On an antiperiodic type boundary value problem for first order linear functional differential equations

Hakl, R.   ; Lomtatidze, A.   ; Šremr, J.

Archivum mathematicum, Tome 38 (2002) no. 2, pp. 149-160

Abstract (VO)

Nonimprovable, in a certain sense, sufficient conditions for the unique solvability of the boundary value problem \[ u^{\prime }(t)=\ell (u)(t)+q(t),\qquad u(a)+\lambda u(b)=c \] are established, where $\ell :C([a,b];R)\rightarrow L([a,b];R)$ is a linear bounded operator, $q\in L([a,b];R)$, $\lambda \in R_+$, and $c\in R$. The question on the dimension of the solution space of the homogeneous problem \[ u^{\prime }(t)=\ell (u)(t),\qquad u(a)+\lambda u(b)=0 \] is discussed as well.

Détail

Classification : 34K13
Keywords: linear functional differential equation; antiperiodic type BVP; solvability and unique solvability

Bibliographie

[1] Azbelev N. V., Maksimov V. P., Rakhmatullina L. F.: Introduction to the theory of functional differential equations. Nauka, Moscow, 1991, in Russian. | MR | Zbl

[2] Azbelev N. V., Rakhmatullina L. F.: Theory of linear abstract functional differential equations and aplications. Mem. Differential Equations Math. Phys., 8 (1996), 1–102. | MR

[3] Bravyi E.: A note on the Fredholm property of boundary value problems for linear functional differential equations. Mem. Differential Equations Math. Phys. 20 (2000), 133–135. | MR | Zbl

[4] Bravyi E., Hakl R., Lomtatidze A.: Optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations. Czechoslovak Math. J., to appear. | MR | Zbl

[5] Bravyi E., Hakl R., Lomtatidze A.: On Cauchy problem for the first order nonlinear functional differential equations of non–Volterra’s type. Czechoslovak Math. J., to appear. | MR

[6] Bravyi E., Lomtatidze A., Půža B.: A note on the theorem on differential inequalities. Georgian Math. J. 7 4 (2000), 627–631. | MR | Zbl

[7] Hakl R.: On some boundary value problems for systems of linear functional differential equations. Electron. J. Qual. Theory Differ. Equ. 10 (1999), 1–16. | MR | Zbl

[8] Hakl R., Kiguradze I., Půža B.: Upper and lower solutions of boundary value problems for functional differential equations and theorems on functional differential inequalities. Georgian Math. J. 7 3 (2000), 489–512. | MR

[9] Hakl R., Lomtatidze A.: A note on the Cauchy problem for first order linear differential equations with a deviating argument. Arch. Math. (Brno) 38 1 (2002), 61–71. | MR | Zbl

[10] Hakl R., Lomtatidze A., Půža B.: On a periodic boundary value problem for the first order scalar functional differential equation. J. Math. Anal. Appl., submitted.

[11] Hakl R., Lomtatidze A., Půža B.: On periodic solutions of first order linear functional differential equations. Nonlinear Anal. 49 7 (2002), 929–945. | MR | Zbl

[12] Hakl R., Lomtatidze A., Půža B.: New optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations. Math. Bohem., to appear. | MR | Zbl

[13] Hakl R., Lomtatidze A., Šremr J. : On an antiperiodic type boundary value problem for first order nonlinear functional differential equations of non–Volterra’s type. Differential Integral Equations, submitted. | Zbl

[14] Hakl R., Lomtatidze A., Šremr J.: On a periodic type boundary value problem for first order linear functional differential equations. Nonlinear Oscillations, submitted.

[15] Hakl R., Lomtatidze A., Šremr J.: On a periodic type boundary value problem for first order nonlinear functional differential equations. Nonlinear Anal., to appear. | Zbl

[16] Hale J.: Theory of functional differential equations. Springer–Verlag, New York-Heidelberg-Berlin, 1977. | MR | Zbl

[17] Kiguradze I.: On periodic solutions of first order nonlinear differential equations with deviating arguments. Mem. Differential Equations Math. Phys. 10 (1997), 134–137. | Zbl

[18] Kiguradze I.: Initial and boundary value problems for systems of ordinary differential equations I. Metsniereba, Tbilisi, 1997, in Russian. | MR

[19] Kiguradze I., Půža B.: On boundary value problems for systems of linear functional differential equations. Czechoslovak Math. J. 47 2 (1997), 341–373. | MR | Zbl

[20] Kiguradze I., Půža B.: On periodic solutions of systems of linear functional differential equations. Arch. Math. (Brno) 33 3 (1997), 197–212. | MR

[21] Kiguradze I., Půža B.: Conti–Opial type theorems for systems of functional differential equations. Differentsial’nye Uravneniya 33 2 (1997), 185–194, in Russian. | MR

[22] Kiguradze I., Půža B.: On boundary value problems for functional differential equations. Mem. Differential Equations Math. Phys. 12 (1997), 106–113. | MR | Zbl

[23] Kiguradze I., Půža B.: On periodic solutions of nonlinear functional differential equations. Georgian Math. J. 6 1 (1999), 47–66. | MR

[24] Kiguradze I., Půža B.: On periodic solutions of systems of differential equations with deviating arguments. Nonlinear Anal. 42 2 (2000), 229–242. | MR

[25] Kolmanovskii V., Myshkis A.: Introduction to the theory and applications of functional differential equations. Kluwer Academic Publishers, 1999. | MR | Zbl

[26] Mawhin J.: Periodic solutions of nonlinear functional differential equations. J. Differential Equations 10 (1971), 240–261. | MR | Zbl

[27] Schwabik S., Tvrdý M., Vejvoda O.: Differential and integral equations: boundary value problems and adjoints. Academia, Praha, 1979. | MR | Zbl