Citation: Jahfar T K, Chithra A V. Central vertex join and central edge join of two graphs[J]. AIMS Mathematics, 2020, 5(6): 7214-7233. doi: 10.3934/math.2020461
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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)=0and∫ηξx(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)=0and∫π0x(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,y∈R.
(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|:f∈F(t,x)}≤|m(t)|+λ2|x|. |
Remark 1. From the assumptions (i)-(iv) we can deduce that (see [1], [5] and [6]) there exists f∈F(t,x), such that
(v) f:I×R⟶R is measurable in t for every x,y∈R 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(η−ξ)≠−2,λ∈R.
(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(η−s)22f(s,λx(s))ds−∫ξ0(ξ−s)22f(s,λx(s))ds]−∫t0(t−s)f(s,λx(s))ds, | (2.2) |
where A=(η−ξ)[2+λ(η−ξ)]≠0.
Proof. Integrating both sides of Eq (2.1) twice, we obtain
x(t)−x(0)−tx′(0)=−∫t0(t−s)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=∫ηξ∫τ0(τ−s)f(s,λx(s))dsdτ,(η−ξ)x(0)+(η−ξ)22λx(0)=∫ξ0∫ηξ(τ−s)dτf(s,λx(s))ds+∫ηξ∫ηs(τ−s)dτf(s,λx(s))ds,(η−ξ)[2+λ(η−ξ)]2x(0)=∫ξ0[(η−s)22−(ξ−s)22]f(s,λx(s))ds+∫ηξ(η−s)22f(s,λx(s))ds=∫η0(η−s)22f(s,λx(s))ds−∫ξ0(ξ−s)22f(s,λx(s))ds |
and we can get
x(0)=2(η−ξ)[2+λ(η−ξ)][∫η0(η−s)22f(s,λx(s))ds−∫ξ0(ξ−s)22f(s,λx(s))ds]=2A[∫η0(η−s)22f(s,λx(s))ds−∫ξ0(ξ−s)22f(s,λx(s))ds]. | (2.5) |
Substituting (2.5) into (2.4), we obtain
x′(0)=2λA[∫η0(η−s)22f(s,λx(s))ds−∫ξ0(ξ−s)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(η−s)22f(s,λx(s))ds−∫ξ0(ξ−s)22f(s,λx(s))ds]−∫t0(t−s)f(s,λx(s))ds. |
To complete the proof, differentiate equation (2.2) twice, we obtain
x′(t)=2λA[∫η0(η−s)22f(s,λx(s))ds−∫ξ0(ξ−s)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(η−s)22f(s,λx(s))ds−∫ξ0(ξ−s)22f(s,λx(s))ds], |
and
λx(0)=2λA[∫η0(η−s)22f(s,λx(s))ds−∫ξ0(ξ−s)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(η−ξ)+λ(η2−ξ2)A=(η−ξ)[2+λ(η−ξ)]A=1, |
from that, we obtain as before
∫ηξx(τ)dτ=∫ηξ2(1+λτ)Adτ[∫η0(η−s)22f(s,λx(s))ds−∫ξ0(ξ−s)22f(s,λx(s))ds]−∫ηξ∫τ0(τ−s)f(s,λx(s))dsdτ,=∫η0(η−s)22f(s,λx(s))ds−∫ξ0(ξ−s)22f(s,λx(s))ds−∫η0(η−s)22f(s,λx(s))ds+∫ξ0(ξ−s)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 x∈C[0,π] of the nonlocal boundary value problem (2.1) and (1.2). Moreover, from Remark 1, then there exists at least one solution x∈C[0,π] of the nonlocal boundary value problem (1.1)-(1.2).
Proof. Define the set Qr⊂C[0,π] by
Qr={x∈C:∥x∥≤r},r≥2(1+|λ|π)π2+π‖m‖L1|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(η−s)22f(s,λx(s))ds−∫ξ0(ξ−s)22f(s,λx(s))ds]−∫t0(t−s)f(s,λx(s))ds. |
Let x∈Qr, we have
|Tx(t)|≤2(1+|λ|t)|A|[∫η0(η−s)22|f(s,λx(s))|ds+∫ξ0(ξ−s)22|f(s,λx(s))ds|]+∫t0(t−s)|f(s,λx(s))|ds,≤2(1+|λ|π)π2|A|∫π0{|m(s)|+λ2|x(s)|}ds+π∫π0{|m(s)|+λ2|x(s)|}ds,≤[2(1+|λ|π)π2|A|+π]{‖m‖L1+λ2π‖x‖},≤2(1+|λ|π)π2+π|A|{‖m‖L1+λ2πr}≤r, |
and we have
2(1+|λ|π)π2+π|A|‖m‖L1≤r(1−2(1+|λ|π)π2+π|A|λ2π). |
Then T:Qr→Qr and the class {Tx}⊂Qr is uniformly bounded in Qr.
