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Research article Special Issues

Asymptotic behavior of ground states for a fractional Choquard equation with critical growth

  • Received: 09 December 2020 Accepted: 17 January 2021 Published: 29 January 2021
  • MSC : 35J50, 35Q40, 58E05

  • In this paper, we are concerned with the following fractional Choquard equation with critical growth:

    (Δ)su+λV(x)u=(|x|μF(u))f(u)+|u|2s2uinRN,

    where s(0,1), N>2s, μ(0,N), 2s=2NN2s is the fractional critical exponent, V is a steep well potential, F(t)=t0f(s)ds. Under some assumptions on f, the existence and asymptotic behavior of the positive ground states are established. In particular, if f(u)=|u|p2u, we obtain the range of p when the equation has the positive ground states for three cases 2s<N<4s or N=4s or N>4s.

    Citation: Xianyong Yang, Qing Miao. Asymptotic behavior of ground states for a fractional Choquard equation with critical growth[J]. AIMS Mathematics, 2021, 6(4): 3838-3856. doi: 10.3934/math.2021228

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  • In this paper, we are concerned with the following fractional Choquard equation with critical growth:

    (Δ)su+λV(x)u=(|x|μF(u))f(u)+|u|2s2uinRN,

    where s(0,1), N>2s, μ(0,N), 2s=2NN2s is the fractional critical exponent, V is a steep well potential, F(t)=t0f(s)ds. Under some assumptions on f, the existence and asymptotic behavior of the positive ground states are established. In particular, if f(u)=|u|p2u, we obtain the range of p when the equation has the positive ground states for three cases 2s<N<4s or N=4s or N>4s.



    The Sturm-Liouville problem arises within many areas of science, engineering and applied mathematics. It has been studied for more than two decades. Many physical, biological and chemical processes are described using models based on it (see [1,2,3], [8], [9] and [11]).

    For the homogeneous Sturm-Liouville problem with nonlocal conditions you can see [2], [9] and [11,12,13,14,15]. For the nonhomogeneous equation see [7]. In [7] the authors studied the nonhomogeneous Sturm-Liouville boundary value problem of the differential equation

    x(t)+m(t)=λ2x(t),t(0,π),

    with the conditions

    x(0)=0,x(ξ)+λx(ξ)=0,ξ(0,π].

    Here, we are concerned, firstly, with the nonlocal problem of the nonlinear differential inclusion

    x(t)F(t,λx(t)),a.e.t(0,π), (1.1)

    with the nonlocal conditions (η>ξ)

    x(0)λx(0)=0andηξx(τ)dτ=0,ξ[0,π),η(0,π]. (1.2)

    For

    h(t,λ)+λ2x(t)=f(t,λx(t))F(t,λx(t)),

    we study the existence of multiple solutions (eignevalues and eignefunctions) of the nonhomogeneous Sturm-Liouville problem of the differential equation

    x(t)+h(t,λ)=λ2x(t),t(0,π), (1.3)

    with the conditions (1.2).

    The special case of the nonlocal condition (1.2)

    x(0)λx(0)=0andπ0x(τ)dτ=0, (1.4)

    will be considered.

    Consider the nonlocal boundary value problem of the nonlinear differential inclusion (1.1)-(1.2) under the following assumptions.

    (ⅰ) The set F(t,x) is nonempty, closed and convex for all (t,x)[0,1]×R×R.

    (ⅱ) F(t,x) is measurable in t[0,1] for every x,yR.

    (ⅲ) F(t,x) is upper semicontinuous in x and y for every t[0,1].

    (ⅳ) There exist a bounded measurable function m:[0,1]R and a constant λ, such that

    F(t,x)=sup{|f|:fF(t,x)}|m(t)|+λ2|x|.

