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Research article

Elastic transformation method for solving ordinary differential equations with variable coefficients

  • Aiming at the problem of solving nonlinear ordinary differential equations with variable coefficients, this paper introduces the elastic transformation method into the process of solving ordinary differential equations for the first time. A class of first-order and a class of third-order ordinary differential equations with variable coefficients can be transformed into the Laguerre equation through elastic transformation. With the help of the general solution of the Laguerre equation, the general solution of these two classes of ordinary differential equations can be obtained, and then the curves of the general solution can be drawn. This method not only expands the solvable classes of ordinary differential equations, but also provides a new idea for solving ordinary differential equations with variable coefficients.

    Citation: Pengshe Zheng, Jing Luo, Shunchu Li, Xiaoxu Dong. Elastic transformation method for solving ordinary differential equations with variable coefficients[J]. AIMS Mathematics, 2022, 7(1): 1307-1320. doi: 10.3934/math.2022077

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  • Aiming at the problem of solving nonlinear ordinary differential equations with variable coefficients, this paper introduces the elastic transformation method into the process of solving ordinary differential equations for the first time. A class of first-order and a class of third-order ordinary differential equations with variable coefficients can be transformed into the Laguerre equation through elastic transformation. With the help of the general solution of the Laguerre equation, the general solution of these two classes of ordinary differential equations can be obtained, and then the curves of the general solution can be drawn. This method not only expands the solvable classes of ordinary differential equations, but also provides a new idea for solving ordinary differential equations with variable coefficients.



    The topic of boundary value problems is an interesting area of research in view of its applications in applied and technical sciences. In the recent years, the class of nonlocal fractional order boundary value problems involving different fractional derivatives (such as Riemann-Liouville, Caputo, etc.) received an overwhelming interest from many researchers. For the details of a variety of nonlocal single-valued and multivalued boundary value problems involving different types of fractional order derivative operators, we refer the reader to the text [1], articles [2,3,4,5,6,7] and the references cited therein. There has been shown a great enthusiasm in developing the existence theory for Hilfer, ψ-Hilfer and (k,ψ) Hilfer type fractional differential equations equipped with different types of boundary conditions, for instance, see [8,9,10,11,12,13,14,15,16].

    Nonlocal boundary conditions are found to be more plausible and practical in contrast to the classical boundary conditions in view of their applicability to describe the changes happening within the given domain. Closed boundary conditions are found to be of great help in describing the situation when there is no fluid flow along the boundary or through it. The free slip condition is also a type of the closed boundary conditions which describes the situation when there is a flow along the boundary, but there is no flow perpendicular to it. Such conditions are also useful in the study of sandpile model [17,18], honeycomb lattice [19], deblurring problems [20], closed-aperture wavefield decomposition in solid media [21], vibration analysis of magneto-electro-elastic cylindrical composite panel [22], etc.

    Now we review some works on the boundary value problems with closed boundary conditions. In [23], the authors studied the single-valued and multivalued fractional boundary value problems with open and closed boundary conditions. A three-dimensional Neumann boundary value problem with a generalized boundary condition in a domain with a smooth closed boundary was discussed in [24]. For some interesting results on impulsive fractional differential equations with closed boundary conditions, see the articles [25,26].

    The objective of the present work is to investigate a new class of mixed nonlinear boundary value problems involving a right Caputo fractional derivative, mixed Riemann-Liouville fractional integral operators, and multipoint variant of closed boundary conditions. In precise terms, we consider the following fractional order nonlocal and nonlinear problem:

    CDαTy(t)+λIρTIσ0+h(t,y(t))=f(t,y(t)),tJ:=[0,T], (1.1)
    y(T)=mı=1(piy(ξi)+Tqiy(ξi)),Ty(T)=mı=1(riy(ξi)+Tviy(ξi)), (1.2)

    where CDαT denote the right Caputo fractional derivative of order α(1,2], IρT and Iσ0+ represent the right and left Riemann-Liouville fractional integral operators of orders ρ,σ>0 respectively, f,h:[0,T]×RR are given continuous functions and λ,pi,qi,ri,viR,i{1,2,3,...,m}, and ξi(0,T). Notice that the integro-differential Eq (1.1) contains the usual and mixed Riemann-Liouville integrals type nonlinearities. The boundary conditions (1.2) can be interpreted as the values of the unknown function and its derivative at the right end-point T of the interval [0,T] are proportional to a linear combination of these values at arbitrary nonlocal positions ξi(0,T). Physically, the nonlocal multipoint closed boundary conditions provide a flexible mechanism to close the boundary at arbitrary positions in the given domain instead of the left end-point of the domain.

    Here we emphasize that much of the literature on fractional differential equations contains the left-sided fractional derivatives and there are a few works dealing with the right-sided fractional derivatives. For instance, the authors in [27,28] studied the problems involving the right-handed Riemann–Liouville fractional derivative operators, while a problem containing the right-handed Caputo fractional derivative was considered in [29]. The problem studied in the present paper is novel in the sense that it solves an integro-differential equation with a right Caputo fractional derivative and mixed nonlinearities complemented with a new concept of nonlocal multipoint closed boundary conditions. The results accomplished for the problems (1.1) and (1.2) will enrich the literature on boundary value problems involving the right-sided fractional derivative operators. The present work is also significant as it produces several new results as special cases as indicated in the last section.

    The rest of the paper is arranged as follows. In Section 2, we present an auxiliary lemma which is used to transform the given nonlinear problem into a fixed-point problem. Section 3 contains the main results and illustrative examples. Some interesting observations are presented in the last Section 4.

