On the solvability of some multi-point boundary value problems

Gupta, Chaitan P., Ntouyas, Sotiris K., and Tsamatos, Panagiotis Ch.. "On the solvability of some multi-point boundary value problems." Applications of Mathematics 41.1 (1996): 1-17. <http://eudml.org/doc/32934>.

@article{Gupta1996,
abstract = {Let $f\colon [0,1]\times \mathbb \{R\}^\{2\} \rightarrow \mathbb \{R\}$ be a function satisfying Caratheodory’s conditions and let $e(t)\in L^\{1\}[0,1]$. Let $\xi _\{i\}, \tau _\{j\}\in (0,1)$, $ c_\{i\},a_\{j\}\in \mathbb \{R\}$, all of the $c_\{i\}$’s, (respectively, $a_\{j\}$’s) having the same sign, $i=1,2,\ldots ,m-2$, $j=1,2,\ldots ,n-2$, $0 < \xi _\{1\}<\xi _\{2\}<\ldots <\xi _\{m-2\}<1$, $0 < \tau _\{1\}<\tau _\{2\}<\ldots <\tau _\{n-2\}<1$ be given. This paper is concerned with the problem of existence of a solution for the multi-point boundary value problems \begin\{align*\} x^\{\prime \prime \}(t)=f(t, x(t),x^\{\prime \}(t))+e(t),\qquad t\in (0,1)E \\ x(0)=\sum \limits \_\{i=1\}^\{m-2\} c\_\{i\}x^\{\prime \}(\xi \_\{i\}),\qquad x(1)=\sum \limits \_\{j=1\}^\{n-2\} a\_\{j\}x(\tau \_\{j\}) BC\_\{mn\}\end\{align*\} and \begin\{align*\} x^\{\prime \prime \}(t)=f(t, x(t),x^\{\prime \}(t))+e(t),\qquad t\in (0,1)E\\ x(0)=\sum \limits \_\{i=1\}^\{m-2\} c\_\{i\}x^\{\prime \}(\xi \_\{i\}),\qquad x^\{\prime \}(1)=\sum \limits \_\{j=1\}^\{n-2\} a\_\{j\}x^\{\prime \}(\tau \_\{j\}), BC\_\{mn\}^\{\prime \} \end\{align*\} Conditions for the existence of a solution for the above boundary value problems are given using Leray-Schauder Continuation theorem.},
author = {Gupta, Chaitan P., Ntouyas, Sotiris K., Tsamatos, Panagiotis Ch.},
journal = {Applications of Mathematics},
keywords = {multi-point boundary value problems; four point boundary value problems; Leray-Schauder Continuation theorem; a priori bounds; multipoint boundary value problem; Leray-Schauder continuation theorem},
language = {eng},
number = {1},
pages = {1-17},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the solvability of some multi-point boundary value problems},
url = {http://eudml.org/doc/32934},
volume = {41},
year = {1996},
}

TY - JOUR
AU - Gupta, Chaitan P.
AU - Ntouyas, Sotiris K.
AU - Tsamatos, Panagiotis Ch.
TI - On the solvability of some multi-point boundary value problems
JO - Applications of Mathematics
PY - 1996
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 41
IS - 1
SP - 1
EP - 17
AB - Let $f\colon [0,1]\times \mathbb {R}^{2} \rightarrow \mathbb {R}$ be a function satisfying Caratheodory’s conditions and let $e(t)\in L^{1}[0,1]$. Let $\xi _{i}, \tau _{j}\in (0,1)$, $ c_{i},a_{j}\in \mathbb {R}$, all of the $c_{i}$’s, (respectively, $a_{j}$’s) having the same sign, $i=1,2,\ldots ,m-2$, $j=1,2,\ldots ,n-2$, $0 < \xi _{1}<\xi _{2}<\ldots <\xi _{m-2}<1$, $0 < \tau _{1}<\tau _{2}<\ldots <\tau _{n-2}<1$ be given. This paper is concerned with the problem of existence of a solution for the multi-point boundary value problems \begin{align*} x^{\prime \prime }(t)=f(t, x(t),x^{\prime }(t))+e(t),\qquad t\in (0,1)E \\ x(0)=\sum \limits _{i=1}^{m-2} c_{i}x^{\prime }(\xi _{i}),\qquad x(1)=\sum \limits _{j=1}^{n-2} a_{j}x(\tau _{j}) BC_{mn}\end{align*} and \begin{align*} x^{\prime \prime }(t)=f(t, x(t),x^{\prime }(t))+e(t),\qquad t\in (0,1)E\\ x(0)=\sum \limits _{i=1}^{m-2} c_{i}x^{\prime }(\xi _{i}),\qquad x^{\prime }(1)=\sum \limits _{j=1}^{n-2} a_{j}x^{\prime }(\tau _{j}), BC_{mn}^{\prime } \end{align*} Conditions for the existence of a solution for the above boundary value problems are given using Leray-Schauder Continuation theorem.
LA - eng
KW - multi-point boundary value problems; four point boundary value problems; Leray-Schauder Continuation theorem; a priori bounds; multipoint boundary value problem; Leray-Schauder continuation theorem
UR - http://eudml.org/doc/32934
ER -