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
[1] | Md. Asaduzzaman, Md. Zulfikar Ali . Existence of positive solution to the boundary value problems for coupled system of nonlinear fractional differential equations. AIMS Mathematics, 2019, 4(3): 880-895. doi: 10.3934/math.2019.3.880 |
[2] | Yujun Cui, Chunyu Liang, Yumei Zou . Existence and uniqueness of solutions for a class of fractional differential equation with lower-order derivative dependence. AIMS Mathematics, 2025, 10(2): 3797-3818. doi: 10.3934/math.2025176 |
[3] | Mengyu Wang, Xinmin Qu, Huiqin Lu . Ground state sign-changing solutions for fractional Laplacian equations with critical nonlinearity. AIMS Mathematics, 2021, 6(5): 5028-5039. doi: 10.3934/math.2021297 |
[4] | Iman Ben Othmane, Lamine Nisse, Thabet Abdeljawad . On Cauchy-type problems with weighted R-L fractional derivatives of a function with respect to another function and comparison theorems. AIMS Mathematics, 2024, 9(6): 14106-14129. doi: 10.3934/math.2024686 |
[5] | Eleonora Amoroso, Giuseppina D'Aguì, Valeria Morabito . On a complete parametric Sturm-Liouville problem with sign changing coefficients. AIMS Mathematics, 2024, 9(3): 6499-6512. doi: 10.3934/math.2024316 |
[6] | Ravi Agarwal, Snezhana Hristova, Donal O'Regan . Integral presentations of the solution of a boundary value problem for impulsive fractional integro-differential equations with Riemann-Liouville derivatives. AIMS Mathematics, 2022, 7(2): 2973-2988. doi: 10.3934/math.2022164 |
[7] | Kangqun Zhang . Existence and uniqueness of positive solution of a nonlinear differential equation with higher order Erdélyi-Kober operators. AIMS Mathematics, 2024, 9(1): 1358-1372. doi: 10.3934/math.2024067 |
[8] | Saima Rashid, Abdulaziz Garba Ahmad, Fahd Jarad, Ateq Alsaadi . Nonlinear fractional differential equations and their existence via fixed point theory concerning to Hilfer generalized proportional fractional derivative. AIMS Mathematics, 2023, 8(1): 382-403. doi: 10.3934/math.2023018 |
[9] | Alessandra Jannelli, Maria Paola Speciale . On the numerical solutions of coupled nonlinear time-fractional reaction-diffusion equations. AIMS Mathematics, 2021, 6(8): 9109-9125. doi: 10.3934/math.2021529 |
[10] | Sinan Serkan Bilgici, Müfit ŞAN . Existence and uniqueness results for a nonlinear singular fractional differential equation of order $ \sigma\in(1, 2) $. AIMS Mathematics, 2021, 6(12): 13041-13056. doi: 10.3934/math.2021754 |
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,βi≥0 for i=1,2 and α1α2+α1β2+α2β1>0; f∈C((0,1)×[0,∞),R) and f≥−M, for M>0,∀t∈[0,1],u≥0 (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 f≥−M, 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<α≤3, 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)≤f≤z(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(n−2)(0)=0,Dp0+u(t)|t=1=m∑i=1aiDq0+u(t)|t=ξi, |
where λ is a positive parameter, α∈(n−1,n],n∈N,n≥3,ξi∈R for all i=1,...m,(m∈N),0<ξ1<ξ2<⋯<ξm<1,p,q∈R,p∈[1,n−2],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)≤f≤z(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(t−τ))=0, t∈(0,1)∖{τ},x(t)=η(t), t∈[−τ,0],x′(0)=x′′(0)=⋯=x(n−2)(0)=0, n≥3,x(n−2)(1)=0, |
where n−1<α≤n,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 ρ∈C(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, φ2∈L(0,1) are positive, h1, h2∈C(R+0,R+) are nondecreasing, g∈C(R+0,R+) is nonincreasing, R+0=[0,+∞), and
v(t)={1, t∈(0,τ],(t−τ)α−2n+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⋯,n−2,Dγ0+u(0)=0,q(1)Dγ0+u(1)=∫10q(t)Dγ0+u(t)dη0(t),Dα00+u(1)=p∑i=1∫10Dαi0+u(t)dηi(t), |
where β∈(1,2],γ∈(n−1,n],n∈N,n≥3,p∈N,αi∈R,i=0,1⋯p,0≤α1<α2<⋯<αp≤α0<γ−1,α0≥1,λ>0,q:[0,1]→(0,∞) is a continuous function, f∈C((0,1)×[0,∞),R) may be singular at t=0 and/or t=1, and there exist the functions ξ,ϕ∈C((0,1),[0,∞)), φ∈C((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(n−2)(0)=u(n−2)(1)=0,n≥3, | (1.1) |
where n−1<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 ω∈C(0,1)∩L(0,1) with ω(t)>0, t∈(0,1) and ∫10(1−s)α−2ω(s)ds<+∞, such that f(t,u)>−ω(t), for t∈(0,1), u∈R.
