Now we establish existence result of positive, concave, and pseudosymmetric solutions and its monotone iterative scheme for BVP (9), (10).

Proof.
Let
(48)w0(t)={22at,0≤t≤1+η2,22a(1+η-t),1+η2≤t≤1,w1(t)={∫0t∫τ(1+η)/2ϕp-1(∫s(1+η)/2q(r) ×f(r,w0(r),w0′(r))dr∫s(1+η)/2)ds dτ, 0≤t≤1+η2,∫0η∫τ(1+η)/2ϕp-1(∫s(1+η)/2q(r) ×f(r,w0(r),w0′(r))dr∫s(1+η)/2)ds dτ, -∫t1∫(1+η)/2τϕp-1(∫(1+η)/2sq(r) ×f(r,w0(r),w0′(r))dr∫s(1+η)/2)ds dτ, 1+η2≤t≤1.
Then w1(t)∈C1[0,(1+η)/2]∩C1[(1+η)/2,1].

Next we prove that
(49)limt→((1+η)/2)-w1(t)=limt→((1+η)/2)+w1(t),(50)limt→((1+η)/2)-w1′(t)=limt→((1+η)/2)+w1′(t).
In fact, from (H2) it follows that
(51)limt→((1+η)/2)+w1(t)=∫0η∫τ(1+η)/2ϕp-1 ×(∫s(1+η)/2q(r)f(r,22ar,22a)dr)ds dτ -∫(1+η)/21∫(1+η)/2τϕp-1 ×(∫(1+η)/2sq(r)f(r,22a(1+η-r),-22a)dr)ds dτ=(∫0(1+η)/2+∫(1+η)/2η) ×∫τ(1+η)/2ϕp-1(∫s(1+η)/2q(r)f(r,22ar,22a)dr)ds dτ -∫(1+η)/21∫(1+η)/2τϕp-1 ×(∫(1+η)/2sq(r) ×f(r,22a(1+η-r),-22a)dr)ds dτ=∫0(1+η)/2∫τ(1+η)/2ϕp-1 ×(∫s(1+η)/2q(r)f(r,22ar,22a)dr)ds dτ +∫(1+η)/2η∫τ(1+η)/2ϕp-1 ×(∫s(1+η)/2q(r)f(r,22ar,22a)dr)ds dτ -∫(1+η)/21∫(1+η)/2τϕp-1 ×(∫(1+η)/2sq(r) ×f(r,22a(1+η-r),-22a)dr)ds dτ=∫0(1+η)/2∫τ(1+η)/2ϕp-1 ×(∫s(1+η)/2q(r)f(r,22ar,22a)dr)ds dτ=limt→((1+η)/2)-w1(t).
Then (49) holds. Equation (50) can be obtained in a similar way. Thus from (49) and (50), it follows that
(52)w1(t)∈C1[0,1].
We note that for t∈[0,(1+η)/2],
(53)0≤w1(t)=∫0t∫τ(1+η)/2ϕp-1 ×(∫s(1+η)/2q(r)f(r,22ar,22a)dr)ds dτ≤∫0t∫τ(1+η)/2ϕp-1(∫s(1+η)/2q(r)ϕp(aA)dr)ds dτ≤aA∫0t∫τ(1+η)/2ϕp-1(∫0(1+η)/2q(r)dr)ds dτ≤aAA2t=w0(t),
and for t∈[(1+η)/2,1],
(54)w1(t)=∫0η∫τ(1+η)/2ϕp-1 ×(∫s(1+η)/2q(r)f(r,22ar,22a)dr)ds dτ -∫t1∫(1+η)/2τϕp-1 ×(∫(1+η)/2sq(r) ×f(r,22a(1+η-r),-22a)dr)ds dτ≤∫0η∫τ(1+η)/2ϕp-1(∫s(1+η)/2q(r)ϕp(aA)dr)ds dτ +∫t1∫(1+η)/2τϕp-1(∫(1+η)/2sq(r)ϕp(aA)dr)ds dτ≤aA∫0η∫τ(1+η)/2ϕp-1(∫0(1+η)/2q(r)dr)ds dτ +aA∫t1∫(1+η)/2τϕp-1(∫(1+η)/21q(r)dr)ds dτ≤aA∫0η∫τ(1+η)/2A2ds dτ +aA∫t1∫(1+η)/2τϕp-1(∫0(1+η)/2q(r)dr)ds dτ≤aA∫0ηA2dτ+aA∫t1∫(1+η)/2τA2ds dτ≤aAA2η+aAA2(1-t)=w0(t).