    Remark 1. From the assumptions (i)-(iv) we can deduce that (see [1], [5] and [6]) there exists fF(t,x), such that

    (v) f:I×RR is measurable in t for every x,yR and continuous in x for t[0,1] and there exist a bounded measurable function m:[0,π]R and a constant λ2 such that

    |f(t,x)||m(t)|+λ2|x|,

    and f satisfies the nonlinear differential equation

    x(t)=f(t,λx(t)),a.e.t(0,π). (2.1)

    So, any solution of (2.1) is a solution of (1.1).

    (ⅵ) λ(ηξ)2,λR.

    (ⅶ)

    2(1+|λ|π)π2+π|A|λ2π<1.

    For the integral representation of the solution of (2.1) and (1.2) we have the following lemma.

    Lemma 2.1. If the solution of the problem (2.1) and (1.2) exists, then it can be represented by the integral equation

    x(t)=2(1+λt)A[η0(ηs)22f(s,λx(s))dsξ0(ξs)22f(s,λx(s))ds]t0(ts)f(s,λx(s))ds, (2.2)

    where A=(ηξ)[2+λ(ηξ)]0.

    Proof. Integrating both sides of Eq (2.1) twice, we obtain

    x(t)x(0)tx(0)=t0(ts)f(s,λx(s))ds (2.3)

    and using the assumption x(0)λx(0)=0, we obtain

    x(0)=1λx(0). (2.4)

    The assumption ηξx(τ)dτ=0 implies that

    x(0)ηξdτ+x(0)ηξτdτ=ηξτ0(τs)f(s,λx(s))dsdτ,(ηξ)x(0)+(ηξ)22λx(0)=ξ0ηξ(τs)dτf(s,λx(s))ds+ηξηs(τs)dτf(s,λx(s))ds,(ηξ)[2+λ(ηξ)]2x(0)=ξ0[(ηs)22(ξs)22]f(s,λx(s))ds+ηξ(ηs)22f(s,λx(s))ds=η0(ηs)22f(s,λx(s))dsξ0(ξs)22f(s,λx(s))ds

    and we can get

    x(0)=2(ηξ)[2+λ(ηξ)][η0(ηs)22f(s,λx(s))dsξ0(ξs)22f(s,λx(s))ds]=2A[η0(ηs)22f(s,λx(s))dsξ0(ξs)22f(s,λx(s))ds]. (2.5)

    Substituting (2.5) into (2.4), we obtain

    x(0)=2λA[η0(ηs)22f(s,λx(s))dsξ0(ξs)22f(s,λx(s))ds]. (2.6)

    Now from (2.3), (2.5) and (2.6), we obtain

    x(t)=2(1+λt)A[η0(ηs)22f(s,λx(s))dsξ0(ξs)22f(s,λx(s))ds]t0(ts)f(s,λx(s))ds.

    To complete the proof, differentiate equation (2.2) twice, we obtain

    x(t)=2λA[η0(ηs)22f(s,λx(s))dsξ0(ξs)22f(s,λx(s))ds]t0f(s,λx(s))ds,

    and

    x(t)=f(t,λx(t)),a.e.t(0,T).

    Now

    x(0)=2λA[η0(ηs)22f(s,λx(s))dsξ0(ξs)22f(s,λx(s))ds],

    and

    λx(0)=2λA[η0(ηs)22f(s,λx(s))dsξ0(ξs)22f(s,λx(s))ds].

    From that, we get x(0)λx(0)=0.

    Now, to ensure that ηξx(τ)dτ=0,

    we have

    ηξ2(1+λt)A=2(ηξ)+λ(η2ξ2)A=(ηξ)[2+λ(ηξ)]A=1,

    from that, we obtain as before

    ηξx(τ)dτ=ηξ2(1+λτ)Adτ[η0(ηs)22f(s,λx(s))dsξ0(ξs)22f(s,λx(s))ds]ηξτ0(τs)f(s,λx(s))dsdτ,=η0(ηs)22f(s,λx(s))dsξ0(ξs)22f(s,λx(s))dsη0(ηs)22f(s,λx(s))ds+ξ0(ξs)22f(s,λx(s))ds=0.

