts. In Section , we present the notion of the so-called space Cδ (X) and study equation (.) on this space with the fixed point theorem, as the additional pro- duction, we obtain the corresponding results for the global mild solutions of the fractional differential equations as follows: cDq(x(t)–m(t, x(t))) = A(x(t)–m(t, x(t))) + h(t, x(t)), t >, t (.) x() = x,
and the global solutions of the following fractional differential equations: cDqx(t)=h(t, x(t)), t >, t (.) x() = x,
where A is the infinitesimal generator of an analytic semigroup of linear operators
{T(t)}t≥ in X. Finally, three examples are given to illustrate our main results.
2Preliminaries We introduce some terminology. Throughout this paper, X denotes a Banach space with norm ·and L(X) the Banach space of all linear and bounded operators on X.Wewrite
Br(x, Z) to denote the closed ball with center at x and radius r > in a Banach space Z,and
C(R+, X)thespaceofallX-valued continuous functions on R+ with the supremum norm
x∞ = sup{x(t) : t ≥ } for any x ∈ C(R+, X). We recall the following basic definitions and properties of the fractional calculus theory. For more details see [].
Definition . ([]) The fractional integral of order q with the lower limit zero for a func- tion f ∈ L[, ∞)isdefinedas t q q– Jt f (t)= (t – s) f (s) ds, t >, provided the right side is point-wise defined on [, ∞), where (·) is the gamma func- tion. Wang and Li Advances in Difference Equations (2016) 2016:186 Page 3 of 15 Definition . ([]) The Riemann-Liouville derivative of order q with the lower limit zero for a function f ∈ C[, ∞)canbewrittenas t L q d –q Dt f (t)= (t – s) f (s) ds, t >, Definition . ([]) The Caputo derivative of order q for a function f ∈ C[, ∞)canbe written as c q L q Dt f (t)= Dt f (t)–f () , t >, c q dq c q where Dt := dtq .Moreover, Dt c =,wherec is a constant. ∈ ∈ ∞ αc α c α α Remark . ([]) If α (, ), f C[, ), then (Jt Dt f )(t)=f (t)–f () and ( Dt Jt f )(t)= f (t). We need the following lemma concerning on a fixed point theorem (see [], Theo- rem ..). Lemma . Let D be a convex, bounded, and closed subset of a Banach space Z and F : D → Dbeacondensingmap. Then F has a fixed point in D. Assuming that is a nonempty subset of the space C(R+, X), we review the concept of attractivity of solutions of equation (.). Definition . ([]) We say that solutions of equation (.)arelocally attractive if there exists a ball B(x, C(R+, X)) such that for arbitrary solutions x(t)andy(t)ofequation(.) belonging to B(x, C(R+, X)) ∩ , lim x(t)–y(t) = (.) t→∞ holds. Remark . When the limit (.)isuniformwithrespecttothesetB(x, C(R+, X)) ∩ , i.e. when for each ε >thereexistsT >suchthatx(t)–y(t) < ε for all solutions x, y ∈ B(x, C(R+, X)) ∩ of equation (.)andforanyt ≥ T,wewillsaythatsolutions of equation (.)areuniformly locally attractive (or equivalently that solutions of (.)are asymptotically stable). Lemma . ([]) () For all τ, θ >–, we have t sθ (t – s)τ ds = tθ+τ+B(τ +,θ +). () For all λ, κ, >, for t ≥ , we have t κ sλ–(t – s)κ–e–(t–s) ds ≤ max , –λ (κ) + –κ tλ–, λ where B(·, ·) is the beta function. Wang and Li Advances in Difference Equations (2016) 2016:186 Page 4 of 15 3 Main results In this section, we study the existence and attractivity of global solutions for equation (.), equation (.), and equation (.), respectively. 