abstract:In this paper, we study the existence of solution for the fourth-order three-point boundary value problem having the following formwhere η ∈ (0, 1), α, β ∈ R, f ∈ C([0, 1] × R, R), and f (t, 0) = 0. We give sufficient conditions that allow us to obtain the existence of solution. And by using the LeraySchauder nonlinear alternative we prove the existence of at least one solution of the posed problem. As an application, we also given some examples to illustrate the results obtained.

In this paper, we explore the existence of nontrivial solution for the fifth-order three-point boundary value problem of the form u (5) Under certain growth conditions on the non-linearity f and using Leray-Schauder nonlinear alternative, we prove the existence of at least one solution of the posed problem. Furthermore, the obtained results are further illustrated by mean of some examples.

In this paper, we study the existence of positive solutions for boundary value problem of sixth-order elastic beam equation of the form −u (6) The boundary conditions describe the deformation of an elastic beam simply supported at left and clamped at right by sliding clamps. We give sufficient conditions that allow us to obtain the existence of positive solution. The main tool used in the proof is the Leray-Schauder nonlinear alternative and Leray-Schauder fixed point theorem. As an application, we also give example to illustrate the results obtained.

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In this research collection, we estimate the existence of the unique solution for the boundary value problem of nonlinear fractional [Formula: see text]-difference equation having the given form [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] represents the Caputo-type nonclassical [Formula: see text]-derivative of order [Formula: see text]. We use well-known principal of Banach contraction, and Leray–Schauder nonlinear alternative to vindicate the unique solution existence of the given problem. Regarding the applications, some examples are solved to justify our outcomes.