Research article

The existence of positive solutions for high order fractional differential equations with sign changing nonlinearity and parameters

  • Received: 15 June 2023 Revised: 17 July 2023 Accepted: 18 July 2023 Published: 08 September 2023
  • MSC : 34A08, 34A12, 34B18

  • By constructing an auxiliary boundary value problem, the difficulty caused by sign changing nonlinearity terms is overcome by means of the linear superposition principle. Using the Guo-Krasnosel'skii fixed point theorem, the results of the existence of positive solutions for boundary value problems of high order fractional differential equation are obtained in different parameter intervals under a more relaxed condition compared with the existing literature. As an application, we give two examples to illustrate our results.

    Citation: Luchao Zhang, Xiping Liu, Zhensheng Yu, Mei Jia. The existence of positive solutions for high order fractional differential equations with sign changing nonlinearity and parameters[J]. AIMS Mathematics, 2023, 8(11): 25990-26006. doi: 10.3934/math.20231324

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  • By constructing an auxiliary boundary value problem, the difficulty caused by sign changing nonlinearity terms is overcome by means of the linear superposition principle. Using the Guo-Krasnosel'skii fixed point theorem, the results of the existence of positive solutions for boundary value problems of high order fractional differential equation are obtained in different parameter intervals under a more relaxed condition compared with the existing literature. As an application, we give two examples to illustrate our results.



    Fractional differential equation theory comes with fractional calculus and is an abstract form of many engineering and physical problems. It has been widely used in system control, system identification, grey system theory, fractal and porous media dispersion, electrolytic chemistry, semiconductor physics, condensed matter physics, viscoelastic systems, biological mathematics, statistics, diffusion and transport theory, chaos and turbulence and non-newtonian fluid mechanics. Fractional differential equation theory has attracted the attention of the mathematics and natural science circles at home and abroad, and has made a series of research results. It has become one of the international hot research directions and has very important theoretical significance and application value.

    As an important research area of fractional differential equation, boundary value problems have attracted a great deal of attention in the last ten years, especially in terms of the existence of positive solutions, and have achieved a lot of results (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]). When the nonlinear term changes sign, the research on the existence of positive solutions progresses slowly, and relevant research results are not many (see [21,22,23,24,25,26,27,28,29,30,31,32,33]).

    In [21], using a fixed point theorem in a cone, Agarwal et al. obtained the existence of positive solutions for the Sturm-Liouville boundary value problem

    {(p(t)u(t))+λf(t,u(t))=0,t(0,1),α1u(0)β1p(0)u(0)=0,α2u(1)+β2p(0)u(1)=0,

    where λ>0 is a parameter, p(t)C((0,1),[0,)), αi,βi0 for i=1,2 and α1α2+α1β2+α2β1>0; fC((0,1)×[0,),R) and fM, for M>0,t[0,1],u0 (M is a constant).

    In [22], Weigao Ge and Jingli Ren studied the Sturm-Liouville boundary value problem

    {(p(t)u(t))+λa(t)f(t,u(t))=0,t(0,1),α1u(0)β1p(0)u(0)=0,α2u(1)+β2p(0)u(1)=0,

    where a(t)0 and λ>0 is a parameter. They removed the restriction fM, using Krasnosel'skii theorem, obtained some new existence theorems for the Sturm-Liouville boundary value problem.

    In [23], Weigao Ge and Chunyan Xue studied the same Sturm-Liouville boundary value problem again. Without the restriction that f is bounded below, by the excision principle and area addition principle of degree, they obtained three theorems and extended the Krasnosel'skii's compression-expansion theorem in cones.

    In [25], Yongqing Wang et al. considered the nonlinear fractional differential equation boundary value problem with changing sign nonlinearity

    {Dα0+u(t)+λf(t,u(t))=0,t(0,1),u(0)=u(0)=u(1)=0,

    where 2<α3, λ>0 is a parameter, Dα0+ is the standard Riemann-Liouville fractional derivative. f is allowed to change sign and may be singular at t=0,1 and r(t)fz(t)g(x) for some given nonnegative functions r,z,g. By using Guo-Krasnosel'skii fixed point theorem, the authors obtained the existence of positive solutions.

