Let (X=C([0,1],R)) be the Banach space of real-valued continuous functions on ([0,1]) endowed with norm (| x | =max_{t in [0,1]}| x(t) | ).

Throughout this paper, we make the following assumption on the singularity of nonlinear function (f(t,x(t))) in (1.1):

  1. (H1)

    (f(t,x(t))) has a singularity at (t=0) and (t=1), that is,

    $$ lim_{tto 0^{+}} f(t,cdot )=infty ,qquad lim _{t to 1^{-}} f(t,cdot )=infty . $$

Moreover, there exist constants (0<theta _{1}<1) and (0<theta _{2}<1) such that (t^{theta _{1}}(1-t)^{theta _{2}}f(t,x(t))) is continuous on ([0,1]).

Based on condition (H1), we know that there is a positive constant (M_{0}) such that

$$ biglvert t^{theta _{1}}(1-t)^{theta _{2}}f bigl(t,x(t) bigr) bigrvert leq M_{0}, quad x in X,tin [0,1]. $$

(3.1)

Let (lambda =frac{3}{2+gamma ^{3}}). By Lemma 2.2 the operator (A:Xrightarrow X) can be represented as

$$begin{aligned} (Ax) (t) =&frac{1}{Gamma (alpha )} int _{0}^{t}(t-tau )^{alpha -1}f bigl( tau ,x(tau ) bigr),dtau +frac{lambda t^{2}}{Gamma (alpha +1)} int _{0}^{1}(1- tau )^{alpha }f bigl(tau ,x( tau ) bigr),dtau \ &{}-frac{lambda t^{2}}{Gamma (alpha )} int _{0}^{1}(1-tau )^{ alpha -1}f bigl(tau ,x( tau ) bigr),dtau \ &{}- frac{lambda t^{2}}{Gamma (alpha +1)} int _{0}^{gamma }(gamma – tau )^{alpha }f bigl( tau ,x(tau ) bigr),dtau . end{aligned}$$

(3.2)

Then the solutions of problem (1.1) include the FPs of A.

Lemma 3.1

Suppose (0<theta _{1}<1) and (0<theta _{2}<1). Then the integral operator J defined as

$$ J(t)= int _{0}^{t} (t-tau )^{alpha -1}tau ^{-theta _{1}}(1-tau )^{- theta _{2}},dtau ,quad tin [0,1] $$

has the following specifications:

  1. (1)

    (lim_{tto 0^{+}} J(t)=0);

  2. (2)

    (| J(t)-J(t_{0})| <(alpha -1)B(1-theta _{1},alpha – theta _{2}-1)| t-t_{0}|) for all (t,t_{0}in [0,1]),

where (B(cdot ,cdot )) denotes the beta function.

Proof

(1) By Lemma 2.6, for any (p_{1}>1), (p_{2}>1), (p_{3}>1) such that (frac{1}{p_{1}}+frac{1}{p_{2}}+frac{1}{p_{3}}=1), (0< p_{1}theta _{1}<1), and (0< p_{2}theta _{2}<1), we have

$$begin{aligned} J(t)leq{}& biggl[ int _{0}^{t}tau ^{-p_{1}theta _{1}},dtau biggr]^{1/p_{1}} biggl[ int _{0}^{t}(1-tau )^{-p_{2}theta _{2}},dtau biggr]^{1/p_{2}} bigg[ biggl[ int _{0}^{t}(t- tau )^{p_{3}(alpha -1)},dtau biggr]^{1/p_{3}} \ ={}& biggl[frac{1}{1-p_{1}theta _{1}}tau ^{1-p_{1}theta _{1}} bigg| _{0}^{t} biggr]^{1/p_{1}} biggl[- frac{1}{1-p_{2}theta _{2}}(1-tau )^{1-p_{2}theta _{2}}bigg| _{0}^{t} biggr]^{1/p_{2}} \ &{}cdot biggl[- frac{1}{1+p_{3}(alpha -1)}(t-tau )^{1+p_{3}(alpha -1)}bigg| _{0}^{t} biggr]^{1/p_{3}} \ ={}&frac{1}{sqrt[p_{1}]{1-p_{1}theta _{1}}} frac{1}{sqrt[p_{2}]{1-p_{2}theta _{2}}} frac{1}{sqrt[p_{3}]{1+p_{3}(alpha -1)}} sqrt[p_{1}]{t^{1-p_{1}theta _{1}}} \ &{}cdot sqrt[p_{2}]{1-(1-t)^{1-p_{2}theta _{2}}}cdot sqrt[p_{3}]{t^{1+p_{3}(alpha -1)}}. end{aligned}$$