    Let us begin this section with some definitions [30].

    Definition 2.1. The left and right Riemann-Liouville fractional integrals of order β>0 for gL1[a,b], existing almost everywhere on [a,b], are respectively defined by

    Iβa+g(t)=ta(ts)β1Γ(β)g(s)dsandIβbg(t)=bt(st)β1Γ(β)g(s)ds.

    Definition 2.2. For gACn[a,b], the right Caputo fractional derivative of order β(n1,n],nN, existing almost everywhere on [a,b], is defined by

    CDβbg(t)=(1)nbt(st)nβ1Γ(nβ)g(n)(s)ds.

    In the following lemma, we solve a linear variant of the fractional integro-differential equation (1.1) supplemented with multipoint closed boundary conditions (1.2).

    Lemma 2.1. Let H,FC[0,T] and Δ0. Then the linear problem

    {CDαTy(t)+λIρTIσ0+H(t)=F(t),tJ:=[0,T],y(T)=mı=1(piy(ξi)+Tqiy(ξi)),Ty(T)=mı=1(riy(ξi)+Tviy(ξi)),0<ξi<T, (2.1)

    is equivalent to the integral equation

    y(t)=Tt(st)α1Γ(α)[F(s)λIρTIσ0+H(s)]ds+b1(t){mı=1piTξi(sξi)α1Γ(α)[F(s)λIρTIσ0+H(s)]dsTmı=1qiTξi(sξ)α2Γ(α1)[F(s)λIρTIσ0+H(s)]ds}+b2(t){mı=1riTξi(sξi)α1Γ(α)[F(s)λIρTIσ0+H(s)]dsTmi=1viTξi(sξi)α2Γ(α1)[F(s)λIρTIσ0+H(s)]ds}, (2.2)

    where

    b1(t)=1Δ(tS6S7TS9+T),b2(t)=1Δ[(1S1)t+S2+TS4T],Δ=(S11)(S7+TS9T)S6(S2+TS4T),S1=mı=1pi,S2=mı=1piξi,S3=mı=1piAi,S4=mı=1qi,S5=mı=1qiBi,S6=mı=1ri,S7=mı=1riξi,S8=mı=1riAi,S9=mı=1vi,S10=mı=1viBi,Ai=IαT[F(ξi)λIρTIσ0+H(ξi)],Bi=Iα1T[F(ξi)λIρTIσ0+H(ξi)]. (2.3)

    Proof. Applying the right fractional integral operator IαT to the integro-differential equation in (2.1), we get

    y(t)=IαTF(t)λIα+ρTIσ0+H(t)c0c1t, (2.4)

    where c0 and c1 are unknown arbitrary constants. Using (2.4) in the nonlocal closed boundary conditions of (2.1), we obtain

    {(S11)c0+(S2+TS4T)c1=S3+TS5,S6c0+(S7+TS9T)c1=S8+TS10, (2.5)

    where Si,i=1,,10, are given in (2.3).

    Solving the system (2.5) for c0 and c1, we find that

    c0=1Δ[(S7+TS9T)(S3+TS5)(S2+TS4T)(S8+TS10)],c1=1Δ[S6(S3+TS5)+(S11)(S8+TS10)],

    where Δ is given in (2.3). Substituting the above values of c0 and c1 in (2.4) together with the notation (2.3), we obtain the solution (2.2). The converse of this lemma can be obtained by direct computation. This completes the proof.

    This section is devoted to our main results concerning the existence and uniqueness of solutions for the problems (1.1) and (1.2).

    In order to convert the problems (1.1) and (1.2) into a fixed point problem, we define an operator V:XX by using Lemma 2.1 as follows:

    Vy(t)=Tt(st)α1Γ(α)[f(s,y(s))λIρTIσ0+h(s,y(s))]ds+b1(t){mı=1piTξi(sξi)α1Γ(α)[f(s,y(s))λIρTIσ0+h(s,y(s))]dsTmı=1qiTξi(sξ)α2Γ(α1)[f(s,y(s))λIρTIσ0+h(s,y(s))]ds}+b2(t){mı=1riTξi(sξi)α1Γ(α)[f(s,y(s))λIρTIσ0+h(s,y(s))]dsTmi=1viTξi(sξi)α2Γ(α1)[f(s,y(s))λIρTIσ0+h(s,y(s))]ds},tJ, (3.1)

    where X=C([0,T],R) denotes the Banach space of all continuous functions from [0,T]R equipped with the norm y=sup{|y(t)|:t[0,T]}. Notice that the fixed point problem Vy(t)=y(t) is equivalent to the boundary value problems (1.1) and (1.2) and the fixed points of the operator V are its solutions.

    In the forthcoming analysis, we use the following estimates:

    Tt(st)α+ρ1Γ(α+ρ)Iσ0+ds=Tt(st)α+ρ1Γ(α+ρ)s0(su)σ1Γ(σ)dudsTσ(Tt)α+ρΓ(σ+1)Γ(α+ρ+1),Tξi(sξi)α+ρ1Γ(α+ρ)Iσ0+ds=Tξi(sξi)α+ρ1Γ(α+ρ)s0(su)σ1Γ(σ)dudsTσ(Tξi)α+ρΓ(σ+1)Γ(α+ρ+1),

    where we have used uσTσ,ρ,σ>0.