    In [28], J. Henderson and R. Luca studied the existence of positive solutions for a nonlinear Riemann-Liouville fractional differential equation with a sign-changing nonlinearity

    {Dα0+u(t)+λf(t,u(t))=0,t(0,1),u(0)=u(0)=u(n2)(0)=0,Dp0+u(t)|t=1=mi=1aiDq0+u(t)|t=ξi,

    where λ is a positive parameter, α(n1,n],nN,n3,ξiR for all i=1,...m,(mN),0<ξ1<ξ2<<ξm<1,p,qR,p[1,n2],q[0,p], Dα0+ is the standard Riemann-Liouville fractional derivative. With the restriction that f may be singular at t=0,1 and r(t)fz(t)g(t,x) for some given nonnegative functions r,z,g, applying Guo-Krasnosel'skii fixed point theorem, the existences of positive solutions are obtained.

    In [31], Liu and Zhang studied the existence of positive solutions to the boundary value problem for a high order fractional differential equation with delay and singularities including changing sign nonlinearity

    {Dα0+x(t)+f(t,x(tτ))=0, t(0,1){τ},x(t)=η(t), t[τ,0],x(0)=x(0)==x(n2)(0)=0, n3,x(n2)(1)=0,

    where n1<αn,n=[α]+1,Dα0+ is the standard Riemann-Liouville fractional derivative. The restriction on the nonlinearity f is as follows: there exists a nonnegative function ρC(0,1)L(0,1), ρ(t)0, such that f(t,x)ρ(t) and φ2(t)h2(x)f(t,v(t)x)+ρ(t)φ1(t)(g(x)+h1(x)), for  (t,x)(0,1)×R+, where φ1, φ2L(0,1) are positive, h1, h2C(R+0,R+) are nondecreasing, gC(R+0,R+) is nonincreasing, R+0=[0,+), and

    v(t)={1, t(0,τ],(tτ)α2n+1,t(τ,1).

    By Guo-krasnosel'skii fixed point theorem and Leray-Schauder's nonlinear alternative theorem, some existence results of positive solutions are obtained, respectively.

    In [33], Tudorache and Luca considered the nonlinear ordinary fractional differential equation with sequential derivatives

    {Dβ0+(q(t)Dγ0+u(t))=λf(t,u(t)),t(0,1),u(j)(0)=0,j=0,1,n2,Dγ0+u(0)=0,q(1)Dγ0+u(1)=10q(t)Dγ0+u(t)dη0(t),Dα00+u(1)=pi=110Dαi0+u(t)dηi(t),

    where β(1,2],γ(n1,n],nN,n3,pN,αiR,i=0,1p,0α1<α2<<αpα0<γ1,α01,λ>0,q:[0,1](0,) is a continuous function, fC((0,1)×[0,),R) may be singular at t=0 and/or t=1, and there exist the functions ξ,ϕC((0,1),[0,)), φC((0,1)×[0,),[0,)) such that ξ(t)f(t,x)ϕ(t)φ(t,x),t(0,1),x(0,) with 0<10ξ(s)ds<,0<10ϕ(s)ds<. By the Guo-Krasnosel'skii fixed point theorem, the existence of positive solutions are obtained.

    As can be seen from the above research results, fixed point theorems are still common tools to solve the existence of positive solutions to boundary value problems with sign changing nonlinearity, especially the Guo-Krasnosel'skii fixed point theorem. In addition, for boundary value problems of ordinary differential equations, Weigao Ge et al. removed the restriction that the nonlinear item bounded below. However, for fractional boundary value problems, from the existing literature, there are still many restrictions on nonlinear terms.

    Our purpose of this paper is to establish the existence of positive solutions of boundary value problems (BVPs for short) of the nonlinear fractional differential equation as follows

    {Dα0+u(t)+λf(t,u(t))=0,t(0,1),u(0)=u(0)=u(n2)(0)=u(n2)(1)=0,n3, (1.1)

    where n1<α<n, λ>0, f:[0,1]×[0,+)R is a known continuous nonlinear function and allowed to change sign, and Dα0+ is the standard Riemann-Liouville fractional derivative.