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(x−t)α−1φ(t)dt,x>a, |
(Iαb−φ)(x)def=1Γ(α)∫bx(t−x)α−1φ(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(n−α)(ddx)n∫xa(x−t)n−α−1f(t)dt,n=[α]+1, |
(Dαb−f)(x)=(−1)nΓ(n−α)(ddx)n∫bx(t−x)n−α−1f(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 P⊂E is called a cone if the following two conditions are satisfied:
(1) if x∈P and μ≥0, then μx∈P;
(2) if x∈P and −x∈P, then x=0.
Every cone P⊂E induces the ordering in E given by x1≤x2 if and only if x2−x1∈P.
Lemma 2.1. (see [3]) Let α>0, assume that u,Dα0+u∈C(0,1)∩L1(0,1), then,
Iα0+Dα0+u(t)=u(t)+C1tα−1+C2tα−2+⋯+Cntα−n |
holds for some Ci∈R,i=1,2,…,n, where n=[α]+1.
Lemma 2.2. Let y∈C[0,1] and n−1<α<n. Then, the following BVPs
{Dα0+u(t)+y(t)=0,0<t<1,u(0)=u′(0)⋯=u(n−2)(0)=u(n−2)(1)=0,n≥3 | (2.1) |
has a unique solution
u(t)=∫10G(t,s)y(s)ds, |
where
G(t,s)=1Γ(α){tα−1(1−s)α−n+1−(t−s)α−1,0≤s≤t≤1,tα−1(1−s)α−n+1,0≤t≤s≤1. | (2.2) |
Proof. From Definitions 2.1 and 2.2, Lemma 2.1, we know
u(t)=−Iα0+y(t)+C1tα−1+C2tα−2+⋯+Cntα−n=−1Γ(α)∫t0(t−s)α−1y(s)ds+C1tα−1+C2tα−2+⋯+Cntα−n, |
where Ci∈R,i=1,2⋯n.
From u(0)=u′(0)⋯=u(n−2)(0)=0, we get Ci=0,i=2,3⋯n, such that
u(n−2)(t)=−1Γ(α−n+2)∫t0(t−s)α−n+1y(s)ds+C1Γ(α)Γ(α−n+2)tα−n+2,u(n−2)(1)=−1Γ(α−n+2)∫10(1−s)α−n+1y(s)ds+C1Γ(α)Γ(α−n+2). |
From u(n−2)(1)=0, we get C1=1Γ(α)∫10(1−s)α−n+1y(s)ds, so that
u(t)=−1Γ(α)∫t0(t−s)α−1y(s)ds+tα−1Γ(α)∫10(1−s)α−n+1y(s)ds=1Γ(α)∫t0[tα−1(1−s)α−n+1−(t−s)α−1]y(s)ds+1Γ(α)∫1ttα−1(1−s)α−n+1y(s)ds=∫10G(t,s)y(s)ds. |
The proof is completed.
Lemma 2.3. Let n−1<α<n. The function G(t,s) defined by (2.2) is continuous on [0,1]×[0,1] and satisfies 0≤G(t,s)≤G(1,s) and G(t,s)≥tα−1G(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 0≤G(t,s)≤G(1,s).
