On the existence of positive solutions for the bending elastic beam equations☆
Section snippets
Introduction and main results
The deformations of an elastic beam in equilibrium state, whose two ends are simply supported, can be described by fourth-order ordinary differential equation boundary value problem (BVP)where is continuous [3], [4], and the u″ in f is the bending moment term which represents bending effect. Owing to its importance in physics, the existence of solutions to this problem has been studied by many authors, see [1], [2], [3], [4],
Preliminaries
We denote the maximum norm of C(I) by ∥u∥. Let C+(I) be the cone of all nonnegative functions in C(I). We choose an equivalent norm in Banach space C2(I) byGiven , we consider the linear boundary value problem (LBVP)Let G(t, s) be the Green’s function to the linear boundary value problemwhich is explicitly expressed byIt is easy to see that G(t, s) has
Proof of the main results
Proof of Theorem 1 We show that the mapping A defined by (10) has a non-zero fixed point by counting of the fixed point index in cone K in C2(I). Choose , where r0 is the constant in assumption (F1). We prove that for and . In fact, if there exist and such that , then by definition of A, u0(t) satisfies differential equationand boundary conditionSince , from assumption (F1) and
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Existence and uniqueness results for the bending elastic beam equations
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2010, Applied Mathematics and ComputationCitation Excerpt :In fact, it is well known that the mathematical modelling of important questions in different fields of research, such as mechanical engineering, control systems, economics, computer science and many others, leads naturally to the consideration of nonlinear differential equations. In particular, the deformations of an elastic beam in an equilibrium state, whose two ends are simply supported, can be described by fourth-order boundary value problems and, also for this reason, the existence and multiplicity of solutions for this kind of problems have been widely investigated (see, for instance, [1,3,5–9] and references therein). It is worth noticing that, since the parameters p(t), q(t), r(t) are variable, the study of (Sλ) allows us to investigate fourth-order problems with equations in a complete form, that is, explicitly depending also on u‴ and u′ (see Theorem 1.1 below).
A monotone iterative technique for solving the bending elastic beam equations
2010, Applied Mathematics and ComputationCitation Excerpt :All the known results of [1–16] are not applicable to this equation, we will use our new results to prove that the equation has a positive solution.
A study on solvability of the fourth-order nonlinear boundary value problems
2023, International Journal of Nonlinear Sciences and Numerical Simulation
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Research supported by NNSF of China (10271095) and the NSF of Gansu Province (ZS031-A25-003-Z).