    In this paper, by the Guo-Krasnosel'skii fixed point theorem, the sufficient conditions for the existence of positive solutions for BVPs (1.1) are obtained under a more relaxed condition compared with the existing literature, as follows. Throughout this paper, we suppose that the following conditions are satisfied.

    H0: There exists a known function ωC(0,1)L(0,1) with ω(t)>0, t(0,1) and 10(1s)α2ω(s)ds<+, such that f(t,u)>ω(t), for t(0,1), uR.

    This paper is organized as follows. In Section 2, we introduce some definitions and lemmas to prove our major results. In Section 3, some sufficient conditions for the existence of at least one and two positive solutions for BVPs (1.1) are investigated. As applications, some examples are presented to illustrate our major results in Section 4.

    In this section, we give out some important definitions, basic lemmas and the fixed point theorem that will be used to prove the major results.

    Definition 2.1. (see[1]) Let φ(x)L1(a,b). The integrals

    (Iαa+φ)(x)def=1Γ(α)xa(xt)α1φ(t)dt,x>a,
    (Iαbφ)(x)def=1Γ(α)bx(tx)α1φ(t)dt,x<a,

    where α>0, are called the Riemann-Liouville fractional integrals of the order α. They are sometimes called left-sided and right-sided fractional integrals respectively.

    Definition 2.2. (see[1]) For functions f(x) given in the interval [a,b], each of the expressions

    (Dαa+f)(x)=1Γ(nα)(ddx)nxa(xt)nα1f(t)dt,n=[α]+1,
    (Dαbf)(x)=(1)nΓ(nα)(ddx)nbx(tx)nα1f(t)dt,n=[α]+1

    is called Riemann-Liouville derivative of order α, α>0, left-handed and right-handed respectively.

    Definition 2.3. (see [2]) Let E be a real Banach space. A nonempty, closed, and convex set PE is called a cone if the following two conditions are satisfied:

    (1) if xP and μ0, then μxP;

    (2) if xP and xP, then x=0.

    Every cone PE induces the ordering in E given by x1x2 if and only if x2x1P.

    Lemma 2.1. (see [3]) Let α>0, assume that u,Dα0+uC(0,1)L1(0,1), then,

    Iα0+Dα0+u(t)=u(t)+C1tα1+C2tα2++Cntαn

    holds for some CiR,i=1,2,,n, where n=[α]+1.

    Lemma 2.2. Let yC[0,1] and n1<α<n. Then, the following BVPs

    {Dα0+u(t)+y(t)=0,0<t<1,u(0)=u(0)=u(n2)(0)=u(n2)(1)=0,n3 (2.1)

    has a unique solution

    u(t)=10G(t,s)y(s)ds,

    where

    G(t,s)=1Γ(α){tα1(1s)αn+1(ts)α1,0st1,tα1(1s)αn+1,0ts1. (2.2)

    Proof. From Definitions 2.1 and 2.2, Lemma 2.1, we know

    u(t)=Iα0+y(t)+C1tα1+C2tα2++Cntαn=1Γ(α)t0(ts)α1y(s)ds+C1tα1+C2tα2++Cntαn,

    where CiR,i=1,2n.

    From u(0)=u(0)=u(n2)(0)=0, we get Ci=0,i=2,3n, such that

    u(n2)(t)=1Γ(αn+2)t0(ts)αn+1y(s)ds+C1Γ(α)Γ(αn+2)tαn+2,u(n2)(1)=1Γ(αn+2)10(1s)αn+1y(s)ds+C1Γ(α)Γ(αn+2).

    From u(n2)(1)=0, we get C1=1Γ(α)10(1s)αn+1y(s)ds, so that

    u(t)=1Γ(α)t0(ts)α1y(s)ds+tα1Γ(α)10(1s)αn+1y(s)ds=1Γ(α)t0[tα1(1s)αn+1(ts)α1]y(s)ds+1Γ(α)1ttα1(1s)αn+1y(s)ds=10G(t,s)y(s)ds.

