Periodic boundary value problems for impulsive integro-differential equations of mixed type

The solutions of initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces

In this paper, we prove the existence of solutions of initial value problems for nth order nonlinear impulsive integro-differential equations of mixed type on an infinite interval with an infinite number of impulsive times in Banach spaces. Our results are obtained by introducing a suitable measure of noncompactness.

PERIODIC BOUNDARY VALUE PROBLEMS FOR NONLINEAR IMPULSIVE INTEGRODIFFERENTIAL EQUATIONS OF MIXED TYPE IN BANACH SPACES

Sufficient conditions for the existence of solutions of the Picard boundary value problem for the second order nonlinear integro-differential equation are established. We allow $G$ to grow linearly, superlinearly or sublinearly in our obtained results, see Theorem 2.1 and Theorem 2.2. Examples are presented to illustrate the efficiency of our theorems. 2000 Mathematics Subject Classification. 34B10, 34B15

Integral boundary value problems for first order integro-differential equations with deviating arguments

The theorem of the existence and uniqueness of the solution of the boundary value problem for the equation in partial derivatives of the fourth order with variable coefficients containing the product of the mixed parabolic-hyperbolic operator and the differential operator of the oscillation string with discontinuous conditions of gluing in the pentagon to the plane is proved. By the method of reducing the order of equations, the solvability of the boundary value problem is reduced to the solution of the Tricomi problem for the mixed parabola-hyperbolic equation with variable coefficients and discontinuous gluing conditions. Solving this problem is reduced to the solution of Fredholm’s integral equation of the second order relative to the trace of the derivative function on y along the line of variation of the equation type. In the hyperbolic part of the domain, the representation of the solution of the problem for the hyperbolic equation with the smallest terms was obtained by using the Riemann function method. In the parabolic part of the domain, the solution of the first boundary value problem for the parabolic equation with the smallest terms is obtained by the method of successive approximations and the Green’s function. As a result, the solution of the problem is realized by the method of solving the Gursa problem and the first boundary value problem for the equation of string oscillation.

In this paper, we discuss the existence of weak solutions for a nonlinear boundary value problem of fractional q-difference equations in Banach space. Our analysis relies on the Mönch's fixed-point theorem combined with the technique of measures of weak noncompactness. MSC:26A33, 34B15.

A unique solution of initial value problems for first order impulsive integro-differential equations of mixed type in Banach spaces

A study of an impulsive four-point nonlocal boundary value problem of nonlinear fractional differential equations

Positive solutions to a multi-point boundary value problem with delay

In this paper, we investigate the existence and multiplicity results of positive solutions for a boundary value problem of nonlinear fractional differential equations. The differential operator is taken in the Riemann-Liouville sense and the nonlinear term depends on the fractional derivative of an unknown function. We derive first the associated Green’s function and prove its properties. Consequently, the considered problem is deduced to an equivalent integral equation. Next, by means of the Leggett-Williams fixed point theory, we obtain the existence and multiplicity results of positive solutions.

In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable; therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type; then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.

Second-order periodic boundary value problems on time scales

In this paper, we investigate the existence of solutions to the boundary value problems for the ordinary integro-differential equations with three-point boundary conditions. Existence and uniqueness results are obtained by using the well-known fixed point theorems.

Initial value problems for nonlinear impulsive integro-differential equations in Banach space