For 0≤s≤t≤1,
∂G(t,s)∂t=1Γ(α)(α−1)(t−s)α−2[tα−2(1−s)α−n+1tα−2(1−st)α−2−1]≥1Γ(α)(α−1)(t−s)α−2[(1−s)3−n−1]≥0(n≥3). |
For 0≤t≤s≤1, obviously, ∂G(t,s)∂t≥0. Such that, G(t,s) is an increasing function of t and satisfies 0≤G(t,s)≤G(1,s).
At last, we prove that G(t,s) satisfies G(t,s)≥tα−1G(1,s).
For 0≤s≤t≤1,
G(t,s)−tα−1G(1,s)=1Γ(α)[tα−1(1−s)α−n+1−(t−s)α−1]−tα−1Γ(α)[(1−s)α−n+1−(1−s)α−1]=1Γ(α)[(t−ts)α−1−(t−s)α−1]≥0. |
For 0≤t≤s≤1,
G(t,s)G(1,s)=tα−1(1−s)α−n+1(1−s)α−n+1−(1−s)α−1≥tα−1(1−s)α−n+1(1−s)α−n+1=tα−1. |
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 P⊂X be a cone in X. Assume Ω1,Ω2 are open subsets of X with 0∈Ω1⊂¯Ω1⊂Ω2. Let F:P→P be a comletely continuous operator such that either
1) ‖Fx‖≤‖x‖,x∈P∩∂Ω1,‖Fx‖≥‖x‖,x∈P∩∂Ω2; or
2) ‖Fx‖≥‖x‖,x∈P∩∂Ω1,‖Fx‖≤‖x‖,x∈P∩∂Ω2;
holds. Then, F has a fixed point in P∩(¯Ω2∖Ω1).
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 P⊂E by
P={x∈E:x(t)≥tα−1‖x‖,t∈I}. |
Lemma 3.1. Let λ>0, ω∈C(0,1)∩L(0,1) with ω(t)>0 on (0,1), and n−1<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(n−2)(0)=v(n−2)(1)=0,n≥3 | (3.1) |
has a unique solution
v(t)=λ∫10G(t,s)ω(s)ds | (3.2) |
and
0≤v(t)≤λtα−1M, | (3.3) |
where
M=1Γ(α)∫10(1−s)α−n+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
0≤v(t)=λ∫10G(t,s)ω(s)ds=λtα−1Γ(α)∫10(1−s)α−n+1ω(s)ds−λ1Γ(α)∫t0(t−s)α−1ω(s)ds≤λtα−1Γ(α)∫10(1−s)α−n+1ω(s)ds=λtα−1M. | (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(n−2)(0)=x(n−2)(1)=0,n≥3. | (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:P→E be the operator defined by
Tx(t):=λ∫10G(t,s)g(s,x(s)−v(s))ds. | (3.7) |
Then, T:P→P 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]|λ∫10G(t,s)g(s,x(s)−v(s))ds|=λ∫10G(1,s)g(s,x(s)−v(s))ds. |
So that, for t∈[0,1],
Tx(t)=λ∫10G(t,s)g(s,x(s)−v(s))ds≥tα−1λ∫10G(1,s)g(s,x(s)−v(s))ds=tα−1‖Tx‖. |
Thus, T(P)⊂P.
As the continuity and nonnegativeness of G(t,s) and H0 implies T is a continuous operator.
Let Ω⊂P be bounded, there exists a positive constant r>0, such that |x|≤r, for all x∈Ω. Set M0=max0≤x≤r,t∈I∣f(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 x∈Ω and t∈[0,1], we have
|Tx(t)|=|λ∫10G(t,s)g(s,x(s)−v(s))ds|≤λ(M0∫10G(1,s)ds+∫10G(1,s)ω(s)ds)≤λ(M0∫10G(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 ∣t1−t2∣≤δ, implies |G(t1,s)−G(t2,s)|<ελ(M0+∫10ω(s)ds), for s∈[0,1].
