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

A unique solution of initial value problems for first order impulsive 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

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

We study an initial value problem for a class of integro-differential equations of Volterra type in a real Banach space. Using method of upper and lower solutions and M¨ onch and Von Harten theorem, we obtain an existence theorem of coupled quasi-solutions, which is an extension of those established by Y. Chen and W. Zhuang in [1].

The paper is focused on solvability of an inverse boundary problem with an unknown coefficient which depends on time for a second-order parabolic equation with non-classical boundary conditions. The idea of the problem is that together with the solution it is required to determine the unknown coefficient. The problem is considered in the rectangular area. The paper introduces a classical solution of the set problem. At first, an auxiliary inverse boundary problem is examined and the equivalence (in some sense) of the original problem is proved. First, we apply a method of variable separation to analyze the auxiliary inverse boundary problem. Then, we examine a spectral problem for an ordinary second-order differential equation with integral conditions. Having used a formal scheme of the method of variable separation, the solution of a direct boundary problem (in case of specified unknown function) resolves itself into solution of Cauchy problem. After that the solution is limited to the solution of a countable system of integro-differential equations in Fourier coefficients. In its turn, the last system regarding unknown Fourier coefficients is recorded in the form of an integro-differential equation in the desired solution. Using relevant additional conditions of the auxiliary inverse boundary problem, we obtain a system of two nonlinear integral equations for defining unknown functions. Thus, the solution of the auxiliary inverse boundary problem comes down to the system of two nonlinear integro-differential equations in unknown functions. The specific Banach space is designed. Then, in the sphere made of the Banach space we with the help of contracted mapping prove the solvability of the nonlinear integro-differential equations set, which is a unique solution of the additional inverse boundary problem. Using the equivalence of problems, it is concluded about existence and uniqueness of a classical solution of the original problem.

This work deals with the existence of solutions for a class of nonlinear nonlocal fractional functional differential equations of neutral type in Banach spaces. In particular, we prove the existence of solutions with the assumptions that the nonlinear parts satisfy locally Lipschitz like conditions and closed linear operator \(-A(t)\) generates analytic semigroup for each \(t \ge 0\). We also investigate global existence of solution and study the continuous dependence of solution on initial data. We conclude the article with an application to the developed results.

Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis

Abstract. In a Banach space, we study an operator equation with a monotone operator T. The operator is an operator from a Banach space to its conjugate, and T=AC, where A and C are operators of some classes. The considered problem belongs to the class of ill-posed problems. For this reason, an operator regularization method is proposed to solve it. This method is constructed using not the operator T of the original equation, but a more simple operator A, which is B-monotone, B=C−1. The existence of the operator B is assumed. In addition, when constructing the operator regularization method, we use a dual mapping with some gauge function. In this case, the operators of the equation and the right-hand side of the given equation are assumed to be perturbed. The requirements on the geometry of the Banach space and on the agreement conditions for the perturbation levels of the data and of the regularization parameter are established, which provide a strong convergence of the constructed approximations to some solution of the original equation. An example of a problem in Lebesgue space is given for which the proposed method is applicable.

Properties of solutions of nonlinear integral equations of evolution type in Banach spaces

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

In this paper, we will discuss some results on the existence and uniqueness of mild and strong solution of initial value problem of fractional order subjected to non-local conditions, by using the Banach fixed point theorem and the theory of strongly continuous cosine family under Caputo sense. Furthermore, we also prove that solution of Nonlinear Fractional Volterra Integrodifferential Equations and Nonlinear Fractional Mixed Integrodifferential Equations With Nonlocal Conditions is unique. Moreover, examples demonstrate the validity of the obtained main result and we obtain the solution for an equation, and proved that this solution is unique.

On linear Volterra equations in Banach spaces

In this paper we study subordination principles for fractional differential equations of Sobolev type in Banach space. With the help of the theory of Sobolev type resolvent families (known also as propagation family) as well as these subordination principles, we obtain the existence of mild solutions for this kind of equations. We study simultaneously the case 0 < α < 1 and 1 < α < 2 for the Caputo and Riemann-Liouville fractional derivatives.

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