    In the sequel, we set

    Ω1=1Γ(α+1){Tα+¯b1[mı=1|pi|(Tξi)α+αTmı=1|qi|(Tξi)α1]+¯b2[mı=1|ri|(Tξi)α+αTmı=1|vi|(Tξi)α1]},Ω2=|λ|TσΓ(σ+1)Γ(α+ρ+1){Tα+ρ+¯b1[mı=1|pi|(Tξi)α+ρ+(α+ρ)Tmı=1|qi|(Tξi)α+ρ1]+¯b2[mı=1|ri|(Tξi)α+ρ+(α+ρ)Tmı=1|vi|(Tξi)α+ρ1]}, (3.2)

    where

    ¯b1=maxt[0,T]|b1(t)|,¯b2=maxt[0,T]|b2(t)|.

    In the following, Krasnosel'skii's fixed point theorem [31] is applied to prove our first existence result for the problems (1.1) and (1.2).

    Theorem 3.1. Assume that:

    (H1) There exists L>0 such that |f(t,x)f(t,y)|L|xy|,t[0,T],x,yR;

    (H2) There exists K>0 such that |h(t,x)h(t,y)|K|xy|,t[0,T],x,yR;

    (H3) |f(t,y)|δ(t) and |h(t,y)|θ(t), where δ,θC([0,T],R+).

    Then, the problems (1.1) and (1.2) has at least one solution on [0,T] if Lγ1+Kγ2<1, where

    γ1=TαΓ(α+1),γ2=|λ|Tα+ρ+σΓ(σ+1)Γ(α+ρ+1). (3.3)

    Proof. Introduce the ball Bη={yX:yη}, with

    ηδΩ1+θΩ2. (3.4)

    Now we verify the hypotheses of Krasnosel'skii's fixed point theorem in three steps by splitting the operator V:XX defined by (3.1) on Bη as V=V1+V2, where

    V1y(t)=Tt(st)α1Γ(α)[f(s,y(s))λIρTIσ0+h(s,y(s))]ds,tJ,V2y(t)=b1(t){mı=1piTξi(sξi)α1Γ(α)[f(s,y(s))dsλIρTIσ0+h(s,y(s))]dsTmı=1qiTξi(sξ)α2Γ(α1)[f(s,y(s))λIρTIσ0+h(s,y(s))]ds}+b2(t){mı=1riTξi(sξi)α1Γ(α)[f(s,y(s))λIρTIσ0+h(s,y(s))]dsTmi=1viTξi(sξi)α2Γ(α1)[f(s,y(s))λIρTIσ0+h(s,y(s))]ds},tJ.

    (i) For y,xBη, we have

    V1y+V2xsupt[0,T]{Tt(st)α1Γ(α)[|f(s,y(s))|+|λ|IρTIσ0+|h(s,y(s))|]ds+|b1(t)|{mı=1|pi|Tξi(sξi)α1Γ(α)[|f(s,x(s))|+|λ|IρTIσ0+|h(s,x(s))|]ds+Tmı=1|qi|Tξi(sξi)α2Γ(α1)[|f(s,x(s))|+|λ|IρTIσ0+|h(s,x(s))|]ds}+|b2(t)|{mı=1|ri|Tξi(sξi)α1Γ(α)[|f(s,x(s))|+|λ|IρTIσ0+|h(s,x(s))|]ds+Tmı=1|vi|Tξi(sξi)α2Γ(α1)[|f(s,x(s))|+|λ|IρTIσ0+|h(s,x(s))|]ds}}δsupt[0,T]{Tt(st)α1Γ(α)ds+|b1(t)|[mı=1|pi|Tξi(sξi)α1Γ(α)ds+Tmı=1|qi|Tξi(sξi)α2Γ(α1)ds]+|b2(t)|[mı=1|ri|Tξi(sξi)α1Γ(α)ds+Tmı=1|vi|Tξi(sξi)α2Γ(α1)ds]}+θ|λ|supt[0,T]{Tt(st)α+ρ1Γ(α+ρ)Iσ0+ds+|b1(t)|[mı=1Tξi(sξi)α+ρ1Γ(α+ρ)Iσ0+ds+Tmı=1|qi|Tξi(sξi)α+ρ2Γ(α+ρ1)Iσ0+ds]+|b2(t)|[mı=1|ri|Tξi(sξi)α+ρ1Γ(α+ρ)Iσ0+ds+Tmı=1|vi|Tξi(sξi)α+ρ2Γ(α+ρ1)Iσ0+ds]}δsupt[0,T]{(Tt)αΓ(α+1)+|b1(t)|[mı=1|pi|(Tξi)αΓ(α+1)+Tmı=1|qi|(Tξi)α1Γ(α)]+|b2(t)|[mı=1|ri|(Tξi)αΓ(α+1)+Tmı=1|vi|(Tξi)α1Γ(α)]}+θ|λ|TσΓ(σ+1)supt[0,T]{(Tt)α+ρΓ(α+ρ+1)ds+|b1(t)|[mı=1(Tξi)α+ρΓ(α+ρ+1)+Tmı=1|qi|(Tξi)α+ρ1Γ(α+ρ)]+|b2(t)|[mı=1|ri|(Tξi)α+ρΓ(α+ρ+1)+Tmı=1|vi|(Tξi)α+ρ1Γ(α+ρ)]}δΓ(α+1){Tα+¯b1[mı=1|pi|(Tξi)α+αTmı=1|qi|(Tξi)α1]+¯b2[mı=1|ri|(Tξi)α+αTmı=1|vi|(Tξi)α1]}+θ|λ|TσΓ(σ+1)Γ(α+ρ+1){Tα+ρ+¯b1[mı=1|pi|(Tξi)α+ρ+(α+ρ)Tmı=1|qi|(Tξi)α+ρ1]+¯b2[mı=1|ri|(Tξi)α+ρ+(α+ρ)Tmı=1|vi|(Tξi)α+ρ1]}βΩ1+θΩ2<η,

    where we used (3.4). Thus V1y+V2xBη.

