Existence for a Fourth-Order Boundary Value Problem under a Two-Parameter Nonresonance Condition

Nonresonance conditions for fourth order nonlinear boundary value problems

“…We will be able to prove the existence of a solution under the assumption that the point (g(t, x, y), h(t,x, y, z, w)) always lies in a rectangle with sides parallels to the axes and which does not intersect y eigenline. This kind of result c be considered as an extension of the results of Y. Yang [2] [5], C. Fabry and F. Munyamarere [6], Y. Yang [22]. Unfortunately, the existence conditions obtned with the decomposition (1.9) e not nice those obtnl with (1.8).…”

“…This lmt condition can be clearly read a non-interference condition of tlm nonlinearity with respect to the spectrum of the operator u u(), subject to the boundary conditions (1.4). Recently, M. Del Pino and R. Manevich [5] have extcnded Y. Yang's [22] result.…”

Existence and uniqueness theorems for some fourth‐order nonlinear boundary value problems

“…We remark that the existence and uniqueness theorems obtained in this paper for the boundary value problems (1.1), when particularized to the case when G in (1.1) is independent of y and f (x,y,y ,y ,y ) = f (x) gives new existence theorems for the problems studied in [3]. We would like to refer the reader to [1,2,4,5,6,7,8] and references therein for related works on fourth-order boundary value problems.…”

Existence of Positive Solutions for an Elastic Curved Beam Equation

“…Boundary-value problems for ordinary differential equations arise in different areas of applied mathematics and physics and the existence and multiplicity of positive solutions for such problems has become an important area of investigation in recent years; we refer the reader to [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and the references therein. For example, the deformations of an elastic beam in the equilibrium state can be described as a boundary value problem of some fourth-order differential equations.…”

“…al. [5], Ma and Wang [6], Aftabizadeh [7], Yang [8], Del Pino and Manasevich [9] (see also the references therein). All of those results are based on the LeraySchauder continuation method, topological degree and the method of lower and upper solutions.…”