CUBO A Mathematical Journal Vol.11, No¯ 03, (115–124). August 2009 Bounded Solutions and Periodic Solutions of Almost Linear Volterra Equations Muhammad N. Islam And Youssef N. Raffoul Department of Mathematics, University of Dayton, Dayton, OH 45469-2316, USA emails: muhammad.islam@notes.udayton.edu, youssef.raffoul@notes.udayton.edu ABSTRACT This article addresses boundedness and periodicity of solutions of certain Volterra type equations. These equations are studied under a set of assumptions on the functions involved in the equations. The equations will be called almost linear when these as- sumptions hold. RESUMEN Este artículo es concerniente a acotomiento y periocidad de ciertas ecuaciones de tipo Volterra. Estas ecuaciones son estudiadas bajo un conjunto de condiciones sobre las funciones envolvidas en las ecuaciones. Las ecuaciones serán llamadas casi lineales cuando estas condiciones sean válidas. Key words and phrases: Volterra integral equation, integrodifferential equation, resolvent, Kras- noselsii’s fixed point theorem, bounded solution, periodic solution. Math. Subj. Class.: 45D05,45J05. 116 Muhammad N. Islam And Youssef N. Raffoul CUBO 11, 3 (2009) 1 Introduction. Consider the following scalar equations: x(t) = a(t) + ∫ t 0 C(t,s)g(x(s))ds, t ≥ 0, (1.1) and x ′ (t) = a(t)h(x(t)) + ∫ t −∞ C(t,s)g(x(s))ds + p(t), t ∈ (−∞,∞). (1.2) We assume that the functions h and g are continuous and that there exist positive constants H, H∗, G, G∗ such that |h(x) − Hx| ≤ H∗, (1.3) and |g(x) − Gx| ≤ G∗. (1.4) Equations (1.1) and (1.2) will be called almost linear if (1.3) and (1.4) hold. In [2] Burton in- troduced this concept of almost linear equations and studied certain important properties of the resolvent kernel of a linear Volterra equation. Throughout this paper we assume a(t) in (1.1) is continuous for t ≥ 0, and a(t),p(t) in (1.2) are continuous for −∞ < t < ∞. Also, we assume that C(t,s) in (1.1) is continuous for 0 ≤ s ≤ t < ∞, and C(t,s) in (1.2) is continuous for −∞ < s ≤ t < ∞. In Section 2 we obtain the boundedness of solutions of (1.1) using the respective resolvent kernels. In Section 3 we study (1.2) and show the existence of a periodic solution by employing Krasnoselskii’s fixed point theorem. The literature on the resolvent is massive. However, for many interesting results on resolvents of Volterra integral and integrodifferential equations we refer to [1], [3], [4], [6–8], [10–16], [18] and [19]. Burton [8] contains a large number of existing studies on the resolvents of Volterra integral equations which also includes many recent works related to the resolvent. On Krasnoselskii’s fixed point theorem and it’s application in integral equations we refer the reader to [5], [9] and [17]. 2 On Solutions of (1.1). We rewrite (1.1), x(t) = a(t) + ∫ t 0 C(t,s)[g(x(s)) − Gx(s)]ds + ∫ t 0 C(t,s)Gx(s)ds. (2.1) Let A(t) = a(t) + ∫ t 0 C(t,s)[g(x(s)) − Gx(s)]ds (2.2) CUBO 11, 3 (2009) Almost Linear Volterra Equations 117 and B(t,s) = GC(t,s). Then (2.1) becomes x(t) = A(t) + ∫ t 0 B(t,s)x(s)ds. (2.3) Let R(t,s) be the resolvent kernel associated with (2.3). Then R(t,s) exists and satisfies R(t,s) = −B(t,s) + ∫ t s R(t,u)B(u,s)du. (2.4) Then any solution x(t) of (2.3) satisfies x(t) = A(t) − ∫ t 0 R(t,s)A(s)ds. (2.5) Theorem 2.1 Assume a(t) is bounded for t ≥ 0. Also assume sup t≥0 ∫ t 0 |C(t,s)|ds < ∞, (2.6) and sup t≥0 ∫ t 0 |R(t,s)|ds < ∞. (2.7) Then any solution x(t) of (1.1) is bounded. Proof. From (2.2),using (1.4) and (2.6) we obtain |A(t)| ≤ |a(t)| + G∗ ∫ t 0 |C(t,s)|ds < ∞. Therefore from (2.5) and (2.7), we get |x(t)| ≤ |A(t)| + ∫ t 0 |R(t,s)||A(s)|ds < ∞. This concludes the proof of Theorem 2.1. Assume a′(t) and Ct(t,s) both exist and are continuous. Now differentiating (1.1), one gets x ′ (t) = a ′ (t) + C(t,t)g(x(t)) + ∫ t 0 Ct(t,s)g(x(s))ds (2.8) = a ′ (t) + C(t,t)[g(x(t)) − Gx(t)] + ∫ t 0 Ct(t,s)[g(x(s)) − Gx(s)]ds +C(t,t)Gx(t) + ∫ t 0 