Nonlinear Nonlocal Boundary Value Problems Research Articles

Applications of generalized trigonometric functions with two parameters

Generalized trigonometric functions (GTFs) are simple generalization of the classical trigonometric functions. GTFs are deeply related to the $p$-Laplacian, which is known as a typical nonlinear differential operator, and there are a lot of works on GTFs concerning the $p$-Laplacian. However, few applications to differential equations unrelated to the $p$-Laplacian are known. We will apply GTFs with two parameters to nonlinear nonlocal boundary value problems without $p$-Laplacian. Moreover, we will give integral formulas for the functions, e.g. Wallis-type formulas, and apply the formulas to the lemniscate function and the lemniscate constant.

Chebyshev Wavelet collocation method for solving a class of linear and nonlinear nonlocal boundary value problems

This study proposes the Chebyshev Wavelet Colocation method for solving a class of rth-order Boundary-Value Problems (BVPs) with nonlocal boundary conditions. This method is an extension of the Chebyshev wavelet method to the linear and nonlinear BVPs with a class of nonlocal boundary conditions. In this study, the method is tested on second and fourth-order BVPs and approximate solutions are compared with the existing methods in the literature and analytical solutions. The proposed method has promising results in terms of the accuracy.

On nonlinear boundary value problems for higher order functional differential equations

Abstract For higher order nonlinear functional differential equations, sufficient conditions for the solvability and unique solvability of some nonlinear nonlocal boundary value problems are established.

Bounded Solutions for Nonlocal Boundary Value Problems on Lipschitz Manifolds with Boundary

Abstract We consider nonlinear nonlocal boundary value problems associated with fractional operators (including the fractional p-Laplace and the regional fractional p-Laplace operators) and subject to general (fractional-like) boundary conditions on bounded domains with Lipschitz boundary. Under suitable conditions on the nonlinearities of our system, we establish the existence of bounded solutions and provide explicit L ∞ ${L^{\infty}}$ -estimates of solutions which are optimal with respect to the inhomogeneous “sources” present in the system. As application, these results are shown to apply to a class of nonlinear nonlocal equations for the Dirichlet fractional p-Laplacian and regional fractional p-Laplace with a dissipative nonlinearity, and to a class of semilinear nonlocal boundary value problems with fractional Wentzell–Robin boundary conditions corresponding to the so-called fractional Wentzell Laplacian.

An efficient numerical algorithm based on Haar wavelet for solving a class of linear and nonlinear nonlocal boundary-value problems

Based on Haar wavelet a new numerical method for the solution of a class of nth-order boundary-value problems (BVPs) with nonlocal boundary conditions is proposed. This new method is an extension of the Haar wavelet method (Siraj-ul-Islam et al. in Math Comput Model 50:1577---1590, 2010; Int J Therm Sci 50:686---697, 2011) from BVPs with local boundary conditions to BVPs with a class of nonlocal boundary conditions. A major advantage of the proposed method is that it is applicable to both linear and nonlinear BVPs. The method is tested on BVPs of third- and fourth-order. The numerical results are compared with an existing method in the literature and analytical solutions. The numerical experiments demonstrate the accuracy and efficiency of the proposed method.

Nonlinear nonlocal problems for second-order differential equations singular with respect to the phase variable

For second-order differential equations singular with respect to the phase variable, we obtain in a sense optimal criteria for the existence and uniqueness of positive solutions of nonlinear nonlocal boundary value problems.

POSITIVE SOLUTIONS FOR A SYSTEM OF SINGULAR SECOND ORDER NONLOCAL BOUNDARY VALUE PROBLEMS

Sucient conditions for the existence of positive solutions for a coupled system of nonlinear nonlocal boundary value problems of the type ix 00 (t) = f(t,y(t)), t 2 (0,1), iy 00 (t) = g(t,x(t)), t 2 (0,1), x(0) = y(0) = 0, x(1) = fix(·), y(1) = fiy(·), are obtained. The nonlinearities f,g : (0,1) ◊ (0,1) ! (0,1) are con- tinuous and may be singular at t = 0, t = 1, x = 0, or y = 0. The parameters ·, fi satisfy · 2 (0,1), 0 < fi < 1/·. An example is provided to illustrate the results.

Linear and nonlinear nonlocal boundary value problems for differential-operator equations

This study focuses on nonlocal boundary value problems (BVPs) for linear and nonlinear elliptic differential-operator equations (DOEs) that are defined in Banach-valued function spaces. The considered domain is a region with varying bound and depends on a certain parameter. Some conditions that guarantee the maximal Lp -regularity and Fredholmness of linear BVPs, uniformly with respect to this parameter, are presented. This fact implies that the appropriate differential operator is a generator of an analytic semigroup. Then, by using these results, the existence, uniqueness and maximal smoothness of solutions of nonlocal BVPs for nonlinear DOEs are shown. These results are applied to nonlocal BVPs for regular elliptic partial differential equations, finite and infinite systems of differential equations on cylindrical domains, in order to obtain the algebraic conditions that guarantee the same properties.