Research article Special Issues

Existence and uniqueness of a positive solutions for the product of operators

  • In this paper, we prove the existence of a positive solution for some equations involving multiplication of concave (possibly nonlinear) operators. Also, we provide a successively sequence to approximate the solution for such equations. This kind of the solution is necessary for quadratic differential and integral equations.

    Citation: Golnaz Pakgalb, Mohammad Jahangiri Rad, Ali Salimi Shamloo, Majid Derafshpour. Existence and uniqueness of a positive solutions for the product of operators[J]. AIMS Mathematics, 2022, 7(10): 18853-18869. doi: 10.3934/math.20221038

    Related Papers:

    [1] Xiulin Hu, Lei Wang . Positive solutions to integral boundary value problems for singular delay fractional differential equations. AIMS Mathematics, 2023, 8(11): 25550-25563. doi: 10.3934/math.20231304
    [2] Shaoyuan Xu, Li Fan, Yan Han . Fixed points of generalized $ \varphi $-concave-convex operators with mixed monotonicity and applications. AIMS Mathematics, 2024, 9(11): 32442-32462. doi: 10.3934/math.20241555
    [3] Yitao Yang, Dehong Ji . Properties of positive solutions for a fractional boundary value problem involving fractional derivative with respect to another function. AIMS Mathematics, 2020, 5(6): 7359-7371. doi: 10.3934/math.2020471
    [4] Kirti Kaushik, Anoop Kumar, Aziz Khan, Thabet Abdeljawad . Existence of solutions by fixed point theorem of general delay fractional differential equation with $ p $-Laplacian operator. AIMS Mathematics, 2023, 8(5): 10160-10176. doi: 10.3934/math.2023514
    [5] Chengbo Zhai, Yuanyuan Ma, Hongyu Li . Unique positive solution for a p-Laplacian fractional differential boundary value problem involving Riemann-Stieltjes integral. AIMS Mathematics, 2020, 5(5): 4754-4769. doi: 10.3934/math.2020304
    [6] Gennaro Infante, Serena Matucci . Positive solutions of BVPs on the half-line involving functional BCs. AIMS Mathematics, 2021, 6(5): 4860-4872. doi: 10.3934/math.2021285
    [7] Ridha Dida, Hamid Boulares, Bahaaeldin Abdalla, Manar A. Alqudah, Thabet Abdeljawad . On positive solutions of fractional pantograph equations within function-dependent kernel Caputo derivatives. AIMS Mathematics, 2023, 8(10): 23032-23045. doi: 10.3934/math.20231172
    [8] Saud Fahad Aldosary, Mohamed M. A. Metwali . Solvability of product of $ n $-quadratic Hadamard-type fractional integral equations in Orlicz spaces. AIMS Mathematics, 2024, 9(5): 11039-11050. doi: 10.3934/math.2024541
    [9] Abdelouaheb Ardjouni . Positive solutions for nonlinear Hadamard fractional differential equations with integral boundary conditions. AIMS Mathematics, 2019, 4(4): 1101-1113. doi: 10.3934/math.2019.4.1101
    [10] Shaoyuan Xu, Yan Han, Qiongyue Zheng . Fixed point equations for superlinear operators with strong upper or strong lower solutions and applications. AIMS Mathematics, 2023, 8(4): 9820-9831. doi: 10.3934/math.2023495
  • In this paper, we prove the existence of a positive solution for some equations involving multiplication of concave (possibly nonlinear) operators. Also, we provide a successively sequence to approximate the solution for such equations. This kind of the solution is necessary for quadratic differential and integral equations.



    In this paper we prove some fixed point theorems for the problem

    x=A(x)+B(x)C(x). (1.1)

    However, these kind of theorems are related to some "quadratic" problems. Let us mention the quadratic integral equation

    x(t)=g(t,x(t))+u(t,x(t))10K(t,s)f(s,x(s))ds. (1.2)

    Special cases of Eq (1.2) were investigated in connection with the applications of some kind of problems in the theories of radiative transfer, neutron transport, and the kinetic theory of gases [4]. A more general problem (motivated by some practical interests in plasma physics) was investigated in [21]. See [11,24] for other applications.

    So far, two methods have been proposed to solve Eq (1.1). In the former, the measure of non-compactness technique (see [2,8,9,10,14,17]) is used to prove the existence of a solution for Eq (1.1), and in the latter, Dhage [13] used the combining Schauder's fixed point theorem and Banach's contraction principle to prove the existence of a solution for Eq (1.1), also see [5]. Some authors have combined the two methods and have proved the existence of a solution for Eq (1.1).

    In this paper, we prove the existence of a positive solution for the Eq (1.1) in which the operators A, B, and C are concave (or convex) or monotone. Also, we give a successively sequence to approximate it. But what is our motivation to prove the existence of a positive solution for the Eq (1.1) when the operators A, B, and C are concave (or convex) or monotone? The mentioned methods have not provided a way to approximate the solution for the Eq (1.1). Also, in the case that the operators A, B, and C are concave (or convex) or monotone, we do not have to suppose the continuity, compactness, and upper-lower assumptions for the operators A, B, and C. These assumptions play an important roles in order to prove the existence of positive solutions for nonlinear differential and integral equations and they are difficult to verify for real problems. Furthermore, there exist more extensively applied of the positive solution of nonlinear differential and integral equations in practical issues (see [3,12,28,29,30,31]).