Then, for all x∈Ω:
|Tx(t1)−Tx(t2)|=|λ∫10G(t1,s)g(s,x(s)−v(s))ds−λ∫10G(t2,s)g(s,x(s)−v(s))ds|=|λ∫10(G(t1,s)−G(t2,s))g(s,x(s)−v(s))ds|≤λ∫10|G(t1,s)−G(t2,s)||g(s,x(s)−v(s))|ds≤λ∫10|G(t1,s)−G(t2,s)|(M0+ω(s))ds<λ∫10ελ(M0+∫10ω(s)ds)(M0+ω(s))ds≤λελ(M0+∫10ω(s)ds)∫10(M0+ω(s))ds=ε. |
Hence, T(Ω) is equicontinuous. By Arzelà-Ascoli theorem, we have T:P→P 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)≥tα−1|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→+∞inft∈Iηf(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)=supt∈I,0≤x≤r1g(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,(α−1)Γ(α+1)R1L}, where ∫10G(1,s)ds=1(α−1)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∗, t∈I let Ω1={x∈E:‖x‖<R1}. Suppose x∈P∩∂Ω1, we obtain
R1≥x(t)−v(t)≥tα−1‖x‖−λtα−1M>tα−1R1−R1Mtα−1M>0. |
So that
g(t,x(t)−v(t))≤g1(R1)=L |
and
Tx(t)=λ∫10G(t,s)g(s,x(x)−v(s))ds≤λ∫10G(1,s)g(s,x(s)−v(s))ds≤λ∗∫10G(1,s)g1(R1)ds=λ∗L∫10G(1,s)ds<(α−1)Γ(α+1)R1LL(α−1)Γ(α+1)=R1. |
Therefore,
‖Tx‖<‖x‖, x∈P∩∂Ω1. |
Step 2:
From H1, we know that
limx→+∞inft∈Iηg(t,x)x=limx→+∞inft∈Iηf(t,x)+ω(t)x=+∞. |
Then, there exists R2>(1+η1−α)R1>R1, such that for all t∈Iη, when x>R21+η1−α,
g(t,x)>δx, |
where δ>1+η1−αλN>0, N=∫1ηG(1,s)ds.
Let Ω2={x∈E:‖x‖<R2}, for all x∈P∩∂Ω2, t∈Iη we have
x(t)−v(t)≥tα−1R2−λtα−1M>tα−1R2−λ∗tα−1M≥tα−1R2−tα−1R1≥ηα−1(R2−R1)=R21+η1−α>0. |
So that
g(t,x(t)−v(t))>δ(x(t)−v(t))>δR21+η1−α |
and
‖Tx‖=maxt∈Iλ∫10G(t,s)g(s,x(s)−v(s))ds=λ∫10G(1,s)g(s,x(s)−v(s))ds>λ∫1ηG(1,s)g(s,x(s)−v(s))ds>λδR21+η1−α∫1ηG(1,s)ds=λδR21+η1−αN>λ1+η1−αλNR21+η1−αN=R2. |
Thus, ‖Tx‖>‖x‖, for x∈P∩∂Ω2.
Therefore, by the Lemma 2.4, the BVPs (3.6) has at least one positive solution x∈P∩(¯Ω2∖Ω1), and R1≤‖x‖≤R2. 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→+∞inft∈Iηf(t,x)=+∞;
H3:limx→+∞supt∈If(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→+∞inft∈Iηg(t,x)=limx→+∞inft∈Iη(f(t,x)+ω(t))=+∞, |
such that for the above σ, there exists X>0, when x>X, for all t∈Iη, we obtain
g(t,x)>σ. |
Let λ∗=max{Nηα−1M,XM}, R1=2λMη1−α, where λ>λ∗. Let Ω1={x∈E:‖x‖<R1}, if x∈P∩∂Ω1, t∈Iη, we have
x(t)−v(t)≥tα−1R1−λtα−1M=ηα−1R1−λM=ηα−1⋅2λMη1−α−λM=λM>λ∗M≥X, |
such that
g(t,x(t)−v(t))>σ |
and
‖Tx‖=maxt∈Iλ∫10G(t,s)g(s,x(s)−v(s))ds=λ∫10G(1,s)g(s,x(s)−v(s))ds>λ∫1ηG(1,s)g(s,x(s)−v(s))ds=λNσ=2λMNN=2λM>R1=‖x‖. |
Hence, ‖Tx‖>‖x‖, x∈P∩∂Ω1.