In what follows we show that the class {Tx}, x∈Qr is equicontinuous. For t1,t2∈[0,π],t1<t2 such that |t2−t1|<δ, we have
Tx(t2)−Tx(t1)=2(1+λt2)A[∫η0(η−s)22f(s,λx(s))ds−∫ξ0(ξ−s)22f(s,λx(s))ds]−∫t20(t2−s)f(s,λx(s))ds−2(1+λt1)A[∫η0(η−s)22f(s,λx(s))ds−∫ξ0(ξ−s)22f(s,λx(s))ds]−∫t10(t2−s)f(s,λx(s))ds,|Tx(t2)−Tx(t1)|=|2(1+λt2)A[∫η0(η−s)22f(s,λx(s))ds−∫ξ0(ξ−s)22f(s,λx(s))ds]−∫t20(t2−s)f(s,λx(s))ds−2(1+λt1)A[∫η0(η−s)22f(s,λx(s))ds−∫ξ0(ξ−s)22f(s,λx(s))ds]+∫t10(t1−s)f(s,λx(s))ds|,≤2|λ|(t2−t1)A[∫η0(η−s)22|f(s,λx(s))|ds+∫ξ0(ξ−s)22|f(s,λx(s))|ds]+(t2−t1)∫t10|f(s,λx(s))|ds+π∫t2t1|f(s,λx(s))|ds,≤2|λ|(t2−t1)π2A∫π0|f(s,λx(s))|ds+(t2−t1)∫π0|f(s,λx(s))|ds+π∫t2t1|f(s,λx(s))|ds,≤2|λ|(t2−t1)π2A{‖m‖L1+λ2‖x‖}+(t2−t1){‖m‖L1+λ2‖x‖}+π∫t2t1{|m(s)|+λ2|x(s)|}ds. |
Hence the class of function {Tx}, x∈Qr is equicontinuous. By Arzela-Ascolis [4] Theorem, we found that the class {Tx} is relatively compact.
Now we prove that T:Qr→Qr is continuous.
Let {xn}⊂Qr, such that xn→x0∈Qr, then
Txn(t)=2(1+λt)A[∫η0(η−s)22f(s,λxn(s))ds−∫ξ0(ξ−s)22f(s,λxn(s))ds]−∫t0(t−s)f(s,λxn(s))ds, |
and
limn→∞Txn(t)=limn→∞{2(1+λt)A[∫η0(η−s)22f(s,λxn(s))ds−∫ξ0(ξ−s)22f(s,λxn(s))ds]−∫t0(t−s)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
limn→∞Txn(t)=2(1+λt)A[∫η0(η−s)22limn→∞f(s,λxn(s))ds−∫ξ0(ξ−s)22limn→∞f(s,λxn(s))ds]−∫t0(t−s)limn→∞f(s,λxn(s))ds,=2(1+λt)A[∫η0(η−s)22f(s,λx0(s))ds−∫ξ0(ξ−s)22f(s,λx0(s))ds]−∫t0(t−s)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 x∈C[0,π].
Consequently the nonlocal boundary value problem (2.1)-(1.2) has at least one solution x∈C[0,π]. Moreover, from Remark 1, then there exists at least one solution x∈C[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 x∈C[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 x∈C[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 x∈C[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),t∈J. A minimal solution s(t) can be defined in a similar way by reversing the above inequality i.e. x(t)≥s(t),t∈J.
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(η−s)22f(s,λx(s))ds−∫ξ0(ξ−s)22f(s,λx(s))ds]−∫t0(t−s)f(s,λx(s))ds,y(t)≥2(1+λt)A[∫η0(η−s)22f(s,λy(s))ds−∫ξ0(ξ−s)22f(s,λy(s))ds]−∫t0(t−s)f(s,λy(s))ds |
where one of them is strict.
