Existence and uniqueness results for the bending of an elastic beam equation at resonance

In this paper, we deal with the existence and uniqueness of the solutions of two-point boundary value problem of fourth-order ordinary differential equation: u4(t)=f(t,u(t),u′(t)), t∈[0,1], u(0)=u′(0)=u′′(1)=u′′′(1)=0, where f:[0,1]×R2→R is a continuous function. The problem describes the static deformation of an elastic beam whose left end-point is fixed and right is freed, which is called slanted cantilever beam. Under some weaker assumptions, we establish a new maximum principle by the perturbation of positive operator and construct the monotone iterative sequence of the lower and upper solutions, and, based on this, we obtain the existence and uniqueness results for the slanted cantilever beam.

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Multi-term fractional differential equations with nonlocal boundary conditions

Open Mathematics ◽

10.1515/math-2018-0127 ◽

2018 ◽

Vol 16 (1) ◽

pp. 1519-1536

Author(s):

Bashir Ahmad ◽

Najla Alghamdi ◽

Ahmed Alsaedi ◽

Sotiris K. Ntouyas

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Existence And Uniqueness ◽

Fixed Point Theorems ◽

Nonlocal Boundary ◽

Nonlocal Boundary Conditions ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Fractional Differential ◽

The Given

AbstractWe introduce and study a new kind of nonlocal boundary value problems of multi-term fractional differential equations. The existence and uniqueness results for the given problem are obtained by applying standard fixed point theorems. We also construct some examples for demonstrating the application of the main results.

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BSDEs with polynomial growth generators

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953300000216 ◽

2000 ◽

Vol 13 (3) ◽

pp. 207-238 ◽

Cited By ~ 41

Author(s):

Philippe Briand ◽

René Carmona

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Existence And Uniqueness ◽

Polynomial Growth ◽

Probabilistic Representation ◽

Existence And Uniqueness Results ◽

State Variable ◽

Cahn Equation ◽

Uniqueness Results ◽

Terminal Time

In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in the state variable. We deal with the case of a fixed terminal time, as well as the case of random terminal time. The need for this type of extension of the classical existence and uniqueness results comes from the desire to provide a probabilistic representation of the solutions of semilinear partial differential equations in the spirit of a nonlinear Feynman-Kac formula. Indeed, in many applications of interest, the nonlinearity is polynomial, e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrödinger equations.

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Nontrivial solutions of non-autonomous dirichlet fractional discrete problems

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0051 ◽

2020 ◽

Vol 23 (4) ◽

pp. 980-995

Author(s):

Alberto Cabada ◽

Nikolay Dimitrov

Keyword(s):

Existence And Uniqueness ◽

Point Boundary ◽

Nontrivial Solutions ◽

Fractional Difference ◽

Nonlinear Part ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Fractional Difference Equation ◽

Discrete Problems ◽

Series Of Functions

AbstractIn this paper, we introduce a two-point boundary value problem for a finite fractional difference equation with a perturbation term. By applying spectral theory, an associated Green’s function is constructed as a series of functions and some of its properties are obtained. Under suitable conditions on the nonlinear part of the equation, some existence and uniqueness results are deduced.

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