On the other hand, from H3, we know that there exists ε0=(α−1)Γ(α+1)2λ>0, R0>R1, for t∈[0,1], x>R0, f(t,x)<ε0x holds.
Because of f∈C([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={x∈E:‖x‖<R2}, for x∈P∩∂Ω2 and t∈[0,1], we have
x(t)−v(t)≥tα−1R2−λtα−1M=tα−1(R2−λM)≥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‖=maxt∈Iλ∫10G(t,s)g(s,x(s)−v(s))ds=λ∫10G(1,s)g(s,x(s)−v(s))ds≤λ∫10G(1,s)(¯M+ε0x(s)+ω(s))ds≤λε0R2∫10G(1,s)ds+λ∫10G(1,s)(¯M+ω(s))ds≤λε0R21(α−1)Γ(α+1)+λΓ(α)(¯M+∫10ω(s)ds)<λR2(α−1)Γ(α+1)(α−1)Γ(α+1)2λ+R22=R2=‖x‖. |
So, we get
‖Tx‖<‖x‖, x∈P∩∂Ω2. |
Hence, from Lemma 2.4, we know that the operator T has at least one fixed point x, which satisfies x∈P∩(¯Ω2∖Ω1) and R1≤‖x‖≤R2. 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(u2−esint−3t−2e)=0,u(0)=u′(0)=u′(1)=0. | (4.1) |
where α=52, f(t,u)=u2−esint−2e, ω(t)=esin10−1√10t−12+12e.
Let η=10−1, then,
g(t,u)=u2−esint+esin10−1√10t−12+10e, |
g1(r)=supt∈Iη,0≤u≤rg(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, (α−1)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(e−tu23−t−10)=0,u(0)=u′(0)=u′(1)=0. | (4.2) |
where α=73, f(t,u)=e−tu23−t−10, ω(t)=t−23+10.
Let η=0.3, such that
g(t,u)=e−tu23+t−23−t, |
and M=8.52480, N=0.219913, σ=2MN=77.5290, Nηα−1M=0.128451, R1=2λMη−43=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.
[1] | S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: theory and applications, Switzerland: Gordon and Breach Science Publishers, 1993. |
[2] |
R. I. Avery, A. C. Peteson, Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl., 42 (2001), 313–322. https://doi.org/10.1016/S0898-1221(01)00156-0 doi: 10.1016/S0898-1221(01)00156-0
![]() |
[3] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, New York: Elsevier Science, 2006. |
[4] |
H. Weitzner, G. M. Zaslavsky, Some applications of fractional equations, Commun. Nonlinear Sci., 8 (2003), 273–281. https://doi.org/10.1016/S1007-5704(03)00049-2 doi: 10.1016/S1007-5704(03)00049-2
![]() |
[5] |
Z. B. Bai, H. S. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311 (2005), 495–505. https://doi.org/10.1016/j.jmaa.2005.02.052 doi: 10.1016/j.jmaa.2005.02.052
![]() |
[6] |
M. El-Shahed, Positive solutions for boundary value problem of nonlinear fractional differential equation, Abstr. Appl. Anal., 2007 (2007), 010368. https://doi.org/10.1155/2007/10368 doi: 10.1155/2007/10368
![]() |
[7] |
H. R. Lian, P. G. Wang, W. G. Ge, Unbounded upper and lower solutions method for Sturm-Liouville boundary value problem on infinite intervals, Nonlinear Anal. Theor., 70 (2009), 2627–2633. https://doi.org/10.1016/j.na.2008.03.049 doi: 10.1016/j.na.2008.03.049
![]() |
[8] |
C. F. Shen, H. Zhou, L. Yang, Existence and nonexistence of positive solutions of a fractional thermostat model with a parameter, Math. Method. Appl. Sci., 39 (2016), 4504–4511. https://doi.org/10.1002/mma.3878 doi: 10.1002/mma.3878
![]() |
[9] |
L. C. Zhang, W. G. Zhang, X. P. Liu, M. Jia, Existence of positive solutions for integral boundary value problems of fractional differential equations with p-Laplacian, Adv. Differ. Equ., 2017 (2017), 36. https://doi.org/10.1186/s13662-017-1086-5 doi: 10.1186/s13662-017-1086-5
![]() |
[10] |