    The proof is completed.

    Lemma 2.3. Let n1<α<n. The function G(t,s) defined by (2.2) is continuous on [0,1]×[0,1] and satisfies 0G(t,s)G(1,s) and G(t,s)tα1G(1,s) for t,s[0,1].

    Proof. From the definition (2.2), it's easy to know G(t,s) is continuous on [0,1]×[0,1]. Next, we prove that G(t,s) satisfies 0G(t,s)G(1,s).

    For 0st1,

    G(t,s)t=1Γ(α)(α1)(ts)α2[tα2(1s)αn+1tα2(1st)α21]1Γ(α)(α1)(ts)α2[(1s)3n1]0(n3).

    For 0ts1, obviously, G(t,s)t0. Such that, G(t,s) is an increasing function of t and satisfies 0G(t,s)G(1,s).

    At last, we prove that G(t,s) satisfies G(t,s)tα1G(1,s).

    For 0st1,

    G(t,s)tα1G(1,s)=1Γ(α)[tα1(1s)αn+1(ts)α1]tα1Γ(α)[(1s)αn+1(1s)α1]=1Γ(α)[(tts)α1(ts)α1]0.

    For 0ts1,

    G(t,s)G(1,s)=tα1(1s)αn+1(1s)αn+1(1s)α1tα1(1s)αn+1(1s)αn+1=tα1.

    The proof is completed.

    At the end of this section, we present the Guo-Krasnosel'skii fixed point theorem that will be used in the proof of our main results.

    Lemma 2.4. (see [34]) Let X be a Banach space, and let PX be a cone in X. Assume Ω1,Ω2 are open subsets of X with 0Ω1¯Ω1Ω2. Let F:PP be a comletely continuous operator such that either

    1) Fxx,xPΩ1,Fxx,xPΩ2; or

    2) Fxx,xPΩ1,Fxx,xPΩ2;

    holds. Then, F has a fixed point in P(¯Ω2Ω1).

    By a positive solution of BVPs (1.1), we mean a function u:[0,1][0,+) such that u(t) satisfies (1.1) and u(t)>0 for t(0,1).

    Let Banach space E=C[0,1] be endowed with x=maxt[0,1]|x(t)|. Let I=[0,1], define the cone PE by

    P={xE:x(t)tα1x,tI}.

    Lemma 3.1. Let λ>0, ωC(0,1)L(0,1) with ω(t)>0 on (0,1), and n1<α<n. Then, the following boundary value problem of fractional differential equation

    {Dα0+v(t)+λω(t)=0,0<t<1,v(0)=v(0)=v(n2)(0)=v(n2)(1)=0,n3 (3.1)

    has a unique solution

    v(t)=λ10G(t,s)ω(s)ds (3.2)

    and

    0v(t)λtα1M, (3.3)

    where

    M=1Γ(α)10(1s)αn+1ω(s)ds.

    Proof. From Lemma 2.2, let y(t)=λω(t), we have (3.2) immediately. In view of Lemma 2.3, we obtain

    0v(t)=λ10G(t,s)ω(s)ds=λtα1Γ(α)10(1s)αn+1ω(s)dsλ1Γ(α)t0(ts)α1ω(s)dsλtα1Γ(α)10(1s)αn+1ω(s)ds=λtα1M. (3.4)

    From (3.4), (3.3) holds.

    The proof is completed.

    Lemma 3.2. Suppose that v=v(t) is the solution of BVPs (3.1) and define the function g(t,u(t)) by

    g(t,u(t))=f(t,u(t))+ω(t). (3.5)

    Then, u(t) is the solution of BVPs (1.1), if and only if x(t)=u(t)+v(t) is the solution of the following BVPs

    {Dα0+x(t)+λg(t,x(t)v(t))=0x(0)=x(0)=x(n2)(0)=x(n2)(1)=0,n3. (3.6)

    And when x(t)>v(t), u(t) is a positive solution of BVPs(1.1).