    The start of proving the existence of a positive solution for differential and integral equations can be found in the Picard investigation (see [25], p.129–138). Authors in [18,19,20] generalized theorems for abstract operator equations with special positive operators called u0-concave. After that, ordered concavity (convexity) and α-concavity (convexity) were introduced by Amann [1] in 1976 and Potter [26] in 1977. In [7,22,23,32,33,34], some others type of concave operators were investigated.

    The paper is organized as follows: In Section 2, we introduce some of the preliminaries needed for the next sections. In Section 3, we prove some existential results for the Eq (1.1). Furthermore, we provide some examples that satisfy the main results. In Section 4, we prove the existence of positive solutions for nonlinear quadratic integral equations by theorems provided in the main results section. Section 5 is devoted to concluding and proposing new ideas.

    Throughout this paper, we assume that E is a real Banach algebra. Which means, E is a real Banach space in which an operation of multiplication is defined, subject to the following properties (for all x,y,zE,λR):

    (1) (xy)z=x(yz);

    (2) x(y+z)=xy+xz and (x+y)z=xz+yz;

    (3) λ(xy)=(λx)y=x(λy);

    (4) ||xy||||x||||y||.

    Now let us recall the concepts of cone and partial order for a Banach algebra. A subset P of E is called a cone of E if

    (1) P is a non-empty closed and θP;

    (2) λP+γPP for all non-negative real numbers λ,γ;

    (3) P2=PPP;

    (4) P(P)={θ},

    where θ denotes the null of E. For a given cone PE, we can define a partial ordering with respect to P by xy if and only if yxP. x<y will stand for xy and xy. The cone P is called normal if there is a number M>m0 such that for all x,yE,

    θxy||x||M||y||.

    The last positive number satisfying the above inequality is called the normal constant of P. In the following, we always assume that P is a cone in E and, is the partial ordering with respect to P. We call such space ordered Banach algebra and denote it by (OBA).

    If x1,x2E, the set [x1,x2]={xE|x1xx2} is called the order interval between x1 and x2. An operator A:EE is called increasing (decreasing) if xy implies AxAy (AxAy) where x,yE. For h>θ (i.e. hθ and hθ), set

    Ph={x|xE,λ(x)>0,μ(x)>0,s.t.λ(x)hxμ(x)h}.

    It is easy to notice that PhP.

    Lemma 2.1. ([15]) The two following assumptions are equivalent:

    (1) P is a normal cone,

    (2) xnznyn (n=1,2,3,...) and ||xnx||0, ||ynx||0, imply that ||znx||0.

    Definition 2.1. ([16]) Let α be a real number such that 0α<1. An operator A:PP is called an α-concave ((-α)-convex) if it satisfies,

    A(tx)tαAx(A(tx)tαAx),t(0,1),xP. (2.1)

    Theorem 2.2. ([6]) Assume that P is a normal cone and the operator T satisfies:

    (D1) T:PhPh is an increasing self-map in Ph;

    (D2) For any xPh and t(0,1), there exists β(t)(0,1) such that T(tx)tβ(t)Tx;

    (D3) For every x0P, there is a constant l0 suchthat x0[θ,lh].

    Then, operator equation x=Tx+x0 has a unique solution in Ph.

    Now the main results could be stated and proved.

    Theorem 3.1. Let P be a normal cone, A:PP is an increasing α-concave operator, B:PP is an increasing γ1-concave operator, and C:PP is anincreasing γ2-concave operator such that γ1+γ2=γ1. Also, suppose that

    (i) there exists h>θ such that hhPh, and Ah,Bh,ChPh;

    (ii) there exists a constant δ0>0 such that for all xP, we have Axδ0BxCx.

    Then, the operator Eq (1.1) has a unique solution x in Ph. Moreover, for the constructing successivelysequence yn=Ayn1+Byn1Cyn1(n=1,2,) and for any initial value y0Ph, we have ynx as n.

    Proof. Since Ah,Bh,ChPh, there exist constants λ1,λ2,μ1,μ2,υ1,υ2>0 such that λ1hAhλ2h,μ1hBhμ2h,υ1hChυ2h. We have

    λ1h+μ1υ1hhAh+BhChλ2h+μ2υ2hh.

    By (i), there exist r,s>0 such that rhhhsh. We get

    (λ1+μ1υ1r)hAh+BhCh(λ2+μ2υ2s)h.

    Hence, we can write K1hAh+BhChJ1h, where K1=λ1+μ1υ1r and J1=λ2+μ2υ2s. Thus, Ah+BhChPh. Define the operator T=A+BC by Tx=Ax+BxCx. Then T:PP and ThPh. Next, we show that T:PhPh. By (2.1), for any t(0,1) and xP, we have

    A(1tx)1tαAx,B(1tx)1tγ1Bx,C(1tx)1tγ2Cx.

    For any xPh, we can choose a sufficiently small number t0(0,1) such that

    t0hx1t0h. (3.1)

    Note that T:PP is an increasing self-map and by (3.1),

    Tx=Ax+BxCxA(1t0h)+B(1t0h)C(1t0h)1tα0Ah+1tγ0BhChλ2tα0h+μ2υ2tγ0hh=J2h,

    where J2=λ2tα0+μ2υ2tγ0s. Also,

    Tx=Ax+BxCxA(t0h)+B(t0h)C(t0h)tα0Ah+tγ0BhChλ1tα0h+μ1υ1tγ0hh=K2h,

    where K2=λ1tα0+μ1υ1tγ0r. Thus TxPh. Henece, T:PhPh. Moreover, A:PhPh,B:PhPh and C:PhPh. In the following, we show that for any t(0,1), there exists β0(t)(α,1) with respect to t, such that for all xPh,

    T(tx)tβ0(t)Tx,t(0,1). (3.2)

    By (ii), there exists δ0>0 such that Axδ0BxCx. Consider the following function:

    f(t)=tβttαtβ,t(0,1),whereβ(α,1).

