There has also been a great emphasis on studying fractional differential equations supplemented with integral boundary conditions; for instance, see [27 32].
Many researchers have studied the existence theory for nonlinear fractional differential equations with a variety of boundary conditions; for instance, see the papers [6 21], and the references therein.
The literature on the topic includes the existence and uniqueness results related to classical, periodic/anti-periodic, nonlocal, multi-point, and integral boundary conditions, for instance, see [1 18].
Since the work of Fonda [6], some investigation has been devoted to the nonresonance condition of (1.1) by studying the asymptotic interaction of the ratio 2 F ( t, s ) s 2, where F ( t, s ) = ∫ 0 s f ( t, τ ) d τ, with the spectrum of − x ″ under different boundary conditions; for instance, see [10] for the periodic boundary condition, [16] for the two-point boundary condition.
As (srightarrow1^), problem (1) becomes the elliptic Kirchhoff equation -M biggl( int_{Omega}|nabla u|^{2},dx biggr)Delta u=f x,u), quad xinOmega, (3) where (Omegasubsetmathbb{R}^{N} ) is a smooth domain, and u satisfies some boundary conditions; see, for instance, [1, 2] for more information about Eq. (3).
Owing to the importance of fourth-order two-point boundary value problems in describing a large class of elastic deflection, there is a wide literature that deals with the existence and multiplicity results for such a problem with different boundary conditions (see, for instance, [3 8] and the references therein).
For more details of boundary value problems involving integral boundary conditions; see for instance, [25 34] and references therein.
Various boundary conditions are considered, and the present results represent the first instance in which, the effects of boundary conditions for this class of problems, have been reported in open literature.
The technique, called adaptive subdomain modeling, concurrently analyzes any number of child domains, with each instance corresponding to a unique design or failure scenario, in addition to a full-scale parent domain providing the boundary conditions for its children.
