Chandrasekhar quadratic and cubic integral equations via Volterra-Stieltjes quadratic integral equation
In this paper, the following quadratic integral equation of Volterra-Stieltjes type is studied: \[ x(t)=a(t)+\int\limits_{0}^{\varphi(t)}f_1(t,s,x(s))d_sg_1(t,s)\int\limits_{0}^{1}f_2(t,s,x(s))d_sg_2(t,s), \quad t\in[0,1]. \] The authors discuss the existence of solutions of this equation in the class of continuous functions and the continuous dependence of its unique solution on the functions \(g_1\), \(g_2\) and \(\varphi\). The obtained results are then applied to the fractional-order quadratic integral equation \[ x(t)=a(t)+I^{\alpha}f_1(t,s,x)\cdot\int\limits_{0}^{1}\frac{t}{t+s}f_2(t,s,x(s))ds \] and to the Chandrasekhar quadratic and cubic integral equations \[ x(t)=a(t)+x^p(t)\cdot\int\limits_{0}^{1}\frac{t}{t+s}a(s)x(s)ds, \quad p=1,2. \]
Volterra-Stieltjes type integral equations
scientific article; zbMATH DE number 7353411
scientific article