Existence, Uniqueness, and Stability Solutions of Nonlinear System of Integral Equations

The aim of this work is to study the existence, uniqueness, and stability solutions of a new nonlinear system of integral equation by using Picard approximation (successive approximation) method and Banach fixed point theorem. The study of such nonlinear integral equations is more general and leads us to improve to extend the result of Butris. Theorems on the existence and uniqueness of a solution are established under some necessary and sufficient conditions on closed and bounded domains (compact spaces)

Buckling of a nonlinear elastic rod

AbstractThe buckling of a pin-ended slender rod subjected to a horizontal end load is formulated as a nonlinear boundary value problem. The rod material is taken to be governed by constitutive laws which are nonlinear with respect to both bending and compression. The nonlinear boundary value problem is converted to a suitable integral equation to allow the application of bounded operator methods. By treating the integral equation as a bifurcation problem, the branch points (critical values of load) are determined and the existence and form of nontrivial solutions (buckled states) in the neighborhood of the branch points is established. The integral equation also affords a direct attack upon the question of uniqueness of the trivial solution (unbuckled state). It is shown that, under certain conditions on the material properties, only the trivial solution is possible for restricted values of the load. One set of conditions gives uniqueness up to the first branch point

A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator

In this paper, we propose a generalized Gronwall inequality through the fractional integral with respect to another function. The Cauchy-type problem for a nonlinear differential equation involving the psi-Hilfer fractional derivative and the existence and uniqueness of solutions are discussed. Finally, through generalized Gronwall inequality, we prove the continuous dependence of data on the Cauchy-type problem.1118710

Nonlinear Pseudoparabolic Equations and Variational Inequalities

The aim of this thesis is to prove existence and uniqueness of weak solutions for some types of quasilinear and nonlinear pseudoparabolic equations and for some types of quasilinear and nonlinear variational inequalities. The pseudoparabolic equations are characterized by the presence of mixed third order derivatives. Here the existence theory for degenerate parabolic equations is extended to the pseudoprabolic case, and degenerate pseudoparabolic equations with nonlinear integral operator are treated. Furthermore, quasilinear equations, posed on time intervals of the form (-\infty,T], are considered. Some nonlinear pseudoparabolic equations are obtained as reduced form of systems of equations. To show existence, the Galerkin and Rothe methods are used. The system of the degenerate equations is solved using the monotonicity and gradient assumptions on the nonlinear function. The discretization along characteristics is applied to equations with convection. The existence of solutions of variational inequalities is proved by a penalty method; here an inequality is replaced by an equation with an added penalty operator. The uniqueness follows from the monotonicity of the differential operators. In the case of nonlinear pseudoparabolic equations, the uniqueness can be shown for regular solutions only. The needed regularity is shown for two dimensional domains

One dimensional nonlinear integral operator with Newton–Kantorovich method

The Newton–Kantorovich method (NKM) is widely used to find approximate solutions for nonlinear problems that occur in many fields of applied mathematics. This method linearizes the problems and then attempts to solve the linear problems by generating a sequence of functions. In this study, we have applied NKM to Volterra-type nonlinear integral equations then the method of Nystrom type Gauss–Legendre quadrature formula (QF) was used to find the approximate solution of a linear Fredholm integral equation. New concept of determining the solution based on subcollocation points is proposed. The existence and uniqueness of the approximated method are proven. In addition, the convergence rate is established in Banach space. Finally illustrative examples are provided to validate the accuracy of the presented method