Ct(t,s)Gx(s)ds. 118 Muhammad N. Islam And Youssef N. Raffoul CUBO 11, 3 (2009) Let F(t) = a ′ (t) + C(t,t)[g(x(t)) − Gx(t)] + ∫ t 0 Ct(t,s)[g(x(s)) − Gx(s)]ds. (2.9) Then (2.8) becomes x ′ (t) = GC(t,t)x(t) + ∫ t 0 GCt(t,s)x(s)ds + F(t). (2.10) Let B(t,s) = GCt(t,s), A(t) = GC(t,t). Then (2.10) becomes x ′ (t) = A(t)x(t) + ∫ t 0 B(t,s)x(s)ds + F(t), x(0) = a(0). (2.11) Let Z(t,s) be the resolvent kernel associated with (2.11). Then Z(t,s) exists and satisfies Zs(t,s) = −Z(t,s)A(s) − ∫ t s Z(t,u)B(u,s)du, Z(t,t) = 1. (2.12) Then from the variation of parameters formula, any solution x(t) of (2.11) has the form x(t) = Z(t, 0)a(0) + ∫ t 0 Z(t,s)F(s)ds. (2.13) Theorem 2.2 Assume a′(t) is bounded. Also assume sup t≥0 ∫ t 0 |Ct(t,s)|ds < ∞, (2.14) and sup t≥0 ∫ t 0 |Z(t,s)|ds < ∞. (2.15) In addition, we assume that |C(t,t)| and |Z(t, 0)| are bounded. Then any solution x(t) of (2.11) is bounded. Proof. Applying (1.4) and (2.14) in (2.9), we get |F(t)| ≤ |a′(t)| + |C(t,t)|G∗ + ∫ t 0 |Ct(t,s)|G∗ds < ∞. Therefore from (2.13) one obtains |x(t)| ≤ |Z(t, 0)||a(0)| + ∫ t 0 |Z(t,s)||F(s)|ds < ∞. CUBO 11, 3 (2009) Almost Linear Volterra Equations 119 This concludes the proof of Theorem 2.2. Properties in (2.7) and (2.15) are known as integrability properties of resolvent. Conditions to ensure (2.7) can be found in [11], [14] and [18], and conditions to ensure (2.15) can be found in [10], [12], [13] and [19]. 3 Periodic Solutions of (1.2) In this section we investigate the existence of a periodic solution of (1.2) using Krasnoselskii’s fixed point theorem. We start with a statement of Krasnoselskii’s fixed point theorem. Theorem Krasnoselskii [17]. Let K be a closed convex non-empty subset of a Banach space M. Suppose that A and B map K into M such that (i) x,y ∈ K, implies Ax + By ∈ K, (ii) A is continuous and AK is contained in a compact subset of M, (iii) B is a contraction mapping. Then there exists z ∈ K with z = Az + Bz. In this section we assume that sup −∞ 0 such that a(t + T ) = a(t), p(t + T ) = p(t), C(t + T,s + T ) = C(t,s). (3.4) 120 Muhammad N. Islam And Youssef N. Raffoul CUBO 11, 3 (2009) We assume that ∫ T 0 a(t)dt 6= 0. (3.5) Let M be the complete metric space of continuous T -periodic functions φ : (−∞,∞) → (−∞,∞) with the supremum metric. Then, for any positive constant m the set PT = {f ∈ M : ||f|| ≤ m}, (3.6) is a closed convex subset of M. Let k(t) = p(t) + ∫ t −∞ C(t,s) [ g(x(s)) − Gx(s) ] ds + ∫ t −∞ C(t,s)Gx(s)ds. Then we may write (3.3) as x ′ (t) − Ha(t)x(t) = −Ha(t)x(t) + a(t)h(x(t)) + k(t). (3.7) Assume (3.4) and (3.5) hold. Multiply both sides of (3.7) with e−H ∫ t 0 a(s)ds and then integrate both sides from t − T to t, to obtain x(t)[e −H ∫ t t−T a(s)ds − 1]e−H ∫ t−T 0 a(s)ds = ∫ t t−T [ − Ha(u)x(u) + a(u)h(x(u)) + k(u) ] e −H ∫ u 0 a(s)ds du. Now, multiplying both sides by eH ∫ t−T 0 a(s)ds , we get x(t)[e −H ∫ t t−T a(s)ds − 1] = ∫ t t−T [ − Ha(u)x(u) + a(u)h(x(u)) + k(u) ] e −H ∫ u t−T a(s)ds du. Due to the periodicity of a(t) we note that e−H ∫ t t−T a(s)ds = e −H ∫ T 0 a(s)ds . Substituting k by the expression given earlier and then dividing by e −H ∫ t t−T a(s)ds − 1, we arrive at x(t) = 1 e −H ∫ T 0 a(s)ds − 1 { ∫ t t−T a(u)[h(x(u)) − Hx(u)]e−H ∫ u t−T a(s)ds du + ∫ t t−T ∫ u −∞ C(u,s)[g(x(s)) − Gx(s)] ds e−H ∫ u t−T a(s)ds du + ∫ t t−T ∫ u −∞ C(u,s)Gx(s) ds e −H ∫ u t−T a(s)ds du + ∫ t t−T p(u)e −H ∫ u t−T a(s)ds du } . (3.8) Define mappings A and B from PT into M as follows. CUBO 11, 3 (2009) Almost Linear Volterra Equations 121 For φ ∈ PT , (Aφ)(t) = 1 e −H ∫ T 0 a(s)ds − 1 { ∫ t t−T a(u)[h(φ(u)) − Hφ(u)]e−H ∫ u t−T a(s)ds du + ∫ t t−T ∫ u −∞ C(u,s)[g(φ(s)) − Gφ(s)] ds e−H ∫ u t−T a(s)ds du } and for ψ ∈ PT , (Bψ)(t) = 1 e −H ∫ T 0 a(s)ds − 1 { ∫ t t−T ∫ u −∞ C(u,s)Gψ(s) ds e −H ∫ u t−T a(s)ds du + ∫ t t−T p(u)e −H ∫ u t−T a(s)ds du } . It can easily be verified that both (Aφ)(t) and (Bψ)(t) are T -periodic and continuous in t. Assume sup −∞