Solutions for a class of nonlinear Volterra integral and integro-differential equation using cyclic "Equation missing" -contraction

Abstract

We establish the existence and uniqueness of solutions for a class of nonlinear Volterra integral and integro-differential equations using fixed-point theorems for a new variant of cyclic (ϕ, ψ, θ )-contractive mappings. Nontrivial examples are given to support the usability of our results. MSC: 47H10; 54H25

Best Proximity Points of Generalized Semicyclic Impulsive Self-Mappings: Applications to Impulsive Differential and Difference Equations

“…The existence of the limits (41) and (42) follows from (34) in Theorem 9 and the above background Result 4 [11] since, for each ∈ for any ∈ , there is a finite = ( ) ∈ Z 0+ such that lim inf → ∞ + ( ) ∈ +1 with = sup ∈ ( ), ∀ ∈ so that the limits (41) exist (note that = 1, ∀ ∈ if : ⋃ ∈ → ⋃ ∈ is acyclic impulsive self-mapping). The limit (42) exists from the background Results 1 and 5 of [11] with ∈ and +1 = ( ) ∈ +1 , ∀ ∈ being unique best proximity points of : ⋃ ∈ → ⋃ ∈ in and +1 ; ∀ ∈ since ( , ) is also a ( , ‖‖) uniformly convex Banach space for the norm-induced metric and the subsets of , ∀ ∈ are nonempty, closed and convex. The limiting set ( , (1)…”

“…Recent results about best proximity points concerning psi-Geraghty contractions and on cyclic orbital contractions are obtained in [36,37], respectively. On the other hand, it turns out that fixed point theory is a useful tool to study the stability of differential and difference equations and dynamic systems [38][39][40][41][42]. Some worked examples are given in the sequel concerning the global feedback stabilization and the stability of the equilibrium points [43][44][45][46], linked with fixed points and best proximity points of impulsive and timedelayed differential equations.…”