3.1 The case of fractional integral equations Motivated by the work in [], for any δ >,wedefinethespace ∈ x(t) Cδ (X)= x C(R+, X): lim = t→∞ eδt –δt endowed with the norm x δ = supt≥ e x(t) . We recall the following results of compactness of this spaces. ⊂ Lemma . ([]) AsetB Cδ (X) is relatively compact in Cδ (X) if and only if: (a) B is equicontinuous; –δt (b) limt→∞ e x(t) =, uniformly for x ∈ B; (c) the set B(t)={x(t):x ∈ B} is relatively compact in X, for every t ≥ . Let r > , and in the following let V(r, g)denotethesetV(r, g)={t → g(t, x(β(t))) : x ∈ Br(, C(R+, X))}. We study the fractional integral equation (.) with the following conditions: (H) α, β, γ , η : R+ → R+ is continuous, η is nondecreasing on R+, α(t) →∞, η(t) →∞, as t →∞,andα(t), η(t), γ (t) ≤ t. (H) The function f (t, x(α(t))) is continuous with respect to t on [, +∞) and there exists a continuous function Lf (t) such that f (t, ψ)–f (t, ψ) ≤ Lf (t)ψ – ψ, for ψ, ψ ∈ C(R+, X), ∗ ∞ where Lf = supt≥ Lf (t)<, limt→∞ Lf (t)=,andsupt≥ f (t,) < . (H) The function g(t, x(β(t))) is continuous with respect to t on [, +∞) and for every a >, g(t, ·):X → X is continuous for t ∈ [, a].ThesetV(r, g) is an equicontinuous subset of C(R+, X) and there exists a function ζ : R+ → R+ such that g(t, x)≤ζ (t) ˜ ∞ and supt∈[,a] ζ (t)=ζa < , for every a >. (H) The operator K(·, s) is continuous in the uniform operator topology for all s ∈ R+ and k = K(t, s)L(X) < ∞. (H) For any a >,thefunctionh :[,a] × X → X satisfies the following conditions: (a) the function h(t, ·):X → X is continuous a.e. t ∈ [, a]; (b) the function h(·, x):[,a] → X is strongly measurable for every x ∈ X; · ∈ ≤ (c) there exists ν( ) Lloc(R+) such that h(t, x) ν(t) x ,thefunction → ν(s) R s (t–s)–q belongs to L ([, t], +) and η(t) ν(s)e–δ(t–s) lim ζ (t) ds =; →∞ –q t (t – s) { δs ∈ ∈ } (d) for every t >and r >,theset K(t, s)h(s, e z):s [, t), z Br(, Cδ (X)) is relatively compact in X. Wang and Li Advances in Difference Equations (2016) 2016:186 Page 5 of 15 Inthefollowing,wechooseaconstantδ >suchthat η(t) –δ(t–s) ∗ ζ (t)k ν(s)e L sup ds f + –q <. (.) t≥ (q) (t – s) Theorem . Let the assumptions (H)-(H) be satisfied, then there exists a solution for equation (.) on the space Cδ (X). ∈ M Proof For x Cδ (X), we consider the operator of the form (Mx)(t)=(Fx)(t)+(Gx)(t), where g(t, x(β(t))) η(t) K(t, s)h(s, x(γ (s))) Fx t f t x α t Gx t ds ( )( )= , ( ) ,()( )= –q . (q) (t – s) From our assumptions, it is easy to see that Gx ∈ C(R+, X)and kζ (t)e–δt η(t) ν(s)eδγ(s)e–δγ(s)x(γ (s)) e–δt Gx t ≤ ds ( )( ) –q (q) (t – s) kζ (t) η(t) ν(s)e–δ(t–s) ≤ ds ·x → t →∞ –q δ , , (q) (t – s) G ∈ G and we conclude that x Cδ (X)and is a function from Cδ (X)toCδ (X). Applying condition (H), we get (Fx)(t) f (t,) ≤ f t, x α(t) – f (t,) + f (t,) ≤ L (t)xδ + . eδt eδt f eδt F F F ≤ ∗ F Hence, is Cδ (X)-valued. Moreover, x – y δ Lf x – y δ, which implies that is a contraction on Cδ (X). G { } → Now, we show that is continuous. Let xn n∈N be a sequence in Cδ (X)suchthatxn x →∞ in Cδ (X)asn , that is, for arbitrary ε >suchthat xn – x δ < ε for sufficient large n. We can see –δt e (Gxn)(t)–(Gx)(t) kg(t, x (β(t))) – g(t, x(β(t)))·x η(t) ν(s)e–δ(t–s) ≤ n n δ ds –q (q) (t – s) kζ (t) η(t) h(s, x (γ (s))) – h(s, x(γ (s))) n ds + δt –q (q)e (t – s) = I(t)+I(t). From condition (H)(c) there exists T >,fort ≥ η(t) ≥ T such that kζ (t) η(t) ν(s)e–δ(t–s) ε ds < .