    It is easy to prove that f is non-negative in (0,1). Especially, for any t(0,1) we have tβ>t and tα>tβ. Furthermore, for fixing t(0,1), we have limβ1f(t)=0. So, there exists β0(t)(α,1) with respect to t such that

    tβ0(t)ttαtβ0(t)δ0,t(0,1).

    Hence, we have

    Axδ0BxCxtβ0(t)ttαtβ0(t)BxCx,t(0,1),xPh.

    Then, we can get

    tαAx+tBxCxtβ0(t)Ax+tβ0(t)BxCx,t(0,1),xPh.

    Consequently, for any t(0,1) and xPh we have

    T(tx)=A(tx)+B(tx)C(tx)tαAx+tγBxCxtαAx+tBxCxtβ0(t)Ax+tβ0(t)BxCx,t(0,1),xPh.

    Therefore,

    T(tx)tβ0(t)T(x),t(0,1),xPh.

    Let x0=θ. Application of Theorem 2.2 implies that the equation Tx=x has a unique solution x in Ph. It can be concluded that the operator Eq (1.1) has a unique solution x in Ph. Now we can construct the successively sequence yn=Ayn1+Byn1Cyn1(n=1,2,) for any initial point y0Ph. Since y0Ph and Ty0Ph, we can choose a sufficiently small number t0(0,1) such that

    t0y0Ty01t0y0. (3.3)

    Note that 0<β0(t0)<1, and we can also take a positive integer k such that

    k>11β0(t0). (3.4)

    Put u0=tk0y0,v0=1tk0y0. Evidently, u0,v0Ph and u0y0v0. By the monotonicity of T, we have Tu0Tv0. Furthermore, by combining (3.2) and (3.3) we have

    Tu0=T(tk0y0)=T(t0tk10y0)tβ0(t0)0T(t0tk20y0)tβ0(t0)0tβ0(t0)0T(tk20y0)(tβ0(t0)0)kTy0(tβ0(t0)0)kt0y0=tkβ0(t0)+10y0. (3.5)

    By (3.4), one can obtains that kβ0(t0)+1<k. Thus

    tkβ0(t0)+10>tk0. (3.6)

    Therefore, Tu0tkβ0(t0)+10y0>tk0y0=u0. By (3.2),

    T(1tx)1tβ0(t)T(x),t(0,1),xPh.

    Thus,

    Tv0=T(1tk0y0)=T(1t01tk0y0)1tβ0(t0)0T(1tk10y0)=1tβ0(t0)0T(1t01tk20y0)1tβ0(t0)01tβ0(t0)0T(1tk20y0)1(tβ0(t0)0)kT(y0)1tkβ0(t0)+10y0. (3.7)

    The application of (3.6) implies Tv01tkβ0(t0)+10y01tk0y0=v0. Thus, u0Tu0Tv0v0. For n=1,2,, let un=Tun1,vn=Tvn1. Then, unynvn (n=1,2,). Similar to the proof of Theorem 1.3 of [35], there exists yPh such that Ty=y and limnun=limvn=y. Then, by Lemma 2.1, yny(n). Since the fixed point of the operator T in Ph is unique, we have x=y.

    Example 3.1. Consider the Banach space E=C[0,1] with the supremum norm. E is a Banach algebra by the multiplication (xy)(t)=x(t)y(t). Assume that P={x(t)C[0,1]|x(t)0,t[0,1]}. Then E is an (OBA) by the cone P. Let us define operators A,B,C:PP as the following

    A(x)=x+1,B(x)=1x+1,C(x)=1.

    Assume that h(t)=1. We can prove that all of the assumptions of Theorem 3.1 are satisfied and the operator T=A+BC has a unique positive solution.

    Example 3.2. Consider the Banach space E=L[0,1] with the L norm. E is a Banach algebra by the multiplication (xy)(t)=x(t)y(t). Assume that P={x(t)L[0,1]|x(t)0,t[0,1]}. Then E is an (OBA) by the cone P. For any xP and t[0,1], let us define

    A(x)=1x+1,B(x)=x+1,C(x)=1,h(t)=1.

    It is easy to prove that for any xP, A(x),B(x),C(x)P. Also we have h>0. Therefore, A,B,C,h:PP. It is easily noticed that, A is an α-concave operator (for some 0<α<1), B is a γ1 concave operator (for some 0γ1<1), and C is a γ2 concave operator (for some 0γ2<1) such that γ1+γ11. Now let us consider that

    x(t)={nt=1n,0t[0,1]{1n}n.

    Then, x(t)P and for t=1n we have

    A(x(1n))=1x(1n)+1=1n+1,B(x(1n))=x(1n)+1=n+1.

    Therefore, there is no δ0>0 such that for all t[0,1] we have Ax(t)δ0Bx(t)Cx(t). Hence, the assumption (ii) of Theorem 3.1 is not satisfied. It is easy to prove that the rest of the assumptions of Theorem 3.1 are satisfied and T=A+BC has a unique positive solution.

    Example 3.3. Consider E, that is defined in Example 3.1. Let us define operators A,B,C:PP as the following

    A(x)=11+sin(x),B(x)=2+sin(x),C(x)=2+cos(x).

    Suppose that h(t)=π4. We have A,B,C:PP and hh>0. Also, for all xP, AxBxCx. None of the functions A,B, and C are concave. But the operator T=A+BC has a unique positive solution.