Since (J(t)geq 0), and (lim_{tto 0^{+}}(sqrt[p_{1}]{t^{1-p_{1}theta _{1}}} cdot sqrt[p_{2}]{1-(1-t)^{1-p_{2}theta _{2}}}cdot sqrt[p_{3}]{t^{1+p_{3}(alpha -1)}})=0), we get

$$ lim_{tto 0^{+}} J(t)=0. $$

(2) By the expression of (J(t)) we easily get

$$begin{aligned} J'(t)& =(alpha -1) int _{0}^{t} (t-tau )^{alpha -2}tau ^{-theta _{1}}(1- tau )^{-theta _{2}},dtau \ & leq (alpha -1) int _{0}^{1} (1-tau )^{alpha -2}tau ^{-theta _{1}}(1- tau )^{-theta _{2}},dtau \ &=(alpha -1) int _{0}^{1} (1-tau )^{alpha -theta _{2}-2}tau ^{- theta _{1}},dtau \ & =(alpha -1)B(1-theta _{1},alpha -theta _{2}-1). end{aligned}$$

Hence the mean value theorem gives us

$$ biglvert J(t)-J(t_{0}) bigrvert =J'(xi ) vert t-t_{0} vert < (alpha -1)B(1- theta _{1},alpha – theta _{2}-1) vert t-t_{0} vert , $$

where the number ξ is between t and (t_{0}). □

Lemma 3.2

Let (2<alpha leq 3), and let (g:(0,1)rightarrow R) be a continuous function such that (lim_{tto 0^{+}} g(t)=infty ) and (lim_{tto 1^{-}} g(t)=infty ). Suppose that there exist two constants (0<theta _{1}<1) and (0<theta _{2}<1) such that (t^{theta _{1}}(1-t)^{theta _{2}}g(t)) is continuous in ([0,1]). Then the function

$$begin{aligned} G(t):={}&frac{1}{Gamma (alpha )} int _{0}^{t} (t-tau )^{alpha -1} g( tau ),d tau +frac{lambda t^{2}}{Gamma (alpha +1)} int _{0}^{1}(1- tau )^{alpha }g(tau ),dtau \ &{} -frac{lambda t^{2}}{Gamma (alpha )} int _{0}^{1}(1-tau )^{ alpha -1}g(tau ),dtau -frac{lambda t^{2}}{Gamma (alpha +1)} int _{0}^{gamma }(gamma -tau )^{alpha }g(tau ) ,dtau end{aligned}$$

is continuous in ([0,1]).

Proof

Based on the expression of (G(t)), we easily find (G(0)=0). As (t^{theta _{1}}(1-t)^{theta _{2}}g(t)) is continuous in ([0,1]), there is a positive constant (M_{1}) such that (| t^{theta _{1}}(1-t)^{theta _{2}}g(t) | leq M_{1}) for all (tin [0,1]). For all (t_{0} in [0,1]), we will prove the continuity of (G(t)) in three cases.

(a) (t_{0}=0), (t in [0,1]). We have

$$begin{aligned}& biglvert G(t)-G(0) bigrvert \& quad = bigglvert frac{1}{Gamma (alpha )} int _{0}^{t} (t- tau )^{alpha -1} tau ^{-theta _{1}}(1-tau )^{-theta _{2}} tau ^{ theta _{1}}(1-tau )^{theta _{2}} g(tau ),dtau \& qquad {}+frac{lambda t^{2}}{Gamma (alpha +1)} int _{0}^{1}(1-tau )^{ alpha } tau ^{-theta _{1}}(1-tau )^{-theta _{2}} tau ^{theta _{1}}(1- tau )^{theta _{2}} g(tau ),dtau \& qquad {}-frac{lambda t^{2}}{Gamma (alpha )} int _{0}^{1}(1-tau )^{ alpha -1} tau ^{-theta _{1}}(1-tau )^{-theta _{2}} tau ^{ theta _{1}}(1-tau )^{theta _{2}} g(tau ),dtau \& qquad {}-frac{lambda t^{2}}{Gamma (alpha +1)} int _{0}^{gamma }(gamma – tau )^{alpha } tau ^{-theta _{1}}(1-tau )^{-theta _{2}} tau ^{ theta _{1}}(1-tau )^{theta _{2}} g(tau ),dtau biggrvert \& quad leq frac{M_{1}}{Gamma (alpha )} int _{0}^{t} (t-tau )^{alpha -1} tau ^{-theta _{1}}(1-tau )^{-theta _{2}},dtau + frac{lambda M_{1}t^{2}}{Gamma (alpha +1)} int _{0}^{1}(1-tau )^{ alpha } tau ^{-theta _{1}}(1-tau )^{-theta _{2}},dtau \& qquad {} +frac{lambda M_{1} t^{2}}{Gamma (alpha )} int _{0}^{1}(1-tau )^{ alpha -1} tau ^{-theta _{1}}(1-tau )^{-theta _{2}},dtau + frac{lambda M_{1} t^{2}}{Gamma (alpha +1)} int _{0}^{gamma }( gamma -tau )^{alpha } tau ^{-theta _{1}}(1-tau )^{-theta _{2}},dtau \& quad leq frac{M_{1}}{Gamma (alpha )}J(t)+ frac{lambda M_{1}t^{2}}{Gamma (alpha +1)} int _{0}^{1}(1-tau )^{ alpha -theta _{2}} tau ^{-theta _{1}},dtau + frac{lambda M_{1} t^{2}}{Gamma (alpha )} int _{0}^{1}(1-tau )^{ alpha -theta _{2}-1} tau ^{-theta _{1}},dtau \& qquad {} +frac{lambda M_{1} t^{2}}{Gamma (alpha +1)} int _{0}^{1}(1- tau )^{alpha -theta _{2}} tau ^{-theta _{1}},dtau \& quad = frac{M_{1}}{Gamma (alpha )}J(t)+ frac{2lambda M_{1}t^{2}}{Gamma (alpha +1)}B(1-theta _{1}, alpha – theta _{2}+1) +frac{lambda M_{1} t^{2}}{Gamma (alpha )}B(1- theta _{1}, alpha – theta _{2} ) \& rightarrow{} 0quad (trightarrow t_{0}=0). end{aligned}$$

