Some New Existence Results for a Class of Four Point Nonlinear Boundary Value Problems

2019 ◽  
Vol 3 (1) ◽  
Author(s):  
Nazia Urus ◽  
Amit K. Verma ◽  
Mandeep Singh
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Iterative Scheme ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Existence Results ◽  
Point Boundary ◽  
Nonlinear Boundary Value Problems ◽  
Monotone Iterative

In this paper we consider the following class of four point boundary value problems—y"(x) = f (x, y), 0 less than x lessthan 1, y'(0) = 0, y(1) = 1y(1) + 2)7(2)’where 1, 2  0 lesstahn 1, 2 less than 1, and f (x, y), is continuous in one sided Lipschitz in y. We propose a monotone iterative scheme and show that under some sufficient conditions this scheme generates sequences which converges uniformly to solution of the nonlinear multipint boundary value problem.

scholarly journals On the reduction of a nonlinear Noetherian differential-algebraic boundary-value problem to a noncritical case

2019 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Nonlinear Oscillations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Differential Algebraic Equations ◽  
Nonlinear Boundary Value Problems ◽  
Actual Problem ◽  
Nonlinear Boundary Value

The study of the differential-algebraic boundary value problems was established in the papers of K. Weierstrass, M.M. Lusin and F.R. Gantmacher. Works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, M.O. Perestyuk, V.P. Yakovets, O.A. Boichuk, A. Ilchmann and T. Reis are devoted to the systematic study of differential-algebraic boundary value problems. At the same time, the study of differential-algebraic boundary-value problems is closely related to the study of nonlinear boundary-value problems for ordinary differential equations, initiated in the works of A. Poincare, A.M. Lyapunov, M.M. Krylov, N.N. Bogolyubov, I.G. Malkin, A.D. Myshkis, E.A. Grebenikov, Yu.A. Ryabov, Yu.A. Mitropolsky, I.T. Kiguradze, A.M. Samoilenko, M.O. Perestyuk and O.A. Boichuk. The study of the nonlinear differential-algebraic boundary value problems is connected with numerous applications of corresponding mathematical models in the theory of nonlinear oscillations, mechanics, biology, radio engineering, the theory of the motion stability. Thus, the actual problem is the transfer of the results obtained in the articles and monographs of S. Campbell, A.M. Samoilenko and O.A. Boichuk on the nonlinear boundary value problems for the differential algebraic equations, in particular, finding the necessary and sufficient conditions of the existence of the desired solutions of the nonlinear differential algebraic boundary value problems. In this article we found the conditions of the existence and constructed the iterative scheme for finding the solutions of the weakly nonlinear Noetherian differential-algebraic boundary value problem. The proposed scheme of the research of the nonlinear differential-algebraic boundary value problems in the article can be transferred to the nonlinear matrix differential-algebraic boundary value problems. On the other hand, the proposed scheme of the research of the nonlinear Noetherian differential-algebraic boundary value problems in the critical case in this article can be transferred to the autonomous seminonlinear differential-algebraic boundary value problems.

scholarly journals Existence of Multiple Positive Solutions for Even Order Multi-Point Boundary Value Problems

2007 ◽  
Vol 14 (4) ◽  
pp. 775-792
Author(s):  
Youyu Wang ◽  
Weigao Ge
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Even Order

Abstract In this paper, we consider the existence of multiple positive solutions for the 2𝑛th order 𝑚-point boundary value problem: where (0,1), 0 < ξ 1 < ξ 2 < ⋯ < ξ 𝑚–2 < 1. Using the Leggett–Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The associated Green's function for the above problem is also given.

scholarly journals On the relation between interior critical points of positive solutions and parameters for a class of nonlinear boundary value problems

2002 ◽  
Vol 31 (12) ◽  
pp. 751-760
Author(s):  
G. A. Afrouzi ◽  
M. Khaleghy Moghaddam
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Critical Points ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlinear Boundary Value Problems ◽  
Nonnegative Solutions ◽  
Nonlinear Boundary Value

