Existence, Uniqueness, and Regularity of Solutions to Semilinear Retarded Differential Equations

In the present work, we consider a semilinear retarded differential equation in a Banach space. We first establish the existence and uniqueness of a mild solution and then prove its regularity under different additional conditions. Finally, we consider some applications of the abstract results.

The existence and uniqueness results for (1.1) may also be applied to the particular cases, namely, the semilinear retarded functional differential equation u (t) + Au(t) = f t,u t , t ∈ (0,T], and a retarded differential equation with a nonlocal history condition where g : Ꮿ 0 → X, and x ∈ X.In this case, we may take h(ψ)(s) ≡ g(ψ) and φ(s Presently, we will be more concerned with the applications of the results for the case (1.4), though we present the analysis here for the more general case (1.1).The linear case of (1.3), in which f (t,φ) = Lφ with a bounded linear operator L : Ꮿ T → X, is recently considered by Bátkai et al. [3] using the theory of perturbed Hille-Yosida operators.
The theory of functional differential equations with the history conditions of the type considered in (1.1) and (1.4) may be applied to the epidemic population dynamic models.For such related works, we refer to Alaoui [1] and references cited therein.
For the earlier works on existence, uniqueness, and stability of various types of solutions of differential and functional differential equations with nonlocal conditions, we refer to Byszewski and Akca [5], Byszewski and Lakshmikantham [6], Byszewski [4], Balachandran and Chandrasekaran [2], Lin and Liu [7], and references cited in these papers.
and u satisfies (1.1) on [−τ, T].We first prove the existence of a mild solution of (1.1) on [−τ,t 0 ] for some 0 < t 0 ≤ T and then prove that this mild solution can be extended uniquely to a mild solution of (1.1) either on the whole [−τ,T] or to the maximal interval [−τ,t max ), 0 < t max ≤ T, and since t max < ∞, we obtain We then establish the regularity of this mild solution under different additional assumptions.Finally, we consider some applications of the results obtained.

Proof of Theorem 2.1
We first establish the existence of a mild solution u on [−τ,t 0 ], 0 < t 0 ≤ T, and then prove its unique continuation.For a fixed R > 0 and any 0 < T ≤ T, let Choose 0 < t 0 ≤ T such that for 0 < t ≤ t 0 , we have where and the constants M ≥ 1, ω ≥ 0 are such that For ψ ∈ ᐃ(χ,t 0 ) R , it may be verified easily that Fψ ∈ ᐃ(χ,t 0 ) R .Let ψ 1 ,ψ 2 ∈ ᐃ(χ,t 0 ) R .Then (3.5) Since t 0 M 0 < 1, it follows that there exists a unique u ∈ ᐃ(χ,t 0 ) such that u is a mild solution on [−τ,t 0 ].If t 0 < T, consider the problem where Since the functions f satisfy (A3) on translated interval and (A2) is automatically satisfied in this case, we may proceed as before to get a function w ∈ C([−τ − t 0 ,T − t 0 ];X) such that w is a mild solution of (3.6) on [−τ − t 0 ,t 1 ] for some 0 < t 1 ≤ T − t 0 .This w is unique as h is the identity map.Then is a mild solution of (1.1) on [−τ,t 0 + t 1 ].Continuing this way, we get either a function u ∈ Ꮿ T , unique in ᐃ(χ,T), which is a mild solution of (1.1) on [−τ,T], or u ∈ C([−τ,t max );X), 0 < t max ≤ T, such that for every sufficiently small > 0, there exists a unique u ∈ ᐃ(χ,T − ) with u = u on [−τ,t max − ] and u is a mild solution of (1.1) on [−τ,t max − ].In the later case, (1.7) holds; otherwise, we may extend u beyond t max contradicting the definition of the maximal interval of existence [−τ,t max ) (cf. [8, Theorem 1.4, pages 185-187]).Now, suppose that χ is Lipschitz continuous on [−τ,0] and χ(0) ∈ D(A).We prove that u is Lipschitz continuous on [−τ,T] for the first case.For the second case when u exists on [−τ,t max ), t max ≤ T, the proof can be modified by establishing the Lipschitz continuity of u on every compact subinterval [−τ, T], T < t max .Let t ∈ [−τ,T] and h ≥ 0. If t + h ≤ 0, then for some positive constant L χ .If t ≤ 0 and t + h > 0, then |t| ≤ h and |t + h| ≤ h.In this case, we have for some positive constant C 1 (R) depending on R only.For the case when t > 0, we have Replacing η by η − h, we get for some positive constants C 2 and C 3 (R).Combining (3.9), (3.10), and (3.12), we have, for t ∈ [−τ,T], h ≥ 0, and Application of Gronwall's inequality in (3.14) implies that for some positive constant D(T,R) depending on T and R only.Thus, u is Lipschitz continuous on [−τ,T].Also, for proving the remaining part of Theorem 2.1, we assume the interval of existence [−τ,T].The proof may be modified for the second case by replacing the function u and the interval [−τ,T] with u and [−τ,t max − ], respectively, for every sufficiently small > 0. Now, suppose that u(0) ∈ D(A) and X is reflexive.The function f : [0,T] → X given by f (t) = f t,u t (3.16) is Lipschitz continuous and therefore differentiable a.e. on [0, T] and f is in L 1 ((0,T);X).Consider the Cauchy problem and hence the solutions u and ũ of (1.1) belonging to ᐃ(χ,T) and ᐃ(χ,T), respectively, are different.This completes the proof of Theorem 2.1.

Applications
Theorem 2.1 may be applied to get the existence, uniqueness, and regularity results for For any given x ∈ X (x ∈ D(A) for regularity), χ ∈ Ꮿ T may be chosen as in (g2).