(.) –q (q) (t – s) (ε + x δ) Wang and Li Advances in Difference Equations (2016) 2016:186 Page 6 of 15 ≤ ∈ Moreover, noting that xn δ ε + x δ and from (H), there exists Nε N such that g t, xn β(t) – g t, x β(t) ε(q) ≤ ∈ –δ(t–s) , t [, T], n > Nε , (.) ε T ν(s)e k( + x δ) (t–s)–q ds then ε I (t) < ,fort ∈ [, T], n > N ,(.) ε ≥ ≥ ∈ on the other hand, for t η(t) T, Nε N,notingthat(.)and(.), we have kg(t, x (β(t))) – g(t, x(β(t)))·x T ν(s)e–δ(t–s) I t ≤ n n δ ds ( ) –q (q) (t – s) kζ (t)x η(t) ν(s)e–δ(t–s) n δ ds + –q (q) T (t – s) ε < , then ε I (t) < ,fort ≥ η(t) ≥ T, n > N .(.) ε For I(t), from the Lebesgue dominated convergence theorem and (H)(a), there exists ∈ Nε N such that k T h(s, x (γ (s))) – h(s, x(γ (s))) ε n ds < , n > N ,(.) –q ˜ ε (q) (t – s) ζT then we have ε I (t) < ,fort ∈ [, T], n > N ,(.) ε ≥ ≥ on the other hand, for t η(t) T and n > Nε ,notingthat h t, xn γ (t) – h t, x γ (t) ≤ ν(t) xn γ (t) – x γ (t) + x γ (t) δt ≤ ν(t)e ε +xδ , and (.), (.), we obtain ζ˜ k T h(s, x (γ (s))) – h(s, x(γ (s))) I t ≤ T n ds ( ) δt –q (q)e (t – s) ζ (t)k η(t) h(s, x (γ (s))) – h(s, x(γ (s))) n ds + δt –q (q)e T (t – s) Wang and Li Advances in Difference Equations (2016) 2016:186 Page 7 of 15 ε ζ (t)k η(t) ν(s)e–δ(t–s) ds · ε x < + –q + δ (q) T (t – s) ε < , t ≥ η(t) ≥ T, n > N .(.) ε Now, from (.)and(.), we have –δt G G ∈ ≤ sup e ( xn)(t)–( x)(t) : t [, T], n > max Nε , Nε ε, and from (.)and(.), we obtain –δt G G ≥ ≥ ≤ sup e ( xn)(t)–( x)(t) : t η(t) T, n > max Nε , Nε ε. Now, we can conclude that G is continuous. G Next, we show that is completely continuous. Let r >andBr = Br(, Cδ (X)), we first of all show that the set G(Br) is equicontinuous. For any ε >,t, t ≥ , we may assume that t < t without loss of generality, for x ∈ Br,weget (Gx)(t)–(Gx)(t) keδt rg(t , x(β(t ))) – g(t , x(β(t ))) η(t) ν(s) ≤ ds –q (q) (t – s) δ η(t )–ε e t rζ (t ) K(t , s)–K(t , s)L (X) k + –q + –q – –q (q) (t – s) (t – s) (t – s) η(t) ν(s) ν(s) η(t) ν(s) × ν s ds k ds k ds ( ) + –q + –q + –q . η(t)–ε (t – s) (t – s) η(t) (t – s) Under the conditions (H) and (H), (Gx)(t)–(Gx)(t)→ast → t and ε → , that is, the set G(Br) is equicontinuous. Next, we prove that the set U(t)={Gx(t):x ∈ Br, t ∈ [, a]} is a relatively compact subset of X for every a ∈ (, +∞). For arbitrary ε ∈ (, η(t)), define an operator Gε as follows: g(t, x(β(t))) η(t)–ε K(t, s)h(s, x(γ (s))) G x t ds ( ε )( )= –q . (q) (t – s) Noting that (H)(c) and (H), for x ∈ Br, from the mean value theorem for the Bochner integral (see [], Lemma ..), we obtain ε ζ˜ (t – ) a q– (Gεx)(t) ∈ co (t – s) K(t, s)h(s, x):s ∈ [, t – ε], x ∈ B , (q) r where co(·) denotes the convex hull. Using (H)(d), we infer that the set {Gεx(t):x ∈ Br} is relatively compact in X for arbitrary ε ∈ (, η(t)). Moreover, for x ∈ Br, t ∈ [, a], we get g(t, x(β(t))) η(t) K(t, s)h(s, x(γ (s))) Gx t G x t ds ( )( )–( ε )( ) = –q (q) η(t)–ε (t – s) eδarζ˜ k η(t) ν(s) ≤ a ds –q . (q) η(t)–ε (t – s) Wang and Li Advances in Difference Equations (2016) 2016:186 Page 8 of 15 Hence, there are relatively compact sets arbitrarily close to the set U(t), t ≥ . This proves that the set U(t), t ≥ is relatively compact in X. Moreover, for x ∈ Br,weobtain kζ (t)r η(t) ν(s)e–δ(t–s) e–δt Gx t ≤ ds → t →∞ ( )( ) –q , . (q) (t – s) G Now, from Lemma .,wecanconcludethat (Br) is relatively compact in Cδ (X). Thus, G is completely continuous. M ⊂ Next, we prove that there exists r >suchthat Br Br,whereBr = Br(, Cδ (X)). ∗ ∗ Suppose on the contrary that, for each r >,thereexistx ∈ Br and some t ≥ suchthat ∗ e–δt (Mx∗)(t∗) > r.Then ∗ ∗ ∗ r < e–δt Mx t ∗ ∗ ∗ η(t ) –δ(t –s) ∗ ∗ ∗ ∗ ζ (t )k ν(s)e ∗ ≤ L t x e–δt f t ds · x f δ + , + ∗ –q δ (q) (t – s) ∗ ∗ ∗ η(t ) –δ(t –s) ∗ ζ (t )k ν(s)e ∗ ∗ ≤ L t ds r e–δt f t f + ∗ –q + , .(.) (q) (t – s) Dividing both sides of (.)byr and taking r →∞,weobtain ∗ ∗ ∗ η(t ) –δ(t –s) ∗ ζ (t )k ν(s)e L t ds ≥ f + ∗ –q . (q) (t – s) This contradicts (.). This shows that there exists r >suchthatM is a condensing map from Br into Br. Now from Lemma . we see that the operator M has a fixed point and thus equation (.) has at least one solution on Cδ (X). Moreover, the solutions of equation (.) are lying in the ball Br, and for any solutions x, y of equation (.)andx, y ∈ Br,weobtain –δt e x(t)–y(t) ≤ Lf (t) xδ + yδ kx η(t) ν(s)e–δ(t–s) δ g t x β t g t y β t ds + , ( ) – , ( ) –q (q) (t – s) kζ (t)e–δt η(t) h(s, x(γ (s))) – h(s, y(γ (s))) ds + –q (q) (t – s) rkζ (t) η(t) ν(s)e–δ(t–s) ≤ rL t ds → t →∞ f ( )+ –q , , (q) (t – s) then the solutions of equation (.) are uniformly locally attractive by Definition . (or equivalently that solutions of (.) are asymptotically stable). From the proof of Theorem ., we can immediately study equation (.)onX = R under the following assumptions: (H ) α, β, γ , η : R+ → R+ is continuous, η is nondecreasing on R+, α(t) →∞, η(t) →∞, as t →∞and α(t), η(t), γ (t) ≤ t. Wang and Li Advances in Difference Equations (2016) 2016:186 Page 9 of 15 (H )Thefunctionf (t, x(α(t))) is continuous with respect to t on [, +∞) and there exists a continuous function Lf (t) such that f (t, ψ)–f (t, ψ) ≤ Lf (t)|ψ – ψ|, for ψ, ψ ∈ C(R+, R), ∗ | | ∞ where Lf = supt≥ Lf (t)<, limt→∞ Lf (t)=,andsupt≥ f (t,) < . (H )Thefunctiong(t, x(β(t))) is continuous with respect to t on [, +∞) and for every a >, g(t, ·):R → R is continuous for t ∈ [, a].ThesetV(r, g) is an equicontinuous subset of C(R+, R) and there exists a function ζ : R+ → R+ such that |g(t, x)|≤ζ (t) ˜ ∞ and supt∈[,a] ζ (t)=ζa < , for every a >. (H )ThefunctionK : R+ × R+ → R is continuous and |K(t, s)|≤k. (H ) For every a >,thefunctionh :[,a] × R → R satisfies the following conditions: (a) the function h(t, ·):R → R is continuous a.e. t ∈ [, a]; (b) the function h(·, x):[,a] → R is strongly measurable for every x ∈ R; · ∈ | |≤ | | (c) there exists ν( ) Lloc(R+) such that h(t, x) ν(t) x ,thefunction → ν(s) R s (t–s)–q belongs to L ([, t], +) and η(t) ν(s)e–δ(t–s) lim ζ (t) ds =. →∞ –q t (t – s) We choose δ >suchthat(.) holds, then we have the following result. Theorem . Assume that (H )-(H ) hold, then there exists a solution on the space Cδ (R) for equation (.). Moreover, the solutions of equation (.) are uniformly locally attractive (or, equivalently, the solutions are asymptotically stable). 