(b) (t_{0}in (0,1]), (tin [0,t_{0})). Then

$$begin{aligned}& biglvert G(t)-G(t_{0}) bigrvert \& quad = bigglvert frac{1}{Gamma (alpha )} int _{0}^{t_{0}} (t_{0}-tau )^{alpha -1} g(tau ),dtau – frac{1}{Gamma (alpha )} int _{0}^{t} (t-tau )^{alpha -1} g(tau ),d tau \& qquad {} +frac{lambda (t_{0}^{2}- t^{2})}{Gamma (alpha +1)} int _{0}^{1}(1- tau )^{alpha } g(tau ),d tau + frac{lambda (t_{0}^{2}- t^{2})}{Gamma (alpha )} int _{0}^{1}(1- tau )^{alpha -1} g(tau ),d tau \& qquad {} +frac{lambda (t_{0}^{2}- t^{2})}{Gamma (alpha +1)} int _{0}^{gamma }(gamma -tau )^{alpha } g(tau ) ,dtau biggrvert \& quad leq bigglvert frac{1}{Gamma (alpha )} int _{0}^{t} bigl[(t_{0}-tau )^{ alpha -1}-(t-tau )^{alpha -1} bigr] g(tau ),dtau + frac{1}{Gamma (alpha )} int _{t}^{t_{0}} (t_{0}-tau )^{alpha -1} g( tau ),dtau biggrvert \& qquad {} +frac{lambda M_{1}(t_{0}+ t)(t_{0}- t)}{Gamma (alpha +1)} int _{0}^{1}(1- tau )^{alpha -theta _{2}}tau ^{-theta _{1}},dtau \& qquad {}+ frac{lambda M_{1}(t_{0}+ t)(t_{0}- t)}{Gamma (alpha )} int _{0}^{1}(1- tau )^{alpha -theta _{2}}tau ^{-theta _{1}},dtau \& qquad {} +frac{lambda M_{1}(t_{0}+ t)(t_{0}- t)}{Gamma (alpha +1)} int _{0}^{ gamma }(gamma -tau )^{alpha }tau ^{-theta _{1}}(1-tau )^{- theta _{2}},dtau \& quad leq frac{M_{1}}{Gamma (alpha )} int _{0}^{t} bigl[(t_{0}-tau )^{ alpha -1}-(t-tau )^{alpha -1} bigr] tau ^{-theta _{1}}(1-tau )^{- theta _{2}},dtau \& qquad {} + frac{M_{1}}{Gamma (alpha )} int _{t}^{t_{0}} (t_{0}- tau )^{alpha -1} tau ^{-theta _{1}}(1-tau )^{-theta _{2}},dtau \& qquad {} +frac{2 lambda M_{1}(t_{0}- t)}{Gamma (alpha +1)} int _{0}^{1}(1- tau )^{alpha -theta _{2}}tau ^{-theta _{1}},dtau + frac{2 lambda M_{1}(t_{0}- t)}{Gamma (alpha )} int _{0}^{1}(1- tau )^{alpha -theta _{2}}tau ^{-theta _{1}},dtau \& qquad {} +frac{2lambda M_{1}(t_{0}- t)}{Gamma (alpha +1)} int _{0}^{1}(1- tau )^{alpha }tau ^{-theta _{1}}(1-tau )^{-theta _{2}},dtau \& quad = frac{M_{1}}{Gamma (alpha )} int _{0}^{t_{0}} (t_{0}-tau )^{ alpha -1} tau ^{-theta _{1}}(1-tau )^{-theta _{2}},dtau – frac{M_{1}}{Gamma (alpha )} int _{0}^{t} (t-tau )^{alpha -1} tau ^{-theta _{1}}(1-tau )^{-theta _{2}},dtau \& qquad {} +frac{2 lambda M_{1}(t_{0}- t)}{Gamma (alpha +1)} int _{0}^{1}(1- tau )^{alpha -theta _{2}}tau ^{-theta _{1}},dtau + frac{2 lambda M_{1}(t_{0}- t)}{Gamma (alpha )} int _{0}^{1}(1- tau )^{alpha -theta _{2}}tau ^{-theta _{1}},dtau \& qquad {} +frac{2lambda M_{1}(t_{0}- t)}{Gamma (alpha +1)} int _{0}^{1}(1- tau )^{alpha -theta _{2}}tau ^{-theta _{1}},dtau \& quad = frac{M_{1}}{Gamma (alpha )} bigl[J(t_{0})-J(t) bigr]+ frac{4 lambda M_{1}(t_{0}- t)}{Gamma (alpha +1)}B(1-theta _{1}, alpha -theta _{2}+1) \& qquad {}+frac{2lambda M_{1}(t_{0}- t)}{Gamma (alpha )}B(1-theta _{1}, alpha -theta _{2}+1). end{aligned}$$

