scholarly journals Existence of unbounded positive solutions of boundary value problems for differential systems on whole lines

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3547-3564 ◽  
Author(s):  
Yuji Liu
Keyword(s):  
Banach Space ◽  
Boundary Value Problems ◽  
Nonlinear Operator ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Continuous Operator ◽  
Differential Systems ◽  
Completely Continuous ◽  
Type Boundary ◽  
Laplacian Operators

This paper is concerned with integral type boundary value problems of second order singular differential systems with quasi-Laplacian operators on whole lines. A Banach space and a nonlinear completely continuous operator are defined. By using the Banach space and the nonlinear operator, together with the Schauder?s fixed point theorem, sufficient conditions to guarantee the existence of at least one unbounded positive solution are established. Finally, we present a concrete example to illustrate the efficiency of the main theorem.

scholarly journals SOLVABILITY OF BOUNDARY VALUE PROBLEMS FOR SINGULAR QUASI-LAPLACIAN DIFFERENTIAL EQUATIONS ON THE WHOLE LINE

2012 ◽  
Vol 17 (3) ◽  
pp. 423-446 ◽  
Author(s):  
Yuji Liu
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonnegative Function ◽  
Continuous Operator ◽  
Completely Continuous ◽  
One Dimensional ◽  
Type Boundary ◽  
The One

This paper is concerned with some integral type boundary value problems associated to second order singular differential equations with quasi-Laplacian on the whole line. The emphasis is put on the one-dimensional p-Laplacian term involving a nonnegative function ρ that may be singular at t = 0 and such that . A Banach space and a nonlinear completely continuous operator are defined in this paper. By using the Schauder's fixed point theorem, sufficient conditions to guarantee the existence of at least one solution are established. An example is presented to illustrate the main theorem.

ANTI-PERIODIC TYPE BOUNDARY VALUE PROBLEMS FOR SINGULAR MULTI-TERM FRACTIONAL DIFFERENTIAL EQUATIONS WITH IMPULSE EFFECTS

2013 ◽  
Vol 24 (06) ◽  
pp. 1350047 ◽  
Author(s):  
YUJI LIU
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Continuous Operator ◽  
Weighted Banach Space ◽  
Fractional Differential ◽  
The Fixed Point Theorem ◽  
Type Boundary

Results on the existence of solutions of anti-periodic type boundary value problems for singular multi-term fractional differential equations with impulse effects are established. We first transform the problem into a hybrid system, then construct a weighted Banach space and a completely continuous operator, and finally, we use the fixed point theorem in the Banach space to prove the main results. An example is given to illustrate the efficiency of the main theorems.

scholarly journals The existence of three positive solutions to integral type BVPs for singular second ODEs with one-dimensional p-Laplacian

2011 ◽  
Vol 27 (2) ◽  
pp. 239-248
Author(s):  
YUJI LIU ◽  
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Integral Type ◽  
Singular Differential Equations ◽  
One Dimensional ◽  
Type Boundary ◽  
The One

This paper is concerned with the integral type boundary value problems of the second order singular differential equations with one-dimensional p-Laplacian. Sufficient conditions to guarantee the existence of at least three positive solutions are established. An example is presented to illustrate the main results. The emphasis is put on the one-dimensional p-Laplacian term [ρ(t)Φ(x 0 (t))]0 involved with the function ρ, which makes the solutions un-concave. Furthermore, f, g, h and ρ may be singular at t = 0 or t = 1.

On Some Boundary Value Problems with Integral Conditions for Functional Differential Equations

1995 ◽  
Vol 2 (2) ◽  
pp. 165-188
Author(s):  
I. Kiguradze ◽  
D. Chichua
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Continuous Operator ◽  
Integer Part ◽  
Integral Conditions ◽  
Differentiable Functions ◽  
Integrable Functions ◽  
Functional Differential

Abstract For the functional differential equation u (n)(t) = ƒ(u)(t) we have established the sufficient conditions for solvability and unique solvability of the boundary value problems and where n ≥ 2, m is the integer part of , ci ∈ R, and ƒ is the continuous operator acting from the space of (n – 1)-times continuously differentiable functions given on an interval [0, +∞[ into the space of locally Lebesgue integrable functions.

scholarly journals Kronecker Product of matrices and their applications to self-adjoint two-point boundary value problems associated with first order matrix differential systems

2021 ◽  
Vol 10 (10) ◽  
pp. 25399-25407
Author(s):  
Sriram Bhagavatula ◽  
Dileep Durani Musa ◽  
Murty Kanuri
Keyword(s):  
Boundary Value Problems ◽  
Kronecker Product ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Least Square ◽  
Point Boundary ◽  
Uniqueness Of Solutions ◽  
Differential Systems ◽  
First Order ◽  
Product Of Matrices

In this paper, we shall be concerned with Kronecker product or Tensor product of matrices and develop their properties in a systematic way. The properties of the Kronecker product of matrices is used as a tool to establish existence and uniqueness of solutions to two-point boundary value problems associated with system of first order differential systems. A new approach is described to solve the Kronecker product linear systems and establish best least square solutions to the problem. Several interesting examples are given to highlight the importance of Kronecker product of matrices. We present adjoint boundary value problems and deduce a set of necessary and sufficient conditions for the Kronecker product boundary value problem to be self-adjoint.

scholarly journals Solutions to Dirichlet-Type Boundary Value Problems of Fractional Order in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Jing-jing Tan ◽  
Cao-zong Cheng
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Fixed Point ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Nonlinear Fractional Differential Equation ◽  
Dirichlet Type ◽  
Fractional Differential ◽  
Type Boundary ◽  
Sadovskii Fixed Point Theorem

We consider the boundary value problems with Dirichlet-type boundary conditions of nonlinear fractional differential equation in Banach space. The existence of the solution to the boundary value problems is established. Our analysis relies on the Sadovskii fixed point theorem. As an application, we give an example to demonstrate our results.

On Two-Point Boundary Value Problems for Two-Dimensional Differential Systems with Singularities

2003 ◽  
Vol 10 (3) ◽  
pp. 595-602
Author(s):  
S. Mukhigulashvili
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Differential System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Two Dimensional ◽  
Point Boundary ◽  
Differential Systems

Abstract For a differential system where λ ∈]0, 1[ and ℎ𝑖 :]𝑎, 𝑏[×]0, +∞[×ℝ → [0, +∞[ (𝑖 = 0, 1, 2) are continuous functions, we have established sufficient conditions for the existence of at least one solution satisfying one of the two boundary conditions and

scholarly journals Three Positive Solutions of Multi-point BVPs for Difference Equations with the Nonlinearity Depending on Δ−operator

2012 ◽  
Vol 20 (3) ◽  
pp. 65-82
Author(s):  
Yuji Liu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Banach Spaces ◽  
Positive Solutions ◽  
Difference Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Type Boundary ◽  
Discrete Type

Abstract This article deals with a class of discrete type boundary value problems. Sufficient conditions guaranteeing the existence of at least three positive solutions of this class of boundary value problems are established by using a fixed point theorem in cones in Banach spaces. An example is given to illustrate the main theorem.

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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