3.2 Application to fractional differential equations Motivated by the proof of Theorem ., we can immediately obtain the global existence of a mild solution for the fractional differential equation as follows: cDq(x(t)–m(t, x(t))) = A(x(t)–m(t, x(t))) + h(t, x(t)), t >, t (.) x() = x, where < q <,m(, x()) = , A is the infinitesimal generator of an analytic semigroup of linear operators {T(t)}t≥ in X with T(t)≤M. It is well known that a function x ∈ C(R+, X)and t q– x(t)=Q(t)x + m t, x(t) + (t – s) R(t – s)h s, x(s) ds, t ≥ , (.) is the mild solution of (.), where ∞ q Q(t)= ξq(σ )T t σ dσ and Q(t) ≤ M, ∞ q M R(t)=q σξq(σ )T t σ dσ and R(t) ≤ , (q) Wang and Li Advances in Difference Equations (2016) 2016:186 Page 10 of 15 and ξq is a probability density function defined on (, ∞)(see[]) such that ∞ (nq) ξ (σ )= (–)n–σ n– sin(nπq) ≥ , σ ∈ (, ∞). q π (n –)! n= For more details, we refer to [, ]. Set f (t, x)=Q(t)x +m(t, x(t)), K(t–s)=R(t–s), we can study the existence of solution for equation (.)byTheorem..Tothisend,foreverya >,δ > , we make the following assumptions: (A) The function m(t, x(t)) is continuous with respect to t on [, +∞) and there exists a continuous function mf (t) such that m(t, ψ)–m(t, ψ) ≤ mf (t)ψ – ψ, for ψ, ψ ∈ C(R+, X), ∗ ∞ where mf = supt≥ mf (t)<, limt→∞ mf (t)=,andsupt≥ m(t,) < . (A) For every a >,thefunctionh :[,a] × X → X satisfies the following conditions: (a) the function h(t, ·):X → X is continuous a.e. t ∈ [, a]; (b) the function h(·, x):[,a] → X is strongly measurable for every x ∈ X; ν · ∈ ≤ν (c) there exists ( ) Lloc(R+) such that h(t, x) (t) x ,thefunction ν(s) t ν(s)e–δ(t–s) → R lim →∞ s (t–s)–q belongs to L ([, t], +) and t (t–s)–q ds =; { δs ∈ ∈ } (d) for every t >and r >,theset T(t – s)h(s, e z):s [, t), z Br(, Cδ (X)) is relatively compact in X. From the continuity of T(t) in the uniform operator topology, (A) and (A), it is easy to check that the assumptions in Theorem . hold. If we choose δ >suchthat t –δ(t–s) ∗ M ν(s)e m sup ds f + –q <, t≥ (q) (t – s) then we obtain the following result on the space Cδ (X). Theorem . Assume that (A) and (A) are satisfied, then there exists a global mild so- lution for (.) on the space Cδ (X). Moreover, the mild solutions of (.) are uniformly locally attractive (or equivalently, the solutions are asymptotically stable). Next, we consider the following fractional differential equation in X = R: cDqx(t)=h(t, x(t)), t >, t (.) x() = x. If h(t, x(t)) is continuous with respect to t on R+,fromRemark., t q– x(t)=x + (t – s) h s, x(s) ds (q) is equivalent to equation (.). For more details see [] and the references therein. For every a >,δ > , assume the function h :[,a] × R → R satisfies the following conditions: Wang and Li Advances in Difference Equations (2016) 2016:186 Page 11 of 15 (a) the function h(t, ·):R → R is continuous a.e. t ∈ [, a]; (b) the function h(·, x):[,a] → R is strongly measurable for every x ∈ R; (c) there exists ν(·) ∈ L (R ) such that |h(t, x)|≤ν(t)|x|,thefunctions → ν(s) loc + (t–s)–q t ν(s)e–δ(t–s) R lim →∞ belongs to L ([, t], +) and t (t–s)–q ds =. Using the results of Theorem ., and choosing δ such that t ν(s)e–δ(t–s) sup ds –q <, t≥ (q) (t – s) we can obtain without proof the following result on the space Cδ (R). Theorem . Assume that (a), (b), (c) are satisfied, then on the space Cδ (R), (.) has a global solution which is uniformly locally attractive (or, equivalently, the solution is asymp- totically stable). 4Examples As applications of our results, we study the following examples. Example . Let X = L([, π]), we consider the following fractional integral equation: t t sin(x(t, ξ)) s– cos(x(s, ξ)) ds x t ξ ( , )= + +t (t +) ( ) t –t–s s e h(s, x( , ξ)) × ≥ ds, t , (.) (t – s) t t –t–s where α(t)=β(t)=t, η(t)= , γ (t)= , K(t, s)=e , q = ,and π t t | s ξ | h t, x , ξ =(+t)– sin x , ξ e– x( , ) dξ. Observe that this equation has the form of equation (.)ifweput x(t)(ξ)=x(t, ξ), t sin(x(t)(ξ)) f t, x α(t) (ξ) = , +t t g t, x β(t) (ξ) = (t +)– s– cos x(s)(ξ) ds, π t | s ξ | h t, x γ (t) (ξ) =(+t)– sin x (ξ) e– x( )( ) dξ. We can easily see t f (t, ψ )–f (t, ψ ) ≤ ψ (t)(ξ)–ψ (t)(ξ) , ψ , ψ ∈ C(R , X), +t + g(t, x) ≤ (t +)– t , h(t, x) < πt– x. Wang and Li Advances in Difference Equations (2016) 2016:186 Page 12 of 15 t t s– lim →∞ ≤ → Obviously, t +t =and +t , the function s belongs to L ([, t], R+) (t–s) and, for any δ > , from Lemma .(), √ t – πt πδ – – –δ(t–s) ≤ → →∞ s (t – s) e ds , t . (t +) (t +) Now, we can see clearly that (H)-(H) in Theorem . hold for x ∈ C(R+, X). Let δ =,wehave √ t πt s– (t – s)– π + sup ds ≤ + sup ≈ . < , t–s t≥ e t≥ (t +) ( ) (t +) then equation (.)hasasolutiononthespaceC (X)byTheorem. and the solutions of (.) are uniformly locally attractive by Theorem .. Example . Let X = L([, π]), we consider the following fractional heat conduction equation: ⎧ ⎪ sin ξ sin ξ ξ ⎪ ∂ ξ t (v(t, )) ∂ ξ t (v(t, )) –t sin v(√t, ) ⎨ (v(t, )– (+t) )= ∂ξ (v(t, )– (+t) )+e ( ), t >, ∂t t ⎪v(t,)– t sin(v(t,)) = v(t, π)– t sin(v(t,π)) =, t >, (.) ⎩⎪ (+t) (+t) v(, ξ)=v. ∩ Totreattheaboveproblem,wedefineD(A)=H ([, π]) H([, π]), Au =–u .Theoper- ator –A is the infinitesimal generator of an analytic semigroup {T(t)}t≥ on X and {T(t)}t> is compact and T(t)≤. For ξ ∈ [, π], we set x(t)(ξ)=v(t, ξ), x(t)(ξ) h(t, x)(ξ)=e–t sin √ , t t sin(x(t)(ξ)) m(t, x)(ξ)= . ( + t) Then the above equation (.) can be reformulated as the abstract equation (.). Clearly, for t >,ψ, ψ ∈ C(R+, X), t m(t, ψ )–m(t, ψ ) ≤ ψ – ψ , ( + t) t t – lim →∞ ≤ ≤ –t where t (+t) =and +t .Moreover, h(t, x) t e x and the function – → s e–s s belongs to L ([, t], R+). Moreover, for any δ > , noting that Lemma .() (t–s) holds, we have √ t t s– e–se–δ(t–s) s– e–(δ–)(t–s) ( ) ds = ds ≤ √ → , t →∞. t (t – s) e (t – s) et t(δ –) Wang and Li Advances in Difference Equations (2016) 2016:186 Page 13 of 15 For δ ∈ (, ], by Lemma .(), we obtain t – –s –δ(t–s) t – –(–δ)s s e e s e ≤ t ds = δt ds δt B , (t – s) e (t – s) e → , t →∞.