scholarly journals Boundary-Value Problems for Weakly Nonlinear Delay Differential Systems

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
A. Boichuk ◽  
J. Diblík ◽  
D. Khusainov ◽  
M. Růžičková
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Linear Part ◽  
Sufficient Conditions ◽  
Nonlinear Boundary Value Problems ◽  
Weakly Nonlinear ◽  
Delay Differential Systems ◽  
Delay Differential

Conditions are derived of the existence of solutions of nonlinear boundary-value problems for systems ofnordinary differential equations with constant coefficients and single delay (in the linear part) and with a finite number of measurable delays of argument in nonlinearity:ż(t)=Az(t-τ)+g(t)+εZ(z(hi(t),t,ε),  t∈[a,b], assuming that these solutions satisfy the initial and boundary conditionsz(s):=ψ(s) if s∉[a,b],  lz(⋅)=α∈Rm. The use of a delayed matrix exponential and a method of pseudoinverse by Moore-Penrose matrices led to anexplicitandanalyticalform of sufficient conditions for the existence of solutions in a given space and, moreover, to the construction of an iterative process for finding the solutions of such problems in a general case when the number of boundary conditions (defined by a linear vector functionall) does not coincide with the number of unknowns in the differential system with a single delay.

scholarly journals Lower and Upper Solutions for Even Order Boundary Value Problems

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 878
Author(s):  
Alberto Cabada ◽  
Lucía López-Somoza
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Periodic Boundary Conditions ◽  
Existence Of Solution ◽  
Lower And Upper Solutions ◽  
Nonlinear Boundary Value Problems ◽  
Even Order

In this paper, we prove the existence of solutions of nonlinear boundary value problems of arbitrary even order using the lower and upper solutions method. In particular, we point out the fact that the existence of a pair of lower and upper solutions of a considered problem could imply the existence of solution of another one with different boundary conditions. We consider Neumann, Dirichlet, mixed and periodic boundary conditions.

scholarly journals On Nonlinear Boundary Value Problems for Functional Difference Equations withp-Laplacian

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Yong Wan ◽  
Yuji Liu
Keyword(s):  
Boundary Value Problems ◽  
Difference Equations ◽  
Nonlinear Boundary ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Functional Difference ◽  
Nonlinear Boundary Value Problems ◽  
Nonlinear Boundary Value ◽  
Functional Difference Equations

Sufficient conditions for the existence of solutions of nonlinear boundary value problems for higher-order functional difference equations withp-Laplacian are established by making of continuation theorems. We allowfto be at most linear, superlinear, or sublinear in obtained results.

Solvability for nonlocal boundary value problems with generalized 𝑝-Laplacian on an unbounded domain

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Marek Matyjasik ◽  
Katarzyna Szymańska-Dȩbowska
Keyword(s):  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Sufficient Conditions ◽  
Integral Boundary Conditions ◽  
Nonlinear Boundary Value Problems ◽  
Integral Boundary ◽  
Positive Half

Abstract This paper is devoted to the existence of solutions for a class of nonlinear boundary value problems with integral boundary conditions and generalized 𝑝-Laplacian on the positive half-line. We establish sufficient conditions to guarantee the existence of solutions in a special function space by using Leray–Schauder-type arguments. Examples are also given to illustrate the main results.

scholarly journals Existence of Solutions for Some Nonlinear Problems with Boundary Value Conditions

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Dionicio Pastor Dallos Santos
Keyword(s):  
Fixed Point ◽  
Continuous Function ◽  
Fixed Point Theorem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Nonlinear Problems ◽  
Nonlinear Boundary Value Problems

We study the existence of solutions for nonlinear boundary value problemsφu′′=ft,u,u′,  lu,u′=0, wherel(u,u′)=0denotes the boundary conditions on a compact interval0,T,φis a homeomorphism such thatφ(0)=0, andf:0,T×R×R→Ris a continuous function. All the contemplated boundary value problems are reduced to finding a fixed point for one operator defined on a space of functions, and Schauder fixed point theorem or Leray-Schauder degree is used.

scholarly journals First-Order Three-Point Boundary Value Problems at Resonance Part III

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Mesliza Mohamed ◽  
Bevan Thompson ◽  
Muhammad Sufian Jusoh
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Implicit Function Theorem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Implicit Function ◽  
Point Boundary ◽  
First Order ◽  
At Resonance

The main purpose of this paper is to investigate the existence of solutions of BVPs for a very general case in which both the system of ordinary differential equations and the boundary conditions are nonlinear. By employing the implicit function theorem, sufficient conditions for the existence of three-point boundary value problems are established.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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