scholarly journals OSCILLATION PROPERTIES FOR NON-CLASSICAL STURM-LIOUVILLE PROBLEMS WITH ADDITIONAL TRANSMISSION CONDITIONS

2021 ◽  
Vol 26 (3) ◽  
pp. 432-443
Author(s):  
Oktay Sh. Mukhtarov ◽  
Kadriye Aydemir
Keyword(s):  
Boundary Value ◽  
Main Difficulty ◽  
Distribution Of Zeros ◽  
Value Transmission ◽  
Transmission Problems ◽  
New Type ◽  
Sturm Liouville ◽  
Left And Right ◽  
New Criteria ◽  
Oscillation Properties

This work is aimed at studying some comparison and oscillation properties of boundary value problems (BVP’s) of a new type, which differ from classical problems in that they are defined on two disjoint intervals and include additional transfer conditions that describe the interaction between the left and right intervals. This type of problems we call boundary value-transmission problems (BVTP’s). The main difficulty arises when studying the distribution of zeros of eigenfunctions, since it is unclear how to apply the classical methods of Sturm’s theory to problems of this type. We established new criteria for comparison and oscillation properties and new approaches used to obtain these criteria. The obtained results extend and generalizes the Sturm’s classical theorems on comparison and oscillation.

Variational methods to the p-Laplacian type nonlinear fractional order impulsive differential equations with Sturm-Liouville boundary-value problem

2021 ◽  
Vol 24 (4) ◽  
pp. 1069-1093
Author(s):  
Dandan Min ◽  
Fangqi Chen
Keyword(s):  
Differential Equation ◽  
Variational Methods ◽  
Impulsive Differential Equation ◽  
Boundary Value ◽  
Impulsive Differential Equations ◽  
Laplacian Operator ◽  
Infinitely Many Solutions ◽  
Critical Point Theorem ◽  
Sturm Liouville ◽  
New Criteria

Abstract In this paper, we consider a class of nonlinear fractional impulsive differential equation involving Sturm-Liouville boundary-value conditions and p-Laplacian operator. By making use of critical point theorem and variational methods, some new criteria are given to guarantee that the considered problem has infinitely many solutions. Our results extend some recent results and the conditions of assumptions are easily verified. Finally, an example is given as an application of our fundamental results.

scholarly journals A Study of the Eigenfunctions of the Singular Sturm–Liouville Problem Using the Analytical Method and the Decomposition Technique

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 415 ◽  
Author(s):  
Oktay Sh. Mukhtarov ◽  
Merve Yücel
Keyword(s):  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Adomian Decomposition Method ◽  
Liouville Problem ◽  
Eigenvalues And Eigenfunctions ◽  
Transmission Problems ◽  
Adomian Decomposition ◽  
Interval Boundary ◽  
Sturm Liouville

The history of boundary value problems for differential equations starts with the well-known studies of D. Bernoulli, J. D’Alambert, C. Sturm, J. Liouville, L. Euler, G. Birkhoff and V. Steklov. The greatest success in spectral theory of ordinary differential operators has been achieved for Sturm–Liouville problems. The Sturm–Liouville-type boundary value problem appears in solving the many important problems of natural science. For the classical Sturm–Liouville problem, it is guaranteed that all the eigenvalues are real and simple, and the corresponding eigenfunctions forms a basis in a suitable Hilbert space. This work is aimed at computing the eigenvalues and eigenfunctions of singular two-interval Sturm–Liouville problems. The problem studied here differs from the standard Sturm–Liouville problems in that it contains additional transmission conditions at the interior point of interaction, and the eigenparameter λ appears not only in the differential equation, but also in the boundary conditions. Such boundary value transmission problems (BVTPs) are much more complicated to solve than one-interval boundary value problems ones. The major difficulty lies in the existence of eigenvalues and the corresponding eigenfunctions. It is not clear how to apply the known analytical and approximate techniques to such BVTPs. Based on the Adomian decomposition method (ADM), we present a new analytical and numerical algorithm for computing the eigenvalues and corresponding eigenfunctions. Some graphical illustrations of the eigenvalues and eigenfunctions are also presented. The obtained results demonstrate that the ADM can be adapted to find the eigenvalues and eigenfunctions not only of the classical one-interval boundary value problems (BVPs) but also of a singular two-interval BVTPs.

scholarly journals Existence and uniqueness of solutions for a mixed p-Laplace boundary value problem involving fractional derivatives

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shuqi Wang ◽  
Zhanbing Bai
Keyword(s):  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solutions ◽  
Novel Approach ◽  
Banach Contraction ◽  
Left And Right ◽  
Banach Contraction Mapping Principle

AbstractIn this article, the existence and uniqueness of solutions for a multi-point fractional boundary value problem involving two different left and right fractional derivatives with p-Laplace operator is studied. A novel approach is used to acquire the desired results, and the core of the method is Banach contraction mapping principle. Finally, an example is given to verify the results.

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