    Proof. In view of Lemma 2.2, if u(t) and v(t) are the solutions of BVPs (1.1) and BVPs (3.1), respectively, we have

    Dα0+(u(t)+v(t))=Dα0+u(t)+Dα0+v(t)=λf(t,u(t))λω(t)=λ[f(t,u(t))+ω(t)]=λg(t,u(t)),

    such that

    Dα0+(u(t)+v(t))+λg(t,u(t))=0.

    Let x(t)=u(t)+v(t), we have u(t)=x(t)v(t) and

    Dα0+x(t)+λg(t,x(t)v(t))=0.

    It is easily to obtain x(0)=x(0)=x(1)=0 from the boundary conditions of BVPs (1.1) and BVPs (3.1).

    Hence, x(t) is the solution of BVPs (3.6).

    On the other hand, if v(t) and x(t) are the solution of BVPs (3.1) and BVPs (3.6), respectively. Similarly, u(t)=x(t)v(t) is the solution of BVPs (1.1). Obviously, when x(t)>v(t), u(t)>0 is a positive solution of BVPs (1.1).

    The proof is completed.

    Lemma 3.3. Let T:PE be the operator defined by

    Tx(t):=λ10G(t,s)g(s,x(s)v(s))ds. (3.7)

    Then, T:PP is comletely continuous.

    Proof. In view of the definition of the function g(t,u(t)), we know that g(t,x(t)v(t))>0 is continuous from the continuity of x(t) and v(t).

    By Lemma 2.3, we obtain

    Tx=maxt[0,1]|λ10G(t,s)g(s,x(s)v(s))ds|=λ10G(1,s)g(s,x(s)v(s))ds.

    So that, for t[0,1],

    Tx(t)=λ10G(t,s)g(s,x(s)v(s))dstα1λ10G(1,s)g(s,x(s)v(s))ds=tα1Tx.

    Thus, T(P)P.

    As the continuity and nonnegativeness of G(t,s) and H0 implies T is a continuous operator.

    Let ΩP be bounded, there exists a positive constant r>0, such that |x|r, for all xΩ. Set M0=max0xr,tIf(t,x(t)v(t)), then,

    |g(t,x(t)v(t))||f(t,x(t)v(t))|+|ω(t)|M0+ω(t).

    So, for xΩ and t[0,1], we have

    |Tx(t)|=|λ10G(t,s)g(s,x(s)v(s))ds|λ(M010G(1,s)ds+10G(1,s)ω(s)ds)λ(M010G(1,s)ds+1Γ(α)10ω(s)ds).

    Hence, T is uniformly bounded.

    On the other hand, since G(t,s)C([0,1]×[0,1]), for ε>0, exists δ>0, for t1,t2[0,1] with t1t2∣≤δ, implies |G(t1,s)G(t2,s)|<ελ(M0+10ω(s)ds), for s[0,1].

    Then, for all xΩ:

    |Tx(t1)Tx(t2)|=|λ10G(t1,s)g(s,x(s)v(s))dsλ10G(t2,s)g(s,x(s)v(s))ds|=|λ10(G(t1,s)G(t2,s))g(s,x(s)v(s))ds|λ10|G(t1,s)G(t2,s)||g(s,x(s)v(s))|dsλ10|G(t1,s)G(t2,s)|(M0+ω(s))ds<λ10ελ(M0+10ω(s)ds)(M0+ω(s))dsλελ(M0+10ω(s)ds)10(M0+ω(s))ds=ε.

    Hence, T(Ω) is equicontinuous. By Arzelà-Ascoli theorem, we have T:PP is completely continuous.

    The proof is completed.

    A function x(t) is said to be a solution of BVPs (3.6) if x(t) satisfies BVPs (3.6). In addition, if x(t)>0, for t(0,1), x(t) is said to be a positive solution of BVPs (3.6). Obviously, if x(t)P, and x(t)0 is a solution of BVPs (3.6), by x(t)tα1|x|, then x(t) is a positive solution of BVPs (3.6). By Lemma 3.2, if x(t)>v(t), u(t)=x(t)v(t) is a positive solution of BVPs (1.1).

    Next, we give some sufficient conditions for the existence of positive solutions.