    (ii) Using (H1) and (H2), it is easy to show that

    V1yV1xsupt[0,T]{Tt(st)α1Γ(α)|f(s,y(s))f(s,x(s))|ds+|λ|Tt(st)α+ρ1Γ(α+ρ)Iσ0+|h(s,y(s))h(s,x(s))|ds}(Lγ1+Kγ2)yx,

    which, in view of the condition Lγ1+Kγ2<1, implies that the operator V1 is a contraction.

    (iii) Continuity of the functions f,h implies that the operator V2 is continuous. In addition, V2 is uniformly bounded on Bη as

    V2ysupt[0,T]{|b1(t)|[mı=1|pi|Tξi(sξi)α1Γ(α)|f(s,y(s))|ds+|λ|mı=1|pi|Tξi(sξi)α+ρ1Γ(α+ρ)Iσ0+|h(s,y(s))|ds+Tmı=1|qi|Tξi(sξi)α2Γ(α1)|f(s,y(s))|ds+|λ|Tmı=1|qi|Tξi(sξi)α+ρ2Γ(α+ρ1)Iσ0+|h(s,y(s))|ds]+|b2(t)|[mı=1|ri|Tξi(sξi)α1Γ(α)|f(s,y(s))|+|λ|mı=1|ri|Tξi(sξi)α+ρ1(α+ρ)Iσ0+|h(s,y(s))|ds+Tmı=1|vi|Tξi(sξi)α2Γ(α1)|f(s,y(s))|ds+|λ|Tmı=1|vi|Tξi(sξi)α+ρ2)Γ(α+ρ1)|h(s,y(s))|ds]}δsupt[0,T]{|b1(t)|[mı=1|pi|Tξi(sξi)α1Γ(α)ds+Tmı=1|qi|Tξi(sξi)α2Γ(α1)ds]+|b2(t)|[mı=1|ri|Tξi(sξi)α1Γ(α)ds+Tmı=1|vi|Tξi(sξi)α2Γ(α1)ds]}+|λ|θsupt[0,T]{|b1(t)|[mı=1|pi|Tξi(sξi)α+ρ1Γ(α+ρ)Iσ0+ds+Tmı=1|qi|Tξi(sξi)α+ρ2Γ(α+ρ1)Iσ0+ds]+|b2(t)|[mı=1|ri|Tξi(sξi)α+ρ1Γ(α+ρ)Iσ0+ds+Tmı=1|vi|Tξi(sξi)α+ρ2)Γ(α+ρ1)Iσ0+ds]}δsupt[0,T]{|b1(t)|[mı=1|pi|(Tξi)αΓ(α+1)+Tmı=1|qi|(Tξi)α1Γ(α)]+|b2(t)|[mı=1|ri|(Tξi)αΓ(α+1)+Tmı=1|vi|(Tξi)α1Γ(α)]+|λ|θTσΓ(σ+1)supt[0,T]{|b1(t)|[mı=1|pi|(Tξi)α+ρΓ(α+ρ+1)+Tmı=1|qi|(Tξi)α+ρ1Γ(α+ρ)]+|b2(t)|[mı=1|ri|(Tξi)α+ρΓ(α+ρ+1)+Tmı=1|vi|(Tξi)α+ρ1)Γ(α+ρ)]}δ(Ω1γ1)+θ(Ω2γ2),

    where Ωi, and γi, i=1,2, are defined in (3.2) and (3.3), respectively. To show the compactness of V2, we fix sup(t,y)[0,T]×Bη|f(t,y)|=¯f, sup(t,y)[0,T]×Bη|h(t,y)|=¯h. Then, for 0<t1<t2<T, we have

    |(V2y)(t2)(V2y)(t1)||b1(t2)b1(t1)|{mı=1|pi|Tξi(sξi)α1Γ(α)[|f(s,y(s))|+|λ|IρTIσ0+|h(s,y(s))|]ds+Tmı=1|qi|Tξi(sξi)α2Γ(α1)[|f(s,y(s))|+|λ|IρTIσ0+|h(s,y(s))|]ds}+|b2(t2)b2(t1)|{mı=1|ri|Tξ(sξ)α1Γ(α)[|f(s,y(s))|+λ|IρTIσ0+|h(s,y(s))|ds]+Tmı=1|vi|Tξi(sξi)α2Γ(α1)[|f(s,y(s))|+|λ|IρTIσ0+|h(s,y(s))|]ds}|S6||t2t1||Δ|{¯fΓ(α+1)[mı=1|pi|(Tξi)α+αTmi=1|qi|(Tξi)α1]+¯h|λ|TσΓ(σ+1)Γ(α+ρ+1)[mı=1|pi|(Tξi)α+ρ+(α+ρ)Tmi=1|qi|(Tξi)α+ρ1]}+|S11||t2t1||Δ|{¯fΓ(α+1)[mı=1|ri|(Tξi)α+αTmi=1|vi|(Tξi)α1]+¯h|λ|TσΓ(σ+1)Γ(α+ρ+1)[mı=1|ri|(Tξi)α+ρ+(α+ρ)Tmi=1|vi|(Tξi)α+ρ1]},

    which tends to zero, independent of y, as t2t1. This shows that V2 is equicontinuous. It is clear from the foregoing arguments that the operator V2 is relatively compact on Bη. Hence, by the Arzelá-Ascoli theorem, V2 is compact on Bη.