    This proves the equivalence between the integral equation (2.2) and the nonlocal boundary value problem (1.1)-(1.2).

    Now, for the existence of at least one continuous solution for the problem of the integral equation (2.2), we have the following theorem.

    Theorem 2.1. Let the assumptions (v)-(vii) be satisfied, then there exists at least one solution xC[0,π] of the nonlocal boundary value problem (2.1) and (1.2). Moreover, from Remark 1, then there exists at least one solution xC[0,π] of the nonlocal boundary value problem (1.1)-(1.2).

    Proof. Define the set QrC[0,π] by

    Qr={xC:∥x∥≤r},r2(1+|λ|π)π2+πmL1|A|[2(1+|λ|π)π2+π]λ2π.

    It is clear that the set Qr is nonempty, closed and convex.

    Define the operator T associated with (2.2) by

    Tx(t)=2(1+λt)A[η0(ηs)22f(s,λx(s))dsξ0(ξs)22f(s,λx(s))ds]t0(ts)f(s,λx(s))ds.

    Let xQr, we have

    |Tx(t)|2(1+|λ|t)|A|[η0(ηs)22|f(s,λx(s))|ds+ξ0(ξs)22|f(s,λx(s))ds|]+t0(ts)|f(s,λx(s))|ds,2(1+|λ|π)π2|A|π0{|m(s)|+λ2|x(s)|}ds+ππ0{|m(s)|+λ2|x(s)|}ds,[2(1+|λ|π)π2|A|+π]{mL1+λ2πx},2(1+|λ|π)π2+π|A|{mL1+λ2πr}r,

    and we have

    2(1+|λ|π)π2+π|A|mL1r(12(1+|λ|π)π2+π|A|λ2π).

    Then T:QrQr and the class {Tx}Qr is uniformly bounded in Qr.

    In what follows we show that the class {Tx}, xQr is equicontinuous. For t1,t2[0,π],t1<t2 such that |t2t1|<δ, we have

    Tx(t2)Tx(t1)=2(1+λt2)A[η0(ηs)22f(s,λx(s))dsξ0(ξs)22f(s,λx(s))ds]t20(t2s)f(s,λx(s))ds2(1+λt1)A[η0(ηs)22f(s,λx(s))dsξ0(ξs)22f(s,λx(s))ds]t10(t2s)f(s,λx(s))ds,|Tx(t2)Tx(t1)|=|2(1+λt2)A[η0(ηs)22f(s,λx(s))dsξ0(ξs)22f(s,λx(s))ds]t20(t2s)f(s,λx(s))ds2(1+λt1)A[η0(ηs)22f(s,λx(s))dsξ0(ξs)22f(s,λx(s))ds]+t10(t1s)f(s,λx(s))ds|,2|λ|(t2t1)A[η0(ηs)22|f(s,λx(s))|ds+ξ0(ξs)22|f(s,λx(s))|ds]+(t2t1)t10|f(s,λx(s))|ds+πt2t1|f(s,λx(s))|ds,2|λ|(t2t1)π2Aπ0|f(s,λx(s))|ds+(t2t1)π0|f(s,λx(s))|ds+πt2t1|f(s,λx(s))|ds,2|λ|(t2t1)π2A{mL1+λ2x}+(t2t1){mL1+λ2x}+πt2t1{|m(s)|+λ2|x(s)|}ds.

    Hence the class of function {Tx}, xQr is equicontinuous. By Arzela-Ascolis [4] Theorem, we found that the class {Tx} is relatively compact.

    Now we prove that T:QrQr is continuous.

