Boundary value problems for some fourth order ordinary differential equations

We consider beam equations of the form with boundary conditions corresponding either to a simply-supported or a clamped beam. The function α is supposed to be nonnegative and we assume that the underlying linear problem has zero as its first eigenvalue. We give a unified treatment of the existence of solutions for both boundary conditions, using variational or degree-theoretic methods.

[1]  Jean Mawhin,et al.  Critical point theory and nonlinear differential equations , 1986 .

[2]  C. P. Gupta,et al.  Existence and uniqueness theorems for the bending of an elastic beam equation , 1988 .

[3]  P. Rabinowitz.  Minimax methods in critical point theory with applications to differential equations , 1986 .

[4]  A. R. Aftabizadeh.  Existence and uniqueness theorems for fourth-order boundary value problems , 1986 .

[5]  Resonance problems with nonlinearity interfering with eigenvalues of higher orderheigen , 1987 .

[6]  J. L. Paul,et al.  ELEMENTARY CRITICAL POINT THEORY AND PERTURBATIONS OF ELLIPTIC BOUNDARY VALUE PROBLEMS AT RESONANCE. , 1976 .

[7]  J. Mawhin,et al.  Necessary and sufficient conditions for the solvability of a nonlinear two-point boundary value problem , 1985 .

[8]  Riaz A. Usmani.  A uniqueness theorem for a boundary value problem , 1979 .

[9]  Yong Yang.  Fourth-order two-point boundary value problems , 1988 .

[10]  C. P. Gupta,et al.  Existence and uniqueness results for the bending of an elastic beam equation at resonance , 1988 .

[11]  J. Leblanc.  THÈSE DE 3ÈME CYCLE , 1978 .

[12]  D. G. Figueiredo,et al.  Lectures on the ekeland variational principle with applications and detours , 1989 .

[13]  Jean Mawhin,et al.  Critical points of convex perturbations of some indefinite quadratic forms and semi-linear boundary value problems at resonance , 1986 .

[14]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[15]  M. Ramos,et al.  Vanational elliptic problems involving noncoercive functionals , 1989, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.