    In view of the foregoing arguments (i)–(iii), the hypotheses of the Krasnosel'skii's fixed point theorem [31] are satisfied. Hence, the operator V1+V2=V has a fixed point, which implies that the problems (1.1) and (1.2) has at least one solution on [0,T]. The proof is finished.

    Remark 3.1. Interchanging the roles of the operators V1 and V2 in the previous result, the condition Lγ1+Kγ2<1 changes to the following one:

    L(Ω1γ1)+K(Ω2γ2)<1,

    where Ω1,Ω2 and γ1,γ2 are defined in (3.2) and (3.3) respectively.

    The following existence result relies on Leray-Schauder nonlinear alternative [32].

    Theorem 3.2. Suppose that the following conditions hold:

    (H4) There exist continuous nondecreasing functions ϕ1,ϕ2:[0,)(0,) such that (t,y)[0,1]×R, |f(t,y)|ω1(t)ϕ1(y) and |h(t,y)|ω2(t)ϕ2(y), where ω1,ω2C([0,T],R+);

    (H5)There exists a constant M>0 such that

    Mω1ϕ1(M)Ω1+ω2ϕ2(M)Ω2>1.

    Then, the problems (1.1) and (1.2) has at least one solution on [0,T].

    Proof. We firstly show that the operator V:XX defined by (3.1) is completely continuous.

    (i) V maps bounded sets into bounded sets in X.

    Let yBr={yX:yr}, where r is a fixed number. Then, using the strategy employed in the proof of Theorem 3.1, we obtain

    Vyω1ϕ1(r)Γ(α+1){Tα+¯b1[mı=1|pi|(Tξi)α+αTmı=1|qi|(Tξi)α1]+¯b2[mı=1|ri|(Tξi)α+αTmı=1|vi|(Tξi)α1]}+|λ|Tσω2ϕ2(r)Γ(σ+1)Γ(α+ρ+1){Tα+ρ+¯b1[mı=1|pi|(Tξi)α+ρ+(α+ρ)Tmı=1|qi|(Tξi)α+ρ1]+¯b2[mı=1|ri|(Tξi)α+ρ+(α+ρ)Tmı=1|vi|(Tξi)α+ρ1]}=ω1ϕ1(r)Ω1+ω2ϕ2(r)Ω2<.

    (ii) V maps bounded sets into equicontinuous sets.

    Let 0<t1<t2<T and yBr. Then, we obtain

    |Vy(t2)Vy(t1)||Tt2(st2)α1(st1)α1Γ(α)f(s,y(s))ds+t2t1(st1)α1Γ(α)f(s,y(s))dsλTt2(st2)α+ρ1(st1)α+ρ1Γ(α+ρ)Iσ0+h(s,y(s))dsλt2t1(st1)α+ρ1Γ(α+ρ)Iσ0+h(s,y(s))ds|+|b1(t2)b1(t1)|{|mı=1piTξi(sξ)α1Γ(α)[f(s,y(s))λIρTIσ0+h(s,y(s))]ds|+|Tmı=1qiTξi(sξi)α2Γ(α1)[f(s,y(s))λIρTIσ0+h(s,y(s))]ds|}+|b2(t2)b2(t1)|{|mı=1riTξi(sξ)α1Γ(α)[f(s,y(s))λIρTIσ0+h(s,y(s))]ds|+|Tmı=1viTξi(sξi)α2Γ(α1)[f(s,y(s))λIρTIσ0+h(s,y(s))]ds|}|Tt2(st2)α1(st1)α1Γ(α)f(s,y(s))ds+t2t1(st1)α1Γ(α)f(s,y(s))ds|+|λTt2(st2)α+ρ1(st1)α+ρ1Γ(α+ρ)Iσ0+h(s,y(s))ds+λt2t1(st1)α+ρ1Γ(α+ρ)Iσ0+h(s,y(s))ds|+|S6||t2t1|Δ|{|mı=1piTξi(sξi)α1Γ(α)[f(s,y(s))λIρTIσ0+h(s,y(s))]ds|+|Tmı=1qiTξi(sξi)α2Γ(α1)[f(s,y(s))λIρTIσ0+h(s,y(s))]ds|}+|S11||t2t1|Δ{|mı=1riTξi(sξi)α1Γ(α)[f(s,y(s))λIρTIσ0+h(s,y(s))]ds|+|Tmı=1viTξi(sξi)α2Γ(α1)[f(s,y(s))λIρTIσ0+h(s,y(s))]ds|}ω1(t)Φ1(r)Γ(α+1){|(Tt2)α(Tt1)α|+2|(t2t1)α|+|t2t1||Δ|[|S6|(mı=1|pi|(Tξi)α+αTmı=1|qi|(Tξi)α1)+|S11|(mı=1|ri|(Tξi)α+αTmı=1|vi|(Tξi)α1)]}+|λ|Tσω2(t)ϕ2(r)Γ(σ+1)Γ(α+ρ+1){|(Tt2)α+ρ(Tt1)α+ρ+2|t2t1|α+ρ+|t2t1||Δ|[|S6|(mı=1|pi|(Tξi)α+ρ+(α+ρ)Tmı=1|qi|(Tξi)α+ρ1)+|S11|(mı=1|ri|(Tξi)α+ρ+(α+ρ)Tmı=1|vi|(Tξi)α+ρ1)]}.

    Notice that the right-hand side of the above inequality tends to 0 as t2t1, independent of yBr. Thus, it follows by the Arzelá–Ascoli theorem that the operator V:XX is completely continuous.