    Let {xn}Qr, such that xnx0Qr, then

    Txn(t)=2(1+λt)A[η0(ηs)22f(s,λxn(s))dsξ0(ξs)22f(s,λxn(s))ds]t0(ts)f(s,λxn(s))ds,

    and

    limnTxn(t)=limn{2(1+λt)A[η0(ηs)22f(s,λxn(s))dsξ0(ξs)22f(s,λxn(s))ds]t0(ts)f(s,λxn(s))ds}.

    Now, we have

    f(s,xn(s))f(s,x0(s))asn,

    and

    |f(s,λxn(s))|m(s)+λ2|xn|L1[0,π],

    then applying Lebesgue Dominated convergence theorem [4], we obtain

    limnTxn(t)=2(1+λt)A[η0(ηs)22limnf(s,λxn(s))dsξ0(ξs)22limnf(s,λxn(s))ds]t0(ts)limnf(s,λxn(s))ds,=2(1+λt)A[η0(ηs)22f(s,λx0(s))dsξ0(ξs)22f(s,λx0(s))ds]t0(ts)f(s,λx0(s))ds=F(x0).

    Then Txn(t)Tx0(t). Which means that the operator T is continuous.

    Since all conditions of Schauder theorem [4] are hold, then T has a fixed point in Qr, then the integral equation (2.2) has at least one solution xC[0,π].

    Consequently the nonlocal boundary value problem (2.1)-(1.2) has at least one solution xC[0,π]. Moreover, from Remark 1, then there exists at least one solution xC[0,π] of the nonlocal boundary value problem (1.1)-(1.2).

    Now, we have the following corollaries

    Corollary 1. Let λ2x(t)=f(t,λx(t))F(t,λx(t)). Let the assumptions of Theorem 2.1 be satisfied. Then there exists at lease one solution xC[0,π] of

    x(t)=λ2x(t),t(0,T).

    with the nonlocal condition (1.2). Moreover, from Remark 1, there exists at lease one solution xC[0,π] of the problem (1.1)-(1.2).

    Corollary 2. Let the assumptions of Theorem 2.1 be satisfied. Then there exists a solution xC[0,π] of the problem (2.1) and (1.4).

    Proof. Putting ξ=0 and η=π and applying Theorem 2.1 we get the result.

    Taking J=(0,π). Here, we study the existence of maximal and minimal solutions of the problem (2.1) and (1.2) which is equivalent to the integral equation (2.2).

    Definition 3.1. [10] Let q(t) be a solution x(t) of (2.2) Then q(t) is said to be a maximal solution of (2.2) if every solution of (2.2) on J satisfies the inequality x(t)q(t),tJ. A minimal solution s(t) can be defined in a similar way by reversing the above inequality i.e. x(t)s(t),tJ.

    We need the following lemma to prove the existence of maximal and minimal solutions of (2.2).

    Lemma 3.2. Let f(t,x) satisfies the assumptionsin Theorem 2.1 and let x(t),y(t) be continuous functions on J satisfying

    x(t)2(1+λt)A[η0(ηs)22f(s,λx(s))dsξ0(ξs)22f(s,λx(s))ds]t0(ts)f(s,λx(s))ds,y(t)2(1+λt)A[η0(ηs)22f(s,λy(s))dsξ0(ξs)22f(s,λy(s))ds]t0(ts)f(s,λy(s))ds

    where one of them is strict.

    Suppose f(t,x) is nondecreasing function inx. Then

    x(t)<y(t),tJ. (3.1)

    Proof. Let the conclusion (3.1) be false; then there exists t1 such that

    x(t1)=y(t1),t1>0

    and

    x(t)<y(t),0<t<t1.

    From the monotonicity of the function f in x, we get

    x(t1)2(1+λt1)A[η0(ηs)22f(s,λx(s))dsξ0(ξs)22f(s,λx(s))ds]t10(ts)f(s,λx(s))ds,<2(1+λt1)A[η0(ηs)22f(s,λy(s))dsξ0(ξs)22f(s,λy(s))ds]t10(ts)f(s,λy(s))ds<y(t1).