    Example 3.4. Consider the Banach space E=C[0,1] with the supremum norm. Let us consider the multiplication

    (fg)(t)=x0f(t)g(xt)dt

    for any x,yE. E is a Banach algebra([27]). Assume that, P={x(t)C[0,1]|x(t)0,t[0,1]}. Then, E is an (OBA) by the cone P. Now suppose that, h(t)=1>0. We have hh(x)=x. Then, hhPh.

    Example 3.5. Consider E that is defined in Example 3.4. Let h(t)=1>0. Let us define the operators A,B,C:PP as the following

    A(x)=x+1,B(x)=1x+1+1,C(x)=1.

    We can prove that all of the assumptions of Theorem 3.1, except hh>0, are satisfied. Operator T=A+BC has no positive solution.

    Let the operators A,B,C:PP be defined as the following

    A(x)=x+1,B(x)=1,C(x)=12.

    It is easy to prove that all of the assumptions of the Theorem 3.1, except the assumption hh>0, are satisfied. The equation T=A+BC has a unique positive solution.

    Lemma 3.2. Assume that, P is a normal cone and the operator A satisfies the following conditions:

    (D1) A:PhPh is decreasing in Ph;

    (D2) For any xPh and t(0,1), there exists α(t)(0,1) such that A(tx)tα(t)Ax.

    Then, there exist u0,v0Ph such that u0<v0, u0Av0Au0v0.

    Proof. Since AhPh, we can select a sufficiently small number t0 such that

    t0hAh1t0h. (3.8)

    Note that 0<α(t0)<1, and we can choose a positive integer k such that

    k>11α(t0). (3.9)

    Set u0=tk0h and v0=1tk0h. Evidently, u0,v0Ph and u0<v0. By the monotonicity of A, Au0Av0. Furthermore, by (D2) and (3.8) we have,

    Au0=A(tk0h)=A(t0tk10h)tα(t0)0A(tk10h)tkα(t0)0A(t0h)tkα(t0)01t0h=tkα(t0)10h. (3.10)

    By (3.10), we get that kα(t0)+1<k. Thus,

    tkα(t0)10<tk0. (3.11)

    Hence,

    Au0tkα(t0)10h<tk0h<v0.

    By (D2),

    tα(t)A(x)A(xt)t(0,t),xPh. (3.12)

    Thus,

    Av0=A(1tk0h)=A(1t01tk10h)tα(t0)0A(1tk10h)=tα(t0)0A(1t01tk20h)tα(t0)0tα(t0)0A(1tk20h)tkα(t0)0A(1t0h)=tkα(t0)+10h.

    Application of (3.9) implies that Av0tkα(t0)+10htk0h=u0. So we have

    u0Av0Au0v0. (3.13)

    Theorem 3.3. Assume that P is normal cone, the operator T satisfies (D1) and(D2) of Lemma 3.2, and there is a constant l0 such that x0[θ,lh].

    Then, the operator equation Tx+x0=x has a unique solutionin Ph.

    Proof. For all xPh, we have TxPh. Then, there exist real numbers λ,μ>0 such that λhTxμh. Thus,

    λhTx+x0(μ+l)h. (3.14)

    Hence,

    Tx+x0Ph,xPh. (3.15)

    Define the operator F by,

    Fx=Tx+x0,xPh. (3.16)

    By (3.15), and considering the monotonicity of the operator T, the operator F:PhPh is decreasing. Furthermore, for all xPh and t(0,1), we have

    F(tx)=T(tx)+x0tα(t)T(x)+tα(t)x0tα(t)F(x). (3.17)

    Lemma 3.2 implies that, there exist u0,v0Ph such that

    u0<v0,u0Fv0Fu0v0. (3.18)

    Construct the successively sequences

    un=Fun1,vn=Fvn1,n=1,2,.

    By the monotonicity of F, we have v1=Fv0Fu0=u1. Similarly, we have

    u0v1u2v2u1v0. (3.19)

    By continuing this process, for n=1,2,, we get

    u0v1u2u2nv2n+1u2n+1v2nv2u1v0. (3.20)

    Therefore, {u2n}, {v2n+1} are the increasing and {u2n+1}, {v2n} are decreasing sequences. By (3.20), for n=1,2,, we have

    u2nv2n,v2n+1u2n+1. (3.21)

    Assume that,

    t2n=sup{t|tv2nu2n},t2n+1=sup{t|tu2n+1v2n+1}.

    Thus, for n=1,2,, we have u2nt2nv2n and v2n+1t2n+1v2n+1. Then,

    u2n+1u2nt2nv2nt2nv2n+1n=1,2,,
    v2mv2m+1t2m+1u2m+1t2m+1u2mm=1,2,.

    Therefore, tn+1tn, i.e. tn is an increasing sequence such that tn(0,1]. If tnt as n, then t=1. Otherwise, 0<t<1. We distinguish two cases:

    Case (i): There exists an integer N such that tN=t. In this case, we know tn=t, for all nN. So for nN, we have

    u2n+1=Fu2nF(tv2n)(t)α(t)F(v2n)=(t)α(t)v2n+1,

    i.e. (t)α(t)u2n+1v2n+1. By the definition of t2n+1, we have t2n+1=t(t)α(t)>t, which is a contradiction.