    The conclusion of the Leray-Schauder nonlinear alternative [32] will be applicable once it is shown that there exists an open set UC([0,T],R) with yνVy for ν(0,1) and yU. Let yC([0,T],R) be such that y=νVy for ν(0,1). As argued in proving that the operator V is bounded, one can obtain that

    |y(t)|=|νVy(t)||ω1(t)|ϕ(y)Ω1+|ω2(t)|ψ(y)Ω2,

    which can be written as

    yω1ϕ(y)Ω1+ω2ψ(y)Ω21.

    On the other hand, we can find a positive number M such that yM by assumption (H5). Let us set

    W={yX:y<M}.

    Clearly, W contains a solution only when y=M. In other words, we cannot find a solution yW satisfying y=νVy for some ν(0,1). In consequence, the operator V has a fixed point y¯W, which is a solution of the problems (1.1) and (1.2). The proof is finished.

    Here we apply Banach contraction mapping principle to establish the uniqueness of solutions for the problems (1.1) and (1.2).

    Theorem 3.3. If the conditions (H1) and (H2) hold, then the problems (1.1) and (1.2) has a unique solution on [0,T] if

    LΩ1+KΩ2<1, (3.5)

    where Ω1 and Ω2 are defined in (3.2).

    Proof. In the first step, we show that VBκBκ, where Bκ={yX:yκ} with

    κf0Ω1+h0Ω21(LΩ1+KΩ2),f0=supt[0,T]|f(t,0)|,h0=supt[0,T]|h(t,0)|.

    For yBκ and using the condition (H1), we have

    |f(t,y)|=|f(t,y)f(t,0)+f(t,0)||f(t,y)f(t,0)|+|f(t,0)|Ly+f0Lr+f0. (3.6)

    Similarly, using (H2), we get

    |h(t,y)|Kr+h0. (3.7)

    In view of (3.6) and (3.7), we obtain

    Vysupt[0,T]|Vy(t)|supt[0,T]{Tt(st)α1Γ(α)[|f(s,y(s))|+|λ|IρTIσ0+|h(s,y(s))|]ds+|b1(t)|{mı=1piTξi(sξi)α1Γ(α)[|f(s,y(s))|+|λ|IρTIσ0+|h(s,y(s))|]ds+Tmı=1|qi|Tξi(sξi)α2Γ(α1)[|f(s,y(s))|+|λ|IρTIσ0+|h(s,y(s))|]ds}+|b2(t)|{mı=1riTξi(sξi)α1Γ(α)[|f(s,y(s))|+|λ|IρTIσ0+|h(s,y(s))|]ds+Tmi=1|vi|Tξi(sξi)α2Γ(α1)[|f(s,y(s))|+|λ|IρTIσ0+|h(s,y(s))|]ds}}(Lr+f0)supt[0,T]{Tt(st)α1Γ(α)ds+|b1(t)|[mi=1|pi|Tξi(sξi)α1Γ(α)ds+Tmı=1|qi|Tξi(sξi)α2Γ(α1)ds]+|b2(t)|[mi=1|ri|Tξi(sξi)α1Γ(α)ds+Tmı=1|vi|Tξi(sξi)α2Γ(α1)ds]}+|λ|(Kr+h0)supt[0,T]{Tt(st)α+ρ1Γ(α+ρ)Iσ0+ds+|b1(t)|[mi=1|pi|Tξi(sξi)α+ρ1Γ(α+ρ)Iσ0+ds+Tmı=1|qi|Tξi(sξi)α+ρ2Γ(α+ρ1)Iσ0+ds]+|b2(t)|[mi=1|ri|Tξi(sξi)α+ρ1Γ(α+ρ)Iσ0+ds+Tmı=1|vi|Tξi(sξi)α+ρ2Γ(α+ρ1)Iσ0+ds]}(Lr+f0)Γ(α+1){Tα+¯b1[mı=1|pi|(Tξi)α+αTmı=1|qi|(Tξi)α1]+¯b2[mı=1|ri|(Tξi)α+αTmı=1|vi|(Tξi)α1]}+Tσ|λ|(Kr+h0)Γ(σ)Γ(α+ρ+1){Tα+ρ+¯b1[mı=1|pi|(Tξi)α+ρ+(α+ρ)Tmı=1|qi|(Tξi)α+ρ1]+¯b2[mı=1|ri|(Tξi)α+ρ+(α+ρ)Tmı=1|vi|(Tξi)α+ρ1]}=(Lr+f0)Ω1+(Kr+h0)Ω2<κ,

    which implies that \mathcal{V}y \in \mathcal{B}_\kappa , for any y \in \mathcal{B}_\kappa . Therefore, \mathcal{V}\mathcal{B}_\kappa\subset\mathcal{B}_\kappa .

    Next, we prove that \mathcal{V} is a contraction. For that, let x, y \in \mathcal{X} and t\in[0, T] . Then, by the conditions (H_1) and (H_2) , we obtain