    This contradicts the fact that x(t1)=y(t1);then

    x(t)<y(t),tJ.

    Theorem 3.2. Let the assumptions of Theorem 2.1 besatisfied. Furthermore, if f(t,x) is nondecreasing function inx, then there exist maximal and minimal solutions of (2.2).

    Proof. Firstly, we shall prove the existence of maximal solution of (2.2). Let ϵ>0 be given. Now consider the integral equation

    xϵ(t)=2(1+λt)A[η0(ηs)22fϵ(s,λxϵ(s))dsξ0(ξs)22fϵ(s,λxϵ(s))ds]t0(ts)fϵ(s,λxϵ(s))ds, (3.2)

    where

    fϵ(t,xϵ(t))=f(t,xϵ(t))+ϵ.

    Clearly the function fϵ(t,xϵ) satisfies assumption (v) and

    |fϵ(t,xϵ)||m(t)|+λ2|x|+ϵ|m1(t)|+λ2|x|,|m1(t)|=|m(t)|+ϵ.

    Therefore, Equation (3.2) has a continuous solution xϵ(t) according to Theorem 2.1.

    Let ϵ1 and ϵ2 be such that 0<ϵ2<ϵ1<ϵ. Then

    xϵ1(t)=2(1+λt)A[η0(ηs)22fϵ1(s,λxϵ1(s))dsξ0(ξs)22fϵ1(s,λxϵ1(s))ds]t0(ts)fϵ1(s,λxϵ1(s))ds,=2(1+λt)A[η0(ηs)22(f(s,λxϵ1(s))+ϵ1)dsξ0(ξs)22(f(s,λxϵ1(s))+ϵ1)ds]t0(ts)(f(s,λxϵ1(s))+ϵ1)ds,>2(1+λt)A[η0(ηs)22(f(s,λxϵ1(s))+ϵ2)dsξ0(ξs)22(f(s,λxϵ1(s))+ϵ2)ds]t0(ts)(f(s,λxϵ1(s))+ϵ2)ds, (3.3)
    xϵ2(t)=2(1+λt)A[η0(ηs)22(f(s,λxϵ2(s))+ϵ2)dsξ0(ξs)22(f(s,λxϵ2(s))+ϵ2)ds]t0(ts)(f(s,λxϵ2(s))+ϵ2)ds. (3.4)

    Applying Lemma 3.2, then (3.3) and (3.4) imply that

    xϵ2(t)<xϵ1(t)fortJ.

    As shown before in the proof of Theorem 2.1, the family of functions xϵ(t) defined by Eq (3.2) is uniformly bounded and of equi-continuous functions. Hence by the Arzela-Ascoli Theorem, there exists a decreasing sequence ϵn such that ϵn0 as n, and limnxϵn(t) exists uniformly in I. We denote this limit by q(t). From the continuity of the function fϵn in the second argument, we get

    x(t)=limnxϵn(t)=2(1+λt)A[η0(ηs)22f(s,λq(s))dsξ0(ξs)22f(s,λq(s))ds]t0(ts)f(s,λq(s))ds,

    which proves that q(t) is a solution of (2.2).

    Finally, we shall show that q(t) is maximal solution of (2.2). To do this, let x(t) be any solution of (2.2). Then

    xϵ(t)=2(1+λt)A[η0(ηs)22fϵ(s,λxϵ(s))dsξ0(ξs)22fϵ(s,λxϵ(s))ds]t0(ts)fϵ(s,λxϵ(s))ds,=2(1+λt)A[η0(ηs)22(f(s,λxϵ(s))+ϵ)dsξ0(ξs)22(f(s,λxϵ(s))+ϵ)ds]t0(ts)(f(s,λxϵ(s))+ϵ)ds,>2(1+λt)A[η0(ηs)22f(s,λxϵ(s))dsξ0(ξs)22f(s,λxϵ(s))ds]t0(ts)f(s,λxϵ(s))ds. (3.5)

    Applying Lemma 3.2, then (2.2) and (3.5 imply that

    xϵ(t)>x(t)fortJ.

