In this study, we prove the of existence of solutions of a convolution Volterra integral equation in the space of the Lebesgue integrable function on the set of positive real numbers and with the standard norm defined...In this study, we prove the of existence of solutions of a convolution Volterra integral equation in the space of the Lebesgue integrable function on the set of positive real numbers and with the standard norm defined on it. An operator P was assigned to the convolution integral operator which was later expressed in terms of the superposition operator and the nonlinear operator. Given a ball B<sub>r</sub> belonging to the space L it was established that the operator P maps the ball into itself. The Hausdorff measure of noncompactness was then applied by first proving that given a set M∈ B r the set is bounded, closed, convex and nondecreasing. Finally, the Darbo fixed point theorem was applied on the measure obtained from the set E belonging to M. From this application, it was observed that the conditions for the Darbo fixed point theorem was satisfied. This indicated the presence of at least a fixed point for the integral equation which thereby implying the existence of solutions for the integral equation.展开更多
In this paper, four kinds of integral equations of convolution type are solved, in which the reflection occurs, that is, besides the unknown f(t),f(-t) is also appeared. Moreover, it is mentioned that the methods or s...In this paper, four kinds of integral equations of convolution type are solved, in which the reflection occurs, that is, besides the unknown f(t),f(-t) is also appeared. Moreover, it is mentioned that the methods or solution for two of them are still effective when translation shifts, i.e., f(t+lambda(j)) or/and f(-t-mu(j)), occur in addition.展开更多
In this paper,we introduce and study a(p,q)-Mellin transform and its corresponding convolution and inversion.In terms of applications of the(p,q)-Mellin transform,we solve some integral equations.Moreover,a(p,q)-analo...In this paper,we introduce and study a(p,q)-Mellin transform and its corresponding convolution and inversion.In terms of applications of the(p,q)-Mellin transform,we solve some integral equations.Moreover,a(p,q)-analogue of the Titchmarsh theorem is also derived.展开更多
文摘In this study, we prove the of existence of solutions of a convolution Volterra integral equation in the space of the Lebesgue integrable function on the set of positive real numbers and with the standard norm defined on it. An operator P was assigned to the convolution integral operator which was later expressed in terms of the superposition operator and the nonlinear operator. Given a ball B<sub>r</sub> belonging to the space L it was established that the operator P maps the ball into itself. The Hausdorff measure of noncompactness was then applied by first proving that given a set M∈ B r the set is bounded, closed, convex and nondecreasing. Finally, the Darbo fixed point theorem was applied on the measure obtained from the set E belonging to M. From this application, it was observed that the conditions for the Darbo fixed point theorem was satisfied. This indicated the presence of at least a fixed point for the integral equation which thereby implying the existence of solutions for the integral equation.
文摘In this paper, four kinds of integral equations of convolution type are solved, in which the reflection occurs, that is, besides the unknown f(t),f(-t) is also appeared. Moreover, it is mentioned that the methods or solution for two of them are still effective when translation shifts, i.e., f(t+lambda(j)) or/and f(-t-mu(j)), occur in addition.
文摘In this paper,we introduce and study a(p,q)-Mellin transform and its corresponding convolution and inversion.In terms of applications of the(p,q)-Mellin transform,we solve some integral equations.Moreover,a(p,q)-analogue of the Titchmarsh theorem is also derived.