    Case (ii): For all integer n, tn<t. Then,

    v2n+1=F(v2n)F(t12nu2n)=F(tt2nu2nt)=F(tt2nu2nt)(t)α(t)F(tt2nu2n)=(t)α(t)F(u2nt2nt)(t)α(t)(t2nt)(α(t2nt))Fu2n(t)α(t)(t2nt)u2n+1=t2n(t)α(t)1u2n+1. (3.22)

    By the definition of tn, we have t2n+1t2n(t)α(t)1. If n, we get t(t)α(t)>t, which is a contradiction. Thus, limntn=1. For any natural number p, we have

    θu2(n+p)u2nv2(n+p)t2nv2nv2nt2nv2n=(1t2n)v2n(1t2n)v0,
    θv2nv2(n+p)v2nu2nv0u2nv0t2nv0(1t2n)v0.

    Since P is normal, we have

    ||u2(n+p)u2n||N(1t2n)||v0||0(n),
    ||v2nv2(n+p)||N(1t2n)||v0||0(n),

    where N is the normal constant. Hence, we can claim that u2n and v2n are Cauchy sequences. Since E is a complete space, there exist u and v such that u2nu, v2nv as n. By (3.20), we know that u2nuvv2n where u,vPh. Then

    θvuv2nu2n(1t2n)v0.

    Furthermore,

    ||vu||N(1t2n)||v0||0(n).

    Thus, u=v. Let x=u=v. Also, by (3.20), we have

    θv2n+1u2nv2nu2n,
    θu2n+1v2n+1v2nv2n+1.

    Then v2n+1x and u2n+1x as n. By the inequality u2nxv2n for n=1,2,, we have

    v2n+1=Fv2nFxFu2n=u2n+1.

    If n, we get x=Fx. That is, x is a fixed point of F in Ph. In the following, we prove that x is the unique fixed point of F in Ph. Let ˉx be any fixed point of F in Ph. Set r1=sup{r>0|rxˉx1rx}. Evidently, 0<r1< and r1xˉx1r1x. Next, we prove that r11. If 0<r1<1,

    rα(r1)1xF(xr1)ˉx=FˉxF(r1x)rα(r1)1x.

    However, by rα(r1)1xˉx, rα(r1)1r1. Since rα(r1)1>r1, we get a contradiction. Hence, r11 and we get ˉxr1xx. Similarly, we can prove that xˉx, and x=ˉx. Therefore, F has a unique fixed point x in Ph. That is to say, Tx+x0=x has a unique solution in Ph.

    Comment 3.4. In [26], Theorem 3.3 is proved by Hilbert'sprojective metric method where, α is constant function. But, the successively sequence that converges to the fixed pointhas not been obtained.

    Theorem 3.5. Consider that P is a normal cone, A,B:PP aredecreasing operators, and C:PP is a decreasing(-α)-convex operator. Assume that

    (i) there exists h>θ such that AhPh,BhPh and ChPh;

    (ii) hhPh.

    Then, the operator Eq (1.1) has a unique solution x in Ph. Moreover, for the constructing successivelysequence yn=Ayn1+Byn1Cyn1,n=1,2,... and for any initial value y0Ph, we have ynx as n.

    Proof. Since Ah,Bh,ChPh, there exist constants λ1,λ2,μ1,μ2,υ1,υ2>0 such that λ1hAhλ2h,μ1hBhμ2h,υ1hChυ2h. Similar to the proof of the Theorem 3.1, we can prove that, if the operator T=A+BC is defined by Tx=Ax+BxCx. Then, T:PP and ThPh. Next, we show that T:PhPh. By (2.1) and the monotonicity of A,B, we have

    A(1tx)Ax,B(1tx)Bx,C(1tx)tαAx,t(0,1),xP.

    For any xPh, we can select a sufficiently small number t0(0,1) such that

    t0hx1t0h. (3.23)

    T:PP is decreasing and by (3.23) we have,

    Tx=Ax+BxCxA(1t0h)+B(1t0h)C(1t0h)Ah+tα0BhChλ2h+μ2υ2tα0hh.

    By (ii), there exist s>0 such that 1shhhsh. So, we have TxJ2h where J2=λ2+μ2υ21stα0. Also,

    Tx=Ax+BxCxA(t0h)+B(t0h)C(t0h)Ah+tα0BhChλ1h+μ1υ1tα0hh.

    Hence, we have TxK2h, where K2=λ1+μ1υ1tα0s. Thus, TxPh. So, T:PhPh. Moreover, A:PhPh,B:PhPh, and C:PhPh. On the other hand, for any t(0,1) and xPh,

    Ax+tαBxCxtα(Ax+BxCx).

    Then,

    T(tx)=A(tx)+B(tx)C(tx)tαT(x)t(0,1),xPh.

    Therefore, T is the (-α)-convex operator. Let, x0=θ. Application of Theorem 3.3 implies that the equation Tx=x has a unique solution x in Ph. That is, the operator Eq (1.1) has a unique solution x in Ph. Now, we construct successively the sequence yn=Ayn1+Byn1Cyn1(n=1,2,) for any initial point y0Ph. Since y0Ph and Ty0Ph, we can choose a sufficiently small number t0(0,1) such that

    t0y0Ty01t0y0. (3.24)

    Since 0<α(t0)<1, we can also take a positive integer k such that

    k>11α(t0). (3.25)

    Set u0=tk0y0,v0=1tk0y0. Let un+1=Tun, vn+1=Tvn(n=1,2,). By Theorem 3.3, unx and vnx(as n). By (3.25), we have u0y1v0. Let us define yn+1=Tyn. Since T is monotone decreasing and by (3.20), we get

    v2n1y2nu2n1,v2ny2n+1u2n,n=1,2,. (3.26)

    Then by Lemma 2.1, y2nx, y2n+1x as n. Thus, for ε>0, there exists an integer N such that for nN,

    ||y2nx||<ε,||y2n+1x||<ε. (3.27)

    Therefore, (3.27) show that ynx as n.

