An Application to the Existence of Solutions of the Integral Equations

Abstract

Integral equations provide mathematical models of many important problems in the physical sciences and engineering. This paper treats one class of such equations, concentrating on methods involving the use of classical fixed point theorem. The study of integral equations in connection with nonlinear equations has a long history, during which a variety of approaches has emerged. Here, we effectively use a strategy that derives key properties of the solvability of integral equations from previously established results in Hölder spaces. Moreover, our approach leads to solvability of the Fredholm integral equations.

Keywords

• Hölder condition
• Fredholm Integral Equations
• Schauder fixed point theorem

References

1. [1] Banas, J., Nalepa, R., On the space of functions with growths tempered by a modulus of continuity and its applications, J. Funct. Space Appl., (2013), 13 pages. doi:10.1155/2013/820437
2. [2] Berenguer, M.I., Munoz, M.V.F., Guillem, A.I.G., Galan, M.R., Numerical treatment of fixed point applied to the nonlinear Fredholm integral equation, Fixed Point Theory Appl., 2009(2009), 1–8.
3. [3] Caballero, J., Abdalla, M., Sadarangani, K., Solvability of a quadratic integral equation of fredholm type in H¨older spaces, Electron. J. Differ. Eq., 31(2014), 1–10.
4. [4] Caballero, M.J., Nalepa, R., Sadarangani, K., Solvability of a quadratic integral equation of Fredholm type with supremum in Hölder spaces, J. Funct. Space Appl., (2014).
5. [5] Fredholm, E.I., Sur une classe d’equations fonctionnelles, Acta Math., 27(1903), 365–390.
6. [6] Kirk, W.A., Srinavasan, P.S., Veeramani, P., Fixed Points for mapping satisfying cyclical contractive conditions, Fixed Point Theory, 4(2003), 79–89.
7. [7] Pathak, H.K., Khan, M.S., Tiwari, R., A common fixed point theorem and its application to nonlinear integral equations, Comput. Math. Appl. Arch., 53(2007), 961–71.
8. [8] Rasham, T., Shoaib, A., Hussain, N., Arshad, M., Khan, S.U., Common fixed point results for new Ciric-type rational multivalued F-contraction with an application, J. Fixed Point Theory Appl., 20(1)(2018), 1–16.

9. [9] Rus, M.D., A note on the existaence of positive solution of Fredholm integral equations, Fixed Point Theory, 5(2004), 369–377.
10. [10] Schauder, J., Der Fixpunktsatz in Funktionalriiumen, Studia Math., 2(1930), 171–180.