So,
(55)w1(t)≤w0(t)≤22a, t∈[0,1].
Thus,
(56)α(w1):=max0≤t≤1|w1(t)|≤22a.
From assumptions, for t∈[0,(1+η)/2],
(57)|w1′(t)|=∫t(1+η)/2ϕp-1 ×(∫s(1+η)/2q(r)f(r,22ar,22a)dr)ds≤∫t(1+η)/2ϕp-1(∫s(1+η)/2q(r)ϕp(aA)dr)ds≤aA∫t(1+η)/2ϕp-1(∫0(1+η)/2q(r)dr)ds≤aAA2≤22a=|w0′(t)|,
and for t∈[(1+η)/2,1],
(58)|w1′(t)|=|∫(1+η)/2tϕp-1(∫(1+η)/2sq(r) ×f(r,22a(1+η-r),-22a)dr)ds|≤∫(1+η)/2tϕp-1(∫(1+η)/21q(r)ϕp(aA)dr)ds≤aA∫(1+η)/2tϕp-1(∫0(1+η)/2q(r)dr)ds≤aAA2≤22a=|w0′(t)|.
Hence from (57) and (58), we have
(59)β(w1):=max0≤t≤1|w1′(t)|≤22a.
Consequently, from (56) and (59), it follows that
(60)∥w1∥≤2max{α(w1),β(w1)}≤a.
From the proof of Lemma 3, we see that w1 is nonnegative, concave, and pseudosymmetric about η on [0,1], and hence
(61)w1∈P¯a.
Define {wn} as follows:
(62)wn+1=Twn=Tnw1=Tn+1w0, n=0,1,….
Then {wn} is well defined and for n=1, 2, …,
(63)wn+1(t)≤wn(t), |wn+1′(t)|≤|wn′(t)|, t∈[0,1].
In fact, for t∈[0,(1+η)/2],
(64)w2(t)=Tw1(t)=∫0t∫τ(1+η)/2ϕp-1 ×(∫s(1+η)/2q(r) ×f(r,w1(r),w1′(r))dr∫s(1+η)/2)ds dτ≤∫0t∫τ(1+η)/2ϕp-1 ×(∫s(1+η)/2q(r) ×f(r,22ar,22a)dr∫s(1+η)/2)ds dτ=w1(t),(65)|w2′(t)|=|Tw1′(t)|=∫t(1+η)/2ϕp-1 ×(∫s(1+η)/2q(r) ×f(r,w1(r),w1′(r))dr∫s(1+η)/2)ds ≤∫t(1+η)/2ϕp-1 ×(∫s(1+η)/2q(r) ×f(r,22ar,22a)dr∫s(1+η)/2)ds=w1′(t).
For t∈[(1+η)/2,1], since w1, w2∈P¯a, it follows from (64) and (65) that
(66)w2(t)=w2(1+η-t)≤w1(1+η-t)=w1(t),(67)|w2′(t)|=|w2′(1+η-t)|≤|w1′(1+η-t)|=|w1′(t)|.
So from (64)–(67), we have
(68)w2(t)≤w1(t), |w2′(t)|≤|w1′(t)|, t∈[0,1],
that is, (63) holds when n=1. Assume that (63) holds when n=k, that is,
(69)wk+1(t)≤wk(t), |wk+1′(t)|≤|wk′(t)|, t∈[0,1].
Then from Lemma 5, we obtain
(70)wk+2(t)=(Twk+1)(t)≤(Twk)(t)=wk+1(t), t∈[0,1],|wk+2′(t)|=|(Twk+1)′(t)|≤|(Twk)′(t)|=|wk+1′(t)|, t∈[0,1].