(.) t s– e–se–δ(t–s) →∞ Hence, for any δ >,limt ds = . Moreover, (A)(d) is ensured by the (t–s) compactness of {T(t)}t>. Now, (A) and (A) in Theorem . hold for x ∈ C(R+, X). Choosing δ =,from(.), we obtain t – –s –(t–s) ( ) s e e ≤ t ≈ + sup ds + sup . < . ≥ ≥ t t ( ) (t – s) ( ) t e Then the existence and uniformly local attractivity of the global mild solution of (.)on the space C (X) can be obtained by Theorem .. Example . Let X = R, we consider the following fractional differential equation: cD x(t)= (t– +)e–t sin(x(t)), t >, t (.) x() = . Clearly, if x(t) ∈ C(R+, R), the integral equation t h(s, x(s)) x(t)=+ ds ( ) (t – s) is the solution of (.), where h(t, x(t)) = (t– +)e–t sin(x(t)), by simple calculations we (s– +)e–s | |≤ – –t| | → see that h(t, x) (t +)e x and the function s belongs to L ([, t], R+). (t–s) Moreover, for any δ > , noting that Lemma .() holds, we have t t t (s– +)e–se–δ(t–s) s– e–(δ–)(t–s) e–(δ–)(t–s) ds ds ds = t + (t – s) e (t – s) (t – s) ≤ B , +t et → , t →∞.(.) For δ ∈ (, ], by Lemma .(), t t (s– +)e–se–δ(t–s) (s– +)e–(–δ)s ds = δt ds (t – s) e (t – s) ≤ B , +t eδt → , t →∞. Wang and Li Advances in Difference Equations (2016) 2016:186 Page 14 of 15 t (s– +)e–se–δ(t–s) → →∞ Hence, for any δ >, ds , t . It is now obvious that conditions (t–s) (a), (b), (c) in Theorem . hold for x ∈ C(R+, R). By choosing δ =,from(.), we have t – –s –(t–s) (s +)e e ≤ t ≈ sup ds + sup . < . ≥ ≥ t t ( ) (t – s) ( ) t e Then the existence and uniformly local attractivity of the solution of equation (.)onthe space C (X) can be obtained by Theorem .. 5Conclusions We discuss the existence and attractivity of global solutions for a class of nonlinear frac- tional quadratic integral equations in a Banach space X in this paper. The nonlinear fractional-order quadratic integral equation on an unbounded interval is difficult to solve. By employing some necessary restrictions on nonlinear terms and defining a Banach space Cδ (X), we obtain the existence and attractivity results of global solutions on Cδ (X), these strategies are different from that of [–]. As an application, we obtain the correspond- ing results for the global mild solutions of two classes of fractional differential equations. The above mentioned techniques are new even in the case of X = R. Competing interests The authors declare that they have no competing interests. Authors’ contributions Each of the authors, HW and FL, contributed to each part of this study equally and read and approved the final version of the manuscript. Acknowledgements The authors thank the referees for their careful reading of the manuscript and insightful comments, which helped to improve the quality of the paper. This work was supported partly by the NSF of China (11201413, 11471227, 11561077), the NSF of Yunnan Province (2013FB034). Received: 18 February 2016 Accepted: 26 June 2016 References 1. O’Regan, D, Meehan, MM: Existence Theory for Nonlinear Integral and Integrodifferential Equations. Kluwer Academic, Dordrecht (1998) 2. Corduneanu, C: Integral Equations and Applications. Cambridge University Press, Cambridge (1991) 3. Banas,´ J, Dhage, BC: Global asymptotic stability of solutions of a functional integral equation. Nonlinear Anal. 69, 949-1952 (2008) 4. 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