    From the uniqueness of the maximal solution (see [10]), it is clear that xϵ(t) tends to q(t) uniformly in tJasϵ0.

    In a similar way we can prove that there exists a minimal solution of (2.2).

    Here, we study the existence and some general properties of the eigenvalues and eigenfunctions of the problem of the homogeneous equation

    x(t)=λ2x(t),t(0,π), (4.1)

    with the nonlocal condition (1.2).

    Lemma 4.3. The eigenfunctions of the nonlocal boundary value problem (4.1) and (1.2) are in the form of

    xn(t)=cn(sin(π+4πn)t2(η+ξ)+cos(π+4πn)t2(η+ξ)),n=1,2,. (4.2)

    Proof. Firstly, we prove that the eigenvalues are

    λn=π+4πn2(η+ξ),n=1,2,. (4.3)

    The general solution of the problem (4.1) and (1.2) is given by

    x(t)=c1sinλt+c2cosλt. (4.4)

    Differentiating equation (4.4), we obtain

    x(t)=λc1cosλtλc2sinλt.

    Using the first condition, when t=0, we obtain

    c1=c2. (4.5)

    Integrating both sides of (4.4) from ξ to η, we obtain

    c1λcosλξc1λcosλη+c2λsinληc2λsinλξ=0.

    Substituting c1=c2, we obtain

    c1λcosλξc1λcosλη+c1λsinληc1λsinλξ=0. (4.6)

    Multiplying (4.6) by λc1, we obtain

    cosλξcosλη+sinληsinλξ=0,2sinλ(ξ+η)2sinλ(ηξ)2+2sinλ(ηξ)2cosλ(η+ξ)2=0,sinλ(ξ+η)2+cosλ(η+ξ)2=0,tanλ(ξ+η)2=1,λ(ξ+η)2=π4+nπ. (4.7)

    From (4.7), we deduce that

    λn=π+4πn2(η+ξ),n=1,2,.....

    Therefore, from (4.4) we can get

    xn(t)=cn(sin(π+4πn)t2(η+ξ)+cos(π+4πn)t2(η+ξ)),n=1,2,....

    Corollary 3. The eigenfunctions of the nonlocal boundary value problem (4.1) and (1.4) are in the form of

    xn(t)=cn(sin(1+4n)t2+cos(1+4n)t2),n=1,2,..... (4.8)

    Proof. Putting ξ=0 and η=π and applying Lemma 4.3 we obtain the result.

    Now, we study the existence of multiple solutions of the nonhomogeneous problem (1.3) and (1.2). Let x1,x2 be two solutions of the problem (1.3) and (1.2). Let u(t)=x1(t)x2(t), then the function u satisfy the Sturm-Liouville problem

    u(t)=λ2u(t)

    with the nonlocal conditions

    u(0)λu(0)=0andηξu(τ)dτ=0,ξ[0,π),η(0,π].

    So, the values of (eigenvalues) λn for the non zero solution of (4.1) and (1.2) is the same values (eigenvalues) of λn for the multiple solutions (eigenfunctions) of (1.3) and (1.2), i.e.

    λn=π+4πn2(η+ξ),n=1,2,.....

    Theorem 5.3. The multiple solutions (eigenfunctions) xn(t) of the problem (1.3) and (1.2) are given by

    xn(t)=An(sin(π+4πn)t2(η+ξ)+cos(π+4πn)t2(η+ξ))t0sin(π+4πn)(ts)2(η+ξ)π+4πn2(η+ξ)h(s,λ)ds. (5.1)

    Proof. Here we use the variation of parameter method to get the solution of (1.3) and (1.2). Assume that the solutions of (1.3) and (1.2) are given by

    xn(t)=A1cosλt+A2sinλt+xp(t). (5.2)

    So, we have

    x1(t)=cosλt,x2(t)=sinλt.