So by induction (63) holds.

Since T:P¯a→P¯a is completely continuous, it follows that {wn}n=1∞ is relative compact. Then {wn} has a convergent subsequence {wnk} and w*∈P¯a such that
(71)wnk⟶w* (k⟶∞),
that is,
(72)wnk(t)⇉w*(t) (k⟶∞),wnk′(t)⇉w*′(t) (k⟶∞) on [0,1].
While from (63) and the fact for each n=1,2,…, wn′((1+η)/2)=0 and wn′′(t)≤0 on [0,1], it follows that
(73)w1(t)≥w2(t)≥⋯≥wn(t)≥wn+1(t)≥⋯, n=1,2,…, on [0,1],w1′(t)≥w2′(t)≥⋯≥wn′(t)≥wn+1′(t)≥⋯, n=1,2,…, on [0,1+η2],w1′(t)≤w2′(t)≤⋯≤wn′(t)≤wn+1′(t)≤⋯, n=1,2,…, on [1+η2,1].
Hence,
(74)wn(t)⇉w*(t) (n⟶∞),wn′(t)⇉w*′(t) (n⟶∞) on [0,1],
that is,
(75)wn⟶w*(n⟶∞).
This together with the continuity of T and wn+1=Twn, implies that
(76)Tw*=w*.
Also, since
(77)0≤wn(t)≤w0(t) ={22at,0≤t≤1+η2,22a(1+η-t),1+η2≤t≤1,0≤|wn′(t)|≤|w1′(t)|≤22a, t∈[0,1],
we have
(78)0≤w*(t)≤22a, 0≤|w*′(t)|≤22a, t∈[0,1].
Furthermore, we have
(79)w*(t)>0, t∈(0,1].
In fact, from (H3) and w*(t)≢0 on [0,1], we have w*((1+η)/2)>0. Since w*(t) is concave on [0,1], then
(80)w*(t)≥w*((1+η)/2) -0((1+η)/2)-0t=21+ηw*(1+η2)t>0, t∈(0,1+η2].
Consequently from the fact w* is pseudosymmetric on [0,1], we have
(81)w*(t)>0, t∈(0,1].
Let v0(t)≡0 on [0,1], then v0∈P¯a. Set vn+1=Tvn, n=0, 1, 2, …. Then from Lemma 6, it follows that
(82)vn∈P¯a, n=1,2,….
From Lemma 4, we see that {vn}n=1∞ is relative compact, and hence there exists a convergent subsequence {vnk}⊂{vn} and v*∈P¯a such that
(83)vnk⟶v* (k⟶∞),
that is,
(84)vnk(t)⇉v*(t) (k⟶∞) on [0,1],(85)vnk′(t)⇉v*′(t) (k⟶∞) on [0,1].
Since v1=Tv0=T0∈P¯a, then
(86)v1(t)=Tv0(t)=(T0)(t)≥0, t∈[0,1],|v1′(t)|=|(Tv0)′(t)|=|(T0)′(t)|≥0, t∈[0,1].
Thus from Lemma 5,
(87)v2(t)=Tv1(t)≥Tv0(t)=v1(t), t∈[0,1],|v2′(t)|=|(Tv1)′(t)|≥|(Tv0)′(t)|=|v1′(t)|, t∈[0,1].
By induction, it is easy to see that for n=1, 2, …,
(88)vn+1(t)≥vn(t), t∈[0,1],(89)|vn+1′(t)|≥|vn′(t)|, t∈[0,1].
From (84)–(89), we see that
(90)vn(t)⇉v*(t) (n⟶∞),vn′(t)⇉v*′(t) (n⟶∞) on[0,1].
Therefore, vn→v*(n→∞), v*∈P¯a. By the continuity of T and vn+1=Tvn, we have
(91)Tv*=v*.
Again from (H3), we have v*(t)>0 on (0,1].

Since every fixed point of T in P is the solution of BVP (9), (10), then w* and v* are two positive, concave and pseudosymmetric solutions of BVP (9), (10). This completes the proof of the theorem.