Y. H. Liu, X. D. Zhao, H. H. Pang, Positive solutions to a Coupled fractional differential system with p-Laplacian operator, Discrete Dyn. Nat. Soc., 2019 (2019), 3543670. https://doi.org/10.1155/2019/3543670 doi: 10.1155/2019/3543670
![]() |
[11] |
L. C. Zhang, W. G. Zhang, X. P. Liu, M. Jia, Positive solutions of fractional p-Laplacian equations with integral boundary value and two parameters, J. Inequal. Appl., 2020 (2020), 2. https://doi.org/10.1186/s13660-019-2273-6 doi: 10.1186/s13660-019-2273-6
![]() |
[12] |
J. Q. Xu, C. Y. Xue, Uniqueness and existence of positive periodic solutions of functional differential equations, AIMS Mathematics, 8 (2023), 676–690. https://doi.org/10.3934/math.2023032 doi: 10.3934/math.2023032
![]() |
[13] |
A. Lachouri, A. Ardjouni, A. Djoudi, Existence results for nonlinear sequential Caputo and Caputo-Hadamard fractional differential equations with three-point boundary conditions in Banach spaces, Filomat, 36 (2022), 4717–4727. https://doi.org/10.2298/FIL2214717L doi: 10.2298/FIL2214717L
![]() |
[14] |
N. Li, H. B. Gu, Y. R. Chen, BVP for Hadamard sequential fractional hybrid differential inclusions, J. Funct. Space., 2022 (2022), 4042483. https://doi.org/10.1155/2022/4042483 doi: 10.1155/2022/4042483
![]() |
[15] |
J. Zhang, W. Zhang, V. D. Rădulescu, Double phase problems with competing potentials: concentration and multiplication of ground states, Math. Z., 301 (2022), 4037–4078. https://doi.org/10.1007/s00209-022-03052-1 doi: 10.1007/s00209-022-03052-1
![]() |
[16] |
W. Zhang, J. Zhang, Multiplicity and concentration of positive solutions for fractional unbalanced double-phase problems, J. Geom. Anal., 32 (2022), 235. https://doi.org/10.1007/s12220-022-00983-3 doi: 10.1007/s12220-022-00983-3
![]() |
[17] |
K. H. Zhao, Existence and uh-stability of integral boundary problem for a class of nonlinear higher-order Hadamard fractional Langevin equation via Mittag-Leffler functions, Filomat, 37 (2023), 1053–1063. https://doi.org/10.2298/FIL2304053Z doi: 10.2298/FIL2304053Z
![]() |
[18] |
K. H. Zhao, Solvability and GUH-stability of a nonlinear CF-fractional coupled Laplacian equations, AIMS Mathematics, 8 (2023), 13351–13367. https://doi.org/10.3934/math.2023676 doi: 10.3934/math.2023676
![]() |
[19] |
X. P. Liu, M. Jia, A class of iterative functional fractional differential equation on infinite interval, Appl. Math. Lett., 136 (2023), 108473. https://doi.org/10.1016/j.aml.2022.108473 doi: 10.1016/j.aml.2022.108473
![]() |
[20] |
A. D. Gaetano, M. Jleli, M. A. Ragusa, B. Samet, Nonexistence results for nonlinear fractional differential inequalities involving weighted fractional derivatives, Discrete Cont. Dyn. S, 16 (2023), 1300–1322. https://doi.org/10.3934/dcdss.2022185 doi: 10.3934/dcdss.2022185
![]() |
[21] |
R. P. Agrawal, H. L. Hong, C. C. Yeh, The existence of positive solutions for the Sturm-Liouville boundary value problems, Comput. Math. Appl., 35 (1998), 89–96. https://doi.org/10.1016/S0898-1221(98)00060-1 doi: 10.1016/S0898-1221(98)00060-1
![]() |
[22] |
W. G. Ge, J. L. Ren, New existence theorems of positive solutions for Sturm-Liouville boundary value problems, Appl. Math. Comput., 148 (2004), 631–644. https://doi.org/10.1016/S0096-3003(02)00921-9 doi: 10.1016/S0096-3003(02)00921-9
![]() |
[23] |
