Solving the Dirichlet problem for fully fourth order nonlinear differential equation

Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis

“…see [18,Theorem 3.1 Condition c] or [12]. For clamped/free ends, to obtain a contraction for the operator with respect to our ρ as per Banach's contraction mapping theorem, the condition…”

“…On the other hand, the fully nonlinear problem (1.1), (1.5) was analysed in [11][12][13]. The assumptions on f in [11,12] are of a local nature, that is, the domain of f is restricted to closed and bounded sets. While these types of assumptions are quite wide-ranging, the very nature of localized assumptions means that only limited, localized information about solutions can be necessarily obtained.…”

“…rerun the same argument as above for the example holds but (3.3) does not. Hence, the results of[12] do not apply to this f with = Furthermore, we consider the special case subject to clamped/free ends (1.5).Choose = / R 1 2 to form the set F in the statement of Theorem 2.7. Here, choose = / (with the remaining θ i not coming in to play) and a bound M on f over F can be chosen to be = M In addition, we see ∂ /∂ f y is continuous and bounded on F by 8.…”

Existence Results and Numerical Solution of Fully Fourth Order Nonlinear Functional Differential Equations

“…associated with general linear boundary conditions. The method is a further development of our method for nonlinear boundary value problems in [8,9]. In connection to the general functional differential equation u (p) (t) = f (t, u(φ(t))), p ≥ 1 it should be said that in its particular case when φ(t) = αt, 0 < α < 1, the equation is called pantograph equation or proportional delay differential equation.…”

Existence and uniqueness of solutions for a mixed p-Laplace boundary value problem involving fractional derivatives

“…In recent years, fractional-order calculus theory has been widely used in mathematics, science, engineering, etc. As a result, the studies of such equation have gained considerable popularity, see [1][2][3][4][5][6][7][8]. Also, fractional-order mixed differential or integral equation involving different fractional derivatives, such as conformable fractional, Riemann-Liouville, and Caputo, has got a lot of interest, and even fractional-order differential or integral equations with p-Laplace operator have been extensively discussed by more and more researchers [9][10][11][12][13][14].…”