By the second result of Lemma 3.1 we have

$$ biglvert G(t)-G(t_{0}) bigrvert leq M_{1} frac{alpha (alpha -1)+4lambda +2lambda alpha }{Gamma (alpha +1)} cdot B(1-theta _{1},alpha -theta _{2}+1) (t_{0}-t)rightarrow 0(t rightarrow t_{0}). $$

(c) (t_{0}in (0,1)), (tin (t_{0},1]). Since the proof for this case is the same as that in case (b), we omit it. □

Lemma 3.3

Let (2<alpha leq 3), and let (f:(0,1)times Rrightarrow R) be a continuous function satisfying the singularity condition (H1). Then the operator (A:Xrightarrow X) is completely continuous.

Proof

According to Lemma 3.2, (A:Xrightarrow X) is continuous. Let (Dsubset X=C([0,1],R)) be a bounded set, that is, there is a positive constant (L_{1}) such that (| x| leq L_{1} ) for all (xin D).

Relations (3.1) and (3.2) give

$$begin{aligned} vert Ax vert leq& frac{1}{Gamma (alpha )} bigglvert int _{0}^{t}(t- tau )^{alpha -1}f bigl(tau ,x(tau ) bigr),dtau biggrvert + frac{lambda }{Gamma (alpha +1)} bigglvert int _{0}^{1}(1-tau )^{ alpha }f bigl(tau ,x( tau ) bigr),dtau biggrvert \ &{}+frac{lambda }{Gamma (alpha )} bigglvert int _{0}^{1}(1-tau )^{ alpha -1}f bigl(tau ,x( tau ) bigr),dtau biggrvert + frac{lambda }{Gamma (alpha +1)} bigglvert int _{0}^{gamma }(gamma – tau )^{alpha }f bigl( tau ,x(tau ) bigr),dtau biggrvert \ leq& frac{M_{0}}{Gamma (alpha )} int _{0}^{t} (t-tau )^{alpha -1} tau ^{-theta _{1}}(1-tau )^{-theta _{2}},dtau + frac{lambda M_{0}}{Gamma (alpha +1)} int _{0}^{1} (1-tau )^{ alpha } tau ^{-theta _{1}}(1-tau )^{-theta _{2}},dtau \ &{} +frac{lambda M_{0}}{Gamma (alpha )} int _{0}^{1}(1-tau )^{ alpha -1}tau ^{-theta _{1}} (1-tau )^{-theta _{2}},dtau \ &{} + frac{ lambda M_{0}}{Gamma (alpha +1)} int _{0}^{gamma }(gamma – tau )^{alpha }tau ^{-theta _{1}} (1-tau )^{-theta _{2}},dtau \ leq& frac{M_{0}}{Gamma (alpha )} int _{0}^{1} (1-tau )^{alpha – theta _{2}-1} tau ^{-theta _{1}},dtau + frac{lambda M_{0}}{Gamma (alpha +1)} int _{0}^{1} (1-tau )^{ alpha -theta _{2}} tau ^{-theta _{1}},dtau \ &{} +frac{lambda M_{0}}{Gamma (alpha )} int _{0}^{1}(1-tau )^{ alpha -theta _{2}-1}tau ^{-theta _{1}},dtau + frac{ lambda M_{0}}{Gamma (alpha +1)} int _{0}^{1}(1-tau )^{ alpha -theta _{2}}tau ^{-theta _{1}},dtau \ =& frac{(1+lambda )M_{0}}{Gamma (alpha )}B(1-theta _{1},alpha – theta _{2})+ frac{2lambda M_{0}}{Gamma (alpha +1)}B(1-theta _{1}, alpha -theta _{2}+1):=L_{2}, end{aligned}$$

that is, (Vert Ax Vert leq L_{2}), for all (xin D). Thus the operator A is bounded on D. This yields the compactness of A. For every (tin [0,1]), we have

$$begin{aligned} biglvert (Ax)'(t) bigrvert ={}& bigglvert frac{1}{Gamma (alpha -1)} int _{0}^{t}(t- tau )^{alpha -2}f bigl(tau ,x(tau ) bigr),dtau + frac{2lambda t}{Gamma (alpha +1)} int _{0}^{1}(1-tau )^{alpha }f bigl( tau ,x( tau ) bigr),dtau \ &{}-frac{2lambda t}{Gamma (alpha )} int _{0}^{1}(1-tau )^{alpha -1}f bigl( tau ,x(tau ) bigr),dtau -frac{2lambda t}{Gamma (alpha +1)} int _{0}^{gamma }(gamma -tau )^{alpha }f bigl( tau ,x(tau ) bigr),dtau biggrvert \ leq{}& frac{M_{0}}{Gamma (alpha -1)} int _{0}^{t} (t-tau )^{ alpha -2} tau ^{-theta _{1}}(1-tau )^{-theta _{2}},dtau \ &{}+ frac{2lambda M_{0}}{Gamma (alpha +1)} int _{0}^{1} (1-tau )^{ alpha } tau ^{-theta _{1}}(1-tau )^{-theta _{2}},dtau \ & {}+frac{2lambda M_{0}}{Gamma (alpha )} int _{0}^{1}(1-tau )^{ alpha -1}tau ^{-theta _{1}} (1-tau )^{-theta _{2}},dtau \ &{}+ frac{ 2lambda M_{0}}{Gamma (alpha +1)} int _{0}^{gamma }(gamma – tau )^{alpha }tau ^{-theta _{1}} (1-tau )^{-theta _{2}},dtau \ leq{}& frac{M_{0}}{Gamma (alpha -1)} int _{0}^{1} (1-tau )^{ alpha -theta _{2}-2} tau ^{-theta _{1}},dtau + frac{2lambda M_{0}}{Gamma (alpha +1)} int _{0}^{1} (1-tau )^{ alpha -theta _{2}} tau ^{-theta _{1}},dtau \ & {}+frac{2lambda M_{0}}{Gamma (alpha )} int _{0}^{1}(1-tau )^{ alpha -theta _{2}-1}tau ^{-theta _{1}},dtau + frac{ 2lambda M_{0}}{Gamma (alpha +1)} int _{0}^{1}(1-tau )^{ alpha -theta _{2}}tau ^{-theta _{1}},dtau \ ={}& frac{M_{0}}{Gamma (alpha -1)}B(1-theta _{1},alpha -theta _{2}-1)+ frac{4lambda M_{0}}{Gamma (alpha +1)}B(1-theta _{1},alpha – theta _{2}+1) \ & {}+ frac{2lambda M_{0}}{Gamma (alpha )}B(1-theta _{1},alpha – theta _{2}):=L_{3}. end{aligned}$$

Now the following inequality holds for (t_{1},t_{2}in [0,1]) and (t_{1}< t_{2}):

$$ biglvert (Ax) (t_{2})-(Ax) (t_{1}) bigrvert = bigglvert int _{t_{1}}^{t_{2}} (Ax)'(s),ds biggrvert leq L_{3}(t_{2}-t_{1}). $$

Therefore A is equicontinuous on D. Thus, by the Arzelà–Ascoli theorem the operator A is completely continuous on X. □

Now we present and demonstrate our fundamental results. The first result deals with the existence and uniqueness of the solution to problem (1.1).

Theorem 3.1

Let (2<alpha leq 3 ) and (0<theta _{1}), (theta _{2}<1) be constants, and let (f(t,x(t))) satisfy condition (H1) and the following conditions:

  1. (H2)

    There is a function (m(t)in L^{p}([0,1],R^{+})) ((p>1)) such that

    $$ t^{theta _{1}}(1-t)^{theta _{2}} biglvert f(t,x)-f(t,y) bigrvert leq m(t) vert x-y vert . $$

  2. (H3)

    There exist three constants (p_{1}), (p_{2}), (p_{3}) satisfying (p_{1}>1), (p_{2}>1), (p_{3}>1), (0< p_{1}theta _{1}<1), and (frac{1}{p_{1}}+frac{1}{p_{2}}+frac{1}{p_{3}}=1). If

    $$begin{aligned}& Vert m Vert _{p_{3}} frac{1}{sqrt[p_{1}]{1-p_{1}theta _{1}}}[ frac{1+lambda }{Gamma (alpha )} frac{1}{sqrt[p_{2}]{1+p_{2}(alpha -theta _{2}-1)}} \& quad {}+ frac{2lambda }{Gamma (alpha +1)} frac{1}{sqrt[p_{2}]{1+p_{2}(alpha -theta _{2})}} < 1, end{aligned}$$

    (3.3)

then the solution to problem (1.1) is unique.