    Theorem 4.1. Assume that E=C[0,1], P={x(t)C[0,1]|x(t)0,t[0,1]},

    (H1) g(t,x):[0,1]×[0,)[0,),u(t,x):[0,1]×[0,)[0,) and f(t,x):[0,1]×[0,)[0,) are increasing operators with respect to x;

    (H2) there exists h>θ in P such that g(t,h),u(t,h)Ph and hhPh;

    (H3) g(t,x) is an α-concave operator, f(t,x) is a γ1-concave operator, and u(t,x) is a γ2-concave operator with respect to x such that γ1+γ2=γ1;

    (H4) G(t,s) is non-negative for any t,s[0,1]. Also, forany fixed t[0,1], the function G(t,s) is boundedin [0,1] and for any fixed s[0,1], we have G(t,s)Ph;

    (H5) there exists δ0>0 such that for any t[0,1] and any y[0,) we have

    g(t,y)δ0u(t,y)10G(t,s)f(s,y)ds.

    Then the problem

    x(t)=g(t,x(t))+u(t,x(t))10G(t,s)f(s,x(s))ds

    has a unique positive solution x in Ph. Moreover, forany x0Ph and for the constructing successivelysequence

    xn+1(t)=g(t,xn(t))+u(t,xn(t))10G(t,s)f(s,xn(s))ds,n=0,1,2,,

    we have ||xnx||0 as n.

    Proof. Let us define the operators A:PE, B:PE, and C:PE as the following:

    (Ax)(t)=g(t,x(t)),(Bx)(t)=u(t,x(t)),(Cx)(t)=10G(t,s)f(s,x(s))ds.

    It can easily be noticed that x is a solution of the problem (4.1) if x=Ax+BxCx. By (H1) and (H4), we notice that A:PP, B:PP and C:PP. By (H3), for any λ(0,1) and xP, we have

    C(λx)(t)=10G(t,s)f(s,λx(s))dsλγ110G(t,s)f(s,x(s))ds=λγ1C(x)(t).

    Then, C is a γ1-concave operator with respect to x. By (H2), AhPh and BhPh. By (H4), for any s[0,1] there exist λ(s),μ(s)>0 such that

    λ(s)h(t)G(t,s)μ(s)h(t).

    Since G(t,s) is bounded, λ(s),μ(s) are bounded positive real numbers(for any s[0,1]). Therefore,

    Ch(t)=10G(t,s)f(s,h(s))ds10μ(s)h(t)f(s,1)ds=h(t)10μ(s)f(s,1)ds

    and

    Ch(t)=10G(t,s)f(s,h(s))ds10λ(s)h(t)f(s,0)ds=h(t)10λ(s)f(s,0)ds.

    Thus, ChPh. Hence, the condition (i) of Theorem 3.1 is satisfied. By (H2), the condition (iii) of Theorem 3.1 is satisfied and by (H5) the condition (ii) of Theorem 3.1 is satisfied. Then, Theorem 4.1 follows from Theorem 3.1.

    Example 4.1. Assume that E=C[0,1], P={x(t)C[0,1]|x(t)0,t[0,1]}. Let us define

    g(t,x(t))=x(t)1+x(t),u(t,x(t))=x(t)1+x2(t),f(t,x(t))=1+x(t)

    for t[0,1] and xP. It is easy to prove that g is αconcave (for α=12). Also, f is a γ1-concave operator (for γ1=0) and u is a γ2-concave operator (for γ2=1) with respect to x. Suppose that G(t,s)=ets1+ts (for t,s[0,1]). It is easy to see that g, u, f, and G are satisfied in all assumptions of Theorem 4.1 for h(t)=1. Hence, the problem

    x(t)=x(t)1+x(t)+x(t)1+x2(t)10ets1+x(s)1+tsds

    has a unique positive solution.

    Theorem 4.2. Assume that E=C[0,1], P={x(t)C[0,1]|x(t)0,t[0,1]},

    (H1) g(t,x):[0,1]×[0,)[0,),u(t,x):[0,1]×[0,)[0,) and f(t,x):[0,1]×[0,)[0,) are decreasing with respect to x;

    (H2) there exists h>θ in P such that g(t,h),u(t,h)Ph, and hhPh;

    (H3) f(t,x) is (-α)-convex with respect to x;

    (H4) G(t,s) is non-negative for any t,s[0,1], for anyfixed t[0,1], the function G(t,s) is boundedin [0,1], and for any fixed s[0,1], we have G(t,s)Ph.

    Then, the problem

    x(t)=g(t,x(t))+u(t,x(t))10G(t,s)f(s,x(s))ds,

    has a unique positive solution x in Ph. Moreover, forany x0Ph and for the constructing successivelysequence

    xn+1(t)=g(t,xn(t))+u(t,xn(t))10G(t,s)f(s,xn(s))ds,n=0,1,2,

    we have ||xnx||0 as n.

    Proof. Let us define the operators A:PE, B:PE and C:PE as the following:

    (Ax)(t)=g(t,x(t)),(Bx)(t)=u(t,x(t)),(Cx)(t)=10G(t,s)f(s,x(s))ds.

    It is easily noticed that if x=Ax+BxCx, x is a solution of the problem (4.1). By (H1) and (H4), we have A:PP, B:PP and C:PP. By (H3), for any λ(0,1) and xP we have,

    C(λx)(t)=10G(t,s)f(s,λx(s))dsλα10G(t,s)f(s,x(s))ds=λαC(x)(t).

