The theory of differential equations arises from the study of physical phenomena. This field has various applications in science and engineering. The study of qualitative properties for each mathematical model plays an important role, attracting the attention of both theoretical and applied researchers. Normally, the most significant qualitative property to be studied first is the existence and uniqueness of the solutions of each mathematical model. However, proving existence and uniqueness results for mathematical models where the source function has a singularity is a difficult problem and requires many different techniques. In this paper, we establish some new conditions suitable to achieve the unique solution criterion for ordinary first-order differential equations. To obtain the desired results, we have improved the methods that have been used to prove the results in the work of Krasnosel'skii and Krein (Krasnoselskii and Krein, 1956). In addition, we also provide an example to illustrate the theoretical results.

Fractional differential equations are an important branch of mathematics and have been considered under many different fractional derivatives. Among them, differential equations with Riesz-Caputo fractional derivatives have also attracted the attention of many researchers. Studying differential equations that may have singularity coefficients is more difficult than usual because they require several complex techniques. In the present paper, we consider a nonlinear pantograph differential equation where the source function may have a temporal singularity. Using the contraction principle, we prove that the problem has a unique solution under some appropriate conditions. Furthermore, we define a new type of Ulam-Hyers stability and show the main equation of the problem is stable in the mentioned sense. To obtain the main results, a new inequality is proposed and proved. Some examples are constructed to confirm the validity and feasibility of the theoretical results.

In this note, we prove a unique result for a nonlinear differential equation concerning a Nagumo-type source. An example is given to illustrate the theoretical result.

In this paper, we consider the Robin-Dirichlet problem for a system of nonlinear pseudoparabolic equations with viscoelastic term. By the Faedo-Galerkin method, we first prove the existence and uniqueness of solution for the problem. Next, we give a sufficient condition to get the global existence and decay of the weak solution. Finally, by the concavity method, we prove the blow-up result of the solution when the initial energy is negative. Furthermore, we establish here the lifespan of the solution by finding the upper bound and the lower bound for the blow-up time.

This paper is devoted to the study of a Kirchhoff wave equation with a viscoelastic term in an annular associated with homogeneous Dirichlet conditions. At first, by applying the Faedo-Galerkin, we prove existence and uniqueness of the solution of the problem considered. Next, by constructing Lyapunov functional, we establish a sufficient condition such that any global weak solution is general decay as t → +∞.

In this paper, we consider the Dirichlet problem for a wave equation of Kirchhoff- Carrier type with a nonlinear viscoelastic term. It consists of two main parts. In Part 1, we establish existence and uniqueness of a weak solution by applying the Faedo- Galerkin method and the standard arguments of density corresponding to the regularity of initial conditions. In Part 2, we give a sufficient condition for the global existence and exponential decay of the weak solutions by defining a modified energy functional together with the technique of Lyapunov functional.

In this paper, we study the Robin-Dirchlet problem (Pn) for a wave equation with the term 1 n Xn i=1 u2( i−1 n , t), n ∈ N. First, for each n ∈ N, under suitable conditions, we prove the local existence and uniqueness of the weak solution un of (Pn). Next, we prove that the sequence of solutions un of (Pn) converges strongly in appropriate spaces to the weak solution u of the problem (P), where (P) is defined by (Pn) by replacing 1 n Xn i=1 u2( i−1 n , t) by Z 1 0 u2(y, t)dy. The main tools used here are the linearization method together with the Faedo-Galerkin method and the weak compact method. We end the paper with a remark related to a similar problem.

In this paper, we consider the Dirichlet boundary problem for a nonlinear wave equation of Kirchhoff-Carrier-Love type as follow   utt − B ? ∥u(t)∥2 , ∥ux(t)∥2
(uxx + uxxtt) = f(x, t, u, ux, ut, uxt) + Xp i=1 εifi(x, t, u, ux, ut, uxt) for 0 < x < 1, 0 < t < T, u(0, t) = u(1, t) = 0, u(x, 0) = ˜u0(x), ut(x, 0) = ˜u1(x), (1) where ˜u0, ˜u1, B, f, fi (i = 1, · · · , p) are given functions, ε1, · · · , εp are small parameters and ∥u(t)∥2 = Z 1 0 u2 (x, t) dx, ∥ux(t)∥2 = Z 1 0 u2 x (x, t) dx. First, a declaration of the existence and uniqueness of solutions provided by the linearly approximate technique and the Faedo-Galerkin method is presented. Then, by using Taylor’s expansion for the functions B, f, fi, i = 1, · · · , p, up to (N + 1)th order, we establish a high-order asymptotic expansion of solutions in the small parameters ε1, · · · , εp.

In this paper, we consider initial value problems for nonlinear fractional equations where source functions may discontinuous. We obtain the existence and uniqueness of maximal mild solutions of the problem. We also give some appropriate conditions such that mild solutions of the problem blow-up at a finite time. Furthermore, we discuss the continuous dependence of mild solutions of the problem with respect to fractional order.