Proof

For (x,y in X=C([0,1])) and (tin [0,1]), by (H2) we have

$$begin{aligned} biglvert (Ax) (t)-(Ay) (t) bigrvert leq {}&frac{1}{Gamma (alpha )} int _{0}^{t}(t- tau )^{alpha -1} biglvert f bigl(tau ,x(tau ) bigr)-f bigl(tau ,y(tau ) bigr) bigrvert ,dtau \ &{}+frac{lambda }{Gamma (alpha +1)} int _{0}^{1}(1-tau )^{alpha } biglvert f bigl(tau ,x(tau ) bigr)-f bigl(tau ,y(tau ) bigr) bigrvert ,dtau \ &{} +frac{lambda }{Gamma (alpha )} int _{0}^{1}(1-tau )^{alpha -1} biglvert f bigl(tau ,x(tau ) bigr)-f bigl(tau ,y(tau ) bigr) bigrvert ,dtau \ & {}+frac{lambda }{Gamma (alpha +1)}| int _{0}^{gamma }(gamma – tau )^{alpha } biglvert f bigl(tau ,x(tau ) bigr)-f bigl(tau ,y(tau ) bigr) bigrvert ,d tau \ leq{}& frac{1}{Gamma (alpha )} int _{0}^{1}(1-tau )^{alpha -{ theta _{2}}-1}tau ^{-theta _{1}}m(tau ) biglvert x(tau )-y(tau ) bigrvert ,dtau \ &{}+frac{lambda }{Gamma (alpha +1)} int _{0}^{1}(1-tau )^{alpha -{ theta _{2}}}tau ^{-theta _{1}}m(tau ) biglvert x(tau )-y(tau ) bigrvert ,d tau \ &{}+frac{lambda }{Gamma (alpha )} int _{0}^{1}(1-tau )^{alpha -{ theta _{2}}-1}tau ^{-theta _{1}}m(tau ) biglvert x(tau )-y(tau ) bigrvert ,dtau \ &{}+frac{lambda }{Gamma (alpha +1)} int _{0}^{1}(1-tau )^{alpha -{ theta _{2}}}tau ^{-theta _{1}}m(tau ) biglvert x(tau )-y(tau ) bigrvert ,d tau . end{aligned}$$

By (H3) and the Hölder inequality we have

$$begin{aligned}& biglvert (Ax) (t)-(Ay) (t) bigrvert \& quad leq biglVert x(tau )-y(tau ) bigrVert cdot biggl{ frac{1}{Gamma (alpha )} biggl[ int _{0}^{1} tau ^{- theta _{1}p_{1}},dtau biggr]^{1/p_{1}} biggl[ int _{0}^{1}(1-tau )^{(alpha -{ theta _{2}}-1)p_{2}},dtau biggr]^{1/p_{2}} \& qquad {}times biggl[ int _{0}^{1} m^{p_{3}}(tau ),dtau biggr]^{1/p_{3}} \& qquad {} + frac{lambda }{Gamma (alpha +1)} biggl[ int _{0}^{1} tau ^{-theta _{1}p_{1}},dtau biggr]^{1/p_{1}} biggl[ int _{0}^{1}(1-tau )^{(alpha -{theta _{2}})p_{2}},dtau biggr]^{1/p_{2}} biggl[ int _{0}^{1} m^{p_{3}}(tau ),dtau biggr]^{1/p_{3}} \& qquad {} + frac{lambda }{Gamma (alpha )} biggl[ int _{0}^{1} tau ^{-theta _{1}p_{1}},dtau biggr]^{1/p_{1}} biggl[ int _{0}^{1}(1-tau )^{(alpha -{theta _{2}}-1)p_{2}},dtau biggr]^{1/p_{2}} biggl[ int _{0}^{1} m^{p_{3}}(tau ),dtau biggr]^{1/p_{3}} \& qquad {} + frac{lambda }{Gamma (alpha +1)} biggl[ int _{0}^{1} tau ^{-theta _{1}p_{1}},dtau biggr]^{1/p_{1}} biggl[ int _{0}^{1}(1-tau )^{(alpha -{theta _{2}})p_{2}},dtau biggr]^{1/p_{2}} biggl[ int _{0}^{1} m^{p_{3}}(tau ),dtau biggr]^{1/p_{3}} biggr} \& quad = Vert m Vert _{p_{3}} frac{1}{sqrt[p_{1}]{1-p_{1}theta _{1}}} biggl[ frac{1+lambda }{Gamma (alpha )} frac{1}{sqrt[p_{2}]{1+p_{2}(alpha -theta _{2}-1)}} + frac{2lambda }{Gamma (alpha +1)} frac{1}{sqrt[p_{2}]{1+p_{2}(alpha -theta _{2})}} biggr] \& qquad {}times biglVert x( tau )-y(tau ) bigrVert . end{aligned}$$

Noticing (3.3), we conclude that A is a contraction mapping. Thus by Lemma 2.3 it has a unique FP, which is also the unique solution to problem (1.1). □

The second result states the existence of the solution to the BVP (1.1) derived from Lemma 2.4.

