Existence of solutions of integral equations with asymptotic conditions

Abstract

In this work we will consider integral equations defined on the whole real line and look for solutions which satisfy some certain kind of asymptotic behavior. To do that, we will define a suitable Banach space which, to the best of our knowledge, has never been used before. In order to obtain fixed points of the integral operator, we will consider the fixed point index theory and apply it to this new Banach space.Partially supported by Xunta de Galicia (Spain), project EM2014/032 and AIE Spain and FEDER, grants MTM2013-43014-P, MTM2016-75140-P.Supported by FPU scholarship, Ministerio de Educación, Cultura y Deporte, Spain.1 consequence of the lack of compactness of the operator. In all of the cited references the authors solve this problem by means of the following relatively compactness criterion (see [1,13]) which involves some stability condition at ±∞: 1 ([13, Theorem 1]). Let E be a Banach space and ( , E) the space of all bounded continuous functions x : → E. For a set D ⊂ ( , E) to be relatively compact, it is necessary and sufficient that:for any t ∈ ; 2. for each a > 0, the family D a := {x| [−a,a] , x ∈ D} is equicontinuous; 3. D is stable at ±∞, that is, for any ǫ > 0, there exists T > 0 and δ > 0 such that if x(T ) − y(T ) ≤ δ, then x(t) − y(t) ≤ ǫ for t ≥ T and if x(−T ) − y(−T ) ≤ δ, then x(t) − y(t) ≤ ǫ for t ≤ −T , where x and y are arbitrary functions in D.

Existence of Solutions of Integral Equations Defined in Unbounded Domains Via Spectral Theory

“…Then, for t ∈ A, we have Remark 4.15. We note that the previous results could also be formulated for ϕ ([a, +∞)) or ϕ ((−∞, a]) (with obvious notation) for any a ∈ (see [2]).…”

“…We note that the results in the present work could not be applied to this problem (even with the modification to include the term p) as, for instance, condition (C 2 ) does not hold. At the same time, we will show in Section 5 an example which is solved with the method developed in this paper but does not satisfy the hypotheses required in [2].…”

A generalization of the compression cone method for integral operators with changing sign kernel functions

“…Assumptions (H 2a ) and (H 2b ) of Theorem 2.1 can be simplified when the nonlinear part is in sub-linear or super-linear cases [4]. We introduce the following new conditions for Theorem 3.1.…”

On the existence of solutions of some non-linear functional integral equations in Banach algebra with applications

“…Integral equations are an important branch of mathematical analysis, science and equations of such types are applicable in many physical problems such as in the vehicular traffic, the biology, control theory and mathematical physics(see Abdou, 2003;Argyros, 1985;Corduneanu, 1990;Deimling, 1985). Recently the theory of FIE and various kind of functional differential equations are developed effectively and emerged in the field of analysis, engineering, applied mathematics, and nonlinear functional analysis (see Aghajani et al, 2014;Arab, 2016;Cabada et al, 2018;Deepmala, 2013a;Roshan, 2017;Tunc¸, 2010a;Tunç 2010b;Tunc¸, 2016;Tunc¸and Golmankhaneh, 2020;Tunc¸and Tunç, 2018a;Tunç and Tunc¸, 2018b;Tunca nd Tunc¸, 2018c;Tunc¸and Tunç, 2019;Tunc¸, 2020 and references therein). In this article, we prove the existence of solution of the following generalized FIE:…”