    Now, we can get W(x1,x2)=λ. Hence

    xp(t)=cosλtt0sinλsλh(s,λ)ds+sinλtt0cosλsλh(s,λ)ds,

    thus

    xp(t)=t0sinλ(ts)λh(s,λ)ds. (5.3)

    From (5.3) and (5.2), we obtain

    xn(t)=A1sin(π+4πn)t2(η+ξ)+A2cos(π+4πn)t2(η+ξ)t0sin(π+4πn)(ts)2(η+ξ)π+4πn2(η+ξ)h(s,λ)ds. (5.4)

    By using the first condition x(0)λx(0)=0, we get

    A1=A2,

    therefore the multiple solutions of the nonlocal problem (1.3) and (1.2) are given by

    xn(t)=An(sin(π+4πn)t2(η+ξ)+cos(π+4πn)t2(η+ξ))t0sin(π+4πn)(ts)2(η+ξ)(π+4πn)2(η+ξ)h(s,λ)ds,n=1,2,.....

    To complete the proof and to ensure that xn(t) is the solution of (1.3) and (1.2), we firstly prove that

    xn(t)+h(t,λ)=λ2xn(t).

    Differentiating (5.4) twice, we get

    xn(t)=Anπ+4πn2(η+ξ)(cos(π+4πn)t2(η+ξ)sin(π+4πn)t2(η+ξ))t0cos(π+4πn)(ts)2(η+ξ)h(s,λ)ds

    and

    xn(t)=An(π+4πn2(η+ξ))2(sin(π+4πn)t2(η+ξ)cos(π+4πn)t2(η+ξ))g(t)+π+4πn2(η+ξ)t0sin(π+4πn)(ts)2(η+ξ)h(s,λ)ds

    and

    xn(t)+h(t,λ)=An(π+4πn2(η+ξ))2(sin(π+4πn)t2(η+ξ)cos(π+4πn)t2(η+ξ))h(t,λ)+π+4πn2(η+ξ)t0sin(π+4πn)(ts)2(η+ξ)h(s,λ)ds+h(t,λ)=λ2xn(t).

    Also we have x(0)λx(0)=0.

    Example 1. Let h(t,λ)=λ2. Then we find that

    xp(t)=t0sinλ(ts)λλ2ds=cosλt1

    and the multiple solutions of the nonlocal problem (1.3) and (1.2) are given by

    xn(t)=A1sin(π+4πn)t2(η+ξ)+A2cos(π+4πn)t2(η+ξ)+cosλt1.

    Now consider the Riemann integral boundary condition (1.4).

    Corollary 4. The multiple solutions (eigenfunctions) xn(t) of the problem (1.3)-(1.4) are given by

    xn(t)=An(sin(1+4n)t2+cos(1+4n)t2)t0sin(1+4n)(ts)21+4n2h(s,λ)ds.

    Proof. In this special case, we put ξ=0 and η=π and applying Theorem 5.3 we get the result.

    Example 2. Let h(t,λ)=λ2. Then we find that

    xp(t)=t0sinλ(ts)λλ2ds=cosλt1,

    and the solution xn(t) of the problem (1.3)-(1.4) are given by

    xn(t)=An(sin(1+4n)t2+cos(1+4n)t2)+cosλt1.

    Here, we proved the existence of solutions xC[0,π] of the nonlocal boundary value problem of the differential inclusion (1.1) with the nonlocal condition (1.2).

    The maximal and minimal solutions of the problem (1.1)-(1.2) have been proved. The eigenvalues and eigenfunctions of the homogeneous and nonhomogeneous equations (4.1) and (1.3) with the nonlocal condition (1.2) have been obtained. Two examples have been studied to illustrate our results.

    We thank the referees for their constructive remarks and comments on our work which reasonably improved the presentation and the structure of the manuscript.

    The authors declare no conflict of interest.



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