Theorem 3.2

Let (2<alpha leq 3 ) and (0<theta _{1}), (theta _{2}<1) be constants, and let (f(t,x(t))) satisfy conditions (H1)(H3) and the following condition:

$$begin{aligned}& Vert m Vert _{p_{3}} frac{1}{sqrt[p_{1}]{1-p_{1}theta _{1}}} biggl[ frac{lambda }{Gamma (alpha )} frac{1}{sqrt[p_{2}]{1+p_{2}(alpha -theta _{2}-1)}} \& quad {}+ frac{2lambda }{Gamma (alpha +1)} frac{1}{sqrt[p_{2}]{1+p_{2}(alpha -theta _{2})}} biggr]< 1. end{aligned}$$

(3.4)

Then problem (1.1) has a solution.

Proof

We fix a constant

$$ r geq M_{0} biggl[frac{1+lambda }{Gamma (alpha )}B(1-theta _{1}, alpha -theta _{2})+frac{2lambda }{Gamma (alpha +1)}B(1-theta _{1}, alpha -theta _{2}+1) biggr]. $$

Consider a ball (B_{r} ={xin X=C([0,1],R):Vert x Vert leq r}). Define two operators (A_{1}) and (A_{1}) on (B_{r}) as

$$begin{aligned}& (A_{1}x) (t)=frac{1}{Gamma (alpha )} int _{0}^{t}(t-tau )^{alpha -1}f bigl( tau ,x(tau ) bigr),dtau , \& (A_{2}x) (t)=frac{lambda t^{2}}{Gamma (alpha +1)} int _{0}^{1}(1- tau )^{alpha }f bigl(tau ,x( tau ) bigr),dtau \& hphantom{(A_{1}x) (t)=}{}- frac{lambda t^{2}}{Gamma (alpha )} int _{0}^{1}(1-tau )^{alpha -1}f bigl( tau ,x(tau ) bigr),dtau \& hphantom{(A_{2}x) (t)=}{}-frac{lambda t^{2}}{Gamma (alpha +1)} int _{0}^{gamma }(gamma -tau )^{alpha }f bigl( tau ,x(tau ) bigr),dtau . end{aligned}$$

For (x,yin B_{r} ), by (3.1) we can check that

$$begin{aligned} Vert A_{1}x+A_{2}y Vert leq{}& frac{M_{0}}{Gamma (alpha )} int _{0}^{1} (1-tau )^{alpha – theta _{2}-1} tau ^{-theta _{1}},dtau + frac{lambda M_{0}}{Gamma (alpha +1)} int _{0}^{1} (1-tau )^{ alpha -theta _{2}} tau ^{-theta _{1}},dtau \ & {}+frac{lambda M_{0}}{Gamma (alpha )} int _{0}^{1}(1-tau )^{ alpha -theta _{2}-1}tau ^{-theta _{1}},dtau + frac{ lambda M_{0}}{Gamma (alpha +1)} int _{0}^{1}(1-tau )^{ alpha -theta _{2}}tau ^{-theta _{1}},dtau \ ={}&M_{0} biggl[ frac{(1+lambda )}{Gamma (alpha )}B(1-theta _{1}, alpha – theta _{2})+ frac{2lambda }{Gamma (alpha +1)}B(1-theta _{1}, alpha – theta _{2}+1) biggr] \ leq{}& r. end{aligned}$$

So (A_{1}x+A_{2}y in B_{r}). Like in the proof of Theorem 3.1, from (H2), (H3), and (3.4) we can conclude that the operator (A_{2}) is also a contraction mapping. Lemma 3.2 and (H1) ensure the continuity of the operator (A_{1}). For any (x in B_{r}), we have

$$begin{aligned} Vert A_{1}x Vert & leq frac{M_{0}}{Gamma (alpha )} int _{0}^{1} (1-tau )^{alpha -1} tau ^{-theta _{1}}(1-tau )^{-}theta _{2},dtau \ &leq frac{M_{0}}{Gamma (alpha )} int _{0}^{1} (1-tau )^{ alpha -theta _{2}-1} tau ^{-theta _{1}},dtau \ & = frac{M_{0}}{Gamma (alpha )}B(1-theta _{1},alpha -theta _{2}). end{aligned}$$