    \begin{eqnarray*} &&\|\mathcal{V}y-\mathcal{V}x\| = \sup\limits_{t\in [0,T]}\big|(\mathcal{V}y)(t)-(\mathcal{V}x)(t)\big| \\&\leq& \sup\limits_{t\in [0,T]}\bigg\{\int_t^T\frac{(s-t)^{\alpha-1}}{\Gamma (\alpha)}|f(s,y(s))-f(s,x(s))|ds \\&& +|\lambda|\int_t^T \frac{(s-t)^{\alpha+\rho-1}}{\Gamma (\alpha+\rho)}I^{\sigma}_{0+}|h(s,y(s))-h(s,x(s))|ds\\&& +|b_1(t)|\bigg[\sum\limits_{ı = 1}^m|p_i|\bigg(\int_{\xi_i}^T\frac{(s-\xi_i)^{\alpha-1}}{\Gamma(\alpha)}|f(s,y(s))-f(s,x(s))|ds\\&& +|\lambda|\int_{\xi_i}^T\frac{(s-\xi_i)^{\alpha+\rho-1}}{\Gamma(\alpha+\rho)}I^{\sigma}_{0+}|h(s,y(s))-h(s,x(s))|ds\bigg)\\&& +T\sum\limits_{ı = 1}^m|q_i|\bigg(\int_{\xi_i}^T\frac{(s-\xi_i)^{\alpha-2}}{\Gamma(\alpha-1)}|f(s,y(s))-f(s,x(s))|ds\\&& +|\lambda|\int_{\xi_i}^T\frac{(s-\xi_i)^{\alpha+\rho-2}}{\Gamma(\alpha+\rho-1)}I^{\sigma}_{0+}|h(s,y(s))-h(s,x(s))|ds\bigg)\bigg]\\&& +|b_2(t)|\bigg[\sum\limits_{ı = 1}^m|r_i|\bigg(\int_{\xi_i}^T\frac{(s-\xi_i)^{\alpha-1}}{\Gamma(\alpha)}|f(s,y(s))-f(s,x(s))|ds\\&& +|\lambda|\int_{\xi_i}^T\frac{(s-\xi_i)^{\alpha+\rho-1}}{\Gamma(\alpha+\rho)}I^{\sigma}_{0+}|h(s,y(s))-h(s,x(s))|ds\bigg)\\&& +T\sum\limits_{ı = 1}^m|v_i|\bigg(\int_{\xi_i}^T\frac{(s-\xi_i)^{\alpha-2}}{\Gamma(\alpha-1)}|f(s,y(s))-f(s,x(s))|ds\\&& +|\lambda|\int_{\xi_i}^T\frac{(s-\xi_i)^{\alpha+\rho-2}}{\Gamma(\alpha+\rho-1)}I^{\sigma}_{0+}|h(s,y(s))-h(s,x(s))|ds\bigg)\bigg]\Bigg\} \\&\leq&\; \frac{\mathcal{L} }{\Gamma (\alpha+1)}\bigg\{T^{\alpha}+\overline{b}_1\Big[\sum\limits_{ı = 1}^m |p_i|(T-\xi_i)^{\alpha}+\alpha T\sum\limits_{ı = 1}^m|q_i|(T-\xi_i)^{\alpha-1}\Big]\\&& +\overline{b}_2\Big[\sum\limits_{ı = 1}^m|r_i|(T-\xi_i)^{\alpha}+\alpha T\sum\limits_{ı = 1}^m|v_i|(T-\xi_i)^{\alpha-1}\Big] \bigg\}\\&& +\frac{T^\sigma|\lambda|\mathcal{K}}{\Gamma(\sigma+1)\Gamma(\alpha+\rho+1)}\bigg\{T^{\alpha+\rho}+\overline{b}_1\Big[\sum\limits_{ı = 1}^m |p_i|(T-\xi_i)^{\alpha+\rho}+(\alpha+\rho) T\sum\limits_{ı = 1}^m|q_i|(T-\xi_i)^{\alpha+\rho-1}\Big]\\&& +\overline{b}_2\Big[\sum\limits_{ı = 1}^m|r_i|(T-\xi_i)^{\alpha+\rho}+(\alpha+\rho) T\sum\limits_{ı = 1}^m|v_i|(T-\xi_i)^{\alpha+\rho-1}\Big] \bigg\} \|y-x\|\\ \\& = &(\mathcal{L}\Omega_1+\mathcal{K}\Omega_2)\|y-x\|, \end{eqnarray*}

    which shows that \mathcal{V} is a contraction in view of the condition (3.5). Therefore, we deduce by Banach contraction mapping principle that there exists a unique fixed point for the operator \mathcal{V} , which corresponds to a unique solution for the problems (1.1) and (1.2) on [0, T] . The proof is completed.

    In this subsection, we construct examples for illustrating the abstract results derived in the last two subsections. Let us consider the following problem:

    \begin{equation} \left\{\begin{array} {ll} &D^{9/8}_{1-}y(t)+3I^{7/3}_{1-}I^{3/4}_{0+}h(t,y(t)) = f(t,y(t)),\ t\in J: = [0,1],\\ & y(T) = \sum\limits_{ı = 1}^3 p_iy(\xi_i)+\sum\limits_{ı = 1}^3 q_iy'(\xi_i), \, y'(T) = \sum\limits_{ı = 1}^3 r_i y(\xi_i)+\sum\limits_{ı = 1}^3 v_iy'(\xi_i),\; ,\; 0 < \xi_i < 1. \end{array} \right. \end{equation} (3.8)

    Here \alpha = 9/8, \rho = 7/3, \sigma = 3/4, \lambda = 3, \xi_1 = 3/7, \xi_2 = 2/3, \xi_3 = 4/5, p_1 = 1/2, p_2 = 1/3, p_3 = 1/4, q_1 = -2, q_2 = -3, q_3 = -4, r_1 = 1, r_2 = -1, r_3 = 3, v_1 = -2/7, v_2 = -3/7, v_3 = -4/7. Using the given data, it is found that

    \overline{b}_1 = \max\limits_{t\in [0,1]}|b_1(t)| = |b_1(t)|_{t = 1} \approx 0.1112461491, \; \; \; \overline{b}_2 = \max\limits_{t\in [0,1]}|b_2(t)| = |b_2(t)|_{t = 1}\approx 0.3364235041.