    Theorem 3.1. For a given 0<η<1, let Iη=[η,1]. If

    H1: limx+inftIηf(t,x)x=+

    holds, there exists λ>0, for any 0<λ<λ, the BVPs (1.1) has at least one positive solution.

    Proof. By Lemma 3.2, if BVPs (3.6) has a positive solution x(t) and x(t)>v(t), BVPs (1.1) has a positive solution u(t)=x(t)v(t). We will apply Lemma 2.4 to prove the theorem.

    In view of the definition of g(t,u(t)), we have g(t,u(t))0, so that BVPs (3.6) has a positive solution, if and only if the operator T has a fixed point in P.

    Define

    g1(r1)=suptI,0xr1g(t,x),

    where r1>0.

    By the definition of g1(r1) and H1, we have

    limr1+r1g1(r1)=0.

    Then, there exists R1>0, such that

    R1g1(R1)=maxr1>0r1g1(r1).

    Let L=g1(R1), λ=min{R1M,(α1)Γ(α+1)R1L}, where 10G(1,s)ds=1(α1)Γ(α+1).

    In order to apply Lemma 2.4, we separate the proof into the following two steps.

    Step 1:

    For every 0<λ<λ, tI let Ω1={xE:x<R1}. Suppose xPΩ1, we obtain

    R1x(t)v(t)tα1xλtα1M>tα1R1R1Mtα1M>0.

    So that

    g(t,x(t)v(t))g1(R1)=L

    and

    Tx(t)=λ10G(t,s)g(s,x(x)v(s))dsλ10G(1,s)g(s,x(s)v(s))dsλ10G(1,s)g1(R1)ds=λL10G(1,s)ds<(α1)Γ(α+1)R1LL(α1)Γ(α+1)=R1.

    Therefore,

    Tx<x, xPΩ1.

    Step 2:

    From H1, we know that

    limx+inftIηg(t,x)x=limx+inftIηf(t,x)+ω(t)x=+.

    Then, there exists R2>(1+η1α)R1>R1, such that for all tIη, when x>R21+η1α,

    g(t,x)>δx,

    where δ>1+η1αλN>0, N=1ηG(1,s)ds.

    Let Ω2={xE:x<R2}, for all xPΩ2, tIη we have

    x(t)v(t)tα1R2λtα1M>tα1R2λtα1Mtα1R2tα1R1ηα1(R2R1)=R21+η1α>0.

    So that

    g(t,x(t)v(t))>δ(x(t)v(t))>δR21+η1α

    and

    Tx=maxtIλ10G(t,s)g(s,x(s)v(s))ds=λ10G(1,s)g(s,x(s)v(s))ds>λ1ηG(1,s)g(s,x(s)v(s))ds>λδR21+η1α1ηG(1,s)ds=λδR21+η1αN>λ1+η1αλNR21+η1αN=R2.

    Thus, Tx>x, for xPΩ2.

    Therefore, by the Lemma 2.4, the BVPs (3.6) has at least one positive solution xP(¯Ω2Ω1), and R1xR2. From x(t)v(t)>0, we know that BVPs (1.1) has at least one positive solution u(t)=x(t)v(t).

    The proof is completed.

    Theorem 3.2. Suppose

    H2:limx+inftIηf(t,x)=+;

    H3:limx+suptIf(t,x)x=0;

    hold, there exists λ>0, for all λ>λ, the BVPs (1.1) has at least one positive solution.

    Proof. Let σ=2MN. From H2, we have

    limx+inftIηg(t,x)=limx+inftIη(f(t,x)+ω(t))=+,

    such that for the above σ, there exists X>0, when x>X, for all tIη, we obtain

    g(t,x)>σ.

    Let λ=max{Nηα1M,XM}, R1=2λMη1α, where λ>λ. Let Ω1={xE:x<R1}, if xPΩ1, tIη, we have

    x(t)v(t)tα1R1λtα1M=ηα1R1λM=ηα12λMη1αλM=λM>λMX,

    such that

    g(t,x(t)v(t))>σ

    and

    Tx=maxtIλ10G(t,s)g(s,x(s)v(s))ds=λ10G(1,s)g(s,x(s)v(s))ds>λ1ηG(1,s)g(s,x(s)v(s))ds=λNσ=2λMNN=2λM>R1=x.

