New Monch–Krasnosel’skii type fixed point theorems applied to solve neutral partial integrodifferential equations without compactness

Random Fixed Point Theorems and Applications to Random First-Order Vector-Valued Differential Equations

“…They are very often used in the theory of functional equations, including ordinary differential equations, equations with partial derivatives, integral and integrodifferential equations, and optimal control theory. We also highlight that the interplay between fixed point theory and measures of noncompactness is very powerful and fruitful (see for instance [19,[23][24][25][26][27][28][29] and the references therein).…”

Results on Neutral Partial Integrodifferential Equations Using Monch-Krasnosel’Skii Fixed Point Theorem with Nonlocal Conditions

“…In [1,2], the authors studied the hypotheses for the existence of resolvent operators for the abstract integrodifferential equations. Further, in [3][4][5][6][7], the authors discussed the solutions of the existence of nonlinear neutral partial differential equations using different approaches. Lizama et al [8] studied (1) with the nonlocal initial values when φ = 0, and using the fixed point of Sadovskii's technique, derived the solution of existence when the nonlocal condition is compact, and R 1 (•) is continuous with respect to the norm.…”

“…Many authors have proven the existence of the solution for neutral integrodifferential equations with initial and nonlocal conditions. In [9], the authors proved the solutions of neutral functional integrodifferential equations with an initial condition in finite delay, and in [4], the authors proved the existence of the mild solution for a class of neutral partial integrodifferential equations using resolvent operator theory and measure of noncompactness and proved the existence using the Monch-Krasnosel'skii type of fixed point theorem with initial conditions. Motivated by the above two particular articles, we construct a new problem (1) and ( 2) using nonlocal conditions with finite delay and apply the Monch-Krasnosel'skii fixed point technique.…”

Existence of random mild solutions of a random integrodifferential equation with lack of compactness

“…Here, (Ω, Σ, µ) is a complete σ-finite measure space, A : D(A) ⊂ E → E is a closed linear operator on a Banach space (E, ∥.∥), (B(t)) t≥0 is a family of closed linear operators on E having the same domain D(B) ⊃ D(A), independent of t, f : Ω×I ×E → E and u 0 : Ω → E are given functions satisfying some conditions to be specified later. More precisely, we shall be concerned with extending some recent deterministic results from [16][17][18] to the more general stochastic setting. This work is organized as follows.…”