We consider the boundary value problem−u″(x)=λf(u(x)),x∈(0,1);u′(0)=0;u′(1)+αu(1)=0, whereα>0,λ>0are parameters andf∈c2[0,∞)such thatf(0)<0. In this paper, we study for the two casesρ=0andρ=θ(ρis the value of the solution atx=0andθis such thatF(θ)=0whereF(s)=∫0sf(t)dt) the relation betweenλand the number of interior critical points of the nonnegative solutions of the above system.

scholarly journals Existence results for nonlinear boundary value problems

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 609-618 ◽  
Author(s):  
Abdeljabbar Ghanmi ◽  
Samah Horrigue
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Boundary Value Problems ◽  
Point Theorem ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Existence Results ◽  
Nonlinear Boundary Value Problems ◽  
Krasnoselskii Fixed Point Theorem

In the present paper, we are concerned to prove under some hypothesis the existence of fixed points of the operator L defined on C(I) by Lu(t) = ?w0 G(t,s)h(s) f(u(s))ds, t ? I, ? ? {1,?}, where the functions f ? C([0,?); [0,?)), h ? C(I; [0,?)), G ? C(I x I) and (I = [0,1]; if ? = 1, I = [0,?), if ? = 1. By using Guo Krasnoselskii fixed point theorem, we establish the existence of at least one fixed point of the operator L.

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.

VARIATIONAL ITERATION METHOD FOR SOLVING NONLINEAR HIGHER-ORDER INITIAL AND BOUNDARY VALUE PROBLEMS

2009 ◽  
Vol 06 (04) ◽  
pp. 521-555 ◽  
Author(s):  
SYED TAUSEEF MOHYUD-DIN ◽  
MUHAMMAD ASLAM NOOR ◽  
KHALIDA INAYAT NOOR
Keyword(s):  
Boundary Value Problems ◽  
Iteration Method ◽  
Nonlinear Boundary ◽  
Iterative Scheme ◽  
Boundary Value ◽  
Variational Iteration Method ◽  
Higher Order ◽  
Nonlinear Boundary Value Problems ◽  
Variational Iteration ◽  
Adomian’S Polynomials

In this paper, we apply variational iteration method (VIM) and variational iteration method using Adomian's polynomials for solving nonlinear boundary value problems. The proposed iterative scheme finds the solution without any discretization, linearization, perturbation, or restrictive assumptions. Several examples are given to verify the accuracy and efficiency of the method. We have also considered an example where the proposed VIM is not reliable.

scholarly journals Spline solutions for nonlinear two point boundary value problems

1980 ◽  
Vol 3 (1) ◽  
pp. 151-167 ◽  
Author(s):  
Riaz A. Usmani
Keyword(s):  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Relative Performance ◽  
Point Boundary ◽  
Nonlinear Boundary Value Problems ◽  
Numerical Evidence ◽  
Nonlinear Boundary Value

Necessary formulas are developed for obtaining cubic, quartic, quintic, and sextic spline solutions of nonlinear boundary value problems. These methods enable us to approximate the solution of the boundary value problems, as well as their successive derivatives smoothly. Numerical evidence is included to demonstrate the relative performance of these four techniques.

scholarly journals A NOTE ON EXISTENCE RESULTS FOR A CLASS OF THREE-POINT NONLINEAR BVPS

2015 ◽  
Vol 20 (4) ◽  
pp. 457-470 ◽  
Author(s):  
Amit K. Verma ◽  
Mandeep Singh
Keyword(s):  
Boundary Value Problem ◽  
Present Method ◽  
Iterative Scheme ◽  
Existence Results ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Point Boundary ◽  
Monotone Iterative ◽  
Close Form ◽  
User Friendly

This article deals with a computational iterative technique for the following second order three point boundary value problem y''(t) + f(t, y, y' ) = 0, 0 &lt; t &lt; 1, y(0) = 0, y(1) = δy(η), where f(I × R, R), I = [0, 1], 0 &lt; η &lt; 1, δ &gt; 0. We consider simple iterative scheme and develop a monotone iterative technique. Some examples are constructed to show the accuracy of the present method. We show that our technique is quite powerful and some user friendly packages can be developed by using this technique to compute the solutions of the nonlinear three point BVPs whose close form solutions are not known.

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