Thus (A_{1}) is uniformly bounded on (B_{r}). For all (t_{1},t_{2} in [0,1]) such that (t_{1}< t_{2} ), we obtain

$$begin{aligned}& biglvert (A_{1}x) (t_{2})-(A_{1}x) (t_{1}) bigrvert \& quad = frac{1}{Gamma (alpha )} bigglvert int _{0}^{t_{2}}(t_{2}-tau )^{ alpha -1}f bigl(tau ,x(tau ) bigr),dtau – int _{0}^{t_{1}}(t_{1}-tau )^{ alpha -1}f bigl(tau ,x(tau ) bigr),dtau biggrvert \& quad leq frac{1}{Gamma (alpha )} bigglvert int _{0}^{t_{1}} bigl[(t_{2}-tau )^{ alpha -1}-(t_{1}-tau )^{alpha -1} bigr]f bigl(tau ,x( tau ) bigr),dtau + int _{t_{1}}^{t_{2}}(t_{2}- tau )^{alpha -1}f bigl(tau ,x(tau ) bigr),dtau biggrvert \& quad leq frac{M_{0}}{Gamma (alpha )} biggl[ int _{0}^{t_{1}} bigl[(t_{2}- tau )^{alpha -1}-(t_{1}-tau )^{alpha -1} bigr]tau ^{-theta _{1}}(1- tau )^{-theta _{2}},dtau \& qquad {} + int _{t_{1}}^{t_{2}}(t_{2}-tau )^{ alpha -1}tau ^{-theta _{1}}(1-tau )^{-theta _{2}},dtau biggr] \& quad = frac{M_{0}}{Gamma (alpha )} bigl[J(t_{2})-J(t_{1}) bigr]. end{aligned}$$

By Lemma 3.1 we have

$$ biglvert (A_{1}x) (t_{2})-(A_{1}x) (t_{1}) bigrvert = frac{(alpha -1)M_{0}}{Gamma (alpha )}B(1-theta _{1}, alpha – theta _{2}-1) (t_{2}-t_{1}). $$

This means that (A_{1} ) is equicontinuous and relatively compact on (B_{r} ). Accordingly, by the Arzelà–Ascoli theorem (A_{1} ) is compact on (B_{r} ). Accordingly, Lemma 2.4 ensures the existence of a solution for problem (1.1) in ([0,1] ). □

The Schaefer fixed point theorem gives the last result.

Theorem 3.3

Let (2<alpha leq 3) and (0<theta _{1}), (theta _{2}<1) be constants, and let (f(t,x(t))) satisfy conditions (H1) and (3.1). Then problem (1.1) has a solution in ([0,1] ).

Proof

By Lemma 3.3 we know that the operator (A:Xrightarrow X) is completely continuous.

Next, we prove that the set (V={x in C([0,1],R):x=mu Ax,0< mu <1}) is bounded.

Let (x in V). Then (x=mu (Ax)). Thus, for each (tin [0,1]), we have

$$begin{aligned} vert x vert ={}& mu biglvert (Ax) (t) bigrvert \ ={}&mu bigglvert frac{1}{Gamma (alpha )} int _{0}^{t}(t-tau )^{alpha -1}f bigl(tau ,x( tau ) bigr),dtau +frac{lambda t^{2}}{Gamma (alpha +1)} int _{0}^{1}(1- tau )^{alpha }f bigl(tau ,x( tau ) bigr),dtau \ &{} -frac{lambda t^{2}}{Gamma (alpha )} int _{0}^{t}(1-tau )^{ alpha -1}f bigl(tau ,x(tau ) bigr),dtau – frac{lambda t^{2}}{Gamma (alpha +1)} int _{0}^{gamma }(gamma – tau )^{alpha }f bigl( tau ,x(tau ) bigr),dtau biggrvert \ leq{}& M_{0} biggl[frac{1}{Gamma (alpha )} int _{0}^{1}(1-tau )^{ alpha -theta _{2}-1}tau ^{-theta _{1}},dtau + frac{lambda }{Gamma (alpha +1)} int _{0}^{1}(1-tau )^{alpha – theta _{2}}tau ^{-theta _{1}},dtau biggr] \ &{}+frac{lambda }{Gamma (alpha )} int _{0}^{1}(1-tau )^{alpha – theta _{2}-1}tau ^{-theta _{1}},dtau + frac{lambda }{Gamma (alpha +1)} int _{0}^{1}(1-tau )^{alpha – theta _{2}}tau ^{-theta _{1}},dtau ] \ ={} &M_{0} biggl[frac{1+lambda }{Gamma (alpha )}B(1-theta _{1},alpha – theta _{2})+frac{2lambda }{Gamma (alpha +1)}B(1-theta _{1}, alpha – theta _{2}+1) biggr]=L_{2}. end{aligned}$$

Hence we have

$$ Vert x Vert leq L_{2}. $$

This shows that the set V is bounded. Lemma 2.5 ensures the existence of fixed points of A. Accordingly, there is at least one solution to problem (1.1) in ([0,1]). □

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