W. G. Ge, C. Y. Xue, Some fixed point theorems and existence of positive solutions of two-point boundary-value problems, Nonlinear Anal. Theor., 70 (2009), 16–31. https://doi.org/10.1016/j.na.2007.11.040 doi: 10.1016/j.na.2007.11.040
![]() |
[24] |
G. S. Li, X. P. Liu, M. Jia, Positive solutions to a type of nonlinear three-point boundary value problem with sign changing nonlinearities, Appl. Math. Comput., 57 (2009), 348–355. https://doi.org/10.1016/j.camwa.2008.10.093 doi: 10.1016/j.camwa.2008.10.093
![]() |
[25] |
Y. Q. Wang, L. S. Liu, Y. H. Wu, Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity, Nonlinear Anal. Theor., 74 (2011), 6434–6441. https://doi.org/10.1016/j.na.2011.06.026 doi: 10.1016/j.na.2011.06.026
![]() |
[26] |
S. Q. Zhang, Positive solution of singular boundary value problem for nonlinear fractional differential equation with nonlinearity that changes sign, Positivity, 16 (2012), 177–193. https://doi.org/10.1007/s11117-010-0110-8 doi: 10.1007/s11117-010-0110-8
![]() |
[27] |
Z. C. Hao, Y. B. Huang, Existence of positive solutions to nonlinear fractional boundary value problem with changing sign nonlinearity and advanced arguments, Abstr. Appl. Anal., 2014 (2014), 158436. https://doi.org/10.1155/2014/158436 doi: 10.1155/2014/158436
![]() |
[28] |
J. Henderson, R. Luca, Existence of positive solutions for a singular fractional boundary value problem, Nonlinear Anal. Model., 22 (2017), 99–114. https://doi.org/10.15388/NA.2017.1.7 doi: 10.15388/NA.2017.1.7
![]() |
[29] |
J. K. He, M. Jia, X. P. Liu, H. Chen, Existence of positive solutions for a high order fractional differential equation integral boundary value problem with changing sign nonlinearity, Adv. Differ. Equ., 2018 (2018), 49. https://doi.org/10.1186/s13662-018-1465-6 doi: 10.1186/s13662-018-1465-6
![]() |
[30] |
R. P. Agarwal, R. Luca, Positive solutions for a semipositone singular Riemann-Liouville fractional differential problem, Int. J. Nonlin. Sci. Num., 20 (2019), 823–831. https://doi.org/10.1515/ijnsns-2018-0376 doi: 10.1515/ijnsns-2018-0376
![]() |
[31] |
D. Y. Liu, K. M. Zhang, Existence of positive solutions to a boundary value problem for a delayed singular hight order fractional differential equation with a sign-changed nonlinearity, J. Appl. Math. Comput., 3 (2020), 1073–1093. https://doi.org/10.11948/20190190 doi: 10.11948/20190190
![]() |
[32] |
W. X. Wang, Unique positive solutions for boundary value problem of p-Laplacian fractional differential equation with a sign-changed nonlinearity, Nonlinear Anal. Model., 27 (2022), 1110–1128. https://doi.org/10.15388/namc.2022.27.29503 doi: 10.15388/namc.2022.27.29503
![]() |
[33] |
A. Tudorache, R. Luca, Positive solutions for a fractional differential equation with sequential derivatives and nonlocal boundary conditions, Symmetry, 14 (2022), 1779. https://doi.org/10.3390/sym14091779 doi: 10.3390/sym14091779
![]() |
[34] | M. A. Krasnosel'skii, Topological methods in the theory of nonlinear integral equations, New York: Pergamon Press, 1964. |
1. | Luchao Zhang, Xiping Liu, Mei Jia, Zhensheng Yu, Piecewise conformable fractional impulsive differential system with delay: existence, uniqueness and Ulam stability, 2024, 70, 1598-5865, 1543, 10.1007/s12190-024-02017-3 |