scholarly journals On the Convergence of a Nonlinear Boundary-Value Problem in a Perforated Domain

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Yulia Koroleva
Keyword(s):  
Boundary Value Problem ◽  
Small Parameter ◽  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlinear Boundary Value Problem ◽  
Perforated Domain ◽  
Nonlinear Boundary Value Problems ◽  
Nonlinear Boundary Value ◽  
Convergence Of Solution

We consider a family with respect to a small parameter of nonlinear boundary-value problems as well as the corresponding spectral problems in a domain perforated periodically along a part of the boundary. We prove the convergence of solution of the original problems to the solution of the respective homogenized problem in this domain.

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 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 and uniqueness of solutions for a class of fractional nonlinear boundary value problems under mild assumptions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Imed Bachar ◽  
Habib Mâagli ◽  
Hassan Eltayeb
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Nonlinear Boundary ◽  
Continuous Solution ◽  
Boundary Value ◽  
Nonlinear Boundary Value Problems ◽  
Uniqueness Of Solutions ◽  
Nonlinear Boundary Value ◽  
Appl Math

AbstractWe deal with the following Riemann–Liouville fractional nonlinear boundary value problem: $$ \textstyle\begin{cases} \mathcal{D}^{\alpha }v(x)+f(x,v(x))=0, & 2< \alpha \leq 3, x\in (0,1), \\ v(0)=v^{\prime }(0)=v(1)=0. \end{cases} $$ { D α v ( x ) + f ( x , v ( x ) ) = 0 , 2 < α ≤ 3 , x ∈ ( 0 , 1 ) , v ( 0 ) = v ′ ( 0 ) = v ( 1 ) = 0 . Under mild assumptions, we prove the existence of a unique continuous solution v to this problem satisfying $$ \bigl\vert v(x) \bigr\vert \leq cx^{\alpha -1}(1-x)\quad\text{for all }x \in [ 0,1]\text{ and some }c>0. $$ | v ( x ) | ≤ c x α − 1 ( 1 − x ) for all  x ∈ [ 0 , 1 ]  and some  c > 0 . Our results improve those obtained by Zou and He (Appl. Math. Lett. 74:68–73, 2017).

scholarly journals Existence of solutions for fourth-order nonlinear boundary value problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mingzhu Huang
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Nonlinear Boundary ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Lower Solution ◽  
Nonlinear Boundary Value Problems ◽  
Nonlinear Boundary Value ◽  
Quasilinearization Technique

AbstractIn this paper, we discuss the existence and approximation of solutions for a fourth-order nonlinear boundary value problem by using a quasilinearization technique. In the presence of a lower solution α and an upper solution β in the reverse order $\alpha \geq \beta $ α ≥ β , we show the existence of (extreme) solution.

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