    Hence, Tx>x, xPΩ1.

    On the other hand, from H3, we know that there exists ε0=(α1)Γ(α+1)2λ>0, R0>R1, for t[0,1], x>R0, f(t,x)<ε0x holds.

    Because of fC([0,1]×[0,+),R), let ¯M=max(t,x)I×[0,R0]{f(t,x)}, then, for t[0,1], x[0,+), f(t,x)¯M+ε0x holds.

    Let R2>max{R0,λM,2λ(¯M+10ω(s)ds)Γ(α)}, Ω2={xE:x<R2}, for xPΩ2 and t[0,1], we have

    x(t)v(t)tα1R2λtα1M=tα1(R2λM)0.

    So that,

    g(t,x(t)v(t))=f(t,x(t)v(t))+ω(t)¯M+ε0(x(t)v(t))+ω(t)¯M+ε0x(t)+ω(t).

    Therefore,

    Tx=maxtIλ10G(t,s)g(s,x(s)v(s))ds=λ10G(1,s)g(s,x(s)v(s))dsλ10G(1,s)(¯M+ε0x(s)+ω(s))dsλε0R210G(1,s)ds+λ10G(1,s)(¯M+ω(s))dsλε0R21(α1)Γ(α+1)+λΓ(α)(¯M+10ω(s)ds)<λR2(α1)Γ(α+1)(α1)Γ(α+1)2λ+R22=R2=x.

    So, we get

    Tx<x, xPΩ2.

    Hence, from Lemma 2.4, we know that the operator T has at least one fixed point x, which satisfies xP(¯Ω2Ω1) and R1xR2. From x(t)v(t)>0, we know that BVPs (1.1) has at least one positive solution u(t)=x(t)v(t).

    The proof is completed.

    In this section, we provide two examples to demonstrate the applications of the theoretical results in the previous sections.

    Example 4.1. Consider the following BVPs

    {D520+u+λ(u2esint3t2e)=0,u(0)=u(0)=u(1)=0. (4.1)

    where α=52, f(t,u)=u2esint2e, ω(t)=esin10110t12+12e.

    Let η=101, then,

    g(t,u)=u2esint+esin10110t12+10e,
    g1(r)=suptIη,0urg(t,u)=r2+10e,

    and

    limr+rg1(r)=limr+rr2+10e=0,

    R1=10e, L=g1(R1)=20e, M=16.7716, N=0.26667, R1M=0.310866, (α1)Γ(α+1)R1L=0.478069, λ=0.310866, R2=170.086, N=0.196967.

    We can check that the condition of Theorem 3.1 is satisfied. Therefore, there exists at least one positive solution.

    Example 4.2. Consider the following BVPs

    {D730+u+λ(etu23t10)=0,u(0)=u(0)=u(1)=0. (4.2)

    where α=73, f(t,u)=etu23t10, ω(t)=t23+10.

    Let η=0.3, such that

    g(t,u)=etu23+t23t,

    and M=8.52480, N=0.219913, σ=2MN=77.5290, Nηα1M=0.128451, R1=2λMη43=84.8957λ>10.9049.

    We can check that the conditions of Theorem 3.2 are satisfied. Therefore, there exists at least one positive solution.

    In this paper, the constraint on the nonlinear term is weakened to f(t,u)>ω(t)(where ω(t)>0). Under similar conditions, by constructing an auxiliary boundary value problem and using the principle of linear superposition, the difficulty caused by sign-change of nonlinear terms is overcome. Under the condition of singularity of nonlinear terms, the existence conclusions of positive solutions are obtained based on the Guo-Krasnosel'skii fixed point theorem.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This research was supported by National Natural Science Foundation of China (Grant No. 12371308). The authors would like to thank the anonymous reviewers and the editor for their constructive suggestions on improving the presentation of the paper.

    The authors declares that they have no competing interest.



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