Using Hausdorff measure of noncompactness and a fixed-point argument we prove the existence of mild solutions for the semilinear integrodifferential equation subject to nonlocal initial conditions $u^{\prime }(t)=Au(t)+{\int }_0^{t}B(t-s)u(s)ds+f(t,u(t))$, $t\in [\mathrm{0,1}]$, $u(0)=g(u)$, where $A:D(A)\subseteq X\to X$, and for every $t\in [\mathrm{0,1}]$ the maps $B(t):D(B(t))\subseteq X\to X$ are linear closed operators defined in a Banach space $X$. We assume further that $D(A)\subseteq D(B(t))$ for every $t\in [\mathrm{0,1}]$, and the functions $f:[\mathrm{0,1}]\times X\to X$ and $g:C([\mathrm{0,1}];X)\to X$ are $X$-valued functions which satisfy appropriate conditions.

The existence results of positive ω-periodic solutions are obtained for the second-order functional differential equation with multiple delays $u″(t)+a(t)u(t)=f(t,u(t),u(t-{\tau }_1(t)),\dots ,u(t-{\tau }_n(t)))$, where $a(t)\in C(ℝ)$ is a positive ω-periodic function, $f:ℝ\times {[0,+\infty )}^{n+1}\to [0,+\infty )$ is a continuous function which is ω-periodic in t, and ${\tau }_1(t),\dots ,{\tau }_n(t)\in C(ℝ,[0,+\infty ))$ are ω-periodic functions. The existence conditions concern the first eigenvalue of the associated linear periodic boundary problem. Our discussion is based on the fixed-point index theory in cones.

This paper deals with the existence of continuous bounded solutions for a rather general nonlinear integral equation of Volterra type and discusses also the existence and asymptotic stability of continuous bounded solutions for another nonlinear integral equation of Volterra type. The main tools used in the proofs are some techniques in analysis and the Darbo fixed point theorem via measures of noncompactness. The results obtained in this paper extend and improve essentially some known results in the recent literature. Two nontrivial examples that explain the generalizations and applications of our results are also included.

By means of the fixed-point theorem in the cone of strict-set-contraction operators, we consider the existence of a nonlinear multi-point boundary value problem of fractional integro-differential equation in a Banach space. In addition, an example to illustrate the main results is given.

We develop a method to obtain approximate solutions of nonlinear system of partial differential equations with the help of Sumudu decomposition method (SDM). The technique is based on the application of Sumudu transform to nonlinear coupled partial differential equations. The nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of three examples, and results of the present technique have close agreement with approximate solutions obtained with the help of Adomian decomposition method (ADM).

The first measure of noncompactness was defined by Kuratowski in 1930 and later the Hausdorff measure of noncompactness was introduced in 1957 by Goldenštein et al. These measures of noncompactness have various applications in several areas of analysis, for example, in operator theory, fixed point theory, and in differential and integral equations. In particular, the Hausdorff measure of noncompactness has been extensively used in the characterizations of compact operators between the infinite-dimensional Banach spaces. In this paper, we present a brief survey on the applications of measures of noncompactness to the theory of infinite system of differential equations in some $\mathrm{BK}$ spaces $c_0,c,{\ell }_p(1\le p\le \infty )$ and $n(\varphi )$.

We investigate some existence results for the solutions to impulsive fractional differential equations having closed boundary conditions. Our results are based on contracting mapping principle and Burton-Kirk fixed point theorem.

We obtain some new estimates for the error of Simpson integration rule, which develop available results in the literature. Indeed, we introduce three main estimates for the residue of Simpson integration rule in $L^1[a,b]$ and $L^{\infty }[a,b]$ spaces where the compactness of the interval $[a,b]$ plays a crucial role.

We establish exact solutions for the Schrödinger-Boussinesq System $i{u}_t+u_{xx}-a\mathrm{uv}=0$, $v_{tt}-v_{xx}+v_{xxxx}-b{({\left|u\right|}^{\mathrm{2}})}_{xx}=0$, where $a$ and $b$ are real constants. The ($G^{\prime }/G$)-expansion method is used to construct exact periodic and soliton solutions of this equation. Our work is motivated by the fact that the ($G^{\prime }/G$)-expansion method provides not only more general forms of solutions but also periodic and solitary waves. As a result, hyperbolic function solutions and trigonometric function solutions with parameters are obtained. These solutions may be important and of significance for the explanation of some practical physical problems.

Using Mönch fixed point theorem, this paper proves the existence and controllability of mild solutions for nonlinear mixed type integrodifferential functional evolution equations with nonlocal conditions in Banach spaces, some restricted conditions on a priori estimation and measure of noncompactness estimation have been deleted, our results extend and improve many known results. As an application, we have given a controllability result of the system.