Existence of Multiple Solutions for Second-Order Problem with Stieltjes Integral Boundary Condition

Hu, Qing-Qing; Yan, Baoqiang

doi:10.1155/2021/6632236

Cited by 7 publications

(7 citation statements)

References 41 publications

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“…So, we have proved that under assumption (8) coupled to the uniqueness of solution of Problem (2), Problem (1) has at least one solution given by expression (9).…”

Section: Explicit Expression Of the Solution Of Problem (1)mentioning

confidence: 91%

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Green's function related to a n order linear differential equation coupled to arbitrary linear non local boundary conditions

Cabada,

López-Somoza,

Yousfi

2021

Preprint

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In this paper we obtain the explicit expression of the Green's function related to a general n order differential equation coupled to non-local linear boundary conditions. In such boundary conditions, a n dimensional parameter dependence is also assumed. Moreover, some comparison principles are obtained.The explicit expression depends on the value of the Green's function related to the two-point homogeneous problem, that is, we are assuming that when all the parameters involved on the boundary conditions take the value zero then the problem has a unique solution which is characterized by the corresponding Green's function g. The expression of the Green's function G of the general problem is given as a function of g and the real parameters considered at the boundary conditions.It is important to show that, in order to ensure the uniqueness of solutions of the linear considered problem, we must assume a non resonant additional condition on the considered problem, which depends on the non local conditions and the corresponding parameters.We point out that the assumption of the uniqueness of solutions of the two-point homogeneous problem is not a necessary condition to ensure the solution of the general case. Of course, in this situation the expression we are looking for must be obtained in a different manner.To show the applicability of the obtained results, a particular example is given.

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“…So, we have proved that under assumption (8) coupled to the uniqueness of solution of Problem (2), Problem (1) has at least one solution given by expression (9).…”

Section: Explicit Expression Of the Solution Of Problem (1)mentioning

confidence: 91%

“…Such difficulty increases with the non-local operators on the boundary. One of the most common non-local boundary conditions are given as integral equations (some of them in the Stieltjes sense) and has been applied to different situations as fourth order beam equations [4], second order problems [9] or fractional equations [5,7].…”

Section: Introductionmentioning

confidence: 99%

Green's function related to a n order linear differential equation coupled to arbitrary linear non local boundary conditions

Cabada,

López-Somoza,

Yousfi

2021

Preprint

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show abstract

“…In particular integral boundary conditions have been widely considered in many works in the recent literature. For this topic, we refer the reader to [8,9,13,14,16,18] (for integral boundary conditions in second and fourth order ODEs) or [1,2,7,11,12,17] (for fractional equations) and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Constant sign Green's function of a second order perturbed periodic problem

Cabada¹,

López-Somoza²,

Yousfi³

2022

Preprint

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In this paper we are interested in obtaining the exact expression and the study of the constant sign of the Green's function related to a second order perturbed periodic problem coupled with integral boundary conditions at the extremes of the interval of definition.To obtain the expression of the Green's function related to this problem we use the theory presented in [10] for general non-local perturbed boundary value problems. Moreover, we will characterize the parameter's set where such Green's function has constant sign. To this end, we need to consider first a related second order problem without integral boundary conditions, obtaining the properties of its Green's function and then using them to compute the sign of the one related to the main problem.

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“…Such difficulty increases with non-local operators on the boundary. Some of the most common non-local boundary conditions are given as integral equations (some of them in the Stieltjes sense) and have been applied to different situations as fourth-order beam equations [11], second-order problems [12] or fractional equations [13,14]. The concept of generalized Green's function appears in the resonance case and also on partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

Green’s Function Related to a n-th Order Linear Differential Equation Coupled to Arbitrary Linear Non-Local Boundary Conditions

2021

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In this paper, we obtain the explicit expression of the Green’s function related to a general n-th order differential equation coupled to non-local linear boundary conditions. In such boundary conditions, an n dimensional parameter dependence is also assumed. Moreover, some comparison principles are obtained. The explicit expression depends on the value of the Green’s function related to the two-point homogeneous problem; that is, we are assuming that when all the parameters involved on the boundary conditions take the value zero then the problem has a unique solution, which is characterized by the corresponding Green’s function g. The expression of the Green’s function G of the general problem is given as a function of g and the real parameters considered at the boundary conditions. It is important to note that, in order to ensure the uniqueness of the solution of the considered linear problem, we must assume a non-resonant additional condition on the considered problem, which depends on the non-local conditions and the corresponding parameters. We point out that the assumption of the uniqueness of the solution of the two-point homogeneous problem is not a necessary condition to ensure the existence of the solution of the general case. Of course, in this situation, the expression we are looking for must be obtained in a different manner. To show the applicability of the obtained results, a particular example is given.

show abstract

Existence of Multiple Solutions for Second-Order Problem with Stieltjes Integral Boundary Condition

Cited by 7 publications

References 41 publications

Green's function related to a n order linear differential equation coupled to arbitrary linear non local boundary conditions

Green's function related to a n order linear differential equation coupled to arbitrary linear non local boundary conditions

Constant sign Green's function of a second order perturbed periodic problem

Green’s Function Related to a n-th Order Linear Differential Equation Coupled to Arbitrary Linear Non-Local Boundary Conditions

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