    In consequence, we get \Omega_1\approx 2.517580993, \Omega_2\approx 0.3543113654 ( \Omega_1 , \Omega_2 are defined in (3.2)).

    (i) For illustrating Theorem 3.1, we consider the functions

    \begin{equation} f(t,y) = \frac{m_1}{2t+25}\Big(\frac{y^2}{1+y^2}+\cos 3t+1\Big), \; \; h(t,y) = \frac{m_2}{3\sqrt{t^2+64}}\bigg(2\tan^{-1}y+\sin t+e^{-t/2}\bigg), \end{equation} (3.9)

    where m_1 and m_2 are finite positive real numbers. Observe that

    |f(t,y)|\leq \delta(t) = \frac{m_1(2+\cos 3t)}{2t+25}, \; \; \; \; \; |h(t,y)|\leq \theta(t) = \frac{m_2(\pi+\sin t+e^{-t/2})}{3\sqrt{t^2+64}},

    and f(t, y) and h(t, y) respectively satisfy the conditions (H_1) and (H_2) with \mathcal{L} = 2m_1/25 and \mathcal{K} = m_2/24. Moreover, \gamma_1 \approx0.9438765902 and \gamma_2 \approx 0.2972831604. By the condition \mathcal{L}\gamma_1+\mathcal{K}\gamma_2 < 1 , we get

    \begin{equation} 0.0755101272 m_1+0.0123867984 m_2 < 1 \end{equation} (3.10)

    For the values of m_1 and m_2 satisfying the inequality (3.10), the hypothesis of Theorem 3.1 is satisfied. Hence, it follows by the conclusion of Theorem 3.1 that the problem (3.8) with f(t, y) and h(t, y) given in (3.9) has at least one solution on [0, 1]. If the values m_1 and m_2 do not satisfy the inequality (3.10), then Theorem 3.1 does not guarantee the existence of at least one solution to the problem (3.8) with f(t, y) and h(t, y) given in (3.9) for such values of m_1 and m_2 .

    (ii) In order to illustrate Theorem 3.2, we take the following functions (instead of (3.9)) in the problem (3.8):

    \begin{equation} f(t,y) = \frac{e^{-3t}}{t^2+3}\big[\sin y+1/5\big], \; \; \; \; \; h(t,y) = \frac{2}{7\sqrt{t^3+1}}\Big(\frac{|y|}{1+|y|}|y|+\pi/4\Big). \end{equation} (3.11)

    Observe that the assumption (H_4) is satisfied as |f(t, y)| \le \omega_1(t) \phi_1(\|y\|) and |h(t, y)| \le \omega_2(t) \phi_2(\|y\|), where \omega_1(t) = e^{-3t}/(t^2+3) , \phi_1(\|y\|) = (\|y\|+1/5), \omega_2(t) = 2/(7\sqrt{t^3+1}) , \phi_2(\|y\|) = (\|y\|+\pi/4). It is easy to see that \| \omega_1\| = 1/3 and \omega_2\| = 2/7. By the condition ( H_5 ), we find that M > 4.151876169 . Thus, all the conditions of Theorem 3.2 are satisfied and hence the problem (3.8) with f(t, y) and h(t, y) given by (3.11) has at least one solution on [0, 1].

    (iii) The conditions (H_1) and (H_2) are respectively satisfied by f(t, y) and h(t, y) defined in (3.9) with \mathcal{L} = 2m_1/25 and \mathcal{K} = m_2/24. By the condition (3.5), we have

    \begin{equation} 0.20140647944 m_1+0.0147629736 m_2 < 1. \end{equation} (3.12)

    Clearly, all the assumptions of Theorem 3.3 hold true with the values of m_1 and m_2 satisfying the inequality (3.12). In consequence, the problem (3.8) with f(t, y) and h(t, y) given in (3.11) has a unique solution on [0, 1]. In case, we take m_1 = m_2 = m in (3.9), then the condition (3.12) implies the existence of a unique solution for the problem at hand for m < 4.62600051. One can notice that Theorem 3.1 does not guarantee the existence of a unique solution to the problem (3.8) with f(t, y) and h(t, y) given in (3.9) for the values of m_1 and m_2, which do not satisfy the inequality (3.12).

    In this study, we discussed the existence and uniqueness of solutions under different assumptions for a boundary value problem involving a right Caputo fractional derivative with usual and mixed Riemann-Liouville integrals type nonlinearities, equipped with nonlocal multipoint version of the closed boundary conditions. Our results are not only new in the given configuration, but also yield some new results as special cases. Here are some examples.

    ● If \lambda = 0 in (1.1), then our results correspond to the fractional differential equation {^CD^{\alpha}_{T-}}y(t) = f(t, y(t)) with the boundary conditions (1.2).

    ● In case, we take q_i = 0, r_i = 0, \forall i = 1, \dots, m in the results of this paper, we obtain the ones for the Eq (1.1) supplemented with boundary conditions: y(T) = \sum\limits_{ı = 1}^m p_iy(\xi_i), \; \; y'(T) = \sum\limits_{ı = 1}^m v_iy'(\xi_i).

    ● We get the results for the Eq (1.1) complemented with boundary conditions: y(T) = T\sum\limits_{ı = 1}^m q_iy'(\xi_i), \; \; Ty'(T) = \sum\limits_{ı = 1}^m r_i y(\xi_i) by taking p_i = 0, v_i = 0, \, \forall i = 1, \dots, m in the obtained results.

    The Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU), Jeddah, Saudi Arabia, has funded this project under grant No. (KEP-PhD: 35-130-1443).

    The authors declare no conflict of interest.



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