On a Fredholm-Volterra integral equation

Abstract

"In this paper we give conditions in which the integral equation $$x(t)=\displaystyle\int_a^c K(t,s,x(s))ds+\int_a^t H(t,s,x(s))ds+g(t),\ t\in [a,b],$$ where $a<c<b$, $K\in C([a,b]\times [a,c]\times \mathbb{B},\mathbb{B})$, $H\in C([a,b]\times [a,b]\times \mathbb{B},\mathbb{B})$, $g\in C([a,b],\mathbb{B})$, with $\mathbb{B}$ a (real or complex) Banach space, has a unique solution in $C([a,b],\mathbb{B})$. An iterative algorithm for this equation is also given."

Existence and Uniqueness of Solutions of Hammerstein-Type Functional Integral Equations

“…Proof. For the proof, it is sufficient to prove the EUSs of HTFIE (12) We describe a mapping T from C([. ], B) to C([.…”

“…to admit a unique solution in the space C([0, b], B), where B is a Banach space. Filip and Rus [12] also gave an iterative algorithm for IE (7).…”

“…To the best knowledge of the authors of this paper, according to the above literature information and relative international databases, the EUSs of HTFIE (12) in a Banach space via the technique called Burton's progressive contractions, the Chebyshev norm, and complete metric have not been discussed up to now. However, Tunç and Tunç [26] and Castro and Simões [36] obtained some new results with regard to the Ulam stabilities of certain-delay Hammerstein-type integral equations in the bounded and infinite interval cases.…”

“…The considered delay of Hammerstein-type integral equations and the topic of Tunç and Tunç [26] and Castro and Simões [36] are not related to those of this paper. Indeed, HTFIE (12) is a new mathematical model and it can be seen that HTFIE (1) is different from all the above mathematical models as IEs and IDEs and those can be found in the international data base. Next, from the above information, it can also be seen that direct fixed-point mappings, classical fixed-point theorems via suitable mappings of a closed convex set in a Banach space, iterative algorithms, fibre contraction principle, etc., have been utilized to provide the existence and uniqueness of solutions of the considered IEs and IDEs.…”

“…The remaining sections of this work include the following contents: The first main outcome of this study, i.e., Theorem 1, is put forward for HTFIE (12) in Section 2. Section 3 includes an application result of the Burton method, i.e., Theorem 2, to the case of functional HTFIEs.…”