    Hence, C is an (-α)-convex operator. By (H2), we have AhPh and BhPh. By (H4), for any s[0,1], there exist λ(s) and μ(s)>0 such that

    λ(s)h(t)G(t,s)μ(s)h(t).

    Since G(t,s) is bounded, λ(s) and μ(s) are bounded positive real numbers (for any s[0,1]). Therefore, we have

    Ch(t)=10G(t,s)f(s,h(s))ds10μ(s)h(t)f(s,0)ds
    =h(t)10μ(s)f(s,0)ds

    and

    Ch(t)=10G(t,s)f(s,h(s))ds10λ(s)h(t)f(s,1)ds
    =h(t)10λ(s)f(s,1)ds.

    Thus, ChPh. Hence, the assumption (i) of Theorem 3.5 is satisfied. By (H2), the assumption (iii) of Theorem 3.5 is satisfied. Then, Theorem 4.2 follows from Theorem 3.5.

    Example 4.2. Assume that E=C[0,1], P={x(t)C[0,1]|x(t)0,t[0,1]}. Let us define,

    g(t,x(t))=11+x2(t),u(t,x(t))=arccot(x(t)4),f(t,x(t))=11+x(t),

    for t[0,1] and xP. It can be proved that f is (-α)-convex (for α=12). Also u,g are convex sub-homogeneous in x. Suppose that G(t,s)=ets1+ts(for t,s[0,1]). We can see that g, u, f, and G are satisfied in all assumptions of Theorem 4.2 for h(t)=1. Then the problem

    x(t)=11+x2(t)+arccot(x(t)4)10ets1+ts11+x(t)ds

    has a unique positive solution.

    In this paper, we firstly proved the existence of a positive solution for the Eq (1.1) and approximated it by the constructing successively sequence, where A is an α-concave operator, B:PP is an increasing γ1-concave operator and C:PP is an increasing γ2-concave operator such that γ1+γ2=γ1.

    Secondly, we proved the existence of a positive solution for the Eq (1.1) and approximated it by the constructing successively sequence, where A,C are decreasing operators and C is a (-α)-convex operator.

    Thirdly, we proved the existence a positive solution for some nonlinear integral equations and approximated it by the constructing successively sequence(especially in the case of quadratic integral equation).

    Remark 5.1. It is suggested that the Theorems 3.1 and 3.5 be proved without the assumption hh>θ, and also the Theorem 3.1 be proved without assumption (ii). Another interesting topic can be the comparison of the results of Theorems 3.1 and 3.5 with the results of theorems that are proven by the measure of non-compactness [2] and Dhage's techniques [13].

    The authors would like to thank the anonymous referee for his/her comments that helped us improve this article.

    Authors state no conflicts of interest.



