Periodic boundary value problems for second order integro-differential equations of volterra type

Hu, Shouchuan; Lakshmikantham, V.

doi:10.1080/00036818608839591

Cited by 10 publications

(5 citation statements)

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“…(II) a Predholm integral operator of the type (1.3) [Su}(t) = f G(s, t, u(s))ds, [2] where G(a, t, r) is continuous for 0 ^ a, t ^ 1 and r £ i ; (III) a 'delay' operator of the type Equations of the type (1.1) with S as in (1.2) have recently been considered by Aftabizadeh and Leela [1] for Dirichlet boundary conditions and by Hu and Lakshmikanthan [6] for periodic boundary conditions, using monotone iterative techniques. Based on fixed point argument, Agarwal in [2] has discussed existence of solutions to the problem (1.1) with 5 as in (1.3).…”

Section: Jcmentioning

confidence: 99%

“…Then the problem (1.1) has at least one solution.PROOF: For k G N, define f k (t, x 0 , x u x 2 ) as f(t, x 0 , x u x 2 ) + zi/Jfe. Each fk satisfies (a) of Theorem 3.1 and hence for every k, there is at least one solution[6] u k e C 2 [0, 1] to the problem uW=f k(t,Su,u,u u 6 B. As in the proof of .Theorem 3.1, we now obtain a bound for «* in ||-|| 2 -norm, namely ||ujfe|| 2 ^ N, where N is independent of k. Continuity of / and 5 and the Ascoli-Arzela theorem imply the existence of a subsequence Ukj which converges to a solution of the problem (1.1).…”

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confidence: 98%

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Topological transversality: applications to differential equations

Sengadir¹,

Pani

1994

Bull. Austral. Math. Soc.

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In this paper, existence results for both integro-differential and functional differential equations are discussed using topological transversality arguments. As applications, third and fourth order boundary value problems are considered. For third order problems, an example has been cited to show that our results cover a wider class of problems than Theorem 2.3 of D.J. O'Regan, Topological transversality: Applications to third order boundary value problems, SIAM J. Math. Anal. 18 (1987)

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Section: Jcmentioning

confidence: 99%

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confidence: 98%

Topological transversality: applications to differential equations

Sengadir¹,

Pani

1994

Bull. Austral. Math. Soc.

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“…The authors developed the monotone iterative method for (1.1) based on a comparison result. As it is pointed out in [6], the method of [7] is not applicable to the general situation. Erbe and Guo studied problem (1.1) with f continuous, and k continuous and positive.…”

Section: -U"(t)--f(tu(t)ku(t))u(o)--u(en)u'(o)--u'(2n) (11)mentioning

confidence: 99%

“…Problem (1.1) is studied in [8] under the assumption that f(t,u,v) is continuous and increasing in v, and in [7] for f continuous and K of Volterra type. Following the ideas of [6] we study (1.1) …”

Section: -U"(t)--f(tu(t)ku(t))u(o)--u(en)u'(o)--u'(2n) (11)mentioning

confidence: 99%