On the Weak Solutions of a Delay Composite Functional Integral Equation of Volterra-Stieltjes Type in Reflexive Banach Space

Differential and integral equations in reflexive Banach spaces have gained great attention and hve been investigated in many studies and monographs. Inspired by those, we study the existence of the solution to a delay functional integral equation of Volterra-Stieltjes type and its corresponding delay-functional integro-differential equation in reflexive Banach space E. Sufficient conditions for the uniqueness of the solutions are given. The continuous dependence of the solutions on the delay function, the initial data, and some others parameters are proved.

Belonging of Laplace-Stieltjes integrals to convergence classes

For positive continuous functions $\alpha$ and $\beta$ increasing to $+\infty$ on $[x_0,+\infty)$ and the Laplace--Stieltjes integral $I(\sigma)=\int\limits_{0}^{\infty}f(x)e^{x\sigma}dF(x),\,\sigma\in{\Bbb R}$, a generalized convergence $\alpha\beta$-class is defined by the condition $$\int\limits_{\sigma_0}^{\infty}\dfrac{\alpha(\ln\,I(\sigma))}{\beta(\sigma)}d\sigma<+\infty.$$ Under certain conditions on the functions $\alpha$, $\beta$, $f$, and $F$, it is proved that the integral $I$ belongs to the generalized convergence $\alpha\beta$-class if and only if $\int\limits_{x_0}^{\infty}\alpha'(x)\beta_1 \left(\dfrac1{x}\ln\dfrac1{f(x)}\right)<+\infty,\,\beta_1(x)= \int\limits_{x}^{+\infty}\dfrac{d\sigma}{\beta(\sigma)}$. For a positive, convex on $(-\infty,\,+\infty)$ function $\Phi$ and the integral $I$, a convergence $\Phi$-class is defined by the condition $\int\limits_{\sigma_0}^{\infty}\dfrac{\Phi'(\sigma)\ln\,I(\sigma)}{\Phi^2(\sigma)}d\sigma<+\infty$, and it is proved that under certain conditions on $\Phi$, $f$ and $F$, the integral $I$ belongs to the convergence $\Phi$-class if and only if $\int\limits_{x_0}^{\infty}\dfrac{dx}{\Phi'\left(({1/x)\ln\,(1/f(x))}\right)}<+\infty$. Conditions are also found for the integral of the Laplace--Stieltjes type $\int\limits_{0}^{\infty} f(x)g(x\sigma)dF(x)$ to belong to the generalized convergence $\alpha\beta$-class if and only if the function $g$ belongs to this class.

On the growth of series in system of functions and Laplace-Stieltjes integrals

For a regularly convergent in ${\Bbb C}$ series $A(z)=\sum\nolimits_{n=1}^{\infty}a_nf(\lambda_nz)$ in the system ${f(\lambda_nz)}$, where$f(z)=\sum\nolimits_{k=0}^{\infty}f_kz^k$ is an entire transcendental function and $(\lambda_n)$is a sequence of positive numbers increasing to $+\infty$, it isinvestigated the relationship between the growth of functions $A$ and $f$ in terms of a generalized order. It is proved that if$a_n\ge 0$ for all $n\ge n_0$, $\ln \lambda_n=o\big(\beta^{-1}\big(c\alpha(\frac{1}{\ln \lambda_n}\ln \frac{1}{a_n})\big)\big)$ for each $c\in (0, +\infty)$ and $\ln n=O(\Gamma_f(\lambda_n))$ as $n\to\infty$ then $\displaystyle\varlimsup\limits_{r\to+\infty}\frac{\alpha(\ln M_A(r))}{\beta(\ln r)}=\varlimsup\limits_{r\to+\infty}\frac{\alpha(\ln M_f(r))}{\beta(\ln r)},$ where $M_f(r)=\max\{|f(z)|\colon |z|=r\}$, $\Gamma_f(r):=\frac{d\ln M_f(r)}{d\ln r}$ and positive continuous on $(x_0, +\infty)$ functions $\alpha$and $\beta$ are such that $\beta((1+o(1))x)=(1+o(1))\beta(x)$, $\alpha(c x)=(1+o(1))\alpha(x)$ and$\frac{d\beta^{-1}(c\alpha(x))}{d\ln x}=O(1)$ as $x\to+\infty$ for each $c\in(0, +\infty)$.\A similar result is obtained for the Laplace-Stieltjes type integral $I(r) = \int\limits_{0}^{\infty}a(x)f(rx) dF(x)$.

Relative Growth of Series in Systems of Functions and Laplace—Stieltjes-Type Integrals

For a regularly converging-in-C series A(z)=∑n=1∞anf(λnz), where f is an entire transcendental function, the asymptotic behavior of the function Mf−1(MA(r)), where Mf(r)=max{|f(z)|:|z|=r}, is investigated. It is proven that, under certain conditions on the functions f, α, and the coefficients an, the equality limr→+∞α(Mf−1(MA(r)))α(r)=1 is correct. A similar result is obtained for the Laplace–Stiltjes-type integral I(r)=∫0∞a(x)f(rx)dF(x). Unresolved problems are formulated.

On Chandrasekhar functional integral inclusion and Chandrasekhar quadratic integral equation via a nonlinear Urysohn–Stieltjes functional integral inclusion

We investigate the existence of solutions for a nonlinear integral inclusion of Urysohn–Stieltjes type. As applications, we give a Chandrasekhar quadratic integral equation and a nonlinear Chandrasekhar integral inclusion.

Chandrasekhar quadratic and cubic integral equations via Volterra-Stieltjes quadratic integral equation

In this work, we study the existence of one and exactly one solution x ∈ C [ 0 , 1 ] x\in C\left[0,1] , for a delay quadratic integral equation of Volterra-Stieltjes type. As special cases we study a delay quadratic integral equation of fractional order and a Chandrasekhar cubic integral equation.