1,218 research outputs found

    Existence of positive solutions to multi-point third order problems with sign changing nonlinearities

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    In this paper, the authors examine the existence of positive solutions to a third-order boundary value problem having a sign changing nonlinearity. The proof makes use of fixed point index theory. An example is included to illustrate the applicability of the results

    Positive operators and maximum principles for ordinary differential equations

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    We show an equivalence between a classical maximum principle in differential equations and positive operators on Banach Spaces. Then we shall exhibit many types of boundary value problems for which the maximum principle is valid. Finally, we shall present extended applications of the maximum principle that have arisen with the continued study of the qualitative properties of Green’s functions

    Existence and iteration of monotone positive solutions for third-order nonlocal BVPs involving integral conditions

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    This paper is concerned with the existence of monotone positive solution for the following third-order nonlocal boundary value problem u′′′(t)+f(t,u(t),u′(t))=0, 0<t<1u^{\prime \prime \prime }\left(t\right) +f\left( t,u\left( t\right), u^{\prime}\left( t\right)\right) =0,\, 0<t<1; u(0)=0,u\left( 0\right) =0, au′(0)−bu′′(0)=α[u],au^{\prime}\left( 0\right)-b u^{\prime\prime}\left( 0\right)=\alpha[u], cu′(1)+du′′(1)=β[u],c u^{\prime}\left( 1\right)+d u^{\prime\prime}\left( 1\right)=\beta[u], where f∈C([0,1]×R+×R+,R+)f\in C([0,1]\times R^{+}\times R^{+}, R^{+}), α[u]=∫01u(t)dA(t)\alpha[u]=\int_{_{0}}^{1}u(t)dA(t) and β[u]=∫01u(t)dB(t)\beta[u]=\int_{_{0}}^{1}u(t)dB(t) are linear functionals on C[0,1]C[0,1] given by Riemann-Stieltjes integrals. By applying monotone iterative techniques, we not only obtain the existence of monotone positive solution but also establish an iterative scheme for approximating the solution. An example is also included to illustrate the main results

    Three point boundary value problems for ordinary differential equations, uniqueness implies existence

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    We consider a family of three point n − 2, 1, 1 conjugate boundary value problems for nth order nonlinear ordinary differential equations and obtain conditions in terms of uniqueness of solutions imply existence of solutions. A standard hypothesis that has proved effective in uniqueness implies existence type results is to assume uniqueness of solutions of a large family of n−point boundary value problems. Here, we replace that standard hypothesis with one in which we assume uniqueness of solutions of large families of two and three point boundary value problems. We then close the paper with verifiable conditions on the nonlinear term that in fact imply global uniqueness of solutions of the large family of three point boundary value problems

    The upper and lower solution method for nonlinear third-order three-point boundary value problem

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    This paper is concerned with the following nonlinear third-order three-point boundary value problem \left\{ \begin{array}{l} u^{\prime \prime \prime }(t)+f\left( t,u\left( t\right) ,u^{\prime}\left(t\right) \right) =0,\, t\in \left[ 0,1\right], \\ u\left( 0\right) =u^{\prime }\left( 0\right) =0,\, u^{\prime}\left( 1\right) =\alpha u^{\prime }\left( \eta \right),\label{1.1} \end{array} \right. where 0<η<10<\eta <1 and 0≤α<1.0\leq \alpha <1. A new maximum principle is established and some existence criteria are obtained for the above problem by using the upper and lower solution method

    A survey on stationary problems, Green's functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions

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    In this paper, we present a survey of recent results on the Green's functions and on spectrum for stationary problems with nonlocal boundary conditions. Results of Lithuanian mathematicians in the field of differential and numerical problems with nonlocal boundary conditions are described. *The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/2014)

    Existence and Monotone Iteration of Positive Pseudosymmetric Solutions for a Third-Order Four-Point BVP with -Laplacian

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    We study the existence and monotone iteration of solutions for a third-order four-point boundary value problem with -Laplacian. An existence result of positive, concave, and pseudosymmetric solutions and its monotone iterative scheme are established by using the monotone iterative technique. Meanwhile, as an application of our result, an example is given

    Iterative technique for third-order differential equation with three-point nonlinear boundary value conditions

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    In this paper, we study the existence of extremal solutions for a nonlinear third-order differential equation with three-point nonlinear boundary value conditions. By means of the method of upper and lower solutions and different monotone iterative techniques, the sufficient conditions which guarantee the existence of extremal solutions are given. An example illustrates the main results
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