Suppose f(t,x) is nondecreasing function inx. Then
x(t)<y(t),t∈J. | (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(η−s)22f(s,λx(s))ds−∫ξ0(ξ−s)22f(s,λx(s))ds]−∫t10(t−s)f(s,λx(s))ds,<2(1+λt1)A[∫η0(η−s)22f(s,λy(s))ds−∫ξ0(ξ−s)22f(s,λy(s))ds]−∫t10(t−s)f(s,λy(s))ds<y(t1). |
This contradicts the fact that x(t1)=y(t1);then
x(t)<y(t),t∈J. |
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(η−s)22fϵ(s,λxϵ(s))ds−∫ξ0(ξ−s)22fϵ(s,λxϵ(s))ds]−∫t0(t−s)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(η−s)22fϵ1(s,λxϵ1(s))ds−∫ξ0(ξ−s)22fϵ1(s,λxϵ1(s))ds]−∫t0(t−s)fϵ1(s,λxϵ1(s))ds,=2(1+λt)A[∫η0(η−s)22(f(s,λxϵ1(s))+ϵ1)ds−∫ξ0(ξ−s)22(f(s,λxϵ1(s))+ϵ1)ds]−∫t0(t−s)(f(s,λxϵ1(s))+ϵ1)ds,>2(1+λt)A[∫η0(η−s)22(f(s,λxϵ1(s))+ϵ2)ds−∫ξ0(ξ−s)22(f(s,λxϵ1(s))+ϵ2)ds]−∫t0(t−s)(f(s,λxϵ1(s))+ϵ2)ds, | (3.3) |
xϵ2(t)=2(1+λt)A[∫η0(η−s)22(f(s,λxϵ2(s))+ϵ2)ds−∫ξ0(ξ−s)22(f(s,λxϵ2(s))+ϵ2)ds]−∫t0(t−s)(f(s,λxϵ2(s))+ϵ2)ds. | (3.4) |
Applying Lemma 3.2, then (3.3) and (3.4) imply that
xϵ2(t)<xϵ1(t)fort∈J. |
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 ϵn→0 as n→∞, and limn→∞xϵn(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)=limn→∞xϵn(t)=2(1+λt)A[∫η0(η−s)22f(s,λq(s))ds−∫ξ0(ξ−s)22f(s,λq(s))ds]−∫t0(t−s)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(η−s)22fϵ(s,λxϵ(s))ds−∫ξ0(ξ−s)22fϵ(s,λxϵ(s))ds]−∫t0(t−s)fϵ(s,λxϵ(s))ds,=2(1+λt)A[∫η0(η−s)22(f(s,λxϵ(s))+ϵ)ds−∫ξ0(ξ−s)22(f(s,λxϵ(s))+ϵ)ds]−∫t0(t−s)(f(s,λxϵ(s))+ϵ)ds,>2(1+λt)A[∫η0(η−s)22f(s,λxϵ(s))ds−∫ξ0(ξ−s)22f(s,λxϵ(s))ds]−∫t0(t−s)f(s,λxϵ(s))ds. | (3.5) |
Applying Lemma 3.2, then (2.2) and (3.5 imply that
xϵ(t)>x(t)fort∈J. |
From the uniqueness of the maximal solution (see [10]), it is clear that xϵ(t) tends to q(t) uniformly in t∈Jasϵ→0.
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)=λc1cosλt−λc2sinλt. |
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
c1λcosλξ−c1λcosλη+c2λsinλη−c2λsinλξ=0. |
Substituting c1=c2, we obtain
c1λcosλξ−c1λcosλη+c1λsinλη−c1λsinλξ=0. | (4.6) |
Multiplying (4.6) by λc1, we obtain
cosλξ−cosλη+sinλη−sinλξ=0,2sinλ(ξ+η)2sinλ(η−ξ)2+2sinλ(η−ξ)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)=0and∫ηξu(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)(t−s)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)=−cosλt∫t0sinλsλh(s,λ)ds+sinλt∫t0cosλsλh(s,λ)ds, |
thus
xp(t)=−∫t0sinλ(t−s)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)(t−s)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)(t−s)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
x″n(t)+h(t,λ)=−λ2xn(t). |
Differentiating (5.4) twice, we get
x′n(t)=An−π+4πn2(η+ξ)(cos(−π+4πn)t2(η+ξ)−sin(−π+4πn)t2(η+ξ))−∫t0cos(−π+4πn)(t−s)2(η+ξ)h(s,λ)ds |
and
x″n(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)(t−s)2(η+ξ)h(s,λ)ds |
and
x″n(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)(t−s)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(t−s)λλ2ds=cosλt−1 |
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)+cosλt−1. |
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)(t−s)2−1+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(t−s)λλ2ds=cosλt−1, |
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)+cosλt−1. |
Here, we proved the existence of solutions x∈C[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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