    [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 602–709. https://doi.org/10.1137/1018114 doi: 10.1137/1018114
    [2] J. Banaś, K. Sadarangani, Solutions of some functional-integral equations in Banach algebra, Math. Comput. Model., 38 (2003), 245–250. https://doi.org/10.1016/S0895-7177(03)90084-7 doi: 10.1016/S0895-7177(03)90084-7
    [3] A. Boscaggin, G. Feltrin, F. Zanolin, Uniqueness of positive solutions for boundary value problems associated with indefinite φ-Laplacian-type equations, Open Math., 19 (2021), 163–183. https://doi.org/10.1515/math-2021-0003 doi: 10.1515/math-2021-0003
    [4] S. Chandrasekhar, Radiative transfer, New York: Dover Publications, 1960.
    [5] F. Chouia, T. Moussaoui, Some fixed point theorems in ordered Banach spaces and application, Appl. Math. E-Notes, 19 (2019), 433–444.
    [6] Y. Cheng, T. Carson, M. B. M. Elgindi, A note on the proof of the Perron-Frobenius theorem, Appl. Math., 3 (2012), 1697–1701. https://doi.org/10.4236/am.2012.311235 doi: 10.4236/am.2012.311235
    [7] C. Zhai, C. Guo, On α-convex operators, J. Math. Anal. Appl., 316 (2006), 556–565. https://doi.org/10.1016/j.jmaa.2005.04.064 doi: 10.1016/j.jmaa.2005.04.064
    [8] M. Cichoń, M. M. A. Metwali, On a fixed point theorem for the product of operators, J. Fixed Point Theory Appl., 18 (2016), 753–770. https://doi.org/10.1007/s11784-016-0319-7 doi: 10.1007/s11784-016-0319-7
    [9] M. Cichoń, M. M. A. Metwali, On monotonic integrable solutions for quadratic functional integral equations, Mediterr. J. Math., 10 (2013), 909–926. https://doi.org/10.1007/s00009-012-0218-0 doi: 10.1007/s00009-012-0218-0
    [10] K. Cichoń, M. Cichoń, M. M. A. Metwali, On some fixed point theorems in abstract duality pairs, Rev. Union Math. Argent., 61 (2020), 249–266. https://doi.org/10.33044/revuma.v61n2a04 doi: 10.33044/revuma.v61n2a04
    [11] M. Cichoń, M. M. A. Metwali, On the Banach algebra of integral-variation type Holder spaces and quadratic fractional integral equations, Banach J. Math. Anal., 16 (2022), 34. https://doi.org/10.1007/s43037-022-00188-4 doi: 10.1007/s43037-022-00188-4
    [12] C. Cowan, A. Razani, Singular solutions of a Lane-Emden system, Discrete Cont. Dyn. Syst., 41 (2021), 621–656. http://dx.doi.org/10.3934/dcds.2020291 doi: 10.3934/dcds.2020291
    [13] B. C. Dhage, On some variants of Schauder's fixed point principle and applications to nonlinear integral equations, J. Math. Phys. Sci., 25 (1988), 603–611.
    [14] G. Garcia, G. Mora, A fixed point result in Banach algebras based on the degree of nondensifiability and applications to quadratic integral equations, J. Math. Anal. Appl., 472 (2019), 1220–1235. https://doi.org/10.1016/j.jmaa.2018.11.073 doi: 10.1016/j.jmaa.2018.11.073
    [15] D. Guo, V. Lakshmikantham, Nonlinear problems in abstract cones, Academic Press, 1988.
    [16] S. Hong, Fixed points for mixed monotone multivalued operators in Banach spaces with applications, J. Math. Anal. Appl., 156 (2008), 333–342. https://doi.org/10.1016/j.jmaa.2007.03.091 doi: 10.1016/j.jmaa.2007.03.091
    [17] A. Jeribi, B. Krichen, B. Mefteh, Fixed point theory in WC-Banach algebras, Turk. J. Math., 40 (2016), 283–291. https://doi.org/10.3906/mat-1504-42 doi: 10.3906/mat-1504-42
    [18] M. A. Krasnoselskii, L. A. Ladyzhenskii, The scope of the concept of a u0-concave operator, Izv. Vyssh. Uchebn. Zaved. Mat., 5 (1959), 112–121.
    [19] M. A. Krasnosel'skii, L. A. Ladyzhenskii, The structure of the spectrum of positive nonhomogeneous operators, Tr. Mosk. Mat. Obs., 3 (1954), 321–346.
    [20] M. A. Krasnoselskii, P. P. Zabreiko, Geometrical methods of nonlinear analysis, Moscow, 1975.
    [21] M. Kunze, On a special class of nonlinear integral equations, J. Integral Equ. Appl., 7 (1995), 329–350. https://doi.org/10.1216/jiea/1181075882 doi: 10.1216/jiea/1181075882
    [22] K. Li, J. Liang, T. J. Xiato, A fixed point theorem for convex and decreasing operators, Nonlinear Anal., 63 (2005), 206–209. https://doi.org/10.1016/j.na.2004.12.014 doi: 10.1016/j.na.2004.12.014
    [23] Z. D. Liang, W. X. Wang, S. J. Li, On concave operators, Acta Math. Sinica, 22 (2006), 577–582. https://doi.org/10.1007/s10114-005-0687-1 doi: 10.1007/s10114-005-0687-1
    [24] Mohamed M. A. Metwali, Solvability of Gripenberg's equations of fractional order with perturbation term in weighted Lp-spaces on R+, Turk. J. Math., 22 (2022), 481–498. https://doi.org/10.3906/mat-2106-84 doi: 10.3906/mat-2106-84
    [25] E. Picard, Traite d'analyse, Tome Ⅲ, Paris: Gauthier-Villars, 1908.
    [26] A. J. B. Potter, Applications of Hilbert's projective metric to certain classes of non-homogenous operators, Q. J. Math., 28 (1977), 93–99. https://doi.org/10.1093/qmath/28.1.93 doi: 10.1093/qmath/28.1.93
    [27] W. Rudin, Functional analysis, 2 Eds., New York: McGraw-Hill, 1991.
    [28] Y. Sang, Y. Ren, Nonlinear sum operator equations and applications to elastic beam equation and fractional differential equation, Bound. Value Probl., 2019 (2019), 49. https://doi.org/10.1186/s13661-019-1160-x doi: 10.1186/s13661-019-1160-x
    [29] Y. Sang, L. He, Existence of an approximate solution for a class of fractional multi-point boundary value problems with the derivative term, Bound. Value Probl., 2021 (2021), 20. https://doi.org/10.1186/s13661-021-01497-7 doi: 10.1186/s13661-021-01497-7
    [30] S. Song, L. Zhang, B. Zhou, N.Zhang, Existence-uniqueness of positive solutions to nonlinear impulsive fractional differential systems and optimal control, Bound. Value Probl., 2020 (2020), 162. https://doi.org/10.1186/s13661-020-01461-x doi: 10.1186/s13661-020-01461-x
    [31] Y. Yang, D. Ji, Properties of positive solutions for a fractional boundary value problem involving fractional derivative with respect to another function, AIMS Math., 5 (2020), 7359–7371. https://doi.org/10.3934/math.2020471 doi: 10.3934/math.2020471
    [32] C. Zhai, Li. Wang, φ-(h,e)-concave operators and applications, J. Math. Anal. Appl., 454 (2017), 571–584. https://doi.org/10.1016/j.jmaa.2017.05.010 doi: 10.1016/j.jmaa.2017.05.010
    [33] C. Zhai, F. Wang, Properties of positive solutions for the operator Ax=λx and applications to fractional diffeential equations with integral boundary conditions, Adv. Differ. Equ., 2015 (2015), 366. https://doi.org/10.1186/s13662-015-0704-3 doi: 10.1186/s13662-015-0704-3
    [34] C. Zhai, C. Yang, X. Zhang, Positive solutions for nonlinear operator equations and several classes of applications to functional equations, Math. Z., 266 (2010), 43–63. https://doi.org/10.1007/s00209-009-0553-4 doi: 10.1007/s00209-009-0553-4
    [35] C. Zhai, C. Yang, C. M. Guo, Positive solutions of operator equation on ordered Banach spaces and applications, Comput. Math. Appl., 56 (2008), 3150–3156. https://doi.org/10.1016/j.camwa.2008.09.005 doi: 10.1016/j.camwa.2008.09.005
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1803) PDF downloads(83) Cited by(0)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog